text stringlengths 4 2.78M |
|---|
---
abstract: 'Explicit relations among moduli of the Heterotic and Type IIB string theories in 8 dimensions are obtained. We identify the BPS states responsible for gauge enhancements in the type IIB theory and their dual partners in the Heterotic theory compactified with and without Wilson lines. The masses of BPS states in Type IIB string theory compactified on the base space of a elliptically fibred $K3$ are computed explicitly for the special cases in which the complex structure of the fibre is constant, ie, for constant scalar fields backgrounds.'
---
20.5cm 18.0cm -0.4in 1.5in
‘=11 addtoreset[equation]{}[section]{}
stequation
‘=11
\#1
=by60 = \#1[[bsphack@filesw [ gtempa[auxout[ ]{}]{}]{}gtempa @nobreak esphack]{} eqnlabel[\#1]{}]{} eqnlabel vacuum \#1
\#1 [[hep-th/\#1]{}]{} \#1(\#2)\#3 [Nucl. Phys. [**B\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Phys. Rept.[**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Phys. Lett. [**\#1B**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Phys. Rev. Lett.[**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Phys. Rev. [**D\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Ann. Phys. [**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Rev. Mod. Phys. [**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Comm. Math. Phys. [**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Mod. Phys. Lett. [**A\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Int. J. Mod. Phys. [**A\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Class. Quant. Grav. [**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Adv. Math. [**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [Invent. Math. [**\#1**]{} (\#2) \#3 ]{} \#1(\#2)\#3 [J. High Energy Phys. [**\#1**]{} (\#2) \#3 ]{}
8[**W\_8**]{} 9[**W\_9**]{} 8[**A\_8**]{} 9[**A\_9**]{} ł 8E8[E\_8E\_8]{} 32[Spin(32)/Z\_2]{} /2 ł
[ [hep-th/9909178]{}]{}\
[ DAMTP–1999–130 ]{}\
0.5cm
1.0cm
[M.C. Daflon Barrozo.[^1]]{}
0.5 cm [*Department of Applied Mathematics and Theoretical Physics\
Cambridge University, Cambridge, England*]{}
1.0cm
September, 1999
1.0cm
Introduction
============
Over the past couple of years non-perturbative aspects of compactifications of Type IIB theory have been studied first in terms of F-theory [@vafa1] compactifications and later in an explicitly stringy form by means of including non-perturbative $(p,q)$ 7-brane configurations in the background. The former has been particularly important in the analysis of the duality with the Heterotic string.
Compactifications of F-theory on elliptic Calabi-Yau two-folds ($K3$), three-folds and four-folds have been argued to be dual to certain compactifications of the Heterotic string to 8, 6 and 4 dimensions. The simplest case to consider is the compactifications of F-theory to 8 dimensions on an elliptic $K3$. This theory is believed to be dual to the Heterotic string on $T^2$. The $ADE$ pattern of gauge symmetry enhancement in the Heterotic string [@witten1; @ginsparg] is reproduced in F-theory by the pattern of collapsible holomorphic two spheres in $K3$ [@vafa2]. The location in the moduli space where the symmetries occur are provided by the zeros of the discriminant of the $K3$ surface. This establishes the duality at a geometrical level.
The other approach that has emerged [@sen1; @johansen; @gab1; @gab2; @imamura; @zwiebach1; @zwiebach2; @zwiebach3; @dewolfe] is type IIB theory compactified on a sphere in the presence of non-local 7-branes which extend in the uncompactified directions and appear as singular points on the sphere. The presence of the $ADE$ series of algebras arising on 7-branes configurations were explored in refs[@johansen; @gab1; @gab2; @imamura; @zwiebach1] where it was shown how $(p,q)$-strings and string junctions stretched between the 7-branes correspond to vector bosons of the eight dimensional gauge theory. Other algebras organised in terms of the conjugacy classes of $SL(2,Z)$ have been identified in terms of the strings and junctions connecting the 7-branes [@zwiebach2; @zwiebach3; @dewolfe]. Some connections among elements in the Type IIB theory and their duals in the Heterotic theory in 8 dimensions have appeared in the literature[@imamura2]. However all identifications have been at a qualitative level, in the sense that the masses of the gauge bosons that are supposed to become massless at specific points in the moduli space to generate the relevant gauge group had not been checked explicitly. We will compute explicitly the masses of the relevant gauge bosons in many examples in this paper.
The masses of BPS $(p,q)$-strings have been computed explicitly in two cases. The first was the work of Sen [@sen1] who used the results of Seiberg-Witten for $N=2$ supersymmetric $SU(2)$ gauge theory with 4 quark flavours. He computed the masses of BPS states in the limit where the 24 7-branes are grouped into four groups of 8 branes yielding an $SO(8)^4$ symmetry. In [@sen2] Sen introduced a new mass formula which showed how the masses above originated from the mass formula of open strings stretched between 7-branes. Later, Lerche and Stieberger [@lerche1] suggested a generalisation of the formula used by Sen in terms of a contribution by the fundamental period of the implicit F-theory $K3$. They argued that this factor would not have any effect in the rigid limit considered by Sen but would provide the correct normalisation in the general case. The masses of the BPS gauge fields responsible for the geometrical enhancement of $U(1)^2_G \rightarrow SU(3)_G$ were computed and an exact match with the Heterotic string mass formula was obtained. However, only one example and with no Wilson lines was considered.
In this paper we use the mass formula of ref[@lerche1] to compute the mass of various gauge bosons potentially responsible for several gauge enhancements in the type IIB side. This takes the description of non perturbative Type IIB theory to a more quantitative level. We also compare the masses of the BPS gauge bosons in Type IIB with their Heterotic dual partners and find complete agreement. We analyse examples with both zero and non zero Wilson lines. The explicit map of the geometrical and Wilson lines moduli in the Heterotic theory to the moduli describing the position of the 7-branes in the sphere is obtained in several examples.
We focus on the branches of moduli space where the massless scalar fields in Type IIB are constant. There are three such branches in the moduli space [@sen1; @mukhi]. We refer to them as Branches 0, I and II. Branch 0 exist for any constant value of the scalar fields and allow for only one symmetry group, namely, $SO(8)^4$. Branches I and II require special values for the scalar fields and allow for a more complex set of gauge symmetries. The moduli space of these branches have dimensions 1, 5, and 8, respectively. We will consider, in particular, one-parameter families of curves living in their multidimensional moduli spaces. Once we define a one-parameter family in one of the branches of Type IIB by choosing a pattern of gauge enhancement we compute the mass of the BPS gauge bosons responsible for the symmetries. The next step is to determine the dual family in the Heterotic theory that must have not only the same pattern of gauge enhancements but also the masses of the BPS gauge bosons must match everywhere in the moduli space (See Fig 1). This procedure is carried out for a number of examples and the explicit duality maps between the dual families are obtained.
This paper is organised as follows. In section 2 we review results for the Heterotic theory compactified on $T^2$. We also obtain an expression for masses of BPS states in the presence of general Wilson lines that generalises the standard holomorphic expression in terms of the Kahler structure, $T$, and complex structure, $U$, of the torus. This expression will play a fundamental role in establishing the duality map in Section 4. In section 3 we review the basics of type IIB compactifications on the sphere in the presence of 7-branes in order to establish our conventions. In section 4 we compute the masses of a number of BPS states in the two branches of constant $\tau$ on the type IIB side and compare with the masses of the Heterotic theory duals obtained in Section 2. Explicit maps between the Heterotic and Type IIB moduli are derived. In section 5 we present our conclusions. In Appendix A we show how to generalise some hypergeometric relations used in the paper. And finally, in Appendix B we obtain the explicit relation among certain moduli in the Heterotic theory that are required elsewhere in the paper.
Heterotic String on $T^2$
=========================
Consider the Heterotic String compactified on a torus, $T^2$, down to 8 dimensions along directions $x^8$ and $x^9$. In this section it will not be necessary to specify which Heterotic theory we are working with.
The expression for the left and right moving momenta with Wilson lines is given by
$$(P_{Ri} | P_{Li}) = (p_i - (g_{ij} - B_{ij})\frac{w ^j}{\alpha '} | p_i +
(g_{ij} + B_{ij})\frac{w ^j}{\alpha '};\; \sqrt{\frac{2}{\alpha
'}}{\bf q})$$
where $m_i$ and $w^i$ are the KK momentum and winding numbers, respectively, along $x^8$ and $x^9$. We define $p_i = m_i -\1/2 {\bf A}_i\cdot ({\bf A}_kw ^k) - {\bf A}_i\cdot
{\bf Q}$, ${\bf Q}$ is an element of either of the lattices $\Gamma
^{16}$ or $\Gamma ^8\bigoplus \Gamma ^8$ and ${\bf q} = {\bf Q} + w^k {\bf A_k}$.
The mass spectrum is given by M\_h\^2 = + +(N\_L - 1) + (N\_R - c\_R) where $N_L$, $N_R$ are the left-, right-moving oscillator numbers, and $c_R= 0 , \1/2$ depending on whether the right-moving fermions are periodic $(R)$ or anti-periodic $(NS)$. We must impose the level matching condition to obtain the physical states + (N\_L - 1)= + (N\_R - c\_R), BPS states are given by the additional requirement that $N_R =
c_R$ [@harvey]. Therefore for states that are physical and BPS saturated we must have P\_L\^2 - P\_R\^2 = (1- N\_L), or /2 Q\^2 + m\_iw\^i = 1-N\_L. \[1\](summation convention). In this case we can write for the mass formula M\_h\^2 &=& P\_[Ri]{}g\^[ij]{}P\_[Rj]{}\
&=& (p\_i - (g\_[il]{} - B\_[il]{}))g\^[ij]{}(p\_j - (g\_[jk]{} - B\_[jk]{})) . ł[1a]{}
The massless states with $N_L = 1$ include the 8-dimensional metric, antisymmetric tensor, dilaton and gauge fields which are the Cartan generators of the gauge group. We also have two scalars in the adjoint of the gauge group which are the Wilson lines. In the $N_L=0$ sector we have massless gauge fields associated to the roots of the underlying gauge group. They must satisfy P\_R\^2&=&0\
P\_L\^2&=& /2 Q\^2 + m\_iw\^i = 1. \[2\]
The zero winding numbers sector gives the roots of the subgroup of $E_8\times E_8$ or $SO(32)$ which is left unbroken by the Wilson lines. Further gauge fields will appear in the non-zero winding numbers sector for special values of the geometrical moduli of the torus, ie, $g_{ij}$ and $B_{ij}$. We discuss these cases in more detail in later sections.
One of the goals of this paper is to use the duality between the Heterotic strings on $T^2$ and Type IIB compactified on the base space of a elliptic fibred $K3$. When comparing Heterotic string states with the dual states in type IIB it turns out to be extremely convenient to rewrite the Heterotic mass formula in a holomorphic or anti-holomorphic form. To do so we introduce the parameters U=U\_1 + iU\_2 &=& + i\
\[2mm\] \[3\] T\^0=T\^0\_1+iT\^0\_2 &=& + i \[4\] where $g = g_{88}g_{99} - g_{89}^2$ and $B= B_{89}$. $T^0$ is the Kahler structure and $U$ the complex structure of the torus. When dealing with Wilson lines it becomes very convenient to introduce a modified version of the Kahler structure. In the presence of Wilson lines we redefine [^2] U &=& U\^0 \[4a\]\
T &=& ( - + ) + i (1 + ) \[5\]\
Z &=& /2 (-) - i \[6\]\
n\_i &=& m\_i - [**A\_i**]{} . \[7\] Note that $T \rightarrow T^0$ and $Z \rightarrow
0$ as we turn off the Wilson lines. The mass formula, eq(\[1a\]), becomes, in terms of the new variables, M\_[BPS]{}=|n\_8 - U n\_9 + T w\^9 + (TU + Z)w\^8| \[8\] Note that this expression for the mass of BPS states is valid for any Wilson line. This mass formula turns out to be a very useful way of rewriting the standard expression, eq(\[1a\]), for compactifications of the Heterotic string on $T^2$, particularly when there are Wilson lines. It will play a major role in what follows.
Type IIB on $S^2_s$
===================
Let us consider Type IIB theory compactified on a two-sphere, $S^2$, in the presence of 24 parallel 7-branes which appear as points on $S^2$. We refer to this punctured sphere as, $S^2_s$. The theory possesses different strings labelled $(p,q)$ according to how it is charged with respect to the $RR$ and $NSNS$ antisymmetric fields. The 7-branes of the theory are also labelled $(p,q)$, according to what $(p,q)$-string can end on it.
In this convention the elementary string is $(1,0)$ and a D-string is $(0,1)$. Each of the 24 7-branes has an associated branch cut depending on its type. We follow the conventions of [@gab2]. We use three basic types of branes, ${\bf A}$, ${\bf B}$ and ${\bf
C}$, whose corresponding $(p,q)$ labels are $(1,0)$, $(1,-1)$ and $(1,1)$ respectively. Across the corresponding cuts the labels $(p,q)$ and $U$ change according to the monodromy matrices =& (1,0): K\_A = & T\^[-1]{} = ,\
[**B**]{}=& (1,-1): K\_B = & S T\^[2]{} = ,\
[**C**]{}=& (1,1): K\_C = & T\^[2]{}S = , where $S$ is the matrix S = .
All branes have their branch cut going upwards vertically. ${\bf A}$-branes are represented by heavy dots, ${\bf B}$-branes by empty boxes and ${\bf C}$-branes by empty circles (see Fig. 1).
A formula for the mass of $(p,q)$-strings in the background described above has been derived by Sen [@sen1; @sen2] and is given by
M\_[IIB]{}(p,q)=\_C |((z))\^2\_[i=1]{}\^[24]{}(z-z\_i)\^[-1/12]{}(p+q(z)) dz| . ł[9]{} For more general backgrounds there has been a suggestion [@lerche1] that this mass formula should be generalised in order to include additional F-theory data. Specifically, the authors of ref[@lerche1] suggested that the mass formula should include the fundamental period of $K3$, $\omega _0$, so that the mass formula becomes
M\_[BPS]{}\^[IIB]{}= \_[z\_i]{}\^[z\_j]{} |(p+q(z) )((z) )\^2(z)\^[-]{}dz|, ł[10]{} where = \_[i=1]{}\^[24]{}(z-z\_i). Note that $\Delta= 4f^3 + 27g^2$ is the discriminant of the elliptic fibre, defined from the polynomial equation, $W(x,y,\xi)= y^2 - x^3 - f(\xi)x - g(\xi) =
0$, defining the elliptic $K3$ surface. The $z_i$’s are the positions of the 24 branes on the sphere, $S^2_s$.
The periods of a fibred $K3$ are given by
\_i =\_[\_i]{}. ł[11]{}
Branches of constant $\tau $
----------------------------
From the expression for the j-function of the elliptic fibre j()= , we see that there are three branches of constant $\tau$ [@sen1; @mukhi]. We have constant $\tau$ if $f^3 = \alpha g^2$, $f=0\; \& \; g\neq 0$ (Branch I) and $f\neq 0\; \& \; g=0$ (Branch II). It is important that for each of these cases we can factorise the data in the periods of $K3$ in a part depending on the elliptic fibre from that depending on the base only. We summarise the results in Table 1.
---------------------------------- ---------------------------------------- ----------------------------------------------------------------------------------
$f^3 =\alpha g^2$ $\Rightarrow \Delta = (4\alpha +27)g^2 $\omega _i = \int_{\gamma _i}\frac{dv}{(v^3+\alpha
$ ^{1/2}v+1)^{1/2}}\int\frac{d\xi}{g^{1/6}}
=A(\tau)\int\frac{d\xi}{\Delta
^{1/12}}$
\[2mm\] $f\neq 0 \; \& \; g=0$ $\Rightarrow \Delta = 4f^3$ $\omega _i = \int_{\gamma _i}\frac{dv}{(v(v^2+1))^{1/2}}\int\frac{d\xi}{f^{1/4}}
=A(i)\int\frac{d\xi}{\Delta
^{1/12}}$
\[2mm\] $f = 0 \; \& \; g\neq 0$ $\Rightarrow \Delta = 27g^2$ $\omega _i = \int_{\gamma _i}\frac{dv}{(v^3+1)^{1/2}}\int\frac{d\xi}{g^{1/6}}
=A(e^{\frac{i\pi }{3}})\int\frac{d\xi}{\Delta
^{1/12}}$
---------------------------------- ---------------------------------------- ----------------------------------------------------------------------------------
: Branches of constant $\tau$. $A(\tau)$ is a constant carrying information on the elliptic fibre.
For the three branches of constant $\tau$ we have an expression for the periods of $K3$ in terms of the discriminant, $\Delta$, which is similar to the one we have in the numerator of the the mass formula for $(p,q)$-strings. Actually, the only subtle point in these expressions in this case are the limits of integration in the periods. In fact, we can write for constant $\tau$ M\^[IIB]{}(p,q) =| | .
In the next section we compute the masses of BPS gauge fields and study how they map under the Heterotic and Type IIB theories duality in the branches I and II. The branch where $f^3 = \alpha g^2$ only has $SO(8)^4$ symmetry and we will not consider it in this paper.
The Duality Map
===============
In this section we will consider the masses of BPS states in Type IIB theory in two branches of constant $\tau$, namely, Branch I and II. And then we identify the dual BPS states in the Heterotic string and compare their masses. This will allow us to identify explicitly the duality map for some moduli of the two theories.
Branch I - ${\bf \tau=e^{i\pi/3}}$
----------------------------------
In Branch I, $f=0 \;\; \& \;\; g\neq 0 \rightarrow \tau = e^{\frac{i\pi
}{3}}$, there are only 9 degrees of freedom. The 24 branes join up in 12 non local pairs of branes. The two branes forming each pair can only move together. We will refer to them as a dynamical unit. Each dynamical unit is formed by an ${\bf AC}$ pair. Their positions on $S^2_s$ are related though. In fact, since the zeros of the discriminant indicate the position of the branes on $S^2_s$ and $g(\xi)$ is a polynomial of degree 12, we have $\Delta =
27g^2 = 0$. This equation has 12 degrees of freedom. Moding out by $SL(2,{\bf C})$ eliminates another 3 complex degrees of freedom giving relations among the position of the branes on the sphere.
These singularities can collide and yield a gauge enhancement at the points where the discriminant of $K3$ vanishes. The pattern of gauge enhancement in each of the branches of constant $\tau$ in Type IIB has been studied in detail in ref[@gab1]. The basic gauge enhancements that appear when $\tau= e^{\frac{i\pi}{3}}$ due to dynamical units colliding at the same point is given by U(1)\^2 SU(3); U(1)\^3 SO(8); U(1)\^4 E\_6; U(1)\^5 E\_8.
In ref[@gab1] the authors qualitatively identified the gauge fields responsible for the symmetries above. In ref[@zwiebach1] a systematic procedure based in string junctions was developed giving further evidence for the identification of these gauge fields. However no explicit check of the mass for this gauge fields and its relation to the heterotic duals has been obtained until recently. In [@lerche1] (see also [@lerche2; @lerche3]) the masses of the gauge fields responsible for the enhancement of $E_8\times
E_8\times U(1)_G\times U(1)_G \rightarrow E_8\times E_8\times SU(3)_G$ were computed. They were shown to be identical to the mass of the heterotic duals.
The mass for BPS $(p,q)$-strings in this branch is given by M\_[IIB]{}(p,q) = | | . ł[12]{}
We now compute the mass of BPS gauge fields in both the Heterotic and Type IIB theories. We start by reviewing the results of ref[@lerche1]. We will then turn on Wilson lines and analyse how it effects the mass for the BPS gauge bosons.
### a) The case with no Wilson lines: ${\bf E_8\times E_8 \times U(1)_G^2 \rightarrow E_8\times E_8
\times SU(3)_G}$ {#a-the-case-with-no-wilson-lines-bf-e_8times-e_8-times-u1_g2-rightarrow-e_8times-e_8times-su3_g .unnumbered}
The duality between the Heterotic string and F-theory can take two forms. Starting from the $E_8\times E_8$ theory we have to go through M-theory by means of a $9-11$-flip from the Heterotic on $T^2$ to Type IA on $S^1\times S^1/Z_2$ and then by T-duality to Type IIB on $T^2/Z_2$ which in turn is the same as the weak coupling limit of F-theory on $K3$. For the $Spin(32)/Z_2$ Heterotic theory on $T^2$ we start by S-duality to Type I and them by two T-dualities to Type IIB on $T^2/Z_2$ and subsequently to F-theory on $K3$. The gauge fields we are considering in this sub-section are not charged under the Cartan of either $E_8\times E_8$ or $SO(32)$. Therefore, it does not matter which theory we start from since the mass is the same for the $SU(3)$ gauge bosons. The authors in ref[@lerche1] considered the $E_8\times E_8$ Heterotic string since in this case the $E_8\times E_8$ is obtained without any Wilson line.
Let us consider the $E_8 \times E_8$ Heterotic string compactified on a torus down to 8 dimensions as in Section 1. We turn off all Wilson lines such that the moduli that specify the torus can be combined into two complex scalars U= U\_1 + iU\_2 &=& +i\
T\^0 = T\^0\_1 + iT\^0\_2 &=& +i At the special point in the moduli space of the Heterotic string where we set the geometric moduli $U =e^{i\pi /3}$ the BPS mass formula for gauge fields, eq(\[8\]), is given by[^3] M\_h\^2 = ł[12a]{} We, of course, still have to impose the level matching condition (LM), eq(\[2\]), which for states not charged under the Cartan of $E_8\times E_8$ is given by m\_8w\^8 + m\_9w\^9 =1 . ł[13]{}
If we now approach with the parameter $T^0$ the point $T^0=U$ we have + i &=& + i = /2 + i\
&& B =1 , g\_[12]{} =1 & g\_[11]{}=g\_[22]{}=R\^2\_c=2 . ł[14]{} Therefore, we are at the well know point of gauge enhancement of two geometrical $U(1)_G$’s to $SU(3)_G$. We need six gauge fields to realise this enhancement. The gauge fields in the Heterotic string together with their quantum numbers are given in Table 2.
------------------------- --------- --------- --------- --------- --------- ------------------------------------------- --------- ---------
${\bf V}$ $w^1$ $w^2$ $m_1$ $m_2$ $p_{R}$ $p_{L}$ $N_{R}$ $N_{L}$
\[2mm\] ${\bf V^{1,2}}$ 0 $\mp 1$ 0 $\mp 1$ 0 $\pm i \sqrt{2}$ $\1/2$ $0$
\[2mm\] ${\bf V^{3,4}}$ $\mp 1$ $\pm 1$ $\mp 1$ 0 0 $\pm \sqrt{2} \left( $\1/2$ $0$
\frac{\sqrt{3}}{2} + \frac{i}{2} \right)$
\[2mm\] ${\bf V^{5,6}}$ $\pm 1$ 0 $\pm 1$ $\pm 1$ 0 $\pm \sqrt{2} \left( $\1/2$ $0$
\frac{\sqrt{3}}{2} - \frac{i}{2} \right)$
\[2mm\]
------------------------- --------- --------- --------- --------- --------- ------------------------------------------- --------- ---------
: Additional $SU(3)$ gauge bosons.
Note that all states in Table 2 satisfy the level matching condition, eq(\[13\]).
Having fixed the value for the quantum numbers identifying the gauge bosons we can rewrite the formula for their mass, (\[12a\]), as in Table 3.
------------------------- ------------------------------------ -----------------------------------------
${\bf V}$ $\f{4}{\alpha '} M_h = |P_{R}|$ LM
\[2mm\] ${\bf V^{1,2}}$ $\frac{|T - U|}{2T^0_2U_2} $ $2T^0_2U_2 = (U- \bar{T})(\bar{U} - T)$
\[2mm\] ${\bf V^{3,4}}$ $\frac{|1 +T(U - 1)|}{2T^0_2U_2} $ $2T^0_2U_2 =|1 +
\bar{T}(U-1)|^2 $
\[2mm\] ${\bf V^{5,6}}$ $\frac{|1 +U(T - 1)|}{2T^0_2U_2} $ $2T^0_2U_2 =
|1 + \bar{U}(T-1)|^2 $
\[2mm\]
------------------------- ------------------------------------ -----------------------------------------
: Mass of the $SU(3)$ gauge bosons.
Furthermore, using the fact that for $U=e^{i\pi /3}$ we have and $U^2-U+1=0$. We can rewrite the numerators and denominators in the expressions in Table 3 as follows |1+T(U-1)| &=& |(1-U)(T-U)|\
|1+U(T-1)| &=& |U(T-U)|\
|1+T|[U]{}+T| &=& |U\^4(U\^2 +T)|\
|T-|[U]{}| &=&|T+U\^2|\
|1+U(|[T]{}-1)|&=& |U\^2(U\^2+T)| With this result we can write the mass of the 6 $SU(3)$ gauge fields as in Table 4. All six gauge fields become massless as $T^0\rightarrow U$.
---------------------------------------------------------------------------------------------------
${\bf V}$ $\f{4}{\alpha '} M_h = |P_{R}|$ LM
------------------------- ----------------------------------- -------------------------------------
${\bf V^{1,2}}$ $\frac{|T - U|}{2T^0_2U_2} $ $2T_2U_2 = |(\bar{T} -U)(T + U^2)|$
\[2mm\] ${\bf V^{3,4}}$ $\frac{|(1-U)(T-U)|}{2T^0_2U_2} $ $2T_2U_2 =|\bar{U}^2(T+U^2)|^2 $
\[2mm\] ${\bf V^{5,6}}$ $\frac{|U(T - U)|}{2T^0_2U_2} $ $2T_2U_2 =
|\bar{U}(T+U^2)|^2 $
\[2mm\]
---------------------------------------------------------------------------------------------------
: Mass of the additional $SU(3)$ gauge bosons.
\
In F-theory the $K3$ surface we are looking for is the one that has the gauge group $E_8\times E_8\times U(1)^2$ and such that it depends on only one moduli to have it enhanced to $SU(3)$. Such a surface is given by : y\^2+x\^3 +fx + g = y\^2+x\^3+z\^5(z-1)(z-z\_s)=0 This surface describes a fibred $K3$ and the fibre has the discriminant given by & = & 4f\^3+27g\^2\
& = & 27g\^2\
& = & 27z\^[10]{}(z-1)\^2(z-z\_s)\^2 The points where the discriminant vanishes give the position of 7-branes on $S^2_s$. For this particular expression the gauge group is given by a fibre $II*$[@vafa2] at $0$ and $\infty$(this point will appear in another patch of $S^2_s$). And fibre $II$ at $1$ and $z_s$ yielding $E_8\times E_8\times U(1)^2$ symmetry, respectively. The important point about this $K3$ is that it depends only on one complex parameter, namely, $z_s$. The point of enhancement to $SU(3)$ is when we have $z_s \rightarrow 1$ therefore the distance $z_s - 1$ must be related to the expression $T^0 - U$ in the heterotic side.
In the equivalent Type IIB picture we want to compute explicitly the mass for a $(p,q)$-string stretching between the two points of $U(1)$ gauge symmetry, ie, $1$ and $z_s$ (see Fig 2). First a simple analysis of the moduli tells us that there are 6 strings that can end on the non-local branes sitting at that points. In our conventions these strings are, up to global $SL(2,Z)$ monodromies, ${\bf V^{1,2}}=\pm (1,0), {\bf V^{3,4}}=\pm (1,1) \,\, \& \,\,{\bf
V^{5,6}}= \pm (0,1)$. This fixes the tension in the mass formula. The integral in the numerator of the mass formula can be rewritten in terms of hypergeometric functions after a simple change of variables. We obtain = \_1\^[z\_s]{}|\^[-]{}|= .
The periods of $K3$, using eq(\[11\]), are also expressible in terms of hypergeometric functions. In fact, w\_0 &=& -2 (-1)\^[2/3]{} \_[2,1]{}\[1/6, 1/6, 1, z\_s\])A(e\^[i/3]{})\
w\_1 &=& 2(-1)\^[5/6]{}(-z\_s)\^[-1/6]{}F\_[2,1]{}\[1/6,1/6,1,\]A(e\^[i/3]{}) .
Through a non-trivial hypergeometric transformation we can rewrite ${\cal L}$ in terms of the periods $w_i$ as follows &=&- (e\^[-2i /3]{} w\_0 + w\_1)\
&=& -e\^[i/3]{} w\_0 + w\_1 Following [@lerche1] we divide the original expression for the mass, eq(\[9\]), by the fundamental period $w_0$ and use the flat coordinate for $K3$, $T_{{\tiny IIB}} = w_1/w_0$. The result is M\_[IIB]{}(p,q) &=&|||(p+q)(e\^[i/3]{} - T\_[[IIB]{}]{})|\
&=& |||(p+q)(- T\_[[IIB]{}]{})| ł[14a]{}
Using the $(p,q)$ charges of the strings in Fig. 2 we find complete agreement among the BPS gauge fields responsible for the enhancement of $E_8\times E_8\times U(1)_G^2
\rightarrow E_8\times E_8\times SU(3)_G$ in the Heterotic string and Type IIB by mapping U\_h &&\
T\^0\_h && T\_[[IIB]{}]{} The overall constants are absorbed in the relation between the metrics of the two theories.
We now consider the case with Wilson lines turned on.
### b) An example with non-zero Wilson line:\
${\bf E_8\times E_8\times SU(3)_G \rightarrow E_8\times
U(1)^2_G\times G_1\times G_2}$ {#b-an-example-with-non-zero-wilson-line-bf-e_8times-e_8times-su3_g-rightarrow-e_8timesu12_gtimes-g_1times-g_2 .unnumbered}
In Fig 3 we represent the configuration we will consider in this subsection from the Type IIB perspective. We break one of the $E_8$’s by moving away an integer number of dynamical units. It is not necessary to specify in detail what the breaking is at infinity. The elliptic $K3$ surface equivalent to this configuration in Type IIB is given by y\^2 + x\^3 + z\^5(z-1)(z-z\_s)(z-M)\^n=0 . ł[14aa]{} We have put five dynamical units at $z=0$ forming a $E_8$, one dynamical units at $z=z_s$ and another one at $z=1$ forming $U(1)_G^2$. At $M$ we have a block formed by $n \le 5$ dynamical units that have moved away from infinity.
As before we want to compute =\_[z\_s]{}\^1\^[-]{} =\_[z\_s]{}\^1z\^[-]{}(z-1)\^[-]{}(z-z\_s)\^[-]{}(z-M)\^[-]{} . ł[um]{} For the periods we have similarly w\_0 &=& \_[0]{}\^[z\_s]{} z\^[-]{}(z-1)\^[-]{}(z-z\_s)\^[-]{}(z-M)\^[-]{}\
w\_1 &=& \_[0]{}\^[1]{} z\^[-]{}(z-1)\^[-]{}(z-z\_s)\^[-]{}(z-M)\^[-]{} ł[dois]{}
To write ${\cal L}$ in terms of the periods as we did before it turns out to be convenient this time to Taylor expand ${\cal L}$ and the periods in a series in $z/M$. We can now write[^4] =\_[l=0]{}\^ C\_l(n) \_[z\_s]{}\^1 z\^[l-]{}(z-1)\^[-]{}(z-z\_s)\^[-]{} . ł[tres]{} where $C_l(n)$ are some numerical coefficients. We also obtain similar expressions for the periods. The point is that we can now rewrite the integrand in terms of hypergeometric functions and apply one of Kummer’s relations to each element in the sum separately as we show in appendix A. It turns out that we end up with the following relation among ${\cal L}$ and the periods = w\_0- w\_1 . Dividing by $w_0$ we obtain for the mass formula of BPS states stretching between the 7 branes sitting at $z=1$ and those at $z=z_s$ M\_[IIB]{} = |||(p+q)(-T\_[[IIB]{}]{})| .\
where we introduced once again $T_{{\tiny IIB}}$. Note that $T_{{\tiny
IIB}}$ incorporates all the information on the positions of the branes as we move them around very much as $T_h$ does with Wilson lines. We start now to explore how they are connected.
[**Heterotic String Duals**]{}
Consider a smooth modification of the background of the Heterotic string considered before by turning on Wilson line moduli. The moduli now becomes T &=& ( + - ) + i (1 + )\
&=& T\^0 + ( - ) + i () . ł[14b]{}
Let us assume further that the geometrical moduli remain fixed to their values as in eq(\[14\]). The BPS mass for the $SU(3)_G$ states is now given by M\_h = | m\_8 - Um\_9 + Tw\^9 + (TU + Z)w\^8|
Let us analyse what are the conditions for the $SU(3)_G$ gauge bosons to become massless again. We surely expect this to be smoothly related to the Wilson lines parameters. In fact, we have\
[**i)**]{} For ${\bf V^{1,2}}$:
w\^8=0, m\_8=0, m\_9 = w\^9 . We have for the mass of this states M\_h = |m\_9(U-T)| . ł[15]{} Level matching condition, eq(\[2\]), implies that $m_9w^9 =1$, ie $w^9=\pm
1 = m_9$, and this will determine, as before, the $p$ charge of the $(p,q)$-string in Type IIB. Note that $T$ depends on the Wilson line parameter now. So to obtain a massless state we must have, $T\rightarrow U$, or from eq(\[14b\]) &=& 0 = [**0**]{}\
ł[16]{} [**A\_9**]{} &=&0 . ł[17]{}
${\bf ii)}$ For ${\bf V^{3,4}}$
m\_8 = w\^8 = - w\^9, m\_9 =0 we obtain for the mass M\_h &=& |w\^8(1+T - TU -Z)|\
&\^[T=U]{}& |w\^8(1+U - U\^2 -Z)|\
&\^[U\^2-U+1=0]{}& |w\^8||Z| Z=0 . ł[17a]{} Note again that level matching requires $w^8= \pm 1= m_8 \rightarrow
w^9 = \mp 1$. Let’s analyse the condition on $Z$. Using $U= e^{\f{i\pi
}{3}}$ in eq(\[6\]) we obtain Z= ([**A\_9A\_9**]{} +2 [**A\_9A\_8**]{} - 2[**A\_8A\_8**]{}) + i ([**A\_9A\_8**]{} - ) = 0 . ł[18]{} The imaginary part is equal to zero due to eq(\[17\]). For the real part we have = 0 = [**0**]{} . These states become massless only when we turn off the Wilson lines, ${\bf A_i} ={\bf 0}$, as expected. Since in this case we are back to the situation of [@lerche1] with the geometrical parameters fixed to their critical values. And Finally
${\bf iii)}$ For ${\bf V^{5,6}}$: w\^9 = 0, w\^8 = m\_8 = m\_9 Level matching requires $w^8 = \pm 1 = m_8= m_9$. Their mass is given by M\_h &\^[T=U]{}& |w\_8(1-U + TU -Z)|\
&=& |w\_8||Z| = 0 .
So for the geometrical parameters fixed to their critical values, $B=1$, $R_c^2 = g_{22}=g_{11}=2$ and $g_{12}=1$, we see, by comparing eq(4.22) with eq(4.26), eq(4.29) and eq(4.33), that the relative separation parameter of the two dynamical units on Type IIB responsible for the geometrical enhancement is mapped to the Wilson line moduli by U\_h &&\
T\_h && T\_[IIB]{} . ł[18a]{}
If we now consider the more general case when we allow for the geometrical parameters to be different from their critical values the six gauge bosons will become massless again when e\^ = U &=& T\
/2 + i = + i &=& ( + - ) + i (1 + ) ł[19]{} If we were to turn on a Wilson line and still keep the $SU(3)_G$ symmetry we would have to tune the geometrical parameters such that the critical values would be shifted as follows B &=& 1 + [**A\_9 A\_8**]{} -\
R\_c\^2 &=& 2 -. Note that the constraints in $Z$, eq(\[17a\]), would still be in place but now in the more general form &=&\
&=& [**A\_8A\_8**]{} .
We have shown that by introducing the parameters $U$ and $T$ as in eq(\[5\]) and by writing the Heterotic mass formula as in eq(\[8\]) the map between Heterotic string BPS states and their duals in Type IIB theory are immediately obtained, eq(\[18a\]). In the next section we extend this analysis to the other non-trivial branch of constant coupling in Type IIB theory.
Branch II - ${\bf \tau =i}$
----------------------------
For branch II, $f\neq 0 \;\; \& \;\; g= 0 \rightarrow \tau =i$, and there are 5 complex degrees of freedom. In fact, $f(\xi )$ is a polynomial of degree 8 and we have to mod out the $SL(2,{\bf C})$ symmetry. The 24 branes join up forming 8 groups with two mutually local branes and a non-local one. In our conventions a dynamical unit is (${\bf AAC}$). In general, we have $SU(2)^8$ gauge group. The other possible gauge groups are
$$\begin{aligned}
\begin{array}{lll}
SU(2)^8 & E_7\times SU(2)^5 & E_7\times E_7\times SU(2)^2\\
SO(8)\times SU(2)^6 & E_7\times SO(8)\times SU(2)^3 &
E_7\times E_7\times SO(8)\\
SO(8)^2\times SU(2)^4 & E_7\times SO(8)^2\times SU(2) &\\
SO(8)^3\times SU(2)^2 & & \\
SO(8)^4 && \\
\nn
\end{array}\end{aligned}$$
Therefore, we see that in Branch II we have two basic gauge enhancements depending on how many dynamical units collide at the same point. We have (SU(2)U(1))\^2 SO(8); & (SU(2)U(1))\^3 E\_7 ;
The mass for BPS $(p,q)$-strings in this branch is given by M\_[IIB]{}(p,q) =| |
Let us consider the map of IIB theory in Branch II to the $\spin32$ Heterotic String on $T^2$. It is convenient to start with the following Wilson lines: &=& (\^4;\^4;0\^2;0\^2;0\^4)\
9 &=& (0\^4;\^4;0\^2;0\^2;\^4) . \[21\] These Wilson lines break the gauge group to $SO(8)^4$. The semicolon separation will become clear below.
As part of our duality map we set as before U\_h . In the Heterotic side this implies that U\_h = + i == i. This fixes two of the geometrical moduli, namely, $g_{89}=0$ and $g_{88}=g_{99} = R^2$. This is why in our redefinition of the geometrical parameters in Heterotic string with Wilson lines we left $U_{h}$ unchanged so that this mapping remains the same.
We analyse the map between the Heterotic and type IIB theories in two examples. Both examples correspond to a one-parameter family in the respective moduli spaces.
Once we define a one-parameter family in the Heterotic theory we present a one-parameter family in the Type IIB theory that we argue is the dual family. There is, obviously, an infinity number of one-parameter families in a 5 dimensional space as the moduli space we are dealing with here. Nevertheless, we manage to identify one potential family in the Type IIB theory. The condition for this family to be the dual family of the one in the Heterotic theory is that they both have the same pattern of gauge enhancements taking place in dual points in the moduli spaces. And the fact that the mass of BPS gauge bosons are the same every where in moduli space. We verify this to be the case in both examples we consider. We then write explicitly the duality map between the moduli of the two theories.
We start with the enhancement:
### [**a)**]{} ${\bf (SU(2)\times U(1))^2\times SO(8)^3
\rightarrow E_7^2\times SO(8)}$ {#a-bf-su2times-u12times-so83rightarrow-e_72times-so8 .unnumbered}
To achieve this enhancement we have to move in the Wilson lines moduli space by adding to the Wilson lines above the following pieces &=& [**W\_8**]{} + (0\^4;0\^4;a\^2,0\^2;0\^4)\
[**A\_9**]{} &=& [**W\_9**]{} + (0\^4;0\^4;0\^2,a\^2;0\^4) , where $a$ is a real positive parameter. It is immediate to see that the effect of the perturbation is to break the third $SO(8)_{(3)}$, ie, the one occupying the third block of four slots in the lattice vector above, to $(SU(2)\times U(1))_{(1)}\times (SU(2)\times U(1))_{(2)}$. We anticipate the map with type IIB by picturing this breaking in Fig 4.
Note that for $a = \frac{1}{2}$ we have $(SU(2)\times U(1))_{(1)} \rightarrow
SO(4)_{(1)}$ and similarly $(SU(2)\times U(1))_{(2)} \rightarrow
SO(4)_{(2)}$ since the following vectors become massless (we will concentrate on one of the enhancements to $E_7$ only since the process is identical for both) \^[(1)]{} = (0\^4;0\^4;+1,+1;0\^2;0\^4) . ł[23]{} We also have states in the vector representation of $SO(4)_{(1)}
\times SO(8)_{(1)}$ becoming massless when $a=1/2$[^5] \^[(2)]{}=(\_;0\^4;\_;0\^2;0\^4) = (8\_v;1,4;1;1) . ł[24]{} These gauge bosons enhance $(SO(4)_{(1)}\times SO(8)_{(1)})
\rightarrow SO(12)_{(1)}$. Analogously, we also have $(SO(4)_{(2)}\times SO(8)_{(4)}) \rightarrow SO(12)_{(2)}$. The mass of this states as they approach $a=\1/2$ are listed in Table 5.
If this model is to represent a configuration of branes in Type IIB in Branch II there can not be a $SO(12)$. But we are not finished yet. We have to check the states with winding numbers.
If we rewrite the expression for the heterotic momenta using $U_h = i$ we can write for the mass[^6], eq(\[1a\]), M\_h\^2 = P\_[iR]{}\^2 = (p\_8 - (g\_[88]{}w\^8 - Bw\^9))\^2g\^[88]{} + (p\_9 - (g\_[99]{}w\^9 + Bw\^8))\^2g\^[99]{} and using $g^{88} = g_{99}/g = 1/g_{88} = R^{-2} = g^{99}$ we write M\_h\^2 = P\_[iR]{}\^2 = (p\_8 - (R\^2w\^8 - Bw\^9))\^2R\^[-2]{} + (p\_9 - (R\^2w\^9 + Bw\^8))\^2R\^[-2]{} For massless states we must have p\_8 &=& ((R\^2w\^8 - Bw\^9)) \[24a\]\
p\_9 &=& (R\^2w\^9 + Bw\^8))
Substituting this in the formula for level matching condition for gauge bosons, $P_L^2 =4/\alpha ' $, we obtain R\^2((w\^8)\^2 + (w\^9)\^2) = ’ (1 - ) This equation requires that we have $q^2 \le 2$. Defining $\Lambda = q
+ \lambda$, $\lambda \in \Gamma^{16}$ such that $\Lambda^2$ is smallest. We obtain R\^2((w\^8)\^2 +(w\^9)\^2) = ’ (1 - )
We separate the winding states in odd and even values for the sake of the analysis. Furthermore, it is easy to see that for odd winding numbers only $w^i=\pm 1$ contribute with massless states. For $w^9 =\pm 1$ and $w^8=0$ we obtain ${\bf q} = {\bf Q} \pm
(0^4;\f{1}{2}^{4};0^2;(\1/2)^6)$. Choosing[^7] = (\_;(/2)\^2;(/2)\^4) ł[25]{} we obtain = (\_;0\^2;0\^4), ł[26]{} These are the spinors ${\bf 32}_{\pm 1}$ of $SO(12)_{(1)}$. The subscript is the $U(1)_{w^9}$ charge associated to winding along $x^9$-direction. This gauge vectors become massless at the critical radius $R_c^2 = \f{\alpha '}{4}$. It turns out that it is convenient to write the spinors in terms of the gauge groups $SO(8)_{(1)} \times
(SU(2)\times U(1))_{(1)}$, ie, ${\bf 32}_{\pm 1} = ({\bf 8_s};{\bf 1};{\bf
1};{\bf 1};{\bf 1})_{\pm \1/2} + ({\bf 8_c};{\bf 1};{\bf
2};{\bf 1};{\bf 1})$. For even winding numbers, only $w^i=\pm 2$ contribute. In this case we take ${\bf Q} = \pm (0^4;(1)^{4};0^2;(1)^6)$ and $\Lambda^2=0$, these are singlets, $({\bf 1,1,1,1,1})_{\pm 2}$. These singlets become massless at the same radius as the spinors. So for $R_c^2
=\f{\alpha '}{4}$ the spinors and singlets enhance $SO(12)_{(1)}\times U(1)_{w^9}$ to $E_7^{(1)}$. Analogously, gauge fields with quantum numbers $w^9=0$ and $w^8=\pm 1,\pm 2$ enhance $SO(12)_{(2)} \times U(1)_{w^8}$ to $E_7^{(2)}$.
Note that from eq(\[24a\]) when $w^8=0$ and $w^9 = \pm 1,\pm 2$, as above, we have a constraint in the value of the anti-symmetric field $B$. A straight forward analysis[^8] of that equation shows that we must have $B \in 2Z$ (this only fixes $B$ up to an $SL(2,Z)$ transformation). Therefore, we choose $B=0$. This fixes the real part in the mass formula, eq(\[8\]), as in Table 5.
Consider now the other two free moduli. We have the Wilson line parameter, $a$, and the radius, $R^2= g_{88}=g_{99}$. As mentioned before, we will consider only one-parameter families. So we have to fix one of this parameters. We know that when we are at the critical radius as $a\rightarrow 1/2$ we obtain the enhancements described above. Therefore, we choose the radius to be a function of $a$ so that at $a=1/2$ we have $R^2 =R_c^2= \alpha ' /4$. Of course there are an infinity number of ways of achieving this but ultimately we want a match between the masses of BPS states in the dual theories. The analysis carried out in Appendix B determines that we set for the specific choice of parameters in the Type IIB side to discussed below R\^2 = ’ (a(1-a) - (/2 -a)) .
It is now possible to write the mass formula for all the BPS states we have been considering in this section. We write their mass in terms of the mass formula, eq(\[8\]). The results are summarised in Table 5.
---------------------------------------------------------------------------------------------------
${\bf Q}$ $M_h$
---------------------------------------------------------- ----------------------------------------
$(\pm (0^4;0^4;+1,+1;0^2;0^4)$ $\f{4}{\alpha '}|2(\f{1}{2} -a)|$
$({\bf 8_v};{\bf 1};{\bf 2};{\bf 1};{\bf 1})_{0}$ $\f{4}{\alpha '}|(\f{1}{2} -a)|$
$({\bf 8_v};{\bf 1};{\bf 1};{\bf 1};{\bf 1})_{\pm 1}$ $\f{4}{\alpha '}|(\f{1}{2} -a)|$
$({\bf 8_c};{\bf 1};{\bf 2};{\bf 1};{\bf 1})_{0}$ $\f{4}{\alpha '}| i(\f{1}{2} -a)|$
$({\bf 8_s};{\bf 1};{\bf 1};{\bf 1};{\bf 1})_{\pm \1/2}$ $\f{4}{\alpha '}|(-1+i)(\f{1}{2} -a)|$
$({\bf 1};{\bf 1};{\bf 1};{\bf 1};{\bf 1})_{\pm 2} $ $\f{4}{\alpha
'}|(2 i(\f{1}{2} -a)|$
---------------------------------------------------------------------------------------------------
: Additional gauge bosons for $SO(8)_{(1)}\times(SU(2)\times
U(1))_{(1)}\times U(1)_{w^9}\rightarrow E_7^{(1)}$
[**Type IIB duals**]{}
In type IIB theory starting from an $SO(8)^4$ configuration we move the two dynamical units forming one of the $SO(8)$s in perpendicular directions toward two $SO(8)$s (see Fig. 4). As each of the dynamical units approach one of the $SO(8)$s a number of $(p,q)$-strings will become massless. We identify this strings by their symmetry properties and compute their mass near the point they become massless.
For the $SU(2)\times U(1)$ in Type IIB we have ${\bf AAC}={\bf
BAA}$. The two vectors in the adjoint of $SU(2)$ correspond to the ${\bf A}\rightarrow {\bf A}$ $(1,0)$-string with both orientations. The enhancement to $SO(4)$ occurs when strings (prongs) like the ones in Fig. 6 become massless. They are equivalent to the vectors ${\bf Q}^{(1)}= \pm (0^4;0^4;+1,+1;0^2;0^4)$. The $\pm$ correspond to orientation.
For the states in the vector representation, ${\bf Q}^{(2)}
=({\bf 8_v},{\bf 1},{\bf 4};{\bf 1};{\bf 1})$, of $SO(8)_{(1)}\times
SO(4)_{(1)}$ we separate it in two parts in terms of $SO(8)_{(1)}\times (SU(2)\times U(1)^2)_{(1)}$. One where the permutations have the same signs, ${\bf Q}_a^{(2)} = (\pm
\underbrace{+1;0^3}_{\mbox{p}};0^4;\underbrace{+1,0}_{\mbox{p}};0^2;0^4)$ and the other where the permutations have opposite signs, ${\bf Q}_b^{(2)} =\pm
(\underbrace{-1;0^3}_{\mbox{p}};0^4;\underbrace{+1,0}_{\mbox{p}};0^2;0^4)$. We identify ${\bf Q}_a^{(2)}$ with the strings (prongs) in Fig 6 and ${\bf Q}_b^{(2)}$ with Fig 7.
For the spinors we identify ${\bf q}_a= ({\bf 8_c};{\bf 1};{\bf 2};{\bf
1};{\bf 1})$ with the configurations exemplified in Fig 8. And ${\bf
q}_b= ({\bf 8_c};{\bf 1};{\bf 2};{\bf 1};{\bf 1})$ is identified with Fig 9.
We now compute the mass for each of this configurations. The $K3$ surface equivalent to the configuration of branes in Fig. 4 is given by : y\^2 + x\^3 + x(z\^[2]{}(z-1)\^[2]{}(z-z\_s)) We obtain for the integral of the discriminant of this $K3$ = \_1\^[z\_s]{} \^[-]{} = i(1-z\_s)\^ F\_1(/2 , /2 , , 1-z\_s)
For the periods we obtain w\_0 &=& - (-1)\^(z\_s)\^ F\_1(/2 , /2 , ,z\_s)\
w\_1 &=& - i (-1)\^(z\_s)\^[-]{} F\_1( , /2 , 1,) . By means of a non-trivial Hypergeometric transformation we can write = w\_0 + i w\_1 And we obtain for the mass of a $(p,q)$-string M\_[IIB]{}(p,q)= |||(p + iq)(i)(i-T\_[[IIB]{}]{})| We can now compare the masses of BPS gauge fields in both theories. The results are summarised in Table 6. It is clear that the explicit duality map is given by U\_h &&\
|(/2 -a)|&&|i(i-T\_[[IIB]{}]{})| . ł[map4]{}
${\bf V}$ $\f{\alpha '}{4} M_h$ $(\f{A(i)}{\eta (i)^2})M_{IIB}$
---------------------------------------------------------- --------------------------- ---------------------------------- -- -- --
$(\pm (0^4;0^4;+1,+1;0^2;0^4)$ $|(2)(\f{1}{2} -a)|$ $|(2)[i(i-T_{{\tiny IIB}})]|$
$({\bf 8_v};{\bf 1};{\bf 2};{\bf 1};{\bf 1})_{0}$ $|(\f{1}{2} -a)|$ $|[i(i-T_{{\tiny IIB}})]|$
$({\bf 8_c};{\bf 1};{\bf 2};{\bf 1};{\bf 1})_{0}$ $|(i)(\f{1}{2} -a)|$ $|(i)[i(i-T_{{\tiny IIB}})]|$
$({\bf 8_s};{\bf 1};{\bf 1};{\bf 1};{\bf 1})_{\pm \1/2}$ $|(-1 + i)(\f{1}{2} -a)|$ $|(-1+i)[i(i-T_{{\tiny IIB}})]|$
$({\bf 1};{\bf 1};{\bf 1};{\bf 1};{\bf 1})_{\pm 2}$ $|(2i)(\f{1}{2} -a)|$ $|(2i)[i(i-T_{{\tiny IIB}})]|$
: Heterotic-IIB map for the gauge bosons of $SO(8)_{(1)}\times(SU(2)\times
U(1))_{(1)}\times U(1)_{w^9}\rightarrow E_7^{(1)}$.
In the next sub-section we consider the vector bosons responsible for the enhancement of $SU(2)\times U(1) \rightarrow SO(8)$.
### [**b)**]{} ${\bf SO(8) \rightarrow SU(2)\times U(1)}$ {#b-bf-so8-rightarrow-su2times-u1 .unnumbered}
In the previous section we identified the $(p,q)$-strings configurations that are responsible for the enhancement to $E_7$. To complete the symmetry enhancements possible in the this branch we here consider the enhancement $(SU(2)\times U(1))^2 \rightarrow SO(8)$. We choose, for convenience, to do this through $E_7^2\times (SU(2)\times
U(1))\times (SU(2)\times U(1)) \rightarrow E_7^2 \times SO(8)$ (see Fig. 10). We choose to put one of the $E_7$’s at $z = \infty$ and the other one at $z=0$. We also put one of the $SU(2)\times U(1)$ at $z=1$ and the other at $z=z_s$.
The mass of a $(p,q)$-string stretching from one dynamical unit (${\bf
AAC}$) to the other is of the form M(p,q) = |||(p + iq)| With the configuration above we obtain for the integral of the discriminant = e\^[3i/4]{}(1-z\_s)\^[1/2]{}\^2 F[1,2]{}(3/4,3/4,3/2,1-z\_s) For the $K3$ periods we obtain w\_0 &=& e\^[i/2]{} F\_[1,2]{}(1/4,1/4,1,z\_s) .\
w\_1 &=& e\^[- i /4]{} (-z\_s\^[-1/4]{})F\_[1,2]{}(1/4,1/4,1,1/z\_s). We now use one of Kummer’s relations for Hypergeommetric functions to write = e\^[i/2]{}w\_0 - w\_1 . Therefore we have for the mass of BPS $(p,q)$-strings in this background M\_[BPS]{}\^[IIB]{}(p,q) = |||(p + iq)(i-T\_[[IIB]{}]{} )| .
In the heterotic side we have the vectors below becoming massless to form the ${\bf Adj}$ of $SO(8)$. In the respective figures we draw the $(p,q)$-strings we identify as the dual pairs in the IIB side. \^[(1)]{} &=& (0\^4,+1,+1,0\^2;0\^8) (0\^4;0\^2;+1,+1;0\^8) ł[27]{}\
[**Q**]{}\^[(2)]{} &=& (0\^4;-1,0;+1,0;0\^8) ł[28]{}\
[**Q**]{}\^[(3)]{} &=& (0\^4;+1,0;+1,0;0\^8) ł[29]{}
In Table 7 we list the masses and quantum numbers of this gauge fields in both theories. The duality map for the moduli space is given by U\_h & &\
|(1+i)(/2 -a)| && |i-T\_[[IIB]{}]{} | . ł[map5]{}
$Q $ $m$ $\f{\alpha '}{4}M_h$ $p$ $q$ $\f{A(i)}{\eta (i)^2}M_{IIB}$
--------------------------- --------- -------------------------------- ------ ----- ------------------------------- -- --
$\pm (0^4,+1,+1,0^2;0^8)$ $\pm 1$ $|(1-i)[(1+i)(\f{1}{2} -a)]|$ $-1$ $1$ $|(1-i)(i-T_{{\tiny IIB}} )|$
$\pm (0^4,0^2,+1,+1;0^8)$ $\pm 1$ $|(1-i)[(1+i)(\f{1}{2} -a)]| $ $-1$ $1$ $|(1-i)(i-T_{{\tiny IIB}} )|$
$\pm (0^4,+1,0,+1,0;0^8)$ $\mp 1$ $|(i)[(1+i)(\f{1}{2} -a)]|$ $0$ $1$ $|(i)(i-T_{{\tiny IIB}} )|$
$\pm (0^4,-1,0,+1,0;0^8)$ $0$ $|[(1+i)(\f{1}{2} -a)]|$ $1$ $0$ $|(i-T_{{\tiny IIB}} )|$
: BPS gauge fields enhancing $(SU(2)\times U(1)) \times
(SU(2)\times U(1)) \rightarrow SO(8)$ in the Heterotic and IIB. Also $m_8=m_9=m$ and $w^8=w^9=w=0$.
Conclusion
==========
In this paper we have analysed the duality between the Heterotic string in 8 dimensions and Type IIB theory in the base space of an elliptically fibred $K3$. The duality map between the relevant BPS states was given in detail in the branches of moduli space of Type IIB with constant coupling.
In Type IIB the flat coordinate, $T_{IIB}=w_0/w_1$, defined in terms of the periods of the underlying $K3$ geometry encompasses all the information on the position of the 7-branes in the background when the elliptic fibre has constant complex structure. It is therefore the natural coordinate to be used when analysing the map with Wilson lines on the Heterotic string side. In fact, redefining the Kahler structure of the torus, $T_h$, in the Heterotic theory to include information on the Wilson lines. We have found that it becomes the natural coordinate to account for the effects of Wilson lines to the mass of BPS states.
For the the case of the $Spin(32)/Z_2$ Heterotic string the masses of BPS states for specific values of the Wilson lines were analysed in detail. In particular, we considered the enhancements: $SO(8)\times SU(2)\times U(1)\times
U(1)_G \rightarrow E_7$ and $(SU(2)\times U(1))^2 \times SO(8)$. In both cases we identified the dual BPS string junctions responsible for the enhancement on the Type IIB side and computed their masses finding complete agreement between the two sets of states.
It would be interesting to consider the case with non constant coupling in Type IIB. Work in this direction is under way.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Matthias Gaberdiel for explaining to me many aspects of his works and for suggestions. I thank also Michael Green for his continuous support and valuable comments and explanations. I would also like to thank W. Lerche for correspondence. I have enjoyed very helpful conversations with Fernando Quevedo, Tathagata Dasgupta and Pierre Vanhove.
I would like to acknowledge financial support from CNPq (Brazilian Ministry of Science) through a PhD. Scholarship. I also acknowledge partial financial support from the Cambridge Overseas Trust.
Generalising The Hypergeometric Relation Among ${\cal L}$, $w_0$ and $w_1$.
===========================================================================
Several times in this paper we had to rewrite the distance ${\cal L}$ in terms of the periods of $K3$. In most cases when we have only four groups of dynamical branes in $S^2_s$ we can rewrite the integrals in terms of hypergeometric functions. It turns out then that the integrals in this form are related by means of the Kummer’s relations[@kummer]. However, when the blocks break apart and we have more then four points in the sphere we cannot write then as hypergeometric functions anymore. Nevertheless, we show that by Taylor expanding the integrand we can write the integrals as a sum of integrals that in turn can be related to hypergeometric functions. We can then apply Kummer’s relations to each element of the sum in separate and sum up the series again to obtain the relation we look for. This generalises the hypergeometric relations to a more general set of integrals. In this appendix we present the details of this calculations for the configuration considered in Section 4.1.
We start with the length of the string stretched from $z=1$ and $z=z_s$ \_[z\_s]{}\^1\^[-]{} =\_[z\_s]{}\^1 z\^[-]{}(z-1)\^[-]{}(z-z\_s)\^[-]{}(z-M)\^[-]{} . ł[uma]{} we now Taylor expand $(z-M)^{-\f{n}{6}}$ as (z-M)\^[-]{} = \_[l=0]{}\^C\_l(n)z\^l ł[umaa]{} where $C_l(n)$ are standard numerical coefficients. If we now plug this back in eq(\[uma\]) we have = \_[l=0]{}\^ C\_l(n) \_[z\_s]{}\^1 z\^[l-]{}(z-1)\^[-]{}(z-z\_s)\^[-]{} . ł[tresa]{} This integral can be written in terms of hypergeometric functions by doing a simple change of variables, ie, $z\rightarrow
(z-z_s)/(1-z_s)$. We obtain =\_[l=0]{}\^ C\_l(n)( F\_[2,1]{}\[,(-l),, 1-z\_s\]) . ł[umaaa]{} Similarly, we obtain for the periods, eq(4.17) and eq(4.18), w\_0 &=& \_[l=0]{}\^ C\_l(n)(z\_s F\_[2,1]{}\[,(+l),(1+l),z\_s\])\
w\_1 &=& -\_[l=0]{}\^ C\_l(n)((-1)\^[5/6]{}(-z\_s)\^[-]{} F\_[2,1]{}\[,(+l),(1+l), \]) .
Using the following Kummer relation[@kummer] e\^[i5/6]{}(1-z\_s)\^[2/3]{}F\_[2,1]{}\[,-l,,1-z\_s\] &=& z\_sF\_[2,1]{}\[,+l,1+l,z\_s\] +\
&=& e\^[i5/6]{}(-z\_s)\^[-]{}F\_[2,1]{}\[,+l,1+l,\] . we can write each element in the series representation of ${\cal L}$ in terms of the respective elements in the series representation of $w_0$ and $w_1$. Plugging this result back in the sum, eq(\[umaaa\]), and reexpressing the sum in its closed form, we arrive at the desired relation among ${\cal L}$ and $w_0$ and $w_1$, ie, = w\_0 - w\_1 .
It is clear that this procedure can be applied to any distribution of the dynamical units on the sphere by Taylor expanding an appropriate number of terms in the expression for ${\cal L}$, $w_0$ and $w_1$.
Fixing Moduli in Branch II
==========================
In Section 4.2 the first example of gauge enhancement in Branch II to be analysed was $SO(8)^3\times (SU(2)\times U(1))^2 \rightarrow E_7^2\times SO(8)$. To obtain this enhancement the following Wilson lines were turned on A\_8 &=& (/2\^4,/2\^4,a,a,0\^2,0\^4)\
A\_9 &=& (0\^4,/2\^4,0\^2,a,a,/2\^4) with $a=0$ and $a=1/2$ being the critical values for the Wilson lines parameter. This was enough to determine the masses of BPS gauge bosons transforming in the vector and adjoint representations of the gauge groups. With this information we fixed part of the map with the BPS states on the Type IIB theory. We saw also that to obtain the full gauge enhancement we needed BPS states transforming in the spinor and singlet representations of the gauge groups. However, for these states to become massless we need to tune not only the parameter in the Wilson lines but the geometric moduli as well. We want to determine the value of the geometric moduli, $R$ and $B$, such that the masses of the BPS states responsible for the enhancement above match with those in Type IIB. We will concentrate in three specific examples of gauge fields here. The quantum numbers of these states will be given explicitly. The results easily generalise to all other states.
We will concentrate on the spinor with quantum numbers $w^9=+1$ and $w^8=0$. And the singlet with quantum numbers, $w^9=+2$ and $w^8=0$. The other quantum numbers will be determined below.
Recall that the critical radius was, eq(4.53), determined in terms of [^9] = (/2\^4, 0\^4,/2\^2,a-/2,a-/2,0\^4) as $R_c^2/2 =1-\Lambda^2/2= a(1-a)$. Was the radius to be set in this form we would have the spinors and singlets all massless for all values of $a$. However this would generate enhancements that have no equivalent in Branch II on Type IIB. But we know that when $a=1/2$ we must have the appropriate enhancement. One way to guarantee that this is the case is to set $R^2/2=1/4$. But it turns out that this choice does not give the right expression for the masses of the spinor and singlets. This means we would be specifying a one-parameter family with the right gauge enhancements but not the dual family of the BPS states we identified on Type IIB. The masses of BPS states on both sides must match as well as the gauge enhancement pattern.
To obtain the correct BPS masses in the Heterotic theory we impose the condition that it matches the ones on Type IIB. This will fix $R$ and $B$.
The analysis naturally separate in two parts. The real part of the mass formula depends on $B$ and $a$ only. The constraint described above will fix $B$ in terms of $a$. The imaginary part depends on the radius, $R$, and $a$ only. The match with the mass formulas on type IIB fixes the relation between the two moduli. The imaginary part is given by Im(M\_h) = m\_9 - A\_9Q - w\^9( + (1+a\^2)). where we used Im(T) &=& +\
&=& + (1+a\^2) .
For the real part we obtain Re(M\_h) = m\_8 - A\_8Q +w\^9( -/2) . where Im(T) &=& -\
&=& - /2 .
Now we require that the imaginary part of the masses for both the spinors and singlets to be equal to $(1/2 - a)$ in order to agree with the map for the vectors as in Table 6. We choose three specific BPS gauge bosons to carry out the analysis explicitly. The results can be easily extended to all other gauge fields. The representatives we choose are Q\^1 &=& (0\^4,-1\^4,0\^2,-1\^6) ([**1,1,1,1,1**]{})\
Q\^2 &=& (+/2\^4,-/2\^4,+/2\^2,-/2\^6) ([**8\_s,1,1,1,1**]{})\
Q\^3 &=& (+/2,-/2\^3,-/2\^4,+/2, -/2,-/2\^6) ([**8\_c,1,2,1,1**]{}) .
First of all we see from eq(B.5) that for this gauge bosons to be massless at $a=1/2$ we must have $m_9=-1$ for the spinors and $m_9=-2$ for the singlet. Furthermore, requiring agreement with the imaginary part of the masses on the Type IIB theory we obtain = 2a -a\^2 - = - ( -a) . This fixes the radius as a function of the Wilson line parameter.
The requirement that the real part is as in Table 6 yields for $B$ Re(M\_h)(Q\^1)&=&m\_8+B|\_[a=/2]{}=0\
Re(M\_h)(Q\^2)&=&(m\_8-1+)|\_[a=/2]{}=0\
Re(M\_h)(Q\^3)&=&(m\_8+1+)|\_[a=/2]{}=0 . These equations imply that $B\in 2Z$. We choose $B=0$ (up to $SL(2,Z)$ transformations). The $m_8$ quantum number for the representatives as well as the expression for the real part of the masses away for $a=1/2$ are also fixed. We summarise the results in table 8.
$Q$ $m_8$ $m_9$ $w^8$ $w^9$ $Re(M_h)$ $Im(M_h)$
------- ------- ------- ------- ------- ------------- -------------
$Q^1$ $0$ $-2$ $0$ $2$ $0$ $(\1/2 -a)$
$Q^2$ $1$ $-1$ $0$ $1$ $-(\1/2-a)$ $(\1/2 -a)$
$Q^3$ $-1$ $-1$ $0$ $1$ $0$ $(\1/2 -a)$
: The quantum numbers and masses of the singlet and spinor representatives.
[99]{}
C. Vafa, [*Evidence for F-theory*]{}, Nucl.Phys. B469 (1996) 403-418, 9602022;
M. Bershadsky, K. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, C. Vafa, [*Geometric Singularities and Enhanced Gauge Symmetries*]{}, Nucl.Phys. B481 (1996) 215-252, 9605200;
K.S. Narain, M.H. Sarmadi , E. Witten, [*A Note on the Toroidal Compactification of Heterotic String Theory*]{}, Nucl.Phys.B279:369,1987;
P. Ginsparg, [*Comment on the Toroidal Compactification of Heterotic Superstrings*]{}, Phys.Rev.D35:648,1987;
P. Aspinwall, [*Enhanced Gauge Symmetries and K3 Surfaces*]{}, Phys. Lett. B357 (1995) 329,9507012 ;
A. Sen, [*F-theory and Orientifolds*]{}, Nucl.Phys. B475 (1996) 562-578;
A. Johansen, [*A comment on BPS states in F-theory in 8 dimensions*]{}, Phys.Lett.[**B395**]{} (1997) 36-41, hep-th/9608186.
M.R.Gaberdiel, B.Zwiebach, [*Exceptional groups from open strings*]{}, Nucl.Phys.[**B518**]{} (1998) 151, hep-th/9709013.
M.R.Gaberdiel, T.Hauer, B.Zwiebach, [*Open string-string junction transitions*]{}, Nucl.Phys.[**B525**]{} (1998) 117, hep-th/9801205.
Y.Imamura, [*$E_8$ Flavour Multiplets*]{}, Phys.Rev.[**D58**]{} (1998) 106005, hep-th/9802189.
O.DeWolfe and B.Zwiebach, [*String Junctions for Arbitrary Lie Algebra Representations*]{}, Nucl.Phys. B541 (1999) 509-565, hep-th/9804210;
O. DeWolfe, T. Hauer, A. Iqbal, B. Zwiebach, [*Uncovering the Symmetries on \[p,q\] 7-branes: Beyond the Kodaira Classification*]{}, 9812028 ;
O. DeWolfe, T. Hauer, A. Iqbal, B. Zwiebach, [*Uncovering Infinite Symmetries on \[p,q\] 7-branes: Kac-Moody Algebras and Beyond*]{},9812209 ;
O. DeWolfe, [*Affine Lie Algebras, String Junctions and 7-Branes*]{}, Nucl.Phys. B550 (1999) 622-637, 9809026 ;
Y. Imamura, [*String Junctions and Their Duals in Heterotic String Theory*]{}, Prog.Theor.Phys. 101 (1999) 1155-1164, 9901001 ; D. O’Driscoll, [*Non-Perturbative Structure in Heterotic Strings from Dual F-Theory Models*]{}, Phys.Lett. B454 (1999) 240-246, 9901028 ;
W. Lerche, S. Stieberger, [*Prepotential, Mirror Map and F-Theory on K3*]{}, Adv.Theor.Math.Phys. 2 (1998) 1105-1140, 9804176 ;
G. Lopes Cardoso, G Curio, D Lust, T Mohaupt, [*On the Duality between the Heterotic String and F-Theory in 8 Dimensions*]{}, Phys.Lett. B389 (1996) 479-484, 9609111;
A. Dabholkar, J. A. Harvey, [*Non renormalisation of the Superstring Tension.*]{}, Phys.Rev.Lett.63:478,1989;
A. Sen, [*BPS states in a Three Brane Probe*]{}, Phys.Rev. D55 (1997) 2501-2503;
W. Lerche, S. Stieberger, N. P. Warner, [*Quartic Gauge Couplings from K3 Geometry*]{}, hep-th/9811228;
W. Lerche, S. Stieberger, [*On the Anomalous and Global Interactions of Kodaira 7-Planes*]{}, hep-th/9903232;
K. Dasgupta, S. Mukhi, [*F-Theory at Constant Coupling*]{}, Phys.Lett. B385 (1996) 125-131, 9606044 ;
G. Lopes Cardoso, D. Lust, T. Mohaupt, [*Threshold Corrections and Symmetry Enhancement in String Compactifications*]{}, Nucl.Phys. B450 (1995) 115-173, 9412209 ;
O. Bergman, M. R. Gaberdiel, G. Lifschytz, [*String Creation and Heterotic-Type I’ Duality*]{},Nucl.Phys. B524 (1998) 524-544, 9711098 ;
A. Erdelyi, [*Higher Transcendental Functions - Bateman Manuscript Project*]{}, McGraw-Hill, 1953;
[^1]: E-mail address: [M.C.D.Barrozo@damtp.cam.ac.uk]{}
[^2]: A similar expression has been considered in compactifications of the Heterotic string on $Z_N$ orbifolds [@mohaupt]. However, only for the case of very specific Wilson lines.
[^3]: We set $\alpha_h
' = 2$ in this section.
[^4]: We present the details in Appendix A
[^5]: Here $p$ stands for all permutations within the bracket
[^6]: This is basically two copies of the 9 dimension case reviewed in [@gab3].
[^7]: Where (e \# -) stands for an even number of minus signs in the permutations.
[^8]: In Appendix B we present some details.
[^9]: In this section $\alpha_h' = 2$.
|
---
abstract: 'We discuss medium corrections of the nucleon-nucleon (NN) cross sections and their influence on direct reactions at intermediate energies $\gtrsim 50$ MeV/nucleon. The results obtained with free NN cross sections are compared with those obtained with a geometrical treatment of Pauli-blocking and with NN cross sections obtained with Dirac-Bruecker methods. We show that medium corrections may lead to sizable modifications for collisions at intermediate energies and that they are more pronounced in reactions involving weakly bound nuclei.'
address:
- '$^{(1)}$Department of Physics and Astronomy, Texas A$\&$M University-Commerce, Commerce, Texas 75429-3011, USA[^1]'
- '$^{(2)}$Department of Physics, Akdeniz University, TR-07058 Antalya, Turkey[^2]'
author:
- 'M. Karakoc$^{(1,2)}$, and C. Bertulani$^{(1)}$'
title: Medium effects in direct reactions
---
Introduction
============
Obtaining a valid optical potential for direct reactions is very crucial for reactions such as nucleon knockout at intermediate and high energies [@HRB91] ($\gtrsim 50$ MeV/nucleon). A microscopic method to deduce optical potentials is based on the construction of the potentials using an effective nucleon-nucleon (NN) interaction, or cross section (e.g. those of Ref. [@Ra79]). This technique is often used to construct the real part of an optical potential and with its imaginary part assumed rescaled in strength to better reproduce experimental data on elastic scattering, or total reaction cross sections. The real and imaginary parts of the potential can also be constructed independently as in Refs. [@trache01; @trache02], where the procedure starts from a NN effective interaction with independent real and imaginary parts. It has also been shown that one can use nucleon-nucleon cross sections as the microscopic input [@HRB91], instead of nucleon-nucleon interactions. In this case, an effective treatment of Pauli-blocking on nucleon-nucleon scattering is needed, as it manifests through medium density dependence. In fact, it is well known that a complete numerical modeling of heavy-ion central collision dynamics requires to account for medium effects on the nucleon-nucleon cross sections [@Ber01]. The main goal in central collisions is to explore the nuclear equation of state (EOS) by studying global collective variables describing the collision process.
Medium modifications of NN scattering have smaller effects in direct reactions since generally low nuclear densities are probed. Although, no comparison with experimental data was supplied, a first work on this effect in knockout reactions was presented in Ref. [@BC10]. In this contribution, we report recent progress on studies of medium modifications in knockout reactions. We will report on medium effects in the NN cross section for the description of knockout reactions by means of (a) a geometrical treatment of Pauli-blocking and a (b) Dirac-Brueckner treatment. A comparison of our calculations to a large number of published experimental data is shown, and full results will be published else where [@KBBT12]. The aim of this project is to obtain more accurate spectroscopic factors that will lead to better understanding nuclear structure and to check and improve the credibility of the use of knockout reactions as an indirect methods for nuclear astrophysics.
Medium effects
==============
Nucleon-nucleon cross sections
------------------------------
In the literature, there are different fits to the free (total) nucleon-nucleon cross sections, such as those in Refs. [@BC10; @JTWT93]. In this work, we have used the parametrization from the Ref. [@BC10] which is obtained using the experimental data from Particle Data Group [@pdgxnn]. For practical reasons, the free nucleon-nucleon cross sections are separated in three energy intervals, by means of the expressions $$\sigma_{pp}=\left\{
\begin{array}
[c]{c}%
19.6+{4253/ E} -{ 375/ \sqrt{E}}+3.86\times 10^{-2}E \\
({\rm for }\ E < 280\ {\rm MeV}) \\ \; \\
32.7-5.52\times 10^{-2}E+3.53\times 10^{-7}E^3 \\
- 2.97\times 10^{-10}E^4 \\
({\rm for }\ 280\ {\rm MeV}\le E < 840\ {\rm MeV}) \\ \; \\
50.9-3.8\times 10^{-3}E+2.78\times 10^{-7}E^2 \\
+1.92\times 10^{-15} E^4 \\
({\rm for}\ 840 \ {\rm MeV} \le E \le 5 \ {\rm GeV})
\end{array}
\right.
\label{signn1}$$ for proton-proton collisions, and $$\sigma_{np}=
\left\{
\begin{array}
[c]{c}%
89.4-{2025/ \sqrt{E}}+{19108/ E}-{43535/ E^2}
\\
({\rm for }\ E < 300\ {\rm MeV}) \\ \; \\
14.2+{5436/ E}+3.72\times 10^{-5}E^2-7.55\times 10^{-9}E^3
\\
({\rm for }\ 300\ {\rm MeV}\le E < 700\ {\rm MeV}) \\ \; \\
33.9+6.1\times 10^{-3}E-1.55\times 10^{-6}E^2 \\
+1.3\times 10^{-10}E^3\\
({\rm for}\ 700 \ {\rm MeV} \le E \le 5 \ {\rm GeV})
\end{array}
\right.
\label{signn2}$$ for proton-neutron collisions. $E$ is the projectile laboratory energy. The coefficients in the above equations have been obtained by a least square fit to the nucleon-nucleon cross section experimental data over a variety of energies, ranging from 10 MeV to 5 GeV.
Most practical studies of medium corrections of nucleon-nucleon scattering are done by considering the effective two-nucleon interaction in infinite nuclear matter, or G-matrix, as a solution of the Bethe-Goldstone equation [@GWW58] $$\langle\mathbf{k}|\mathrm{G}(\mathbf{P},\rho_1,\rho_2)|\mathbf{k}_{0}\rangle
=\langle\mathbf{k}|\mathrm{v}_{NN}|\mathbf{k}_{0}\rangle
-\int{\frac
{d^{3}k^{\prime}}{(2\pi)^{3}}}{\frac{\langle\mathbf{k}|\mathrm{v}%
_{NN}|\mathbf{k^{\prime}}\rangle Q(\mathbf{k^{\prime}},\mathbf{P}%
,\rho_1,\rho_2)\langle\mathbf{k^{\prime}}|\mathrm{G}(\mathbf{P},\rho_1,\rho_2)|\mathbf{k}%
_{0}\rangle}{E(\mathbf{P},\mathbf{k^{\prime}})-E_{0}-i\epsilon}}\,
\label{10}%$$ with $\mathbf{k}_{0}$, $\mathbf{k}$, and $\mathbf{k^{\prime}}$ the initial, final, and intermediate relative momenta of the NN pair, ${\bf k}=({\bf k}_1-{\bf k}_2)/2$ and ${\bf P}=({\bf k}_1+{\bf k}_2)/2$. If energy and momentum is conserved in the binary collision, ${\bf P}$ is conserved in magnitude and direction, and the magnitude of ${\bf k}$ is also conserved. $\mathrm{v}_{NN}$ is the nucleon-nucleon potential. $E$ is the energy of the two-nucleon system, and $E_{0}$ is the same quantity on-shell. Thus $
E(\mathbf{P},\mathbf{k})=e(\mathbf{P}+\mathbf{k})+e(\mathbf{P}-\mathbf{k}%
)$, with $e$ the single-particle energy in nuclear matter. It is also implicit in Eq. that the final momenta ${\bf k}$ of the NN-pair also lie outside the range of occupied states.
Eq. (\[10\]) is density-dependent due to the presence of the Pauli projection operator $Q$, defined by $$Q(\mathbf{k},\mathbf{P},\rho_1,\rho_2)=\left\{
\begin{array}
[c]{c}%
1,\ \ \ \mathrm{if}\ \ \ k_{1,2}>k_{F1,F2}\\
0,\ \ \ \ \ \mathrm{otherwise.}%
\end{array}
\right.
\label{12}%$$ with $k_{1,2}$ the magnitude of the momenta of each nucleon. $Q$ prevents scattering into occupied intermediate states. The Fermi momenta $k_{F1,F2}$ are related to the proton and neutron densities by means of the zero temperature density approximation, $k_{Fi}=(3\pi^2 \rho_i/2)^{1/3}$. For finite nuclei, one usually replaces $\rho_i$ by the local densities to obtain the local Fermi momenta. This is obviously a rough approximation, but very practical and extensively used in the literature.
Only by means of several approximations, Eq. can be related to nucleon-nucleon cross sections. If one neglects the medium modifications of the nucleon-mass, and scattering through intermediate states, the medium modification of the NN cross sections can be accounted for by the geometrical factor $Q$ only, that is, $$\sigma_{NN}(k,\rho_1,\rho_2)=\int {\frac{d\sigma_{NN}^{free}}{d\Omega}} Q(k,P, \rho_1,\rho_2) d\Omega ,
\label{snngeo}$$ where $Q$ is now a simplified geometrical condition on the available scattering angles for the scattering of the NN-pair to unoccupied final states.
After this point, the geometrical treatment of Pauli corrections can be performed using the isotropic NN scattering approximation because the numerical calculations can be largely simplified if we assume that the free nucleon-nucleon cross section is isotropic. In this case, a formula which fits the numerical integration of the geometrical model reads [@BC10] $$\begin{aligned}
\sigma_{NN}(E,\rho_p,\rho_t) &=&\sigma_{NN}^{free}(E)\frac{1}{1+{1.892\left(\frac{2\rho_<}{\rho_0}\right)\left(\frac{|\rho_p-\rho_t|}{\tilde{\rho}\rho_0}\right)^{2.75}}}\nonumber \\
&\times&
\left\{
\begin{array}
[c]{c}%
\displaystyle{1-\frac{37.02 \tilde{\rho}^{2/3}}{E}}, \ \ \ {\rm if} \ \ E>46.27 \tilde{\rho}^{2/3}\\ \, \\
\displaystyle{\frac{E}{231.38\tilde{\rho}^{2/3}}},\ \ \ \ \ {\rm if} \ \ E\le 46.27 \tilde{\rho}^{2/3}\end{array}
\right.
\label{VM1}\end{aligned}$$ where $E$ is the laboratory energy in MeV, $\tilde{\rho}=(\rho_p+\rho_t)/\rho_0$, $\rho_<={\rm min} (\rho_p,\rho_t)$, $\rho_{i=p, t}$ is the local density of nucleus $i$, and $\rho_0=0.17$ fm$^{-3}$.
The Brueckner method goes beyond this simple treatment of Pauli blocking, generating medium effects from nucleon-nucleon potentials, such as the Bonn potential. An example is the work presented in Ref. [@LM:1993; @LM:1994], where a practical parametrization was given, which we will from now on refer as Brueckner theory. It reads[^3] $$\begin{aligned}
\sigma_{np} & = \left[ 31.5 +0.092\left| 20.2-E^{0.53}\right|^{2.9}\right] \frac{1+0.0034E^{1.51} \rho^2}{1+21.55\rho^{1.34}} \nonumber\\
\sigma_{pp} & = \left[ 23.5 +0.00256\left( 18.2-E^{0.5}\right)^{4.0}\right] \frac{1+0.1667E^{1.05} \rho^3}{1+9.704\rho^{1.2}}. \label{brueckner}\end{aligned}$$ A modification of the above parametrization was done in Ref. [@Xian98], which consisted in combining the free nucleon nucleon cross sections parametrized in Ref. [@Cha90] with the Brueckner theory results of Ref. [@LM:1993; @LM:1994]. Their parametrization, which tends to reproduce better the nucleus-nucleus reaction cross sections, is $$\begin{aligned}
\sigma_{np} &= \left[ -70.67-18.18\beta^{-1}+25.26\beta^{-2}+113.85\beta\right]
\times \frac{1+20.88E^{0.04} \rho^{2.02}}{1+35.86\rho^{1.9}} \nonumber\\
\sigma_{pp} &= \left[ 13.73-15.04\beta^{-1}+8.76\beta^{-2}+68.67\beta^{4}\right]
\times \frac{1+7.772E^{0.06} \rho^{1.48}}{1+18.01\rho^{1.46}},
\label{pheno}\end{aligned}$$ where $\beta=\sqrt{1-1/\gamma^2}$ and $\gamma=E{\rm[MeV]}/931.5+1$. We will denote Eq. as the phenomenological parametrization.
The differences between the parametrization of the Brueckner, Eq. , the geometrical Pauli blocking, Eq. , and the phenomenological one, Eq. , are visible. Figure \[signpe\] is an example of that, where the varied parameterizations of proton-neutron cross sections are presented as a function of the laboratory energy. The solid line is the parametrization of the free $\sigma_{pn}$ cross section given by Eq. . The other curves include medium effects for symmetric nuclear matter for $\rho=\rho_0/4$, where $\rho_0=0.17$ fm$^{-3}$. The dashed curve includes the geometrical effects of Pauli blocking, as described by Eq. . The dashed-dotted curve is the result of using the Brueckner theory, Eq. , and the dotted curve is the phenomenological parametrization, Eq. . The large departure of results of the Brueckner parametrization above 300 MeV is not physical since Eq. is valid only under 300 MeV (pion production threshold) [@LM:1993; @LM:1994]. On the other hand, the differences at lower energies are physical and Pauli-blocking effectively reduces the in-medium [*np*]{} cross section. This is not so explicit in the phenomenological parametrization.
The above interpretation cannot be extended to the [*pp*]{} cross sections, which are shown in Figure . Here it is seen that the geometrical Pauli-blocking correction decreases the cross section much more than in the other cases. Some important differences are also clearly visible at larger energies, $E\gtrsim 100$ MeV/nucleon. We now study the impact of these different methods on direct reactions at intermediate energies.
Total reaction cross-sections
-----------------------------
As we mentioned before, obtaining a valid optical potential in knockout reactions [@HRB91] and various direct reactions is crucial. One way to test the optical potentials is to reproduce total reaction cross sections. As elastic scattering data at intermediate energies are scarce, for knockout reactions a proper test this can be done by calculating total reaction cross sections for the core and the valence particle, separately. The total reaction cross-sections can be obtained in the framework of the eikonal approximation as follows $$\sigma_R=2\pi \int db \ b \left[ 1- \left| S(b)\right|^2 \right],$$ where $S$ is the eikonal $S$-matrices. The relation between optical potentials and $S$-matrices is given by $$S_i(b)=\exp[i\chi(b)]=\exp\left[-\frac{i}{\hbar v}\int_{-\infty}^\infty U_{iT}({\bf r})dz\right], \label{sib}$$ where $r=\sqrt{b^2+z^2}$, and $U_{iT}$ is the particle($i$)-target($T$) optical potential. A semiclassical probabilistic approach has been followed to calculate the cross sections and other observables in direct reactions as described in Refs. [@HM85; @HBE96], and a relation has been established between the optical potential and the nucleon-nucleon scattering amplitude in Ref. [@HRB91]. This relation is frequently mentioned in the literature as the “t-$\rho\rho$ approximation". “Experimentally deduced" optical potentials are often not available from elastic and inelastic scattering involving radioactive nuclei. Therefore, the t-$\rho\rho$ approximation is one of the most practical techniques to obtain optical potentials. In this approximation, the eikonal phase becomes $$\chi(b)=\frac{1}{k_{NN}}\int_{0}^{\infty}dq\ q\
\rho_{p}\left( q\right) \rho_{t}\left( q\right) f_{NN}\left(
q\right) J_{0}\left( qb\right)
\ ,\label{eikphase}%$$ where $\rho_{p,t}\left( q\right) $ is the Fourier transform of the nuclear densities of the projectile and target, and $f_{NN}\left( q\right) $ is the high-energy nucleon-nucleon scattering amplitude at forward angles, which can be parametrized as $$f_{NN}\left( q\right) =\frac{k_{NN}}{4\pi}\sigma_{NN}\left( i+\alpha
_{NN}\right) \exp\left( -\beta_{NN}q^{2}\right) \ .
\label{fnn}%$$
![The total reaction cross section of the p+$\;^{12}$C taken from Ref. [@CAR96]. The curves are calculated with the free NN cross sections from Ref. [@JTWT93] (solid), with a geometrical account of Pauli blocking (dashed), a phenomenological fit from Ref. [@Xian98] (dotted), and a correction from Brueckner theory (dashed-dotted). The triangle-dotted curve is calculated with the same free NN cross sections as in Ref. [@JTWT93], but with an another HFB calculation [@Bei74] for the $^{12}$C ground state density.[]{data-label="totsigR"}](totsigR.eps)
One often neglects nuclear medium effects in the experimental analysis of knockout reactions, as pointed out in Ref. [@BC10]. However, their importance has been well known for a long time in the study of elastic and inelastic scattering, as well as of total reaction cross sections [@HRB91; @Ra79]. In these situations, a systematic analysis of the medium effects has been presented in Ref. [@HRB91], and it has shown that the effects becomes larger at lower energies, where Pauli blocking strongly reduces the nucleon-nucleon cross sections in the medium.
The p + $^{12}$C total reaction cross sections in the energy range of 20-100 MeV/nucleon shown in Figure \[totsigR\] presents the justification of these statements, where the experimental data taken from the Ref. [@CAR96]. The cross sections were calculated from the Eqs. (\[sib\],\[eikphase\],\[fnn\]) and $^{12}$C density from a Hartree-Fock-Bogoliubov calculation (HFB) [@Bei74]. Various different calculations are shown in Figure \[totsigR\]. The result of Eq. with the free nucleon-nucleon cross sections and the carbon matter density from a HFB calculation [@Nus12] is represented by the solid curve, whereas the triangle-dotted curve (the triangles are not data, but used for better visibility) uses a different HFB density [@Bei74], consistent with the calculations presented in Ref. [@trache02]. As expected, that the agreement between the two calculations is very good.
The same calculation procedure, but this time including medium corrections for the nucleon-nucleon cross section, has been performed to obtain the other curves in Figure \[totsigR\]. It is evident that the results are very different than the former. The medium effects with various different are shown with the dotted, dashed-dotted, and dashed curves, which they correspond to the calculations with phenomenological, Brueckner, and Pauli geometrical methods, respectively. Obviously, medium effects modify the results, yielding a closer reproduction of the data. But the large experimental error bars do not allow a fair judgment of which model reproduces better the data.
Nucleon knockout reactions
--------------------------
Momentum distributions of the projectile-like residues in one-nucleon knockout are a measure of the spatial extent of the wavefunction of the struck nucleon, while the cross section for the nucleon removal scales with the occupation amplitude, or probability (spectroscopic factor), for the given single-particle configuration in the projectile ground state. The longitudinal momentum distributions are given by (see, e.g., Refs. [@HBE96; @BH04; @BG06]) $$\begin{aligned}
\frac{d\sigma_{\mathrm{str}}}{dk_{z}} & = (C^2S) \frac{1}{2\pi%
}\frac{1}{2l+1}\sum_{m}\int_{0}^{\infty}d^{2}b_{n} \left[ 1-\left\vert
S_{n}\left( b_{n}\right) \right\vert ^{2}\right] \nonumber\\
& \times \int_{0}^{\infty}%
d^{2}b_c \ \left\vert S_{c}\left( b_{c}\right) \right\vert ^{2}\left\vert \int_{-\infty}^{\infty}dz \exp\left[ -ik_{z}z\right]
\psi_{lm}\left( \mathbf{r}\right) \right\vert ^{2},\label{strL}%\end{aligned}$$ where $k_{z}$ represents the longitudinal component of $\mathbf{k}_{c}$ (final momentum of the core of the projectile nucleus) and $(C^2S)$ is the spectroscopic factor, and $\psi_{lm}\left(\mathbf{r}\right)$ is the wavefunction of the core plus (valence) nucleon system $(c+n)$ in a state with single-particle angular momentum $l,m$.
### $^{12}$[C]{}($^{17}$[C]{},$^{16}$[B)X]{} at [35 MeV/u]{}\
![Transverse momentum distributions for the $^{12}$C($^{17}$C,$^{16}$B)X system at 35 MeV/u. Solid lines represent calculations including medium corrections. Dashed lines stem from calculations that do not include medium corrections. The data are taken from Ref. [@LEC09].[]{data-label="c17c12"}](c17c12.eps)
The one-proton removal reaction, $^{12}$C($^{17}$C,$^{16}$B)X, from $^{17}$C at 35 MeV/nucleon has been measured with the aim to explore the low-lying structure of the unbound $^{16}$B nucleus. In Ref. [@LEC09], the unbound $^{16}$B nucleus is assumed to be a $d$-wave neutron decay from $^{15}$B+$n$ system. Here, we have focused to study the consequences of medium corrections on the transverse momentum distribution of the $^{16}$B fragment following the same assumptions as in Ref. [@LEC09]. The configuration mixing of the proton removed from $^{17}$C is assumed to be $$\begin{aligned}
&&|^{17}\text{C}\rangle = \alpha_1|^{16}\text{B}(0^-)\otimes \pi 1p_{3/2}\rangle \nonumber \\
&+& \alpha_2|^{16}\text{B}(3^-_1)\otimes \pi1p_{3/2}\rangle + \alpha_3|^{16}\text{B}(2^-_1)\otimes \pi1p_{3/2}\rangle \nonumber \\
&+& \alpha_4|^{16}\text{B}(2^-_2)\otimes \pi1p_{3/2}\rangle + \alpha_5|^{16}\text{B}(1^-_1)\otimes \pi1p_{3/2}\rangle \nonumber \\
&+& \alpha_6|^{16}\text{B}(3^-_2)\otimes \pi1p_{3/2}\rangle,\end{aligned}$$ where $\alpha_i$ is the spectroscopic amplitude for a core-single particle configuration $i=(c\otimes nlj)$.
In Ref. [@LEC09] a good agreement between data and calculated transverse momentum distributions was achieved using the spectroscopic factors from a shell-model calculation with the WBP interaction [@WAR92]. However, they have obtained a theoretical result of 24.7 mb for total cross section which diverges from the measured cross-section, 6.5(1.5) mb. In Ref. [@GAD04], an explanation is proposed to this large discrepancy as due to a reduction of the spectroscopic factor by 70% for strongly bound nucleon systems. The theoretical estimates for the cross section with the reduction at the spectroscopic factor becomes 7.5 mb, in reasonable accordance with the data. We do not challenge the assumptions of Ref. [@LEC09], and we use the same configuration mixing and spectroscopic factors as in [@LEC09]. The proton binding potential parameters are given in Ref. [@KBBT12], which are adjusted to obtain the effective separation energies. Here, as it is shown in Figure \[c17c12\], we find that medium corrections change the cross sections by 5% which is rather small to explain the observed difference with the total cross sections.
### $^9$[Be(]{}$^{11}$[Be,]{}$^{10}$[Be)X]{} at [60 MeV/u]{}\
![Longitudinal momentum distributions of for the reaction $^9$Be($^{11}$Be,$^{10}$Be) at 60 MeV/nucleon. Solid lines represent calculations including medium corrections. Dashed lines stem from calculations that do not include medium corrections. The data is taken from Ref. [@AUM00].[]{data-label="be11be9"}](be11be9.eps)
In order to further understand the medium effects on knockout reactions, we consider the $^9$Be($^{11}$Be,$^{10}$Be)X system at 60 MeV/u which can be modeled by a core plus valence system with the assumption $^{10}$Be$_{gs}(0^+)+{\rm n}$ in $2s_{1/2}$ orbital for the ground state of $^{11}$Be$_{gs}(1/2^+)$ ($S_n=0.504$ MeV). Here we use the same Woods-Saxon potential parameters for the bound state as published in Ref. [@HBE96]: ($R_0=2.70$ fm, $a_0=0.52$ fm). In Figure \[be11be9\] and Table \[cross\] we present our results for the the neutron removal longitudinal momentum distribution of 60 MeV/nucleon $^{11}$Be projectiles incident on $^9$Be targets. We find that medium corrections for this system change the cross sections by 50% which is quite big.
It has been show that $^{17}$C has a small “effective" size and that $^{11}$Be has a large effective size among the low energy systems studied in Ref. [@KBBT12]. Therefore, the wavefunctions of weakly bound systems extend far within the target where the nucleon-nucleon cross sections are strongly modified by the medium. Momentum distributions and nucleon removal cross sections in knockout reactions are thus expected to change appreciably with the inclusion of medium corrections of nucleon-nucleon cross section. Such corrections are also expected to play a more significant role for loosely-bound systems.
------------------------------ ------- ----------- ------- -----------
$\sigma_{-1n}$
$=\sigma_{dif}+\sigma_{str}$ Full no medium Full no medium
Strip. \[mb\] 7.56 5.63 122.5 164.1
Diff. \[mb\] 18.42 19.15 49.6 97.3
Total \[mb\] 25.98 24.78 172.1 261.4
------------------------------ ------- ----------- ------- -----------
: The cross sections calculated for the systems, $^{12}$C($^{17}$C,$^{16}$B) at 35 MeV/nucleon and $^9$Be($^{11}$Be,$^{10}$Be) at 60 MeV/nucleon.[]{data-label="cross"}
Summary
=======
In this small report, we have explored the importance of the medium modifications of the nucleon-nucleon cross sections on direct reactions, and particularly on knockout reactions. It has been shown that the effects are noticeable at low energies. Nonetheless, we have noticed that medium effects do not lead to sizable modifications on the shapes of momentum distributions. We have shown this explicitly by comparing our results with a large number of available experimental data in Ref. [@KBBT12]. As expected on physics grounds, these corrections are larger for experiments at lower energies, around 50 MeV/nucleon, and for weakly bound nuclei.
Medium effects in knockout reactions have been frequently ignored in the past. We show that they have to be included in order to obtain a better accuracy of the extracted spectroscopic factors. Although these conclusions might not come as a big surprise, they have not been properly included in many previous experimental analyses.
This work was partially supported by the US-DOE grants DE-SC004972 and DE-FG02-08ER41533 and DE-FG02-10ER41706, and by the Research Corporation.
References
==========
[30]{}
M. Hussein, R. Rego and C.A. Bertulani, Physics Reports 201, 279 (1991).
L. Ray, Phys. Rev. C 20, 1857 (1979).
L. Trache, F. Carstoiu, C.A. Gagliardi, and R.E. Tribble, Phys. Rev. Lett. 87, 271102 (2001).
L. Trache, F. Carstoiu, A.M. Mukhamedzhanov, and R.E. Tribble, Phys. Rev. C 66, 035801 (2002).
C.A. Bertulani, J. Phys. G 27 (2001) L67.
C.A. Bertulani and C. De Conti, Phys. Rev. C 81, 064603 (2010).
M. Karakoc, A. Banu, C.A. Bertulani, and L. Trache, submitted for publication.
S. John, L.W. Townsend, J.W. Wilson, and R.K. Tripathi, Phys. Rev. C 48, 766 (1993).
Particle Data Group. J. Phys. G 33, 1 (2006).
L.C. Gomes, J.D. Walecka, and V.F. Weisskopf, Ann. Phys. (N.Y.) 3, 241 (1958).
G. Q. Li and R. Machleidt, Phys. Rev. C 48, 1702 (1993).
G. Q. Li and R. Machleidt, Phys. Rev. C 49, 566 (1994).
C. Xiangzhou, F. Jun, S. Wenqing, Ma Yugang, W. Jiansong, and Ye Wei, Phys. Rev. C58, 572 (1998).
S. K. Charagi and S. K. Gupta, Phys. Rev. C 41, 1610 (1990).
M.S. Hussein, K.W. McVoy, Nucl. Phys. A 445, 124 (1985).
K. Hencken, G. Bertsch and H. Esbensen, Phys. Rev. C 54, 3043 (1996).
R. F. Carlson, At. Data Nucl. Data Tables 63, 93 (1996).
M. Beiner and R.J. Lombard, Ann. Phys. N.Y. 86, 262 (1974);\
F. Carstoiu and R.J. Lombard, ibid. 217, 279 (1992).
http://www.nscl.msu.edu/ brown/resources/resources.html
C.A. Bertulani and P.G. Hansen, Phys. Rev. C 70, 034609 (2004).
C.A. Bertulani and A. Gade, Comp. Phys. Comm. 175, 372 (2006).
J.-L. Lecouey [*et al.*]{}, Phys. Lett. B 672, 6 (2009).
E.K. Warburton and B.A. Brown, Phys. Rev. C 46, 923 (1992).
A. Gade [*et al.*]{}, Phys. Rev. Lett. 93, 042501 (2004).
T. Aumann [*et al.*]{}, Phys. Rev. Lett. 84, 35 (2000).
[^1]: mesut.karakoc@tamuc.edu
[^2]: carlos.bertulani@tamuc.edu
[^3]: The misprinted factor 0.0256 in Ref. [@LM:1994] has been corrected to 0.00256.
|
---
abstract: |
This paper enhances the pricing of derivatives as well as optimal control problems to a level comprising risk. We employ nested risk measures to quantify risk, investigate the limiting behavior of nested risk measures within the classical models in finance and characterize existence of the risk-averse limit. As a result we demonstrate that the nested limit is unique, irrespective of the initially chosen risk measure. Within the classical models risk aversion gives rise to a stream of risk premiums, comparable to dividend payments. In this context, we connect coherent risk measures with the Sharpe ratio from modern portfolio theory and extract the Z-spreada widely accepted quantity in economics to hedge risk. By involving the Z-spread we demonstrate that risk-averse problems are conceptually equivalent to the risk-neutral problem.
The results for European option pricing are then extended to risk-averse American options, where we study the impact of risk on the price as well as the optimal time to exercise the option.
We also extend Merton’s optimal consumption problem to the risk-averse setting.
**Keywords:** Risk measures, Optimal control, BlackScholes
**Classification:** 90C15, 62P05, 90C39, 91G20
author:
- 'Alois Pichler[^1]'
- 'Ruben Schlotter[^2]'
bibliography:
- 'Literatur-book.bib'
- 'Literatur-paper2.bib'
title: '**Quantification of Risk in Classical Models of Finance**'
---
\[sec:Introduction\]Introduction
================================
This paper studies discrete classical models in finance under risk aversion and their behavior in a high-frequency setting. Using nested risk measures we first study risk aversion in the multiperiod model.
We develop risk aversion in a discrete time and discrete space setting and find an important consistency property of nested risk measures. This consistency property, termed *divisibility*, is crucial in high-frequency trading environments. For this, our study of risk-averse models extends to continuous time processes as well. This very property allows consistent decision making, i.e., decisions, which are independent of individually chosen discretizations or trading frequencies. Our results also give rise to a generalized BlackScholes framework, which incorporates risk aversion in addition.
@Riedel2004-R has introduced risk measures in a dynamic setting. Later, @Cheridito2004 study risk measures for bounded càdlàg processes and @cher2006-R also discuss risk measures in a discrete time setting. @Ruszczynski-R introduce nested risk measures, for which @Philpott2013 provide an economic interpretation as an insurance premium on a rolling horizon basis. For a recent discussion on risk measures and dynamic optimization we refer to @DeLara2015-R. Applications can be found in @PhilpottMatos or @Maggioni2012, e.g., where stochastic dual dynamic programming methods are addressed, see also @Guigues2012.
Divisibility is an indispensable prerequisite in defining an infinitesimal generator based on discretizations. This generator, called *risk generator*, constitutes the risk averse assessment of the dynamics of the underlying stochastic process. Using the risk generator we characterize the existence of the risk-averse limit of discrete pricing models. For coherent risk measures the risk generator constitutes a nonlinear operator, comparable to the classical infinitesimal generator but with an additional term, accounting for risk, which takes the form $$s_{\rho}\,\left|\sigma\,\partial_{x}\,\cdot\right|.$$ Here, $s_{\rho}$ is a scalar expressing the risk aversion and $\sigma$ is the volatility of the diffusion process describing the asset price. It turns out that the risk generator does not dependent on the risk measure, which is employed to set up the generator. This surprising feature has important conceptual implications, as evaluating a risk measure is often an optimization problem itself. As well we derive that the scaling quantity $s_{\rho}$ allows the economic interpretation of a Sharpe ratio and $s_{\rho}\cdot\sigma$ is the Z-spread.
Using the risk generator we derive a nonlinear BlackScholes equation, which we relate to the BlackScholes formula for dividend paying stocks proposed by @Merton1973. Moreover we relate risk-averse pricing models to foreign exchange options models as in @Garman1983. Nonlinear Black Scholes equations have been discussed previously in @Barles1998 and @Sevcovic2016 in the context of modeling transaction costs. There, the nonlinearity is in the second derivative. In contrast, risk aversion leads to drift uncertainty and causes nonlinearity in the first derivative.
For coherent risk measures we derive an explicit solution for the European option pricing problem. We show that risk aversion expressed via coherent risk measures can be interpreted either as an extra dividend payment or capital injection. Furthermore we relate risk-aversion with a change of currency as in the foreign exchange option model. The amount of the dividend payment or, equivalently, the interest rate in the risk-averse currency, is given by a multiple of the Sharpe ratio and the volatility of the underlying stock. This ratio, which expresses risk aversion, arises for any coherent risk measure and does not depend on a specific market model such as the BlackScholes model.
Using a free boundary formulation we extend the analysis from European to American option pricing. For the BlackScholes option pricing of European and American options, risk-aversion naturally leads to a bid-ask spread, which we quantify explicitly.
Similarly we extend the Merton optimal consumption problem to a risk-averse setting. We elaborate on the optimal controls and show that risk-aversion reduces the investment in risky assets and increases consumption. We observe the same pattern as for European and American options, i.e., risk-aversion corrects the drift of the underlying market model. For all classical models discussed here, the risk averse assessment still allows explicit pricing and control formulae.
Preliminaries on risk measures
==============================
Recall the definition of *law invariant, coherent risk measures* $\rho\colon L\to\mathbb{R}$ defined on some vector space $L$ of $\mathbb{R}$-valued random variables first. They satisfy the following axioms introduced by @Artzner1999-R.
1. \[enu:Monotonicity\]Monotonicity: $\rho(Y)\le\rho(Y^{\prime})$, provided that $Y\le Y^{\prime}$ almost surely;
2. \[enu:equivariance\]Translation equivariance: $\rho(Y+c)=\rho(Y)+c$ for $c\in\mathbb{R}$;
3. \[enu:Convexity\] Convexity: $\rho\big((1-\lambda)\,Y+\lambda\,Y^{\prime}\big)\le(1-\lambda)\,\rho(Y)+\lambda\,\rho(Y^{\prime})$ for $0\le\lambda\leq1$;
4. \[enu:Homogeneous\]Positive homogeneity: $\rho(\lambda\,Y)=\lambda\,\rho(Y)$ for $\lambda\ge0$;
<!-- -->
1. \[enu:LawInvariance\]Law invariance: $\rho(Y)=\rho(Y^{\prime})$, whenever $Y$ and $Y^{\prime}$ have the same law, i.e., $P(Y\le y)=P(Y^{\prime}\le y)$ for all $y\in\mathbb{R}$.
The expectation ($\rho(Y)=\operatorname{{\mathds E}}Y$) is the *risk-neutral* risk measure, satisfying all axioms above. In contrast to the risk-neutral setting, the risk averse buyer and seller have an opposite perception of risk. We shall refer to $\rho(Y)$ as the seller’s ask price and to $-\rho(-Y)$ as the buyer’s bid price. Note, that $-\rho(-Y)\le\rho(Y)$.[^3]
Nested risk measures
--------------------
We consider a filtered probability space $\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathcal{T}},P\right)$ and associate $t\in\mathcal{T}$ with *stage* or *time*. For the discussion of risk in a dynamic setting we introduce nested risk measures corresponding to the evolution of risk over time. Nested risk measures are compositions of conditional risk measures (cf. @PflugRomisch2007-R). Following @Ruszczynski-R, we introduce conditional risk measures $\rho^{t}$, conditioned on the sigma algebra $\mathcal{F}_{t}$, as $$\rho^{t}\left(Y\left|\mathcal{F}_{t}\right.\right):=\operatorname*{ess\,sup}_{Q\in\mathcal{Q}}\operatorname{{\mathds E}}_{Q}\left[Y\left|\,\mathcal{F}_{t}\right.\right],\label{eq:condRM}$$ where $\mathcal{Q}$ is a convex set of probability measures absolutely continuous with respect to $P$ (cf. also @Delbaen02coherentrisk-R). The conditional risk measures $\rho^{t}$ satisfy conditional versions of the Axioms \[enu:Monotonicity\]\[enu:LawInvariance\] above. For further details we refer the interested reader also to @Shapiro2014 [Section 6.8.2]. For the essential supremum of a set of random variables as in we refer to @Karatzas1998 [Appendix A].
We now introduce nested risk measures in discrete time.
\[def:nRM\] The *nested risk measure*, nested at times $t_{0}<\dots<t_{n}$, is $$\rho^{t_{0}:t_{n}}(Y):=\rho^{t_{0}}\left(\dots\rho^{t_{n}}(Y\mid\mathcal{F}_{t_{n}})\dots\mid\mathcal{F}_{t_{0}}\right),\label{eq:17}$$ where $(\rho^{t_{i}})_{i=0}^{n}$ is a family of conditional risk measures. For a partition $\mathcal{P}=\left(t_{0},t_{1},\dots,t_{n}\right)$ we denote the nested risk measure also by $\rho^{\mathcal{P}}(Y)$.
Similar as above, we distinguish the buyer and seller perspective and consider the bid price $$-\rho^{t_{0}:t_{n}}(-Y):=-\rho^{t_{0}}\left(\dots\rho^{t_{n}}(-Y\mid\mathcal{F}_{t_{n}})\dots\mid\mathcal{F}_{t_{0}}\right),$$ as well as the ask price in .
Nested risk measures for discrete processes
-------------------------------------------
To illustrate key properties of nested risk measures as defined in we discuss the binomial model, well-known from finance by employing the mean semi-deviation, a coherent risk measure satisfying all Axioms \[enu:Monotonicity\]\[enu:LawInvariance\] above.
\[def:mSD\]The mean semi-deviation risk measure of order $p\geq1$ and $Y\in L^{p}$ at level $\beta\in[0,1]$ is $$\operatorname{\mathsf{SD}}_{p,\beta}(Y):=\operatorname{{\mathds E}}Y+\beta\left\Vert \left(Y-\operatorname{{\mathds E}}Y\right)_{+}\right\Vert _{p}.$$
Consider the stochastic process $S=(S_{0},\dots,S_{T})$ with initial state $S_{0}$. The process $S$ models a stock in stochastic finance over time. The discrete stock price changes according to $P\left(S_{t+\Delta t}=S_{t}\cdot e^{\pm\sigma\sqrt{\Delta t}}\right)=p_{\pm}$, where $$p:=p_{+}:=\frac{e^{r\Delta t}-e^{-\sigma\sqrt{\Delta t}}}{e^{\sigma\sqrt{\Delta t}}-e^{-\sigma\sqrt{\Delta t}}}\;\text{and}\;p_{-}:=1-p_{+}.$$ This setting describes the risk free risk measure, because $\operatorname{{\mathds E}}S_{t+\Delta t}=pS_{t}e^{\sigma\sqrt{\Delta t}}+(1-p)S_{t}e^{-\sigma\sqrt{\Delta t}}=S_{t}e^{r\Delta t}$, where $r$ is the risk free interest rate.
We can evaluate various classical coherent risk measures for this binomial model explicitly. The following remark considers the mean semi-deviation for the binomial model as well as the nested mean semi-deviation for the $n$-period model.
\[rem:risk-binom\]Consider the one stage setting in Figure \[fig:single\] first. The risk-averse bid price for the stock $S_{\Delta t}$ employing the mean semi-deviation $\operatorname{\mathsf{SD}}_{1,\beta}$ of order $1$ with risk level $\beta$ in the Bernoulli model is $$\begin{aligned}
-\operatorname{\mathsf{SD}}_{1,\beta}(-S_{\Delta t}) & =\operatorname{{\mathds E}}S_{\Delta t}-\beta\operatorname{{\mathds E}}\left(-S_{\Delta t}+\operatorname{{\mathds E}}S_{\Delta t}\right)_{+}\\
& =pS_{0}e^{\sigma\sqrt{\Delta t}}+(1-p)Se^{-\sigma\sqrt{\Delta t}}-\beta\,p(1-p)\left(Se^{\sigma\sqrt{\Delta t}}-Se^{-\sigma\sqrt{\Delta t}}\right).\end{aligned}$$ We introduce the new probability weights $$\begin{aligned}
\widetilde{p} & :=p\big(1-\beta(1-p)\big)\label{eq:15}\end{aligned}$$ so that $$-\operatorname{\mathsf{SD}}_{1,\beta}(-S_{\Delta t})=\widetilde{\operatorname{{\mathds E}}}S_{\Delta t}.$$ We now repeat this observation in $n$ stages and consider an $n$-period binomial model with step size $\Delta t=\frac{T}{n}$, cf. Figure \[fig:multi\]. The nested risk measure is $$-\operatorname{\mathsf{SD}}_{1,\beta}^{0:T}(-S_{T})=-\operatorname{\mathsf{SD}}_{1,\beta}\left(\dots\operatorname{\mathsf{SD}}_{1,\beta}\left(S_{T}\mid S_{T-\Delta t}\right)\dots\right)=\widetilde{\operatorname{{\mathds E}}}S_{T},$$ where the last expectation is with respect to the probability measure $$\widetilde{P}\left(S_{T}=S_{0}e^{\sigma\left(2k\sqrt{\Delta t}-n\sqrt{\Delta t}\right)}\right)=\binom{n}{k}\widetilde{p}^{k}(1-\widetilde{p})^{n-k},\qquad k=0,\dots,n.$$ It follows from inversion and the central limit theorem that $$\frac{\frac{1}{\sigma}\log\frac{S_{T}}{S_{0}}+n\sqrt{\Delta t}}{2\sqrt{\Delta t}}\to\mathcal{N}\left(n\widetilde{p},n\widetilde{p}(1-\widetilde{p})\right),\label{eq:18}$$ the limit is normally distributed. For this model to be non-degenerate it is inevitable that $\tilde{p}\to\frac{1}{2}$ as $n\to\infty$ and it is important to note that this forces specific choices of $\beta$ in .
In what follows we develop the setting in Remark \[rem:risk-binom\] from an economic perspective and elaborate a rigorous mathematical solution to the question of convergence in . We characterize the risk measures for which the risk-averse model converges by involving a new consistency property. This consistency property is naturally formulated in continuous time and related to a risk-averse analogue of the infinitesimal generator in dynamic optimization.
The risk-averse limit of discrete option pricing models
=======================================================
Most well-known coherent risk measures in the literature as the Average Value-at-Risk, the Entropic Value-at-Risk as well as the mean semi-deviation involve a parameter which accounts for the degree of risk aversion. As Remark \[rem:risk-binom\] elaborates, the nested risk-averse binomial model does not necessarily lead to a well-defined limit. It is essential to relate the coefficient of risk aversion of the conditional risk measures to its time period. We therefore introduce re-parameterized families of coherent risk measures which we call *divisible*. The divisibility property is central in discussing the limiting behavior of risk-averse economic models.
\[def:divisibility\]Let $Y\sim\mathcal{N}(0,1)$ be a standard normal random variable. A family of coherent measures of risk $\rho=\left\{ \rho_{\Delta t}\colon\Delta t>0\right\} $, parameterized by $\Delta t$, is called *divisible* if the limit $$\lim_{\Delta t\downarrow0}\,\frac{\rho_{\Delta t}(\sqrt{\Delta t}\cdot Y)}{\Delta t}=s_{\rho}\label{eq:17-1}$$ exists for some $s_{\rho}\ge0$.
The (conditional) expectation is divisible with $s_{\mathbb{E}}=0$. We provide some further examples of divisible families.
The Entropic Value-at-risk at level $\kappa\ge0$ is given by $$\operatorname{\mathsf{EV@R}}_{\kappa}(Y):=\inf_{x>0}\,\frac{1}{x}(\kappa+\log\operatorname{{\mathds E}}e^{xY}).$$ The family $\left\{ \operatorname{\mathsf{EV@R}}_{\beta\cdot\Delta t}\colon\Delta t>0\right\} $ is divisible with $s_{\operatorname{\mathsf{EV@R}}_{\beta}}=\sqrt{2\beta}$. For a comprehensive discussion on the Entropic Value-at-Risk in this context we refer the reader to @PichlerSchlotter2019a [Proposition 9].
For completeness we provide an additional family of divisible risk measures and remark that any convex combination of them is divisible too.
\[lem:gauss\]The family $$\left\{ \operatorname{\mathsf{SD}}_{p,\beta;\Delta t}:=\operatorname{\mathsf{SD}}_{p,\beta\cdot\sqrt{\Delta t}}\right\} ,\quad\Delta t>0,$$ of mean semi-deviations is divisible with limit $$s_{\operatorname{\mathsf{SD}}_{p,\beta}}=\beta\left(2\pi\right)^{-\frac{1}{2p}}2^{\frac{1}{2}-\frac{1}{2p}}\cdot\Gamma\left(\frac{p+1}{2}\right)^{\frac{1}{p}}.$$
Let $Y\sim\mathcal{N}(0,\Delta t)$, then $$\begin{aligned}
\operatorname{{\mathds E}}Y_{+}^{p} & =\int_{\mathbb{R}}\max(y,0)\cdot y^{p}\cdot\frac{1}{\sqrt{2\pi\Delta t}}e^{-\frac{y^{2}}{2\Delta t}}\,\mathrm{d}y=\frac{1}{\sqrt{2\pi\Delta t}}\int_{0}^{\infty}y^{p}\cdot e^{-\frac{y^{2}}{2\Delta t}}\,\mathrm{d}y.\end{aligned}$$ Employing the Gamma function, the latter integral is $$\begin{aligned}
\frac{1}{\sqrt{2\pi\Delta t}}\int_{0}^{\infty}y^{p}\cdot e^{-\frac{y^{2}}{2\Delta t}}\,\mathrm{d}y & =\frac{1}{\sqrt{2\pi}}\left(2^{\frac{p-1}{2}}\Gamma\left(\frac{p+1}{2}\right)\Delta t^{\frac{p}{2}}\right).\end{aligned}$$ Taking the $p$-th root and multiplying by $\beta\sqrt{\Delta t}$ we obtain $$\frac{\operatorname{\mathsf{SD}}_{p,\beta\sqrt{\Delta t}}(Y)}{\Delta t}=\beta\left(2\pi\right)^{-\frac{1}{2p}}2^{\frac{1}{2}-\frac{1}{2p}}\cdot\Gamma\left(\frac{p+1}{2}\right)^{\frac{1}{p}},$$ showing the assertion.
We now extend nested risk measures to continuous time and demonstrate that the extension is well-defined for divisible families of risk measures. As a result we show that the risk-averse binomial option pricing model converges exactly for divisible families of risk measures.
\[def:nRM2\]Let $T>0$, $t\in[0,T)$ and let $\rho^{\mathcal{P}}$ be divisible for every partition $\mathcal{P}\subset[t,T]$, cf. Definition \[def:nRM\]. The *nested risk measure $\rho^{t:T}$ in continuous time for a random variable $Y$* is $$\rho^{t:T}\left(Y\left|\,\mathcal{F}_{t}\right.\right):=\lim_{\mathcal{P}\subset[t,T]}\,\rho^{\mathcal{P}}\left(Y\left|\,\mathcal{F}_{t}\right.\right)\qquad\text{almost surely},\label{eq:CnRM}$$ where the almost sure limit is among all partitions $\mathcal{P}\subset[t,T]$ with mesh size $\left\Vert \mathcal{P}\right\Vert $ tending to zero for those random variables $Y$, for which the limit exists.
The following proposition evaluates the nested mean semi-deviation for the Gaussian random walk, the basic building block of diffusion processes.
\[prop:randomWalk\]Let $W=(W_{t})_{t\in\mathcal{P}}$ be a Wiener process evaluated on the partition $\mathcal{P}$. For the family of conditional risk measures $\left(\operatorname{\mathsf{SD}}_{p,\beta_{t_{i}}\cdot\sqrt{\Delta t_{i}}}^{t_{i}}\right)_{t_{i}\in\mathcal{P}}$, the nested mean semi-deviation is $$\operatorname{\mathsf{SD}}_{p,\beta}^{0:T}(W_{T})=\sum_{i=0}^{n-1}\beta_{t_{i}}\Delta t_{i}\cdot\left(2\pi\right)^{-\frac{1}{2p}}2^{-\frac{1}{2}}\Gamma\left(\frac{p+1}{2}\right)^{\frac{1}{p}}.\label{eq:D-W}$$
Note that $Z:=W_{t_{i+1}}-W_{t_{i}}\sim\mathcal{N}(0,t_{i+1}-t_{i})$ and the conditional mean semi-deviation is (using conditional translation equivariance \[enu:equivariance\]) $$\begin{aligned}
\operatorname{\mathsf{SD}}_{p,\beta_{t_{i}}\cdot\sqrt{\Delta t_{i}}}^{t_{i}}\left(W_{t_{i+1}}\left|\,W_{t_{i}}\right.\right) & =W_{t_{i}}+\operatorname{\mathsf{SD}}_{p,\beta_{t_{i}};\sqrt{\Delta t_{i}}}^{t_{i}}\left(W_{t_{i+1}}-W_{t_{i}}\left|\,W_{t_{i}}\right.\right).\end{aligned}$$ Furthermore the conditional expectation is zero as Brownian motion has independent and stationary increments with mean zero and thus, with Lemma \[lem:gauss\], $$\begin{aligned}
\operatorname{\mathsf{SD}}_{p,\beta_{t_{i}}\cdot\sqrt{\Delta t_{i}}}^{t_{i}}\left(W_{t_{i+1}}\left|\,W_{t_{i}}\right.\right) & =W_{t_{i}}+\beta_{t_{i}}\Delta t_{i}\cdot\left(2\pi\right)^{-\frac{1}{2p}}2^{-\frac{1}{2}}\Gamma\left(\frac{p+1}{2}\right)^{\frac{1}{p}}.\end{aligned}$$ Iterating as in Definition \[def:nRM\] shows $$\begin{aligned}
\operatorname{\mathsf{SD}}_{p,\beta}^{0:T}(W_{T}) & =\sum_{i=0}^{n-1}\beta_{t_{i}}\Delta t_{i}\cdot\left(2\pi\right)^{-\frac{1}{2p}}2^{-\frac{1}{2}}\Gamma\left(\frac{p+1}{2}\right)^{\frac{1}{p}},\end{aligned}$$ the assertion.
For constant risk level $\beta$ we obtain $$\operatorname{\mathsf{SD}}_{p,\beta}^{0:T}(W_{T})=\sum_{i=0}^{n-1}\Delta t_{i}\cdot\beta\cdot\left(2\pi\right)^{-\frac{1}{2p}}2^{-\frac{1}{2}}\Gamma\left(\frac{p+1}{2}\right)^{\frac{1}{p}}=T\cdot s_{\operatorname{\mathsf{SD}}_{p,\beta}},$$ so that the accumulated risk grows linearly in time.
The risk generator
------------------
This section addresses nested risk measures for Itô processes. Furthermore, we characterize convergence under risk using a natural condition involving normal random variables and introduce a nonlinear operator, the *risk generator,* which also allows discussing risk-averse optimal control problems.
It is well-known that the binomial model in Figure \[fig:multi\] converges to the geometric Brownian motion. We therefore discuss Itô process $(X_{s})_{s\in\mathcal{T}}$ solving the stochastic differential equation $$\begin{aligned}
\mathrm{d}X_{s} & =b(s,X_{s})\,\mathrm{d}s+\sigma(s,X_{s})\,\mathrm{d}W_{s},\quad s\in\mathcal{T},\label{eq:SDE}\\
X_{t} & =x\nonumber \end{aligned}$$ for $\mathcal{T}=[t,\,T]$. We assume that $X$ following is well-defined and refer to @Oksendal2003 [Theorem 5.2.1] for sufficient conditions. We further assume that $s\mapsto\sigma(s,X_{s})$ is Hölder continuous for some $\gamma\in(0,\nicefrac{1}{2})$.
We introduce the *risk generator* for families of divisible coherent risk measures. The risk generator describes the momentary evolution of the risk of the stochastic process.
\[def:RiskGen\]Let $X=(X_{t})_{t}$ be a continuous time process and $(\rho_{\Delta t})_{\Delta t}$ be a family of divisible risk measures. The *risk generator* *based on* $(\rho_{\Delta t})_{\Delta t}$ is $$\mathcal{R}_{\rho}\Phi(t,x):=\lim_{\Delta t\downarrow0}\frac{1}{\Delta t}\Bigl(\rho_{\Delta t}\bigl(\Phi(t+\Delta t,X_{t+\Delta t})\left|\,X_{t}=x\right.\bigr)-\Phi(t,x)\Bigr)\label{eq:riskGen}$$ for those functions $\Phi\colon\mathcal{T}\times\mathbb{R}\to\mathbb{R}$, for which the limit exists.
Using the ideas from Proposition \[prop:randomWalk\] we obtain explicit expressions for the risk generator for Itô diffusion processes.
\[prop:RGen\]Let $X$ be the solution of and let the family $(\rho_{\Delta t})_{\Delta t}$ be divisible. For $\Phi\in C^{2}(\mathcal{T}\times\mathbb{R})$ such that $\Phi(t,X_{t})\in L^{p}$ the risk generator based on $(\rho_{\Delta t})_{\Delta t}$ is given by the nonlinear differential operator $$\begin{aligned}
\mathcal{R}_{\rho}\Phi(t,x) & =\left(\Phi_{t}+b\,\Phi_{x}+\frac{\sigma^{2}}{2}\Phi_{xx}+s_{\rho}\cdot\left|\sigma\,\Phi_{x}\right|\right)(t,x).\label{eq:21}\end{aligned}$$
By assumption, $\Phi\in C^{2}(\mathcal{T}\times\mathbb{R})$ and hence we may apply Itô’s formula. For convenience and ease of notation we set $f_{1}(t,x):=\left(\Phi_{t}+b\Phi_{x}+\frac{\sigma^{2}}{2}\Phi_{xx}\right)(t,x)$ and $f_{2}(t,x):=\left(\sigma\Phi_{x}\right)(t,x)$. In this setting, Eq. rewrites as $$\begin{aligned}
\mathcal{R}_{\rho}\Phi(t,x) & =\lim_{\Delta t\downarrow0}\,\frac{1}{\Delta t}\rho_{\Delta t}\left[\left.\int_{t}^{t+\Delta t}f_{1}(s,X_{s})\mathrm{d}s+\int_{t}^{t+\Delta t}f_{2}(s,X_{s})\mathrm{d}W_{s}\right|\,X_{t}=x\right].\end{aligned}$$ To show for each fixed $(t,x)$ it is enough to show that $$\left|\mathcal{R}_{\rho}\Phi(t,x)-f_{1}(t,x)-s_{\rho}\left|f_{2}(t,x)\right|\right|\leq0.$$ Using convexity of the risk measure together with the triangle inequality we have $$\begin{aligned}
0\leq & \lim_{\Delta t\downarrow0}\,\left|\rho_{\Delta t}\left[\left.\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{1}(s,X_{s})\mathrm{d}s-f_{1}(t,x)\right|\,X_{t}=x\right]\right|\nonumber \\
& +\lim_{\Delta t\downarrow0}\left|\rho_{\Delta t}\left[\left.\frac{1}{h}\int_{t}^{t+\Delta t}f_{2}(s,X_{s})\mathrm{d}W_{s}-s_{\rho}\left|f_{2}(t,x)\right|\right|\,X_{t}=x\right]\right|.\label{eq:??}\end{aligned}$$ We continue by looking at each term separately. Note that $s\mapsto f_{1}(s,X_{s})-f_{1}(t,x)$ is continuous almost surely and hence the mean value theorem for definite integrals implies that there exists a $\xi\in[t,t+\Delta t]$ such that $$\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{1}(s,X_{s})\mathrm{d}s-f_{1}(t,x)=f_{1}(\xi,X_{\xi})-f_{1}(t,x),\quad\text{almost surely}.$$ From continuity of $\rho$ in the $L^{p}$ norm we may conclude $$\lim_{\Delta t\downarrow0}\,\frac{1}{h}\rho_{\Delta t}\left(\left.\left|\int_{t}^{t+\Delta t}f_{1}(s,X_{s})-f_{1}(t,x)\,\mathrm{ds}\right|\,\right|X_{t}=x\right)=0.$$
Note that the stochastic integral term in can be bounded by $$\begin{aligned}
\rho_{\Delta t}\left[\left.\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{2}(s,X_{s})\mathrm{d}W_{s}\right|\,X_{t}=x\right] & \leq\rho_{\Delta t}\left[\left.\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{2}(s,X_{s})-f_{2}(t,x)\mathrm{d}W_{s}\right|\,X_{t}=x\right]\\
& +\,\rho_{\Delta t}\left[\left.\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{2}(t,x)\mathrm{d}W_{s}\right|\,X_{t}=x\right],\end{aligned}$$ where $\rho_{\Delta t}\left[\left.\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{2}(t,x)\mathrm{d}W_{s}\right|\,X_{t}=x\right]$ converges to $s_{\rho}\left|f_{2}(t,x)\right|$ and hence $$\begin{aligned}
\eqref{eq:??} & \leq\lim_{\Delta t\downarrow0}\left|\rho_{\Delta t}\left[\left.\frac{1}{\Delta t}\int_{t}^{t+\Delta t}f_{2}(s,X_{s})-f_{2}(t,x)\mathrm{d}W_{s}\right|\,X_{t}=x\right]\right|.\end{aligned}$$ Furthermore, the stochastic integral $M_{\Delta t}:=\int_{t}^{t+\Delta t}f_{2}(s,X_{s})-f_{2}(t,x)\mathrm{d}W_{s}$ is a continuous martingale with $M_{0}=0$ and hence the divisibility of $\rho$ shows that there exists a constant $C_{1}$ independent of $\Delta t$ such that $$\rho_{\Delta t}(M_{\Delta t})\leq C_{1}\sqrt{\Delta t}\cdot\left\Vert M_{\Delta t}\right\Vert _{p}.$$ Applying the BurkholderDavisGundy inequality implies the upper bound $$\begin{aligned}
\rho_{\Delta t}(M_{\Delta t}) & \leq\widetilde{C}_{1}\sqrt{\Delta t}\left[\operatorname{{\mathds E}}\left|\int_{t}^{t+\Delta t}\left(f_{2}(s,X_{s})-f_{2}(t,x)\right)^{2}\mathrm{d}s\right|^{\frac{p}{2}}\right]^{\frac{1}{p}}\end{aligned}$$ for some constant $\widetilde{C}_{1}$ depending on $p$. Because the diffusion coefficient $\sigma$ is Hölder continuous with exponent $\gamma$ and $\Phi_{x}$ is continuous it follows that $f_{2}$ is locally Hölder continuous in $t$, i.e., there exists a $C_{2}>0$ and a $\gamma<\frac{1}{2}$ such that $$\left|f_{2}(s,X_{s})-f_{2}(t,X_{t})\right|\leq C_{2}\left|t-s\right|^{\gamma}$$ almost surely and hence $$\begin{aligned}
\operatorname{{\mathds E}}\left|\int_{t}^{t+\Delta t}\left(f_{2}(s,X_{s})-f_{2}(t,x)\right)^{2}\mathrm{d}s\right|^{\frac{p}{2}} & \leq\left(\int_{t}^{t+\Delta t}C_{2}\left|s-t\right|^{2\gamma}\mathrm{d}s\right)^{\frac{p}{2}}=\left(C_{2}\frac{\Delta t^{2\gamma+1}}{2\gamma+1}\right)^{\frac{p}{2}}.\end{aligned}$$ We conclude $$\begin{aligned}
\rho_{\Delta t}(M_{\Delta t}) & \leq\widetilde{C}_{1}\sqrt{\Delta t}\left(C_{2}\frac{\Delta t^{2\gamma+1}}{2\gamma+1}\right)^{\frac{1}{2}}=\widetilde{C}_{1}\Delta t^{1+\gamma}\sqrt{\frac{C_{2}}{2\gamma+1}},\end{aligned}$$ such that $\frac{1}{h}\rho_{\Delta t}(M_{\Delta t})$ vanishes, which concludes the proof.
For random variables $Y$ of the form $$Y=\int_{t}^{T}c(s,X_{s})\,\mathrm{d}s+\Psi(X_{T}),$$ where $X$ is a Itô diffusion process based on Brownian motion, the limit exists as a consequence of Definition \[def:divisibility\] as well as the arguments in the proof of Proposition \[prop:RGen\] above.
The next proposition relates the convergence of the binomial model under risk to the risk generator.
Denote by $S^{n}$ the $n$-period binomial tree model and suppose that the sequence of binomial processes $(S^{n})_{n}$ converges to an Itô process $X$ in distribution. The risk-averse discrete binomial model in Remark \[rem:risk-binom\] converges if and only if the family of nested risk measures is divisible.
Let $(\rho_{\Delta t})_{\Delta t}$ be a divisible family of risk measures, then Proposition \[prop:RGen\] shows that the risk generator exists for the diffusion process $X$ and hence $$\rho_{\Delta t}(X_{t+\Delta t}\mid X_{t})-X_{t}=c_{\rho}\cdot\Delta t+o(\Delta t)\label{eq:condRisk}$$ for some constant $c_{\rho}\in\mathbb{R}$. Moreover, applying Itô’s lemma to implies divisibility. In addition, existence of the limit of risk-averse binomial models as in Remark \[rem:risk-binom\] is equivalent to $$\rho_{\Delta t}\left(S_{t+\Delta t}^{n}\mid S_{t}^{n}\right)-S_{t}^{n}=c_{\rho}\cdot\Delta t+o(\Delta t),$$ for $n$ large enough. It follows from Fatou’s lemma that $$\rho^{0:T}(X_{T})\leq\lim_{n\to\infty}\rho^{0:T}(S_{T}^{n})$$ and hence the convergence of the risk-averse binomial model implies the existence of the risk generator.
For the converse, note that $X_{0}=S_{0}$ and hence $$\begin{aligned}
\lim_{n\to\infty}\rho_{\Delta t}\left(S_{\Delta t}^{n}-S_{0}\right) & =\lim_{n\to\infty}\rho_{\Delta t}\left(S_{\Delta t}^{n}+X_{\Delta t}-X_{\Delta t}-X_{0}\right)\\
& \leq\lim_{n\to\infty}\rho_{\Delta t}\left(S_{\Delta t}^{n}-X_{\Delta t}\right)+\rho_{\Delta t}\left(X_{\Delta t}-X_{0}\right).\end{aligned}$$ The latter term satisfies $$\rho_{\Delta t}\left(X_{\Delta t}-X_{0}\right)=c_{\rho}\cdot\Delta t+o(\Delta t)$$ by assumption. For the first term notice that $(S_{\Delta t}^{n}-X_{\Delta t})_{n}$ tends to zero in distribution and hence also converges in probability. Moreover, $\left(\rho_{\Delta}(S_{\Delta t}^{n}-X_{\Delta t})\right)_{n}$ is uniformly bounded and hence $$\lim_{n\to\infty}\rho_{\Delta t}\left(S_{\Delta t}^{n}-X_{\Delta t}\right)=0,$$ showing the assertion.
Dynamic programming
-------------------
This section introduces risk-averse dynamic equations using nested risk measures. In what follows we consider the value function involving nested risk measures defined by $$V(t,x):=\rho^{t:T}\left(e^{-r(T-t)}\,\Psi(X_{T})\mid X_{t}=x\right).\label{eq:riskValue}$$ Here, $r$ is a discount factor and $\Psi$ a given terminal payoff function. The structure of nested risk measures allows extending the dynamic programming principle to the risk-averse setting.
\[lem:DPP\]Let $(t,x)\in[0,T)\times\mathbb{R}$ and $\Delta t>0$, then it holds that $$V(t,x)=\rho^{t:T}\left(\left.e^{-r\Delta t}V(t+\Delta t,X_{t+\Delta t})\right|X_{t}=x\right).\label{eq:Principle}$$
By definition of the risk-averse value function it holds $$V(t+\Delta t,X_{t+\Delta t})=\rho^{t+\Delta t:T}\left(e^{-r(T-t-\Delta t)}\Psi(X_{T})\mid X_{t+\Delta t}\right)$$ and hence the definition of nested risk measure gives $$\begin{aligned}
\rho^{t:t+\Delta t}\left(\left.e^{-r\Delta t}V(t+\Delta t,X_{t+\Delta t})\right|X_{t}=x\right) & =\rho^{t:T}\left(e^{-r(T-t)}\Psi(X_{T})\mid X_{t}=x\right),\end{aligned}$$ which shows the assertion.
To derive the dynamic equations for $V$ we consider in the form $$0=\frac{1}{\Delta t}\rho^{t:t+\Delta t}\left(\left.e^{-r\Delta t}V(t+\Delta t,X_{t+\Delta t})-V(t,x)\right|X_{t}=x\right)\label{eq:DPP2}$$ for $\Delta t\to0$. The following theorem employs the risk generator to obtain dynamic equations for the risk-averse value function .
\[thm:ValuePDE\]The value function solves the terminal value problem $$\begin{aligned}
V_{t}(t,x)+b(t,x)V_{x}(t,x)+\frac{\sigma^{2}(t,x)}{2}V_{xx}(t,x)+s_{\rho}\left|\sigma(t,x)\cdot V_{x}(t,x)\right|-rV(t,x) & =0,\label{eq:5}\\
V(T,x) & =\Psi(x),\nonumber \end{aligned}$$ provided that $V\in C^{2}$ in a neighborhood of $(t,x)$.
Let $(t,x)\in[0,T]\times\mathbb{R}$ be fixed. Similarly to the risk neutral case we define $$Y_{s}:=e^{-r(s-t)}V(s,X_{s}),\qquad s\geq t.$$ The process $Y_{s}$ satisfies the Itô formula $$\begin{aligned}
Y_{t+\Delta t}= & Y_{t}+\int_{t}^{t+\Delta t}e^{-r(s-t)}\left(V_{t}+b\cdot V_{x}+\frac{\sigma^{2}}{2}V_{xx}-rV\right)(s,X_{s})\mathrm{d}s\\
& +\int_{t}^{t+\Delta t}e^{-r(s-t)}\sigma(s,X_{s})\cdot V_{x}(s,X_{s})\mathrm{d}W_{s}.\end{aligned}$$ As $\sigma$ is Hölder continuous it follows that $$\lim_{\Delta t\downarrow0}\,\frac{1}{\Delta t}\rho^{t:t+\Delta t}\left(\left.\int_{t}^{t+\Delta t}e^{-r(s-t)}\sigma\cdot V_{x}\mathrm{d}W_{s}\right|X_{t}=x\right)=s_{\rho}\cdot\left|\sigma\cdot\partial_{x}V\right|.$$ Following the lines of the proof of Proposition \[prop:RGen\] implies $$\begin{aligned}
0 & =\lim_{\Delta t\downarrow0}\,\frac{1}{\Delta t}\rho^{t:t+\Delta t}\left(\left.e^{-r\Delta t}Y_{t+\Delta t}-Y_{t}\right|X_{t}=x\right)\\
& =\lim_{\Delta t\downarrow0}\,\frac{1}{\Delta t}\rho^{t:t+\Delta t}\left[\int_{t}^{t+\Delta t}e^{-r(s-t)}\left(V_{t}+b\cdot V_{x}+\frac{\sigma^{2}}{2}V_{xx}-rV\right)\mathrm{d}s+\int_{t}^{t+\Delta t}e^{-r(s-t)}\sigma\cdot V_{x}\mathrm{d}W_{s}\right]\\
& =V_{t}(t,x)+b(t,x)\cdot V_{x}(t,x)+\frac{\sigma^{2}(t,x)}{2}V_{xx}(t,x)+s_{\rho}\left|\sigma(t,x)\cdot V_{x}(t,x)\right|-rV(t,x),\end{aligned}$$ which demonstrates the assertion.
The dynamic programming principle and Theorem \[thm:ValuePDE\] are usually considered in an environment involving controls $u$. This extends to the risk-averse setting as well. The proofs are similar and we thus state the result only. The value function $$V(t,x):=\inf_{u}\,\rho^{t:T}\left(\int_{t}^{T}c(s,X_{s}^{u},u_{s})\,\mathrm{d}s+\Psi(X_{T}^{u})\right)$$ with diffusion process $X_{T}^{u}$ governed by an adapted control process $u$ satisfies the HamiltonJacobiBellman equation $$\begin{aligned}
\inf_{u}\left\{ V_{t}(\cdot)+b(\cdot,u)V_{x}(\cdot)+\frac{\sigma^{2}(\cdot,u)}{2}V_{xx}(\cdot)+s_{\rho}\left|\sigma(\cdot,u)\cdot V_{x}(\cdot)\right|-rV(\cdot)+c(\cdot,u)\right\} & =0,\\
V(T,\cdot) & =\Psi(\cdot).\end{aligned}$$ This is , but with an extra infimum among all controls $u$. We resume this discussion in Section \[sec:Merton\] below. For a general overview on stochastic optimal control and controlled processes in a risk neutral context we refer the interested reader to @Fleming2006.
\[sec:EU\]Pricing of options under risk
=======================================
The previous section discusses a discrete, risk-averse binomial option pricing problem and subsequently derives characterizations for the risk-averse limit to exist. In this section we study the risk-averse value functions of the limiting process of the binomial tree process, i.e., the geometric Brownian motion. In the risk-averse setting we find again explicit formulae. The resulting explicit pricing formulae lead us to interpret risk aversion as dividend payments and to relate the risk level $s_{\rho}$ to the Sharpe ratio.
Consider a market with one riskless asset, e.g., a bond and a risky asset, usually a stock. The return of the riskless asset is constant and denoted by $r$. As usual in the classical BlackScholes framework, the underlying stock $S$ is modeled by a geometric Brownian motion following the stochastic differential equation $$\begin{aligned}
\mathrm{d}S_{t} & =r\,S_{t}\,\mathrm{d}t+\sigma\,S_{t}\,\mathrm{d}W_{t}\label{eq:geom}\end{aligned}$$ with initial condition $S_{0}=s_{0}$.
The risk-averse BlackScholes model
----------------------------------
Similarly as above we distinguish the risk-averse value function $$V(t,x):=-\rho_{t:T}\left[-e^{-r(T-t)}\,\Psi(S_{T})\mid S_{t}=x\right]\label{eq:BSM-bid}$$ for the *bid price* and the corresponding value function for the *ask price* given by $$\widetilde{V}(t,x):=\rho_{t:T}\left[e^{-r(T-t)}\,\Psi(S_{T})\mid S_{t}=x\right].\label{eq:BSM-ask}$$ Notice that the discount rate $r$ is the same as in the dynamics of the stock $S=(S_{t})_{t}$. In the risk-neutral setting the bid and ask prices coincide.
Theorem \[thm:ValuePDE\] shows that the risk-averse value function of the bid price satisfies the PDE with terminal condition $$\begin{aligned}
V_{t}(t,x)+r\,x\,V_{x}(t,x)+\frac{\sigma^{2}\,x^{2}}{2}V_{xx}(t,x)-s_{\rho}\cdot\left|\sigma\,x\cdot V_{x}(t,x)\right|-r\,V(t,x) & =0,\label{eq:PDE-bid}\\
V(T,x) & =\Psi(x),\nonumber \end{aligned}$$ where $\Psi(x)$ is the payoff function. Similarly, the following PDE representation the value function $\widetilde{V}$ describing the ask price derives as $$\begin{aligned}
\widetilde{V}_{t}(t,x)+r\,x\,\widetilde{V}_{x}(t,x)+\frac{\sigma^{2}\,x^{2}}{2}\widetilde{V}_{xx}(t,x)+s_{\rho}\cdot\left|\sigma\,x\cdot\widetilde{V}_{x}(t,x)\right|-r\,\widetilde{V}(t,x) & =0,\label{eq:PDE-ask}\\
\widetilde{V}(T,x) & =\Psi(x).\nonumber \end{aligned}$$ Notice that and differ only in the sign of the nonlinear term, showing again that in the risk-neutral setting (i.e., $s_{\rho}=0)$ the bid and ask prices coincide. We have the following explicit solution of and for the price of the call option.
\[prop:risky-BSM-2\]Let $\Psi(x):=\max(x-K,0)$, define the auxiliary functions (cf. @Schachermayer+Delbaen-R [Section 4.4]) $$d_{1}^{\pm}\coloneqq\frac{1}{\sigma\sqrt{T-t}}\cdot\left[\log\left(\frac{x}{K}\right)+\left(r\pm s_{\rho}\,\sigma+\frac{1}{2}\sigma^{2}\right)(T-t)\right],\qquad d_{2}^{\pm}\coloneqq d_{1}^{\pm}-\sigma\sqrt{T-t}\label{eq:7}$$ and the value functions $$V^{\pm}(t,x):=xe^{\pm s_{\rho}\sigma(T-t)}\Phi(d_{1}^{\pm})-Ke^{-r(T-t)}\cdot\Phi(d_{2}^{\pm}).\label{eq:valueCall}$$ Then $V^{+}$ solves the risk-averse BlackScholes PDE for the ask price, while $V^{-}$ solves , the corresponding PDE for the bid price; further, we have that $V^{-}\le V^{+}$.
We can solve the problem for the European put option similarly.
\[prop:risky-BSM\]Let $\Psi(x):=\max(K-x,0)$ and define the value functions $$V^{\mp}(t,x):=Ke^{-r(T-t)}\cdot\Phi(-d_{2}^{\mp})-xe^{\mp s_{\rho}\sigma(T-t)}\Phi(-d_{1}^{\mp}),\label{eq:valuePut}$$ with $d_{1}^{\pm}$ and $d_{2}^{\pm}$ as in Proposition \[prop:risky-BSM-2\]. Then $V^{-}$ solves the risk-averse BlackScholes PDE and $V^{+}$ solves , respectively. Note that $V^{+}\le V^{-}$.
Plugging the derivatives into the PDE and shows the assertion.
Rationale of risk aversion in the new formulae
----------------------------------------------
#### Nature of the risk level $s_{\rho}$.
Proposition \[prop:risky-BSM-2\] and \[prop:risky-BSM\] show that the value function for the risk-averse European option pricing problem can be identified with the risk neutral problem, where the stock pays dividends at rate $s_{\rho}\,\sigma$. In case of the bid price of a European call option the dividend payments are given by $s_{\rho}\,\sigma$. Similarly, the dividend payments for the bid price for a European put option are $-s_{\rho}\,\sigma$, thus negative. For an increasing risk level $s_{\rho}$, the bid price for the put and the call price decrease. This monotonicity reverses for the ask price.
The value functions and can also be interpreted within the framework of the GarmanKohlhagen model on foreign exchange options. In this sense the terms $\pm s_{\rho}\,\sigma$ represent the interest rate in the risk-averse currency.
#### Illustration of the risk level $s_{\rho}$.
Figure \[fig:prices\] shows the put and call prices for different values of $s_{\rho}$. For this illustration we choose $T=1$, $S_{0}=1$ with strike $K=1.2$, the interest rate is $r=3\,\%$ and the volatility is $\sigma=15\,\%$.
Figure \[fig:spread\] exhibits the bid-ask spread, which is present in the risk-averse situation.
#### Discussion of the risk level $s_{\rho}$.
The introduction outlines that $s_{\rho}$ is related to the Sharpe ratio, a specific reward-to-variability ratio introduced by William F. Sharpe. The Sharpe ratio is $$\frac{r-r_{\text{free}}}{\sigma},$$ where $r$ is the mean return of an asset with volatility $\sigma$ and $r_{\text{free}}$ is the risk free interest rate. Comparing units in we see that $s_{\rho}\,\sigma$ is an interest rate and hence $s_{\rho}$ has unit $$\frac{\text{interest}}{\text{volatility}},$$ the same unit as the Sharpe ratio. For the risk free return $r$ of the market (see Equation ) and the risk averse interest $r_{\text{averse}}$ the investor expects, we equate $$s_{\rho}=\frac{r-r_{\text{averse}}}{\sigma}$$ with $s_{\rho}$ as in above. Notice that $r_{\text{averse}}$ should not exceed $r$ and may be negative so that $s_{\rho}$ is always positive. It follows that the risk-aversion coefficient $s_{\rho}$ has the structure of a Sharpe ratio. Furthermore, $s_{\rho}\sigma$ is the Z-spread for the risk-averse investor. Regarding the sign of $s_{\rho}\sigma$, notice that the bid price of the European call option is increasing in the interest rate and hence the risk neutral interest rate decreases to $r-s_{\rho}\sigma$. The bid price for the European put option is decreasing with respect to the interest rate and hence the interest rate increases to $r+s_{\rho}\sigma$. The sign changes again when considering the respective ask prices.
Consistency with the discrete model
-----------------------------------
We return to the binomial model with risk-averse probabilities. The preceding sections show that the risk level $\beta$ for the mean semi-deviation risk measure needs to be proportional to $$\sqrt{\Delta t}.$$ In this case we obtain the risk-averse probabilities $$\widetilde{p}=p(1-\beta\sqrt{\Delta t}p)=\frac{1}{2}+\frac{2r-\beta\sigma-\sigma^{2}}{4\sigma}\sqrt{\Delta t}+\frac{\beta(-2r+\sigma^{2})}{4\sigma}\Delta t+\mathcal{O}(\Delta t^{\frac{3}{2}})$$ and following the standard arguments we obtain the distribution for the stock $S_{T}$ in the limit as $$S_{T}=S_{0}\exp\left\{ T\left(r-\frac{\beta\sigma}{2}-\frac{\sigma^{2}}{2}\right)+\sigma W_{T}\right\} .$$ Recall from Lemma \[lem:gauss\] that $s_{\rho}$, for the mean semi deviation, is $\frac{\beta}{\sqrt{2\pi}}$. However the binomial model converges to a process with dividends $\frac{\beta}{2}$. The discrepancy in the scaling factor is in line with the discontinuity of risk measures with respect to convergence in distribution, described in @BaeuerleMueller2006-R [Theorem 4.1].
\[sec:US\]Pricing of American options under risk
------------------------------------------------
In the risk-averse setting explicit formulae for European option prices in the BlackScholes model are available. This is surprising given the initial nonlinear PDE formulation . Similarly we may reformulate the risk-averse American option pricing problem and in what follows we introduce the risk-averse optimal stopping problem for American put options and introduce the value functions.
Again we assume that the stock $S$ follows the geometric Brownian motion . Here, the risk-averse bid price of an American option is given by $\sup_{\tau\in[0,T]}\,-\rho^{0:\tau}\left[-e^{-r\tau}\,\Psi(S_{\tau})\right]$, where $\Psi(\cdot)$ denotes the payoff function and the supremum is among all stopping times with $\tau\in[0,T]$. The ask price is given by $\sup_{\tau\in[0,T]}\,\rho^{0:\tau}\left[e^{-r\tau}\,\Psi(S_{\tau})\right]$. We can further define the value functions $$V(t,x):=\sup_{\tau\in[t,T]}\,-\rho^{t:\tau}\left[-e^{-r(\tau-t)}\,\Psi(S_{\tau})\mid S_{t}=x\right]$$ for the bid price and $$\widetilde{V}(t,x):=\sup_{\tau\in[t,T]}\,\rho^{t:\tau}\left[e^{-r(\tau-t)}\,\Psi(S_{\tau})\mid S_{t}=x\right]$$ for the ask price. For ease of notation we only discuss the bid price for American put options, the arguments for the ask price are analogous. Analogously to the risk-neutral setting we obtain the free boundary problem for the optimal exercise boundary $t\mapsto L(t)$. $$\begin{aligned}
V_{t}(t,x)+rxV_{x}(t,x)+\frac{\sigma^{2}x^{2}}{2}V_{xx}(t,x)-s_{\rho}\sigma x\left|V_{x}\right|= & rV(t,x) & \text{for }x\geq L(t),\label{eq: freeDiffu}\\
V(t,x)= & (K-x)_{+} & \text{for }0\leq x<L(t),\\
V_{x}(t,x)= & -1 & \text{for }x=L(t),\label{eq:freeBee}\\
V(T,x)= & (K-x)_{+}\nonumber \\
L(T)= & K\nonumber \\
\lim_{x\to\infty}V(t,x)= & 0 & \text{for }0\leq t\leq T.\label{eq:8}\end{aligned}$$ For an overview on American options and free boundary problems in general we refer to @Peskier2006. The following result follows with standard arguments for American options.
The value function $$V(t,x)=\sup_{\tau\in[t,T]}\,-\rho^{t:\tau}\left[-e^{-r(\tau-t)}(K-S_{\tau})_{+}\mid S_{t}=x\right]\label{eq:US-value}$$ solves the free boundary problem .
Similarly to European options, risk-aversion reduces to a modification of the drift term and the standard American put option problem is recovered where the underlying stock pays dividends. First we notice that $$-s_{\rho}\sigma x\left|V_{x}\right|\leq-s_{\rho}\,\sigma\,x\,y\,V_{x},\qquad y\in[-1,1],$$ and hence for each $(t,x)\in\mathcal{C}$ $$\begin{aligned}
V_{t}(t,x)+rxV_{x}(t,x) & +\frac{\sigma^{2}x^{2}}{2}V_{xx}(t,x)-s_{\rho}\sigma x\left|V_{x}\right|\\
=\inf_{y\in[-1,1]} & \left\{ V_{t}(t,x)+\left(r-s_{\rho}\sigma y\right)xV_{x}(t,x)+\frac{\sigma^{2}x^{2}}{2}V_{xx}(t,x)\right\} .\end{aligned}$$ For $x\geq L(t)$, the American option is not exercised and the same arguments as for the European options show that the infimum over all constraints is attained at $y\equiv-1$. The first line of the free boundary formulation is thus equal to $$V_{t}(t,x)+\left(r+s_{\rho}\sigma\right)xV_{x}(t,x)+\frac{\sigma^{2}x^{2}}{2}V_{xx}(t,x)=rV(t,x)\qquad\text{for }x\geq L(t).$$ Consequently, we deduce that the value function $$V(t,x):=\sup_{\tau\in[t,T]}\,\operatorname{{\mathds E}}\left[e^{-r(\tau-t)}\Psi\left(S_{\tau}\right)\mid S_{t}=x\right]$$ solves the free boundary problem , where the state process is given by $$\begin{aligned}
\mathrm{d}S_{s} & =\left(r+s_{\rho}\,\sigma\right)S_{s}\,\mathrm{d}s+\sigma\,S_{s}\,\mathrm{d}W_{s},\\
S_{t} & =x.\end{aligned}$$
Numerical illustration {#numerical-illustration .unnumbered}
----------------------
Consider the geometric Brownian motion $$\begin{aligned}
\mathrm{d}S_{t} & =0.03S_{t}\,\mathrm{d}t+0.15S_{t}\,\mathrm{d}W_{t},\qquad0<t\leq1,\\
S_{0} & =1.\end{aligned}$$ The strike price in the next Figure \[fig:3\] is $K=1$. We consider the optimal stopping region for different risk levels $s_{\rho}$. A risk-averse option buyer (bid price) would generally exercise earlier, he accepts less profits due to his risk aversion. Compared with the risk neutral investor, the accumulating construction of nested risk measures ensures that the risk aware option buyer prefers exercising prematurely rather than delayed exercise.
The reverse is true for the option holder (ask price), where the investor waits longer.
In the risk neutral case it is never optimal to exercise an American call option before expiry. However, this is only the case if the underlying asset does not pay dividends (see, for instance, @Shreve2010 [Chapter 8.5] for details). As risk-aversion expressed with coherent risk measures can be represented by dividend paying stocks we conclude that it is generally optimal to exercise the call option early. Figure \[fig:stopCall\] shows the optimal exercise boundary for the risk-averse call option with strike $K=1$ and initial value $S_{0}=1$.
![\[fig:stopCall\]optimal stopping regions for different risk-levels (call option)](USCallEx)
The Merton problem\[sec:Merton\]
================================
The preceding sections demonstrate that classical option pricing models generalize naturally to a risk-averse setting by employing nested risk measures. In what follows we demonstrate that the classical Merton problem, which allows an explicit solution in specific situations, as well allows extending to the risk-averse situation.
Consider a risk-less bond $B$ satisfying the ordinary differential equation $\mathrm{d}B_{t}=r\,B_{t}\,\mathrm{d}t$ and a risky asset $S$ driven by the stochastic differential equation $$\mathrm{d}S_{t}=\mu S_{t}\,\mathrm{d}t+\sigma S_{t}\,\mathrm{d}W_{t}.$$ We are interested in the optimal fraction $\pi_{t}$ of the wealth $w_{t}$ one should invest in the risky asset. Consider the wealth process $$\mathrm{d}w_{t}=\left[\left(\pi_{t}\mu+(1-\pi_{t})r\right)w_{t}-c_{t}\right]\mathrm{d}t+\pi_{t}\,\sigma\,w_{t}\,\mathrm{d}W_{t},$$ where $c_{t}$ is the rate of consumption. Merton employs the power utility function $u(x)=\frac{x^{1-\gamma}}{1-\gamma}$. We consider the risk-averse objective function $$R(t,x):=\sup_{\pi,c}\,-\rho^{t:T}\left(-\int_{t}^{T}e^{-\varrho(s-t)}u(c_{s})\mathrm{d}s-\epsilon^{\gamma}\,e^{-\varrho(T-t)}\,u(w_{T})\mid w_{t}=x\right).$$ Surprisingly, $R$ has a closed form solution and, moreover, the optimal portfolio allocation is $$\pi^{*}=\max\left(\frac{\mu-r-s_{\rho}\,\sigma}{\sigma^{2}\,\gamma},0\right).$$ We observe again that risk aversion leads to a modified drift term $r+s_{\rho}\sigma$ in place of $r$. The optimal portfolio allocation $\pi^{*}$ is a decreasing function of $s_{\rho}$. This is in line with the usual economic perception, as increasing risk-aversion corresponds to less investments into the risky asset. The optimal consumption is given by $$c_{t}^{*}(x)=\frac{x\,\nu}{1+(\nu\,\epsilon-1)e^{-\nu(T-t)}},$$ where $\nu$ is a constant depending on the model parameters. Consumption generally increases with risk aversion as the value of immediate consumption offsets the present value of uncertain wealth in the future.
In what follows we derive the optimal value function $R$ and verify the optimal portfolio allocation $\pi^{*}$ and optimal consumption $c^{*}$ given above.
The following result is a consequence of Proposition \[prop:RGen\].
The optimal discounted value function $R$ satisfies the following HamiltonJacobiBellman equation $$\begin{aligned}
0 & =\max_{\pi,c}\,\left[R_{t}+\left[\left(\pi_{t}\mu+(1-\pi_{t})r\right)x-c_{t}\right]R_{x}+\frac{\sigma^{2}\pi^{2}x^{2}}{2}R_{xx}+e^{-\varrho t}u(c_{t})-s_{\rho}\left|\sigma\pi_{t}xR_{x}\right|\right],\label{eq:HJB}\\
R(T,x) & =\frac{\epsilon^{\gamma}e^{-\varrho T}}{1-\gamma}x^{1-\gamma}.\nonumber \end{aligned}$$
The HamiltonJacobiBellman equation allows for explicit optimal controls, the following proposition outlines them.
In the risk-averse setting, the optimal controls are given by $$\begin{aligned}
\pi_{t}^{*} & (x)=-\frac{(\mu-r)R_{x}}{\sigma^{2}xR_{xx}}+\frac{s_{\rho}\sigma\left|R_{x}\right|}{\sigma^{2}xR_{xx}},\qquad c_{t}^{*}(x)=\left(e^{\varrho t}R_{x}\right)^{-\frac{1}{\gamma}}.\end{aligned}$$ The Hamilton-Jacobi-Bellman equation rewrites as $$\begin{aligned}
0 & =R_{t}-\frac{\left((\mu-r)^{2}+s_{\rho}^{2}\sigma^{2}\right)R_{x}^{2}}{2\sigma^{2}R_{xx}}+\frac{s_{\rho}R_{x}\left|R_{x}\right|}{\sigma R_{xx}}+rxR_{x}+\frac{\gamma e^{-\frac{\varrho t}{\gamma}}}{1-\gamma}R_{x}^{\frac{\gamma-1}{\gamma}},\label{eq:HJB-2}\\
R(T,x) & =\frac{\epsilon^{\gamma}e^{-\varrho T}}{1-\gamma}x^{1-\gamma}.\nonumber \end{aligned}$$
The preceding proposition derives first order conditions for the fraction $\pi_{t}^{*}$ and consumption rate $c_{t}^{*}$. Employing the HamiltonJacobiBellman equations we obtain nonlinear second order partial differential equations for the optimally controlled value function. Surprisingly, this nonlinear equation has an explicit solution too.
\[thm:Merton\]The PDE has the explicit solution $$R(t,x)=e^{-\varrho t}\left(\frac{1+(\nu\,\epsilon-1)e^{-\nu(T-t)}}{\nu}\right)^{\gamma}\frac{x^{1-\gamma}}{1-\gamma},$$ where $\nu:=\frac{\varrho}{\gamma}-r\frac{1-\gamma}{\gamma}-\frac{1-\gamma}{\gamma^{2}}\left(\frac{\left((\mu-r)^{2}+s_{\rho}^{2}\sigma^{2}\right)}{2\sigma^{2}}-\frac{s_{\rho}}{\sigma}\right)$. Moreover, the optimal controls are $$\begin{aligned}
{1}
\pi^{*} & =\frac{\mu-s_{\rho}\,\sigma-r}{\sigma^{2}\gamma}\text{ and}\\
c_{t}^{*}(x) & =\frac{x\,\nu}{1+(\nu\,\epsilon-1)e^{-\nu(T-t)}}.\end{aligned}$$
We recall the PDE , $$\begin{aligned}
0 & =R_{t}-\frac{\left((\mu-r)^{2}+s_{\rho}^{2}\sigma^{2}\right)R_{x}^{2}}{2\sigma^{2}R_{xx}}+\frac{s_{\rho}R_{x}\left|R_{x}\right|}{\sigma R_{xx}}+rxR_{x}+\frac{\gamma e^{-\frac{\varrho t}{\gamma}}}{1-\gamma}\left(R_{x}\right)^{\frac{\gamma-1}{\gamma}},\\
R(T,x) & =e^{-\varrho T}\epsilon^{\gamma}\frac{x^{1-\gamma}}{1-\gamma},\end{aligned}$$ and choose the ansatz $R(t,x)=e^{-\varrho t}f(t)^{\gamma}\frac{x^{1-\gamma}}{1-\gamma}$. In this case the partial derivatives are given by $$\begin{aligned}
R_{t} & =e^{-\varrho t}\left(-\varrho f(t)^{\gamma}+\gamma f(t)^{\gamma-1}f^{\prime}(t)\right)\frac{x^{1-\gamma}}{1-\gamma},\\
R_{x} & =e^{-\varrho t}f(t)^{\gamma}x^{-\gamma},\\
R_{xx} & =-\gamma e^{-\varrho t}f(t)^{\gamma}x^{-\gamma-1}.\end{aligned}$$ The terminal condition for our Merton problem is $v(T,x)=\epsilon^{\gamma}e^{-\varrho T}\frac{x^{1-\gamma}}{1-\gamma}$ hence $f(T)=\epsilon>0$. Setting $C_{1}:=-\frac{\left((\mu-r)^{2}+s_{\rho}^{2}\sigma^{2}\right)}{2\sigma^{2}}$ and $C_{2}:=\frac{s_{\rho}}{\sigma}$ for ease of notation we substitute the derivatives in the PDE and obtain the following ordinary differential equation for $f$; $$\begin{aligned}
f^{\prime}(t) & =f(t)\left(\frac{\varrho}{\gamma}-r\frac{1-\gamma}{\gamma}+\frac{1-\gamma}{\gamma^{2}}\left(C_{1}+C_{2}f^{\gamma}\right)\right)-1.\label{eq:ODE1}\end{aligned}$$ For $\nu$ as defined in Theorem \[thm:Merton\], the general solution of the ordinary differential equation is $$f(t)=\frac{1+(\nu\epsilon-1)e^{-\nu(T-t)}}{\nu},$$ which is positive. The optimal value function thus is $$\begin{aligned}
R(t,x) & =e^{-\varrho t}\left(\frac{1+(\nu\epsilon-1)e^{-\nu(T-t)}}{\nu}\right)^{\gamma}\frac{x^{1-\gamma}}{1-\gamma}.\end{aligned}$$ We assumed that $\pi\geq0$ and hence the optimal control is $\pi_{t}^{*}=\max\left(\frac{(\mu-r)-s_{\rho}\sigma}{\sigma^{2}\gamma},0\right).$ The optimal consumption process is $c_{t}^{*}=\frac{x\nu}{1+(\nu\epsilon-1)e^{-\nu(T-t)}},$ which concludes the proof.
The following Figure \[fig:OC\] illustrates the optimal consumption $c^{*}$ as a function of the risk level $s_{\rho}$ for $\varrho=0.1$, $\gamma=0.4$, $r=0.01$, $\mu=0.1$, $\sigma=0.3$ and $\epsilon=0.1$. The time horizon is $T=4$ and we consider the wealth $w_{0}=1$. Note that $s_{\rho}$ can take only values smaller than $\frac{\mu-r}{\sigma}$ as otherwise $\pi^{*}<0$.
![\[fig:OC\]optimal consumption](consumRisk)
Summary
=======
This paper introduces risk awareness in classical financial models by introducing nested risk measures. We demonstrate that classical formulae, which are of outstanding importance in economics, are explicitly available in the risk-averse setting as well. This includes the binomial option pricing model, the BlackScholes model as well as the Merton optimal consumption problem.
We also give an explicit Z-spread, which reflects risk awareness. The Z-spread involves the volatility of the risky asset and a constant $s_{\rho}$, which derives from nesting risk measures. The results thus provide an economic verification of the Z-spread by thorough risk management employing coherent risk measures.
To aid the discussion on risk-averse value functions we extend nested risk measures from a discrete time setting to continuous time. This allows us to derive a risk generator expressing the momentary dynamics of our model under risk aversion. We show that for every coherent risk measure the risk generator is of the same form, implying that in continuous time there is only one nested risk measure. Moreover, a constant $s_{\rho}$ expresses risk aversion which we associate with the Sharpe ratio.
[^1]: Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project-ID 416228727 SFB 1410
[^2]: Both authors: Technische Universität Chemnitz, 09126 Chemnitz, Germany. Contact: <ruben.schlotter@math.tu-chemnitz.de>
[^3]: The inequality $0=\rho(Y-Y)\leq\rho(Y)+\rho(-Y)$ implies that $$-\rho(-Y)\leq\rho(Y).$$
|
---
abstract: 'The core mass of Saturn is commonly assumed to be 10–25 ${M_{\oplus}}$ as predicted by interior models with various equations of state (EOSs) and the *Voyager* gravity data, and hence larger than that of Jupiter (0–$10\:{M_{\oplus}}$). We here re-analyze Saturn’s internal structure and evolution by using more recent gravity data from the *Cassini* mission and different physical equations of state: the ab initio LM-REOS which is rather soft in Saturn’s outer regions but stiff at high pressures, the standard Sesame-EOS which shows the opposite behavior, and the commonly used SCvH-i EOS. For all three EOS we find similar core mass ranges, i.e. of 0–$20\:{M_{\oplus}}$ for SCvH-i and Sesame EOS and of 0–$17\:{M_{\oplus}}$ for LM-REOS. Assuming an atmospheric helium mass abundance of 18%, we find maximum atmospheric metallicities, of $7\times$ solar for SCvH-i and Sesame-based models and a total mass of heavy elements, $M_Z$ of 25–$30{M_{\oplus}}$. Some models are Jupiter-like. With LM-REOS, we find $M_Z=16$–$20{M_{\oplus}}$, less than for Jupiter, and ${\mbox{$Z_{\rm atm}$}}\lesssim 3\times$ solar. For Saturn, we compute moment of inertia values $\lambda=0.2355(5)$. Furthermore, we confirm that homogeneous evolution leads to cooling times of only $\sim 2.5$ Gyr, independent on the applied EOS. Our results demonstrate the need for accurately measured atmospheric helium and oxygen abundances, and of the moment of inertia for a better understanding of Saturn’s structure and evolution.'
address:
- 'Institute of Physics, University of Rostock, D-18051 Rostock, Germany'
- |
University of California, Santa Cruz, USA\
*Accepted to Icarus, April 2013*
author:
- Nadine Nettelmann
- 'Robert P[ü]{}stow'
- Ronald Redmer
bibliography:
- 'ms-refs.bib'
title: Saturn layered structure and homogeneous evolution models with different EOSs
---
Saturn ,Saturn interior ,Saturn atmosphere
Introduction
============
Saturn is the planet with the lowest mean density in the solar system. Since the mechanisms that can inflate exoplanets with observed overlarge radii do not hold for the outer planet Saturn, one might thus intuitively think of Saturn as having a smaller core and smaller overall metallicity than Jupiter. However, quantitative estimates on the core mass and on the total heavy element enrichment solely come from interior model calculations, and the same modeling approach applied to both planets just predicts the opposite: an about two times larger maximum core mass and heavy element enrichment for Saturn [@SG04; @GuiGau07]. A higher envelope metallicity of Saturn is also supported by the measured atmospheric C:H ratios, which is $\sim 9\times$ solar for Saturn ([@Fletcher+09]; scaled to the solar system abundance data of @Lodders03) but only 3–5$\times$ solar for Jupiter [@Atreya+03].
Certainty about the present core mass and envelope metallicity is desirable because these parameters contain information —albeit not necessarily uniquely [@Helled+10; @Boley+11]— on the formation environment, i.e. on the protosolar disk, and on the process of formation.
Models by @SG04, hereafter SG04, are often considered the standard of what we know today about Saturn’s present internal structure in terms of core mass and heavy element enrichment ([e.g., @Alibert+05; @DodRob+08]), for mainly two reasons. First, these models have been computed for various physical equations of state(EOS) for Saturn’s likely main constituents H and He that also give acceptable solution for Jupiter’s interior and evolution (the EOSs SCvH-i, LM-H4, LM-SOCP). Independent on the EOS, the possible core mass range was found to be $\sim 10$–$25{M_{\oplus}}$, while for Jupiter $\sim 0$–$10{M_{\oplus}}$. Second, a wide range of input parameters was accounted for such as a the position of an internal layer boundary that separates a helium-poor, outer from a helium-rich, inner envelope. However, SG04 computed constant metallicity envelope models only, an assumption that tremendously restricts the resulting range of interior models.
In earlier models by Gudkova & Zharkov (1999) and @Gui99, the metallicity was allowed to vary across the internal layer boundary. As a consequence, zero-core mass models with high heavy element enrichment in the deep envelope were found for both Jupiter and Saturn.
The new *Cassini* gravity data with their tight observational error bars, and also long-term observational data of the Saturnian system [@J+06; @AS07] raised hope to better constrain Saturn’s internal structure. Surprisingly, the most recent Saturn models based on those gravity data cover an even bigger, minimum core mass range of $\sim 0$–$30{M_{\oplus}}$ [@AS07; @Helled+09a; @Helled11]. Therefore, [@Helled11] suggests to measure the axial moment of inertia as an additional constraint. Her models, however, employ *empirical* pressure-density relations that may reach out of the realm of physical EOS which agree with the available experimental data (see, e.g. SG04; @Holst+12).
Our Saturn models are the first that are based on both physical equations of state and the *Cassini* data. Not is it the purpose of this work to better constrain the core mass: this cannot be achieved within the standard three-layer modeling approach, which is adopted in this work. Instead, we here investigate the overall behavior of core mass, atmospheric metallicity, and deep envelope metallicity on the input parameters: we vary the position of an internal layer boundary in order to recall its influence on the core mass, see also @GuiGau07; we exchange the EOS of the envelope material (LM-REOS, SCvH-i EOS, Sesame EOS), and we adopt two different periods of rotation of 10h 32m and 10h 39m. In lack of accurate observations, we make predictions on the possible helium and heavy element mass fractions in Saturn’s atmosphere in dependence on the $J_4$ value and the uncertainty in the rotational period. Our results on the atmospheric helium abundance can serve as constraints for future models of He-sedimentation in Saturn, as long as Saturn’s atmospheric He:H$_2$ ratio is not accurately measured.
Observations of young stellar systems and protostellar disks commonly point to formation of the giant planets within a few Myr [@Strom+93], implying a billions-of-years-old planet should have the same age as its host star. However, homogeneous evolution calculations for Saturn, which are mainly based on the SCvH-i EOS, generally yield cooling times of 2–3 Gyr [@Saumon+92; @Fortney+11], about only *half* of the age of the Sun. This implies a higher luminosity of present Saturn than it should have if the underlying assumption of homogeneous evolution would hold. Despite the obvious failure of this assumption, we here adopt it once more in order to investigate the influence of the EOS on the cooling time.
In Section \[sec:m\_modeling\] we describe our modeling procedure. Section \[sec:m\_obs\] is devoted to a detailed description of the observational data, and Section \[sec:m\_eos\] to the applied EOSs. Our results are presented in Section \[sec:results\]. In Section \[sec:r\_struc1\] we investigate the influence on different H-He-EOS on Saturn’s structure and in Section \[sec:r\_struc2\] of the atmospheric He abundance and rotation rate. In Section \[sec:r\_nmoi\] we give the values for the non-dimensional moment of inertia. Section \[sec:r\_evol\] contains the cooling curves. Section \[sec:discussion\] includes a discussion on the implications for Saturn’s formation process (\[sec:d\_mcoreform\]), on the applicability of the three-layer assumption in the presence of He rain (\[sec:d\_Herain\]), and a summary of our main findings (\[sec:d\_summary\]).
Methods {#sec:methods}
=======
Planetary structure modeling {#sec:m_modeling}
----------------------------
For understanding the interior of giant gas planets like Saturn it is necessary to consider the gravitational field of the planet. The shape of the field is influenced by different effects. Saturn for instance has primarily the form of an ellipsoid due to its rapid rotation, which can be seen from the rather high ratio of centrifugal to gravitational forces, $q=\omega^2\,R^3_{\rm eq}/(G\,M)$, where $\omega$ is the angular velocity, $R_{\rm eq}$ is the equatorial radius, and $M$ the total mass. For Saturn, $q\sim 0.155$ with an uncertainty of 0.004 due to the uncertainty in the rotation period and equatorial radius (see Section \[sec:m\_obs\]), for Jupiter, $q=0.089$, and for the Sun, $q=0.00002$. Tidal forces caused by the gravity of the moons or the parent star can also change the form of a planet’s gravity field. While this effect can be important for close-in exoplanets it is tiny for Saturn and has not been measured yet for any giant planet in the solar system. To assess the rotationally induced deformation, the gravity field $\Phi^{(e)}$ exterior to the mass $M$ is expanded into a series of Legendre polynomials $P_{2n}$, where the expansion coefficients $J_{2n}$ are the gravitational moments at the equatorial reference radius ${\mbox{$R_{\rm eq}$}}$, $$\label{eq:J2n}
J_{2n}=-\frac{1}{MR_{\rm eq}^{2n}}\int\! d^3r\,\rho(r,\theta)r^{2n}P_{2n}(t)\:.$$ Being integrals of the internal mass distribution over the volume enclosed within the geoid of equatorial radius ${\mbox{$R_{\rm eq}$}}$, the $J_{2n}$ can be written as depth-dependent functions $J_{2n}(l)$ whose values increase continuously from the center outward until the observed values $J_{2n}^{(\rm obs)}$ are reached at the geoid’s mean radius $l={\mbox{$R_{\rm m}$}}$. As a measure for the contribution $dJ_{2n}$ of a shell at $l$ and extension $dl$ to $J_{2n}^{(\rm obs)}$ we can define the normalized contribution function $$\label{eq:c2n}
c_{2n}(l) = \frac{(dJ_{2n}/dl)|_{l}}{\int \! dl'\, (dJ_{2n}/dl')}\:.$$ For modeling Saturn we use the same method and code as in @N+12 for Jupiter. We adopt the standard three-layer structure with two envelopes and a core. The composition of each of the envelopes is diverted into the three components hydrogen, helium, and heavy elements, whereas the core consists of heavy elements only. The helium mass fractions and the metallicities (i.e. the heavy element mass fractions) are parameterized by $Y_1$, $Z_1$ and $Y_2$, $Z_2$ for the outer and the inner envelope, respectively. This implies the assumption of homogeneous envelopes. The transition between them occurs at the transition pressure ${\mbox{$P_{\mbox{\tiny1-2}}$}}$ which is a free parameter. As observational constraints we take into account ${\mbox{$R_{\rm eq}$}}$, $\omega$, the total mass , the temperature $T_1$ at the 1 bar level of the planet, and the lowest order moments $J_2$ and $J_4$.
For given values of $Y_1$ and of the mean helium abundance $Y$, $Y_2$ is adjusted to fit $Y$, while $Z_1$ and $Z_2$ are adjusted to fit $J_2$ and $J_4$. Mass conservation is then ensured by the choice of the core mass ${\mbox{$M_{\rm core}$}}=m({\mbox{$R_{\rm core}$}})$.
Observational constraints {#sec:m_obs}
-------------------------
While the *Cassini* mission could provide tight constraints on Saturn’s gravity field, there are still important remaining uncertainties, in particular in Saturn’s period of rotation, equatorial radius, and the atmospheric helium abundance.
#### Period of rotation
Prior to the *Cassini* observations, Saturn’s period of rotation was taken to be 10h 39m 24s, the detected periodicity in the kilometric radio emissions of Saturn’s magnetic field as measured by the Voyager I and II spacecraft [@DK81]. *Cassini* however revealed a prolongation of this period by several minutes within just 20 years; thus the observed magnetic field modulations may not reflect the rotation of Saturn’s deep interior [@Gurnett+07]. On the other hand, while alternative methods of deriving the rotation rate from observed wind speeds make assumptions that may not hold true, such as the minimum energy of the zonal winds or a minimum height of isobar-surfaces relative to computed geoid surfaces [@AS07; @Helled+09b], that alternative methods just suggest similar values of $\sim$ 10h 32m. We therefore use these values as the uncertainty in Saturn’s real solid body rotation period and compute interior models for both periods, i.e. for 10h 32m and 10h 39m. Note that we neglect here the uncertainty to Saturn’s structure from the possibility of differential rotation on cylinders. On the other hand, all observational wind data can well be reproduced by the assumption of solid-body rotation [@Helled+09b] and the effect of zonal winds on Jupiter’s structure have been shown to be negligible if their penetration depth is limited to 1000 km [@Hubbard99].
#### Equatorial radius
Our method of interior modeling requires all outer boundary conditions to be provided at the same pressure level. Although the Voyager I, II, and Pioneer 11 observational data refer to the isobar-surface at 100 mbar [@Lindal+85], we prefer to use the 1 bar level as outer boundary. In particular, we use the equatorial 1 bar radius of $R_{\rm eq}=60,268$ km [@Lindal+85; @GuiGau07]. This radius is a computed one of a geoid that rotates with a solid body period (System III period) of 10h 39m 24s plus an additional, latitude ($\phi$)-dependent component according to the observed zonal wind speeds. The difference in radius at the equator to a reference geoid for a rigidly rotating Saturn in hydrostatic equilibrium —the appropriate radius to constrain interior models— is $\sim 120$ km at the 100 mbar level. The exact difference at the 1 bar level is not provided (see [@Lindal+85] for details), but we expect it to be of same size, and therefore would expect the ${\mbox{$R_{\rm eq}$}}$ value of a reference geoid at 1 bar to be $\sim 100$ km smaller than the given value in @Lindal+85. Neglecting the described inconsistency, we here use this ${\mbox{$R_{\rm eq}$}}$ value, and we use it independently on the period of rotation value. To our awareness, the only consistent reference systems for both limiting periods of rotation (10h 39m 24s and 10h 32m 35s) are presented in @Helled11. Her reference system values are listed in Table \[tab:obs\] for comparison. The surface radius ${\mbox{$R_{\rm Sat}$}}$ is the mean radius of the reference geoid with the equatorial radius $R_{\rm eq}$.
#### Gravitational coefficients
In order to get the gravity field ($J_2$, $J_4$, $J_6$) at the outer boundary as defined by the equatorial 1 bar radius of 60,268 km, we scale the values of @J+06, which are $J'_2=16290.71(0.27)$, $J'_4=-935.8(2.8)$, and $J'_6=86.1(9.6)$. We use the scaling relation $J_{2n}' {\mbox{$R_{\rm eq}$}}^{'\,2n} = J_{2n} {\mbox{$R_{\rm eq}$}}^{\,2n}$, where ${\mbox{$R_{\rm eq}$}}^{'\,2n}=60,330$ km is the equatorial reference radius in @J+06. The gravitational coefficients of @J+06 are based on, but not limited to, Pioneer11, Voyager, and *Cassini* tracking data as well as long-term Earth-based and HST astrometry data. They have significantly reduced error bars compared to the Voyager era data [@Campbell+89]. Since the ${\mbox{$R_{\rm eq}$}}$ value used in this work is larger than that in @Helled11, the scaled, absolute $J_{2n}$ values are smaller. Under variation of the $J_4$ value within its $6\:\sigma$ error bars (Section \[sec:r\_struc2\]), we also cover the $J_4$ values used in @Helled11, see Table \[tab:obs\]. Therefore, the obtained sets of solutions can reasonably be compared to each other.
#### Mean helium abundance
For the protosolar cloud where the Sun and the giant planets formed of, Bahcall & Pinsonneault (1995) calculate a mean helium abundance of 27.0 to 27.8% by mass, depending mainly on the inclusion of helium and heavy element diffusion into solar evolution models. We require our models to have and mean helium abundance $Y=0.2750(1)$ by mass with respect to the H/He subsystem. By particle numbers, this corresponds to He:H$_2\::=\:N_{\rm He}/N_{\rm H_2} = 0.1732$.
#### Atmospheric helium abundance
Combined *Voyager* infrared spectrometer (IRIS) data and Voyager radio occultation (RSS) temperature profile data revealed a depletion of He in the atmospheres of both Jupiter and Saturn compared to the protosolar value [@Conrath+84]. In particular, a modest depletion He:H$_2=0.110\pm 0.032$ was found in Jupiter, and a strong depletion He:H$_2=0.034\pm 0.024$ in Saturn. These particle ratios correspond to atmospheric mass mixing ratios $Y_{\rm atm,\, J}=0.18\pm 0.04$ for Jupiter and $Y_{\rm atm,\, S }=0.06\pm 0.05$ for Saturn. However, the He abundance detector (HAD) aboard the Galileo probe measured *in situ* Jupiter’s atmospheric He:H$_2$ to be $0.157\pm 0.003$ [@Zahn+98], corresponding to $Y_{\rm atm,\, J}=0.238 \pm 0.006$. Because of the discrepancy to the former results, the Voyager data for Jupiter and Saturn were re-examined by @CG00. They found the Voyager data for Jupiter could be made consistent with the Galileo data if the temperature profile obtained by Voyager RSS measurements were shifted by 2 K towards colder temperatures. Because of a possible systematic error of the RSS data, @CG00 developed an inversion algorithm to infer the He:H ratio and the temperature profile from the Saturnian IRIS spectra alone. Their results suggest a significantly larger He abundance $Y_{\rm atm,\, S}=0.18$–0.25, in agreement with the early Pioneer IR data based determination of $0.182\pm 0.005$ (He:H$_2=0.111\pm 3\%$) [@OrtonIng80]. Unfortunately, the method of @CG00 cannot be checked by application to Jupiter due to disturbing NH$_3$ cloud formation in the spectral range of interest in Jupiter’s slightly warmer atmosphere.
@Kerley04a instead suggests to trust the ratio of the He abundances for the two planets as derived originally from the Voyager IRIS and RSS data. With $({\mbox{He:H$_2$}})_{\rm S} = 0.31(1\pm 1.1)\times({\mbox{He:H$_2$}})_{\rm J}$ and $m_{\rm He}=4$ g/mol, $m_{\rm H_2}=2$ g/mol, this would give $$Y_{\rm atm, \,S}
= \frac{({\mbox{He:H$_2$}})_{\rm S}}{({\mbox{He:H$_2$}})_{\rm J}}
\:\frac{m_{\rm H_2} + m_{\rm He}({\mbox{He:H$_2$}})_{\rm J}}{m_{\rm H_2} + m_{\rm He}({\mbox{He:H$_2$}})_{\rm S}} \:Y_{\rm atm,\, J}\:,$$ which is in numbers $$\label{eq:Y_Kerley}
Y_{\rm atm, \,S} = 0.353\,(1\pm 2)\times 0.238 = 0.084\,(1\pm 2)\:.$$ We thus compute Saturn models for different helium abundances $Y_{\rm atm}$ between 0.10 and 0.18. Because of the assumed convection below the 1 bar level, $Y_1 = Y_{\rm atm}$.
#### Atmospheric temperature
Saturn’s atmospheric temperature was determined to be $135\pm 4$ K at the 1 bar level and $145\pm 4$ K at 1.3 bars [@Lindal+85]. For our model calculations we assume a slightly higher 1 bar temperature of 140 K but also compute single models with ${\mbox{ $T_{\rm 1\,bar}$ }}=135$ K. Note that the physical temperature is not a direct observable. In particular, the *observational constraint* depends on an assumed composition [@Lindal+85]. It is the height-dependent refractivity which was measured during egress and ingress of the Voyager II spacecraft. From these data and the assumed composition, the mass density and thus the particle number density can be derived. Integration of the equation of hydrostatic equilibrium, $dP/dh=-g\,\rho$, over height $h$ in the atmosphere allows to relate density to pressure. Finally, the thermal equation of state for the assumed composition with mean molecular weight $\bar{\mu}$ yields the temperature $T(P,\rho)$, which is proportional to $\bar{\mu}P/\rho$. Since the above cited value was derived for the low $Y$ value of 0.06 [@Lindal+85], a revision of that value may also require a revision of the atmospheric temperature determination towards higher temperatures. Table \[tab:obs\] summarizes the used constraints.
Planetary evolution modeling
----------------------------
We compute the cooling time of Saturn almost exactly as in @N+12 for Jupiter. In particular, out of the set of possible structure models for present Saturn, we pick one model, implying known values of the structure parameters ${\mbox{$M_{\rm core}$}}$, $Y_2$, $Z_1$, and $Z_2$. Keeping these parameter values constant, we then generate a series of models with increased values. For these models, we also keep the angular momentum, $L$ conserved and store the energy of rotation, $E_{\rm rot}=L\omega /2$. The higher ${\mbox{ $T_{\rm 1\,bar}$ }}$, the warmer the interior and hence the larger the planet radius $R_p$ and the higher the luminosity. Finally, is mapped onto time by integrating the cooling equation $$\label{eq:cool}
dt = -\frac{\int_{{\mbox{$M_{\rm core}$}}}^{{\mbox{$M_{\rm Sat}$}}}dm\:T(t)\,ds(t) + {\mbox{$M_{\rm core}$}}c_v dT_{\rm core}(t) + dE_{\rm rot}(t)}
{4\pi R^2(t)\,\sigma\, (T_{\rm eff}^4(t)-T_{\rm eq}^4(t)) - L_{\rm radio}(t)}\quad.$$ from present time $\tau_{\odot}=4.56$ Gyr backwards. In Equation (\[eq:cool\]), parameters that change with time are written as a function of time, although most of them do not depend on time explicitly; in fact, only the solar luminosity $L_{\odot}$, which is proportional to $T_{\rm eq}^4$ and assumed to increase linearly with time, and the luminosity $L_{\rm core}$ from the decay of radioactive elements in the core do so. Unlike in @N+12 where Jupiter’s $L_{\rm radio}$ was kept constant, we here include its time-dependence as in @N+11, because Saturn’s ${\mbox{$M_{\rm core}$}}/M$ ratio can be $\sim 6$ times larger than Jupiter’s and thus special core contributions potentially be non-negligible. The relation between $T_{\rm eff}$ and is adapted from @Guillot+95 with constant $K=1.565$. As determines $R_p$ and $T_{\rm eff}$, the map ${\mbox{ $T_{\rm 1\,bar}$ }}\mapsto t$ includes the more familiar maps $R_p\mapsto t$ and $T_{\rm eff}\mapsto t$. We then repeat the described procedure to compute the cooling curve for another structure model of present Saturn in order to learn about, e.g., the effect of the chosen EOS on the cooling time.
Equations of state {#sec:m_eos}
------------------
Saturn’s mantle is believed to mainly consist of hydrogen and helium, and some amount of heavier atoms or molecules which we call *heavy elements*. For each of these three components we apply a separate EOS. To obtain an EOS for the mantle material, we linearly mix the single-component EOS. Saturn models are then computed for different sets of equations of state for Saturn’s mantle: LM-REOS (see [@N+08; @N+12] and references therein), SCvH EOS for hydrogen and helium by @SCvH95 where we mimic heavy elements by scaling the density of the He EOS by a factor of 3/2, and the Sesame EOS.
The Sesame-5251 hydrogen EOS [@SESAME] is the deuterium EOS 5263 scaled in density as developed by Kerley in 1972. It is based on the chemical picture and built upon the assumption of three phases: a molecular solid phase, an atomic solid phase, and a fluid phase which takes into account chemical equilibrium between molecules and atoms and ionization equilibrium between atoms, protons and electrons. A completely revised version [@Kerley03] includes fits to more recent shock compression data resulting into a larger compressibility at $\sim 0.5\:$Mbar, and a smaller one at $\sim 10\:$Mbar. We here apply the earlier version as it shows stronger deviations to the SCvH EOS and the LM-R EOS, which allows to attribute differences in the resulting Saturn models more clearly to properties of the EOS. For application to planetary models, the Sesame H EOS is linearly mixed with the helium EOS of @Kerley04b and with the water EOS H$_2$O-REOS.
### Core material
The cores of giant planets are usually assumed to consist of ices and rocks as a relict from their time of formation [@Mizuno80; @Miller+11]. As limiting cases, we either compute models with pure water cores using the H$_2$O-REOS [@N+08; @French+09], or with pure rocky cores using the $P-\rho$ relation of @HM89. This approach ignores the possibility of an eroded core that would also contain some hydrogen and helium.
### Comparison of the applied hydrogen EOSs
Even if the heavy elements in Saturn would have the low molecular weight of helium, still at least 70% of the particles in Saturn would be hydrogen. Therefore, among the equations of state for planetary materials, that of hydrogen is expected to have the biggest influence on the resulting structure models. In Figure \[fig:hugos\] we compare the above described hydrogen EOSs with experimental deuterium shock compression data. For a certain experimental set-up and given initial conditions $(\rho_0, T_0)$, different final compression ratios $\rho/\rho_0$ and pressures can be achieved depending on the velocity of the flyer plate upon impact, be it accelerated by a gas gun, magnetic pressure, laser light, or explosives. The first-shock end states follow the Hugoniot-relation $$\label{eq:hugo}
u-u_0 = 0.5(P-P_0)(\rho_0^{-1}-\rho^{-1})\:,$$ where initial pressure $P_0(\rho_0,T_0)$ and initial internal energy $u_0(\rho_0,T_0)$ are derived from the EOS. Figure \[fig:hugos\] shows experimental gas gun data from @Nellis+83, Sandia Z machine data from @Knudson+04, the modified Omega laser data from @KnudDesj09, and spherical compression data using explosives from @Boriskov+05. By scaling the initial density, theoretical hydrogen EOSs can reasonably be compared to deuterium experimental data, although some differences between D and H become non-negligible in the molecular region, which are probed by the gas gun data, due to differences in the molecular vibrational states; see @Holst+12 for a detailed discussion. Figure \[fig:hugos\] also shows the theoretical hydrogen Hugoniot curves for H-REOS.2 ([@Holst+12], with additional data points by A. Becker *pers. comm*), for the H-SCvH-i EOS, and for the H-Sesame EOS.
![\[fig:hugos\] (Color online) Theoretical hydrogen Hugoniot curves (*solid, black*: H-REOS.2, *short-dashed*: H-SCvH-i, *long-dashed*: Sesame-5251) and experimental shock data (*grey filled squares*: SNL Z-pinch, *open squares*: modified omega laser, *circles*: explosives, *diamonds*: gas gun). The *dot-dot-dashed orange curve (S)* shows part of the Saturn adiabat. ](./f1_hugoniotsH.eps){width="48.00000%"}
Obviously, the theoretical hydrogen EOSs differ substantially from each other. Along the Hugoniot states, the Sesame EOS, which precedes even the gas gun data, is relatively stiff at $P\lesssim 1$ Mbar but the most compressible one at $\sim 5$ Mbars. Conversely, the ab initio H EOS is relatively compressible below 1 Mbar compared to the Sesame and SCvH-i EOS, and to the spherical-shock compression data, with a maximum compressibility at $P\sim 0.5$ Mbar where dissociation occurs. At higher pressures of 1 to 3 Mbar, the ab initio EOS runs nearly through the experimental central values which indicate a low compression ratio of 4.25, whereas SCvH-i EOS agrees well with the data up to 0.8 Mbars but then turns to a large maximum compressibility at $\sim 1$ Mbar where in the underlying SCvH-ppt EOS the plasma phase transition occurs. These properties lead to systematic differences in the resulting Saturn models.
Results {#sec:results}
=======
Structure models with different equations of state {#sec:r_struc1}
--------------------------------------------------
In this Section we focus on the effect of the different EOSs (Sesame, SCvH, LM-REOS) on the resulting Saturn models. The presented models have been calculated for $Y_1=0.18$, $2\pi/\omega=$10h 39m, and the default values of the other observational constraints as given in Table \[tab:obs\]. Figure \[fig:mcZZ\_P12\] shows the results.
![\[fig:mcZZ\_P12\] (Color online) Resulting Saturn structure models for different equations of state for the mantle (*solid, black*: LM-REOS; *long-dashed, blue*: Sesame EOS; *dashed, green*: SCvH-i EOS) and for the core (*thick lines*: rocky cores, *symbols connected by thin lines*: water cores), and different surface temperatures (*open symbols*: ${\mbox{ $T_{\rm 1\,bar}$ }}=135$ K). For given other input parameters (see Section \[sec:m\_modeling\] for details), each displayed model is uniquely defined by its ${\mbox{$P_{\mbox{\tiny1-2}}$}}$-value ($x$-axis). Note that the two upper panels have a common $y$-axis which changes scale at $Z_1=Z_2=7\times Z_{\rm solar}$, where $Z_{\rm solar}=1.5\%$. One model is highlighted for each EOS by a filled circle in panel (c). The *red diamond* shows the measured $9.12\times$ solar C/H ratio.](./f2_mcZZ.eps){width="40.00000%"}
For each of the considered mantle EOS, the parameters $Z_1$, $Z_2$, ${\mbox{$M_{\rm core}$}}$, and $M_Z$ behave similarly with ${\mbox{$P_{\mbox{\tiny1-2}}$}}$. The deeper the layer boundary, the higher become $Z_1$ and $Z_2$ and, as a response, the lower becomes ${\mbox{$M_{\rm core}$}}$, while $M_Z$ remains nearly constant within $\sim 5{M_{\oplus}}$. Zero-core mass models are possible simply by putting the layer boundary sufficiently deep into the planet. The maximum core mass is determined by the condition that neither $Z_1$ nor $Z_2$ must become negative. It is $15{M_{\oplus}}$ for rocky cores and about $20{M_{\oplus}}$ for water cores.
For given ${\mbox{$P_{\mbox{\tiny1-2}}$}}$, water cores are up to 50% heavier than rock cores because low-density material requires a larger volume. Displaced envelope material must be replaced by core material, hence the core becomes heavier. Note that water core models that approach the limit ${\mbox{$M_{\rm core}$}}\to 0$ have been computed for LM-REOS mantle EOS only, but they exist for the other mantle EOS as well.
The core mass responds weakly to the surface temperature. The found slight decrease by $\leq 2{M_{\oplus}}$ can intuitively be explained by the colder and thus denser envelopes. However, the colder outer envelope requires $\Delta Z_1\gtrsim 0.01$ less heavy elements to match $J_4$, but up to $\Delta Z_2=0.05$ *more* heavy elements in the inner envelope. Thus the found insensitivity of ${\mbox{$M_{\rm core}$}}$, and also of $M_Z$, to the surface temperature appears to be a more complex compensation of different effects in Saturn.
The general behavior of the solutions can be understood with the help of Figure \[fig:contribJ2n\]. It shows the sensitivity of the gravitational moments to the internal mass distribution as parameterized by the contribution functions $c_{2n}$, see Eq. (\[eq:c2n\]). While they have been computed for a particular model, i.e. the LM-REOS based Saturn model S12-3a as highlighted in Figure \[fig:mcZZ\_P12\]c, their properties are the same for all three-layer models. As it is well knowm [@ZT78] the higher the order of the gravitational harmonic, the farther out are the locations of mean and maximum sensitivity and the more pronounced is the latter one. $J_2$ and $J_4$ are most sensitive at pressures of $\sim 0.5$ and $0.1$ Mbar, respectively. At the 1 Mbar level, the sensitivity of $J_4$ has dropped to $\sim 30$% of its maximum value. $J_4$ is almost insensitive to the mass below 3 Mbars, a typical pressure for the outer/inner envelope boundary of LM-REOS based Saturn models. Therefore, $Z_1$ changes little with ${\mbox{$P_{\mbox{\tiny1-2}}$}}$ for ${\mbox{$P_{\mbox{\tiny1-2}}$}}> 1$ Mbar. As the sensitivity of $J_2$ is similar to that of $J_4$ and just slightly shifted to higher pressures, the computed $J_2$ value is affected by the value of $Z_1$ as well. But because $J_2$ is to be adjusted by the $Z_2$ value, i.e. by the mass distribution interior to ${\mbox{$P_{\mbox{\tiny1-2}}$}}$ where its sensitivity is low, small changes in $J_2$ for adjustment require strong changes in $Z_2$, see Figure \[fig:mcZZ\_P12\]b.
![\[fig:contribJ2n\] (Color online) Contribution functions of the gravitational harmonics $J_2$ (*thick solid, blue*), $J_4$ (*short-dashed, red*), $J_6$ (*dot-dashed, green*), and $J_8$ (*long-dashed, orange*) for the Saturn model S12-3a. The *bottom x-axis* scales linearly with mean radius; the *top x-axis* shows the radii where the pressures of 0.01–50 Mbar occur. Layer boundaries are clearly seen in the function $c_0$ (*thin solid, black*). *Diamonds* show the radius where half of the final $J_{2n}$ value is reached.](./f3_contribJ2n.eps){width="48.00000%"}
#### LM-REOS based models
Using LM-REOS, we find solutions for $1.2 \leq {\mbox{$P_{\mbox{\tiny1-2}}$}}\leq 5$ Mbars. For all of these models, $Z_1$ is nearly constant no matter what the values of $Z_2$ and ${\mbox{$M_{\rm core}$}}$ are. This is because with the layer boundary so deep inside, $J_4$ is little sensitive to the mass distribution there. As $Z_2$, in contrast, covers a wide range of 0–60%, there are solutions with $Z_1=Z_2$, unlike for Jupiter [@N+12].
For LM-REOS based models the layer boundary has to be put rather deep inside the planet in order to nearly suppress the influence of the matter interior to ${\mbox{$P_{\mbox{\tiny1-2}}$}}$ to the values of $J_2$ and $J_4$. Otherwise, say for ${\mbox{$P_{\mbox{\tiny1-2}}$}}< 1$ Mbar, the rise in $J_2$ could no longer be compensated for by a lower $Z_2$ value because $Z_2$ already goes to zero. This behavior is a direct consequence of the higher compressibility of the H EOS at sub-Mbar pressures (Figure \[fig:hugos\]). The total mass of heavy elements is 16–$20\:{M_{\oplus}}$, less than predicted for Jupiter with LM-REOS ($\sim 30{M_{\oplus}}$, [@N+12]). However, Saturn’s $M_Z$ corresponds to an overall enrichment $Z_p=10$–15 , higher than Jupiter’s ($\sim 6\:{\mbox{$Z_{\odot}$}}$).
#### Sesame based models
Sesame EOS based models have the layer boundary between 0.2 and 2.5 Mbar. With ${\mbox{$P_{\mbox{\tiny1-2}}$}}\to 0.2$ Mbar, the sensitivity of $J_4$ rises strongly (Figure \[fig:contribJ2n\]) so that $Z_1$ has to decrease rapidly in order to not produce too high $|J_4|$ values. There is no overlap of the functions $Z_1({\mbox{$P_{\mbox{\tiny1-2}}$}})$ and $Z_2({\mbox{$P_{\mbox{\tiny1-2}}$}})$, i.e. Sesame EOS based Saturn models require a higher metallicity in the inner than in the outer envelope. Because of the stiffness of the H-Sesame EOS up to 1 Mbar (Figure \[fig:hugos\]), the layer boundary can be farther out and the $Z_1$ values as well as the total mass of heavy elements can be up to 2.5 times higher than for LM-REOS based models. In fact, the layer boundary *must* be farther out because the rise in $Z_2$ for increasing ${\mbox{$P_{\mbox{\tiny1-2}}$}}$ is accompanied by a rapid decrease in ${\mbox{$M_{\rm core}$}}$, which must not become negative, a direct effect of the higher compressibility of the H-Sesame EOS at higher pressures up to 10 Mbar (Figure \[fig:hugos\]).
#### SCvH-i based models
From the gross compressibility behavior of the Hugoniot up to 100 GPa one would expect the SCvH-i EOS based solutions to fall between those for the other two EOSs. This in fact happens with respect to the parameters $Z_2$, ${\mbox{$M_{\rm core}$}}$, and $M_Z$. In particular, the just slightly higher compressibilities of SCvH-i compared to Sesame EOS below 100 GPa suggest just slighty *lower* $Z_1$ values than for that EOS. However, the $Z_1$ values are relatively *higher* than for the Sesame EOS based models. This can only be due to higher temperatures along the SCvH-i Saturn adiabat, as higher temperatures at given pressure level reduce the mass density, which allows for more heavy elements to be added. We will encounter the same argument again in Section \[sec:r\_struc2\]. In contrast to SG04, we do not find models with $Z_1=Z_2$. Presumably, this stems from the application of the more recent, accurate $J_{2n}$ data.
Both Sesame and SCvH-i EOS based models have $M_Z=25$–$30\:{M_{\oplus}}$, a similar amount as Jupiter may have ($\sim 15$–$40{M_{\oplus}}$; SG04). With $Z_p=17$–$21~{\mbox{$Z_{\odot}$}}$, this is a larger enrichment than in comparable models for Jupiter (3–8 ). Note that SG04 use an earlier value ${\mbox{$Z_{\odot}$}}^{('89)}=0.019$.
#### Internal profiles
For each of the EOS, one selected interior profile is shown in Figure \[fig:profilesS\]. The temperatures along the SCvH-i Saturn adiabat in the outer envelope are indeed higher than for the Sesame adiabat. For the LM-REOS based adiabat, the onset of dissociation occurs at $\sim 0.7{\mbox{$R_{\rm Sat}$}}$ where the temperature gradient flattens. A typical value of the density is 2 g cm$^{-3}$ in the inner envelope, 15 g cm$^{-3}$ in a rocky core, and 8 g cm$^{-3}$ in a water core, while in the outer envelope the density changes by four orders of magnitude.
![\[fig:profilesS\] (Color online) Interior density and temperature profiles of the Saturn models that are highlighted in Fig. \[fig:mcZZ\_P12\]c by filled circles.; *solid, black*: model with LM-REOS, *dashed, green*: SCvH-i, and *long-dashed, blue*: with Sesame EOS. These models have an isothermal rock core. Between 0.95 and 1 , the density changes by three orders of magnitude (not displayed), and between 0.95 and the boundary to the inner envelope, by one order of magnitude.](./f4_profileS.eps){width="48.00000%"}
Concluding, for each of the considered EOS, similar values of the metallicity in the outer envelope (e.g., $3\times$ solar), in the inner envelope (e.g. 10–$30\times$ solar), and of the core (e.g., 0–$15\:{M_{\oplus}}$ for rocky or 0–$18\:{M_{\oplus}}$ for water cores) are possible. It depends mainly on the position of an internal compositional gradient (for instance in form of a layer boundary), which values are adopted. In contrast, the different EOSs require different locations of that gradient, and LM-REOS predicts a lower total mass of heavy elements than the other two EOSs.
LM-REOS based structure models with different helium abundances, rotation rates, and $J_4$ values {#sec:r_struc2}
-------------------------------------------------------------------------------------------------
The resulting atmospheric heavy element enrichment of $\sim 3\times$ solar of the LM-REOS based Saturn models is rather low compared to the measured carbon enrichment of $9\times$ solar. Therefore, we investigate in this Section qualitatively the effect of the assumed atmospheric helium abundance, of the rotation rate, and of the $J_4$ value on the resulting outer envelope metallicity. As we are aiming to get higher $Z_1$ values than for the models in Section \[sec:r\_struc1\], we only consider higher $|J_4|$ values, up to the $6\sigma$ observational uncertainty (any $|J_{2n}|$ increases with the mass density in the sensitive region), lower $Y_1$ values ($Z_1$ decreases linearly with $Y_1$) of respectively 0.16 and 0.1; but we consider a higher rotation rate (to face a possible reality), although it will require a reduction in the heavy element content of the outer part of a planet. For all of these models, the transition pressure is at 300 GPa and the core consists of rocks. At temperatures between 140 and 300 K, the analytic Van-der-Waals-gas EOS is used for H$_2$ and He, and the ideal gas EOS for molecular H$_2$O, but this has no relevant effect on the resulting adiabats.
Figure \[fig:ZZ\_J4\] shows the resulting enrichment factors for the metallicity in the outer envelope –to be compared with the (potentially) measured abundances– and also in the inner envelope.
![\[fig:ZZ\_J4\] Resulting outer (*solid lines*) and inner envelope metallicities (*dashed lines*) in terms of the solar metallicity $Z_{\odot}=0.015$ for different He/(He+H) mass ratios of 0.10, 0.16, 0.18 and different periods of rotation of 10h 39m and 10h 32m as labeled, and different given $J_4$ values up to the $6\sigma$ uncertainty (*x-axis*). Some of the numerically possible models with $Z_2$ close to zero are not displayed. ](./f5_ZZ_J4.eps){width="40.00000%"}
Obviously, the influence of $J_4$ on $Z_1$ is weak. Only for extremely high $|J_4|$ values of $9.6\times 10^{-4}$ the effect becomes of same size as the effect of a slower rotation by 7 minutes, which is $\Delta Z_1\sim +1Z_{\odot}$. Note that $\Delta Z_1/Z_1 > 20\%$ constitutes a reather big amplification of the 1% relative uncertainty in the rotation rate.
In case a high rotation rate of 10h 32m should proof true, the heavy element enrichment would decrease down to $2\times$ solar for a helium abundance ${\mbox{$Y_{\rm atm}$}}=0.18$. Note that we do not adjust the planet radius to the rotation rate. Interestingly, the rotation rate mainly affects the inner envelope metallicity. This can be explained by the response of $J_2$ on a change in $Z_1$ and $Z_2$ as described in \[apx:J2Z1Z2\].
The biggest effect on $Z_1$ can be achieved by lowering the helium abundance, where $\Delta Z_1\sim 1.5\%$ of heavy elements can be added for $\Delta Y_1\sim 2\%$ of helium. It is $|\Delta Z_1|/|\Delta Y_1| < 1$ because the less helium there is, the higher the specific heat of the material, and thus the lower must be the temperature along the adiabat to keep the entropy constant. Colder adiabats give denser envelopes and thus allow for less heavy elements to be put into.
Despite the wide range of considered parameter values, with $Z_1\lesssim 6\ Z_{\odot}$ the possible atmospheric heavy element abundance remains clearly below the 9fold enrichment of carbon. Our LM-REOS based models therefore predict O/H to be less than $9\times$ solar.
Moment of Inertia {#sec:r_nmoi}
-----------------
![\[fig:nmoi\] (Color online) (a) Nondimensional axial moment of inertia—core mass relation for a subset of the structure models of Fig. \[fig:mcZZ\_P12\] using the different EOSs LM-REOS (*solid, black*), SCvH-i (*dashed, green*), Sesame (*long-dashed, blue*) with rock cores (*lines*) or water cores (*symbols*); (b)$^1$: Core mass range and $\lambda$-range; *thin dotted:* from Ref. @Helled11 for a period of rotation of 10h39m24s and scaling by $R_{\rm eq}^{-2}$ (R. Helled, *pers. comm. 2013*); *thick dotted:* same as thin dotted but scaled by $R_{\rm mean}^{-2}$; *Arrows* (vertical position has no meaning) Radau-Darwin approximation, see text for details.](./f6_nmoiMc2.eps){width="48.00000%"}
@Helled11 showed that an observational determination of Saturn’s axial moment of inertia, $C$ would impose an additional constraint on Saturn’s core mass, and that the necessary measurements can be provided by the *Cassini* extended extended mission. Figure \[fig:nmoi\] presents our results for Saturn’s nondimensional moment of inertia, $\lambda=C/{\mbox{$M_{\rm Sat}$}}{\mbox{$R_{\rm m}$}}^2$ for a representative subset of the models of Fig. \[fig:mcZZ\_P12\]. As pointed out by and in agreement with @Helled11, we find different $\lambda$ values for different core mass values for models that all meet the observed $J_2$ and $J_4$ values within their tight $1\sigma$ bounds. In particular, for a fixed core EOS (rocks or water) and mantle EOS (SCvH, Sesame, or LM-REOS), and not too small core mass values ($\gtrsim 10{M_{\oplus}}$), the relation between ${\mbox{$M_{\rm core}$}}$ and $\lambda$ becomes unique. However, due to the uncertainties in the core and envelope EOS, a measurement of $\lambda$ could further, but not unambiguously constrain the core mass. Moreover, our physical EOS based, adiabatic models yield a very narrow possible range for $\lambda$, that even does not overlap with the prediction by @Helled11, see Figure \[fig:nmoi\]b. Therefore, a measured $\lambda$ value would definitely be of great value for discriminating between competing Saturn models.
As moment of inertia measurements for gas giant planets are challenging, the Radau-Darwin approximation is often used for an estimate. It expresses $\lambda$ in terms of the easier accessible $J_2$ and the expansion parameter $m=\omega^2 {\mbox{$R_{\rm m}$}}^3/GM$, and becomes exact only in the limit of constant density bodies. Using ${\mbox{$R_{\rm m}$}}=58201$ km, $P=10\rm h\,39m$, and the values of Table \[tab:obs\], we calculate $\lambda^{(RD)}=0.2283$ (solid arrow in Figure \[fig:nmoi\]b). The more consistent values of Table 1 in @Helled11 suggest $\lambda^{(RD)}=0.2292$ (dashed arrow in Figure \[fig:nmoi\]b). Interestingly, our interior model based values are $\sim 3\%$ higher than $\lambda^{(RD)}$ indicating that higher-order deformations ($J_4$) play a non-negligible role in the internal mass distribution. Note[^1] that different scalings of $\lambda$ by either the equatorial radius as in @Helled11 (R. Helled, *pers.comm. 2013*), or by the mean radius changes the value of Saturn’s $\lambda$ by 7.2%.
Homogeneous Evolution {#sec:r_evol}
---------------------
Homogeneous evolution implies a constant mean molecular weight in every mass shell of the planet with time, while inhomogeneous evolution also allows for vertical mass transport such as He rain or core erosion. In the considered case of homogeneous evolution, we get cooling times $\tau_{\rm Sat}$ of 2.56 Gyr for the LM-REOS model, 2.36 Gyr (SCvH-i EOS model), and 2.31 Gyr (Sesame EOS model). Neglecting the three contributions from angular momentum conservation, change of the energy of rotation, and from the time-dependence of the irradiation yields 0.05 Gyr longer cooling times for Saturn. While with $\tau_{\rm Sat}\sim 2.5$ Gyr, the cooling time comes out significantly shorter than the age of the solar system of 4.56 Gyr —commonly believed to also be the age of the planets within an uncertainty of a few Myr according to circumstellar disk observations [@Strom+93]— it is in agreement with previous calculations. For instance, using the SCvH-i EOS, @Guillot+95 compute $2.6\pm 0.2$ Gyr and @FH03 2.1 Gyr for an adiabatic, homogeneously evolving Saturn.
Our results show once more that the short cooling time of a homogeneously evolving Saturn is essentially independent on details of the model assumptions such as the size of the core. In other words, we confirm the well-known evidence for a real excess luminosity compared to the predicted luminosity from homogeneous evolution [@Pollack+77; @Saumon+92; @FH03].
Summary and Discussion {#sec:discussion}
======================
Gross features
--------------
We have applied different EOSs to compute structure and evolution models for Saturn within the standard approach of a layered interior with only few layers that cool down homogeneously with time. Because of the large applied input parameter space we have selected combinations of parameters that we believe yield a reliable estimate of the overall uncertainty in key internal structure parameters. In particular, with LM-REOS and assuming $Y_1=0.18$ and $2\pi/\omega=$10h 39m we find ${\mbox{$M_{\rm core}$}}= 0$–$17{M_{\oplus}}$, $M_Z=16$–$20{M_{\oplus}}$, $Z_{\rm atm}\lesssim 3\times$ solar, and $\tau_S=$2.6 Gyr; with Sesame-EOS we find ${\mbox{$M_{\rm core}$}}=0$–$20{M_{\oplus}}$, $Z_{\rm atm}\leq 7\times$ solar, $M_Z=26$–$30{M_{\oplus}}$, and $\tau_S=$2.3 Gyr, while SCvH EOS based models have values in between. The value of $Z_{\rm atm}$ of the LM-REOS based models can be lifted up to a factor of two if $Y_1$ is lowered down to 0.10. With $\tau_S=2.3$–2.6 Gyr, the cooling time is significantly shorter than the age of the solar system, pointing to a failure of the cooling model that is to be sought beyond the uncertainties in Saturn’s composition or the EOS.
We encourage measurements of Saturn’s moment of inertia and of the atmospheric abundance of helium and oxygen for discriminating between the wide range of possible Saturn models and for probing the underlying EOS in certain pressure ranges.
O:H ratio
---------
The O:H ratio could in principle be derived from brightness temperature measurements at $\sim 1$ m wavelengths using LOFAR (D. Gautier, *pers. comm.*), which is a set of ground-based radio antennas in western Europe. From a measured O:H, in addition to the already measured C:H, the atmospheric metallicity can be estimated and compared with the $Z_1$ values of the Saturn structure models. For this purpose, we show possible O:H– relations in Fig. \[fig:ZOH\]. For a given O:H, has been computed as the sum of the heavy element particle abundances times their atomic weight, divided by the sum over all element abundances times their weight, where we assume He:H = 0.052 (${\mbox{$Y_{\rm atm}$}}=0.18$), C:H$=0.912\times$ solar, and $2\times$ (or $4\times$, $6\times$) solar abundances of the elements {N, P, S, Ne, Ar, Kr, Xe, Mg, Al, Ca}, using the solar system abundances of @Lodders03. As Jupiter’s atmosphere is observed to be strongly depleted in Ne, which may also be the case for Saturn if caused by He sedimentation [@WilMil10], we also compute ${\mbox{$Z_{\rm atm}$}}$ with Ne:H=0.
Obviously, models with $Z_1=\sim 3{\mbox{$Z_{\odot}$}}$ would imply a low O:H of only 2x solar; $Z_1\sim 6\times {\mbox{$Z_{\odot}$}}$, the upper limit of the LM-REOS based Saturn models, would imply O:H = 6–8$\times$ solar. Higher values are possible with the Sesame or the SCvH-i EOS. Concluding, a measured O:H could tremendously help to discriminate between the various Saturn models.
![\[fig:ZOH\]Relation between the atmosperic O:H ratio and the atmosheric metallicity, ${\mbox{$Z_{\rm atm}$}}$ in solar units, for which different element abundances are assumed: elements apart from C and O are 2 times solar (*thick solid*), 4x solar (*dashed*), and 6x solar (*dot-dot-dashed*). *Thin lines*: the same, respectively, but with zero-Neon abundances. C:H is $9.12\times$ solar and O:H is displayed on the *x-axis*.](./f8_ZatmOH.eps){width="48.00000%"}
Core mass and formation {#sec:d_mcoreform}
-----------------------
The possible core mass values of Jupiter and Saturn persistently attract attention, as the hope to infer the planet formation process from the ”face value” of the present core mass continues to exist. Leading candidates for possible formation processes are the core accretion scenario, where the gaseous envelope is accreted onto a heavy-element core [@Alibert+05; @DodRob+08; @Kobayashi+12], and the gravitational disk instability scenario, where the planet would form through self-contraction of a gaseous cloud ([e.g. @HelledSchubert08]).
Let us assume the initial core mass of Saturn was $\gtrsim 15{M_{\oplus}}$. What could that tells us? Such a heavy core exceeds the maximum core mass of $8 {M_{\oplus}}$ found by @HelledSchubert08 and thus would rule out a disk-instability-kind formation for Saturn. On the other hand, it would allow for a comfortably short time scale for core accretion formation [@DodRob+08; @Kobayashi+12], even for a non-zero abundance of grains in the protoplanetary envelope [@DodRob+08].
If the initial core mass of Saturn was $\sim 7{M_{\oplus}}$, then both formation processes could have let to the present Saturn. Given Saturn’s high total heavy element mass of 16–30${M_{\oplus}}$, disk-instability formation would indicate a massive protosolar disk [@HelledSchubert09], as well as an early presence of planetesimals before formation was completed [@HelledSchubert08].
In the case of an initial core mass of $< 1{M_{\oplus}}$, again both scenarios could be possible. Here, core accretion formation would require the absence of grains in the envelope in order to reduce the gas pressure induced by warm temperatures in an opaque medium [@HoriIkoma10].
As main paths toward better constrained core mass properties, we support Helled’s (2011) suggestion of a moment of inertia measurement, and emphasize the need for a better understanding of Saturn’s envelope structure.
Three-layer models and helium rain in Saturn {#sec:d_Herain}
--------------------------------------------
The over and over confirmed finding of a too short cooling time for homogeneously evolving Saturn models, regardless of details in the structure models, or of the underlying EOS used, and the simultaneously repeated mentioning of a high likelihood for an inhomogeneously evolving Saturn as a result of He sedimentation (e.g., [@Pollack+77; @SS77b; @Saumon+92; @Guillot+95; @FH03]) cast doubt on the usefulness of Saturn models that ignore this process.
The most recent theories of H/He phase separation predict demixing under relevant planetary conditions to occur when hydrogen metallizes under high pressures while the helium atoms are still neutral [@Morales+09; @Lorenzen+09; @Lorenzen+11]. In Jupiter, metallization occurs rather far out in the planet at $\sim 0.9 R_J$ where the pressure is only 0.5 Mbar [@French+12], but the lowest pressure that could be achieved in H/He demixing calculations so far is 1 Mbar [@Lorenzen+11]. Indeed, the ab initio data based H/He phase diagram suggests demixing in Saturn at least within 1–5 Mbar [@Lorenzen+11], which corresponds to the region beneath $\sim 2/3\: {\mbox{$R_{\rm Sat}$}}$. Let us thus divide Saturn’s mantle into three regions: an outer region down to 1 Mbar ($\sim 2/3{\mbox{$R_{\rm Sat}$}}$), an innermost helium-rich region where the sedimented helium dissolves again in its surrounding, possibly a helium-layer [@FH03], and a middle region where the $P-T$ conditions favor H/He phase separation.
In the middle region, the helium abundance at given $P,T$ will reduce until the remaining He atoms become miscible: the helium abundance will follow the demixing curve [@SS77b]. In $P$–$\rho$-space, such an inhomogeneous region could simply be described by a smoothed layer boundary. In $P$–$T$-space, the effect of an inhomogeneity could be significant, as it may be correlated with a strongly superadiabatic temperature gradient [@SS77b], which would require higher envelope metallicities and affect the derived core mass [@LC12]. Also important may be the induced differential rotation from sinking droplets that conserve their angular momentum. @Cao+12 showed that a tiny differential rotation would suffice to drive the magnetic field and explain its observed dipolarity.
In case the sedimented helium dissolves in the lower region, that part is well represented by an inner envelope with enhanced abundance as in our models. But in case the helium rains down to the core, a preferred solution to explain Saturn’s high luminosity [@FH03], the upper part of what we count to be core material may in fact be helium. Thus, Saturn’s maximum core mass could be lower than predicted by our models.
In the outer region we would mainly see the depletion in helium, because helium atoms from the upper regions are transported repeatedly by convection down into the immiscibility region, where a fraction of them gets lost into the deep through sedimentation. Whatever happens to the compositional and temperature gradient deep in the planet, the outer, miscible envelope should remain homogeneous and adiabatic. Therefore, our results for the atmospheric helium and heavy element abundances are certainly caused or influenced by He-sedimentation, but are not expected to alter when He sedimentation would be explicitly accounted for.
Why would we expect a (dis)continuous heavy element distribution?
-----------------------------------------------------------------
Standard three-layer models with heavy element discontinuity benefit from two additional parameters ($Z_2$, ) that can be used to fit the observational constraints. However, those models would even more benefit from a physical justification. At least, the assumption of a sharp layer boundary between two convective, homogeneous layers offers the advantage of a self-consistent picture, where upward particle transport across would be inhibited but not necessarily the heat transport. In contrast, a continuous, inhomogeneous heavy element distribution as suggested by @LC12 might lead to a semi-convective boundary layer with reduced heat transport. Whether such a picture can explain the luminosities of both Jupiter and Saturn remains to be shown.
A possible origin for an inhomogeneous heavy element distribution could be the erosion of an initially big core, with subsequent small-scale layer formation with small compositional gradients [@LC12], and finally merging of the layers to a single one as seen in hydrodynamic simulations of fluids with both temperature and compositional gradients [@Wood+12]. Thus, a high, homogeneous metallicity in the inner envelope could be the result of an eroded core where the core material went through stages of layer merging until the compositional gradient got large enough to stop merging with what we now see as an outer envelope.
Summarized main findings {#sec:d_summary}
------------------------
- The H/He EOS strongly influences the atmospheric metallicity, ${\mbox{$Z_{\rm atm}$}}$ and the possible position of an internal layer boundary, but has little influence on the core mass and the cooling time. We find ${\mbox{$M_{\rm core}$}}=0$–$20{M_{\oplus}}$ and $\tau_{\rm Sat}\sim 2.5$ Gyr.
- The total mass of heavy elements in Saturn can be less (LM-REOS) or equal (Sesame, SCvH-i EOS) to that in Jupiter, while the averaged enrichment is larger than that of comparable Jupiter models.
- Our LM-REOS based Saturn models predict ${\mbox{$Z_{\rm atm}$}}\sim 3\times$ solar ($\lesssim 6\times$) for an atmospheric helium abundance ${{\mbox{$Y_{\rm atm}$}}}=18\%$ (${\mbox{$Y_{\rm atm}$}}=10\%$) by mass. The corresponding predicted maximum O:H ratio is $\sim 2 \times$ solar ($8\times$).
- For Saturn, we calculate a non-dimensional axial moment of inertia $\lambda=0.235$ to 0.236.
We thank Daniel Gautier for illuminating conversations on abundance measurements, Johannes Wicht and Ravit Helled for interesting discussions, Andreas Becker for computing the high-temperature extension of the H-REOS.2 Hugoniot curve, and Winfried Lorenzen for discussions on H-He demixing. The work presented in this paper is supported by the “Deutsche Forschungsgemeinschaft” (DFG) within the SFB 652 and the project RE 882/11.
Erratum Jupiter-paper
=====================
In the Jupiter-II paper by @N+12 on Jupiter structure and homogeneous evolution models it was stated that including the energy of rotation in the thermal evolution leads to a reduced luminosity with time and thus a *longer* cooling time. The latter statement must be corrected for a *shorter* cooling time: the energy of rotation will *not* be released by radiation at a later time. Therefore, Jupiter’s cooling time is *decreased* by 0.2 Gyr (and not prolonged by 0.2 Gyr as stated in that paper). The given final value for cooling time of 4.41 Gyr, when in addition the time-dependence of $L_{\odot}$ is included, remains valid.
Estimate $\Delta Z_2$ as a function of $\Delta Z_1$ {#apx:J2Z1Z2}
===================================================
We here derive an estimate for the necessary change in $Z_2$ in response to a change $\Delta Z_1$ when $J_2$ is to be kept unchanged. According to Equation \[eq:J2n\] we can approximate $J_2$ by $$\label{eq:J2approx}
J_2\approx V_{L1}\,\bar{\rho}_{1}\, r^4(\bar{\rho}_1) + V_{L2}\,\bar{\rho}_{2}\,r^4(\bar{\rho}_2),$$ where $\bar{\rho}_i$, $i=1,2$, is the mean density of layer No. $i$, and $V_{Li}$ its volume. The contribution from the small, central core is neglected. According to the additive volume law for mixtures we can write $\rho^{-1}=(1-Z)\rho_{\rm H,He}^{-1} + Z\rho_Z^{-1}$ and thus $$\label{eq:drhodZ}
\frac{d\rho}{dZ} = \rho^2\left(\rho^{-1}_{\rm H,He} - \rho_Z^{-1}\right)\quad,$$ so that $\Delta \rho_1\sim \bar{\rho}_1^2\,\Delta Z_1$ and $\Delta \rho_2\sim \bar{\rho}_2^2\,\Delta Z_2$. Because the value of $J_2$ is an observational constraint, $\Delta J_2$ must be zero under the perturbations $\Delta \bar{\rho}_1$ and $\Delta \bar{\rho}_2$. With Equations (\[eq:J2approx\]) and (\[eq:drhodZ\]) we thus have $$0 = \Delta J_2 \approx V_{L1}\:\Delta Z_1\: \bar{\rho}^2_1\: r^4(\bar{\rho}_1)
+ V_{L2}\:\Delta Z_2\: \bar{\rho}_2^2 \: r^4(\bar{\rho}_2)\quad,$$ hence $$\label{eq:factors}
\Delta Z_2 = -\Delta Z_1 \left(\frac{V_{L1}}{V_{L2}}\right)\left(\frac{r(\bar{\rho}_1)}{r(\bar{\rho}_2)}\right)^4
\left(\frac{\bar{\rho_1}}{\bar{\rho}_2}\right)^2\:.$$ For a typical three-layer Saturn model as shown in Figure \[fig:profilesS\], $r({\mbox{$P_{\mbox{\tiny1-2}}$}})\sim 0.5 {\mbox{$R_{\rm Sat}$}}$, leading to $V_{L1}/V_{L2}\sim 1/0.5^3= 2^3$, $r(\bar{\rho}_1)=0.7{\mbox{$R_{\rm Sat}$}}$, $r(\bar{\rho}_2)=0.35 {\mbox{$R_{\rm Sat}$}}$, leading to $\left(r(\bar{\rho}_1)/r(\bar{\rho}_2)\right)^4\sim 2^4$, $\bar{\rho}_1\sim 0.7\:\rm g\,cm^{-3}$, $\bar{\rho}_1\sim 2.05\:\rm g\,cm^{-3}$ leading to $\left(\bar{\rho}_1/\bar{\rho}_2\right)^2 \sim (1/3)^2$, and thus $\Delta Z_2/\Delta Z_1 \approx -2^7/3^2 \approx -16$, in reasonable agreement with $\Delta Z_2/\Delta Z_1 \approx -10$ as seen in Figure \[fig:ZZ\_J4\].
[^1]: modified after acceptance of this paper
|
---
abstract: |
This paper provides a sample of a LaTeX document which conforms to the formatting guidelines for ACM SIG Proceedings. It complements the document *Author’s Guide to Preparing ACM SIG Proceedings Using LaTeX$2_\epsilon$ and BibTeX*. This source file has been written with the intention of being compiled under LaTeX$2_\epsilon$ and BibTeX.
The developers have tried to include every imaginable sort of “bells and whistles", such as a subtitle, footnotes on title, subtitle and authors, as well as in the text, and every optional component (e.g. Acknowledgments, Additional Authors, Appendices), not to mention examples of equations, theorems, tables and figures.
To make best use of this sample document, run it through LaTeXand BibTeX, and compare this source code with the printed output produced by the dvi file.
author:
- |
Ben Trovato\
\
\
\
G.K.M. Tobin\
\
\
\
Lars Th[ø]{}rv[ä]{}ld\
\
\
\
- |
Lawrence P. Leipuner\
\
\
\
Sean Fogarty\
\
\
\
Charles Palmer\
\
\
\
bibliography:
- 'sigproc.bib'
date: 30 July 1999
subtitle: '\[Extended Abstract\] '
title: 'A Sample [ACM]{} SIG Proceedings Paper in LaTeX Format'
---
\[complexity measures, performance measures\]
Introduction
============
The *proceedings* are the records of a conference. ACM seeks to give these conference by-products a uniform, high-quality appearance. To do this, ACM has some rigid requirements for the format of the proceedings documents: there is a specified format (balanced double columns), a specified set of fonts (Arial or Helvetica and Times Roman) in certain specified sizes (for instance, 9 point for body copy), a specified live area (18 $\times$ 23.5 cm \[7“ $\times$ 9.25”\]) centered on the page, specified size of margins (2.54cm \[1“\] top and bottom and 1.9cm \[.75”\] left and right; specified column width (8.45cm \[3.33“\]) and gutter size (.083cm \[.33”\]).
The good news is, with only a handful of manual settings[^1], the LaTeX document class file handles all of this for you.
The remainder of this document is concerned with showing, in the context of an “actual” document, the LaTeX commands specifically available for denoting the structure of a proceedings paper, rather than with giving rigorous descriptions or explanations of such commands.
The [Body]{} of The Paper
=========================
Typically, the body of a paper is organized into a hierarchical structure, with numbered or unnumbered headings for sections, subsections, sub-subsections, and even smaller sections. The command `’134section` that precedes this paragraph is part of such a hierarchy.[^2] LaTeX handles the numbering and placement of these headings for you, when you use the appropriate heading commands around the titles of the headings. If you want a sub-subsection or smaller part to be unnumbered in your output, simply append an asterisk to the command name. Examples of both numbered and unnumbered headings will appear throughout the balance of this sample document.
Because the entire article is contained in the **document** environment, you can indicate the start of a new paragraph with a blank line in your input file; that is why this sentence forms a separate paragraph.
Type Changes and [Special]{} Characters
---------------------------------------
We have already seen several typeface changes in this sample. You can indicate italicized words or phrases in your text with the command `’134textit`; emboldening with the command `’134textbf` and typewriter-style (for instance, for computer code) with `’134texttt`. But remember, you do not have to indicate typestyle changes when such changes are part of the *structural* elements of your article; for instance, the heading of this subsection will be in a sans serif[^3] typeface, but that is handled by the document class file. Take care with the use of[^4] the curly braces in typeface changes; they mark the beginning and end of the text that is to be in the different typeface.
You can use whatever symbols, accented characters, or non-English characters you need anywhere in your document; you can find a complete list of what is available in the *LaTeXUser’s Guide*[@Lamport:LaTeX].
Math Equations
--------------
You may want to display math equations in three distinct styles: inline, numbered or non-numbered display. Each of the three are discussed in the next sections.
### Inline (In-text) Equations
A formula that appears in the running text is called an inline or in-text formula. It is produced by the **math** environment, which can be invoked with the usual `’134begin. . .’134end` construction or with the short form `$. . .$`. You can use any of the symbols and structures, from $\alpha$ to $\omega$, available in LaTeX[@Lamport:LaTeX]; this section will simply show a few examples of in-text equations in context. Notice how this equation: $\lim_{n\rightarrow \infty}x=0$, set here in in-line math style, looks slightly different when set in display style. (See next section).
### Display Equations
A numbered display equation – one set off by vertical space from the text and centered horizontally – is produced by the **equation** environment. An unnumbered display equation is produced by the **displaymath** environment.
Again, in either environment, you can use any of the symbols and structures available in LaTeX; this section will just give a couple of examples of display equations in context. First, consider the equation, shown as an inline equation above: $$\lim_{n\rightarrow \infty}x=0$$ Notice how it is formatted somewhat differently in the **displaymath** environment. Now, we’ll enter an unnumbered equation: $$\sum_{i=0}^{\infty} x + 1$$ and follow it with another numbered equation: $$\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f$$ just to demonstrate LaTeX’s able handling of numbering.
Citations
---------
Citations to articles [@bowman:reasoning; @clark:pct; @braams:babel; @herlihy:methodology], conference proceedings [@clark:pct] or books [@salas:calculus; @Lamport:LaTeX] listed in the Bibliography section of your article will occur throughout the text of your article. You should use BibTeX to automatically produce this bibliography; you simply need to insert one of several citation commands with a key of the item cited in the proper location in the `.tex` file [@Lamport:LaTeX]. The key is a short reference you invent to uniquely identify each work; in this sample document, the key is the first author’s surname and a word from the title. This identifying key is included with each item in the `.bib` file for your article.
The details of the construction of the `.bib` file are beyond the scope of this sample document, but more information can be found in the *Author’s Guide*, and exhaustive details in the *LaTeX User’s Guide*[@Lamport:LaTeX].
This article shows only the plainest form of the citation command, using `’134cite`. This is what is stipulated in the SIGS style specifications. No other citation format is endorsed.
Tables
------
Because tables cannot be split across pages, the best placement for them is typically the top of the page nearest their initial cite. To ensure this proper “floating” placement of tables, use the environment **table** to enclose the table’s contents and the table caption. The contents of the table itself must go in the **tabular** environment, to be aligned properly in rows and columns, with the desired horizontal and vertical rules. Again, detailed instructions on **tabular** material is found in the *LaTeX User’s Guide*.
Immediately following this sentence is the point at which Table 1 is included in the input file; compare the placement of the table here with the table in the printed dvi output of this document.
Non-English or Math Frequency Comments
--------------------- ------------- -------------------
Ø 1 in 1,000 For Swedish names
$\pi$ 1 in 5 Common in math
\$ 4 in 5 Used in business
$\Psi^2_1$ 1 in 40,000 Unexplained usage
: Frequency of Special Characters
To set a wider table, which takes up the whole width of the page’s live area, use the environment **table\*** to enclose the table’s contents and the table caption. As with a single-column table, this wide table will “float" to a location deemed more desirable. Immediately following this sentence is the point at which Table 2 is included in the input file; again, it is instructive to compare the placement of the table here with the table in the printed dvi output of this document.
Command A Number Comments
----------------------- ---------- --------------------
`’134alignauthor` 100 Author alignment
`’134numberofauthors` 200 Author enumeration
`’134table` 300 For tables
`’134table*` 400 For wider tables
Figures
-------
Like tables, figures cannot be split across pages; the best placement for them is typically the top or the bottom of the page nearest their initial cite. To ensure this proper “floating” placement of figures, use the environment **figure** to enclose the figure and its caption.
This sample document contains examples of **.eps** and **.ps** files to be displayable with LaTeX. More details on each of these is found in the *Author’s Guide*.
As was the case with tables, you may want a figure that spans two columns. To do this, and still to ensure proper “floating” placement of tables, use the environment **figure\*** to enclose the figure and its caption.
Note that either [**.ps**]{} or [**.eps**]{} formats are used; use the `’134epsfig` or `’134psfig` commands as appropriate for the different file types.
Theorem-like Constructs
-----------------------
Other common constructs that may occur in your article are the forms for logical constructs like theorems, axioms, corollaries and proofs. There are two forms, one produced by the command `’134newtheorem` and the other by the command `’134newdef`; perhaps the clearest and easiest way to distinguish them is to compare the two in the output of this sample document:
This uses the **theorem** environment, created by the`’134newtheorem` command:
Let $f$ be continuous on $[a,b]$. If $G$ is an antiderivative for $f$ on $[a,b]$, then $$\int^b_af(t)dt = G(b) - G(a).$$
The other uses the **definition** environment, created by the `’134newdef` command:
If $z$ is irrational, then by $e^z$ we mean the unique number which has logarithm $z$: $${\log e^z = z}$$
Two lists of constructs that use one of these forms is given in the *Author’s Guidelines*.
and don’t forget to end the environment with [figure\*]{}, not [figure]{}!
There is one other similar construct environment, which is already set up for you; i.e. you must *not* use a `’134newdef` command to create it: the **proof** environment. Here is a example of its use:
Suppose on the contrary there exists a real number $L$ such that $$\lim_{x\rightarrow\infty} \frac{f(x)}{g(x)} = L.$$ Then $$l=\lim_{x\rightarrow c} f(x)
= \lim_{x\rightarrow c}
\left[ g{x} \cdot \frac{f(x)}{g(x)} \right ]
= \lim_{x\rightarrow c} g(x) \cdot \lim_{x\rightarrow c}
\frac{f(x)}{g(x)} = 0\cdot L = 0,$$ which contradicts our assumption that $l\neq 0$.
Complete rules about using these environments and using the two different creation commands are in the *Author’s Guide*; please consult it for more detailed instructions. If you need to use another construct, not listed therein, which you want to have the same formatting as the Theorem or the Definition[@salas:calculus] shown above, use the `’134newtheorem` or the `’134newdef` command, respectively, to create it.
A [Caveat]{} for the TeX Expert {#a-caveat-for-the-texexpert .unnumbered}
-------------------------------
Because you have just been given permission to use the `’134newdef` command to create a new form, you might think you can use TeX’s `’134def` to create a new command: *Please refrain from doing this!* Remember that your LaTeX source code is primarily intended to create camera-ready copy, but may be converted to other forms – e.g. HTML. If you inadvertently omit some or all of the `’134def`s recompilation will be, to say the least, problematic.
Conclusions
===========
This paragraph will end the body of this sample document. Remember that you might still have Acknowledgments or Appendices; brief samples of these follow. There is still the Bibliography to deal with; and we will make a disclaimer about that here: with the exception of the reference to the LaTeX book, the citations in this paper are to articles which have nothing to do with the present subject and are used as examples only.
Acknowledgments
===============
This section is optional; it is a location for you to acknowledge grants, funding, editing assistance and what have you. In the present case, for example, the authors would like to thank Gerald Murray of ACM for his help in codifying this *Author’s Guide* and the **.cls** and **.tex** files that it describes.
Headings in Appendices
======================
The rules about hierarchical headings discussed above for the body of the article are different in the appendices. In the **appendix** environment, the command **section** is used to indicate the start of each Appendix, with alphabetic order designation (i.e. the first is A, the second B, etc.) and a title (if you include one). So, if you need hierarchical structure *within* an Appendix, start with **subsection** as the highest level. Here is an outline of the body of this document in Appendix-appropriate form:
Introduction
------------
The Body of the Paper
---------------------
### Type Changes and Special Characters
### Math Equations
#### Inline (In-text) Equations
#### Display Equations
### Citations
### Tables
### Figures
### Theorem-like Constructs
### A Caveat for the TeX Expert {#a-caveat-for-the-texexpert-1 .unnumbered}
Conclusions
-----------
Acknowledgments
---------------
Additional Authors
------------------
This section is inserted by LaTeX; you do not insert it. You just add the names and information in the `’134additionalauthors` command at the start of the document.
References
----------
Generated by bibtex from your .bib file. Run latex, then bibtex, then latex twice (to resolve references) to create the .bbl file. Insert that .bbl file into the .tex source file and comment out the command `’134thebibliography`.
More Help for the Hardy
=======================
The acm\_proc\_article-sp document class file itself is chock-full of succinct and helpful comments. If you consider yourself a moderately experienced to expert user of LaTeX, you may find reading it useful but please remember not to change it.
[^1]: Two of these, the [`’134 numberofauthors`]{} and [`’134 alignauthor`]{} commands, you have already used; another, [`’134 balancecolumns`]{}, will be used in your very last run of LaTeX to ensure balanced column heights on the last page.
[^2]: This is the second footnote. It starts a series of three footnotes that add nothing informational, but just give an idea of how footnotes work and look. It is a wordy one, just so you see how a longish one plays out.
[^3]: A third footnote, here. Let’s make this a rather short one to see how it looks.
[^4]: A fourth, and last, footnote.
|
---
abstract: 'This paper extends the RRT\* algorithm, a recently-developed but widely-used sampling-based optimal motion planner, in order to effectively handle nonlinear kinodynamic constraints. Nonlinearity in kinodynamic differential constraints often leads to difficulties in choosing appropriate distance metric and in computing optimized trajectory segments in tree construction. To tackle these two difficulties, this work adopts the affine quadratic regulator-based pseudo metric as the distance measure and utilizes iterative two-point boundary value problem solvers for computing the optimized segments. The proposed extension then preserves the inherent asymptotic optimality of the RRT\* framework, while efficiently handling a variety of kinodynamic constraints. Three numerical case studies validate the applicability of the proposed method. This paper extends RRT\* algorithms, a sampling-based optimal motion planner, to kinodynamic planning problems. The main focus of this paper is to address the difficulties that arise when the RRT\* is applied to systems having kinodynamic constraints, especially for non-linear cases. This paper suggests an extended RRT\* framework for kinodynamic planning of nonlinear dynamical systems by using an affine quadratic regulator-based metric, great approximation of distance metric under differential constraints, and iterative two-point boundary-value problem solvers. Three case studies numerically verify that the proposed algorithm provides an asymptotically optimal motion trajectory for nonlinear dynamical systems.'
author:
- 'Jung-Su Ha, Han-Lim Choi, and Jeong hwan Jeon[^1][^2]'
bibliography:
- 'IET\_RRT.bib'
title: 'Iterative Methods for Efficient Sampling-Based Optimal Motion Planning of Nonlinear Systems'
---
Sampling-based motion planning, nonlinear optimal control, iterative methods.
Introduction
============
Robotic motion planning designs a trajectory of robot states from a given initial state to a specified goal state through a complex configuration space. While motion planning algorithms can be categorized into two groups: combinatorial and sampling-based approach [@lavalle2011motion], the latter (such as Probabilistic Road Map (PRM) and Rapidly-exploring Random Tree (RRT)) has been successful in practice as their computational advantage over combinatorial methods allows for handling of complex planning environments. In particular, the Rapidly-exploring Random Tree Star (RRT\*) algorithm proposed in [@Karaman11J] is one of the most influential algorithms of this type, as it guarantees probabilistic completeness and asymptotic optimality at the same time. In other words, RRT\* guarantees that if the planning problem is feasible, the probability of the algorithm failing to find a solution reduces to zero as the number of iterations increase, and that the solution asymptotically approaches to the optimal solution; it is an “anytime" algorithm which finds a feasible trajectory quickly and refines the solution for allowed computation time. In addition, RRT\* inherits the key advantage of RRT: it explores the unexplored search space rapidly [@lavalle1998rapidly]. Due to these advantages, the algorithm has been successfully extended to many applications such as differential game and stochastic optimal control problems [@karaman2011incremental; @huynh2012incremental; @huynh2014martingale].
Motion planning is a term used in robotics, which is the design of a trajectory from an initial state to a goal state through a complex configuration space. There are two approaches to the motion planning problems: combinatorial and sampling-based planning [@lavalle2011motion]. Sampling-based algorithms such as Probabilistic Road Map (PRM) and Rapidly-exploring Random Tree (RRT) have been widely used in practice because unlike combinatorial algorithms, it can be adapted to the highly complex planning problems due to its computational advantage. Recently, the RRT\* were proposed, which has properties of probabilistic completeness and asymptotic optimality[@Karaman11J]; that is, if the planning problem has a feasible solution, the probability that the algorithm fails to find a solution goes to zero as the number of iterations increases, and also a solution approaches the *optimal* solution. RRT\* inherits the key advantage of RRT: it explores the unexplored search space rapidly [@lavalle1998rapidly]. In addition, RRT\* is an “anytime" algorithm which finds a feasible trajectory quickly and refines the solution for allowed computation time. Due to its advantages, the algorithm has been successfully extended to many applications such as differential game and stochastic optimal control problems[@karaman2011incremental; @huynh2012incremental; @huynh2014martingale].
The motion planning problem is called an *optimal kinodynamic* motion planning when the objective of planning is to minimize a given cost function defined by the state and control trajectory under system dynamics constraints. [[ Applications of such problem include the automatic car-parking [@kim2010practical] and the trajectory planning of underwater vehicles [@yuan2009optimal] or gantry cranes[@blajer2007motion] in cluttered environment. The problem is challenging because the resulting trajectory should not only satisfies system dynamics but also lies over highly non-convex state space because of the obstacle field. ]{} ]{} Due to the aforementioned properties, RRT\* can provide a good framework for an optimal kinodynamic motion planner, supposed that the following two issues are appropriately addressed. First, the distance metric should be able to take into account kinodynamic constraints of the problem. RRT-based algorithms take advantage of Voronoi bias for rapid exploration of the state space; with a wrong distance metric, the configuration space may not be effectively explored. Second, there should be a way to construct a optimal trajectory segment under kinodynamic constraints for a given cost form, because the RRT\* algorithm improves the quality of solution by refining the segments of the trajectory so that the solution asymptotically converges to the optimal solution.
There have been many attempts to handle the kinodynamic planning problem in the framework of RRT\* by tackling the aforementioned two issues in some ways. The minimum-time/length planning for holonomic and non-holonomic vehicles[@Karaman10; @Karaman11; @karaman2013sampling] have first been addressed in the RRT\* framework; a method tailored to high-speed off-road vehicles taking tight turns [@Jeon11] was also proposed. Several recent researches have been devoted to deal with kinodynamic constraints in the form of linear differential constraint [@Dustin13; @perez2012lqr; @goretkinoptimal]; these work in particular proposed to adopt the optimal control theory for linear systems for cost functions of some linear-[@perez2012lqr; @goretkinoptimal] or affine-quadratic regulator[@Dustin13] (LQR or AQR) type. Despite these recent progresses, the question of a systematic and efficient method to handle generic nonlinear dynamics, which inevitably involves computation of two-point boundary value problems (TPBVPs) solutions in the RRT\* process remains unsettled.
RRT-based algorithms explore the state-space rapidly using the Voronoi bias; the frontier nodes have a broad Voronoi region so a new state is likely extended from the frontier nodes into unexplored region. If the algorithm uses inaccurate distance metric, the algorithm is not able to rapidly extend the tree into whole configuration space. Another issue is to construct the optimal trajectory segment under kinodynamic constraints and given cost.
This work focuses on presenting methodology that can effectively handle nonlinear dynamics in the framework RRT\*. The methodology finds out an optimal trajectory for an affine-quadratic cost functional under nonlinear differential constraints while allowing for rapid exploration of the state space. The AQR-based pseudo metric is proposed as an approximation to the optimum distance under nonlinear differential constraints, and two iterative methods are presented to solve associated TPBVPs efficiently. The proposed extension of RRT\* preserves asymptotic optimality of the original RRT\*, while taking into account a variety of kinodynamic constraints. Three numerical case studies are presented to demonstrate the applicability of the proposed methodology.
While one of the two iterative methods in this paper was first introduced in the authors’ earlier work [@ha2013sa], this article includes more extended description of the methodology, in particular proposing one more iterative algorithm, as well as more diverse/extensive numerical case studies.
Problem Definition
==================
A kinodynamic motion planning problem is defined for a dynamical system $$\dot{x}(t)=f(x(t),u(t)), \label{eq:dyn_nonl}$$ where $x$ denotes the state of the system defined over the state space $\chi \subset \mathbb{R}^n$ and $u$ denotes the control input defined over the control input space $U \subset \mathbb{R}^m$. Let $\chi_{obs} \subset \chi$ and $\chi_{goal} \subset \chi$ be the obstacle region and the goal region where the system tries to avoid and to reach, respectively. Then, the feasible state and input spaces are given by $\chi_{free} \subset \chi\setminus\chi_{obs}$ and $U_{free} \subset U$, respectively. The trajectory is represented as $\pi = (x(\cdot),u(\cdot), \tau)$, where $\tau$ is the arrival time at the goal region, $u:[0,\tau]\rightarrow U$, $x:[0,\tau]\rightarrow\chi$ are the control input and the corresponding state along the trajectory. The trajectory, $\pi_{free}$, for given initial state $x_{init}$ is called feasible if it does not cross the obstacle region and eventually achieves the goal region while satisfying the system dynamics (\[eq:dyn\_nonl\]), i.e., $\pi_{free} = (x(\cdot),u(\cdot), \tau)$, where $u:[0,\tau]\rightarrow U_{free}$, $x:[0,\tau]\rightarrow\chi_{free}$, $x(0) = x_{init}$ and $x(\tau) \in \chi_{goal}$.
In order to evaluate a given trajectory $\pi$, the below form of the cost functional is considered in this paper: $$c(\pi) =\int^{\tau}_{0} \left[1+\frac{1}{2}u(t)^TRu(t) \right]dt. \label{eq:cost}$$ The above cost functional denotes trade-off between arrival time of a trajectory and the expanded control effort. $R$ is user-define value; the cost function penalizes more for the trajectory spending large control effort than late arrival time as $R$ is larger. This type of cost functional is widely used for the kinodynamic planning problems [@Glassman10; @Dustin13].
Then finally, the problem is defined as follows.\
**Problem 1 (Optimal kinodynamic motion planning):** *Given $\chi_{free},~x_{init}$ and $\chi_{goal}$, find a minimum cost trajectory $\pi^* = (x^*(\cdot),u^*(\cdot), \tau^*)$ such that $\pi^* = {\operatornamewithlimits{argmin}}_{\pi\in\Pi_{free}} c(\pi)$, where $\Pi_{free}$ denotes a set of feasible paths.*
Background {#sec:background}
==========
The RRT\* Algorithm
-------------------
This section summarizes the RRT\* algorithm [@Karaman11J] upon which this paper builds an extension to deal with kinodynamic constraints. The RRT\* is a sampling-based algorithm that incrementally builds and produces an optimal trajectory from a specified initial state $x_{init}$ to a specified goal region $X_{goal}$. The overall structure of the algorithm is summarized in Algorithm \[RRT\*\]. It should be pointed out that Algorithm \[RRT\*\] uses more generic notations/terminologies than the original algorithm to better describe the proposed kinodynamic extension within the presented algorithm structure.
$(V,E) \gets (\{x_{init}\},\emptyset);$ $x_{rand}\gets \textsc{Sampling}(\chi_{free});$ $x_{nearest}\gets \textsc{Nearest}(V, x_{rand});$ $x_{new}\gets \textsc{Steer}(x_{nearest}, x_{rand});$ $\pi_{new}\gets\textsc{TPBVPsolver}(x_{nearest},x_{new}); $ $X_{near\_b}\gets \textsc{NearBackward}(V,x_{new});$ $X_{near\_f}\gets \textsc{NearForward}(V,x_{new});$ $c_{min}\gets \textsc{Cost}(x_{nearest})+c(\pi_{new});$ $\pi_{new}'\gets \textsc{TPBVPsolver}(x_{near},x_{new});$ $c'\gets \textsc{Cost}(x_{near})+c(\pi_{new}');$ $c_{min}\gets c';~\pi_{new}\gets \pi_{new}';$ $V\gets V \cup x_{new};~E\gets E\cup\pi_{new};$ $\pi_{near}'\gets \textsc{TPBVPsolver}(x_{new},x_{near});$ $c'\gets \textsc{Cost}(x_{new})+c(\pi_{near}');$ $E\gets (E \setminus \pi_{near}) \cup \pi_{near}';$ replace existing edge $T\gets (V,E);$
$(V,E) \gets (\{x_{init}\},\emptyset);$ $x_{rand}\gets \textsc{Sampling}(\chi_{free});$ $x_{nearest}\gets \textsc{Nearest}(V, x_{rand});$ $x_{new}\gets \textsc{Steer}(x_{nearest}, x_{rand});$ $\pi_{nearest\rightarrow new}\gets\textsc{TPBVPsolver}(x_{nearest},x_{new}); $ $X_{near\_b}\gets \textsc{NearBackward}(V,x_{new});~X_{near\_f}\gets \textsc{NearForward}(V,x_{new});$ $c_{min}\gets \textsc{Cost}(x_{nearest})+c(\pi_{nearest\rightarrow new});~\pi_{par\rightarrow new}\gets \pi_{nearest\rightarrow new};$ $\pi_{near\rightarrow new}\gets \textsc{TPBVPsolver}(x_{near},x_{new});$ $c_{min}\gets \textsc{Cost}(x_{near})+c(\pi_{near\rightarrow new});~\pi_{par\rightarrow new}\gets \pi_{near\rightarrow new};$ $V\gets V \cup x_{new};~E\gets E\cup\pi_{par\rightarrow new};$ $\pi_{new\rightarrow near}\gets \textsc{TPBVPsolver}(x_{new},x_{near});$ $par \gets \textsc{Parent}(x_{near});~E\gets (E \setminus \pi_{par\rightarrow near}) \cup \pi_{new\rightarrow near};$ $T\gets (V,E);$
$(V,E) \gets (\{x_{init}\},\emptyset);$ $x_{rand}\gets \textsc{Sampling}(\chi_{free});$ $x_{nearest}\gets \textsc{Nearest}(V, x_{rand});$ $x_{new}\gets \textsc{Steer}(x_{nearest}, x_{rand});$ $\pi_{nearest\rightarrow new}\gets\textsc{TPBVPsolver}(x_{nearest},x_{new}); $ $X_{near\_b}\gets \textsc{NearBackward}(V,x_{new});$ $X_{near\_f}\gets \textsc{NearForward}(V,x_{new});$ $V\gets V \cup x_{new};~E\gets E\cup\pi_{nearest\rightarrow new};$ $c_{min}\gets \textsc{Cost}(x_{nearest})+c(\pi_{nearest\rightarrow new}); $ $E\gets \textsc{ChooseParent}(X_{near\_b},x_{new},c_{min});$ $E\gets \textsc{Rewire}(X_{near\_f},x_{new});$ $T\gets (V,E);$
$\pi_{par\rightarrow new}\gets \pi_{nearest\rightarrow new};$ $\pi_{near\rightarrow new}\gets \textsc{TPBVPsolver}(x_{near},x_{new});$ $c' \gets \textsc{Cost}(x_{near})+c(\pi_{near\rightarrow new});$ $c_{min}\gets c';$ $\pi_{par\rightarrow new}\gets \pi_{near\rightarrow new};$ $E\gets (E \setminus \pi_{nearest\rightarrow new}) \cup \pi_{par\rightarrow new};$
$c_{new}\gets \textsc{Cost}(x_{new});$ $\pi_{new\rightarrow near}\gets \textsc{TPBVPsolver}(x_{new},x_{near});$ $par \gets \textsc{Parent}(x_{near});$ $E\gets (E \setminus \pi_{par\rightarrow near}) \cup \pi_{new\rightarrow near};$ $E$
At each iteration, the algorithm randomly samples a state $x_{rand}$ from $\chi_{free}$; finds the *nearest* node $x_{nearest}$ in the tree to this sampled state (line 3-4). Then, the algorithm steers the system toward $x_{rand}$ to determine $x_{new}$ that is closest to $x_{rand}$ and stays within some specified distance from $x_{nearest}$; then, add $x_{new}$ to the set of vertices $V$ if the trajectory from $x_{nearest}$ to $x_{new}$ is obstacle-free (lines 5 – 7 and 18). Next, the best parent node for $x_{new}$ is chosen from near nodes in the tree so that the trajectory from the parent to $x_{new}$ is obstacle-free and of minimum cost (lines 8, 10 – 17). After adding the trajectory segment from the parent to $x_{new}$ (line 18), the algorithm rewires the near nodes in the tree so that the forward paths from $x_{new}$ are of minimum cost (lines 9, 19 – 25); Then, the algorithm proceeds to the next iteration.
To provide more detailed descriptions of the key functions of the algorithm:
- <span style="font-variant:small-caps;">Sampling</span>($\chi_{free})$: randomly samples a state from $\chi_{free}$.
- <span style="font-variant:small-caps;">Nearest</span>($V,x$): finds the nearest node from $x$ among nodes in the tree $V$ under a given distance metric *dist$(x_1,x_2)$* representing distance from $x_1$ to $x_2$.
- <span style="font-variant:small-caps;">Steer</span>($x,y$): returns a new state $z\in\chi$ such that $z$ is closest to $y$ among candidates: $$\textsc{Steer}(x,y) := {\operatornamewithlimits{argmin}}_{z\in B^+_{x,\eta}}\textit{dist}(z, y), \nonumber$$ where $B^+_{x,\eta}\equiv\{z\in\chi|\textit{dist(x, z)}\leq\eta\}$ with $\eta$ representing the maximum length of an one-step trajectory forward.
- <span style="font-variant:small-caps;">TPBVPsolver($x_0,x_1$):</span> returns the optimal trajectory from $x_0$ to $x_1$, *without* considering the obstacles.
- <span style="font-variant:small-caps;">ObstacleFree($\pi$)</span>: returns indication of whether or not the trajectory $\pi$ overlaps with the obstacle region $\chi_{obs}$.
- <span style="font-variant:small-caps;">NearBackward($V,x$)</span> and <span style="font-variant:small-caps;">NearForward($V,x$)</span>: return the set of nodes in $V$ that are within the distance of $r_{|V|}$ from/to $x$, respectively. In other words, $$\textsc{NearBackward}(V, x) := \{v\in V|v\in B^-_{x,r_{|V|}}\}, \nonumber$$ $$\textsc{NearForward}(V, x) := \{v\in V|v\in B^+_{x,r_{|V|}}\}, \nonumber$$ where $B^-_{x,r_{|V|}}\equiv\{z\in\chi|\textit{dist}(z, x)\leq r_{|V|}\}$, $B^+_{x,r_{|V|}}\equiv\{z\in\chi|\textit{dist}(x, z)\leq r_{|V|}\}$, and $|V|$ denotes the number of nodes in the tree.
Also, $r_{|V|}$ needs to be chosen such that a ball of volume $\gamma\frac{\log |V|}{|V|}$ is contained by $B^-_{x,r_{|V|}}$ and $B^+_{x,r_{|V|}}$ with large enough $\gamma$. For example, with the Euclidean distance metric this can be defined as $r_{|V|}=\min\{(\frac{\gamma}{\zeta_d}\frac{\log n}{n})^{1/d},\eta\}$ with the constant $\gamma$ [@Karaman11J]; $\zeta_d$ is the volume of the unit ball in $\mathbb{R}^d$.
- $\textsc{Cost}(x)$: returns the cost-to-come for node $x$ from the initial state.
- $\textsc{Parent}(x)$: returns a pointer to the parent node of $x$.
- $c(\pi)$: returns the cost of trajectory $\pi$ defined by (\[eq:cost\]).
Note specifically that when the cost of a trajectory is given by the path length and kinodynamic constraint is involved (as was in the first version presented in [@Karaman11J]), the Euclidean distance can be used as the distance metric $dist(x_1,x_2)$. Thus, $\textsc{NearBackward}(V,x)$ becomes identical to $\textit{NearForward}(V,x)$ because $dist(x,z) = dist(z,x)$, and $\textsc{TPBVPsolver}(x_0,x_1)$ simply returns a straight line from $x_0$ to $x_1$.
The probabilistic completeness and asymptotic optimality of RRT\* algorithm are proven in [@Karaman11J]: if the planning problem has at least one feasible solution, the probability that the algorithm cannot finds the solution goes to zero as the number of iterations increases, and at the same time, the solution is being refined to the optimal one asymptotically. It should be noted that the RRT\* structure in Algorithm \[RRT\*\] can be adopted/extended for *kinodynamic* optimal planning, supposed that the distance metric takes into account the kinodynamic constraints and that the <span style="font-variant:small-caps;">TPBVPsolver</span> computes the optimal trajectory between two states under differential kinodynamic constraints.
Optimal Control with Affine Dynamics {#subsec:OC}
------------------------------------
This section presents a procedure to compute the optimal solution for an affine system with affine-quadratic cost functional, which will be taken advantage of for quantification of the distance metric for generic nonlinear systems later in the paper. Consider an affine system $$\dot{x}(t) = Ax(t)+Bu(t)+c,\label{eq:dyn_l}$$ and the performance index $$J(0) = \int_{0}^{\tau} \left[1+\frac{1}{2}u(t)^TRu(t) \right] dt,\label{eq:perf_ind}$$ with boundary conditions, $$x(0) = x_0, \qquad x(\tau) = x_1 \label{eq:boundary_affine}$$ and *free* final time $\tau$.
The Hamiltonian is given by, $$H = 1+\frac{1}{2}u(t)^TRu(t)+\lambda(t)^T(Ax(t)+Bu(t)+c). \nonumber$$ The minimum principle yields that the optimal control takes the form of: $$u(t)=-R^{-1}B^T\lambda(t), \nonumber$$ with a reduced Hamilitonian system [@lewis1995optimal]: $$\begin{aligned}
\dot{x}(t) &= Ax(t)-BR^{-1}B^T\lambda(t)+c, \notag\\
-\dot{\lambda}(t) &= A^T\lambda(t),\label{eq:TPBVP1} \tag{\textbf{BVP1}}\end{aligned}$$ where $x(0) = x_0,~x(\tau) = x_1$. The solution of (\[eq:TPBVP1\]) with the boundary conditions (\[eq:boundary\_affine\]) is the optimal trajectory from $x_0$ to $x_1$, supposed that the final time $\tau$ is given.
Note that for given $\tau$, the terminal values of $x(t)$ and $\lambda(t)$ in (\[eq:TPBVP1\]) can be expressed: $$x(\tau) = x_1, \lambda(\tau) = -G(\tau)^{-1}(x_1-x_h(\tau)), \label{eq:bc1}$$ where a homogeneous solution of (\[eq:dyn\_l\]), $x_h(\tau)$, and the weighted continuous reachability Gramian, $G(\tau)$, are the final values of the following initial value problem: $$\begin{aligned}
\dot{x}_h(t) &= Ax_h(t)+c,\notag\\
\dot{G}(t) &= AG(t)+G(t)A^T+BR^{-1}B^T,\notag\\
x_h(0) &= x_0, G(0) = 0.\label{eq:IVP1} \tag{\textbf{IVP1}}\end{aligned}$$ With (\[eq:bc1\]), the optimal trajectory can be obtained by integrating (\[eq:TPBVP1\]) backward.
For given final time $\tau$, the performance index of the optimal trajectory can be written as $$C(\tau) = \tau + \frac{1}{2}(x_1-x_h(\tau))^TG(\tau)^{-1}(x_1-x_h(\tau)). \label{eq:cost_tau}$$ and the optimal final time can be expressed as: $$\tau^* = {\operatornamewithlimits{argmin}}_{\tau\geq0} C(\tau). \label{eq:opt_tau}$$ Equation (\[eq:cost\_tau\]) implies that $C(\tau) \geq \tau$ since $G(\tau)$ is positive (semi-)definite. Therefore, $\tau^*$ can be computed by calculating $C(\tau)$ with increasing $\tau$ until $\tau$ equals to the incumbent best cost $\widetilde{C}(\tau) \triangleq \min_{t \in [0,\tau]} C(t)$. With this optimal final time $\tau^*$, the optimal cost can be obtained as: $$C^* = C(\tau^*). \label{eq:opt_cost}$$
Within the RRT\* framework, if the kinodynamic constraints take an affine form, the procedure in this section can be used to define/quantify the distance metric and to compute the optimized trajectory between two nodes in $\textsc{TBPVPsolver}(x_0, x_1)$.
Efficient Distance Metric and Steering Method {#sec:Dis}
=============================================
To take advantage of the explorative property of RRT\*, it is crucial to use an appropriate distance metric in the process. The Euclidean distance, which cannot consider the system dynamics, is certainly not a valid option in kinodynamic motion planning problem – for example, it would fail to find the nearest node and thus would not be able to steer toward the sample node. Thus, a distance metric which appropriately represents the degree of closeness taking into account the dynamic constraints and the underlying cost measure needs to be defined/quantified for kinodynamic version of RRT\*.
Affine Quadratic Regulator-Based Pseudo Metric
----------------------------------------------
An AQR-based pseudo metric is firstly proposed in [@Glassman10] as a distance metric to kinodynamic planning to consider a first-order linear dynamics. In this work, an AQR-based pseudo metric is adopted as an approximate distance measure for problems with nonlinear differential constraints. For kinodynamic planning with cost functional (\[eq:cost\]) and dynamic constraint (\[eq:dyn\_nonl\]), the distance from $x_0$ to $x_1$ is computed as $$\textit{dist}(x_0, x_1) = C^* \equiv \min_{\tau\geq 0}C(\tau). \label{eq:dist}$$ where $C(\tau)$ is calculated from (\[eq:cost\_tau\]) with $$A = \left.\frac{\partial f}{\partial x}\right|_{x=\hat{x}, u = \hat{u}}, ~~ B = \left.\frac{\partial f}{\partial u}\right|_{x=\hat{x}, u = \hat{u}},$$ where $\hat{x}$ and $\hat{u}$ indicate the linearization points that are set to be the initial (or the final) state, i.e., $\hat{x} = x_0$ (or $x_1$), and $\hat{u}=0$, in the framework of RRT\*. In other words, the distance from $x_0$ to $x_1$ is approximated as the cost of the optimal control problem for a linearized system with the same cost functional and boundary conditions.
In this work, an AQR-based pseudo metric is adapted as a distance metric in the RRT\* framework. An exactness of a metric is related to the property of rapid exploration of the RRT\* algorithm: for a randomly chosen sample, the algorithm finds the nearest node by <span style="font-variant:small-caps;">Nearest</span> procedure and expands the tree toward the sample in <span style="font-variant:small-caps;">Steer</span> procedure. Calculation of the exact metric between two states are equivalent to solving nonlinear optimal control problem which is computationally expensive; the algorithm needs to solve such optimal control problems for every pair of states, from the sample state to the nodes in the tree and vice versa. Although the AQR-based pseudo metric does not take into account the nonlinear dynamics, as will be mentioned in next subsection, it can measure the metric for all pairs of states by integrating only $n$ (for $x_h(t)$) + $n(n+1)/2$ (for $G(t)$ which is symmetric) first order ODEs and produces the much more exact degree of closeness than Euclidean distance.
Implementation of AQR Metric in RRT\* Framework
-----------------------------------------------
Based on the AQR metric in (\[eq:dist\]), <span style="font-variant:small-caps;">Nearest</span>($V,x$), <span style="font-variant:small-caps;">NearBackward</span>($V,x$), <span style="font-variant:small-caps;">NearForward</span>($V,x$) and <span style="font-variant:small-caps;">Steer</span>($x,y$) functions can be readily implemented in the RRT\* structure.
- $x_{nearest}\gets \textsc{Nearest}(V, x_{rand})$: First, the system dynamics is linearized at $x_{rand}$. Since $x_{rand}$ is final state, $x_h(t)$ and $G(t)$ are integrated from $t=0, x_h(0) = x_{rand}, G(0) = 0$ in the backward direction, i.e., $t<0$. With integration, the cost from $i$th node $v_i \in V$ at the time $-t$ is calculated as $C_i(-t) = -t - \frac{1}{2}(v_i-x_h(t))^TG(t)^{-1}(v_i-x_h(t))$ while the minimum cost is saved as a distance, $d_i = \min_{t<0}C_i(-t)$. The integration stops when $\min_{i}d_i\geq-t$; since $G(t)$ is a negative (semi-)definite matrix, cost less than $-t$ cannot be found by more integration. Finally, the nearest node, $x_{nearest}\gets v_k$, where $k = {\operatornamewithlimits{argmin}}_{i}d_i$, is returned.
- $X_{near\_b}\gets \textsc{NearBackward}(V,x_{new})$: The procedure is similar to <span style="font-variant:small-caps;">Nearest</span>. First, the system dynamics is linearized at $x_{new}$. $x_h(t)$ and $G(t)$ are integrated from $t=0, x_h(0) = x_{new}, G(0) = 0$ for backward direction, i.e., $t<0$. With integration, the cost for the time $-t$ and for each $v_i \in V$, the corresponding cost is calculated while the minimum cost is saved as a distance, $d_i = \min_{t<0}C_i(-t)$. The integration stops when $-t\geq r_{|V|}$ and the set of backward near nodes, $X_{near\_b} \gets \{v_i\in V|d_i \leq r_{|V|}\}$, are returned.
- $X_{near\_f}\gets \textsc{NearForward}(V,x_{new})$: The procedure is exactly same as <span style="font-variant:small-caps;">NearBackward</span> except for the direction of integration, $t>0$.
- $x_{new}\gets \textsc{Steer}(x_{nearest}, x_{rand})$: First, the system dynamics is linearized at $x_{nearest}$. Then, the optimal trajectory is calculated by procedure in section \[subsec:OC\] for the linearized system. If the resulting cost is less than $\eta$, $x_{rand}$ is returned as a new node. Otherwise, it returns $x'$ such that $x'$ is in the trajectory and the cost to $x'$ from $x_{nearest}$ is $\eta$.
Efficient Solver for the Two-Point Boundary-Value Problem {#sec:TPBVP}
=========================================================
As mentioned previously, the $\textsc{TPBVPsolver}(x_0,x_1)$ function returns the optimal trajectory from $x_0$ to $x_1$. A straight-line trajectory, which is optimal for the problem without dynamic constraints, cannot be a valid solution in general, because it would be not only suboptimal but also likely to violate the kinodynamic constraints. Therefore, this section derives the two-point boundary value problem (TPBVP) involving nonlinear differential constraints and presents methods to compute the solution of this TPBVP. Consider the optimal control problem (OCP) with nonlinear system dynamics in (\[eq:dyn\_nonl\]) to minimize the cost functional in (\[eq:perf\_ind\]) where boundary conditions are given as $ x(0) = x_0,~x(\tau) = x_1$ with free final time $\tau$. The Hamiltonian of this OCP is defined as, $$H(x(t),u(t),\lambda(t)) = 1+\frac{1}{2}u(t)^TRu(t)+\lambda(t)^Tf(x(t),u(t)). \nonumber$$ From the minimum principle: $$H_u^T = Ru(t)+f_u^T\lambda(t) = 0,\nonumber$$ which allows the optimal control to be expressed in terms of $x(t)$ and $\lambda(t)$: $$u(t) = h(x(t),\lambda(t)) \label{eq:con}.$$ Thus, a system of differential equations for the state $x(t)$ and the costate $\lambda(t)$ is obtained: $$\begin{aligned}
\dot{x}(t) &= f(x(t), h(x(t),\lambda(t))), \notag\\
-\dot{\lambda}(t) &= H_x^T = f_x^T\lambda(t)
\label{eq:TPBVP2} \tag{\textbf{BVP2}}\end{aligned}$$ with boundary conditions $ x(0) = x_0,~x(\tau) = x_1$. The system of differential equations in (\[eq:TPBVP2\]) is nonlinear in general and has boundary conditions at the initial and final time; thus, it is called a nonlinear two-point boundary value problem (TPBVP). An analytic solution to a nonlinear TPBVP is generally unavailable because not only it is nonlinear but also its the boundary conditions are split at two time instances; numerical solution schemes have often been adopted for this.
In the present work, two types of numerical *iterative* approaches are presented: the successive approximation (SA) and the variation of extremals (VE) based methods. These two approaches find the solution of a nonlinear TPBVP by successively solving a sequence of more tractable problems: in the SA based method, a sequence of linear TPBVPs are solved and in the VE approach, a sequence of nonlinear initial value problems (IVPs) are iteratively solved, respectively. The main concept of the methods has already been presented to solve a *free final-state & fixed final-time* problem [@Tang05; @kirk2012optimal] but this work proposes their variants that can handle a *fixed final-state & free final-time* problem in order to implemented to the RRT\* framework.
In an iterative method, an initial guess of the solution is necessary and often substantially affects the convergence of the solution. In the proposed RRT\* extension, the optimal trajectory with the linearized dynamics can be a good choice of initial condition, particularly because such trajectory is already available in the RRT\* process by calculating the AQR-based distance metric in <span style="font-variant:small-caps;">NearBackward</span> or <span style="font-variant:small-caps;">NearForward</span>.
Successive Approximation {#sec:SA}
------------------------
Let $\hat{x}$ be a linearization point, which is set to be the initial or the final value of a trajectory segment in the RRT\* implementation; $\hat{u}=0$ be also the corresponding linearization point. By splitting the dynamic equation in (\[eq:dyn\_nonl\]) into the linearized and remaining parts around $\hat{x}$ and $\hat{u}$, $$\dot{x}(t) = Ax(t)+Bu(t)+g(x(t),u(t)),$$ where $A \triangleq \left.\frac{\partial f}{\partial x}\right|_{x=\hat{x}, u = \hat{u}}$, $B \triangleq \left. \frac{\partial f}{\partial u}\right|_{x=\hat{x}, u = \hat{u}}$ and $g(x(t),u(t))\triangleq f(x(t),u(t))-Ax(t)-Bu(t)$, the optimal control and the reduced Hamiltonian system are expressed as, $$u(t) = -R^{-1}B^T\lambda(t)-R^{-1}g_u^T\lambda(t),$$ and $$\begin{aligned}
\dot{x}(t) &= Ax(t)-BR^{-1}B^T\lambda(t)-BR^{-1}g_u^T\lambda(t) \nonumber\\
&~~~ +g(x(t),u(\lambda (t))),\nonumber\\
-\dot{\lambda}(t) &= A^T\lambda(t)+g_x^T\lambda(t), \label{eq:tpbvp_sa}\end{aligned}$$ respectively. The boundary conditions are given by $$x(0) = x_0, x(\tau) = x_1$$ and the final time $\tau$ is *free*. The TPBVP in (\[eq:tpbvp\_sa\]) is still nonlinear, as $g_u^T\lambda$, $g$ and $g_x^T\lambda$ in (\[eq:tpbvp\_sa\]) are *not* necessarily linear w.r.t $x$ or $\lambda$.
Consider the following sequence of TPBVPs: $$\begin{aligned}
\dot{x}^{(k)}(t) &= Ax^{(k)}(t)-BR^{-1}B^T\lambda^{(k)}(t) \nonumber\\
& ~~~-BR^{-1}g_u^{(k-1)T}\lambda^{(k-1)}(t)+g^{(k-1)}, \label{eq:x_seq}\\
-\dot{\lambda}^{(k)}(t) &= A^T\lambda^{(k)}(t)+g_x^{(k-1)T}\lambda^{(k-1)}(t), \label{eq:lamb_seq}\end{aligned}$$ with boundary conditions $x^{(k)}(0)=x_0,~x^{(k)}(\tau)=x_1$, where $g^{(k)}\equiv g(x^{(k)},u^{(k)}),~g_x^{(k)}\equiv g_x(x^{(k)},u^{(k)})$ and $g_u^{(k)} \equiv g_u(x^{(k)},u^{(k)})$. Note that this system of differential equations are linear w.r.t. $x^{(k)}(t)$ and $\lambda^{(k)}(t)$ for given $x^{(k-1)}(t)$ and $\lambda^{(k-1)}(t)$. It can be converted into an initial value problem as follows. From (\[eq:lamb\_seq\]), $\lambda^{(k)}(t)$ can be expressed as: $$\lambda^{(k)}(t) = e^{A^T(\tau-t)}\lambda^{(k)}(\tau)+\lambda_p^{(k)}(t), \label{eq:alg1}$$ where $\lambda_p^{(k)}(t)$ is a solution of $$\dot{\lambda}_p^{(k)}(t) = -A^T\lambda_p^{(k)}(t)-g_x^{(k-1)T}\lambda^{(k-1)}(t), \label{eq:lp}$$ with terminal condition $\lambda_p^{(k)}(\tau) = 0$. Plugging (\[eq:alg1\]) into (\[eq:x\_seq\]) yields $$\begin{aligned}
\dot{x}^{(k)}(t) &= Ax^{(k)}(t)+g^{(k-1)}-BR^{-1}B^Te^{A^T(T-t)}\lambda(\tau)\nonumber\\
&~~-BR^{-1}B^T\lambda_p^{(k)}(t)-BR^{-1}g_u^{(k-1)T}\lambda^{(k-1)}(t).\end{aligned}$$ Then, the final state $x^{(k)}(\tau)$ is expressed as, $$x^{(k)}(\tau) = x^{(k)}_h(\tau)-G(\tau)\lambda^{(k)}(\tau), \label{eq:xf_seq}$$ where $x_h(\tau)$ and $G(\tau)$ are the solution of $$\begin{aligned}
\dot{x_h}^{(k)}(t) &= Ax^{(k)}_h(t)-BR^{-1}B^T\lambda_p^{(k)}(t)\nonumber\\
&~~~-BR^{-1}g_u^{(k-1)T}\lambda^{(k-1)}(t)+g^{(k-1)} \label{eq:sa1}\\
\dot{G}(t) &= AG(t)+G(t)A^T+BR^{-1}B^T,\label{eq:sa2}\\
x^{(k)}_h(0) &= x_0, G(0) = 0. \nonumber\end{aligned}$$ Using (\[eq:xf\_seq\]) and $x^{(k)}(\tau) = x_1$, the final costate value can be obtained: $$\lambda^{(k)}(\tau)=-G(\tau)^{-1}(x_1-x^{(k)}_h(\tau)).$$ Finally, $x^{(k)}(t)$ and $\lambda^{(k)}(t)$ for $t\in[0,\tau]$ are calculated by backward integration of the following differential equation, $$\begin{aligned}
\begin{bmatrix}\dot{x}^{(k)}(t)\\ \dot{\lambda^{(k)}}(t) \end{bmatrix} =& \begin{bmatrix}A & -BR^{-1}B^T\\ 0 & -A^T \end{bmatrix}\begin{bmatrix}x^{(k)}(t)\\ \lambda^{(k)}(t)\end{bmatrix} \label{eq:alg2}\\
&+\begin{bmatrix}-BR^{-1}g_u^{(k-1)T}\lambda^{(k-1)}(t)+g^{(k-1)}\\ -g_x^{(k-1)T}\lambda^{(k-1)}(t)\end{bmatrix}, \nonumber\end{aligned}$$ with given boundary values $x^{(k)}(\tau)$ and $\lambda^{(k)}(\tau) $. Also, the optimal control $u^{(k)}(t)$ can also be computed accordingly: $$u^{(k)}(t) = -R^{-1}B^T\lambda^{(k)}(t)-R^{-1}g_u^{(k-1)T}\lambda^{(k-1)}(t). \label{eq:con_seq}$$
The optimal final time can be found numerically by a gradient descent scheme. The derivative of $J(0)$ for the trajectory at $k$’th iteration w.r.t final time is given as, $$\begin{aligned}
\left[ \frac{dJ(0)}{d\tau}\right]^{(k)} &= 1-\frac{1}{2}\lambda^{(k)}(\tau)^TBR^{-1}B^T\lambda^{(k)}(\tau)\nonumber\\
&~~+\frac{1}{2}\lambda^{(k-1)}(\tau)^Tg_u^{(k-1)}R^{-1}g_u^{(k-1)T}\lambda^{(k-1)}(\tau)\nonumber\\
&~~+\lambda^{(k)}(\tau)^T(Ax_1+g^{(k)}). \label{eq:tau_seq1}\end{aligned}$$ Then, a gradient descent-based update rule of the final time is given as, $$\tau^{(k+1)} = \tau^{(k)} - \eta \left[\frac{dJ(0)}{d\tau} \right]^{(k)}. \label{eq:tau_seq2}$$
With the process described thus far, the TPBVP solver for nonlinear system based on successive-approximation can be summarized as Algorithm \[alg:SA\].
Initialize $x^{(0)},~\lambda^{(0)},~\tau^{(0)}$ and set $k=0$ $k = k+1;$ Update $\tau^{(k)}$ Get $x^{(k)}(t)$ and $\lambda^{(k)}(t)$ Calculate $u(t)$ **return** $\pi \gets (x^{(k)}(\cdot),u(\cdot), \tau^{(k)})$
Variation of Extremals {#sec:VarofEx}
----------------------
The VE approach is a technique to successively find the initial value of the costate $\lambda(0)$ using the Newton-Rapson method [@kirk2012optimal]. The method has been well established, but its implementation into a sampling-based planning is not trivial.
Suppose that the initial value of the costate $\lambda(0)$ and the final value of the state $x(\tau)$ are related via nonlinear function: $$x(\tau)=F(\lambda(0)). \nonumber$$ Then, the initial costate $\lambda(0)$ that leads to $x(\tau) = x_1$ can be approximated successively as follows: $$\lambda^{(k+1)}(0) = \lambda^{(k)}(0) - [P_x(\lambda^{(k)}(0),\tau)]^{-1}(x^{(k)}(\tau)-x_1), \nonumber$$ where $P_x(p^{(k)}(0),t)$ is called the *state influence function matrix* which is the matrix of partial derivatives of the components of $x(t)$, evaluated at $p^{(k)}(0)$, $$P_x(\lambda^{(k)}(0),t)\equiv \begin{bmatrix}\frac{\partial x_1(t)}{\partial \lambda_1(0)} & \cdots & \frac{\partial x_1(t)}{\partial \lambda_n(0)} \\
\vdots & \ddots & \vdots \\ \frac{\partial x_n(t)}{\partial \lambda_1(0)} & \dots & \frac{\partial x_n(t)}{\partial \lambda_n(0)}\end{bmatrix}_{\lambda^{(k)}(0)}. \nonumber$$ Similarly, the *costate influence function matrix* is defined and given by, $$P_\lambda(\lambda^{(k)}(0),t)\equiv \begin{bmatrix}\frac{\partial \lambda_1(t)}{\partial \lambda_1(0)} & \cdots & \frac{\partial \lambda_1(t)}{\partial \lambda_n(0)} \\
\vdots & \ddots & \vdots \\ \frac{\partial \lambda_n(t)}{\partial \lambda_1(0)} & \dots & \frac{\partial \lambda_n(t)}{\partial \lambda_n(0)}\end{bmatrix}_{\lambda^{(k)}(0)}. \nonumber$$
The dynamics of the *influence function matrices* can be derived from the state and costate equation $\dot{x}(t) = H_\lambda^T, \dot{\lambda}(t) = -H_x^T$. Taking the partial derivatives of these equations w.r.t the initial value of the costate yields $$\begin{aligned}
\frac{\partial}{\partial\lambda(0)} \left[\dot{x}(t) \right] &= \frac{\partial}{\partial\lambda(0)} \left[H_\lambda^T \right] \nonumber\\
\frac{\partial}{\partial\lambda(0)}\left[\dot{\lambda}(t)\right] &= \frac{\partial}{\partial\lambda(0)}\left[-H_x^T\right]. \label{eq:infMat}\end{aligned}$$ With the assumption that $\frac{\partial}{\partial\lambda(0)}[\dot{x}(t)]$, $\frac{\partial}{\partial\lambda(0)}[\dot{\lambda}(t)]$ are continuous and by using the chain rule on right had side of (\[eq:infMat\]), the differential equations of the *influence function matrices* are obtained as $$\begin{bmatrix}\dot{P_x}(\lambda^{(k)}(0),t)\\ \dot{P_{\lambda}}(\lambda^{(k)}(0),t) \end{bmatrix} = \begin{bmatrix}\frac{\partial^2 H}{\partial x \partial \lambda} & \frac{\partial^2 H}{\partial \lambda^2}\\ -\frac{\partial^2 H}{\partial x^2} & -\frac{\partial^2 H}{\partial \lambda \partial x} \end{bmatrix}\begin{bmatrix}P_x(t)\\ P_{\lambda}(t)\end{bmatrix},
\label{eq:ve1}$$ where $P_x(\lambda^{(k)}(0),0) = 0, P_{\lambda}(\lambda^{(k)}(0),0) = I$. On the other side, since the final time of the problem is free, the Hamilitonian should vanish: $$H(x(\tau),\lambda(\tau),\tau) = 0. \nonumber$$ The update rule for the initial costate value and the final time can then be obtained as: $$\begin{aligned}
&\begin{bmatrix}\lambda^{(k+1)}(0)\\ \tau^{(k+1)} \end{bmatrix} = \begin{bmatrix}\lambda^{(k)}(0)\\ \tau^{(k)} \end{bmatrix} \nonumber\\
&-\begin{bmatrix}P_x(\lambda^{(k)}(0),\tau^{(k)}) & \frac{\partial x^{(k)}(\tau^{(k)})}{\partial\tau} \\ \frac{\partial H^{(k)}(\tau^{(k)})}{\partial\lambda^{(k)}(0)} & \frac{\partial H^{(k)}(\tau^{(k)})}{\partial \tau} \end{bmatrix}^{-1}\begin{bmatrix}x^{(k)}(\tau^{(k)})-x_1\\ H^{(k)}(\tau^{(k)})\end{bmatrix},
\label{eq:ve2}\end{aligned}$$ where entries of the matrix on the right hand side can be obtained from: $$\begin{aligned}
\frac{\partial H(\tau^{(k)})}{\partial\lambda_0} &= \frac{\partial H}{\partial x} \frac{dx(\tau^{(k)})}{d\lambda(0)} + \frac{\partial H}{\partial \lambda} \frac{d\lambda(\tau^{(k)})}{d\lambda(0)} \nonumber\\
&= -\dot{\lambda}(\tau^{(k)})^TP_{\lambda}(\tau^{(k)})+\dot{x}(\tau^{(k)})^TP_x(\tau^{(k)}),
\label{eq:ve3}\end{aligned}$$
$$\frac{\partial x^{(k)}(\tau^{(k)})}{\partial \tau^{(k)}} = f(x^{(k)}(\tau^{(k)}), h(x^{(k)}(\tau^{(k)}),\lambda^{(k)}(\tau^{(k)}))),
\label{eq:ve4}$$
$$\begin{aligned}
\frac{\partial H(\tau^{(k)})}{\partial\tau^{(k)}} &=& \frac{\partial H}{\partial x} \frac{dx(\tau^{(k)})}{d\tau^{(k)}} + \frac{\partial H}{\partial \lambda} \frac{d\lambda(\tau^{(k)})}{d\tau^{(k)}} \nonumber\\
&=& 0.
\label{eq:ve5}\end{aligned}$$
Finally, the TPBVP solver with the variation of extremals based method can be summarized as Algorithm \[alg:ve\].
$\textmd{Initialize}~\lambda^{(0)}(0),~\tau^{(0)}~\textmd{and set}~k=0$ $k = k+1;$ Integrate $x^{(k)},\lambda^{(k)},P_x,P_{\lambda}$ Calculate $\lambda^{(k+1)}(0),\tau^{(k+1)}$ Calculate $u(t)$ **return** $\pi \gets (x^{(k)}(\cdot),u(\cdot), \tau^{(k)})$
Discussions
-----------
Both methods presented in this section find the solution to (\[eq:TPBVP2\]) iteratively. At each iteration, the SA-based method computes the optimal solution to an linear approximation of the original problem linearized at the solution of the previous iteration, while the VE-based method updates the estimate of the initial costate and the final time. It should be pointed out that these solvers guarantee local (not global) optimality, because the underlying TPBVP is derived as a necessary (not sufficient) condition for optimality. However, this locality would not make a significant impact on the overall motion planning, since the TPBVP itself is posed for optimally linking a small segment of the overall plan.
Compared to an existing approach that treated *linearized* dynamics [@Dustin13] and thus did not require iterative process for solving a TPBVP, the proposed methods need to solve additional $k_{iter}$ first-order ODEs, where $k_{iter}$ is the number of iterations for convergence, in computing a single optimal trajectory segment in the RRT\* framework. However, the proposed schemes can take into account nonlinear kinodynamic constraints, which was not accurately realized in the previous work.
Note also that the two methods exhibit different characteristics from the computational point of view. First, the storage requirement of the VE-based method has an advantage over that of the SA-based method: at every iteration, the VE solver only needs to store 2 $(n \times n)$ matrices, $P_x(t)$ and $P_{\lambda}(t)$, while the SA solver needs to store the trajectories, $\lambda_p^{(k)}(t)$, $x^{(k)}(t)$ and $\lambda^{(k)}(t)$ for $t\in[0,\tau]$ and a $(n \times n)$ matrix, $G(t)$. On the other side, the SA solver demands integration of $4n$ first-order ODEs in (\[eq:lp\]), (\[eq:sa1\]) and (\[eq:alg2\]) at each iteration ($n^2$ first-order ODEs in (\[eq:sa2\]) needs to be integrated only at first iteration and then reused.), while the VE solver needs to integrate $2n(n+1)$ first-order ODEs in (\[eq:ve1\]) and (\[eq:TPBVP2\]) in each iteration. Thus, assuming both methods converge with similar number of iterations, the VE-based method exhibits better memory complexity, while the SA-based one gives better time complexity, in general. However, detailed convergence characteristics such as convergence time might be problem dependent.
Numerical Examples {#sec:numerics}
==================
This section demonstrates the validity of the proposed optimal kinodynamic planning methods on three numerical examples: The first pendulum swing-up example compares the proposed scheme with a state-of-the-art method based on linearized dynamics. The second example on a two-wheeled mobile robot in a cluttered environment demonstrates the asymptotic optimality of the algorithms. The third example on a SCARA type robot arm investigates the characteristics of the optimized plans with varying cost functionals.
Pendulum Swing-Up {#sec:pendulum}
-----------------
The control objective of the pendulum swing-up problem is to put the pendulum in an upright position from its downward stable equilibrium. The dynamics of the system is given as $$I\ddot{\theta} + b\dot{\theta} + mgl_c\sin\theta = u.$$ The angular position and velocity, $x(t) = [\theta(t)~\dot{\theta}(t)]^T$ are the state variables and the torque, $u(t)$ is the control input to the system. The cost of the trajectory is given by (\[eq:cost\]). The initial and goal states are given as $x_{init}=[0~0]^T, x_{goal}=\{[\pi~0]^T,~[-\pi~0]^T\}$. To comparatively investigate the capability of RRT\* variants in handling nonlinearity in dynamics, the proposed iterative methods are compared with kinodynamic-RRT\* [@Dustin13] that takes into account the first-order Taylor approximation of the dynamics.
Fig. \[result2\] illustrates state trajectories in the phase plane; an open-loop and a closed-loop trajectories are depicted as well as the planned trajectory. The open-loop trajectory is generated by using the planned control input trajectory $u(t)$ as a feedfoward control term, while the closed-loop trajectory is produced by adding an LQR trajectory stabilizer [@tedrake2009underactuated] that utilizes feedback control in the form of $u(t) - K \bar{x}$ with LQR gain $K$ where $\bar{x}$ being the deviation from the planned trajectory. It was found that both of the proposed iterative solvers produce very similar results; thus only the result of the SA-based solver is presented herein.
The blue-solid line represents the planned trajectories from the algorithm using the SA-based solver (left) and the solver for linearized dynamics (right); the red-dashed-dot and black-dashed lines denote the open-loop and closed-loop state trajectories, respectively. Note that as the linearization is only valid around the nodes of the RRT\* tree, the optimized trajectory segment with using the linearization-based scheme [@Dustin13] is dynamically infeasible for the original (nonlinear) system as the state moves far away from the nodes. As shown in Fig.\[result2\] (b) and (d), the system may not follow the planned trajectory with open-loop control and even with a feedback trajectory stabilizer; while, as shown in Fig.\[result2\] (a) and (c), the planned trajectories from the algorithm using the iterative TPBVP solver is followed not only by the feedback control but also by the open-loop control.
The control input histories are shown in Fig. \[result4\]. The blue-solid line represents the input from the planning algorithm and the black-dashed line denotes the input with feedback control implementation. In case the planned trajectory is inconsistent with the real dynamics, a large amount of control effort is required in the feedback controller, which results in cost increase. This can be observed in the case where the linearization-based planner is used; note that the control effort for the proposed method is significantly smaller than that of the linearization-based method. Also, Table \[result\_tab1\] represents average costs of ten simulations for varying cost functions with feedback controller; the ‘planned cost’ means the cost returned from the algorithm, and the ‘executed cost’ represents the cost which include control effort from the trajectory stabilization feedback controller. It is shown that the cost achieved by using proposed method is smaller and more consistent with planned cost.
[c|cc|cccccccc]{} &&\
& planned cost & executed cost & planned cost & executed cost\
R=1 & 6.4621 & 6.4727 & 4.5122 & 8.4837\
R=5 & 18.9790 & 18.9929 & 9.7229 & 31.2797\
R=10 & 33.0323 & 33.7034 & 12.9783 & 55.9197\
R=15 & 52.8430 & 52.9865 & 15.1409 & 83.6147\
Two-Wheeled Mobile Robot {#sec:TWMR}
------------------------
The second example addresses design of the trajectory of a two-wheeled mobile robot. The states and inputs of the system are $x = [p_x~p_y~\theta~v~w]^T$ and $u = [F_1~F_2]^T$, where $p_x,p_y$ and $\theta$ represent robot position and orientation, respectively, $v,~w$ denote the linear and angular velocity of the robot, and $F_1,~F_2$ are the force from each wheel. The dynamic equation of the system is given as $$\begin{bmatrix}\dot{p}_x\\ \dot{p}_y\\ \dot{\theta}\\ \dot{v}\\ \dot{w}\end{bmatrix} = \begin{bmatrix}v\cos\theta\\ v\sin\theta\\ w\\ F_1+F_2\\ F_1-F_2\end{bmatrix} = \begin{bmatrix}v\cos\theta\\ v\sin\theta\\ w\\ u_1+u_2\\ u_1-u_2\end{bmatrix}.\nonumber$$ The initial and the goal states are given as $x(0) = [0.5~0.5~\pi/4~1~0]^T$ and $x(\tau) = \{[p_x~p_y~\theta~v~w]^T|23\leq p_x\leq 24,~9\leq p_y\leq 10,~0\leq \theta\leq \pi/2,~0.8\leq v\leq 1.2,~-0.2\leq w\leq 0.2\}$, respectively, with free final time $\tau$. The cost functional is given in (\[eq:cost\]) with $R = 20\text{diag}(1,1)$.
[ccccccccccc]{}
\# of nodes & 300 & 500 & 1000 & 3000 & 5000\
\# of feasible & 81 & 100 & 100 & 100 & 100\
Mean & inf & 21.8125 & 20.5124 & 19.5235 & 19.1839\
Variance & NaN & 2.2570 & 0.7864 & 0.2261 & 0.1687\
As the two proposed iterative methods have also shown similar performances, only the result for the VE-based solver case is depicted in this example. Fig. \[fig:result1\] represents the progression of the RRT\* tree, by projecting the tree in five-dimensional space onto the two-dimensional position space, $(p_x,p_y)\in \mathbb{R}^2$. The red star and the square represent the initial position and the goal region; the thick dark-blue line represents the best trajectory found up to the corresponding progression. Observe that the resulting trajectories connect the initial and the goal state smoothly. Also, as the number of nodes in the tree increases, the tree fills up the feasible state space and finds out a lower-cost trajectory.
Table \[result\_tab2\] reports the result of a Monte-Carlo simulation with 100 trials; it shows the number of trials finding a feasible trajectories, the mean and variance of the minimum costs in each iteration. It is found that the mean and variance of the cost decreases as the number of nodes increases, implying convergence to the same solution. Finally, Fig. \[fig:result2\] depicts the resulting trajectories with VE-based solver and the same linearization-based solver as the first example [@Dustin13]. It is observed that with the solver for linearized dynamics, there is a portion of sideway-skid moving at the middle of trajectory, which is dynamically infeasible for the robot. On the contrary, the result of the proposed solver shows feasible moving in the whole trajectory.
SCARA type Robot Arm
--------------------
![SCARA Robot[]{data-label="fig:scara"}](scara){width="7cm"}
The third example is about generating a motion plan of a three-degree-of-freedom SCARA robot with two rotational joints (represented by $\theta_1$ and $\theta_2$), and one prismatic joint ($\theta_3$) as shown in Fig. \[fig:scara\]. The dynamics of the robot is given as, $$M(\theta)\ddot{\theta} + C(\theta,\dot{\theta})\dot{\theta} +N(\theta,\dot{\theta}) = u, \nonumber$$ where $$\begin{aligned}
M(\theta) = \begin{bmatrix}\alpha+\beta+2\gamma\cos\theta_2 & \beta+\gamma\cos\theta_2 & 0\\ \beta+\gamma\cos\theta_2 & \beta & 0 \\ 0 & 0 & m_3 \end{bmatrix}, \nonumber\\
C(\theta,\dot{\theta}) = \begin{bmatrix}-\gamma\sin\theta_2\dot{\theta}_2 & -\gamma\sin\theta_2(\dot{\theta}_1+\dot{\theta}_2) & 0\\ \gamma\sin\theta_2\dot{\theta}_1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \nonumber\\
N(\theta,\dot{\theta}) = \begin{bmatrix} 0 \\ 0 \\ m_3g \end{bmatrix}\nonumber\end{aligned}$$ and $$\begin{aligned}
\alpha = I_{z1}+r_1^2m_1+l_1^2m_2+l_1^2m_3, \nonumber\\
\beta = I_{z2}+I_{z3}+l_2^2m_3+m_2r_2^2, \nonumber\\
\gamma = l_1l_2m_3+l_1m_2r_2.\nonumber\end{aligned}$$ $m_1,~m_2,~m_3$ and $l_1$, $l_2$ represent mass and length of each link, respectively. Also, $r_1$, $r_2$ denote length between a joint axis and the center of mass of each link and $I_{z1},~I_{z2},~I_{z3}$ represent the moment of inertia about the rotation axis. The system has a six-dimensional state vector, $x=[\theta_1~\theta_2~\theta_3~\dot{\theta}_1~\dot{\theta}_2~\dot{\theta}_3]^T$, and a three-dimensional control inputs, $u = [\tau_1~\tau_2~f_3]^T$ representing the torques of the two rotational joints and the force of the prismatic joint.
The problem is to find the optimal motion trajectory that leads to the end-effector of the robot reaching the goal region (shown as red circle in Figs. \[fig:result3\]-\[fig:result5\]) from its initial state, $x(0) = [\pi/2~0~3~0~0~0]^T$ (the corresponding end-effector position is shown as red star in Figs \[fig:result3\]–\[fig:result4\]), while avoiding collision with obstacles. The cost of the trajectory takes the form of (\[eq:cost\]). With other parameters being fixed, the solutions are obtained with varying input penalty matrix, $R$, to see the effect of cost functional on the resulting trajectories. Two values of $R$ are considered: $$R = 0.05\text{diag}(1, 1, 0.5),~0.05\text{diag}(0.5, 0.5, 10).$$ The proposed VE-based method is implemented to solve nonlinear TPBVPs in the process of RRT\*. There are two homotopy classes for the solutions from the initial state to the final states: the end effector 1) goes over the wall or 2) makes a detour around the wall.
Figs. \[fig:result3\]–\[fig:result4\] show the resulting motions of the robot when 5000 nodes are added into the tree for the two cases; the magenta dashed line and the red circle represent the end-effector trajectory and the goal region, respectively. For the first case ($R = 0.05\text{diag}(1, 1, 0.5)$), which has smaller control penalty on joint 3, the resulting trajectory goes over the wall as shown in Fig. \[fig:result3\]; going over the wall by using cheap control input of joint 3 leads to lower cost in this case. On the other hand, in the second case ($R = 0.05\text{diag}(0.5, 0.5, 10)$) shown in Fig. \[fig:result4\], the proposed algorithm generates the trajectory that makes a detour as the control penalty for the joint 3 is larger; going over the wall is so expensive in this case that the detouring trajectory is chosen as the best one.
Considering the control penalty for joint 1 and 2, the penalty is smaller in the second case than the first one; thus, the duration of the trajectory weighs more in the second case. Fig. \[fig:result5\] shows that the top-views of the SCARA robot motion at each time step in Figs. \[fig:result3\]–\[fig:result4\] and the corresponding control inputs; the motion starts at the position colored in light-green and ends at the position colored in dark-green. It can be seen that the trajectory reaches the goal state more quickly in the second case despite it requires the large amount of input-energy of the joint 1 and 2 (the arrival time of the minimum-cost trajectory is $2.15$ and $1.25$ second, respectively).
Concluding Remarks
==================
In this work, an extension of RRT\* to handle nonlinear kinodynamic differential constraints was proposed. In order to tackle two caveats: choosing a valid distance metric and solving two point boundary value problems to compute an optimal trajectory segment. An affine quadratic regulator (AQR)-based pseudo metric was adopted, and two iterative methods were proposed, respectively. The proposed methods have been tested on three numerical examples, highlighting their capability of generating dynamically feasible trajectories in various settings.
Despite the proposed methods have been focused on implementing to the RRT\* framework, it is expected to help other state-of-the-art sampling-based motion planning algorithms, like RRT\# [@arslan2013use], GR-FMTs [@Jeon2015optimal] or FMT\* [@janson2015fast] to address nonlinear dynamical systems.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by Agency for Defense Development (under in part contract \#UD140053JD and in part contract \#UD150047JD).
[^1]: J.-S. Ha and H.-L. Choi are with the Department of Aerospace Engineering, KAIST, Daejeon, Korea. `{wjdtn1404, hanlimc}@kaist.ac.kr`
[^2]: J. Jeon is with nuTonomy Inc., Cambridge, MA 02142, USA. ` jhjeon@alum.mit.edu`
|
---
abstract: 'Many current approaches to the design of intrusion detection systems apply feature selection in a static, non-adaptive fashion. These methods often neglect the dynamic nature of network data which requires to use adaptive feature selection techniques. In this paper, we present a simple technique based on incremental learning of support vector machines in order to rank the features in real time within a streaming model for network data. Some illustrative numerical experiments with two popular benchmark datasets show that our approach allows to adapt to changing network behaviour and novel attack patterns.'
address: 'Department of Computer Science, Aalto University, Finland; firstname.lastname@aalto.fi'
title: ONLINE FEATURE RANKING FOR INTRUSION DETECTION SYSTEMS
---
network security, intrusion detection, online learning, support vector machine, stochastic gradient descent
Introduction {#sec:intro}
============
The design of efficient intrusion detection systems (IDS) has received considerable attention recently. An IDS aims at identifying and mitigating malicious activities. Most of the current IDSs can be divided into two main categories: signature-based and anomaly-based IDS [@Aaltodoc].
A signature-based IDS tries to detect intrusions by comparing the incoming network traffic to already known attacks, which are stored in the database as signatures. This class of IDS performs well in identifying known attacks but often fail to detect novel (unseen) attacks [@jacobus2015network].
The second category is referred to as anomaly-based IDS. Those IDS models the normal traffic by learning patterns in the training phase. They label deviations from these learned patterns as anomaly or intrusion [@denning1987intrusion]. The implementation of real-time anomaly-based IDS is challenging due to rapidly evolving network traffic behavior and limited amount of computational resources (computation time and memory) [@hu2014online]. Another challenge is the risk of overfitting due to the high-dimensional feature space and the model complexity of IDS. [@shrivas2014ensemble; @sindhu2012decision; @wang2010new; @garcia2009anomaly].
In this paper we will address some of the short comings of existing anomaly-based IDS and present an efficient online feature ranking method. Our approach is based on online (or incremental) training of a support vector machine (SVM) which allows to cope with limited computational resources. The closest to our wok is [@hamed2018network; @xia2009incremental], which applies SVM for feature selection in a static setting. In contrast, our feature ranker finds approximate to the best feature subset in real-time and in the case of streaming network traffic with changing behaviour. Some illustrative numerical experiments indicate that real-time intrusion detection mechanisms using our feature ranker can be trained faster and achieves less error rate compared to offline (batch) feature selection methods.
This paper is organized as follows: The problem setup is formalized in Section \[sec:problem\]. In Section \[sec:Arch\], we detail the proposed online feature ranking method. The results of some illustrative numerical experiments are discussed in Section \[sec:exp\].
Problem Setup {#sec:problem}
=============
We consider a communication network within which information is exchanged using atomic units of data which are referred to as “packets”. These packets consist of a header for control information and the actual payload (user data). We represent a packet by a feature vector $\x = (x_{1}, \cdots,x_{d}) \in \mathbb{R}^{d}$ with $d$ individual features $x_i$ listed in Table \[tab:Extracted features of the network traffic\]. The last feature $x_{d}$ is a dummy feature being constant $x_{d}=1$.
A machine learning based IDS aims at classifying network packets into the classes $\mathcal{Y} = \{ 0,1\}$. Each packet is associated with a binary label $y \in \mathcal{Y}$ with $y=1$ indicating a malicious packet (an “attack”) and $y=0$ for a regular (normal) data packet. We view an IDS as a classifier $h: \mathbb{R}^{d} \rightarrow \mathcal{Y}$ which maps a given packet with feature vector $\x$, to a predicted label $\hat{y} = h(\x) \in \mathcal{Y}$ [@AGentleIntroML]. Naturally, the predicted label $\hat{y}$ should resemble the true class label $y$ as accurate as possible. Thus, we construct the classifier $h$ such that $y \approx h(\vx)$ for any packet with feature vector $\vx$ and true label $y$. In particular, we measure the classification error incurred when classifying a packet with feature $\vx$ and true label $y$ using the classifier $h$ by some loss function $\mathcal{L}((\vx,y),h)$. We will focus on a particular choice for the loss function as discussed in Section \[ssec:support\].
We learn the classifier $h$ in a supervised fashion by relying on a set of $N$ labeled packets $\{\x_i, y_i\}_{i=1}^{N}$ (the training set). In particular, we choose the classifier $h$ by minimizing the empirical risk $$(1/N) \sum_{i=1}^{N} \mathcal{L}((\vx_{i},y_{i}),h)$$ incurred by $h$ on the training data set $\{ \vx_{i},y_{i}\}_{i=1}^{N}$. We will restrict ourselves to linear classifiers $h$, i.e., which have a linear decision boundary. The corresponding hypothesis space, constituted by all such linear classifiers, is $$\label{equ_hypospace_lin_class}
\mathcal{H} = \{ h^{(\vw)}(\mathbf{x}) = \mathcal{I}( \mathbf{w}^{T} \vx \geq 0) \mbox{ for some } \vw \in \mathbb{R}^{d} \}.$$ Here, we used the indicator function $\mathcal{I}(\mbox{``statement''}) \in \{0,1\}$ which is equal to one of the argument is a true statement and equal to zero otherwise. Using the parameterization , learning the optimal classifier for an IDS amounts to finding the optimal weight vector $\vw \in \mathbb{R}^{d}$: $$\label{equ_training_problem_weightvector}
\min_{\vw \in \mathbb{R}^{d}} (1/N) \sum_{i=1}^{N} \mathcal{L}((\vx_{i},y_{i}),h^{(\vw)}).$$
It turns out that most of the relevant information for classifying network packets is often contained in a relatively small number of features due to redundant or irrelevant features [@guyon2003introductionVariable]. Thus, it is beneficial to perform some form of feature selection in order to avoid overfitting and improve the resulting classification performance. Moreover, for an online or real-time IDS, feature selection is instrumental in order to cope with a limited budget of computational resources.
Most existing feature selection methods are based on statistical measures for dependency, such as Fisher score or (conditional) mutual information, and therefore independent of any particular classification method [@FScoreFeatSel; @StructureFeatSelEventLogs]. In contrast, feature ranking methods use the entries $w_{j}$ fo the weight vector $\vw=(w_{1},\ldots,w_{j})$ of a particular classifier, e.g., obtained via solving , as a measure for the relevance of an individual feature $w_{j}$ [@chang2008feature]. We will apply feature ranking based on the weight vector obtained as the solution of for the particular loss function which is underlying the support vector machine (SVM).
General Architecture of The Feature Ranker {#sec:Arch}
==========================================
Figure \[fig:General architecture of the proposed mechanism\] shows the architecture of our system with an ability to rank features of the streaming data in real time based on the weights of the support vector machines (SVM) classifier. For the simplicity of our experiments, we used the same SVM also for the IDS part, but it might be any other intrusion detection mechanism using machine learning techniques.
Before training the feature ranker and the IDS, the first $N$ packets are collected as a historical data. These packets are used for pretraining the feature ranker and the IDS. In our feature ranker, features critical to detect attacks have negative values, while features contributing for the normal traffic have positive values. After waiting another $N$ packets and predicting the label of each packet as normal or attack, the mean squared error $$\frac{1}{N}\sum_{i=1}^{N}(y_i - \tilde{y}_i)^2$$ is measured between the actual labels $y_i $ and the predicted labels $\tilde{y}_i$. If this error is quite high, then the feature ranker and the learning model is retrained by feeding the actual labels to the mechanism. Thus, feature weights are adapted to changes in normal network behaviour or new attack strategies.
Support Vector Machines (SVM) {#ssec:support}
-----------------------------
Our design of an IDS is based on using the particular loss loss function $$\label{equ_reghingeloss}
\mathcal{L}((\vx,y),h) = \big[1- y (\vw^{T} \vx)\big]_{+} + (\lambda/2) {\left\lVert\mathbf{w}\right\rVert}^2,$$ with the shorthand $[z]_{+} = \max \{0,z\}$. The loss function consists of two components: The first component in is known as the hinge loss and underlying the SVM classifier. The second component in amounts to regularize the resulting classifier in order to avoid overfitting to the training data. and a regularization parameter $\lambda > 0$ which allows to avoid overfitting. Inserting into yields the weight of the SVM classifier $$\label{equ_SVM_class}
\hat{\vw}\!=\!\operatorname*{argmin}_{\vw \in \mathbb{R}^{d}} (1/N) \sum_{i=1}^{N} \big[1- y_{i} (\vw^{T} \x_{i})\big]_{+} + (\lambda/2) {\left\lVert\mathbf{w}\right\rVert}^2.$$ Note that is a non-smooth convex optimization problem which precludes the application of basic gradient based optimization methods [@JungFixedPoint]. Instead, as detailed in Section \[ssec:stoch\], we apply the stochastic sub-gradient method to solve in an online fashion.
Once we have learnt the optimal weight vector $\hat{\vw} = (\hat{w}_{1},\ldots,\hat{w}_{d})^{T}$ by solving , we can classify newly arriving packet with feature $\vx$. Therefore, for any testing sample $\x_i$, the SVM classifier $h(\cdot)$ can be written as $$h(\vx) = \mathcal{I} (\hat{\mathbf{w}}^T\vx > 0 ).$$ Thus, the classification result depends on the features $x_{i}$ via the weights $\hat{w}_{i}$. A weight $\hat{w}_{i}$ having large absolute value implies that the feature $x_{i}$ has a strong influence no the classification result [@guyon2003introduction]. It is therefore reasonable to keep only those features $x_{i}$ whose weights $w_{i}$ are largest and discard the remaining features in order to save computational resources and avoid overfitting.
Stochastic Gradient Descent (SGD) {#ssec:stoch}
---------------------------------
A naive implementation of SVM does not scale well due to excessive computational time and memory requirements [@zheng2013online]. In order to cope with the requirements of real-time IDS one typically needs to use online implementations of SVM. A simple modification of the linear SVM method can be done by incorporating gradient methods and adapting optimum weight vector as new data arrives.
Since SVM is trying to solve a convex cost function, we can find a solution by using iterative optimization algorithms. Stochastic gradient descent is one of the optimization algorithm that minimizes the cost function by performing the gradient descent on a single or few examples. in SGD, each $k$-th iteration updates the weights of the learning algorithms including SVM, percepton or linear regression, with respect to the gradient of the loss function $\mathcal{L}$
$$\mathbf{w}^{k+1} = \mathbf{w}^{k} - \alpha \nabla_{\mathbf{w}}\mathcal{L}_n(h(x),y),$$
where $\alpha$ denotes constant learning rate and $n$ is a set of indices randomly sampled from the training set $\{\x_i, y_i\}_{i=1}^{N}$. In our setup, we approximate the gradient by drawing $M$ points uniformly over the training data and perform mini batch iterations of SGD over SVM. Therefore the updates of weight vector $\mathbf{w}$ will become
$$\label{gradupdate}
\mathbf{w}^{k+1} = \mathbf{w}^{k} - \alpha \frac{1}{M}\sum_{j=0}^{M}\nabla_{\mathbf{w}}\mathcal{L}(g(\x)_j, y_j)$$
Although the hinge loss is not differentiable, we can still calculate the subgradient with respect to the weight vector $\mathbf{w}$. The derivative of the hinge loss can be written as: $$\frac{\partial \big[1- y_{i} (\vw^{T} \x_{i})\big]_{+}}{\partial w_k} = \left\{
\begin{array}{ll}
-y_ix_i & \quad y_i(\mathbf{w}_{k}^T \x_{i} + b_{i}) < 1 \\
0 & \text{otherwise}.
\end{array}
\right.$$ Therefore, the gradient update for Equation \[gradupdate\] at $k$-th iteration is calculated as: $$\mathbf{w}^{k+1} = \mathbf{w}^{k} - \alpha \lambda \mathbf{w}_k - \frac{1}{M}\sum_{j=0}^{M}\mathbbm{1}[\, (\mathbf{w}_k^T\x_{j} + b_{j}) < 1 ]\,y_{j}\x_j,$$ where $\mathbbm{1}[\, (\mathbf{w}_k^T\x_{j} + b_{j}) < 1 ]\, $ is an indicator function which results in one when its argument is true, and zero otherwise.
In large scale learning problems, SGD decreases the time complexity, since it does not need all the data points to calculate the loss. Although, it only samples few data point to reach the optimum point, it provides a reasonable solution to the optimization problem in the case of big data [@bottou2010large].
In our proposed feature ranker mechanism, the weight vector of a SVM classifier is optimized in an online fashion using SGD with mini batch size $N$. After predicting the labels of packets in the next mini-batch, the predicted labels are compared to the true classes of packets, which are decided by experts. If the predicted labels highly deviate from the true labels based on the MSE, then the weights are updated according to this $N$ packets. This also enables an online correction of the feature weights, since the weight vector is updated based on the incoming $N$ packets with a possible different connections, applications or attacks.
Experimental Results {#sec:exp}
====================
We verified the performance of the proposed IDS by means of numerical experiements run on a 64bit Ubuntu 16.04 PC with 8 GiB of RAM and CPU of 2.70GHz. Two different datasets, the ISCX IDS 2012 dataset and CIC Android Adware-General Malware, were used for our experiments to test both batch and online algorithms. The ISCX-IDS 2012 dataset [@shiravi2012toward] was generated by the Canadian Institute for Cybersecurity and mostly used for anomaly based intrusion detection systems. This dataset includes seven days of traffic and produces the complete capture of the all network traces with HTTP, SMTP, SSH, IMAP, POP3, and FTP protocols. It contains multi-stage intrusions such as SQL injection, HTTP DoS, DDoS and SSH attacks. The CIC Android Adware-General Malware dataset [@lashkaritowards] is also produced by the same research group. This dataset consists of malwares, including viruses, worms, trojans and bots, on over 1900 Android applications.
Both datasets are provided as pcap files including both malicious network packets and normal packets. We selected different malware types (InfoStl, Dishigy, Zbot, Gamarue) for testing batch (offline) feature ranking. We used the ISCX-IDS 2012 dataset for the online feature ranking, since it has the complete capture of one week network traffic data without any interruption to the overall connection. From each dataset, we transferred raw packets to a file with 43 features using “tshark” network analyzer. Table \[tab:Extracted features of the network traffic\] lists all the features measured with tshark and used in the experiments.
The first set of experiments does not include online incremental learning of SVM weights. It implements the linear SVM classifier with SGD update, as well as other widely used feature selection algorithms, selects 20 features with the highest weight out of 43, and uses them as an input to a single-layer feedforward neural network with 10 hidden neurons and sigmoid activation function. The experiments were performed on various pcap files from the CIC Android dataset having different sparsity, attacks and malwares in order to prove that feature ranking with linear SVM weights in batch (offline) learning provides good performance compared to the other feature selection methods. Table \[tab:Comparison of offline feature selection methods\] shows the experimental results of these feature selection methods by measuring the time spent on extracting the best features from the training set and the accuracy of the classification. It can be seen that linear SVM weights trained with the SGD method is quite fast and generally gives a higher accuracy than the other methods. Therefore, we can suggest that incremental version of the linear SVM approach can be extended to the case of streaming network traffic.
The second experiment was implemented with both offline and online SVM with a linear kernel in the case of streaming network data. Both feature weighting and classification is done by linear SVM in either online or offline fashion. Experiments were performed with the ISCX-IDS 2012 dataset. Figure \[fig:MSE of the streaming test data with a load change\] shows the mean squared error between the true and predicted labels of the streaming data while the network behaviour changes and new type of attacks are fed to the input. Unlike the offline SVM, online SVM changes its weight vector in a sliding window with $N$ packets.The training was implemented in the first $100*N$ packets which spans one day of network activities including injection attacks. The second day comes with a more complex and harder to detect injection attack and the third day also starts with a DoS attack. As can be seen in the figure, our online feature ranker model is able to adapt itself to a change in the network statistics. However, batch learning with SVM performs poorly when the arriving packets shows a completely different behaviour. In conclusion, on-line models quickly adapt to changing behaviour in the network data and achieves a notable improvement on the performance of prediction over batch learning.
The last experiment was also implemented with the ISCX-IDS 2012 dataset by choosing 3 days of network data with different attacks to prove that the importance of features are different for each attack. Figure \[fig:Feature Weights for Different Network Behaviors\] shows the resulting absolute weights of randomly selected 5 features. Figure \[fig:Feature Weights for Different Network Behaviors\] indicates that feature weights change even from positive to negative in each day having different connections and different attack types. These results prove that our feature ranker mechanism works as we desired and adjusts the feature weights with streaming network data.
Conclusion {#sec:conclusion}
==========
In this paper, we proposed a simple method based on incremental learning of linear SVM models to rank or weight features in real time. Experimental results indicated that our method can adjust the importance of any feature based on not only the changing network behaviour but also for novel attacks. In future research, the online feature ranking with a variable chunk size will be incorporated to better fine-tuning of our proposed method. In addition, the weights of the features will be used as input layer weights of neural networks based IDSs, which can be used as real-time and self-trained intrusion detection mechanism.
[10]{}
Buse Atli, “Anomaly-based intrusion detection by modeling probability distributions of flow characteristics,” M.S. thesis, Aalto University, Espoo, Finland, 2017-10-23.
Agustinus Jacobus and Alicia A.E. Sinsuw, “Network packet data online processing for intrusion detection system,” in [*2015 1st International Conference on Wireless and Telematics (ICWT)*]{}. IEEE, 2015, pp. 1–4.
Dorothy E. Denning, “An intrusion-detection model,” , vol. 13, no. 2, pp. 222–232, Feb. 1987.
Weiming Hu, Jun Gao, Yanguo Wang, Ou Wu, and Stephen Maybank, “Online adaboost-based parameterized methods for dynamic distributed network intrusion detection,” , vol. 44, no. 1, pp. 66–82, 2014.
Akhilesh Kumar Shrivas and Amit Kumar Dewangan, “An ensemble model for classification of attacks with feature selection based on kdd99 and nsl-kdd data set,” , vol. 99, no. 15, 2014.
Siva S Sivatha Sindhu, S Geetha, and Arputharaj Kannan, “Decision tree based light weight intrusion detection using a wrapper approach,” , vol. 39, no. 1, pp. 129–141, 2012.
Gang Wang, Jinxing Hao, Jian Ma, and Lihua Huang, “A new approach to intrusion detection using artificial neural networks and fuzzy clustering,” , vol. 37, no. 9, pp. 6225–6232, 2010.
Pedro Garcia-Teodoro, J Diaz-Verdejo, Gabriel Maci[á]{}-Fern[á]{}ndez, and Enrique V[á]{}zquez, “Anomaly-based network intrusion detection: Techniques, systems and challenges,” , vol. 28, no. 1, pp. 18–28, 2009.
Tarfa Hamed, Rozita Dara, and Stefan C Kremer, “Network intrusion detection system based on recursive feature addition and bigram technique,” , vol. 73, pp. 137–155, 2018.
Yong-Xiang Xia, Zhi-Cai Shi, and Zhi-Hua Hu, “An incremental svm for intrusion detection based on key feature selection,” in [*Intelligent Information Technology Application, 2009. IITA 2009. Third International Symposium on*]{}. IEEE, 2009, vol. 3, pp. 205–208.
A. Jung, “A [G]{}entle [I]{}ntroduction to [S]{}upervised [M]{}achine [L]{}earning,” , 2018.
Isabelle Guyon and Andr[é]{} Elisseeff, “An introduction to variable and feature selection,” , vol. 3, no. Mar, pp. 1157–1182, 2003.
S. Ding, “Feature selection based f-score and aco algorithm in support vector machine,” in [*2009 Second International Symposium on Knowledge Acquisition and Modeling*]{}, Nov 2009, vol. 1, pp. 19–23.
M. Hinkka, T. Lehto, K. Heljanko, and A. Jung, “Structural feature selection for event logs,” in [*Business Process Management Workshops. BPM 2017.*]{}, 2017.
Yin-Wen Chang and Chih-Jen Lin, “Feature ranking using linear svm,” in [*Causation and Prediction Challenge*]{}, 2008, pp. 53–64.
A. Jung, “A fixed-point of view on gradient methods for big data,” , vol. 3, 2017.
Isabelle Guyon and Andr[é]{} Elisseeff, “An introduction to variable and feature selection,” , vol. 3, pp. 1157–1182, 2003.
Jun Zheng, Furao Shen, Hongjun Fan, and Jinxi Zhao, “An online incremental learning support vector machine for large-scale data,” , vol. 22, no. 5, pp. 1023–1035, 2013.
L[é]{}on Bottou, “Large-scale machine learning with stochastic gradient descent,” in [*Proceedings of COMPSTAT’2010*]{}, pp. 177–186. Springer, 2010.
Ali Shiravi, Hadi Shiravi, Mahbod Tavallaee, and Ali A Ghorbani, “Toward developing a systematic approach to generate benchmark datasets for intrusion detection,” , vol. 31, no. 3, pp. 357–374, 2012.
Arash Habibi Lashkari, Andi Fitriah A Kadir, Hugo Gonzalez, Kenneth Fon Mbah, and Ali A Ghorbani, “Towards a network-based framework for android malware detection and characterization,” in [*15th International Conference on Privacy, Security and Trust (PST)*]{}, 2017.
|
---
abstract: 'We examine the effects of a global magnetic field and outflow on radiatively inefficient accretion flow (RIAF) in the presence of magnetic resistivity. We find a self-similar solutions for the height integrated equations that govern the behavior of the flow. We use the mixing length mechanism for studying the convection parameter. We adopt a radius dependent mass accretion rate as $\dot{M}=\dot{M}_{out}{(\frac{r}{r_{out}})^{s}}$ with $s> 0$ to investigate the influence of outflow on the structure of inflow where $s$ is a constant and indication the effect of wind. Also, we have studied the radiation spectrum and temperature of CDAFs. The thermal bermsstrahlung emission as a radiation mechanism is taken into account for calculating the spectra emitted by the CDAFs. The energy that powers bremsstrahlung emission at large radii is provided by convective transport from small radii and viscous and resistivity dissipation. Our results indicate that the disc rotates slower and accretes faster, it becomes hotter and thicker for stronger wind. By increasing all component of magnetic field, the disc rotates faster and accretes slower while it becomes hotter and thicker. We show that the outflow parameter and all component of magnetic field have the same effects on the luminosity of the disc. We compare the dynamical structure of the disc in two different solutions (with and without resistivity parameter). We show that only the radial infall velocity and the surface density could changed by resistivity parameter obviously. Increasing the effect of wind increases the disc’s temperature and luminosity of the disc. The effect of magnetic field is similar to the effect of wind in the disc’s temperature and luminosity of the disc, but the influence of resistivity on the observational properties is not evident.'
author:
- Maryam Ghasemnezhad
- Shahram Abbassi
title: The influence of outflow and global magnetic field on the structure and spectrum of resistive CDAFs
---
INTRODUCTION
============
Black hole accretion discs provide the most powerful energy production mechanism in the universe. It is well accepted that many astrophysical objects are powered by black hole accretion. The standard geometrically thin, optically thick accretion disc model can be successfully explain many observational features of X-ray binaries, but it is unable to explain observations of low-luminosity X-ray binaries and AGNs accretion discs. A particular example of such low luminous sources is our galactic center, Sagittarius, with host a $2\times 10^6$ solar mass black hole with luminosity well below the estimated value based on standard model (Melia & Falcke 2001). At low luminosities (less than a few percent of the Edington luminosity), black holes can accrete via advection dominated accretion flows (ADAFs) (Ichimaru 1977, Narayan & Yi 1994, Kato, Fukue & Mineshige 2008 and Yuan & Narayan 2014 for review). In such a flow, radiative losses are small compare to viscously heating because of low particle density of accreting flow at low accretion rate. Consequently, most of the energy released via viscosity is stored as entropy and transport inward with accretion. ADAFs are optically thin, geometrically thick and hot (compare the virial temperature of the gas in the flow) and radiate mostly in X-ray band (see Narayan et al. 1996). In the past decades the ADAFs models have captured great attentions and rapid progress has been made.
At the same time as ADAFs model was introduced, it was realized that they are likely to be unstable against convection in the radial direction. Because of low radiative efficiency in hot accretion flow, since the gas is heated but hardly cools, the entropy increases with decreasing radius. Hot accretion flows are therefore potentially unstable to convection. However, according some numerical simulations (e.g. Stone et al. 1999; Narayan et al. 2000) there are some debates about whether convection exists in hot accretion flow or not. Some of them have clearly shown that convection in an MHD accretion flow likely does not exist (Pen et al. 2003; Narayan et al. 2012; Yuan et al. 2012b). But some other series of numerical simulations reveal that the convection instability likely occurs in hot accretion flows (Igumenshchev, Chen & Abramowicz 1996, Igumenshchev & Abramowicz 1999, 2000, Stone, Pringle & Begelman 1999, Yuan & Bu 2010). Of course, some uncertainties still exist and we can’t conclude the non-existence of convection (see discussion in Yuan & Narayan 2014), thus it is still feasible to study convection and it worth to study CDAFs. Narayan, Igumenshchev & Abramowicz (2000) and Quataert & Gruzinuv (2000) introduced analytical model based on self-similar solution which was called convection dominated accretion flows (CDAFs). In particular, Igumenshchev, & Abramowicz (1999, 2000) have been point out that the ADAFs becomes convectively unstable whenever the viscous parameter $\alpha \le 0.1$. On the other hand, Narayan et. al (2000), Quataret & Gruzinuv (2000) based on self-similar solutions have shown that CDAF consist of a hot plasma about virial temperature and have a flattened time-averaged radial density profile, $\rho\propto r^{-\frac{1}{2}}$, where much flatter than usual ADAFs with $\rho\propto r^{-\frac{3}{2}}$. In CDAFs the most part of the energy which realized in inner most region of accretion flow is transport outward by convection.
Mass loss mechanism (in the form of wind or outflow) is an interesting phenomenon in the structure and evolution of the accretion discs. The existence of wind and outflow has been observationally verified in various astronomical objects like AGNs and YSOs (Whelan et al. 2005, Bally et al. 2007). **On the other hand several numerical simulation have been performed and they clearly confirm the existence of outflow in such systems (Yuan et al. (2012a, 2012b), Narayan et al. (2012), Li, Ostriker & Sunyaev (2013), Yuan et al. (2015), Bu et al. (2016a, 2016b).** In these objects some part of angular momentum of the accretion flow will dissipated outward in the form of wind and jet. For generating outflow various driving forces are proposed, such as thermal, radiative and magnetic field. The wind mechanism has been investigated by many others (Meier 1979, Fukue 1989, Abbassi et al. 2008, 2010, Ghasemnezhad & Abbassi 2016). The effect of magnetic field on the disc were also studied ( see Balbus & Hawley 1998, Kaburaki 2000, Shadmehri & Khajenabi 2005, Abbassi et al. 2008, Ghasemnezhad et al. 2012, 2013, Samadi et al. 2014, 2016, Bu et al. 2009, Soria et al. 1997). The effect of large scale magnetic field on the physical properties of CDAFs with hydrodynamically driven wind have been investigated by Abbassi & Mosallanezhad (2012, here after AM12).
The magnetic field have several effect in the dynamical and observational appearance of the discs such as: the formation of wind/jet, the interaction of discs and black holes and synchrotron emission. The traditional view of the magnetic field in the accretion disc is that the magnetic field is not completely frozen into the accreting matter. The fluid is not a perfect conductor, so the magnetic field advected inward by accretion and diffused by viscosity and resistivity (Guan & Gammie 2009). The resistivity diffusion of magnetic field is important in accretion disc and the simulations of local shearing box have indicated that the resistive dissipation increases the linear growth rate of magneto rotational instability (MRI) (Fleming et al. 2000). It will be interesting to study the effect of resistivity on optically thin ADAFs with convection, outflow and global magnetic field. Faghei & Omidvand (2012, hear after FO12) studied radial self similar solution of accretion flow in the presence of toroidal magnetic field, convection and resistivity. They ignored the effect of outflow and global magnetic field. AM12 studied the self similar solution of CDAFs with a global magnetic field and outflow. We have improved AM12 paper by adding the magnetic resistivity parameter and then have compared two solutions.
The main aim of our present work is highlighting observational consequences of CDAF models, focusing in particular on power spectra. Our results are similar to those of Blandford & Begelman(1999), AM12 who have assumed that a significant fraction of mass in an ADAF would be lost to outflow/wind, rather than accreting onto central object. In ADAFs the importance of outflows can be shown by a radial density profile as ($\rho \propto r^{s-\frac{3}{2}}$) which $(0<s<1)$. The density profile in CDAFs is equivalent to $s=1$ ($\rho \propto r^{-\frac{1}{2}}$). In order to capture many feature of CDAFs we have considered various values of $s$ in this study.
In CDAFs the convection motions transport a luminosity $L_{c}\approx (10^{-3}-10^{-2})\dot{M} c^{2}$ from small to large radii. The most of the energy that transport outward by convection can be radiated from the outer regions of the flow as thermal bremsstrahlung emission which is a function of temperature and density of accreting gas (Igumenshchev, & Abramowicz (2000) , Ball et al. 2001). We assumed the same mechanism in our study.
This paper organized as follows. The basic equations and assumptions are presented in section 2. Self-similar solutions are presented in section 3. The radiation properties of CDAFs are discussed in section 4. We show the result in section 5 and finally we present the summary and conclusion in section 6.
The Basic Equations
===================
We use the cylindrical coordinates $(r, \varphi, z)$ to write the MHD equations of steady state and axi-symmetric ($\frac{\partial}{\partial
\phi}=\frac{\partial}{\partial t}=0$) hot accretion flow around compact black hole of mass $M_{\star}$. Following AM12, we assume a magnetic field with three components $(B_{r}, B_{\varphi},
B_{z})$. We have vertically integrated the equations and then all our physical variables become only a function of radial distances, $r$. Moreover, the disc suppose to have Newtonian gravity in radial direction and also we neglect the self-gravity of the discs and the general relativistic effects. The disc is supposed to turbulent and possesses an effective turbulent viscosity. We adopt $\alpha$ -prescription for viscosity of rotating gas in accretion flow. The convection, outflow and magnetic field and its correspond resistivity are important to transfer of energy and angular momentum in disc.
The equation of continuity gives:
$$\frac{\partial}{\partial r}(r \Sigma V_r )+\frac{1}{2\pi}\frac{\partial \dot{M}_\mathrm{w}}{\partial r}=0$$
where $V_{r}$ is the accretion velocity ($V_{r}<0$) and $\Sigma=2\rho H$ is the surface density at a cylindrical radius $r$. $H$ is the disc half-thickness and $\rho$ is the density. Mass-loss rate by wind/outflow is represented by $\dot{M}_\mathrm{w}$. So
$$\dot{M}_\mathrm{w}=\int 4\pi r'\dot{m}_\mathrm{w}(r')dr',$$
where $\dot{m}_\mathrm{w}(r)$ is the mass-loss per unit area from each disc face. Similar to Blandford & Begelman (1999) and AM12, we write the dependence of accretion rate as follows,
$$\dot{M}=-2\pi r \Sigma V_r =\dot{M}_{out}(\frac{r}{r_{out}})^{s}
$$
where $\dot{M}_{out}$ is the mass accretion rate at the outer adge of the disc ($r_{out}$) (Blandford & Begelman 1999) and $s$ is a constant with order of unity. Considering equation (1-3), we can write $$\dot{m}_\mathrm{w}=\frac{s \dot{M}_{out}}{4 \pi r_{out}^{2}}(\frac{r}{r_{out}})^{s-2}
$$ The equation of motion in the radial direction is:
$$V_r \frac{\partial V_r}{\partial
r}=\frac{V_{\varphi}^{2}}{r}-\frac{GM_{\star}}{r^2}-\frac{1}{\Sigma}\frac{d}{dr}(\Sigma
c_\mathrm{s}^{2})$$
$$-\frac{c_\mathrm{\varphi}^{2}}{r}-\frac{1}{2\Sigma}\frac{d}{dr}(\Sigma c_\mathrm{\varphi}^{2}+\Sigma c_\mathrm{z}^{2})$$
where $V_{\varphi}$, $G$ and $c_\mathrm{s}$ are the rotational velocity of the flow, the gravitational constant and sound speed respectively. The sound speed is defined as $c_\mathrm{s}^{2}=\frac{p_\mathrm{gas}}{\rho}$ where $p_\mathrm{gas}$ is the gas pressure. Following AM12 and Zhang & Dai (2008), we introduce three component of Alfven sound speed $c_\mathrm{r},c_\mathrm{\varphi}$ and $c_\mathrm{z}$ as:
$$c_\mathrm{r,\varphi,z}^{2}=\frac{B_{r,\varphi,z}^{2}}{\textbf{4}\pi\rho}=\frac{2p_\mathrm{mag_{r,\varphi,z}}}{\rho}$$
where $B_{r,\varphi,z}$ and $p_\mathrm{mag_{r,\varphi,z}}$ are three components of magnetic field and magnetic pressure respectively.
By integration over $z$ of the azimuthal equation of motion gives.
$$\Sigma V_r \frac{d}{dr}(rV_\varphi)=-\frac{1}{r}\frac{d}{dr}(J_{vis})-\frac{1}{r}\frac{d}{dr}(J_{con})- \frac{\Omega(lr)^{2}}{2\pi}\frac{d\dot{M_{w}}}{dr}$$
$$+r \sqrt{\Sigma} c_\mathrm{r}\frac{d}{dr}(\sqrt{\Sigma} c_\mathrm{\varphi})+\Sigma c_\mathrm{r} c_\mathrm{\varphi}$$
where $\Omega(=\frac{V_\varphi}{r})$ and are the angular and Keplerian velocities respectively. The third term on the right hand side shows the angular momentum carried a way by wind/outflow materials. Knigge (1999) define the $l$ parameter as the length of the rotational lever-arm that allows we have several types of accretion disc winds models. The parameter $l=0$ corresponds to a non-rotating wind and the angular momentum is not extracted by the wind and the disc losses only mass because of the wind while $l=1$ represents outflowing materials that carries away the specific angular momentum $(r^{2}\Omega)$. This latter would be most fitting value for radiation-driven wind (Proga et al. 1998). Centrifugally driven MHD wind/outflow are correspond to $l>1$ and it would be able to remove a lot of angular momentum of the discs.
The $J_{vis}$ and $J_{con}$ are the viscous and convective angular momentum fluxes respectively that define as follow: $$J_{vis}=-r^{3}\nu
\Sigma\frac{d\Omega}{dr}$$ and $$J_{con}=-\nu_{con}
\Sigma r^{3\frac{(1+g)}{2}}\frac{d}{dr}(\Omega r^{3\frac{(1-g)}{2}})$$
where $\nu$ is the kinematic viscosity coefficient and formalized by Shakura & Sunyaev (1973) as: $$\nu=\alpha c_{s}H$$ where $\alpha$ is a constant less than unity that has called the viscous parameter. Also we have formalized all of the turbulence in our system like convective diffusion and resistivity similar to viscosity turbulence. So, $$\nu_{con}=\alpha_{c} c_{s}H$$ where $\alpha_{c}$ is the dimensionless convective parameter and we get it according the mixing length theory and $g$ is an index for determining the condition for angular momentum transportations. There are several possibilities for transporting of angular momentum by convection (Narayan et al. 2000) which is depends on the magnitude of $g$ parameter. Generally, convection transports angular momentum inward or outward for $g < 0 $ or $g > 0$, respectively while $g = 0$ corresponds to zero angular momentum transportation (Narayan et al. 2000). When $g=1$ the convection behaves like turbulence viscosity but if $g=-\frac{1}{3}$ the convection transport angular momentum inward. In this paper we consider the convective angular momentum flux as: $$J_{con}=-r\nu_{con}
\Sigma\frac{d}{dr}(r^{2}\Omega)$$ where is correspond to $g=-\frac{1}{3}$ and represents that the convective angular momentum flux is oriented down the specific angular momentum gradient. It means that convection tries to drive the system toward a stat of uniform specific angular momentum and consequently it corresponds to an inward angular momentum transportation (Narayan et al. 2000). We have determined the convective turbulence parameter $\alpha_{c}$ be the mixing length approximation. It can be imagined that a convective differentially-rotating fluid include of many independent fluid blobs. following Grossman et al. (1993) the convectively turbulence viscosity is defined as $$\nu_{turb}=\sigma L_{M}$$ where $\sigma$ is the velocity dispersion of the blobs and $L_{M}$ is the characteristic mixing length corresponding to effective mean free path of the blobs. So we can write the $\nu_{con}$ as follows (Lu et al. 2004): $$\nu_{con}=\frac{L_{M}^{2} }{4} (-N^{2}_{eff})^{\frac{1}{2}}$$ Here $N_{eff}$ is the effective frequency of the convective blobs and it will be:
$$N^{2}_{eff}=N^{2}+k^{2}$$
where $N$ and $k$ are the Brunt-Vaisala frequency and epicyclic frequency respectively, which are defined as $$N^{2}=-\frac{1}{\rho}\frac{ dp_{g}}{dr}\frac{d}{dr}\ln(\frac{p^{\frac{1}{\gamma}}_{g}}{\rho})$$ and $$k^{2}=2\Omega ^{2}\frac{d \ln(r^{2}\Omega)}{d lnr}$$ Also the characteristic mixing length $L_{M}$ could be written in terms of the pressure scale height $(H_{p}=-\frac{d r}{d \ln p})$ and the dimensionless mixing length parameter $l_{m}$ as bellow: $$L_{M}=2^{-\frac{1}{4}}l_{M}H_{p}$$
We have adopt $l_{M}=\sqrt{2}$ as it was estimated by Narayan et al. (2000) and Lu et al. (2004). Convection is present whenever $N^{2}_{eff}< 0$. $\alpha_{c}$ can be written in the form similar to normal viscosity as: $$\alpha_{c}=\frac{\nu_{con}}{c_{s} H}$$
By integrating along $z$ of the hydrostatic balance, we have: $$\Omega^{2}_{k} H^{2}-\frac{c_\mathrm{r}}{\sqrt{\Sigma}}\frac{d}{dr}(\sqrt{\Sigma} c_\mathrm{z})H = c^{2}_\mathrm{s}+\frac{1}{2}(c^{2}_\mathrm{r}+c^{2}_\mathrm{\varphi})$$
In order to complete the problem we need to introduce energy equation. We assume the generated energy due to viscosity and resistivity into the volume is balanced by the advection cooling, outward energy by convection and energy loss of outflow $(Q^{adv}+Q^{rad}+Q^{conv}+Q^{wind}=Q^{diss})$. Thus, $$\Sigma V_{r} T \frac{d S}{dr}+\frac{1}{r}\frac{d}{dr}(r F_{con})= f(\nu+g\nu_{con})\Sigma r^{2} (\frac{d \Omega}{dr})^{2}+\frac{\eta}{4\pi}\mathrm{J}^{2}$$ $$-\frac{1}{2}\zeta \dot{m}_\mathrm{w}(r)V_\mathrm{k}^{2}(r)$$
where $F_{con}$, $S$, $f$ and $T$ are the convective energy flux, the specific entropy, advection parameter and temperature respectively. Also we consider $Q^{diss}-Q^{rad}=fQ^{diss}$. Their corresponding relations are: $$F_{con}=-\nu_{con} \Sigma T \frac{d S}{dr}$$ where $$T\frac{d S}{dr}=\frac{1}{\gamma-1}\frac{d c^{2}_{s}}{dr}-\frac{c^{2}_{s}}{\rho}\frac{d \rho}{dr}$$ here $\gamma$ is the specific energy heats, $J=\nabla \times B $ is the current density and $\eta$ is the magnetic diffusivity due to turbulence. The two first term on the right hand side of the energy equation corresponds to the dissipation energy by viscosity, convection and resistivity $Q_{diss}= f(\nu+g\nu_{con})\Sigma r^{2} (\frac{d \Omega}{dr})^{2}+\frac{\eta}{4\pi}\mathrm{J}^{2}$. We can write the magnetic resistivity turbulence in the form of viscosity and convection turbulence as we was stated as: $$\nu=P_{m}\eta=\alpha c_{s} H$$ where $P_{m}$ is the magnetic Prandt number of the turbulence, which adopted to be a constant less than unity, $\eta$ is the magnetic diffusivity (Shadmehri 2004). The last term on the right hand side of the energy equation represents the energy loss due to wind or outflow (Knigge 1999). In our model $\zeta$ is a free and dimensionless parameter. The large $\zeta$ corresponds to more energy extraction from the disc because of wind (Knigge 1999).
Finally since we consider three components of magnetic field, the three components of induction equation can be written as: $$\dot{B}_{r}=0$$
$$\dot{B}_{\varphi}=\frac{d}{dr}[V_\varphi B_r - V_r B_\varphi+\frac{\eta}{r}\frac{d}{dr}(r B_\varphi)]$$
$$\dot{B}_{z}=-\frac{d}{dr}[rV_r B_z -\eta r\frac{d B_z}{dr}]$$
where $\dot{B}_ r,\varphi,z$ is the field escaping/creating rate due to magnetic instability or dynamo effect. Now we have a set of MHD equations that control the structure of magnetized CDAFs. The solutions of these coupled equation are strongly correlated to given values of viscosity, connectivity, magnetic field strength, $\beta_{r, \phi, z}$ and degree of advection $f$. In the next section we will demonstrate the self-similar solution of this MHD equations.
Self-Similar Solutions
======================
The basic equations of our models was discussed in the last section. We use the self similar method to solving above complicated differential equations. This powerful technique is a dimensional analysis and scaling law and is widely used in astrophysical fluid mechanics. Following to AM12, and other similar works (Ghanbari et al 2009 and samadi et al 2014, 2016), self-similarity in the radial direction is assumed: $$\Sigma=c_{0}\Sigma_{out} (\frac{r}{r_{out}})^{s-\frac{1}{2}}$$ $$V_r(r)=-c_1 \sqrt{\frac{G M_{\ast}}{r_{out}}}(\frac{r}{r_{out}})^{-\frac{1}{2}}$$ $$V_\varphi(r)= r \Omega (r) =c_2 \sqrt{\frac{G M_{\ast}}{r_{out}}}(\frac{r}{r_{out}})^{-\frac{1}{2}}$$ $$c_\mathrm{s}^{2}=c_3 \frac{G M_{\ast}}{r_{out}}(\frac{r}{r_{out}})^{-1}$$ $$c_\mathrm{r, \varphi, z}^{2} =\frac{B^{2}_{r, \varphi, z}}{4\pi\rho}= 2 \beta_{r, \varphi, z} c_3 \frac{G M_{\ast}}{r_{out}}(\frac{r}{r_{out}})^{-1}$$ $$H(r)=c_{4}r_{out}(\frac{r}{r_{out}})$$ $$\rho=\frac{1}{2}\frac{c_0 \Sigma_{out}}{c_4}\frac{1}{r_{out}}(\frac{r}{r_{out}})^{s-\frac{3}{2}}$$ $$B_{r,\varphi,z}= 2\sqrt{\frac{\pi\beta_{r, \varphi, z} c_3 c_0 \Sigma_{out} G M_\ast}{c_4}} \frac{1}{r_{out}}(\frac{r}{r_{out}})^{\frac{s}{2}-\frac{5}{4}}$$ where the constant $c_0$, $c_1$, $c_2$, $c_3$ and $c_4$ are dimensionless constants and will be determined later. $\Sigma_{out}$ and $r_{out}$ have been exploited in order to write equations in non-dimensional form. Substituting the above self-similar transformation in the MHD equations of the system, we’ll obtain the following system of coupled ordinary equations, which should be solve to having $c_0$, $c_1$, $c_2$, $c_3$ and $c_4$: $$\dot{m}=c_0 c_1$$
$$-\frac{1}{2}c_1^{2}=c_2^{2}-1-[(s-\frac{3}{2})+\beta_\varphi(s+\frac{1}{2})+(s-\frac{3}{2})\beta_z]c_3$$
$$-\frac{1}{2}c_1 c_2=-\frac{3}{2}(s+\frac{1}{2})(\alpha+g\alpha_c)c_2 c_4 \sqrt{c_3}+(s+\frac{1}{2})c_3\sqrt{B_r B_\varphi}-s l^2 c_1 c_2$$
$$c_4=\frac{1}{2}(s-\frac{3}{2})\sqrt{B_r B_z}+\frac{1}{2}\sqrt{(s-\frac{3}{2})^{2} c^{2}_{3} B_r B_z+4(1+B_r+B_\varphi)c_3}$$
$$(\frac{1}{\gamma-1}+s-\frac{3}{2})[(s-1)\alpha_c c^{\frac{3}{2}}_{3}c_4+c_1 c_3]=-\frac{1}{4}s \zeta c_1$$
$$+f[\frac{9}{4}(\alpha+g\alpha_c)c_4 c^{2}_{2} c^{\frac{1}{2}}_{3} + \frac{1}{2}\frac{\alpha}{P_m}c_4 c^{\frac{3}{2}}_{3}\beta_\varphi (s-\frac{1}{2})^{2}$$
$$+\frac{1}{2}\frac{\alpha}{P_m}c_4 c^{\frac{3}{2}}_{3}\beta_z (s-\frac{5}{2})^{2}-\frac{\alpha}{P_m}c_4 c^{\frac{3}{2}}_{3}\sqrt{\beta_z \beta_\varphi} (s-\frac{5}{2})(s-\frac{1}{2})]$$
$$\alpha_c=\frac{l^{2}_M}{4\sqrt{2 c_3}c_4 (s-\frac{5}{2})^{2}}\sqrt{(s-\frac{5}{2})c_3[(s-\frac{5}{2})\frac{1}{\gamma}-(s-\frac{3}{2})]-c^{2}_{2}}$$
where $\dot{m}$ is the dimensionless mass accretion rate and define as: $$\dot{m}=\frac{\dot{M}_\mathrm{out}}{\pi \Sigma_{out} r_{out}\sqrt{\frac{GM_{\ast}}{r_{out}}}}$$ Also, the field scaping/creating rate $\dot{B}_r,\varphi,z$ is written as follows: $$\dot{B}_{r,\varphi,z}=\dot{B}_{0 r,0\varphi,z}(\frac{r}{r_{out}})^{\frac{s}{2}-\frac{11}{4}}$$ By using of the self similarity solutions, we will have: $$\dot{B}_{r}=0$$ $$\dot{B}_{0\varphi}=\frac{1}{2}(s-\frac{7}{2})\frac{GM_{\ast}}{r^{\frac{5}{2}}_{out}}\sqrt{\frac{4\pi c_0 c_3 \Sigma_{out}}{c_4}}[c_2\sqrt{B_r}+c_1\sqrt{B_\varphi}$$ $$-(s-\frac{1}{2})\frac{\alpha c_4 \sqrt{c_3}}{2 P_m}]$$
$$\dot{B}_{0z}=\frac{1}{2}(s-\frac{3}{2})\frac{GM_{\ast}}{r^{\frac{5}{2}}_{out}}\sqrt{\frac{4\pi c_0 c_3 \Sigma_{out}}{c_4}}[c_1
-(s-\frac{5}{2})\frac{\alpha c_4 \sqrt{c_3}}{2 P_m}]$$
We can solve these equations numerically. The equations reduce to the equations of AM12 without the resistivity parameter $\eta=0$ or ($P_m=\infty$), . Also our equations reduce to the result of FO12 without outflow/wind parameter, radial and vertical magnetic filed.
The radiation properties of CDAFs
=================================
Using self-similar solutions obtained in the pervious section we will able to produce observational appearance of CDAFs. The inner part of accretion discs, where ADAFs or CDAFs conditions is valid, has a very high temperature and is moreover optically thin and magnetized. The relevant radiation processes are synchrotron emission, bremesstrahlung and modified Comptonization. Bremsstrahlung emission is the main cooling process for high temperature $T (>10^{7} K)$ plasma. In this paper, we have supposed that the bremsstrahlung radiation is the only contributor to our spectrum model.\
As we have shown in the last section, the density of gas in our model is : $$\rho=\frac{c_0}{c_4}(\frac{1}{2}\frac{\Sigma_{out}}{r_{out}})(\frac{r}{r_{out}})^{s-\frac{3}{2}}=\frac{c_0}{c_4}\rho_{out} (\frac{r}{r_{out}})^{s-\frac{3}{2}}$$
Following $Ball$ et al. (2001), we employ the Schawrzchild units for the radius, i.e., $R=\frac{r}{R_s}$ and $R_{out}=\frac{r_{out}}{R_s}$. $R_s=\frac{2GM_{\ast}}{c^{2}}=2.95 \times 10^{5} m \ cm$ is the Schawrzchild radius and $c$ and $m$ are light speed and the black hole mass in solar units $(=\frac{M_{\ast}}{M_{\odot}})$ respectively. So we can write the density as: $$\rho=\frac{c_0}{c_4}\rho_{out} R^{\frac{3}{2}-s}_{out} (R)^{s-\frac{3}{2}}=\frac{c_0}{c_4}\rho_0 (R)^{s-\frac{3}{2}}$$ where $\rho_{0}=\rho_{out} R^{\frac{3}{2}-s}_{out}$. As we stated in introduction, for $s=1$ the ADAF(+wind+convection) solutions include of CDAF models.
In this model, we can estimate the temperature of the dics as (Akizuki & Fukue 2006): $$\frac{\Re}{\bar{\mu}}T=c_\mathrm{s}^{2}=c_3\frac{GM_{\star}}{r}$$ where $\Re$ the gas constant and $\bar{\mu}$ the mean molecular weight ($\bar{\mu}=0.5$). So, $$T=c_3\frac{c^{2}\bar{\mu}}{2 \Re}(\frac{r}{R_\mathrm{s}})^{-1}=2.706 \times10^{12}c_3(\frac{r}{R_\mathrm{s}})^{-1}$$ $$=2.706 \times10^{12}c_3(R)^{-1}=T_0 c_3 (R)^{-1}$$ In this formula the coefficient $c_3$ implicitly depends on the wind, magnetic diffusion, magnetic field, advection and viscosity parameters, ($s, \eta (or P_m), \beta_{r,\varphi,z}, f, \alpha$). Total emissivity due the bremsstrahlung emission is (Rybicki & Lightman 1986): $$q_{bremss}(T,\nu)=2.4\times 10^{10} T^{-\frac{1}{2}}\rho^{2}\exp(\frac{-h\nu}{KT}) G_b$$ $$(erg s^{-1} cm^{-3}Hz^{-1})$$ where $G_b$, $K$ and $h$ are the Gaunt factor (and is around $1$), the Boltzmann constant and Planck’s constant respectively. We have ignored a weak frequency-dependent Gaunt factor. As we see the density $\rho$ instead of temperature is the dominant factor in bremsstrahlung emission. As the above equation, the bremsstrahlung emission in CDAFs is a dominant process because of the less steep density profile. Also the spectral structure of CDAFs are similar to the ADAFs with outflow solutions (Ball et al. 2001). Therefore we consider bremsstrahlung X-ray emission from the outer parts of CDAFs. By integrating over the whole frequency, we have: $$q_{bremss}(T)=5 \times 10^{20} T^{\frac{1}{2}}\rho^{2} G_b(erg s^{-1} cm^{-3})$$
The height integration of the bremsstrahlung emissivity is: $$F_{\nu}=2.4 \times 10^{10} T^{-\frac{1}{2}}\rho^{2}\exp(\frac{-h\nu}{KT}) H G_b (erg s^{-1} cm^{-2}Hz^{-1})$$ and The bolometric bremsstrahlung flux is: $$F_{bol}=5 \times 10^{20} T^{\frac{1}{2}}\rho^{2} H G_b(erg s^{-1} cm^{-2})$$ The bremsstrahlung luminosity of a CADF is given by: $$L_{\nu}=2 \int F_\nu 2\pi r dr (erg s^{-1})$$ by using the Schawrzchild units, the bremsstrahlung luminosity will be: $$L_{\nu}=4\pi R^{3}_{s} \int_{1}^{R_{out}} F_\nu R^{2} dR$$ As we mentioned in section 1, convective motions can transport energy from small to large radii and we can write the convective luminosity as $L_C\equiv \epsilon_c \dot{M} c^{2}$, where $\epsilon_c\cong 10^{-2}-10^{-3}$ is the convective efficiency. A fraction $\eta_{c}$ of this energy can be radiated at large radii in the CDAFs and this radiation emitted as thermal bremsstrahlung emission (Ball et al. 2001): $$L_c=\eta_c \epsilon_c \dot{M} c^{2}$$ In this study we assume $\eta_c=1$ which correspond to the all convected energy radiated away. The bolometric bremsstrahlung luminosity of a CDAFs is written as: $$L_c=4\pi R^{3}_{s} \int_{1}^{R_{out}} F_{bol} R^{2} dR$$ by equating this relation to $L_c=\eta_c \epsilon_c \dot{M} c^{2}$ and after some calculations we obtain : $$\rho_0=0.37 (\frac{\eta_c \epsilon_c }{10^{-2}})\frac{10^{-3}}{m R^{2s-\frac{1}{2}}_{out}}\frac{(2s-\frac{1}{2})c_4}{c^{2}_0\sqrt{c_3}}$$
Therefor, by substituting the above expressions for the density $\rho_0$ and temperature of the gas in equation 56, we find the bremsstrahlung spectrum from a CDAFs as follows: $$\nu L_\nu=0.4 \times 10^{19}m \frac{(\eta_c \epsilon_c)^{2}}{R^{4s-1}_{out}}\frac{c_4}{c^{2}_0 c^{\frac{3}{2}}_3}(2s-\frac{1}{2})^{2}$$ $$\int_{1}^{R_{out}} \nu R^{2s-\frac{1}{2}} \exp{(-\frac{1.7\times 10^{-23}\nu R}{c_3})} dR$$ We assume that the accretion flow extends from an outer radius $R_{out}$ down to an inner radius $R_{in}=1$. The bremsstrahlung emission can arises from all radii in the flow in contrast to synchrotron emission and modified Comptonization processes. As we have introduced above $\nu L_{\nu} \propto \nu \exp(\frac{-h\nu}{KT})$, by differentiating respect to $\nu$ from this expression, we can obtain the peak of the bremsstrahlung spectrum occurs at: $$h \nu_{peak}=K T=K \frac{T_0 c_3}{R}$$
As we can see in this equation, the peak of the spectrum is located in the X-ray frequency band.
=3.6in =2.3in
=3.6in =2.2in
=3.6in =2.2in
=3.6in =2.2in
=3.6in =2.2in
=3.6in =2.8in
=3.6in =2.8in
Results
=======
In this section, we will investigate numerically the role of magnetic field parameters, $\beta_{r, \phi, z}, P_{m}$ and also parameters, $s, l$, on the dynamical and observational appearance of CDAF in presence of all components of magnetic field. In this study constant values for some parameters like $\alpha=0.5$, $f=1$, $\gamma=1.01$, $g=-\frac{1}{3}$, $\zeta=1$, $\eta_c=1$ and $\epsilon_c=0.0045$ have been adopted.\
Figures 1,2,3 show the self similar coefficient $c_0$ (the surface density), $c_1$ ( the radial velocity) and $c_2$ (the rotational velocity) as a functions of the three components of magnetic field for different values of wind parameter $s (=0, 0.1, 0.2)$. By adding the all components of magnetic field $(\beta_{r,\varphi,z})$ which indicate the role of magnetic field on the dynamics of accretion disc, we see that the surface density and rotational velocity of the disc gradually increase, although the radial speed decreases. This results are qualitatively consistent with results presented by AM12 and FO12. In these figures we also studied the effect of parameter, $s$ which measures the strength of wind, on physical coefficients. Generally, when the exponent $s$ increases, the flow will rotate slower than that without wind. Also radial and rotational velocities in Figure 1, 2 and 3 represent significant deviations from non-wind solutions. Consequently, CDAFs in the presence of wind rotate more slowly than those without winds and wind leads to enhance accretion velocities. The strong wind causes the strong accretion velocity and reduction of surface density. For the larger values of $\beta_{r,\varphi}$, a small change in $c_2$ is observed.\
Our focus is the investigate the effects of magnetic resistivity (or the Prandtl number $P_m$) and outflow on the structure of the disc. Here, the inverse of Prandtl number specifies the resistivity of the fluid as we stated in section 2. Our solutions switch back to the solution in AM12, in which the magnetic resistivity is not considered (or $P_m=\infty$). Figure 1,2,3,4 and 5 show a comparison of our model and AM12. As is clear in these Figure 1, the influence of magnetic diffusivity on the surface density and accretion velocity is more evident for deferent values of $\beta_{\phi}$. The effects of magnetic diffusion and outflow are almost the same on the structure of our model. These properties confirm the results of Faghei & Mollatayefeh (2012) and FO12.
The vertical thickness profiles are presented as a function of $\beta_{r,\varphi,z}$ for various values of exponent of $s$ in Figure 4. They show that the disc vertical thickness increases with increasing $\beta_{\phi}$, toroidal component, but decreases by increasing the other components. This figure demonstrates that the disc thickness increases by increasing s. Equation 39, is clearly shows that the vertical thickness posses a complicated dependencies with our input parameters. This figure reveal that the magnetic diffusion has not a obvious effect on the thickness of disc.
In figure 5. we have plotted the convection parameter $\alpha_c $ versus the components of magnetic field for several values of s. The results are compatible with AM12. As it is seen, the convective parameter decrease, if the radial and toroidal magnetic field parameter become stronger although by adding z-component of magnetic field the convective parameter $\alpha_c$ increases. Larger values of wind parameter, $s$, cause the strong convection in the disc. It is obviously seen, the effect magnetic resistivity on the convection parameter is more important for the large vertical magnetic field. The results of wind and resistivity on the convection is the same.
In figure 6. the surface temperature ($T_{eff}$ ) is plotted as a function of the dimensionless radius ($\frac{r}{R_s}$). It is obvious that the surface temperature is monotonically decreasing with $\frac{r}{R_s}$. We see that the surface temperature increases by adding wind parameter $s$, top left In Figure 6. In the top right panel and bottom left panel we show that surface temperature increases by increasing toroidal and vertical magnetic field parameters. But the effect of $\beta_{\varphi}$ is more evident. And finally in the bottom right panel we can see the values of magnetic resistivity don’t affect the surface temperature. So the disc is hotter with strong wind or even stronger magnetic field.
The radiation spectrum of the hot CDAF around a $10^{8} M_{\odot}$ black hole is represented in Figure 7 for several values of wind, radial and vertical magnetic field and resistivity parameters. The corresponding values of $\dot{m}(=\frac{\dot{M}}{\dot{M}_{Edd}})$ and $R_{out}$ are $10^{-3.9}$ and $10^{3}$ respectively. Luminosity depends explicitly on some of our input parameters like $s$, $R_{out}$, $m$, $c_4$ ( see equation 60) and implicitly on other parameters. The bremsstrahlung X-ray emission has been adopted since it is dominant mechanism in the outer parts of CDAFs.
As can be seen in figure 7, the maximum of $\nu L_{\nu}$ is highly depends on the given values of wind parameter. Whenever bremsstrahlung emission dominants, the X-ray emission depends directly on $\dot{m}_{out}(=\frac{\dot{M}_{out}}{\dot{M}_{Edd}})$ (see section 3.1 in Quataert & Narayan 1999 and see equation 57 in our paper). As the wind becomes stronger, the inner mass accretion rate $\dot{m}_{in}(=\frac{\dot{M}_{in}}{\dot{M}_{Edd}} = \dot{m}_{out} R_{out}^{-s})$ becomes smaller than $\dot{m}_{out}$, so the importance of bermesstrahlung emission increases. So by adding the wind parameter $(s)$ the maximum of bermesstrahlung luminosity increases. Quataert & Narayan (1999) stated the bremsstrahlung emission produces a peak that extends from a few to a few hundred $kev$. Also we have shown that the peak of power spectrum is located approximately at $ h\nu \approx 41 keV $, which is independent of the strength of winds in the system. This result is in full agreement with perviously study by Quataert & Narayan (1999).
Also, we are plotted the radiation spectrum of CDAFs for different values of all components of magnetic field. The effect of toroidal magnetic field on the radiation spectrum is similar to wind parameter. While the effect of $\beta_{z}$ and $p_m$ on the power spectrum are almost negligible.
Summary and Conclusion
======================
The CDAFs model consistently explain radiation inefficient accretion flow (RIAF) in the presence of convection. Many authors studied the importance of convection in RIAFs (CDAFs model) by means of numerical simulation or even analytically using the mixing length theory. Also observational results and MHD simulation confirm the importance of outflow and magnetic diffusion in CDAFs. Our primary focus in this research is to develop of AM12 solutions by considering magnetic resistivity in the main MHD equations. Folowing FO12, AM12 and Zhang & Dai 2008 we considered power-law function for mass inflow rate and solved the inflow-outflow equations by using self-similar approach in CDAFs regime. Some approximation have been done in order to simplify the main equations. We ignore the relativistic effects, self-gravity of the discs. For viscosity $\alpha$-prescription has been adopted. Our results reduce to AM12 solutions when the effect of magnetic resistivity is neglected.
Consequently our results represent that by increasing all components of magnetic field the surface density and rotational velocity increase although the radial velocity decreases. Also existence of the wind will lead to a significant reduction of surface density as well as rotational velocity and increasing radial velocity. Increasing $\beta_{\phi}$ will increase vertical thickness while it decreases by increasing $\beta_{z, r}$. Additionally the radial velocity and vertical thickness is will increase when outflow becomes important while the surface density and rotational velocity will decrease. Our results shown that the radial and rotational velocity will increase if the magnitude of resistivity increases, while rotational velocity decreases. The influence of magnetic diffusivity on the surface density and accretion infall velocity is more evident in the bigger toroidal magnetic field. These results are generally constant with results presented by AM12, FO12.
We also calculated the continuum spectrum emitted from the discs with assuming bremsstrahlung mechanism. Convection motion in CDAF transports a large amount of energy stored in small radii to large radii. Igumenshchev & Abramowicz (2000) suggested that some or perhaps main part of this energy budget might be radiated a way to the outer regions as a thermal Bremsstrahlung emission. So we used this model in our calculations. Consequently our solutions show that the bolometric luminosity increases as wind/outflow becomes stronger. The maximum of $\nu L_{\nu}$ is strongly changed by different values of wind parameter. On the other hand, increasing $\beta_{\phi}$ make the bolometric luminosity of the disc increase gradually. The other components of magnetic field and magnetic resistive have not a considerably effect on power spectra.
The main feature of the self-similar solution is that it is purely analytic and provides a transparent way of understanding the key properties of an CDAFs. However, the self-similar solution is not valid near the inner or outer boundaries. Consequently for calculating the radiation spectrum which mostly comes from inner regions (where the self-similar solution is invalid), we requires a global solution obtained by solving directly the differential equations of the problem. on the other hand in this paper we only considered the bremsstrahlung emission; while in the inner region of the disks, where the radiation mainly comes from there, the synchrotron radiation and its Comptonization is much more important than bremsstrahlung emission. Thus it worth to investigations the effects of these mechanism of radiation with global solutions.
Although that we have made some simplification and some assumption in order to solve equations analytically, our solutions show explicitly that outflow, large scale magnetic field and its corresponding resistivity can really change dynamical and observational appearance of CDAF. It means in any realistic model these parameters should take into account. This kind of self-similar solution could greatly facilitate testing and interpretations of the numerical simulations and observational evidences.
S. Abbassi acknowledges support from the International Center for Theoretical Physics (ICTP) for a visit through the regular associateship scheme.
Abbassi S., Ghanbari J., Najjar S., 2008, MNRAS, 388, 663 Abbassi S., Ghanbari J., Ghasemnezhad M., 2010, MNRAS, 409, 1113 Abbassi S., Mosallanezhad A., 2012, Ap&SS, 341, 375A Akizuki C., Fukue J., 2006, PASJ, 58, 469 Ball G., Narayan R., Quataert E., 2001, ApJ, 552, 221B
Bally J., Reipurth B., Davis C. J., 2007, Protostars and Planets V, 215 Balbus S., Hawley J.F., 1998, RvMP,70,1
Blandford R. D., Begelman M. C., 1999, MNRAS, 303, L1
Bu D.; Yuan F.; Gan Z.; Yang, X., 2016a, ApJ, 818, 83 Bu D.; Yuan F.; Gan Z.; Yang, X., 2016b, ApJ, 823, 90 Bu D.; Yuan F.; Xie F.; 2009, MNRAS, 392, 325
Faghei K., Omidvand M., 2012, Ap&SS, 341, 363F Faghei K., Mollatayefeh A., 2012, MNRAS, 422, 672 Fleming T.P., Ston J.M., Hawley J.F., 2000, ApJ, 530, 464
Fukue J., 1989, PASJ, 41, 123
Ghanbari J., Abbassi S., Ghasemnezhad M. 2009,MNRAS, 400, 422 Ghasemnezhad M., Khajavi M., Abbassi S., 2012,ApJ,v750 Ghasemnezhad M., Khajavi M., Abbassi S., 2013, Ap &SS, 346, 341G Ghasemnezhad M., Abbassi S., 2016, MNRAS,456, 71G Guan, X., Gammie, C.F.,2009, ApJ, 697, 1901 Grossman, S.A., Narayan R., Arneet D., 1993, ApJ, 407, 284G Ichimaru, S.,1977, ApJ, 214, 840 Igumenshchev I.V., Chen X., Abramowicz M.A., 1996, MNRAS, 278, 236I Igumenshchev I.V., Abramowicz M.A., 1999, MNRAS, 303, 309I Igumenshchev I.V., Abramowicz M.A., 2000, ApJS, 130, 463I Kaburaki O., 2000, ApJ, 660, 1273J Kato S., Fukue J., & Mineshige S., 2008, Black hole accretion discs Knigge C., 1999, MNRAS, 309, 409 Li J.; Ostriker J.; Sunyaev R.; 2013, ApJ, 767, 105 Lu J.F., Li S.L., Gu W.M., 2004, 352, 147 Meier D.L., 1979, ApJ, 233, 664 Melia F., Falcke H., 2001, ARA&A, 39, 309M Narayan R., & Yi, I. 1994, ApJ, 428, L13 Narayan R.; Sadowski A.; Penna R. F.; Kulkarni A. K.; 2012, MNRAS, 426, 3241 Narayan R., MacClintock J.E, Yi I., 1996, ApJ , 457, 821 Narayan, R., Igumenshchev I.V, Abramowicz M.A., 2000, ApJ , 539, 798 Pen U.; Matzner C. D.; Wong S., 2003, ApJ, 596, 207 Proga D., Stone J.M, Drew J.E, 1998, MNRAS, 295, 595 Quataert E., Narayan, R., 1999, ApJ , 520, 298 Quataert E., Gruzinov, A., 2000, ApJ , 539, 809Q Rybicki, G.B., Lightman, A.,P., 1986, radiative processes in Astrophysics Samadi, M., Abbassi, S., 2016, MNRAS, 455, 3381S Samadi, M., Abbassi, S., Khajavi, M., 2014, MNRAS, 437, 3124S Shadmehri M., 2004, Astrophys,J., 612, 1000 Shadmehri M.,Khajenabi F., 2005, MNRAS, 361, 719 Shakura N. I., & Sunyaev R.A., 1973, A&A, 24, 337 Soria R.; Li J.; Wickramasinghe D.T., 1997, ApJ, 487, 769 Stone J.M., Pringle J.E., Begelman M.C., 1999, MNRAS, 310, 1002S Whelan, E.T.; Ray, T.P., Bacciotti F., et al. 2005, Nature, 435, 652W
Yuan F., Narayan R., 2014, ARA&A, 52, 529 Yuan F., Bu D.F, 2010, MNRAS, 408, 1051Y Yuan F.; Wu, M.; Bu, D., 2012a, ApJ, 761, 129 Yuan F.; Bu, D.; Wu, M.; 2012b, ApJ, 761, 130 Yuan F.; Gan Z.; Narayan R.; Sadowski A.; Bu D.; Bai X.; 2015, ApJ, 804, 101
Zhang D., Dai Z.G., 2008, MNRAS, 0805, 3254Z
|
---
abstract: 'For a Frobenius cellular algebra, we prove that if the left (right) dual basis of a cellular basis is again cellular, then the algebra is symmetric. Moreover, some ideals of the center are constructed by using the so-called Nakayama twisted center.'
address: 'School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, P.R. China'
author:
- Yanbo Li
title: Nakayama twisted centers and dual bases of Frobenius cellular algebras
---
[^1]
[^2]
Introduction
============
Frobenius algebras are finite dimensional algebras over a field that have a certain self-dual property. These algebras appear in not only some branches of algebra, such as representation theory, Hopf algebras, algebraic geometry and so on, but also topology, geometry and coding theory, even in the work on the solutions of Yang-Baxter equation. Symmetric algebras including group algebras of finite groups are a large source examples. Furthermore, finite dimensional Hopf algebras over fields are Frobenius algebras for which the Nakayama automorphism has finite order.
Cellular algebras were introduced by Graham and Lehrer in 1996. The theory of cellular algebras provides a linear method to study the representation theory of many interesting algebras. The classical examples of cellular algebras include Hecke algebras of finite type, Ariki-Koike algebras, q-Schur algebras, Brauer algebras, partition algebras, Birman-Wenzl algebras and so on. We refer the reader to [@G; @GL; @Xi1; @Xi2] for details. It is helpful to point out that some cellular algebras are symmetric, including Hecke algebras of finite type, Ariki-Koike algebras satisfying certain conditions et cetera. In [@L1; @L3; @LX], Li and Xiao studied the general theory of symmetric cellular algebras, such as dual bases, cell modules, centers and radicals.
Apart from its own importance, Frobenius algebras are useful because the symmetricity of an algebra is usually not easy to verify. Thus it is meaningful to develop methods that can be used to deal with algebras only known to be Frobenius. In [@L2; @LZ], Li and Zhao investigated Nakayama automorphisms and projective cell modules of Frobenius cellular algebras. In this paper, we will study dual bases and central ideals. Exactly, we prove that there not exists a non-symmetric Frobenius cellular algebra such that the right or left dual basis being cellular. Moreover, some ideals of the center are also constructed by using the so-called Nakayama twisted center.
The paper is organized as follows. We first study Higman ideals and Nakayama twisted centers of Frobenius algebras. Then in Section 3, we detect properties of dual bases and construct ideals of centers for Frobenius cellular algebras. In Section 4, we illustrate some examples of non-symmetric Frobenius cellular algebras and give some remarks.
Higman ideals and Nakayama twisted centers
==========================================
Let $K$ be a field and let $A$ be a finite dimensional $K$-algebra. Let $f:A\times A\rightarrow K$ be a $K$-bilinear form. We say that $f$ is non-degenerate if the determinant of the matrix $(f(a_{i},a_{j}))_{a_{i},a_{j}\in B}$ is not zero for some basis $B$ of $A$. We say $f$ is associative if $f(ab,c)=f(a,bc)$ for all $a,b,c\in A$, and symmetric if $f(a,b)=f(b,a)$ for all $a,b\in A$.
\[2.1\] A $K$-algebra $A$ is called a Frobenius algebra if there is a non-degenerate associative bilinear form $f$ on $A$. We call $A$ a symmetric algebra if in addition $f$ is symmetric.
It is well known that for a Frobenius algebra $A$, there is an automorphism $\alpha$ of $A$ such that $Hom_K(A,K)\,\simeq\,_{\alpha}A_1$ as $A$-$A$-bimodules. This automorphism is unique up to inner automorphisms and is called the Nakayama automorphism. In [@HZ], Holm and Zimmermann proved the following lemma.
[[@HZ Lemma 2.7]]{}\[2.2\] Let $A$ be a finite dimensional Frobenius algebra. Then an automorphism $\alpha$ of $A$ is a Nakayama automorphism if and only if $$f(a, b)=f(\alpha(b), a).$$
Let $A$ be a Frobenius algebra with a basis $B=\{a_{i}\mid
i=1,\ldots,n\}$ and $f$ a non-degenerate associative bilinear form $f$. Define a $K$-linear map $\tau: A\rightarrow R$ by $\tau(a)=f(a,1)$. We call $\tau$ a symmetrizing trace if $A$ is symmetric. Denote by $d=\{d_{i}\mid i=1,\ldots,n\}$ the basis which is uniquely determined by the requirement that $\tau(a_{i}d_{j})=\delta_{ij}$ and $D=\{D_{i}\mid i=i,\ldots,n\}$ the basis determined by the requirement that $\tau(D_{j}a_{i})=\delta_{ij}$ for all $i, j=1,\ldots,n$. We will call $d$ the right dual basis of $B$ and $D$ the left dual basis of $B$, respectively. Then we can define an $R$-linear map $\alpha:A\rightarrow A$ by $\alpha(d_{i})=D_{i}$. It is easy to show that $\alpha$ is a Nakayama automorphism of $A$ by Lemma \[2.2\]. If $A$ is a symmetric algebra, then $\alpha$ is the identity map and then the left and the right dual basis are the same.
Write $a_{i}a_{j}=\sum_{k}r_{ijk}a_{k}$, where $r_{ijk}\in K$. We proved the following lemma in [@L2], which is useful in this section.
[@L2 Lemma 1.3]\[2.3\] Let $A$ be a Frobenius algebra with a basis $B$ and dual bases $d$ and $D$. Then the following hold:
[(1)]{}$a_{i}d_{j}=\sum_{k}r_{kij}d_{k}$;
[(2)]{}$D_{i}a_{j}=\sum_{k}r_{jki}D_{k}.$
It is well known that $\{\sum\limits_{i}d_{i}aa_{i}\mid a\in A\}$ is an ideal of $Z(A)$, see [@CR]. The ideal $H(A)=\{\sum\limits_{i}d_{i}aa_{i}\mid a\in A\}$ is called Higman ideal of $Z(A)$. Note that $H(A)$ is independent of the choice of $\tau$ and the dual bases $a_i$ and $d_i$. Then it is clear that $H(A)=\{\sum\limits_{i}a_{i}aD_{i}\mid a\in A\}$.
A natural problem is to consider sets $\{\sum\limits_{i}a_{i}ad_{i}\mid a\in A\}$ and $\{\sum\limits_{i}D_{i}aa_{i}\mid a\in A\}$. We claim that these two sets are the same. The proof is similar to that $H(A)$ is independent of the choice of the dual bases, which can be found in [@CR]. We omit the details and left it to the reader. From now on, let us write $\{\sum\limits_{i}a_{i}ad_{i}\mid a\in A\}$ by $H_{\alpha}(A)$.
In [@HZ] Holm and Zimmermann introduced Nakayama twisted center $Z_{\alpha}(A)$ for a Frobenius algebra $A$. We will prove that $H_{\alpha}(A)\subset Z_{\alpha}(A).$ Firstly, let us recall the definition of Nakayama twisted centers of Frobenius algebras.
[[@HZ Definition 2.2]]{}\[2.4\] Let $A$ be a Frobenius algebra and $\alpha$ a Nakayama automorphism. The Nakayama twisted center is defined to be $$Z_{\alpha}(A) := \{x\in A \mid xa = \alpha(a)x\,\,\, {\rm for \,\,\,all}\,\, a\in
A\}.$$
It is easy to know that $$Z_{\alpha}(A) = \{x\in A \mid
x\alpha^{-1}(a) = ax\,\,\, {\rm for\,\,\, all}\,\, a\in A\}.$$ Moreover, we denote by $$Z_{\alpha^{-1}}(A) = \{x\in A \mid xa =
\alpha^{-1}(a)x\,\,\, {\rm for\,\,\, all}\,\, a\in A\}.$$
For the Nakayama twisted center of a Frobenius algebra, we have the following result:
\[2.5\] Let $A$ be a Frobenius algebra and let $\alpha$ be an automorphism of $A$. Then for any $x,y\in Z_{\alpha}(A)$, we have\
[(1)]{}$\alpha(x)\in Z_{\alpha}(A)$.\
[(2)]{}$\alpha(xy)=xy$.\
[(3)]{}$Z_{\alpha^{-1}}(A)Z_{\alpha}(A)$ and $Z_{\alpha}(A)Z_{\alpha^{-1}}(A)$ are both ideals of $Z(A)$.
(1)We need to prove that $\alpha(x)a=\alpha(a)\alpha(x)$ for any $a\in A$. In fact, since $\alpha$ is an automorphism of $A$, $\alpha(x)a=\alpha(x)\alpha(\alpha^{-1}(a))=\alpha(x\alpha^{-1}(a))$. Note that $x\in Z_{\alpha}(A)$, then $\alpha(x\alpha^{-1}(a))=\alpha(ax)=\alpha(a)\alpha(x).$
(2)It follows from $x\in Z_{\alpha}(A)$ that $xy=\alpha(y)x$. By (1), $y\in Z_{\alpha}(A)$ implies that $\alpha(y)\in
Z_{\alpha}(A)$. Then $\alpha(y)x=\alpha(x)\alpha(y)=\alpha(xy)$.
(3)$Z_{\alpha^{-1}}(A)Z_{\alpha}(A)\subset Z(A$ is clear by definitions of $Z_{\alpha^{-1}}(A)$ and $Z_{\alpha}(A)$. Moreover, $cZ_{\alpha}(A)\subset Z_{\alpha}(A)$ for all $c\in Z(A)$. Consequently $Z_{\alpha^{-1}}(A)Z_{\alpha}(A)$ is an ideal of $Z(A)$. It is proved similarly for $Z_{\alpha}(A)Z_{\alpha^{-1}}(A)$.
Now let us show that $H_{\alpha}(A)$ is in the Nakayama twisted center of $A$.
\[2.6\] Let $A$ be a Frobenius algebra. Then $H_{\alpha}(A)\subset
Z_{\alpha}(A).$
For arbitrary $a\in A$, denote $\sum\limits_{i}a_{i}ad_{i}$ by $x_{a}$. We prove $x_{a}\alpha^{-1}(b)=bx_{a}$ for all $b\in A$. In fact, for an $a_{j}\in B$, we have from Lemma \[2.3\] that $$x_{a}\alpha^{-1}(a_j)=\sum_{i}a_{i}ad_{i}\alpha^{-1}(a_{j})=\sum_{i}a_{i}a\alpha^{-1}(D_{i}a_{j})
=\sum_{i,k}r_{jki}a_{i}ad_{k},$$ and $$a_jx_{a}=\sum_{i}a_{j}a_{i}ad_{i}=\sum_{i,k}r_{jik}a_{k}ad_{i}.$$ Obviously, the right sides of the above two equations are equal.
It is well known that a Frobenius algebra $A$ is separable if and only of $H(A)=Z(A)$. Then what implies that $H_{\alpha}(A)=
Z_{\alpha}(A)$?
In order to construct ideals of $Z(A)$, we hope to find some elements of $Z_{\alpha^{-1}}(A)$ in view of Lemma \[2.5\]. One option may be the set $$\{\sum\limits_{i}d_{i}a\alpha(a_{i})\mid
a\in A\},$$ which will be denoted by $H_{\alpha^{-1}}(A)$.
\[2.7\] $H_{\alpha^{-1}}(A)\subset Z_{\alpha^{-1}}(A).$
Denote $\sum\limits_{i}d_{i}a\alpha(a_{i})$ by $x_{\alpha(a)}$ for some $a\in A$. Now let us prove that $x_{\alpha(a)}\alpha(b)=bx_{\alpha(a)}$ for any $b\in A$. In fact, for an $a_j\in B$, we have from Lemma \[2.3\] that $$x_{\alpha(a)}\alpha(a_j)=\sum\limits_{i}d_{i}a\alpha(a_{i})\alpha(a_j)
=\sum\limits_{i}d_{i}a\alpha(a_{i}a_j)=\sum_{i,k}r_{ijk}d_{i}a\alpha(a_{k}).$$ $$a_jx_{\alpha(a)}=\sum\limits_{i}a_jd_{i}a\alpha(a_{i})=\sum_{i,k}r_{kji}d_{k}a\alpha(a_{i}).$$ The right sides of the above two equalities are the same and then we complete the proof.
The following lemma reveals some relation among $H(A)$, $H_{\alpha}(A)$, $H_{\alpha^{-1}}(A)$, $Z_{\alpha}(A)$ and $Z_{\alpha^{-1}}(A)$.
\[2.8\] The sets $H_{\alpha^{-1}}(A)Z_{\alpha}(A)$, $Z_{\alpha}(A)H_{\alpha^{-1}}(A)$, $H_{\alpha}(A)Z_{\alpha^{-1}}(A)$, and $Z_{\alpha^{-1}}(A)H_{\alpha}(A)$, which contained in $H(A)$ are all ideals of $Z(A)$.
Firstly, let us consider $H_{\alpha^{-1}}(A)Z_{\alpha}(A)$. It follows from Lemma \[2.5\], \[2.6\] and \[2.7\] that $H_{\alpha^{-1}}(A)Z_{\alpha}(A)\subset Z(A)$. It is easy to know that for any $c\in Z(A)$, $cZ_{\alpha}(A)\in Z_{\alpha}(A)$. This implies that $H_{\alpha^{-1}}(A)Z_{\alpha}(A)$ is an ideal of $Z(A)$. We now prove that $H_{\alpha^{-1}}(A)Z_{\alpha}(A)\subseteq H(A).$ In fact, for any $a\in A$ and $z\in Z_{\alpha}(A)$, we have $$\begin{aligned}
x_{\alpha(a)}z
&=&\sum_{i}d_{i}a\alpha(a_{i})z=\sum_{i}d_{i}a(\alpha(a_{i})z)\\
&=&\sum_{i}d_{i}a(za_i)=\sum_{i}d_{i}(az)a_i\in H(A).\end{aligned}$$
It is proved similarly for other sets.
A natural generalization of $H_{\alpha^{-1}}(A)$ is $$\{\sum_{i}d_{i}a\alpha^{m}(a_{i})\mid a\in A\}.$$ It is easy to check that for all $b\in A$, $$\sum_{i}d_{i}a\alpha^{m}(a_{i})\alpha^{m}(b)=b\sum_{i}d_{i}a\alpha^{m}(a_{i}).$$
Dual bases of Frobenius cellular algebras
=========================================
Cellular algebras were introduced by Graham and Lehrer in 1996. The definition of a cellular algebra is as follows.
[[@GL]]{}\[3.1\] Let $R$ be a commutative ring with identity. An associative unital $R$-algebra is called a cellular algebra with cell datum $(\Lambda, M, C, i)$ if the following conditions are satisfied:
1. The finite set $\Lambda$ is a poset. Associated with each ${\lambda}\in\Lambda$, there is a finite set $M({\lambda})$. The algebra $A$ has an $R$-basis $\{C_{S,T}^{\lambda}\mid S,T\in
M({\lambda}),{\lambda}\in\Lambda\}$.
2. The map $i$ is an $R$-linear anti-automorphism of $A$ with $i^{2}=id$ which sends $C_{S,T}^{\lambda}$ to $C_{T,S}^{\lambda}$.
3. If ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$, then for any element $a\in A$, we have\
$$aC_{S,T}^{\lambda}\equiv\sum_{S^{'}\in
M({\lambda})}r_{a}(S',S)C_{S^{'},T}^{{\lambda}} \,\,\,\,(\mod
A(<{\lambda})),$$ where $r_{a}(S^{'},S)\in R$ is independent of $T$ and where $A(<{\lambda})$ is the $R$-submodule of $A$ generated by $\{C_{S'',T''}^\mu \mid \mu<{\lambda}, S'',T''\in M(\mu)\}$.
If we apply $i$ to the equation in [(C3)]{}, we obtain
[$\rm(C3')$]{} $C_{T,S}^{\lambda}i(a)\equiv\sum_{S^{'}\in
M({\lambda})}r_{a}(S^{'},S)C_{T,S^{'}}^{{\lambda}} \,\,\,\,(\mod A(<{\lambda})).$
It is easy to check that $$C_{S,T}^{\lambda}C_{U,V}^{\lambda}\equiv\Phi(T,U)C_{S,V}^{\lambda}\,\,\,\, (\rm mod\,\,\, A(<{\lambda})),$$ where $\Phi(T,U)\in R$ depends only on $T$ and $U$. We say ${\lambda}\in\Lambda_0$ if there exist $S,T\in M({\lambda})$ such that $\Phi(S,T)\neq 0$.
Let $A$ be a finite dimensional Frobenius cellular $K$-algebra with a cell datum $(\Lambda, M, C, i)$. Given a non-degenerate bilinear form $f$, denote the left dual basis by $D=\{D_{S,T}^{\lambda}\mid S,T\in M({\lambda}),{\lambda}\in\Lambda\}$, which satisfies $$\tau(D_{U,V}^{\mu}C_{S,T}^{{\lambda}})=\delta_{{\lambda}\mu}\delta_{SV}\delta_{TU}.$$ Denote the right dual basis by $d=\{d_{S,T}^{\lambda}\mid S,T\in
M({\lambda}),{\lambda}\in\Lambda\}$, which satisfies $$\tau(C_{S,T}^{{\lambda}}d_{U,V}^{\mu})=\delta_{{\lambda}\mu}\delta_{S,V}\delta_{T,U}.$$ For $\mu\in\Lambda$, let $A_{d}(>\mu)$ be the $K$-subspace of $A$ generated by $$\{d_{X,Y}^\epsilon \mid X,Y\in M(\epsilon),\mu<\epsilon\}$$ and let $A_{D}(>\mu)$ be the $K$-subspace of $A$ generated by $$\{D_{X,Y}^\epsilon \mid X,Y\in M(\epsilon),\mu<\epsilon\}.$$ Note that the $K$-linear map $\alpha$ which sends $d_{X,Y}^\epsilon$ to $D_{X,Y}^\epsilon$ is a Nakayama automorphism of the algebra $A$. The following result is easy, we omit the proof.
Keep notations as above. Then $\{\alpha(C_{S,T}^{{\lambda}})\mid
{\lambda}\in\Lambda, S,T\in M({\lambda})\}$ is a cellular basis if and only if $i\alpha=\alpha i$.
We obtained the following lemma in [@LZ] about structure constants. It will play an important role in this paper.
\[3.2\] For arbitrary ${\lambda},\mu\in\Lambda$ and $S,T,P,Q\in M({\lambda})$, $U,V\in M(\mu)$ and $a\in A$, the following hold:
1. $D_{U,V}^{\mu}C_{S,T}^{{\lambda}}=\sum\limits_{\epsilon\in
\Lambda, X,Y\in
M(\epsilon)}r_{(S,T,{\lambda}),(Y,X,\epsilon),(V,U,\mu)}D_{X,Y}^{\epsilon}.$
2. $D_{U,V}^{\mu}C_{S,T}^{{\lambda}}=\sum\limits_{\epsilon\in
\Lambda, X,Y\in
M(\epsilon)}R_{(Y,X,\epsilon),(U,V,\mu),(T,S,{\lambda})}C_{X,Y}^{\epsilon}.$
3. $aD_{U,V}^{\mu}\equiv \sum\limits_{U'\in
M(\mu)}r_{i(\alpha^{-1}(a))}(U,U')D_{U',V}^{\mu}\,\,\,(\mod
A_{D}(>\mu))$.
4. $D_{P,Q}^{{\lambda}}C_{S,T}^{{\lambda}}=0\,\,\,\, if \,\,\,Q\neq
S.$
5. $D_{U,V}^{\mu}C_{S,T}^{{\lambda}}=0 \,\,\,\,if
\,\,\,\mu\nleq {\lambda}.$
6. $D_{T,S}^{{\lambda}}C_{S,Q}^{{\lambda}}=D_{T,P}^{{\lambda}}C_{P,Q}^{{\lambda}}.$
7. $C_{S,T}^{{\lambda}}d_{U,V}^{\mu}=\sum\limits_{\epsilon\in
\Lambda, X,Y\in
M(\epsilon)}r_{(Y,X,\epsilon),(S,T,{\lambda}),(V,U,\mu)}d_{X,Y}^{\epsilon}.$
8. $C_{S,T}^{{\lambda}}d_{U,V}^{\mu}=\sum\limits_{\epsilon\in
\Lambda, X,Y\in
M(\epsilon)}R_{(U,V,\mu),(Y,X,\epsilon),(T,S,{\lambda})}C_{X,Y}^{\epsilon}.$
9. $d_{U,V}^{\mu}a\equiv \sum\limits_{V'\in
M(\mu)}r_{\alpha(a)}(V,V')d_{U,V'}^{\mu}\,\,\,\,\,\,\,\,\,\,\,(\mod
A_{d}(>\mu))$.
10. $C_{S,T}^{{\lambda}}d_{P,Q}^{{\lambda}}=0\,\, if \,\,T\neq P.$
11. $C_{S,T}^{{\lambda}}d_{U,V}^{\mu}=0 \,\,\,\,if\,\,\,
\mu\nleq {\lambda}.$
12. $C_{S,T}^{{\lambda}}d_{T,P}^{{\lambda}}=C_{S,Q}^{{\lambda}}d_{Q,P}^{{\lambda}}.$
\[3.3\] We have from Lemma \[3.2\] that if $i(d_{S,T}^{{\lambda}})=d_{T,S}^{{\lambda}}$ for arbitrary ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$, then $\{d_{S,T}^{{\lambda}}\mid{\lambda}\in\Lambda,
S,T\in M({\lambda})\}$ is again cellular with respect to the opposite order on $\Lambda$. The similar claim also holds for the left dual basis.
In [@L2], we obtained a result about the relationship among $i$, $\tau$ and $\alpha$. For the convenience of the reader, we copy it here.
[@L2 Theorem 2.5]\[3.4\] Let $A$ be a Frobenius cellular algebra with cell datum $(\Lambda,
M, C, i)$. Then one of the following three statements holds implies that the other two are equivalent:
1. $i(d_{S,T}^{{\lambda}})=d_{T,S}^{{\lambda}}$ and $i(D_{S,T}^{{\lambda}})=D_{T,S}^{{\lambda}}$ for arbitrary ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$.
2. $\alpha=id$, that is, $A$ is a symmetric algebra.
3. $\tau(a)=\tau(i(a))$ for all $a\in A$.
\[3.5\] In the proof of $(1),(3)\Rightarrow (2)$ in [@L2], we only need that $i(D_{S,T}^{{\lambda}})=D_{T,S}^{{\lambda}}$ or $i(d_{S,T}^{{\lambda}})=d_{T,S}^{{\lambda}}$ for arbitrary ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$.
The results of Lemma \[3.4\] can be partially developed as follows.
\[3.6\] If $i(D_{S,T}^{{\lambda}})=D_{T,S}^{{\lambda}}$ for arbitrary ${\lambda}\in\Lambda$ and $S,T\in M({\lambda})$, then $\tau(a)=\tau(i(a))$ for all $a\in A$.
Suppose that $\tau(a)\neq\tau(i(a))$ for some $a\in A$. Then there exists $C_{S,T}^{{\lambda}}$ such that $\tau(C_{S,T}^{{\lambda}})\neq\tau(C_{T,S}^{{\lambda}})$. Let $1=\sum_{\epsilon\in \lambda,\,\, X,Y\in
M(\epsilon)}r_{X,Y,\epsilon}D_{X,Y}^{\epsilon}$. Applying $\alpha^{-1}$ on both sides yields that $1=\sum_{\epsilon\in
\lambda,\,\, X,Y\in
M(\epsilon)}r_{X,Y,\epsilon}d_{X,Y}^{\epsilon}$. Then $$\tau(C_{S,T}^{{\lambda}})=\tau(\sum_{\epsilon\in \lambda,\,\, X,Y\in
M(\epsilon)}r_{X,Y,\epsilon}D_{X,Y}^{\epsilon}C_{S,T}^{{\lambda}})=r_{T,S,{\lambda}},$$ $$\tau(C_{T,S}^{{\lambda}})=\tau(C_{T,S}^{{\lambda}}\sum_{\epsilon\in \lambda,\,\, X,Y\in
M(\epsilon)}r_{X,Y,\epsilon}d_{X,Y}^{\epsilon})=r_{S,T,{\lambda}}.$$ This implies that $r_{T,S,{\lambda}}\neq r_{S,T,{\lambda}}$.
On the other hand, it follows from $i(D_{S,T}^{{\lambda}})=D_{T,S}^{{\lambda}}$ that $$1=i(1)=\sum_{\epsilon\in
\lambda,\,\, X,Y\in
M(\epsilon)}r_{Y,X,\epsilon}D_{X,Y}^{\epsilon}.$$ This forces $r_{X,Y,\epsilon}=r_{Y,X,\epsilon}$ for all $\epsilon\in \Lambda$, $X,Y\in M(\epsilon)$. It is a contradiction. This implies that $\tau(a)=\tau(i(a))$ for all $a\in A$.
Combining Remarks \[3.3\], \[3.5\] with Lemma \[3.6\] yields one main result of this section.
\[3.7\] Let $A$ be a finite dimensional Frobenius cellular algebra with cell datum $(\Lambda, M, C, i)$. If the right (left) dual basis is again cellular, then $A$ is symmetric.
In symmetric case, the condition $\tau(a)=\tau(i(a))$ is equivalent to that the dual basis being cellular. However, in Frobenius case, $\tau(a)=\tau(i(a))$ does not implies the cellularity of dual bases. We will give a counterexample in Section 4.
Now let us study Nakayama twisted centers of Frobenius cellular algebras by using dual bases.
Let $A$ be a finite dimensional Frobenius cellular algebra. For arbitrary ${\lambda}, \,\mu\in \Lambda$, $S,\,T\in M({\lambda})$, $U,\,V\in
M(\mu)$, write $$C_{S,T}^{{\lambda}}C_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in M(\epsilon)}
r_{(S,T,{\lambda}),(U,V,\mu),(X,Y,\epsilon)}C_{X,Y}^{\epsilon},$$
$$D_{S,T}^{{\lambda}}D_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in M(\epsilon)}
R_{(S,T,{\lambda}),(U,V,\mu),(X,Y,\epsilon)}D_{X,Y}^{\epsilon}.$$ Applying $\alpha^{-1}$ on both sides of the above equation, we obtain $$d_{S,T}^{{\lambda}}d_{U,V}^{\mu}=\sum\limits_{\epsilon\in\Lambda,X,Y\in M(\epsilon)}
R_{(S,T,{\lambda}),(U,V,\mu),(X,Y,\epsilon)}d_{X,Y}^{\epsilon}.$$
For any ${\lambda}\in\Lambda$ and $T\in M({\lambda})$, set $$e_{{\lambda}}=\sum_{S\in
M({\lambda})}C_{S,T}^{{\lambda}}d_{T,S}^{{\lambda}}\,\,\,\,{\rm
and}\,\,\,\,L_{\alpha}(A)=\{\sum_{{\lambda}\in\Lambda}r_{{\lambda}}e_{{\lambda}}\mid
r_{{\lambda}}\in R\}.$$
\[3.9\] With the notations as above. Then $H_{\alpha}(A)\subseteq
L_{\alpha}(A)\subseteq Z_{\alpha}(A)$. Moreover, $\dim
L_{\alpha}(A)\geq |\Lambda_0|$.
The proof of $H_{\alpha}(A)\subseteq L_{\alpha}(A)$ is similar to that of [@L1 Theorem 3.2] by using Lemma \[3.2\]. We omit the details here. Now let us prove $L_{\alpha}(A)\subseteq
Z_{\alpha}(A)$. Clearly, we only need to show that $e_{{\lambda}}\alpha^{-1}(C_{U,V}^{\mu})=C_{U,V}^{\mu}e_{{\lambda}}$ for arbitrary ${\lambda},\mu\in\Lambda$ and $U,V\in M(\mu)$.
On one hand, by the definition of $\alpha$ and Lemma \[3.2\], $$\begin{aligned}
e_{{\lambda}}\alpha^{-1}(C_{U,V}^{\mu})&=&\sum_{S\in
M({\lambda})}C_{S,T}^{{\lambda}}\alpha^{-1}(D_{T,S}^{{\lambda}}C_{U,V}^{\mu})\\&=&\sum_{S\in
M({\lambda})}\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(U,V,\mu),(Y,X,\epsilon),(S,T,{\lambda})}C_{S,T}^{{\lambda}}d_{X,Y}^{\epsilon}\\
&=&\sum_{S,Y\in
M({\lambda})}r_{(U,V,\mu),(Y,T,{\lambda}),(S,T,{\lambda})}C_{S,T}^{{\lambda}}d_{T,Y}^{{\lambda}}.\end{aligned}$$ On the other hand, $$\begin{aligned}
C_{U,V}^{\mu}e_{{\lambda}}&=&\sum_{S\in
M({\lambda})}\sum_{\epsilon\in\Lambda,X,Y\in
M(\epsilon)}r_{(U,V,\mu),(S,T,{\lambda}),(X,Y,\epsilon)}C_{X,Y}^{\epsilon}d_{T,S}^{{\lambda}}\\
&=&\sum_{S,X\in
M({\lambda})}r_{(U,V,\mu),(S,T,{\lambda}),(X,T,{\lambda})}C_{X,T}^{{\lambda}}d_{T,S}^{{\lambda}}.\end{aligned}$$ The proof of $\dim L_{\alpha}(A)\geq |\Lambda_0|$ is similar to that of [@L1 Proposition 3.3 (2)].
Then [@L1 Theorem 3.2] becomes a corollary of Lemma \[3.9\].
If we define $e'_{{\lambda}}=\sum_{S\in
M({\lambda})}D_{S,S}^{{\lambda}}C_{S,S}^{{\lambda}}$ and $$L_{\alpha}(A)'=\{\sum_{{\lambda}\in\Lambda}r_{{\lambda}}e'_{{\lambda}}\mid r_{{\lambda}}\in
K\},$$ then there are analogous results on $L_{\alpha}(A)'$. However, $L_{\alpha}(A)\neq L_{\alpha}(A)'$ in general. We will give an example in Section 4.
Certainly, another class of elements of $A$ would not be ignore, that is, $\sum_{S\in M({\lambda})}C_{S,T}^{{\lambda}}D_{T,S}^{{\lambda}}$. Unfortunately, we know from Lemma \[3.2\] that $C_{S,T}^{{\lambda}}D_{T,S}^{{\lambda}}$ is more complex than $D_{T,S}^{{\lambda}}C_{S,T}^{{\lambda}}$ and hence we do not know more properties about them so far.
The next aim of this section is to construct ideals of the center for a Frobenius cellular algebra. We have the following result.
\[3.11\] Let $x\in Z_{\alpha^{-1}}(A)$. Then $\sum_{S\in
M({\lambda})}C_{S,T}^{{\lambda}}xD_{T,S}^{{\lambda}}\in Z(A)$ for arbitrary ${\lambda}\in\Lambda$. Furthermore, $$Z_{{\lambda}}(A):=\{\sum_{S\in
M({\lambda})}C_{S,T}^{{\lambda}}xD_{T,S}^{{\lambda}}\mid x\in
Z_{\alpha^{-1}}(A)\}$$ are ideals of $Z(A)$ with $Z_{{\lambda}}(A)Z_{\mu}(A)=0$ if ${\lambda}\neq\mu$.
It follows from $x\in Z_{\alpha^{-1}}(A)$ that $$\sum_{S\in
M({\lambda})}C_{S,T}^{{\lambda}}xD_{T,S}^{{\lambda}}=e_{{\lambda}}x.$$ Then Lemma \[2.5\] and \[3.9\] imply that $e_{{\lambda}}x\in Z(A)$. Note that $cx, xc\in Z_{\alpha^{-1}}(A)$ for $c\in Z(A)$ and $x\in
Z_{\alpha^{-1}}(A)$. Thus $Z_{{\lambda}}(A)$ is an ideal of $Z(A)$. Moreover, if ${\lambda}\neq\mu$, then Lemma \[3.2\] implies that $Z_{{\lambda}}(A)Z_{\mu}(A)=0$.
Denote by $Z_{\Lambda}(A)$ the central ideal generated by $\cup_{{\lambda}\in\Lambda} Z_{{\lambda}}(A)$. What is the relationship between $Z_{\Lambda}(A)$ and $H(A)$?
Examples
========
As is well known, all symmetric algebras are Frobenius. However, we have rarely found examples of non-symmetric Frobenius cellular algebras. In this section, we will give some such examples. The first one comes from local algebras.
(Nakayama-Nesbitt) Let $K$ be a field with two nonzero elements $u$, $v$ such that $u\neq v$. Let $A\subset M_{4\times 4}(K)$ be a $K$-algebra with a basis $$\{C_{1,1}^1=E_{14}, C_{1,1}^2=E_{13}+uE_{24},
C_{1,1}^3=E_{12}+vE_{34}, C_{1,1}^4=E.\} \eqno(\ast)$$ It is proved that $A$ is a non-symmetric Frobenius algebra. For details, see [@LT]. Moreover, the basis $(\ast)$ is clearly cellular. Thus $A$ is a non-symmetric Frobenius cellular algebra. Note that $A$ is symmetric if $u=v\neq 0$.
Let $K$ be a field and $Q$ be the following quiver $$\xymatrix@C=13mm{
^{1}\bullet \ar@<2.0pt>[dr]^{\alpha_1} & & \bullet^{2}\ar@<2.0pt>[dl]^(0.5){\alpha_2}\\
& \bullet^0\ar@<2.0pt>[ul]^(0.5){\beta_1}\ar@<2.0pt>[ur]^(0.5){\beta_2}\ar@<2.0pt>[d]^(0.5){\beta_3} & \\
& \bullet_{3}\ar@<2.0pt>[u]^(0.5){\alpha_3}
}$$ with relation $\rho$ given as follows: $$\alpha_1\beta_1, \alpha_1\beta_2, \alpha_2\beta_1,
\alpha_2\beta_3, \alpha_3\beta_2, \alpha_3\beta_3,
\beta_1\alpha_1-\beta_2\alpha_2, \beta_2\alpha_2-\beta_3\alpha_3.$$ It is easy to check that $A=K(Q,\rho)$ is a Frobenius algebra with Nakayama permutation $(13)$, that is, $A$ is a non-symmetric Frobenius algebra. We claim that $A$ is also cellular. Here is a cellular basis of it. $$\begin{matrix}
\begin{matrix} e_3 & \alpha_3\\ \beta_3 &
e_0\end{matrix} \,\,;\quad &
\begin{matrix}
e_1 & \alpha_1 & \alpha_1\beta_3\\
\beta_1 & \beta_1\alpha_1 & \beta_2\\
\alpha_3\beta_1 & \alpha_2 & e_2
\end{matrix} \,\,;\quad & \begin{matrix} \alpha_2\beta_2
\end{matrix}.
\end{matrix}$$
Define $\tau$ by\
(1) $\tau(e_{1})=\cdots=\tau(e_{4})=0$;\
(2) $\tau(\alpha_{1}\beta_{3})=\tau(\alpha_{3}\beta_{1})=\tau(\alpha_{2}\beta_{2})=\tau(\beta_1\alpha_1)=1$,\
(3)$\tau(\alpha_{i})=\tau(\beta_{i})=0$, $i=1,\cdots, 4$.\
The right dual basis is $$\begin{matrix}
\begin{matrix} \alpha_3\beta_1 & \beta_1\\ \alpha_3 &
\beta_1\alpha_1\end{matrix} \,\,;\quad &
\begin{matrix}
\alpha_1\beta_3 & \beta_3 & e_3\\
\alpha_1 & e_0 & \alpha_2\\
e_1 & \beta_2 & \alpha_2\beta_2
\end{matrix} \,\,;\quad & \begin{matrix} e_2
\end{matrix}.
\end{matrix}$$ The left dual basis is $$\begin{matrix}
\begin{matrix} \alpha_1\beta_3 & \beta_3\\ \alpha_1 &
\beta_1\alpha_1\end{matrix} \,\,;\quad &
\begin{matrix}
\alpha_3\beta_1 & \beta_1 & e_1\\
\alpha_3 & e_0 & \alpha_2\\
e_3 & \beta_2 & \alpha_2\beta_2
\end{matrix} \,\,;\quad & \begin{matrix} e_2
\end{matrix}.
\end{matrix}$$
Let us give some remarks on this example.
1\. A direct computation yields that $L_{\alpha}(A)$ is a $K$-space of dimension 3 with a basis $\{\alpha_3\beta_1+\beta_1\alpha_1,
\alpha_1\beta_3, \alpha_2\beta_2\}$, that $L_{\alpha}(A)'$ is a $K$-space generated by $\{3\alpha_3\beta_1, 2\alpha_1\beta_3,
\alpha_2\beta_2\}$. This implies that $L_{\alpha}(A)\neq
L_{\alpha}(A)'$. Moreover, if $K$ is of characteristic 3, then $L_{\alpha}(A)'\subsetneqq L_{\alpha}(A)$. Note that $H_{\alpha}(A)\subset L_{\alpha}(A)'\cap L_{\alpha}(A)$. Thus if $K$ is of characteristic not 3, then $H_{\alpha}(A)\neq
L_{\alpha}(A)'$ and $H_{\alpha}(A)\neq L_{\alpha}(A)$.
2\. By a direct computation we obtain that $i\alpha=\alpha i$.
3\. For $S, T, U, V\in M({\lambda})$, $d_{S,T}^{{\lambda}}C_{U,V}^{{\lambda}}$ may not be 0 when $T\neq U$. For example, $d_{13}^2C_{12}^2=\alpha_1$. Furthermore, let ${\lambda}, \mu\in \Lambda$ with $\mu\nleq{\lambda}$, $d_{P,Q}^{\mu}C_{U,V}^{{\lambda}}$ may not be 0. For example, $d_{11}^3C_{32}^2=\alpha_2$.
4\. Define a new involution $i'$ on $A$ by $$\begin{matrix}
i(\alpha_1)=\beta_3 & & i(\alpha_2)=\alpha_2\\
i(\alpha_3)=\beta_1 & & i(e_0)=e_0\\
i(e_1)=e_3 & & i(e_2)=e_2
\end{matrix}.$$ Then the (left) right dual basis is cellular.
The next example can be found in [@KX]. We illustrate it here for the sake of giving some remarks.
Let $K$ be a field. Let us take ${\lambda}\in K$ with ${\lambda}\neq 0$ and ${\lambda}\neq 1$. Let $$A=K<a, b, c, d>/I,$$ where $I$ is generated by $$a^2, b^2, c^2, d^2, ab, ac, ba, bd, ca, cd, db, dc, cb-{\lambda}bc,
ad-bc, da-bc.$$ If we define $\tau$ by $\tau(1)=\tau(a)=\tau(b)=\tau(c)=\tau(d)=0$ and $\tau(bc)=1$ and define an involution $i$ on $A$ to be fixing $a$ and $d$, but interchanging $b$ and $c$, then $A$ is a Frobenius cellular algebra with a cellular basis $$\begin{matrix}
\begin{matrix} bc \end{matrix} ;&
\begin{matrix} a & b\\ c &
d\end{matrix} ; &
\begin{matrix} 1 \end{matrix}.
\end{matrix}$$ The right dual basis is $$\begin{matrix}
\begin{matrix} 1 \end{matrix} ;&
\begin{matrix} d & c\\ b/{\lambda}&
a\end{matrix} ; &
\begin{matrix} bc \end{matrix}.
\end{matrix}$$ The left dual basis is $$\begin{matrix}
\begin{matrix} 1 \end{matrix} ;&
\begin{matrix} d & c/{\lambda}\\ b &
a\end{matrix} ; &
\begin{matrix} bc \end{matrix}.
\end{matrix}$$
1\. Clearly, the matrix associated with $\alpha$ with respect to the right dual basis is $$\left(
\begin{array}
[c]{cccccc}1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1/\lambda & 0 & 0 & 0 \\
0 & 0 & 0 & \lambda & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right).$$ Then the finiteness of the order of $\alpha$ is independent of cellularity of $A$. In fact, let $K=\mathbb{C}$. If ${\lambda}$ is not a root of unity, then the order of $\alpha$ is infinite. Otherwise, the order of $\alpha$ is finite.
2\. The example implies that $\tau(a)=\tau(i(a))$ is not a sufficient condition such that dual bases being cellular in Frobenius case.
3\. We have from this example that in general $Z(A)\cap
Z_{\alpha}(A)\neq \{0\}$. In fact, $bc\in Z(A)$ is clear. On the other hand, $bc=C_{11}^1d_{11}^1\in Z_{\alpha}(A)$.
4\. It is easy to check that $i\alpha\neq \alpha i$ and thus $\{\alpha(C_{S,T}^{{\lambda}})\mid {\lambda}\in\Lambda, S,T\in M({\lambda})\}$ is not cellular.
5\. Note that $d_{1,1}^{2}C_{1,1}^{2}=da$ and $d_{1,2}^{2}C_{2,1}^{2}=bc/{\lambda}$. This implies that $d_{S,T}^{{\lambda}}C_{T,S}^{{\lambda}}\neq d_{S,U}^{{\lambda}}C_{U,S}^{{\lambda}}$ in general.
[**Acknowledgement**]{} The author would like to express his sincere thanks to Chern Institute of Mathematics of Nankai University for the hospitality during his visit. He also acknowledges Prof. Chengming Bai for his kind consideration and warm help.
C.W. Curtis and I. Reiner, [*Representation theory of finite groups and associative algebras*]{}, Interscience, New York, 1964.
M. Geck, [*Hecke algebras of finite type are cellular*]{}, Invent. math., **169** (2007), 501-517.
J.J. Graham and G.I. Lehrer, [*Cellular algebras*]{}, Invent. Math., [**123**]{} (1996), 1-34.
D. G. Higman, [*On orders in separable algebras*]{}, Can. J. Math., [**7**]{} (1955), 509-515.
T. Holm and A. Zimmermann, [*Deformed preprojective algebras of type L: Külshammer spaces and derived equivalences*]{}, J. Algebra, [**346**]{} (2011), 116-146.
S. Koenig and C. C. Xi, [*A self-injective cellular algebra is weakly symmetric*]{}, J. Algebra, [**228**]{} (2000), 51-59.
T.Y. Lam, [*Lectures on modules and rings*]{} GTM 189. Spring-Verlag, New York, 1998.
Yanbo Li, [*Centers of symmetric cellular algebras*]{}, Bull. Aust. Math. Soc., [**82**]{} (2010) 511-522.
Yanbo Li, [*Nakayama automorphisms of Frobenius cellular algebras*]{}, Bull. Aust. Math. Soc., [**86**]{} (2012), 515-524.
Yanbo Li, [*Radicals of symmetric cellular algebras*]{}, Colloq. Math., accepted (2013), 1-18. arxiv: math0806.1532.
Yanbo Li and Z. Xiao, [*On cell modules of symmetric cellular algebra*]{}, Monatsh. Math., [**168**]{} (2012), 49-64.
Yanbo Li and Deke Zhao, [*Projective cell modules of Frobenius cellular algebras*]{}, Monatsh. Math., accepted (2013)
C.C. Xi, [*Partition algebras are cellular*]{}, Compositio math., **119**, (1999), 99-109.
C.C. Xi, [*On the quasi-heredity of Birman-Wenzl algebras*]{}, Adv. Math., **154**, (2000), 280-298.
[^1]: Date: September 20, 2013
[^2]: This work is supported by Fundamental Research Funds for the Central Universities (N110423007) and the Natural Science Foundation of Hebei Province, China (A2013501055).
|
---
abstract: 'Energetic particle irradiation of solids can cause surface ultra-smoothening [@yamada-etal-2001-MSER], self-organized nanoscale pattern formation [@chan-chason-JAP-2007], or degradation of the structural integrity of nuclear reactor components [@baldwin-doerner-NF-2008]. Periodic patterns including high-aspect ratio quantum dots [@facsko-etal-SCIENCE-1999], with occasional long-range order [@ziberi-etal-PRB-2005] and characteristic spacing as small as 7 nm [@wei-etal-2008-CPL], have stimulated interest in this method as a means of sub-lithographic nanofabrication [@cuenat-etal-2005-AM]. Despite intensive research there is little fundamental understanding of the mechanisms governing the selection of smooth or patterned surfaces, and precisely which physical effects cause observed transitions between different regimes [@ziberi-etal-APL-2008; @madi-etal-2008-PRL] has remained a matter of speculation [@davidovitch-etal-PRB-2007]. Here we report the first prediction of the mechanism governing the transition from corrugated surfaces to flatness, using only parameter-free molecular dynamics simulations of single-ion impact induced crater formation as input into a multi-scale analysis, and showing good agreement with experiment. Our results overturn the paradigm attributing these phenomena to the removal of target atoms via sputter erosion. Instead, the mechanism dominating both stability and instability is shown to be the impact-induced redistribution of target atoms that are not sputtered away, with erosive effects being essentially irrelevant. The predictions are relevant in the context of tungsten plasma-facing fusion reactor walls which, despite a sputter erosion rate that is essentially zero, develop, under some conditions, a mysterious nanoscale topography leading to surface degradation. Our results suggest that degradation processes originating in impact-induced target atom redistribution effects may be important, and hence that an extremely low sputter erosion rate is an insufficient design criterion for morphologically stable solid surfaces under energetic particle irradiation.'
author:
- 'Scott A. Norris$^{1,3}$, Juha Samela$^{2}$, Laura Bukonte$^{2}$, Marie Backman$^{2}$, Djurabekova Flyura$^{2}$, Kai Nordlund$^{2}$, Charbel S. Madi$^{3}$, Michael P. Brenner$^{3}$, & Michael J. Aziz$^{3}$'
bibliography:
- 'tagged-bibliography.bib'
title: 'MD-Predicted Phase diagrams for Pattern Formation due to Ion Irradiation'
---
Southern Methodist University, Dallas, TX 75205
2 Department of Physics and Helsinki Institute of Physics, P.O. Box 43, FIN-0014 University of Helsinki, Helsinki, Finland
Harvard School of Engineering and Applied Sciences, Cambridge, MA 02138
At irradiation energies between $10^{2}\,\mathrm{eV}-10^{4}\,\mathrm{eV}$, the irradiation process is dominated by the nuclear collision cascade caused by ion impact [@sigmund-PR-1969; @sigmund-JMS-1973]. Displaced atoms that reach the surface with enough kinetic energy to leave are permanently sputtered away; all other displaced atoms come to rest within the solid or on the surface after phonon emission times of $\sim10^{-12}$ seconds. These processes contribute *prompt* *erosive* [@bradley-harper-JVST-1988; @makeev-etal-NIMB-2002] and *prompt redistributive* [@carter-vishnyakov-PRB-1996; @moseler-etal-SCIENCE-2005; @davidovitch-etal-PRB-2007] components of morphology evolution, respectively, and are collectively denoted $P\left[\mathbf{x}\right]$. For most materials other than elemental metals, the damage resulting from these collisions quickly ($\sim10^{-3}$ seconds) leads to the amorphization of a thin layer of target material. Over much longer time scales ($\sim100$ seconds), mass transport by kinetic relaxation processes causes a *gradual* *relaxational* effect [@bradley-harper-JVST-1988; @umbach-etal-PRL-2001]. Hence, to the prompt term $P\left[\mathbf{x}\right]$ we add a phenomenological term for the gradual relaxation regime denoted $G\left[\mathbf{x}\right]$, assuming a mechanism of ion-enhanced viscous flow, which is expected to predominate in irradiated amorphous materials near room temperature [@umbach-etal-PRL-2001]. The prompt and gradual contributions to the rate of motion of the surface in the normal direction $v_{\mathbf{n}}$ superpose: $$v_{\mathbf{n}}=P\left[\mathbf{x}\right]+G\left[\mathbf{x}\right].\label{eqn: p-and-g}$$
The prompt regime may be characterized using molecular dynamics (MD) simulations [@moseler-etal-SCIENCE-2005; @kalyanasundaram-etal-APL-2008] or experimental methods [@costantini-etal-PRL-2001]. Given data from many impact events, we may obtain the “crater function” $\Delta h\left(\mathbf{x}-\mathbf{x}',\,\theta\right)$ describing the average change in local surface height at a point $\mathbf{x}$ resulting from a single-ion impact at $\mathbf{x}'$, with incidence angle $\theta$. We then upscale the crater function into a continuum partial differential equation for the surface evolution using a multi-scale framework. The theoretical formalism for this process is described elsewhere [@norris-etal-2009-JPCM]; here we provide a brief summary of the important points for the linear case. Given the crater function $\Delta h$ and the flux distribution $I\left(\mathbf{x}\right)$, we write the prompt contribution to surface evolution as a flux-weighted integral of the crater function [@aziz-MfM-2006; @davidovitch-etal-PRB-2007]: $$P\left[\mathbf{x}\right]=\int I\left(\mathbf{x}'\right)\Delta h\left(\mathbf{x}-\mathbf{x}',\,\theta\right)\, d\mathbf{x}'\label{eqn: integral-formulation}$$ A well-known observation in the field is that scale of the craters is much smaller than the scale of the resulting pattern and of the flux distribution. To exploit this fact, we employ a formal multiple-scale analysis, based on a small parameter $\varepsilon$ related to the ratio of impact scales to pattern scales. This formalism allows ready separation of the spatial dependence of the crater function (fast) from that of the surface shape (slow), and leads eventually to an upscaled description of the prompt regime of the form$$P\left[\mathbf{x}\right]=\left(IM^{\left(0\right)}\right)+\varepsilon\nabla_{\mathbf{S}}\cdot\left(IM^{\left(1\right)}\right)+\frac{1}{2}\varepsilon^{2}\nabla_{\mathbf{S}}\cdot\nabla_{\mathbf{S}}\cdot\left(IM^{\left(2\right)}\right)+\dots.\label{eqn: p-result}$$ Here the $\nabla_{\mathbf{S}}$ represent surface divergences, and the (increasing-order) tensors $M^{\left(i\right)}$ are simply the angle-dependent *moments* of the crater function $\Delta h$. This compact formulation is interesting for two reasons. First, the moments are readily obtainable directly via MD simulation, and they converge with far fewer trials than do descriptions of the entire crater function. Second, while atomistic methods have been used in the past to obtain the amplitude of a single term in a PDE obtained via phenomenological modeling [@enrique-bellon-PRL-2000; @moseler-etal-SCIENCE-2005; @zhou-etal-PRB-2008; @headrick-zhou-JPCM-2009], we believe this is the first derivation of an entire PDE from molecular dynamics results.
Equation describes the prompt regime $P\left[\mathbf{x}\right]$. To fully capture the surface dynamics, we add to this a relaxation mechanism $G\left[\mathbf{x}\right]$ associated with ion-enhanced viscous flow [@umbach-etal-PRL-2001]. Together, $P\left[\mathbf{x}\right]$ and $G\left[\mathbf{x}\right]$ completely determine surface morphology evolution via Equation . From this (nonlinear) equation, pattern-forming predictions are then obtained by examining stability of the the *linearized* equation as a function of the laboratory incidence angle $\theta$. The derivation follows that in [@bradley-harper-JVST-1988], and in an appropriate frame of reference one finds that the magnitude of infinitesimal perturbations $h$ away from a flat surface evolve, to leading order in $\varepsilon$, according to the PDE $$\frac{\partial h}{\partial t}=\left(S_{X}\left(\theta\right)\frac{\partial^{2}h}{\partial x^{2}}+S_{Y}\left(\theta\right)\frac{\partial^{2}h}{\partial y^{2}}\right)-B\nabla^{4}h,\label{eqn: linear-stability}$$ where the angle-dependent coefficients $$\begin{aligned}
S_{X}\left(\phi\right) & = I_0 \frac{d}{d\phi}\left[M^{\left( 1 \right)}\left(\phi\right)\cos\left(\phi\right)\right]\\
S_{Y}\left(\phi\right) & = I_0 M^{\left( 1 \right)}\left(\phi\right)\cos\left(\phi\right)\cot\left(\phi\right)
\end{aligned}
.\label{eqn: moment-to-linear-coeff}$$ are determined from the *first* moments obtained via MD, and the constant coefficient $B$ is estimated from independent experiments. The structure of Equation indicates that linear stability is determined strictly by the signs of the calculated coefficients $\left(S_{X},S_{Y}\right)$: for values of $\theta$ where either of these coefficients is negative, linearly unstable modes exist and we expect patterns, whereas for values of $\theta$ where they are both positive we expect flat, stable surfaces.
Existing uses of MD crater data for investigations of surface pattern-forming are entirely numerical in nature [@kalyanasundaram-etal-JPCM-2009], and could be viewed as a scheme for *numerically* integrating Equation \[eqn: integral-formulation\]. In contrast, our *analytical* upscaling of Equation \[eqn: integral-formulation\] illustrates exactly which qualities of the crater – namely, its moments – play the dominant role in surface evolution. Furthermore, this analytical form can be linearized, allowing predictions of stability boundaries, and changes to those boundaries as crater shape is varied. A crucial component of our approach is that the crater function $\Delta h$ – and hence the moments $M^{\left(i\right)}$ – contains the contributions of both erosion and mass redistribution. Whereas these effects have traditionally been treated separately via unrelated phenomenological models, viewing the crater function as fundamental integrates erosion and redistribution into a unified description, allowing both processes to be treated identically and readily separated and compared. Indeed, this approach has permitted us to confirm for the first time conjectures [@aziz-MfM-2006; @davidovitch-etal-PRB-2007; @kalyanasundaram-etal-APL-2008] that the stability of irradiated surfaces could be dominated by redistributive effects, with erosion – long assumed to be the source of roughening – being essentially irrelevant.
We obtain angle-dependent moments from a series of MD simulations, in an environment consisting of an amorphous, 20x20x10 nm Si target consisting of 219,488 atoms created using the Wooten/Winer/Weaire (WWW) method [@wooten-winer-weaire-1985-PRL], and then annealed with the EDIP potential [@bazant-kaxiras-justo-1997-PRB]. This gives an optimized amorphous structure where most of the Si atoms have coordination number 4. The target was then bombarded with Ar at 100 eV and 250 eV. During bombardment, the interaction between Si atoms was again described using the EDIP potential, which gives a good agreement between simulated and experimental sputtering yields [@samela-etal-2007-NIMB], whereas the Ar-Si interaction was a potential calculated for the Ar-Si dimer [@nordlund-etal-NIMB-1997]. The kinetic energy was gradually removed during the simulations from the 1 nm borders of the substrate to prevent it re-entering the impact area via the periodic boundary conditions used in the simulations. The ambient temperature in the simulations was 0 K. The simulation arrangements and their suitability for cluster and ion bombardment simulations are discussed in more detail in the supplement and in Refs. [@nordlund-etal-1998-PRB; @ghaly-nordlund-averback-1999-PMA; @samela-etal-2005-NIMB; @samela-etal-2007-NIMB].
For each energy, two hundred impacts were simulated at each incidence angle in 5-degree increments between 0 and 90 degrees, yielding moments as summarized in Figure 1. From the initial and final atomic positions, moments were obtained using the method described in [@norris-etal-2009-JPCM]. Briefly, by assuming that densities in the amorphous layer attain a steady state (i.e., that defect distributions immediately project to the target surface), we obtain erosive moments by assigning a height loss at each location proportional to the number of sputtered atoms originating from that location, while redistributive moments were obtained by assigning height losses at initial atomic positions and height gains at final atomic positions. For use within our analytical framework, we also fit the moments to Fourier series constrained by symmetry conditions and by the observation that all moments tend to zero at $\theta=\pm90^{\circ}$. For both energies, the redistributive first moments are much larger in magnitude than – and have the opposite sign of – the erosive moments. The implication is that redistributive effects completely dominate erosive effects, except possibly at the highest (grazing) angles where all moments tend to zero.
To corroborate this finding, we calculate in Fig. 2 the coefficients in equation for the 250 eV moments, and compare the pattern wavelengths they predict to experimental observations in the same environmental conditions [@madi-etal-JPCM-2009] (clean linear experimental data at 100 eV are not currently available). The agreement is generally good at intermediate angles, but several discrepancies between theory and experiment should be addressed. First, the small quantitative difference between predicted and observed bifurcation angles – which depends only on the shape of $S_{X}\left(\theta\right)$ – could readily arise from the approximate nature of the classical potential, on which our simulations are based (unlike the bifurcation angle, precise wavelength values depend on the value of $B$, which could only be estimated). Second, the measured moments do not predict a transition to perpendicular modes at the highest angles; this could be due to our neglect of explicit curvature-dependence in the crater function, but additional physical effects such as shadowing and surface channeling – not addressed here – are known to be important at grazing angles. Third, we find no prediction via MD of the experimentally-observed perpendicular-mode ripples at low angles[@madi-etal-2008-PRL]; indeed, our redistribution-dominated continuum PDE is *maximally stable* at low angles, and equations of the general form \[eqn: linear-stability\] are anyway unable to generate the constant-wavelength or “Type I” bifurcation that is observed [@davidovitch-etal-PRB-2007]. These observations, together with the experimental observation that low-angle ripples develop over much longer timescales than their high-angle counterparts [@madi-etal-JPCM-2009], suggest that low-angle perpendicular-mode ripples are not due to crater-function effects at all. It has already been observed [@davidovitch-etal-PRB-2007] that the low-angle Type I bifurcation is consistent with any of several *non-local* physical mechanisms such as *gradual* stress buildup and relaxation [@davidovitch-etal-PRB-2007], or non-local damping [@facsko-etal-PRB-2004]. None of these effects would be captured in the (prompt, local) crater function, and this observation motivates future studies incorporating such physics.
Despite the limitations of our approach, when one considers the lack of any free parameters in the theory, the agreement for the diverging-wavelength or “Type II” bifurcation [@cross-hohenberg-RoMP-1993] near 50 degrees is remarkably good. The agreement remains good even when the erosive coefficients are omitted, and the similar shapes of the redistributive moments at 100 and 250 eV is consistent with the reported [@madi-etal-JPCM-2009] energy-insensitivity of the stability boundary. The most striking aspect of this result is its logical conclusion that erosion effects are essentially irrelevant for determining the patterns: according to Fig. 2 the contributions of redistributive effects to the $S$ coefficients, which determine stability and patterns, are about an order of magnitude greater and opposite in sign. This conclusion overturns the erosion-based paradigm that has dominated the field for two decades [@bradley-harper-JVST-1988] and we suggest its replacement with a redistribution-based paradigm. An important direction for future research is identifying the energy range over which this conclusion holds. Although it is conceivable that erosive effects might become non-negligible at higher ion energies, or for dense metals where heat spikes enhance sputtering yields, it remains to be determined whether erosion is actually important for stability or pattern formation in any physical experiment to date. Preliminary analysis of simulations at 1 keV are consistent with our conclusions from the results at 100 eV and 250 eV, and experimental results at 1 keV lead to the same conclusion [@madi-etal-unpub-2010].
Non-erosive ion impact-induced atom redistribution at surfaces has heretofore not been considered in the design of plasma-facing fusion reactor wall materials, where low sputter yield has been an important design criterion in the selection of tungsten for stable surfaces that must be exposed to large plasma particle fluxes for extended periods. Because the average helium ion energy is only $\sim60\,\mathrm{eV}$ and the threshold energy for sputter removal of tungsten is $\sim100\,\mathrm{eV}$, this material has been considered impervious to the effects of erosion. Within the erosion-based paradigm of pattern formation, the nanoscopic surface morphology that evolves on tungsten surfaces under some such conditions has therefore appeared mysterious [@baldwin-doerner-NF-2008]. However, because the negligibility of erosive effects does not prevent redistributive effects from causing pattern-forming instabilities, as we have shown here through a crater-function analysis, atom redistributive effects may be important contributors to the origin of these mysterious morphologies, and we present in the Supplement a re-analysis of data gathered in [@henriksson-etal-NIMB-2006] that supports this idea. If this conjecture turns out to be correct, then ultimately crater function engineering considerations may provide a more refined materials design criterion than simply a low sputter yield.
S.A.N. and M.P.B. were supported by the National Science Foundation through the Division of Mathematical Sciences, M.P.B. was additionally supported through the Harvard MRSEC and the Kavli Insitute for Bionano Science and Technology at Harvard University. M.J.A. and C.S.M. were supported by Department of Energy grant DE-FG02-06 ER46335. The work of J.S., L.B., M.B., D.F., and K.N. was performed within the Finnish Centre of Excellence in Computational Molecular Science (CMS), financed by The Academy of Finland and the University of Helsinki; grants of computer time from the Center for Scientific Computing in Espoo, Finland, are gratefully acknowledged. We also thank M.J. Baldwin, N. Kalyanasundaram, and H.T. Johnson for helpful discussions.
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to M.J.A. (email: aziz@seas.harvard.edu).
![Fitted angle-dependent moments of the crater function $\Delta h\left(x,y\right)$ as determined from molecular dynamics, for both 100 eV and 250 eV. (a) Zeroeth erosive moment $M^{\left(0\right)}$ (sputter yield times atomic volume). (b,c) First erosive (b) and redistributive (c) moments $M^{\left(1\right)}$. Each of the latter contain components in both the $x$ (downbeam) and $y$ (crossbeam) directions, with the latter expected to be zero from symmetry arguments [@norris-etal-2009-JPCM]. At both energies, redistribution dominates erosion.[]{data-label="fig: moments"}](moment-columns-san.pdf){width="5.5in"}
![ Comparison between predicted and measured wavelength for 250 eV Ar$\to$ Si. Dimensional coefficients were calculated with a flux of $f=3.5\times10^{15}\,\mathrm{ions}/\left(\mathrm{cm}^{2}\mathrm{s}\right)$ for comparison with experiment [@madi-etal-JPCM-2009]. (a,b) Coefficients of $\frac{\partial^{2}h}{\partial x^{2}}$ and $\frac{\partial^{2}h}{\partial y^{2}}$ in the linearized evolution equation \[eqn: linear-stability\] using the experimental flux. These coefficients are dominated by redistributive effects. (c) Comparison of predicted ripple wavelengths with average experimentally-observed wavelengths. Circles / squares indicate experimental patterns with wavevector parallel / perpendicular to the ion beam, and the vertical dashed black lines indicate experimental phase boundaries. On top of this, the solid line indicates our predicted wavelengths (which are all parallel-mode), and the dashed blue line indicates the wavelengths predicted if erosion were neglected entirely.[]{data-label="fig: predicted-wavelengths"}](coeffs-wavelengths-san.pdf){width="5in"}
Simulation Methods
==================
In the simulations, the actual impact induced structural deformations can be detected only in a well relaxed silicon structure. A stress existing within the material as a result of incomplete relaxation of the structure before the simulation can induce displacements of silicon atoms upon impact. For example, our test simulations showed that a structure created by annealing silicon in MD is not dense enough to model the real amorphous silicon. The surface of such a structure collapses upon impact. It is possible to relax the internal stresses by bombarding the surface with silicon atoms before the actual simulation. However, the relaxation will be not uniform. Therefore, we have used the Wooten/Winer/Weaire (WWW) method [@wooten-winer-weaire-1985-PRL].
The WWW method is a computer algorithm that generates realistic random network models of amorphous silicon. In this method, the structure is described with positions of N atoms and a list of bonds between the atoms. After a random switch of two bonds, the structure is relaxed using an interatomic potential [@mousseau-barkema-JPCM-2004; @alfthan-kuronen-kaski-2003-PRB]. In connection to the structural relaxation, the Berendsen pressure control algorithm is used to relax the diagonal components of the pressure tensor to zero [@berendsen-etal-JCP-1984]. The algorithm is computationally demanding and therefore it is possible to fully optimize only relatively small structures of a few thousand atoms. Therefore, the optimized block must be copied and the copies joined to create an optimized silicon structure large enough for impact simulations. Our tests showed that the best approach is to partially optimize a rather large (10x10x10 nm) building block instead of building the structure of very small fully optimized blocks. The latter approach can induce artificial internal shear stresses in spite of the optimization (note that pressure control at periodic boundaries an an orthogonal cell does not necessarily remove the nondiagonal, shear components of the stress tensor). The optimization of a 10x10x10 nm amorphous silicon structure was achieved using a parallelized implementation of the WWW algorithm.
After the WWW optimization phase, the structure and the surface were relaxed in MD using the EDIP potential, before it was used in the actual impact simulations. The structure used in the simulations was built of four identical optimized blocks. The density was $2.5\,\mathrm{g}/\mathrm{cm}^{3}$, which indicates that the structure is dense and not likely to collapse upon impact. Two-thirds of the silicon atoms had four neighbours, the others had five neighbours, which is also a sign of a dense structure. In a perfect amorphous network, all silicon atoms should have four neighbors. However, the test with structures built of smaller blocks that were better optimized than the 10x10x10 nm structure showed that the behavior of the moments as a function of the impact angle are very similar as those reported in the main article.
To simulate ion irradiation of bulk silicon samples, we have used periodic boundary conditions with slab boundary conditions (free boundary in the incoming ion direction, and periodic boundaries in the other two lateral directions). The bottom 1 nm layer of atoms in the simulation cell were held fixed, and temperature scaling was also applied to the atoms in a 1 nm thick layer above it. A border region of 1 nm thickness in the lateral directions was cooled during the simulations. The role of the cooling zones is to prevent shock waves and phonons to re-enter the impact area via the periodic boundaries. The size of the simulation box (20x20x10 nm) was chosen to be large enough to contain not only the area containing the crater but also the area of small deformations which reach 5-7 nm from the impact point.
In addition to the good amorphous silicon structure and cooling of the boundaries, the quality of the repulsive Ar-Si potential affects the outcome of impact simulations. The kinetic energy of the impacting Ar atom is deposited first in relatively strong Ar-Si collisions where the atoms come close each other. The repulsive Ar-Si potential used in this study is calculated using density-functional-theory calculations utilizing a numerical basis sets. This method is shown [@nordlund-etal-NIMB-1997] to give a more accurate repulsive potential than the standard universal ZBL potential [@ziegler-biersack-littmark-1985-SRIM]. The same method was used to create Si-Si short-range repulsive potential which was smoothly joined to the EDIP potential. This ensures that the possible collisions between high-energy primary knock-on silicon atoms are correctly modelled.
The same initial structure was used for all simulations. However, to reduce as much as possible the effect of the initial surface structure on the measured crater statistics, the following steps were taken. First, the impact point was varied randomly over the entire surface, with (periodic) cooling regions dynamically re-assigned for each simulation so as to be maximally distant from the impact point. Second, the azimuthal angle was varied randomly, and only the atoms within 7.5 nm of the impact point were included in the moment calculations (i.e., cooling zones and corners of the square box were ignored). Last, the global average – or “background” – displacement (over all simulations at a given impact angle) was subtracted from each individual displacement before analysis. Because we varied the impact point and azimuthal angle randomly, this global average ought to be zero for a perfect target over enough trials, and subtracting it helps to remove target-specific effects from our measurements. With this setup, 200 simulations at each impact angle were performed, which was sufficient to isolate the background displacement, and to cancel the effect of thermal vibrations which were present in the structure after impact.
A final challenge in analyzing the data arose due to a combination of the amorphous nature of the target with the periodic boundary conditions. On an ideal, very large MD target, the effect of the ion impact would only be felt within a finite distance from the impact point. However, to allow the gathering of data within reasonable time, a limited box size must be used, and the periodic box described above – with cooling zones – appears to be the most physically plausible way of accomplishing this. However, a periodic box always means that one is truly simulating an infinite number of simultaneous side-by-side impacts, and with a small enough domain, these impacts can generate enough co-ordinated momentum transfer to shear the entire target, especially for an amorphous target. Indeed, within our target, the average downbeam displacement of atoms was consistently a linear function of distance from the target’s rigid floor, except near the surface, where larger displacements were concentrated.
To combat the problem of “global shear,” we measured the average downbeam displacement within different annular slices of target parallel to the surface, using inner/outer radii of 2nm/9nm (i.e., away from both the impact and the cooling boundaries). We then fit the *bottom* half of the resulting depth-dependent profile to a line using least squares, and finally subtracted the *extrapolation to the surface* of this fit from the overall displacement field, as illustrated in Figure \[fig: shear-removal\]. The results localize the displacements to within a few nanometers of the surface, which is consistent with measured amorphous layer thicknesses of 3 nm for Si irradiated by Ar at 250 eV [@madi-etal-JPCM-2009]. In the future, we will explore the response of a larger, hybrid target consisting of a 3 nm layer of amorphous Si atop a crystalline (but not rigid) base. However, we believe our existing measurements are within the accuracy level of the other estimates in the paper.
Moment Capture and Fitting
==========================
Here we describe in more detail how we obtain moments, how we fit them, and how we obtain the final linearized evolution equation (4). For each simulated ion impact with our initially flat MD target, we define a co-ordinate system $\left(x,y,z\right)$ centered at the impact point with $z$ pointing normal to the surface, $x$ pointed in the direction of the projected ion path, and $y$ perpendicular to both $x$ and $z$ so as to complete a right-handed co-ordinate system. Hence, the ion is always arriving from the *negative* *$x$*-direction. Hence, the crater function $\Delta h\left(x,y;\,\theta\right)$ describes the height change associated with an impact at the origin, of an ion with indicence angle of $\theta$ from the vertical, coming from the negative $x$-direction.
After impact, we extract the moments $$\begin{aligned}M^{\left(0\right)}\left(\theta\right) & =\iint\Delta h\left(x,y\right)\,\mathrm{d}x\,\mathrm{d}y\\
M_{x}^{\left(1\right)}\left(\theta\right) & =\iint\Delta h\left(x,y\right)x\,\mathrm{d}x\,\mathrm{d}y\\
M_{y}^{\left(1\right)}\left(\theta\right) & =\iint\Delta h\left(x,y\right)y\,\mathrm{d}x\,\mathrm{d}y\end{aligned}$$ using the method described in [@norris-etal-2009-JPCM]. Here, it is important to note that the first moments $M^{\left(1\right)}$ contain both erosive and redistributive components (because the zeroeth moment $M^{\left(0\right)}$ describes mass loss, and because redistribution is mass-conserving, $M^{\left(0\right)}$ has no redistributive component).
Simulation sets were performed at 5-degree increments, and the average of the resulting moments were fit to Fourier functions of the form
$$\begin{aligned}M_{x-odd} & =\sum_{n=1}^{3}a_{n}\sin\left(2n\theta\right)\\
M_{x-even} & =\sum_{n=1}^{3}b_{n}\cos\left(\left(2n-1\right)\theta\right)\end{aligned}
\label{eqn: fittings}$$
These fittings reflect the observation that all moments tend to zero at $\theta=90^{\circ}$, and also their symmetries about $\theta=0$ (because a positive theta indicates an ion beam coming from the negative $x$-direction, moments that are odd/even in $x$ should also be odd/even in $\theta$). As seen in the main text, this method does not produce perfect fits, but the use of simple Fourier modes eliminates potential model-bias, while the restriction to a small number of terms excludes inter-angle noise from the fitted curve.
For our data, the moment fits were given by:$$\begin{aligned}M^{\left(0\right)} & \approx-12.0\cos\left(\theta\right)+13.2\cos\left(3\theta\right)-3.46\cos\left(5\theta\right) & \left[10^{-3}\,\mathrm{nm}^{3}/\mathrm{ion}\right]\\
M_{x,\text{erosive}}^{\left(1\right)} & \approx-10.4\sin\left(2\theta\right)+5.81\sin\left(4\theta\right)-0.516\sin\left(6\theta\right) & \left[10^{-3}\,\mathrm{nm}^{4}/\mathrm{ion}\right]\\
M_{x,\text{redist.}}^{\left(1\right)} & \approx291\sin\left(2\theta\right)-39.6\sin\left(4\theta\right)-2.16\sin\left(6\theta\right) & \left[10^{-3}\,\mathrm{nm}^{4}/\mathrm{ion}\right]\end{aligned}
.$$ The moments $M_{y}^{\left(1\right)}$ are zero to within sampling error, as expected from symmetry considerations.
Analysis: from Moments to Coefficients
=======================================
For the general reduction of moments to (nonlinear) PDE terms $P\left[\mathbf{x}\right]$, we refer to the framework derived in [@norris-etal-2009-JPCM]. However, in the linear case discussed here, it is sufficient to consider the linearization of the first-order term obtained by combining Equations (1) and (3) from the main text: $$v_{\mathbf{n}}^{P}\left(\mathbf{x}\right)\approx\varepsilon\nabla_{\mathbf{S}}\cdot\left(IM^{\left(1\right)}\right)$$ (in the linearization, the zeroeth- and second-moment terms do not contribute to stability). Now, $\nabla_{S}$ indicates a *surface* divergence, and indeed this calculation is most naturally performed in a *local* co-ordinate system associated with the surface normal and projected beam direction. In particular, both the flux $I\left(\phi\right)$ and the moments $M^{\left(i\right)}\left(\phi\right)$ depend on the local incidence angle $\phi$, while the vector $M^{\left(1\right)}\left(\phi\right)$ is observed to always point in the direction $\mathbf{e}_{P}$ of the projected ion beam.
Following [@norris-etal-2009-JPCM], surface velocities at an arbitrary point $\mathbf{x}$ will be calculated in a local co-ordinate system $\left(U,V,W\right)$ centered at $\mathbf{x}$, where $\mathbf{e}_{W}=\mathbf{n}$ corresponds to the surface normal, $\mathbf{e}_{U}=\mathbf{e}_{P}$ corresponds to the *downbeam* direction associated with the projected ion beam, and $\mathbf{e}_{V}=\mathbf{e}_{W}\times\mathbf{e}_{U}$. In this system the surface is described locally by the equation $W=H\left(U,V\right)$, the ion flux is $I=I_{0}\cos\left(\phi\right)$, and the first moment is $M^{\left(1\right)}=f\left(\phi\right)\mathbf{e}_{P}$, where $f\left(\phi\right)=M_{X}^{\left(1\right)}\left(\theta\right)$ as measured from MD. Now, as described in [@norris-etal-2009-JPCM], it is sufficient for the purposes of calculating one surface divergence to approximate $H\left(U,V\right)$ via$$H\approx\frac{1}{2}\left(H_{UU}U^{2}+2H_{UV}UV+H_{VV}V^{2}\right).$$ where $H_{UU}$, $H_{UV}$, and $H_{VV}$ describe the surface curvature at $\mathbf{x}$. All other variable quantities can then be approximated *in the vicinity of $\mathbf{x}$* to first order in$\left(U,V\right)$ via: $$\begin{aligned}\mathbf{n} & \approx\left\langle -\frac{\partial H}{\partial U},\,-\frac{\partial H}{\partial V},\,1\right\rangle \\
\mathbf{e}_{P} & \approx\left\langle 1,\,-\cot\left(\phi_{0}\right)\frac{\partial H}{\partial V},\,\frac{\partial H}{\partial U}\right\rangle \\
\cos\left(\phi\right) & \approx\cos\left(\phi_{0}\right)+\frac{\partial H}{\partial U}\sin\left(\phi_{0}\right)\end{aligned}$$ When we now take the surface divergence $\nabla_{S}=\left(\partial_{U},\,\partial_{V}\right)$ and evaluate at $\left(U,V\right)=\mathbf{0}$ (i.e., at $\mathbf{x}$), we obtain in the local co-ordinate system$$v_{\mathbf{n}}^{P}\left(\mathbf{x}\right)\approx\varepsilon\nabla_{S}\cdot\left(IM^{\left(1\right)}\right)=\varepsilon I_{0}\left[S_{U}\left(\phi\right)H_{UU}+S_{V}\left(\phi\right)H_{VV}\right]\label{eqn: local-coords-evolution}$$ where$$\begin{aligned}S_{U}\left(\phi\right) & =\frac{d}{d\phi}\left[f\left(\phi\right)\cos\left(\phi\right)\right]\\
S_{V}\left(\phi\right) & =f\left(\phi\right)\cos\left(\phi\right)\cot\left(\phi\right)\end{aligned}
.\label{eqn: moment-to-linear-coeff2}$$ While linear in the local co-ordinates, Equation is in general nonlinear in the lab fram. However, for stability studies we need only the linearization of in the lab frame, which in dimensional form is simply$$\left.\frac{\partial h}{\partial t}\right|_{\mathsf{prompt}}= I_{0}\left(S_{X}\left(\theta\right)\frac{\partial^{2}h}{\partial x^{2}}+S_{Y}\left(\theta\right)\frac{\partial^{2}h}{\partial y^{2}}\right)\label{eqn: linearization}$$ because, to linear order, $$\begin{aligned}\frac{\partial^{2}h}{\partial x^{2}} & \approx H_{UU}\\
\frac{\partial^{2}h}{\partial y^{2}} & \approx H_{VV}\\
S_{X}\left(\theta\right) & \approx S_{U}\left(\phi\right)\\
S_{Y}\left(\theta\right) & \approx S_{V}\left(\phi\right)\end{aligned}$$ To expression for the prompt regime we add the linearization of the gradual regime associated with ion enhanced viscous flow [@umbach-etal-PRL-2001], which is a lubrication approximation with the form $$\left.\frac{\partial h}{\partial t}\right|_{\mathsf{gradual}}=-B\nabla^{4}h.$$ Adding the prompt and gradual regimes, we obtain the evolution equation (4) in the main text.
For completeness, we conclude with the functional forms of the coefficients $\left(S_{X},S_{Y}\right)$ associatd with our fittings, which are obtained from equations :$$\begin{aligned}S_{X}^{\mathsf{eros.}} & =-52.1\cos\left(\theta\right)-69.2\cos\left(3\theta\right)+132\cos\left(5\theta\right)-18.1\cos\left(7\theta\right)\\
S_{Y}^{\mathsf{eros.}} & =-50.5\cos\left(\theta\right)+24.7\cos\left(3\theta\right)+21.3\cos\left(5\theta\right)-2.58\cos\left(7\theta\right)\\
S_{X}^{\mathsf{redist.}} & =1450\cos\left(\theta\right)+3760\cos\left(3\theta\right)-1040\cos\left(5\theta\right)-75.6\cos\left(7\theta\right)\\
S_{Y}^{\mathsf{redist.}} & =3520\cos\left(\theta\right)+815\cos\left(3\theta\right)-231\cos\left(5\theta\right)-10.8\cos\left(7\theta\right)\end{aligned}
.$$
Estimation of Viscous Flow coefficient
======================================
The materials parameter $B$ appearing in Equation (4) of the main text is defined [@orchard-ASR-1962] as$$B=\frac{\gamma d^{3}}{3\eta}$$ where $\gamma$ is the surface free energy, $d$ is the thickness of the thin amorphous layer that is being stimulated by the ion irradiation, and $\eta$ is the layer’s viscosity. We assume the surface free energy of amorphous silicon under ion irradiation to be equal to its value in the absence of irradiation; the value of $\gamma=1.36\,\mathrm{J}/\mathrm{m}^{2}$ measured via molecular dynamics simulations by Vauth and Mayr [@vauth-mayr-PRB-2007] happens to be numerically equal to that measured experimentally for single-crystal Si(001) [@eaglesham-etal-PRL-1993]. For the amorphous layer thickness, we directly measured $d\approx3.0\,\mathrm{nm}$ via cross-sectional transmission electron microscopy on samples irradiated at normal incidence and 30 degrees from normal. Finally, we estimate the viscosity of the top amorphous Si layer during irradiation to be $\eta\approx6.2\times10^{8}\,\mathrm{Pa\, sec}$, as shown below.
The reciprocal of viscosity is the fluidity $\phi$, which is generally understood to scale with the flux $f$ , and can be expressed in the form$$\phi=H\times N_{\mathrm{DPAPS}}$$ where $H$ is the radiation-induced fluidity, and $N_{\mathrm{DPAPS}}$ is the average number of displacements per atom per second. Using molecular dynamics simulations, Vauth and Mayr [@vauth-mayr-PRB-2007] report $H=1.04\times10^{-9}\,\left(\mathrm{Pa\, dpa}\right)^{-1}$ at an energy $E=1\,\mathrm{keV}$ and temperature $T=300\mathrm{K}$ – we use this value for our comparison with experiment, with the caveats discussed below. The average number of displacements per atom per second is given by $$N_{\mathrm{DPAPS}}=\frac{\Omega f}{d}N_{\mathrm{recoils}},$$ where $\Omega=.02\,\mathrm{nm}^{3}/\mathrm{atom}$ is the atomic volume of silicon, $f=3.5\times10^{15}\,\mathrm{ions}/\left(\mathrm{cm}^{2}\mathrm{s}\right)$ is the experimental flux in the plane perpendicular to the ion beam, $d\approx3\,\mathrm{nm}$ is the amorphous layer thickness, and $N_{\mathrm{recoils}}$ is the number of recoils generated per ion impact. To estimate $N_{\mathrm{recoils}}$, we use the Kinchin-Pease [@gnaser-BOOK-1999] model for the gross number of Frenkel pairs per incident ion, obtaining $N_{\mathrm{recoils}}=0.8E/E_{D}=6.7$, where $E=250\,\mathrm{eV}$ is the ion beam energy and $E_{D}=15\,\mathrm{eV}$ is the displacement threshold energy of Si [@wallner-etal-JNM-1988]. Taking all of these quantities, we obtain a value of the viscosity of$$\eta=\frac{1}{H\times N_{\mathrm{DPAPS}}}=6.2\times10^{8}\mathrm{Pa\, sec}.\label{eqn: eta-estimate}$$
As discussed above, the value for $\eta$ listed in is associated with Vauth and Mayr’s value of $H=1.04\times10^{-9}\,\left(\mathrm{Pa\, dpa}\right)^{-1}$ at an energy $E=1\,\mathrm{keV}$ and temperature $T=300\mathrm{K}$. In contrast, our experiments were carried out at $E=250\,\mathrm{eV}$, and the irradiated sample is observed to reach temperatures of approximately $450\,\mathrm{K}$. Hence, there is some uncertainty in our value of $\eta$, which translates to uncertainty regarding the vertical position of the theoretical curve in Figure 2 of the main text. For the temperature difference, Vauth and Mayr observe $H$ to increase weakly with increasing substrate temperature, which would shift the curve upward; however, the shift would likely be less than a factor of two. As for the energy difference, in a study of CuTi, another amorphous material, Mayr et al [@mayr-etal-PRL-2003] found $H$ to either increase or decrease with recoil energy depending on the details of the simulations; hence the theoretical curve in Fig. 2 of the main text could shift either upward or downward for MD simulations at 250 eV, with a potential magnitude of perhaps a factor of two.
Preliminary Supportive Data
===========================
We believe the phenomena reported in this paper are applicable to a wide variety of systems. However, the main data sets leading to the discovery of this result both describe low-energy irradiation of amorphous silicon. Therefore, we provide here some preliminary results associated with higher-energy argon irradiation of silicon, and also for the case of helium-irradiated tungsten which was referred to in our conclusion.
For silicon irradiated by argon at 1 keV, a much larger target must be used, with dimensions of 40 x 40 x 10 nm, which makes simulations expensive. However, we have performed 30 simulations at 5-degree increments between 30 and 65 degrees, which spans the experimental transition, and the results so far are consistent with those at 100 and 250 eV: the redistributive contributions to the first moment dominate the erosive contributions.
For tungsten irradiated by helium at 100 eV, *average* displacements are so small that we were not able to isolate a clear signal against background noise after 200 simulations. However, using existing data from earlier work [@henriksson-etal-NIMB-2006], we were able to calculate the *cumulative* displacement field at a single angle after 4,000 simulations. In all of these simulations, not a single tungsten atom was sputtered, yet a downbeam bias in the displacement field is clearly observed in Figure 3. Hence a crater-function approach may enable better engineering of surface stability under conditions where the sputter yield is zero but impact-induced target atom displacements still occur.
![Illustration of the removal of shear at 60 degrees for irradiation at 250 eV. Green dots are the original measurements, and the blue line represents a linear fit to the bottom half of the dots, extrapolated to the surface. The red line is the result of subtracting this extrapolation from the original data, which localizes the displacements to the vicinity of the surface. All data are associated with a 2nm/9nm annulus that masks the impact zone and the cooling boundary zone.[]{data-label="fig: shear-removal"}](60-drift-estimate){width="5in"}
![Cumulative displacement field for 100 eV He $\to$ W after 4,000 impacts. Impinging helium ion comes from upper right, at an angle of 25 degrees from normal. The sticks combine the initial and final position of atoms that were initially in a half a unit cell thick cross section through the simulation cell. The displacements are analyzed for data initially simulated in Ref. [@henriksson-etal-NIMB-2006]. The blue ends of the sticks indicate the initial and the red ends the final positions of the atoms. The surface is located at the top. No W atoms were sputtered, but a clear downbeam bias in the displacement field is seen. []{data-label="fig: cumulative-w"}](cumulative-w-displacements){width="5in"}
|
---
abstract: 'A field in the vacuum state, which is in principle separable, can evolve to an entangled state in a dynamical gravitational collapse. We will study, quantify, and discuss the origin of this entanglement, showing that it could even reach the maximal entanglement limit for low frequencies or very small black holes, with consequences in micro-black hole formation and the final stages of evaporating black holes. This entanglement provides quantum information resources between the modes in the asymptotic future (thermal Hawking radiation) and those which fall to the event horizon. We will also show that fermions are more sensitive than bosons to this quantum entanglement generation. This fact could be helpful in finding experimental evidence of the genuine quantum Hawking effect in analog models.'
author:
- 'Eduardo Martín-Martínez'
- 'Luis J. Garay'
- Juan León
date: 'August 26, 2010'
title: Quantum entanglement produced in the formation of a black hole
---
Introduction
============
Quantum entanglement has been recognized to play a key role in black hole thermodynamics and the fate of information in the presence of horizons; some previous studies were performed in stationary cases, namely the eternal acceleration scenario and the stationary eternal Schwarzschild black hole [@Alsingtelep; @Alicefalls; @AlsingSchul; @Edu2; @Edu3; @Edu4; @Edu5; @Edu6], not addressing issues related with dynamics and time evolution of gravitating quantum fields. On the other hand some studies involving entanglement dynamics in expanding universe scenarios have shown that the interaction with the gravitational field can produce entanglement between quantum field modes [@Ball; @Edu7].
In this paper we analyze the issue of entanglement production in a dynamical gravitational collapse. With this aim, we consider both a bosonic (scalar) and a fermionic (Grassmann scalar) field which initially are in the vacuum state and compute their asymptotic time evolution under the gravitational interaction in a stellar collapse. The vacuum state evolves to an entangled state of modes in the future null infinity (which give rise to Hawking radiation [@Hawking]) and modes that do not reach it because they fall into the forming event horizon.
We will argue that the initial vacuum state in the asymptotic past does not have any physical quantum entanglement, and that it evolves to a state that is physically entangled as a consequence of the creation of the event horizon. This entanglement depends on the mass of the black hole and the frequency of the field modes. In particular, for very small frequencies or very small black holes, a maximally entangled state could be produced.
The entanglement generated in a gravitational collapse thus appears as a quantum resource for non-demolition methods aiming to extract information about the field modes which fall into the horizon from the outgoing Hawking radiation. These methods would be most relevant for cases such as the formation of micro-black holes and the final stages of an evaporating black hole when the mass is getting smaller and, therefore, quantum correlations generated between the Hawking radiation and the infalling modes grow to become even maximal, as we will show.
Earlier works proved that fermions and bosons have qualitatively different behaviors in phenomena such as the Unruh entanglement degradation [@Alicefalls; @Edu4; @Edu5] and the entanglement generation in the background of expanding universes [@Ball; @Edu7]. Here, we will show that for fermions the generation of entanglement due to gravitational collapse is more robust than for bosons. This robustness is more evident from the peak of the thermal spectrum of Hawking radiation towards the ultraviolet.
Previous works in the literature (see for example [@Balbinot; @NavarroSalas; @BalbinotII; @Serenada] among many others) showed that Hawking radiation is correlated with the field state falling into the collapsing star. However neither the analysis of the associated entanglement entropy as a function of the black hole parameters nor the comparison between fermionic and bosonic behavior have been carried out so far. The study of these issues, the nature of the entanglement produced in a gravitational collapse and, more important, its dependence on the nature of the quantum field (bosonic/fermionic) is decisive in order to gain a deeper understanding about quantum entanglement in general relativistic scenarios as it was proven for other setups such as acceleration horizons, eternal black holes and expanding universes [@Alicefalls; @Edu4; @Edu5; @Ball; @Edu7].
Since entanglement is a pure quantum effect, understanding its behavior in these scenarios can well be relevant to discern the genuine quantum Hawking radiation from classical induced emission in black hole analogs [@Unruhan] (see, for example, Ref. [@serena]), where both classical and quantum perturbations obey the same evolution laws. It will also follow from our study that fermionic modes could be more suitable for this task since they are more reliable in encoding entanglement information.
Finally, we will argue that the entanglement between the infalling and the Hawking radiation modes neither existed as a quantum information resource nor could have been acknowledged by any observer before the collapse occurs, namely in the asymptotic past. This is important in order to understand the dynamics of the creation of correlations in the gravitational collapse scenario since these correlations are exclusively due to quantum entanglement, as discussed in the literature [@Balbinot; @NavarroSalas; @BalbinotII; @Serenada].
Gravitational Collapse
======================
In order to analyze the entanglement production induced by gravitational collapse we will consider the Vaidya dynamical solution to Einstein equations (see e.g. Ref. [@NavarroSalas]) that, despite its simplicity, contains all the ingredients relevant to our study. Refinements of the model to make it more realistic only introduce subleading corrections. The Vaidya spacetime (Fig. \[fig:vaidya\]), is conveniently described in terms of ingoing Eddington-Finkelstein coordinates by the metric $${\text{d}}s^2=-\left(1-\frac{2M(v)}{r}\right){\text{d}}v^2+2{\text{d}}v{\text{d}}r+r^2 \,
{\text{d}}\Omega^2,$$ where $r$ is the radial coordinate, $v$ is the ingoing null coordinate, and $M(v)=m\theta(v-v_0)$. For $v_0<v$ this is nothing but the ingoing Eddington-Finkelstein representation for the Schwarzschild metric whereas for $v<v_0$ it is just Minkowski spacetime. This metric represents a radial ingoing collapsing shockwave of radiation. As it can be seen in Fig. \[fig:vaidya\], $v_\textsc{h}=v_0-4m$ represents the last null ray that can escape to the future null infinity $\mathscr{I}^+$ and hence that will eventually form the event horizon.
![Carter-Penrose diagrams for gravitational collapse: Stellar collapse (left) and Vaidya spacetime (right).[]{data-label="fig:vaidya"}](fig-collapse-vaidya){width=".9\columnwidth"}
Let us now introduce two different bases of solutions to the Klein-Gordon equation in this collapsing spacetime. On the one hand, we shall define the ‘in’ Fock basis in terms of ingoing positive frequency modes, associated with the natural time parameter $v$ at the null past infinity $\mathscr{I}^-$, which is a Cauchy surface: $${{u_{{\omega}}^{\text{in}}}}\sim\frac{1}{4\pi r\sqrt{\omega}}e^{-i\omega v}.$$ On the other hand, we can also consider an alternative Cauchy surface in the future to define another basis. In this case, the asymptotic future $\mathscr{I}^+$ is not enough and we also need the future event horizon $\mathscr{H}^+$. The ‘out’ modes defined as being outgoing positive-frequency in terms of the natural time parameter $\eta_{\text{out}}$ at $\mathscr{I}^+$ are $${{u_{{\omega}}^{\text{out}}}}\sim\frac{1}{4\pi r\sqrt{\omega}}e^{-i\omega \eta_{\text{out}}},$$ where ${\eta_{\text{out}}}=v-2r^*_{\text{out}}$ and $r^*_\text{out}
$ is the radial tortoise coordinate in the Schwarzschild region. It can be shown (see e.g. Ref. [@NavarroSalas]) that, for late times ${\eta_{\text{out}}}\to\infty$ at $\mathscr{I}^+$, these modes ${{u_{{\omega}}^{\text{out}}}}$ are concentrated near $v_\textsc{h}$ at $\mathscr{I}^-$ and have the following behavior: $$\label{uout2}
{{u_{{\omega}}^{\text{out}}}}\approx\frac{1}{4\pi r \sqrt{\omega}}
e^{-i\omega\left(v_\textsc{h}-4m\ln\frac{|v_\textsc{h}-v|}{4m}\right)}
\theta(v_\textsc{h}-v).$$ These modes have only support in the region $v < v_\textsc{h}$. This is evident as only the light rays that depart from $v < v_\textsc{h}$ will reach the asymptotic region $\mathscr{I}^+$ since the rest will fall into the forming horizon defined by $v=v_\textsc{h}$. This is the only relevant regime, as far as entanglement production is concerned.
For the ‘hor’ modes defined at $\mathscr{H}^+$, there is no such natural time parameter. A simple way to choose these modes is defining them as the modes that in the asymptotic past $\mathscr{I}^-$ behave in the same way as ${{u_{{\omega}}^{\text{out}}}}$ but defined for $v > v_\textsc{h}$, that is to say, as modes that leave the asymptotic past but do not reach the asymptotic future since they will fall into the horizon. This criterion is the simplest that clearly shows the generation of quantum entanglement between the field in the horizon and the asymptotic region. In any case, since we will trace over all modes at the horizon, the choice of such modes does not affect the result. Therefore, we define the incoming modes crossing the horizon by reversing the signs of $v_\textsc{h}-v$ and $\omega$ in so that, near $\mathscr{I}^-$ these modes are $${{u_{{\omega}}^{{\text{hor}}}}}\sim\frac{1}{4\pi r \sqrt{\omega}}
e^{i\omega\left(v_\textsc{h}-4m\ln\frac{|v_\textsc{h}-v|}{4m}\right)}
\theta(v-v_\textsc{h}).$$
We are now ready to write the annihilation operators of bosonic field modes in the asymptotic past in terms of the corresponding creation and annihilation operators defined in terms of modes in the future: $$\begin{aligned}
\label{ain}
\nonumber{{a_{{\omega'}}^{\text{in}}}}&=&\int d\omega \Big[\alpha^*_{\omega\omega'}
\big({{a_{{\omega}}^{\text{out}}}}-\tanh r_{\omega}\,{{a_{{\omega}}^{{\text{hor}}}}}^\dagger\big)\\
&&+\alpha_{\omega\omega'}e^{i\varphi}\big({{a_{{\omega}}^{{\text{hor}}}}}-\tanh r_{\omega}\,
{{a_{{\omega}}^{\text{out}}}}^\dagger\big)\Big],\end{aligned}$$ where $\tanh r_{\omega} = e^{-4\pi m\omega}$. The precise values of $\varphi$ and $\alpha_{\omega\omega'}$ are not relevant for this analysis.
Hence the vacuum ${\left| {0} \right\rangle}_\text{in}$, annihilated by for all frequencies $\omega'$, acquires the following form in terms of the ‘out-hor’ basis: $${\left| {0} \right\rangle}_\text{in}=N \exp\left(\sum_{\omega}\tanh r_{\omega}\,
{{a_{{\omega}}^{{\text{hor}}}}}^\dagger {{a_{{\omega}}^{\text{out}}}}^\dagger\right)
{\left| {0} \right\rangle}_{\text{hor}}{\left| {0} \right\rangle}_{\text{out}},$$ where $N=\big(\prod_{\omega}\cosh r_{\omega}\big)^{-1}$ is a normalization constant. We can rewrite this state in terms of modes ${\left| {n_{\omega}} \right\rangle}$ with frequency $\omega$ and occupation number $n$ as $$\label{sque}
{\left| {0} \right\rangle}_\text{in}=\prod_{\omega}\frac{1}{\cosh r_{\omega}}
\sum_{n=0}^\infty (\tanh r_{\omega})^{n}{\left| {n_{\omega}} \right\rangle}_\text{hor}
{\left| {n_{\omega}} \right\rangle}_{\text{out}}.$$
Analyzing entanglement
======================
This is a two-mode squeezed state. Therefore, it is a pure entangled state of the modes in the asymptotic future and the modes falling across the event horizon. Given the tensor product structure no entanglement is created between different frequency modes. Hence, we will concentrate the analysis in one single arbitrary frequency $\omega$.
We can compute the entropy of entanglement for this state which is the ultimate entanglement measure for a bipartite pure state, defined as the Von Neumann entropy of the reduced state obtained upon tracing over one of the subsystems of the bipartite state. To compute it we need the partial state $\rho_{\text{out}}={\operatorname{tr}}_{\text{hor}}({\left| {0} \right\rangle}_{\text{in}}\!\!{\left\langle {0} \right|})$, which turns out to be $\rho_{\text{out}}=\prod_\omega\rho_{\text{out},\omega}$, where $$\rho_{\text{out},\omega}=\frac{1}{(\cosh r_{\omega})^2}
\sum_{n=0}^\infty (\tanh r_{\omega})^{2n}
{\left| {n_{\omega}} \right\rangle}_{\text{out}}\!\!{\left\langle {n_{\omega}} \right|}.$$ This is, indeed, a thermal radiation state whose temperature is nothing but the Hawking temperature $(8\pi m)^{-1}$, as it can be easily seen. However, this is only the partial state of the field, not the complete quantum state, which is globally entangled. If we compute the entropy of entanglement $S_{\textsc{e},\omega}={\operatorname{tr}}(\rho_{\text{out},\omega}\log_2
\rho_{\text{out},\omega})$ for each frequency, after some calculations, we obtain $$\begin{aligned}
\nonumber S_{\textsc{e},\omega}&=
\left(\cosh r_\omega\right)^2\log_2\left(\cosh r_\omega\right)^2
\\&-(\sinh r_\omega)^2\log_2\left(\sinh r_\omega\right)^2,\end{aligned}$$ which is displayed in Fig. \[fig2\]. As is a pure state, all the correlations between modes at the horizon and modes in the asymptotic region are due to quantum entanglement.
![Entanglement between bosonic (continuous blue) and fermionic (red dashed) field modes in $\mathscr{H}^+$ and in $\mathscr{I}^+$. The lesser the mass of the star or the mode frequency, the higher the entanglement reached.[]{data-label="fig2"}](figboth){width=".90\columnwidth" height=".5\columnwidth"}
Analogously we can compute the entanglement for fermionic fields. If we consider a spinless Dirac field (either one dimensional or a Grassmann scalar), the analysis is entirely analogous considering now both particle and antiparticle modes. We assume again that the initial state of the field is the vacuum that, after some long nontrivial calculations, can be expressed it in the Fock basis at the asymptotic future and the ‘hor’ modes: $$\begin{aligned}
\label{squef}
{\left| {0} \right\rangle}_\text{in}&=\prod_{\omega}\Big[(\cos\tilde r_{\omega})^2
{\left| {00} \right\rangle}_\text{hor}{\left| {00} \right\rangle}_{\text{out}}
\nonumber\\
&-\frac{\sin 2\tilde r_{\omega}}{2}
\big({\left| {01_\omega} \right\rangle}_\text{hor}{\left| {1_\omega0} \right\rangle}_{\text{out}}
- {\left| {1_\omega0} \right\rangle}_\text{hor}
{\left| {01_\omega} \right\rangle}_{\text{out}}\big)
\nonumber\\
&-(\sin\tilde r_{\omega})^2
{\left| {1_\omega1_\omega} \right\rangle}_\text{hor}
{\left| {1_\omega1_\omega} \right\rangle}_{\text{out}}\Big],\end{aligned}$$ where $\tan \tilde r_\omega= e^{-4\pi m\omega} $. Here, we are using the double Fock basis, the first figure inside each ket representing particles and the second antiparticles.
We can compute the entropy of entanglement of this pure state. The partial density matrix in the asymptotic future $\rho_{\text{out}}={\operatorname{tr}}_{\text{hor}}({\left| {0} \right\rangle}_{\text{in}}\!\!{\left\langle {0} \right|})=
\prod_\omega \rho_{\text{out},\omega}$, is given by $$\begin{aligned}
\rho_{\text{out},\omega}&= (\cos\tilde r_\omega)^4
{\left| {00} \right\rangle}_{\text{out}}\!\!{\left\langle {00} \right|}
\nonumber\\
&+\frac{(\sin 2\tilde r_\omega)^2}{4}
\Big({\left| {1_{\omega}0} \right\rangle}_{\text{out}}\!\!{\left\langle {1_{\omega}0} \right|}
+{\left| {01_{\omega}} \right\rangle}_{\text{out}}\!\!{\left\langle {01_{\omega}} \right|}\Big)
\nonumber\\
&+(\sin \tilde r_\omega)^4 {\left| {1_{\omega}1_{\omega}} \right\rangle}_{\text{out}}\!\!
{\left\langle {1_{\omega}1_{\omega}} \right|},\end{aligned}$$ which is again a thermal state with Hawking temperature $(8\pi
m)^{-1}$, and $$S_{\textsc{e},\omega}\!=\!-2\big[(\cos \tilde r_\omega)^2\log_2
(\cos \tilde r_\omega)^2+
(\sin \tilde r_\omega)^2\log_2(\sin \tilde r_\omega)^2\big],$$ which is also displayed in Fig. \[fig2\].
Figure \[fig2\] shows that the entanglement decreases as the mass of the black hole or the frequency of the mode increase. When comparing bosons with fermions one must have in mind that the entropy of entanglement is bounded by (the logarithm of) the dimension of the partial Hilbert space (‘out’ Fock space in our case). Therefore, due to Pauli exclusion principle, the maximum entropy of entanglement for fermions is $S_{\textsc{e},\omega}=2$, which corresponds to a maximally entangled state. On the other hand, for bosons, the entanglement is distributed among the superposition of all the occupation numbers and the entropy can grow unboundedly, reaching the maximally entangled state in the limit of infinite entropy. In this sense, the entanglement generated in the fermionic case is more useful and robust due to the limited dimension of the Fock space for each fermionic mode.
This result can be traced back to the inherent differences between fermions and bosons. Specifically, it is Pauli exclusion principle which makes fermionic entanglement more reliable. Similar results about realiability of entanglement for fermions were also found in the expanding universe scenarios [@Edu7]. This responds to the high influence of statistics in entanglement behaviour in general relativistic settings as it was investigated in [@Edu4; @Edu5]. On the other hand, Vaidya space-time has all the fundamental features of a stellar collapse and shows how the entanglement is created by the appearance of an event horizon. Hence, in other collapsing scenarios or including the sub-leading grey-body factor corrections, these fundamental statistical differences will not disappear. The qualitatively different behavior of entanglement for bosons and fermions is not an artifact of choosing a particular collapse scenario but is due to fundamental statistical principles.
In the above analysis we have considered plane wave modes, which are completely delocalized. However, an entirely analogous analysis can be easily carried out using very well localized Gaussian states, with the same results about quantum entanglement behavior.
Conclusions
===========
We have shown that the formation of an event horizon generates entanglement. If we start from the vacuum state in the asymptotic past, after the gravitational collapse process is complete we end up with a state in the asymptotic future which shares pure quantum correlations with the field modes which fall into the horizon. One could think that this entanglement was already present before the collapse, arguing that (as proved in [@Vacbell]) the vacuum state of a quantum field can be understood as an entangled state of space-like separated regions. In other words, if we artificially divided the Cauchy surface in which the vacuum state is determined into two parts, we would have a quantum correlated state between the two partitions. In principle we could have done a bipartition of the vacuum state in $\mathscr{I}^-$ such that it would reflect entanglement between the partial state of the vacuum for $v<v_\textsc{h}$ and the corresponding partial state for $v>v_\textsc{h}$. However, it is not until the collapse occurs that we have the information about what $v_\textsc{h}$ is. So, achieving beforehand the right bipartition (trying to argue that the entanglement was already in the vacuum state) would require a complete knowledge of the whole future and, consequently, there is no reason ‘a priori’ to do such bipartition. The entanglement, eventually generated by the collapse, will remain unnoticed to early observers, who are deprived of any means to acknowledge and use it for quantum information tasks. It is well known that if we introduce artificial bipartitions of a quantum system, its description can show entanglement as a consequence of the partition. However, not being associated with a physical bipartition this entanglement does not codify any physical information and, hence, cannot be used to perform any quantum information processes. (One example of this kind of non-useful entanglement is statistical entanglement between two undistinguishable fermions [@sta1]).
Gravitational collapse selects a specific partition of the initial vacuum state by means of the creation of an event horizon. In the asymptotic past there was no reason to consider a specific bipartition of the vacuum state, whereas in the future there is a clear physically meaningful bipartition: What in $\mathscr{I}^-$ was expressed as a separable state, now becomes expressed in terms of modes that correspond to the future null infinity and the ones which fall across the event horizon. This means that gravitational collapse defines a particular physical way to break the arbitrariness of bipartitioning the vacuum into different subsystems. This gravitational production of entanglement would be a physical realization of the potentiality of the vacuum state to be an entangled state and is therefore a genuine entanglement creation process.
We have computed the explicit functional form of this entanglement and its dependence on the mass of the black hole (which determines the surface gravity). For more complicated scenarios (with charge or angular momentum), it will depend on these parameters as well.
For small black holes, the outgoing Hawking radiation tends to be maximally entangled with the state of the field falling into the horizon for both bosons and fermions. This means that if a hypothetical high energy process generates a micro-black hole, a projective measurement carried out on the emitted radiation (as, for instance, the detection of Hawking radiation) will ‘collapse’ the quantum state of the field that is falling into the event horizon and give us certainty about the outcome of possible measurements carried out in the vicinity of the horizon. Furthermore, at least theoretically speaking, the available quantum information resources would be maximum and, therefore, one could perform quantum information tasks such as quantum teleportation with maximum fidelity from the infalling modes to the modes in the asymptotic future $\mathscr{I}^+$ if the observer of the infalling modes managed to dispatch an outgoing classical signal before crossing the horizon. On the other hand, low frequency modes become more entangled than the higher ones. So, the infrared part of the Hawking spectrum would provide more information about the state at the horizon than the ultraviolet.
Arguably, similar conclusions can be drawn for the final stages of an evaporating black hole: As the mass of the black hole diminishes, the temperature of the Hawking radiation spectrum increases, and therefore, the quantum state of the field tends to a maximally entangled one in the limit of $m\rightarrow0$.
We have seen that the entanglement generated in fermionic fields is more robust than for bosons. Although the entropy of entanglement in the zero mass limit is greater in the bosonic case due to the higher dimension of the partial Hilbert space, we have argued that the information is more reliably encoded in the limited Fock space of fermionic fields. Furthermore, as we consider higher frequency modes, fermionic entanglement proves to be much more easily created by the collapse. What is more, the turning point in which the entropy of entanglement for fermions becomes numerically larger than for bosons is actually near the peak of the thermal emission (Fig. \[fig2\]). This means, that, in general, a measurement carried out on Hawking radiation of fermionic particles will give us more information about the near-horizon field state. This might also be useful in analog gravity realizations as we have already discussed, specifically in systems where the field excitations are fermionic (see e.g. Ref. [@Volovik]), which would be, as shown, at an advantage over the bosonic cases. To account for this quantum entanglement in analog experiments one should carry out measurements of the quantum correlations between the emitted thermal spectrum and the infalling modes and detect Bell inequalities violations. This is easier as it gets closer to the maximally entangled case.
Acknowledgements
================
The authors want to thank Carlos Barceló for useful discussions. This work was supported by the Spanish MICINN Projects FIS2008-05705/FIS, FIS2008-06078-C03-03, the CAM research consortium QUITEMAD S2009/ESP-1594, and the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042). E. M-M was partially supported by a CSIC JAE-PREDOC2007 grant.
[20]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , (), .
, , , ****, ().
, , , , (), .
, ****, ().
, , , , , ****, ().
, ** (, ).
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, ** (, ).
|
---
abstract: 'The foundational theory for dusty plasmas is the dust charging theory that provides the dust potential and charge arising from the dust interaction with a plasma. The most widely used dust charging theory for negatively charged dust particles is the so-called orbital motion limited (OML) theory, which predicts the dust potential and heat collection accurately for a variety of applications, but was previously found to be incapable of evaluating the dust charge and plasma response in any situation. Here we report a revised OML formulation that is able to predict the plasma response and hence the dust charge. Numerical solutions of the new OML model show that the widely-used Whipple approximation of dust charge-potential relationship agrees with OML theory in the limit of small dust radius compared with plasma Debye length, but incurs large (order-unity) deviation from the OML prediction when the dust size becomes comparable with or larger than plasma Debye length. This latter case is expected for the important application of dust particles in a tokamak plasma.'
author:
- 'Xian-Zhu Tang'
- Gian Luca Delzanno
title: 'Orbital-motion-limited theory of dust charging and plasma response'
---
Introduction
============
Two years before Irving Langmuir coined the term “plasma” as in plasma physics [@langmuir-pnas-1928], he and Mott-Smith laid down a foundational theory on the charging of a spherical and cylindrical probe in a laboratory plasma [@mott-smith-langmuir-pr-1926], which was necessary for interpreting the measurement of what became later known as Langmuir probes [@hutchinson-book-2005]. In the ensuing decades, the charging and dynamics of solid particulates immersed in plasmas built the foundation for a new discipline in plasma physics – dusty plasmas [@shukla-book-2001; @fortov-book-2010; @morfill-rmp-2009], in which the collective behavior of a group of dust particles in a plasma environment is studied. The physics of both dust in a plasma and dusty plasmas finds applications in space [@whipple-rpp-1981; @verheest-book-2000], astrophysics [@draine-araa-2003; @mendis94], and laboratory [@bouchoule-book-1999; @bacharis10; @krash11]. The fundamental dust-plasma interaction, which is also essential to understand the collective behavior in a dusty plasma, includes (1) charging of the dust by absorption of plasma particles so it becomes subject to the electric field (${\bf E}$) via the electrical force ${Q}_d{\E}$ (${Q}_d$ is the dust charge); (2) heating of the dust via the collection of plasma electron and ion energy fluxes so the dust particulate can melt, evaporate, or simply sublimate; (3) dragging of the dust via a frictional force enhanced by the Coulomb interaction with the flowing background plasma.
To understand the dust dynamics and its change of state, one must resort to a dust charging theory. The most-widely used of such is the Orbital-Motion-Limited (OML) theory, which dates back to the work of Mott-Smith and Langmuir [@mott-smith-langmuir-pr-1926]. In the 1960s, Al’pert et al [@alpert-book-1965] and Laframboise [@laframboise-thesis-1966] completed the current formulation. The OML theory is known to predict accurately the dust potential for a small [@lampe-jpp-2001; @kennedy03] and not so small dust grain [@delzanno04; @willis-etal-psst-2010; @delzanno-etal-ieee-2013], despite its simplifying assumption on collisionless ion orbit that misses the absorption radius effect away from the dust surface [@bohm-etal-book-1949; @allen-ps-1992]. Surprisingly, the existing formulation can not be used for predicting the plasma response, namely solving the plasma potential $\phi$, as shown in Allen et al [@allen-etal-jpp-2000]. As the result, OML theory to date can not predict the fundamental quantity of dust charge. In practice, one has been using an idealized dust charge-potential relationship due to Whipple [@whipple-rpp-1981], who computed the dust capacitance using the conventional Debye shielding potential that does not take into account the constraint of angular momentum conservation in setting the plasma density near the dust surface.
The purpose of this paper is to present a revised OML formulation that, for the first time, is able to predict the plasma response and hence the dust charge. The OML predictions will then be contrasted with the Whipple approximation to elucidate the missing physics in dust charging for a spherical dust of comparable size with the Debye length, which is of special importance to dust transport/survivability in tokamaks [@tang-delzanno-jfe-2007; @delzanno-tang-pop-2014]. This new OML formulation is also important for electron-emitting dust particulates as it provides the basis for extending the OML theory to positively charged dust [@delzanno-tang-submitted-2014].
As an approximation of the Orbital Motion (OM) theory which follows the collisionless particle orbit via conservation of energy and angular momentum, the OML theory will inevitably introduce discrepancies in the evaluation of dust current/heat collection, and in the plasma ion density evaluation. The former will give rise to a discrepancy in dust potential, while the latter in plasma potential and hence dust charge, in comparison with those predicted by the OM theory. The usefulness of the OML formulation is due to its simplicity, and the relatively high accuracy for a variety of applications where the dust size is not large compared with the plasma Debye length.
The rest of the paper is organized as follows. In section \[sec:oml\], we briefly recall the key components of the OML theory and its inability to predict dust charge in its current formulation. A corrected formulation on the OML ion density is shown in section \[sec:corrected-oml\], which completes the OML formulation for calculating both dust potential and dust change. In section \[sec:oml-result\], we apply the new OML model to evaluate the plasma response and the dust charge, and contrast the results with the widely used Whipple approximation. The findings are summarized in section \[sec:summary\].
Background on OML theory\[sec:oml\]
===================================
There are three essential ideas underlying OML theory. First, the collection of plasma electrons and ions by the dust is governed by their collisionless orbit. These are subject to two conservation laws: energy \[$E=m_\alpha \left (v_r^2 + v_t^2\right)/2 + q_\alpha\phi(r)$\] conservation and angular momentum ($J=m v_t r$) conservation. Here $m$ ($q$) and $\phi$ are the mass (charge) of the particles of species $\alpha$ ($\alpha=e,\,i$ labels electrons and ions, respectively) and the plasma potential, while $r$, $v_r$ and $v_t$ are the radial distance, radial velocity and tangential velocity in a spherical reference frame centered on the dust grain. The radial motion of the plasma particles is governed by a one degree-of-freedom Hamiltonian with effective potential $\Phi_{\rm eff},$ $$\begin{aligned}
H = \frac{1}{2} m_\alpha v_r^2 + \Phi_{\rm eff}(r),\,\,\,
\Phi_{\rm eff} \equiv \frac{1}{2} \frac{J^2}{m_\alpha r^2} + q_\alpha\phi.\end{aligned}$$ Second, whether the plasma particle reaches the dust is governed by $\Phi_{\rm eff}.$ For a typical negatively charged dust, $\phi$ is negative and monotonically increasing with $r,$ so that $\Phi_{\rm
eff}$ is positive and monotonically decreasing with $r$ for electrons. As the result, only electrons with $E>\Phi_{\rm
eff}(r=r_d)$ from far away can reach the dust of radius $r_d.$ The final and third idea, which is a simplifying approximation, is what delineates OML from the more complete but rarely used Orbital Motion (OM) theory [@laframboise-thesis-1966; @kennedy03]. The OML theory approximates the $\Phi_{\rm eff}$ for ions also as a monotonic function of $r,$ which results in the simplification that ions with $v_r^2 > - 2 \Phi_{\rm eff}(r_d)/m_i$ from far away can reach and charge the dust. This is a remarkable simplification since one does not need $\phi(r)$ to evaluate the ion and the electron current collected by the dust, which are $$\begin{aligned}
I_e & = - e 4\pi r_d^2 n_{e0} \sqrt{\frac{k_B T_e}{2\pi m_e}} \exp\left(\frac{e\phi_d}{k_B T_e}\right),\\
I_i & = Z e 4\pi r_d^2 Z n_{i0} \sqrt{\frac{k_B T_i}{2\pi m_i}}
\left(1-\frac{Ze\phi_d}{k_B T_i}\right).\end{aligned}$$ Here $e$ is the elementary charge, $k_B$ is the Boltzmann constant, $n_{e0}$ ($n_{i0}$) is the electron (ion) density away from the dust, $n_{e0}=Zn_{i0}$, $T_{e,\,(i)}$ is the electron (ion) temperature, $q_e=-e$, $q_i=Ze$, and $\phi_d=\phi(r_d).$ Setting $I_i+I_e=0$, one can solve for the dust potential as a function of $T_e/T_i,$ ion charge state $Z,$ and $m_e/m_i.$ For an electron-proton plasma with $T_e=T_i,$ OML predicts $\phi_d = -2.5
(k_B T_e/e),$ which is in remarkable agreement with particle-in-cell simulations [@delzanno-etal-ieee-2013] that do not make the OML assumption of ion $\Phi_{\rm eff}$ being monotonic.
The OML approximation of a monotonic ion $\Phi_{\rm eff}$ is known to be violated for ions with a certain range of $J.$ This comes about [@allen-ps-1992] because the centrifugal potential energy $J^2/(2m_i r^2)$ decreases with $r$ at precisely $1/r^2,$ but the electrical potential energy $Ze\phi(r)$ increases at a faster rate (exponential in $r$) in the Debye shielding region, before it eventually asymptotes to $1/r^2$ for large $r.$ This can produce one or multiple extrema in $\Phi_{\rm eff}$ away from the dust surface, Fig. \[fig:om-potential\], which can turn back some ions in a range of $J$ at $r_m>r_d.$ In the literature this is known as the absorption radius (at $r_m>r_d$) effect for certain ions. By neglecting this subtlety, OML approximation would over-estimate the ion current to the dust. Interestingly, for dust size small and even comparable to Debye length, this correction appears to be small and OML prediction of dust potential remains reliable.
![The effective potential $\Phi_{\rm eff}$ is shown for ions in a hydrogen plasma with different values of $J$ or $\alpha\equiv J^2/(2m_ik_B T_e).$ $\Phi_{\rm
eff}$ can have extrema at $r_m>r_d$ that reflects some ions ($\alpha=2.5,3.0$). The OML approximation ignores this absorption radius effect, which can over-estimate the ion current ($\alpha=2.5$) and the ion density between dust surface and the absorption radius ($\alpha=2.5, 3.0$).[]{data-label="fig:om-potential"}](om-phi-profile.eps)
A far more serious problem was identified by Allen, Annaratone, and de Angelis in 2000 [@allen-etal-jpp-2000] that the OML theory can not predict the plasma response to the presence of a dust particle, which requires the solution of the OML Poisson equation for the plasma potential $$\begin{aligned}
\nabla^2\phi = - {\varepsilon_0}^{-1}\left({Ze n_i^{OML} - e
n_e^{OML}}\right), \label{eq:oml-poisson}\end{aligned}$$ where $\varepsilon_0$ is vacuum permittivity. Allen et al. [@allen-etal-jpp-2000] used the well-known OML ion and electron density given by Al’pert, Gurevich, and Pitaevskii in 1965 [@alpert-book-1965],
$$\begin{aligned}
\frac{n_e(z)}{n_{e0}} = &
\frac{1}{2}\left\{
1 + {\rm Erf}\left(\sqrt{\varphi-\varphi_d}\right)
+ \sqrt{1-z^{-2}}\left[1 - {\rm Erf}\left(\sqrt{\frac{\varphi-\varphi_d}{1-z^{-2}}}\right)
\right]
\exp\left[\frac{\varphi-\varphi_d}{z^2-1}\right]
\right\}
\exp \left(\varphi\right).
\label{eq:ne-oml}\\
\frac{n_i(z)}{n_{i0}}
= &
\sqrt{-\frac{Z\beta\varphi}{\pi}}
\left[1 + \sqrt{1 - \frac{\varphi_d}{z^2\varphi}}\right]
+ \frac{e^{-Z\beta\varphi}}{2}
\left[
1 - {\rm Erf}\left(\sqrt{-Z\beta\varphi}\right)\right]
+ \frac{\sqrt{1- z^{-2}}}{2}
e^{- Z\beta\tilde{\varphi}}
\left[ 1 - {\rm Erf}\left(\sqrt{-Z\beta\tilde{\varphi}}\right) \right],
\label{eq:ni-oml-alpert}\end{aligned}$$
where $$\begin{aligned}
& z\equiv r/r_d, \\
& \varphi\equiv e\phi/k_B T_e, \\
& \beta\equiv
T_e/T_i, \\
& \tilde{\varphi} \equiv \left(\varphi -
\varphi_d/z^2\right)/(1-z^{-2}).\end{aligned}$$ The failure of OML theory manifests in an imaginary ion density when $\phi > \phi_d/z^2.$ For the initially faster than $1/r^2$ increase in $\phi(r)$ due to Debye shielding, which is the cause for ion absorption radius at $r_m>r_d$ as noted previously for a negatively charged dust, Allen et al. concluded that the contradiction between $\phi(r)$ and an imaginary $n_i$ in Eq. (\[eq:ni-oml-alpert\]) would be an intrinsic defect which prevents the OML theory from predicting the plasma response. Since the dust charge is related to the normal electric field at the dust surface, which requires the solution of $\phi(r),$ one reaches the inevitable position from Ref. [@allen-etal-jpp-2000] that OML theory can not predict the dust charge, despite its success on dust potential as re-affirmed in Ref. [@lampe-jpp-2001; @willis-etal-psst-2010].
Resolving the OML contradiction between $\phi(r)$ and $n_i(r)$\[sec:corrected-oml\]
===================================================================================
Intuitively it is quite puzzling that the OML simplification of ignoring the absorption radius effect, which contributes a small error in the ion charging current, would produce an imaginary ion density, as revealed in Allen et al.’s analysis. Physically, a maxima in ion $\Phi_{\rm eff}(r)$ at $r_m>r_d$ would pose a barrier that turn back ions with a range of $J.$ Ignoring this effect with the OML approximation should manifest in an over-estimation of the ion density for $r<r_m.$ So why Allen et al. discovered an apparently inherent contradiction in the OML theory between $\phi(r)$ and $n_i^{OML}(r)?$
Following the physical picture just given, one is tempted to conclude that the resolution has to come from a revision of the OML expression for $n_i^{OML}(r)$ as given in Eq. (\[eq:ni-oml-alpert\]). We find that this is indeed the case. The cause is a change in integration bound for the OML ion density when the plasma potential transitions from $\phi(r) < \phi_d/z^2$ to $\phi(r) > \phi_d/z^2.$ To understand this subtlety, which has been evidently elusive for the past five decades, we recall that since OML assumes a monotonically varying $\Phi_{\rm eff}(r),$ the ions at $r$ have a collisionless orbit that will either intercept the dust particle or be reflected by the effective potential before it can reach the dust surface.
In the canonical case that $\lim_{r\rightarrow\infty}\phi(r) = 0,$ the birth energy of the background plasma ion far away must have $$\begin{aligned}
E_0=\frac{1}{2}m_i\left(v_r'^2 + v_t'^2\right) + Ze\phi(r\rightarrow\infty) \ge 0.\end{aligned}$$ If this ion reaches $r,$ it must have, at $r,$ that $$\begin{aligned}
E = \frac{1}{2}m_i\left(v_r^2 + v_t^2\right) + Ze\phi(r) = E_0 \ge 0.\end{aligned}$$ For a negatively charged dust which has $\phi(r)< 0,$ a plasma ion of such unbounded orbit (meaning that the ion orbit connects to infinity) must have higher kinetic energy as it approaches the dust, $$\begin{aligned}
v_r^2 + v_t^2 \ge - {2Ze}\phi(r)/m_i > 0.\label{eq:unbounded-orbit}\end{aligned}$$ As illustrated in Fig. \[fig:oml-integration-bounds\], this is outside a circle in $(v_r,v_t)$ space, which intercepts the $v_r=0$ axis at $$\begin{aligned}
v_t^b = \sqrt{-{2Ze}\phi(r)/m_i}.\label{eq:vt-unbounded}\end{aligned}$$
Not all of these ions can reach the dust surface ($r=r_d$) due to angular momentum conservation. In the OML approximation, the effective potential $\Phi_{\rm eff}(r)$ peaks at $r=r_d,$ $$\begin{aligned}
\Phi_{\rm eff}(r_d) = {J^2}/{(2m_ir_d^2)} + Ze\phi_d. \end{aligned}$$ The ions with $E<\Phi_{\rm eff}(r_d)$ will be reflected by the effective potential before they can reach $r_d.$ At $r>r_d,$ these reflected ions satisfy, after explicitly writing out $E<\Phi_{\rm eff}(r_d),$ $$\begin{aligned}
v_r^2 - \left(z^2 - 1 \right) v_t^2 < {2Ze} \left(\phi_d - \phi\right)/m_i.\end{aligned}$$ For a negatively charged dust with $\phi_d - \phi(r) \le 0,$ the reflected ions are bounded by a parabola in $(v_r,v_t)$ space, $$\begin{aligned}
\left(z^2 - 1 \right) v_t^2 - v_r^2 > {2Ze}\left(\phi - \phi_d\right)/m_i.\end{aligned}$$ As illustrated in Fig. \[fig:oml-integration-bounds\], it intercepts the $v_r=0$ axis at $$\begin{aligned}
v_t^r = \sqrt{\frac{2Ze}{m_i}\frac{\phi-\phi_d}{z^2 - 1}}.\end{aligned}$$ As we shall see, this should be compared with the intercept of Eq. (\[eq:unbounded-orbit\]) with $v_r=0$ axis, i.e. $v_t^b$ in Eq. (\[eq:vt-unbounded\]).
To evaluate the ion density at $r\ge r_d,$ which is the solution of the Vlasov equation, we integrate the Maxwellian distribution, which is assumed for the background plasma far away, in the velocity space region where they are allowed [@alpert-book-1965]. For ions with $v_r<0,$ their population lies in the region of the $(v_r, v_t)$ velocity space given by $$\begin{aligned}
v_r < 0 \,\,\, \& \,\,\,
v_r^2 + v_t^2 \ge - {2Ze} \phi(r)/m_i.\end{aligned}$$ For ions with $v_r>0,$ only the reflected ions are present so they are in a region of the velocity space given by $$\begin{aligned}
& v_r \ge 0 \,\,\, \&\,\,\,
v_r^2 + v_t^2 \ge -{2Ze}\phi(r)/m_i \\
&
\left(z^2 - 1\right) v_t^2
-
v_r^2 > {2Ze}\left(\phi - \phi_d\right)/m_i\end{aligned}$$ These give rise to integration bounds for $v_r<0$ and $v_r>0$ separately. The one for $v_r<0$ is straightforward, see Fig. \[fig:oml-integration-bounds\], and the previous OML form [@alpert-book-1965] is correct.
The one for $v_r>0$ is complicated by the possibility that ${v_t^r}^2 < {v_t^b}^2,$ which implies $$\begin{aligned}
\phi < \phi_d/z^2.\end{aligned}$$ If this is the case, the integration has two zones, separated by a set of critical $v_r^c$ and $v_t^c,$ which are the intercept of the two constraints, $v_r^2 + v_t^2 = -{2Ze}\phi(r)/m_i$ and $\left(z^2 - 1\right) v_t^2
-
v_r^2 = {2Ze}\left(\phi - \phi_d\right)/m_i,$ $$\begin{aligned}
v_r^c = \sqrt{- \frac{2Ze}{m_i}\left(\phi - \frac{\phi_d}{z^2}\right)};\,\,\,
v_t^c = \sqrt{- \frac{2Ze}{m_i}\frac{\phi_d}{z^2}}\end{aligned}$$ As shown in the left-side diagram of Fig. \[fig:oml-integration-bounds\], the integration can be carried out in two zones, depending on whether $v_r > v_r^c$ or not. The lower zone (denoted as I) is $$\begin{aligned}
v_r \in [0, v_r^c] \,\,\, \& \,\,\,
v_t \in [\sqrt{-({2Ze}/{m_i})\phi - v_r^2}, \infty).\end{aligned}$$ The upper zone (denoted as II) is $$\begin{aligned}
v_r \in [v_r^c, \infty) \,\,\, \& \,\,\,
v_t \in \left[\sqrt{ \frac{v_r^2 + \frac{2Ze}{m_i}\left(\phi -\phi_d\right)}{z^2 - 1}}, \infty\right).\end{aligned}$$ The ion density is given by $$\begin{aligned}
n_i(r) =
\frac{n_{i0}}{\sqrt{2\pi}} \left(\frac{m_i}{k_B T_i}\right)^{3/2}
\iint_{{\rm I}+{\rm II}}
e^{ -\frac{m_i\left(v_r^2 + v_t^2\right)+2Ze\phi}{2k_B T_i}
} v_t dv_t dv_r,\end{aligned}$$ which yields Eq. (\[eq:ni-oml-alpert\]), [*now valid only for $\phi(r)<\phi_d/z^2$*]{}.
![The plasma potential $\phi,$ ion density $n_i,$ and electron density $n_e$ are plotted as functions of $(r-r_d)/\lambda_D$ for three cases of $r_d/\lambda_D=1$ (black), $r_d/\lambda_D=0.1$ (red), and $r_d/\lambda_D=0.01$ (green).[]{data-label="fig:oml-plasma-profile"}](oml-profile-normalized.eps)
In the case that ${v_t^r}^2 > {v_t^b}^2,$ i.e., $$\begin{aligned}
\phi > \phi_d/z^2,\end{aligned}$$ the integration is restricted to a single zone given by $$\begin{aligned}
v_r \in [0, \infty) \,\,\, \& \,\,\,
v_t \in \left[\sqrt{ \frac{v_r^2 + \frac{2Ze}{m_i}\left(\phi -\phi_d\right)}{z^2 - 1}}, \infty\right), \label{eq:v-bound-single}\end{aligned}$$ which is illustrated in the right-side diagram of Fig. \[fig:oml-integration-bounds\]. The ion density then takes the form
$$\begin{aligned}
\frac{n_i(z)}{n_{i0}}
=
\sqrt{-\frac{Z\beta\varphi}{\pi}}
+ \frac{e^{- Z\beta\varphi}}{2}
\left[
1 - {\rm Erf}\left(\sqrt{- Z\beta\varphi}\right)\right]
+ \frac{\sqrt{1- z^{-2}}}{2}
e^{- Z\beta\tilde{\varphi}},\,\,\,\,\,\, \phi\ge\phi_d/z^2,
\label{eq:ni-oml-corrected}\end{aligned}$$
where $\phi(z)\ge \phi_d/z^2,$ while Eq. (\[eq:ni-oml-alpert\]) should be used for $\phi(z) < \phi_d/z^2.$ Contrasting Eq. (\[eq:ni-oml-corrected\]) with Eq. (\[eq:ni-oml-alpert\]), one sees that the previously-known imaginary OML ion density for $\phi>\phi_d/z^2$ is removed in the corrected OML theory for $n_i.$
OML prediction of plasma response and dust charge\[sec:oml-result\]
===================================================================
With the corrected OML ion density, the OML Poisson equation (\[eq:oml-poisson\]) can be solved for the plasma response. It is important to note that the radial electric field at the dust surface $E_d=-d\varphi/d z(z=1)$ is related to the dust charge according to Gauss’s law, $$\begin{aligned}
Q_d = 4\pi \varepsilon_0 r_d^2 E_d.\end{aligned}$$
Since the OML theory to date can not predict $Q_d,$ users of the OML charging theory have been using a simple relation for spherical dust capacitance due to Whipple [@whipple-rpp-1981], $$\begin{aligned}
Q_d = 4\pi\varepsilon_0 {r_d}\left(1 + {r_d}/{\lambda_D}\right)\phi_d,\label{eq:wipple-formula}\end{aligned}$$ with $$\begin{aligned}
\lambda_D\equiv\sqrt{\varepsilon_0 T_e/e^2/n_{e0}}\end{aligned}$$ the electron Debye length. Whipple’s idea is to expand the plasma density at $z\gg 1,$ as in the standard Debye shielding calculation. Applying this approach to OML’s Poisson equation, one finds $$\begin{aligned}
& n_e/n_{e0}\approx 1 + \varphi, \\
& n_i/n_{i0}\approx 1 - Z\beta\varphi,\end{aligned}$$ and hence $$\begin{aligned}
\frac{1}{z^2}\frac{d}{d z} \left(z^2\frac{d\varphi}{d z}\right)
=
\frac{r_d^2}{\lambda_D^2} \left(1 + Z\beta\right)\varphi.\end{aligned}$$ Introducing the so-called linearized Debye length $$\begin{aligned}
\lambda_{lin} \equiv {\lambda_D}/{\sqrt{1 + Z \beta}},\end{aligned}$$ and defining the normalized dust radius as $$\begin{aligned}
\hat{r_d} \equiv {r_d}/{\lambda_{lin}},\end{aligned}$$ one finds the solution to the Poisson equation, $$\begin{aligned}
\varphi = (\varphi_d/z)\exp\left[-\hat{r_d}(z-1)\right].\label{eq:oml-debye-shielding}\end{aligned}$$ Hence the dust charge is $$\begin{aligned}
Q_d = 4\pi\varepsilon_0 {r_d} \left(1 +
\frac{r_d\sqrt{1+ZT_e/T_i}}{\lambda_D}\right)\phi_d.
\label{eq:oml-wipple-formula}\end{aligned}$$ Unlike the original Whipple formula \[Eq. (\[eq:wipple-formula\])\] which assumes a uniform ion density, Eq. (\[eq:oml-wipple-formula\]) takes into account the ion density response. It is interesting to note that for a small dust $r_d \ll \lambda_D,$ the Debye shielding contribution, which is usually small, can be enhanced substantially if $Z\gg 1$ and/or $T_e\gg T_i.$ We will call the charge-potential relationship given in Eq. (\[eq:oml-wipple-formula\]) the generalized Whipple approximation, to distinguish it from the well-known Whipple formula in Eq. (\[eq:wipple-formula\]).
The actual OML density for $r-r_d < \lambda_D$ can deviate significantly from the asymptotic expansion valid when $r-r_d \gg
\lambda_D,$ see Fig. \[fig:oml-plasma-profile\]. This is already evident from the analytical form of $n_e$ and $n_i$ in OML theory. Since the dust charge can be alternatively computed by integrating the net charge of the plasma, $$\begin{aligned}
Q_d = - \int (Zn_i - n_e)/{n_{e0}}d^3{\x},\end{aligned}$$ due to overall charge conservation, one suspects that the OML prediction of dust charge can be significantly different from that in Eq. (\[eq:oml-wipple-formula\]). From the numerically evaluated $n_e$ and $n_i$ shown in Fig. \[fig:oml-plasma-profile\], we find that sharper deviation occurs closer to $r_d$ as the dust size becomes smaller, but there is a greater spatial extent of the deviation for large dust size. Since the dust charge corresponds to the spatial integration of electron and ion density, one finds that such deviation from the simple Debye shielding calculation is proportional to $r_d/\lambda_D.$ In Fig. \[fig:oml-scan-dust-size\], we plot $$\begin{aligned}
\Gamma\equiv Q_d/(4\pi \varepsilon_0 r_d \phi_d)-1\end{aligned}$$ as a function of $r_d/\lambda_D.$ The specific case has $Z=1$ and $\beta=T_e/T_i=1,$ so the generalized Whipple approximation, Eq. (\[eq:oml-wipple-formula\]), is simply $$\begin{aligned}
\Gamma = 1.414 r_d/\lambda_D.\end{aligned}$$ The deviation of the OML prediction of dust charge from Whipple approximation can be significantly greater if the dust size is approaching the Debye shielding length, which is consistent with the results of Ref. [@daugherty92]. This is ultimately due to the effect of angular momentum conservation on the plasma density near the dust particle. Obviously a significant correction in $Q_d$ implies a substantial change in the electrical force $Q_d {\bf E},$ which can impact the dust dynamics, for example, in a tokamak reactor [@tang-delzanno-jfe-2007; @delzanno-tang-pop-2014].
summary\[sec:summary\]
======================
In conclusion, we have resolved a long-standing issue in the OML charging theory, and by doing so, obtained a complete OML theory that predicts both dust potential and dust charge. This is enabled by a revised OML ion density formula for the case of $\phi>\phi_d/z^2,$ which is given in Eq. (\[eq:ni-oml-corrected\]). We also provide the first calculation of the plasma potential and the dust charge using the OML theory. Our results show that for applications where the dust particulate radius is much smaller than the plasma Debye length, the Whipple approximation is in good agreement with the OML prediction of the dust charge-potential relationship. When the dust size is comparable to or larger than the Debye shielding length, which is a case of importance to magnetic fusion, we discover significant deviation from the Whipple approximation of the dust charge-potential relationship. This is attributed to the fundamental role of angular momentum conservation in setting the plasma electron and ion density near the dust particle.
![Dust capacitance is shown as a function of $r_d/\lambda_D.$ The deviation from Eq. (\[eq:oml-wipple-formula\]) becomes large when the dust size becomes comparable or greater than the Debye length.[]{data-label="fig:oml-scan-dust-size"}](oml-scan-1.eps)
This research was supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences, under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under contract DE-AC52-06NA25396.
[27]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase 10.1073/pnas.14.8.627) [****, ()](\doibase 10.1103/PhysRev.28.727) @noop [**]{} (, , ) @noop [**]{} (, , ) @noop [**]{} (, , ) [****, ()](\doibase 10.1103/RevModPhys.81.1353) @noop [****, ()]{} @noop [**]{} (, , ) [****, ()](\doibase 10.1146/annurev.astro.41.011802.094840) @noop [****, ()]{} @noop [**]{} (, , ) @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [**]{}, (, ) [****, ()](http://permalink.lanl.gov/object/view?what=info:lanl-repo/isi/000171401000002) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, , ) [****, ()](\doibase 10.1088/0031-8949/45/5/013) @noop [****, ()]{} [****, ()](\doibase 10.1007/s10894-010-9295-x) @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1017/S0022377803002265) @noop [****, ()]{}
The breakdown of previous OML forumlation
=========================================
As pointed out by Allen et al [@allen-etal-jpp-2000], the OML ion density can become imaginary if one is to solve the Poisson equation, Eq. (\[eq:oml-poisson\]), for the plasma potential using the previously known OML ion density formula by Al’pert et al [@alpert-book-1965]. This suggests a fundamental breakdown of the OML theory for plasma potential and hence dust charge calculation. Allen et al investigated this problem from the angle of the OML approximation, i.e., the negelect of the ion absorption radius effect. In the main text, we give a physics argument that ignoring the ion absorption radius effect should only introduce a discrepancy in ion density compared to OM prediction, but not an unphysical imaginary number. Our corrected calculation of the OML ion density removes the possibility of an imaginary density. Here we give a more detailed account of how the imaginary ion density comes about in the previous OML formulation. The objective is to further clarify the contrast between (1) a physical approximation (OML) that introduces quantitative discrepency which vanishes in its limit of applicability, and (2) an invalid theoretical formulation that produces unphysical results.
The condition for $n_i(z)$ to become imaginary (hence unphysical) in Eq. (\[eq:ni-oml-alpert\]) is to have $$\begin{aligned}
\frac{\varphi_d}{z^2\varphi} > 1. \label{eq:cross-over}\end{aligned}$$ For a negatively charged dust ($\varphi<0$ and $\varphi_d<0$), this implies a plasma potential $\varphi(z)$ that rises faster than $\varphi_d/z^2$ with $z-1$ the normalized distance from the dust surface. As long as Debye shielding is in action (i.e. the plasma transport to the dust surface is not intrinsically ambipolar), we know that $\varphi$ would rise exponentially as a function of $z$ within the Debye shielding sphere. The cross-over point after which Eq. (\[eq:cross-over\]) is satisfied can be illustrated using the Debye shielding potential solution of Eq. (\[eq:oml-debye-shielding\]) by setting $\varphi_d/z^2\varphi=1.$ The cross-over point $z_c$ is the solution of $$\begin{aligned}
z_c\exp\left[-\hat{r_d}(z_c -1)\right] = 1. \label{eq:zc}\end{aligned}$$ Here $\hat{r_d}$ is the dust radius normalized against the Debye length and $z-1$ is the radial distance from the dust surface normalized against the dust size. For $\hat{r_d}=0.01,$ one finds $z_c=648$ or 6.48 Deybe length away. If $\hat{r_d}=0.0001,$ we have $z_c=116672$ or 11.7 Debye length away. In other words, the breakdown of the ion density formula, Eq. (\[eq:ni-oml-alpert\]), is the direct result of the Debye shielding physics, which is independent of how small the dust size is compared to the Debye length.
Of course, the Debye shielding potential as given in Eq. (\[eq:oml-debye-shielding\]) is an analytical approximation, so the actual cross-over point $z_c$ would likely differ in its exact location from that predicted by Eq. (\[eq:zc\]). When the OML Poisson equation is numerically solved, the failure of the OML formulation using the ion density given in Eq. (\[eq:ni-oml-alpert\]) manifests in a not-a-number floating point violation for all small $\hat{r_d}$s we have attempted, as one would expect by taking the squared root of a negative number. This is to be contrasted with the corrected OML formulation which produces the numerical solutions shown in section \[sec:oml-result\].
|
---
author:
- 'M. Wittkowski'
- 'P. Kervella'
- 'R. Arsenault'
- 'F. Paresce'
- 'T. Beckert'
- 'G. Weigelt'
date: 'Received …; accepted …'
title: ' VLTI/VINCI observations of the nucleus of NGC 1068 using the adaptive optics system MACAO[^1]'
---
Introduction
============
JD $N$ B(m) $Az$ $\mu^{2} \pm$ stat. IE $\pm$ stat. $\pm$ syst. $V^2 \pm$ stat. $\pm$ syst. Target
------------- ----- ------- ------- --------------------- --------------------------------- ----------------------------- --------
2452948.717 250 45.79 44.87 $0.054 \pm 0.017$ $\it 0.340 \pm 0.009 \pm 0.001$ $0.158 \pm 0.049 \pm 0.001$
2452948.722 214 46.00 45.13 $0.062 \pm 0.031$ $\it 0.340 \pm 0.009 \pm 0.001$ $0.182 \pm 0.091 \pm 0.001$
2452948.761 407 46.62 45.40 $0.319 \pm 0.005$ $0.348 \pm 0.006 \pm 0.001$
2452948.767 438 46.64 45.62 $0.304 \pm 0.005$ $0.331 \pm 0.005 \pm 0.001$
2452948.772 416 46.63 45.80 $0.315 \pm 0.006$ $0.344 \pm 0.007 \pm 0.001$
The Seyfert galaxy NGC 1068 harbors one of the brightest and closest active galactic nuclei (AGN). Active galaxies appear as types 1 and 2, where the spectra of the former exhibit broad and narrow emission lines, and those of the latter show only narrow lines. Antonucci & Miller ([@antonuccimiller]) suggested that the broad-line emission regions are located inside an optically and geometrically thick disk and that central continuum and broad-line photons are scattered into the line-of-sight by free electrons above and below the disk. Depending on the observer’s viewing angle, the broad-line emission region is either obscured or not. This suggestion is now widely accepted and has evolved to the so-called “unified scheme of AGN” (e.g., Antonucci [@antonucci]).
Various theoretical models of the postulated dusty tori have been presented by, for instance, Krolik & Begelmann ([@krolikbegelmann]), Pier & Krolik ([@pierkrolik1]), Granato & Danese ([@granato]), Efstathiou & Rowan Robinson ([@efstathiou1]), Manske et al. ([@manske]), Nenkova et al. ([@nenkova]), and Vollmer et al. ([@vollmer]). Owing to the lack of spatially resolved observations, the models were usually compared to the integrated spectrum of the core of NGC 1068. It turned out that torus models with a wide variety of geometries, spatial extensions, and optical depths are consistent with this spectrum. In addition, the nature of the central emission source itself could not be well constrained because the flux spectrum of the very inner nuclear engine could not be separated from the emission of surrounding material.
Several high-resolution infrared observations of NGC 1068 showing a compact central IR core and surrounding structure were carried out by, for instance, Thatte et al. ([@thatte]), Rouan et al. ([@rouan]), Wittkowski et al. ([@wittkowski]), Weinberger et al. ([@weinberger]), and Bock et al. ([@bock]). Wittkowski et al. ([@wittkowski]) presented both a $K$-band visibility function up to spatial frequencies corresponding to a baseline of 6m, as well as the first 76mas resolution $K$-band image of the nucleus of NGC 1068 obtained by bispectrum speckle interferometry. The compact central IR core was resolved with a Gaussian FWHM of $\sim$30mas (2pc). New $K$-band and $H$-band bispectrum speckle interferometry observations were performed by Weigelt et al. ([@weigelt]).
Very recently, the first interferometers consisting of 8–10m class telescopes started operations, and they have already succeeded in observing AGN with much higher spatial resolutions. The potential of optical/infrared interferometry to investigate AGN was recently discussed by Wittkowski et al. ([@wittkowski2]). Swain et al. ([@swain]) reported the first $K$-band interferometric observations of the Seyfert1 galaxy NGC 4151 obtained with the 85m baseline of the Keck Interferometer. Jaffe et al. ([@jaffe]) reported on the first mid-infrared interferometric observation of the dusty torus of NGC 1068 using VLTI/MIDI.
In the present letter, we report on the first $K$-band long-baseline interferometric observation of NGC 1068.
Observations and data reduction
===============================
The NGC 1068 interferometric data were obtained with the Very Large Telescope Interferometer (VLTI) and the $K$-band commissioning instrument VINCI (Kervella et al. [@kervella00]; Kervella et al. [@kervella03a]), used with the fiber-based beam combiner MONA, on Nov. 4, 2003. The UT2-UT3 baseline with 47m ground length was used. Both telescopes were equipped with the Multi Application Curvature Adaptive Optics (MACAO) system. These data were taken in the framework of the commissioning of MACAO-VLTI.
#### MACAO systems.
For a detailed description of the MACAO systems see Arsenault et al. ([@arsenault]). The MACAO systems are four curvature adaptive optics systems installed at the Coudé focus of each UT (the systems for UTs 1 and 4 will be installed in the fall of 2004 and first quarter of 2005). The Coudé train mirror M8 is replaced by a 60 actuator bimorph deformable mirror in a tip-tilt mount. The mirror M9 is a dichroic which transmits visible light to the wavefront sensor and reflects the wavelength range from 1 to 13$\mu$m to the VLTI recombination laboratory. During commissioning, the two MACAO systems on UTs 2 and 3 showed very similar and consistent results. Strehl ratios of 65% were obtained on bright guide stars ($V<11$) under optical seeing conditions of up to 0.8. On fainter sources with $V$ magnitudes of 14.5 and 16, Strehl ratios of 45% (seeing 0.60) and 25% (seeing 0.55) were obtained, respectively. For use with the single-mode fiber instrument VINCI, a high Strehl ratio ensures a high concentration of light onto the 56mas diameter fiber core of VINCI, hence allowing the interferometric observation of faint sources such as NGC 1068. For both MACAO-VLTI systems the reference source was the nucleus of NGC 1068 itself, and the same control parameters were used for both AO systems. No neutral density filters were used, the main loop gain was 0.50. The optical seeing at the time of observation was $\sim$0.9. It is difficult to estimate the MACAO performance for our NGC 1068 observations since the source is extended and an acquisition image could not be taken.
#### Interferometric data.
1000 interferograms of the nucleus of NGC 1068 were obtained in two series of 500 scans.
The fringe frequency was 216Hz, a compromise between faint object sensitivity and immunity to the atmospheric piston effect. Of the recorded interferograms, 464 were processed successfully by the VINCI data reduction software as described by Kervella et al. ([@kervella03b]). Table \[Cal\_ngc1068\] lists the observational details for the NGC 1068 and the calibrator data, as well as the resulting visibility values. Figure \[wl\_psd\_peak\] shows the background- and noise-corrected average wavelet power spectral density of the NGC 1068 interferograms. The fringes’ power peak around wave number 4500cm$^{-1}$ is not affected by any significant power spectrum bias, despite the faintness of the source. The effective wavelength of the NGC1068 observations is approximately $2.18\,\mu$m. The K5 giant HD20356 ($\theta_{\rm UD} = 1.81 \pm 0.02$mas) from Cohen et al. ([@cohen99]) and Bordé et al. ([@borde02]) was used as calibration star (effective wavelength $\lambda = 2.181\,\mu$m). Observational parameters and data reduction of the calibration star were identical to those of NGC 1068. The systematic error induced by the calibrator on the final visibility values is negligible compared to the statistical error. A calibrated squared visibility of $V^2 = 16.3 \pm 4.3$% for NGC 1068 was obtained for a sky-projected baseline with length $B = 45.839$m and azimuth angle $Az = 44.93\,\deg$ (east of north).
#### Photometric estimate. \[magnitude\_sect\]
Any photometric estimates using a single-mode fiber instrument is, in general, difficult due to the large and rapid fluctuations of the coupling of the object light into the fiber core. However, the MACAO systems keep a large fraction of the object light inside the Airy disk, stabilize the injected flux, and a photometric estimate can be attempted. For commissioning purposes, a number of stars of various $K$-band magnitudes were observed on Nov. 3-5, 2003 with the same VLTI configuration as used for our NGC 1068 observations. The relation between the observed flux values and the $K$-band magnitudes $m_K$ is consistent with the expected exponential function, so that the attempt of an absolute photometric calibration is reasonable. The best-fit relations between $m_K$ and the photometric fluxes $P_A$ and $P_B$ (in ADU/s) are $P_A=3.50\,10^6 \exp(-0.906\ m_K)$ and $P_B=1.46\,10^6 \exp (-0.901\ m_K)$. The residual dispersions on $m_K$ are $\sigma_A = 0.5$mag and $\sigma_B = 0.6$mag.
Since the performance, i.e. the Strehl ratio, of the MACAO systems for our extended source NGC1068 is unknown, but very likely lower than that of the single bright stars (see above), the application of this photometric estimate on NGC1068 can only give us a lower limit of the NGC 1068 flux in our field of view (FOV) of $F_K \ge 130 \pm 60$mJy ($m_K \le 9.2 \pm 0.4$). The size of the $K$-band Airy disk of the UTs with MACAO is 56mas. In the absence of atmospheric turbulence, the FOV of our measurements would correspond exactly to the projection of the single-mode fiber on the sky. As this mode is matched by design to the diffraction pattern of the UTs, the effective FWHM of our FOV would also be 56mas. In practice, however our FOV is slightly larger due to the presence of residual uncorrected speckles, i.e. the limited Strehl ratio.
Discussion
==========
#### Intensity distribution of the $K$-band emission.
Our measured VLTI/VINCI $K$-band squared visibility amplitude of $|V|^2=16.3 \pm 4.3$% at a projected baseline length of $B=45.8$m corresponds to a single-component Gaussian intensity distribution with FWHM $5.0 \pm 0.5$mas (taking into account the VINCI broad-band $K$ filter). With the distance to NGC 1068 of 14.4Mpc (Bland-Hawthorn et al. [@bland]), this corresponds to a linear FWHM of $0.4 \pm 0.04$ pc. This means that the VLTI/VINCI visibility value is consistent with a multi-component model only if at least 40% ($|V|=40.4$%) of the flux (i.e. $\gtrsim 50$mJy) originates from scales $\lesssim$5mas (Gaussian FWHM $\le$6.7mas or $\le$0.51pc at the 3 $\sigma$ level).
The $K$-band visibility values obtained by speckle interferometry at spatial frequencies up to a baseline of 6–10m (Wittkowski et al. [@wittkowski]: SAO 6m telescope; Weinberger et al. [@weinberger]: Keck 10m; Weigelt et al. [@weigelt]: SAO 6m) are consistent with a single-component Gaussian intensity distribution with azimuthally averaged FWHM $\sim$30mas (up to $B=$6m; Wittkowski et al. [@wittkowski]), and also with a larger $\sim$30-50mas component plus a much smaller (unresolved) component (Weinberger et al. [@weinberger], Weigelt et al. [@weigelt]).
The comparison of the visibility measurements obtained by these different methods is difficult since the visibility scales with the total observed flux in each FOV. However, since the speckle measurements show a structure with FWHM $\sim$30–50mas, and the VLTI/VINCI FOV is $\sim$56mas, the total flux observed by VLTI/VINCI is very similar to that of the compact $\sim$ 30–50mas speckle component.
Both, the VLTI/VINCI and the speckle measurements are consistent with a multi-component intensity distribution where $\gtrsim 50$mJy originate from scales $\lesssim$5mas or $\lesssim$0.4pc (VLTI/VINCI), and another part of the flux from larger scales of the order of 40mas or 3pc (speckle). Fig. \[vis\_comp\] shows the visibility models for a 5mas Gaussian matching our VLTI/VINCI measurement, a 30mas Gaussian matching the speckle measurements up to $B=$6m, and two examples for two-component models matching both, VLTI/VINCI and speckle measurements. It illustrates that two-component models are consistent with both measurements only if the small component has a size clearly below $\sim$5mas and a flux contribution of clearly less than the total flux in our FOV.
In the following, we discuss the possible origin of the newly constrained very compact ($\lesssim$5mas) $K$-band component.
#### Sources of very compact $K$-band emission.
Dust at the inner cavity of the torus is heated to near the sublimation temperature of about 1000-1500K and emits thermal $K$-band photons. These photons could escape in a direction other than the equatorial plane into our line-of-sight. Depending on the adopted dust properties and luminosity of the central source, the sublimation radius could be as small as about 0.3pc (Barvainis et al. [@barvainis]; Thatte et al. [@thatte]), or may as well be of the order of 1pc or above (Dopita et al. [@dopita]). Hence, the size of the inner dust cavity seems to be larger than our observed scale of FWHM clearly below $\sim$0.4pc. It may more likely coincide with the scale of the $\sim$40mas ($\sim$3pc) speckle component. However, since the work of Krolik & Begelmann ([@krolikbegelmann]) it has been discussed that the dusty torus very likely does not have a smooth uniform shape but may consist of a large number of clumps (Nenkova at al. [@nenkova]; Vollmer et al. [@vollmer]). Vollmer et al. ([@vollmer]) give a theoretical estimate for the radius of such clumps of $r_\mathrm{Cl} \sim 0.1$pc. Since this size is consistent with our observed scale ($\lesssim$0.4pc), one might speculate that light from distinct clumps may contribute to our observed flux. Such dust clumps can also exist in polar direction at similar distances from the nucleus. Another possibility is that free electrons located above and below the broad-line emission region in the ionization cone at distances of only a fraction of a parsec scatter central light into our line-of-sight.
It has already been discussed by Wittkowski et al. ([@wittkowski]) that a fraction of the $K$-band photons originating from the central engine could reach us directly through only moderate extinction in the near-infrared, despite the Seyfert 2 type of this AGN. The $A_V$ could be as low as $\sim$10m (Bailey et al. [@bailey]), corresponding to $A_K\,\sim$1.2m assuming standard galactic extinction. With the concept of a clumpy torus, the chance of low extinction towards the central source is even larger than for a smooth dust distribution. In this case, an analysis of the separated flux spectrum of only the very compact ($\lesssim$5mas) component should reveal a type 1 spectrum, which is supported by the possible detection of a broad Br$\gamma$ line by Gratadour et al. ([@gratadour]). Non-negligible $K$-band flux contributions of the order of a few hundred milli-Jansky could arise from the central accretion flow (cf. Wittkowski et al. [@wittkowski]; Beckert & Duschl [@beckertduschl]; Weigelt et al. [@weigelt]). Stars are very unlikely to contribute significantly to the $K$-band flux on scales $\lesssim$0.4pc (see e.g., Thatte et al. [@thatte]).
#### Comparison to VLTI/MIDI NGC 1068 and Keck NGC 4151 observations.
It is striking that the two NGC 1068 $K$-band scales found by our VLTI/VINCI observations and by speckle interferometry (Wittkowski et al. [@wittkowski]; Weigelt et al. [@weigelt]) of $\lesssim$0.4pc and about $2\times 4$pc, respectively, are very similar to those of the two NGC 1068 dust components reported by Jaffe et al. ([@jaffe]) based on the $N$-band VLTI/MIDI observations. The latter include a hot ($T>800$K) $0.8 \times (<1)$pc and a warm ($T\sim 320$K) $\sim 2.5 \times 4$pc component. Hot thermal emission from inner substructure of the dusty torus, as well as direct light from the central accretion flow could explain the $K$-band as well as the $N$-band very compact sub-parsec component. It is unlikely, though, that the warm VLTI/MIDI component and the $2\times 4$pc $K$-band component can be explained by the same dust component.
The Keck Interferometer observations of the [*Seyfert1*]{} nucleus of NGC 4151 (Swain et al. [@swain]) showed that the majority of the central ($\sim 3$pc) $K$-band light arises from very small scales of $\le 0.1$pc, probably from the central accretion disk. Our $K$-band observations of the [*Seyfert2*]{} galaxy NGC 1068 show that only part of the central light arises from compact $\lesssim$0.4pc scales while another part arises from larger scales. This difference is in line with the unified scheme which predicts that the central engine of Seyfert2 cores is obscured, which leads to a larger (up to 100%) relative flux contribution from surrounding material.
Conclusion
==========
We have obtained $K$-band interferometric observations of the nucleus of the Seyfert2 galaxy NGC 1068, and hereby show that the correlated magnitude of this AGN up to a baseline of at least $\sim$50m is such that it can be studied with the VLTI at near-infrared wavelengths.
Together with other observations, we conclude that a $K$-band flux of $\gtrsim$50mJy originates from scales clearly smaller than about 5mas or 0.4pc and another part of the flux from larger scales. Our VLTI/VINCI measurement alone sets an upper limit of $\le$6.7mas at the 3$\sigma$ level to the Gaussian FWHM of the very compact component. The origin of this newly constrained small-scale emission can be interpreted as substructure of the dusty torus, as for instance part of a clumpy inner cavity or distinct clumps forming the torus, or as direct emission from the central accretion flow viewed through only moderate extinction in the $K$-band.
The VLTI observations reported here were made possible through the efforts of the whole ESO VLTI and MACAO commissioning teams.
Antonucci, R. R. J., Miller, J. S. 1985, , 297, 621 Antonucci, R. R. J. 1993, , 31, 473 Arsenault, R., et al. 2003, The Messenger, 112, 7 Bailey, J, et al. 1988, , 234, 899 Barvainis, R. 1987, , 320, 537 Beckert, T. & Duschl, W. J. 2002, , 387, 422 Bland-Hawthorn, J., et al. 1997, Ap&SS, 248, 9 Bock, J. J., Neugebauer, G., Matthews, K., et al. 2000, , 120, 2904 Bordé, P., Coudé du Foresto, V., Chagnon, G. & Perrin, G. 2002, , 393, 183 Claret, A. 2000, , 363, 1081 Cohen, M., Walker, R. G., Carter, B., et al. 1999, , 117, 1864 Dopita, M. A., Heisler, C., Lumsden, S., Bailey, J., 1998, , 498, 570 Efstathiou, A. & Rowan Robinson, M. 1995, , 273, 649 Granato, G. L. & Danese, L., 1994, , 268, 235 Gratadour, D., Clénet, Y., Rouan, et al. 2003 , 411, 335 Jaffe, W., Meisenheimer, K., Röttgering, H. J. A. et al. 2004, , submitted Kervella, P., Coudé du Foresto, V., Glindemann, A. & Hofmann, R. 2000, , 4006, 31 Kervella, P., Gitton, Ph. & Ségransan, D., et al. 2003, , 4838, 858 Kervella, P., Ségransan, D. & Coudé du Foresto, V. 2004, , submitted Krolik, J. H. & Begelmann, M. C., 1988, , 329, 702 Manske, V., Henning, T. & Men’shchikov, A. B. 1998, , 331, 52 Nenkova, M., Ivezić, Z. & Elitzur, M. 2002, , 570, L9 Pier, E. A. & Krolik, J.H. 1992, , 401, 99 Rouan, D., Rigaut, F., Alloin, D., et al. 1998, , 339, 687 Swain, M., Vasisht, G., Akeson, R., et al. 2003, , 596, L163 Thatte, N., Quirrenbach, A., Genzel, R. et al. 1997, , 490, 238 Vollmer, B., Beckert, T. & Duschl., W. J. 2004, , 413, 949 Weinberger, A. J., Neugebauer, G. & Matthews, K. 1999, , 117, 2748 Wittkowski, M., Balega, Y., Beckert, T., et al. 1998, , 329, L45 Wittkowski, M., Duschl, W.J., Hofmann, K.-H., et al. 2003, , 4838, 1378 Weigelt, G., Wittkowski, M., Balega, Y.Y., et al., , submitted
[^1]: Based on public commissioning data released from the VLTI (www.eso.org/projects/vlti/instru/vinci/vinci\_data\_sets.html).
|
---
abstract: 'Atom loss resonances in ultracold trapped atoms have been observed at scattering lengths near [*atom-dimer resonances*]{}, at which Efimov trimers cross the atom-dimer threshold, and near [*two-dimer resonances*]{}, at which universal tetramers cross the dimer-dimer threshold. We propose a new mechanism for these loss resonances in a Bose-Einstein condensate of atoms. As the scattering length is ramped to the large final value at which the atom loss rate is measured, the time-dependent scattering length generates a small condensate of shallow dimers coherently from the atom condensate. The coexisting atom and dimer condensates can be described by a low-energy effective field theory with universal coefficients that are determined by matching exact results from few-body physics. The classical field equations for the atom and dimer condensates predict narrow enhancements in the atom loss rate near atom-dimer resonances and near two-dimer resonances due to inelastic dimer collisions.'
author:
- Christian Langmack
- 'D. Hudson Smith'
- Eric Braaten
title: |
Atom Loss Resonances\
in a Bose-Einstein Condensate
---
Nonrelativistic particles whose scattering lengths are large compared to the range of their interactions exhibit universal low-energy behavior [@Braaten:2004rn]. The universal few-body phenomena can include a spectrum of loosely-bound molecules as well as reaction rates of the particles and molecules. In some cases, including identical bosons, the universal behavior is governed by discrete scale invariance. The universal molecules then include sequences of 3-particle clusters (Efimov trimers) [@Efimov70; @Efimov73], 4-particle clusters (universal tetramers) [@Platter:2004qn; @Hammer:2006ct; @vSIG:0810], and clusters with even more particles [@vonStecher:1106] . The universal reaction rates exhibit intricate resonance and interference features [@Braaten:2004rn].
The technology for trapping atoms and cooling them to extremely low temperatures has made the universal low-energy region experimentally accessible. The use of Feshbach resonances to control the scattering length $a$ experimentally makes ultracold atoms an ideal laboratory for universal physics. One particularly dramatic probe of universality is the loss rate of atoms from a trapping potential. Resonance and interference effects in few-body reaction rates can produce local maxima and minima in the loss rate as a function of $a$. The most dramatic signature for a universal $N$-atom cluster with $N \ge 3$ is the resonant enhancement of the $N$-atom inelastic collision rate at a negative $a$ where the cluster crosses the $N$-atom threshold and becomes unbound. We refer to such a scattering length as an [*$N$-atom resonance*]{}. The first observations of an Efimov trimer [@Grimm:0512] and a universal tetramer [@Grimm:0903] and the first evidence for a universal 5-atom cluster [@Grimm:1205] were all obtained using a thermal gas of $^{133}$Cs atoms by tuning $a$ to 3-atom, 4-atom, and 5-atom resonances, respectively. An Efimov trimer has also been observed as an enhancement in the loss rate in a mixture of $^{133}$Cs atoms and dimers [@Grimm:0807] at a positive scattering length $a_*$ where an Efimov trimer crosses the atom-dimer threshold and becomes unbound. We refer to $a_*$ as an [*atom-dimer resonance*]{}. Another dramatic loss feature at positive $a$ is an interference minimum in the 3-atom recombination rate into the shallow dimer, which was also first observed in a thermal gas of $^{133}$Cs atoms [@Grimm:0512]. Many of these loss features have been subsequently observed in ultracold trapped atoms of other elements [@Ferlaino:1108].
There are a few loss features in ultracold atoms that have not yet been related to universal few-body reaction rates. They all appear in systems that were believed to consist of atoms only and no dimers. Narrow enhancements of the loss rate near atom-dimer resonances have been observed in both a Bose-Einstein condensate (BEC) and a thermal gas of $^{39}$K atoms [@Zaccanti:0904] and in both a BEC and a thermal gas of $^7$Li atoms [@Hulet:0911; @Khaykovich:1201]. In a BEC of $^7$Li atoms, narrow enhancements of the loss rate have also been observed at positive values of $a$ near [*two-dimer resonances*]{} [@Hulet:0911], at which universal tetramers cross the dimer-dimer threshold and become unbound. No mechanism has been proposed for a narrow loss feature at a two-dimer resonance in a system consisting of atoms only and no dimers. One proposed mechanism for a narrow loss feature near an atom-dimer resonance $a_*$ in such a system is the [*avalanche mechanism*]{}, in which the 3-body recombination rate into the shallow dimer is amplified by secondary elastic collisions of the outgoing dimer [@Zaccanti:0904]. It was recently shown that the avalanche mechanism cannot produce a narrow loss feature near $a_*$ [@LSB:1205]. This is a consequence of the universal energy dependence of the atom-dimer cross section. Instead of having a narrow peak at $a_*$, the elastic cross section for the energetic dimer from the recombination event has a broad maximum near $4.3\, a_*$. Detailed Monte Carlo simulations of the avalanche process demonstrate that it does not produce any narrow loss features [@LSB:1209].
In this Letter, we propose a new mechanism for narrow loss features near atom-dimer resonances and near two-dimer resonances in a Bose-Einstein condensate of atoms. The mechanism is motivated by the phenomenon of [*atom-molecule coherence*]{}, which involves the coherent transfer of atom pairs between an atom condensate and a coexisting dimer condensate. A small condensate of shallow dimers can be produced coherently by the time-dependent scattering length as it is ramped to the large final value where the atom loss rate is measured. The loss features then arise from the resonant enhancement of inelastic collisions involving dimers from the dimer condensate.
The phenomenon of atom-molecule coherence was discovered by Donley [*et al.*]{} in 2002 using a BEC of $^{85}$Rb atoms [@Weinman:0204]. In these experiments, a pulse in the magnetic field brought the atoms very close to a Feshbach resonance. The atoms were allowed to evolve at a large constant scattering length for a variable holding time, and then a second pulse took the atoms close to the Feshbach resonance again. Subsequent measurements of the number of atoms revealed three distinct components: a “remnant” BEC, a “burst” of relatively energetic atoms, and “missing” atoms that were not detected. The numbers of remnant, burst, and missing atoms all varied sinusoidally with the holding time at the frequency associated with the dimer binding energy. That sinusoidal dependence can be explained by a coexisting condensate of shallow dimers that was created by the first pulse [@KH:0204; @GKB:0209; @DS:0312].
The behavior of atoms and dimers with sufficiently small kinetic and potential energies can be described by a quantum field theory with independent fields for the atoms and dimers. Atom and dimer condensates are described by classical fields $\psi(\bm{r},t)$ and $d(\bm{r},t)$ that are the expectation values of the quantum fields. The quantum field equations can be formulated as coupled integro-differential equations for $\psi$, $d$, and an infinite hierarchy of correlation functions for quantum fluctuations. A typical experiment on the atom loss rate in a Bose-Einstein condensate begins with a stable BEC of atoms with a small positive scattering length. This system is described by a static atom condensate $\psi(\bm{r})$, with $d(\bm{r})=0$ and zero correlation functions. A ramp in the magnetic field produces a time-dependent scattering length with a large final value $a$. During the ramp, the system is described by time-dependent condensates $\psi(\bm{r},t)$ and $d(\bm{r},t)$ and nonzero correlation functions. At the end of the ramp, the dimer condensate $d(\bm{r},0)$ will be nonzero, although it could be very small if the ramp is nearly adiabatic. It could presumably be calculated using the methods developed in Refs. [@KH:0204; @GKB:0209; @DS:0312] to describe atom-molecule coherence. We will not attempt to calculate $d(\bm{r},0)$, but simply take the initial fraction $f_D$ of the atoms that are bound into dimers in the dimer condensate to be an unknown initial condition. We will ask whether observable loss features can be produced by the dimer condensate for a plausibly small fraction $f_D$.
During the subsequent holding time, there are transient effects in the atom and dimer condensates and in the correlation functions that will die away. We will assume that after they have died away, the system can be described by coexisting atom and dimer condensates only and that the correlation functions can be neglected. Atom-molecule coherence will produce oscillations in $\psi(\bm{r},t)$ and $d(\bm{r},t)$ at the frequency $\hbar/(2 \pi ma ^2)$ associated with the dimer binding energy. The time-averaged condensates also decrease with time due to loss processes. The coexisting condensates can be described by a low-energy effective field theory for atoms and dimers whose kinetic and potential energies are small compared to the dimer binding energy. The interaction terms in the classical Hamiltonian density include $$\begin{aligned}
{\cal H}_{\rm int} &=&
\nu_2 d^* d + \frac{\hbar^2 \lambda_2}{4m} (\psi^* \psi)^2
+ \frac{\hbar^2 h_3}{m} (d^* d) (\psi^* \psi)
\nonumber
\\
&& + \frac{\hbar^2 f_4}{4m} (d^* d)^2
+ \frac{\hbar^2 \lambda_3}{36m} (\psi^* \psi)^3
+ \ldots .
\label{eq:H-eff}\end{aligned}$$ The coefficients of the interaction terms are universal functions of $a$ and the complex Efimov parameter $\kappa_* \exp(i \eta_*/s_0)$, which can be interpreted as the binding wavenumber of an Efimov trimer in the unitary limit $a = \pm \infty$ [@Braaten:2004rn]. The small positive parameter $\eta_*$ takes into account tightly-bound diatomic molecules (deep dimers), which provide decay channels for Efimov trimers. The coefficients of the interaction terms in Eq. (\[eq:H-eff\]) are constrained by discrete scale invariance to be log-periodic functions of $\kappa_*$, with discrete scaling factor $e^{\pi/s_0}$, where $s_0 \approx 1.00624$ is a universal constant.
The coefficients in Eq. (\[eq:H-eff\]) can be determined by demanding that few-body results in the fundamental theory be reproduced by the effective field theory. The coefficient of $d^* d$ is determined by matching the rest energy of a shallow dimer: $\nu_2 = - \hbar^2/m a^2$. The coefficient of the terms that are 4$^{\rm th}$ order in the fields can be determined by matching elastic scattering amplitudes at threshold: $\lambda_2 = 8 \pi a$, $h_3 = 3 \pi a_{AD}$, and $f_4 = 4 \pi a_D$, where $a_{AD}$ and $a_D$ are the atom-dimer and dimer scattering lengths. The atom-dimer scattering length $a_{AD}$ is $a$ multiplied by a simple log-periodic function of $a \kappa_*$ [@Braaten:2004rn]. In the limit $\eta_*=0$, that function is real and it diverges at the atom-dimer resonance $a_{*} = 0.07076/\kappa_*$. The dimer scattering length $a_D$ is $a$ multiplied by a log-periodic function of $a \kappa_*$ that is complex even if $\eta_*=0$. For $\eta_* = 0$, Deltuva has calculated $a_D$ with several digits of accuracy [@Deltuva:1107]. Its imaginary part has narrow resonant peaks at the two-dimer resonances $a_{1*} \approx 2.196\, a_*$ and $a_{2*} \approx 6.785\, a_*$. These two-dimer resonances were first calculated in Ref. [@vSIG:0810]. For $\eta_*>0$, $a_D$ can be obtained by constructing an analytic fit to Deltuva’s results for $a_D/a$ as a function of $a \kappa_*$ and then carrying out the analytic continuation $\kappa_* \to \kappa_* \exp(i \eta_*/s_0)$. The coefficient of $(\psi^* \psi)^3$ in Eq. (\[eq:H-eff\]) could be determined by matching the elastic 3-atom scattering amplitude in the low-energy limit. By the optical theorem, its imaginary part is proportional to the 3-atom recombination rate coefficient, ${\rm Im} \, \lambda_3 = - (3 m/\hbar) \alpha$, which can be separated into contributions from recombination into the shallow dimer and into deep dimers: $\alpha = \alpha_{\rm shallow} + \alpha_{\rm deep}$. They both have the form $\hbar a^4/m$ multiplied by log-periodic functions of $a$ [@Braaten:2004rn]. In the limit $\eta_*=0$, $\alpha_{\rm deep}=0$ and $\alpha_{\rm shallow}$ has an interference zero at $a_{+} = 0.31649/\kappa_*$.
The time dependence of $\psi(\bm{r},t)$ and $d(\bm{r},t)$ is determined by the classical field equations associated with the interaction Hamiltonian density ${\cal H}_{\rm int}$ in Eq. (\[eq:H-eff\]). The corresponding number densities are $n_A = \psi^* \psi$ and $n_D = d^* d$. The rates at which the numbers $N_A = \int d^3r \, n_A$ and $N_D = \int d^3r \, n_D$ change is determined by the anti-hermitian part of ${\cal H}_{\rm int}$: $$\begin{aligned}
\frac{dN_A}{dt} &=&
\frac{6 \pi \hbar\, {\rm Im} a_{AD}}{m} \!\!\int \!\! d^3r \, n_D n_A
- \frac{\alpha}{2} \!\!\int \!\!d^3r \, n_A^3 ,
\label{eq:dN/dt}
\\
\frac{dN_D}{dt} &=&
\frac{6 \pi \hbar\, {\rm Im} a_{AD}}{m} \!\!\int \!\! d^3r \, n_D n_A
+ \frac{4 \pi \hbar\, {\rm Im} a_D}{m} \!\!\int \!\!d^3r \, n_D^2 .
\nonumber\end{aligned}$$ Since unitarity requires the imaginary parts of $a_{AD}$ and $a_D$ to be negative, these equations imply that $N_A$ and $N_D$ both decrease monotonically with time. This excludes atom-molecule coherence, which involves coherent oscillations between $N_A$ and $N_D$ with angular frequency $E_D/\hbar$. The appropriate interpretation is that the high-frequency variations in the condensates associated with atom-molecule coherence are not resolved within our low-energy effective field theory. It can at best describe number densities $n_A(\bm{r},t)$ and $n_D(\bm{r},t)$ that are averaged over many periods of the atom-dimer oscillations.
To predict the loss rate of atoms during the holding time, we need initial conditions $\psi(\bm{r},0)$ and $d(\bm{r},0)$ that are robust approximate solutions of the classical field equations associated with the hermitian part of ${\cal H}_{\rm int}$ in Eq. (\[eq:H-eff\]). At the start of the holding time, the scattering length has its final value $a$ and there is a specified initial total number of atoms $N_0$, an unknown fraction $f_D$ of which are bound atoms forming the dimer condensate. We consider atoms trapped in a cylindrically symmetric harmonic potential: $V(\bm{r}) = \frac12 m \omega_z^2 [z^2 + \zeta^2 (x^2 + y^2)]$. If $f_D$ is sufficiently small, the effect of the dimer condensate on the atoms is negligible. If the ramp to the final scattering length is slow enough, the atom condensate remains adiabatically in its ground state. We assume $N_A a$ is large enough that the kinetic energy of the atoms is small compared to their potential and interaction energies. The atom condensate then has the familiar Thomas-Fermi density profile: $$n_A(\bm{r}) =
\frac{m}{4 \pi \hbar^2 a} \max \{ 0, \mu_A - V(\bm{r})\}.
\label{eq:TF-atomBEC}$$ The chemical potential is determined by the number $N_A = (1-f_D) N_0$ of unbound atoms: $\mu_A = \frac12 m \omega_z^2 a_z^2 [15 \zeta^2 N_A a/a_z]^{2/5}$, where $a_z = (\hbar/m \omega_z)^{1/2}$.
We next consider the initial condition for $d(\bm{r},0)$. Since the initial number of dimers $f_D N_0/2$ is small compared to $N_0$, their self-interaction energy is negligible but the mean-field energy from the atom condensate can be important. Assuming the dimer condensate produced by the ramp of the scattering length is in its ground state, it satisfies the Schroedinger equation $$\begin{aligned}
\left[ - \frac{\hbar^2}{4m} \bm{\nabla}^2 + 2 V(\bm{r})
+ \frac{3 \pi \hbar^2\, {\rm Re} a_{AD}}{m} n_A(\bm{r}) \right] d(\bm{r})
\nonumber\\
= \mu_D d(\bm{r}).
\label{eq:SchEq-d}\end{aligned}$$ The eigenvalue $\mu_D$ is the chemical potential of the dimer. The normalization of the dimer condensate is determined by the condition $\int\! d^3r \, d^* d = f_D N_0/2$. The total potential energy of the dimer is the sum of the trapping potential $2 V(\bm{r})$ and the mean-field energy. Its minimum is at the origin if ${\rm Re} a_{AD} < \frac83 a$ and near the edge of the atom condensate if ${\rm Re} a_{AD} > \frac83 a$. In these two cases, the dimer condensate is an ellipsoid centered at the origin and an ellipsoidal shell near the edge of the atom condensate, respectively. The boundaries in $a$ for the ellipsoidal shell region are at $a_*$ and $2.86\, a_*$.
![ (Color online) Three-atom recombination rate coefficient $L_3$ as a function of the scattering length $a$. The data points were measured using a BEC of $^{7}$Li atoms [@Hulet:1301]. The arrows point to the narrow loss features identified in Ref. [@Hulet:1301]. The lower and upper parallel curves are the universal result for $L_3$ (with $a_+ = 1402~a_0$ and $\eta_* = 0.038$) and $9.2\, L_3$. The three vertical lines are the universal predictions for $a_*$, $a_{1*}$, and $a_{2*}$ using $a_+$ as input. The additional atom-dimer and dimer-dimer contributions in $L_3^{\rm eff}$ are included in the thinner solid (green) and dashed (red) lines, respectively. In the regions near $a_*$, $a_{1*}$, and $a_{2*}$, the dimer fractions are $f_D = 10^{-5}$, $3 \times 10^{-2}$, and $6 \times10^{-3}$. []{data-label="fig:Rice"}](rice13gray.pdf){width="8.5cm"}
The data on the loss rate of $^7$Li atoms in Ref. [@Hulet:0911] has recently been reanalyzed using a more accurate determination of the parameters of the Feshbach resonance [@Hulet:1301]. Under the assumptions that the system is a pure BEC of atoms and that the loss rate comes from 3-atom recombination only, the rate coefficient $L_3$ is defined by $dN/dt = - (L_3/6) \int \!d^3r\, n_A^3$. The data from Ref. [@Hulet:1301] that was measured in a BEC of $^7$Li atoms is shown in Fig. \[fig:Rice\]. The initial number of atoms was $N_0 = 4 \times 10^5$. Their fit to the universal prediction $L_3 = 3\alpha$ using an adjustable normalization factor is shown in Fig. \[fig:Rice\]. It determines the Efimov parameters $a_+ = 1402~a_0$ and $\eta_* = 0.038$ [@Hulet:0911] and requires a normalization factor of approximately $9.2$. The corresponding universal predictions for the resonances are $a_* = 313~a_0$, $a_{1*} = 688~a_0$, and $a_{2*} = 2127~a_0$. Near each of these three resonances, there is a narrow loss feature where the data are significantly higher than the fitted curve. The positions of the loss features reported in Ref. [@Hulet:1301] are $426~a_0$, $919~a_0$, and $1902~a_0$. As shown in Fig. \[fig:Rice\], they are not as widely spaced as the predicted resonances.
We now consider the effects of a small coexisting dimer condensate. Our initial conditions are determined by $N_0 = 4 \times 10^5$ and an assumed dimer fraction $f_D$, which we expect to be small compared to 1 and to depend strongly on $a$. We take $n_A$ to be the Thomas-Fermi profile in Eq. (\[eq:TF-atomBEC\]). For $n_D = d^* d$, we use a variational approximation for $d$ that reduces the 3-dimensional Schroedinger equation in Eq. (\[eq:SchEq-d\]) to a 1-dimensional equation for a single radial variable. The subsequent time dependence of the total number of atoms $N(t)$ can be obtained by solving Eqs. (\[eq:dN/dt\]). A quantitative comparison with the data would require comparing with the time dependence observed in the experiment. Instead we compare the data for $L_3$ in Fig. \[fig:Rice\] with an effective rate coefficient $L_3^{\rm eff}$ determined by the initial loss rate. It is defined by $dN/dt = - (L_3^{\rm eff}/6)\int\!\! d^3r\, n_{A0}^3$, where the integral is evaluated under the assumption that $f_D=0$: $\int\!\! d^3r\, n_{A0}^3 = (N_0/168 \pi^2 a_z^4 a^2)[15 \zeta^2 N_0 a/a_z]^{4/5}$. In Fig. \[fig:Rice\], the universal prediction for $L_3$ from 3-atom recombination falls an order of magnitude below the data. This allows room for additional contributions from the atom-dimer and dimer-dimer terms in Eqs. (\[eq:dN/dt\]). We would like to determine whether narrow loss features can stand out above the smooth contributions for plausible values of $f_D$.
In Fig.\[fig:Rice\], the terms in $L_{3}^{\rm eff}$ are illustrated by two thin curves, which correspond to 3-atom recombination plus atom-dimer losses and to 3-atom recombination plus dimer-dimer losses. We show these curves near $a_*$, $a_{1*}$, and $a_{2*}$ using different values of $f_D$ chosen to make the narrow loss feature visible: $f_D = 10^{-5}$, $3 \times 10^{-2}$, and $6 \times10^{-3}$, respectively. The curve near $a_*$ that includes the atom-dimer terms in Eqs. (\[eq:dN/dt\]) has a narrow peak at the atom-dimer resonance. The peak is not symmetric, because the dimer condensate changes from an ellipsoid centered at the origin for $a < a_*$ to an ellipsoidal shell near the edge of the atom condensate for $a > a_*$. The curves near $a_{1*}$ and $a_{2*}$ that include the dimer-dimer term in Eqs. (\[eq:dN/dt\]) have narrow peaks at the two-dimer resonances. The behavior near $a_{1*}$ and $a_{2*}$ is different, because near $a_{1*}$ the dimer condensate is an ellipsoidal shell while near $a_{2*}$ it is an ellipsoid. If $f_D$ is large enough near $a_{1*}$ and $a_{2*}$, the peaks can stand out above the 3-atom recombination and atom-dimer contributions. If $f_D$ is too large near $a_+$, the atom-dimer contribution can fill in the interference minimum from 3-atom recombination. For $\eta_* = 0.038$, there is no longer a local minimum near $a_+$ if $f_D > 2.2 \times 10^{-4}$. The dimer fractions illustrated in Fig. \[fig:Rice\] are sufficient to make the atom-dimer and two-dimer resonances stand out in the initial loss rate. Larger fractions would be required to make a significant difference in the integrated loss rates. Nevertheless our results demonstrate that narrow loss features can be produced at the atom-dimer and dimer-dimer resonances with plausibly small values of the dimer fraction.
Our dimer condensate mechanism provides a plausible explanation for the narrow loss feature at an atom-dimer resonance that was observed in a BEC of $^{39}$K atoms [@Zaccanti:0904] and for the narrow loss features near an atom-dimer resonance and near two two-dimer resonances that were observed in a BEC of $^7$Li atoms [@Hulet:0911]. It cannot explain the narrow loss features near atom-dimer resonances that have been observed in thermal gases of $^{39}$K atoms [@Zaccanti:0904] and $^7$Li atoms [@Hulet:0911; @Khaykovich:1201]. These loss features were observed at relatively small scattering lengths, so they could be associated with nonuniversal effects. In a BEC, the dimer fraction $f_D$ would depend sensitively on the detailed form of the ramp that brings the scattering length to its final value $a$. This sensitivity could be exploited to test our mechanism. The fraction $f_D$ could be amplified by pulsing the magnetic field very close to the Feshbach resonance before measuring the loss rate, as in the experiments on atom-molecule coherence [@Weinman:0204]. The loss rates at atom-dimer and dimer-dimer resonances would increase linearly and quadratically with the amplification factor, respectively.
In summary, we have proposed a new mechanism for narrow atom loss features at atom-dimer resonances and at two-dimer resonances in a BEC of ultracold atoms. The positions of these features are determined by universal few-body physics and thus provide additional tests of universality. There could be similar loss features where universal 5-atom clusters cross the atom-dimer-dimer threshold. The strengths of all these loss features are determined by the many-body physics of Bose-Einstein condensates and open a new window into the remarkable phenomenon of atom-molecule coherence.
We thank R. Hulet for valuable discussions. This research was supported in part by a joint grant from the Army Research Office and the Air Force Office of Scientific Research.
[99]{}
E. Braaten and H.-W. Hammer, Phys. Rept. [**428**]{}, 259 (2006) \[arXiv:cond/mat0410417\].
V. Efimov, Phys. Lett. [**33B**]{}, 563 (1970).
V. Efimov, Nucl. Phys. A [**210**]{}, 157 (1973).
L. Platter, H.W. Hammer and U.-G. Meissner, Phys. Rev. A [**70**]{}, 052101 (2004) \[cond-mat/0404313\].
H.-W. Hammer and L. Platter, Eur. Phys. J. A [**32**]{}, 113 (2007) \[nucl-th/0610105\].
J. von Stecher, J.P. D’Incao, and C.H. Greene, Nature Physics [**5**]{}, 417 (2009) \[arXiv:0810.3876\].
J. von Stecher, Phys. Rev. Lett. [**107**]{}, 200402 (2011) \[arXiv:1106.2319\].
T. Kraemer et al., Nature [**440**]{}, 315 (2006) \[arXiv:cond-mat/0512394\].
F. Ferlaino et al., Phys. Rev. Lett. [**102**]{}, 140401 (2009) \[arXiv:0903.1276\].
A. Zenesini et al., arXiv:1205.1921.
S. Knoop et al., Nature Physics [**5**]{}, 227 (2009) \[arXiv:0807.3306\].
F. Ferlaino et al., Few-Body Syst. [**51**]{}, 113 (2011) \[arXiv:1108.1909\].
M. Zaccanti et al., Nature Physics [**5**]{}, 586 (2009) \[arXiv:0904.4453\].
S.E. Pollack, D. Dries, and R.G. Hulet, Science [**326**]{}, 1683 (2009) \[arXiv:0911.0893\].
O. Machtey et al., Phys. Rev. Lett. [**108**]{}, 210406 (2012) \[arXiv:1201.2396\].
C. Langmack, D.H. Smith, and E. Braaten, Phys. Rev. A [**86**]{}, 022718 (2012) \[arXiv:1205.2683\].
C. Langmack, D.H. Smith, and E. Braaten, Phys. Rev. A [**87**]{}, 023620 (2013) \[arXiv:1209.4912\].
E.A. Donley, N.R. Claussen, S.T. Thompson, and C.E. Wieman, Nature [**417**]{}, 529 (2002) \[arXiv:cond-mat/0204436\].
S.J.J.M.F. Kokkelmans and M.J. Holland, Phys. Rev. Lett. [**89**]{}, 180401 (2002) \[arXiv:cond-mat/0204504\].
T. Köhler, T. Gasenzer, and K. Burnett, Phys. Rev. A [**67**]{}, 013601 (2003) \[arXiv:cond-mat/0209100\].
R.A. Duine and H.T.C. Stoof, Phys. Rep. [**396**]{}, 115 (2004) \[arXiv:cond-mat/0312254\].
A. Deltuva, Phys. Rev. A [**84**]{}, 022703 (2011) \[arXiv:1107.3956\].
P. Dyke, S.E. Pollack, and R.G. Hulet, arXiv:0302.0281.
|
---
author:
- |
[B. Yüzbaş[i]{}$^1$[^1] and M. Arashi$^2$]{}\
*$^{1}$Department of Econometrics, Inonu University, Malatya, Turkey\
*$^{2}$Department of Statistics, Shahrood University of Technology, Iran**
title: '**Double shrunken selection operator**'
---
> [*Abstract:*]{} The least absolute shrinkage and selection operator (LASSO) of Tibshirani (1996) is a prominent estimator which selects significant (under some sense) features and kills insignificant ones. Indeed the LASSO shrinks features lager than a noise level to zero. In this paper, we force LASSO to be shrunken more by proposing a Stein-type shrinkage estimator emanating from the LASSO, namely the Stein-type LASSO. The newly proposed estimator proposes good performance in risk sense numerically. Variants of this estimator have smaller relative MSE and prediction error, compared to the LASSO, in the analysis of prostate cancer data set.
>
> [*Key words and phrases:*]{} Double shrinking; Linear regression model; LASSO; MSE; Prediction error; Stein-type shrinkage estimator
>
> [*AMS Classification:*]{} 62G08, 62J07, 62G20
Introduction
============
It is well-known that the least squares estimator (LSE) in the linear regression model, is unbiased with minimum variance. However, dealing with sparse linear models, it is deficient from prediction accuracy and/or interpretation. As a remedy, one may use the least absolute shrinkage and selection operator (LASSO) estimator of Tibshirani (1996). It defines a continuous shrinking operation that can produce coefficients that are exactly “zero" and is competitive with subset selection and ridge regression retaining good properties of both the estimators. LASSO simultaneously estimates and selects the coefficients of a given linear regression model. Recently, Saleh and Raheem (2015) have proposed an improved LASSO estimation technique based on Stein-rule, where they use uncertain prior information on parameters of interest. See Saleh (2006) for a comprehensive overview on shrinkage estimation with uncertain prior information. Saleh and Raheem (2015) illustrated superiority of a set of LASSO-based shrinkage estimators over the classical LASSO estimator. However, in this paper, we have a different look to improve the LASSO.
In this paper, we present a Steinian LASSO-type estimator by double shrinking the features. Specifically, following James and Stein (1961) and Stein (1981), we propose a set of Stein-type LASSO estimators. We will illustrate how the proposed set of estimators perform well compared to the LASSO. In all comparisons, we use the ${\textnormal{L}}_2$-risk measure of closeness, i.e., for any estimator $\widehat{\boldsymbol{\theta}}$ of the vector-parameter ${\boldsymbol{\theta}}$, the ${\textnormal{L}}_2$-loss function is given by ${\mathcal{L}}({\boldsymbol{\theta}};\widehat{\boldsymbol{\theta}})=\|\widehat{\boldsymbol{\theta}}-{\boldsymbol{\theta}}\|^2$ and the associated ${\textnormal{L}}_2$-risk is evaluated by ${\textnormal{E}}\left[{\mathcal{L}}({\boldsymbol{\theta}};\widehat{\boldsymbol{\theta}})\right]$.
In what follows, we propose the set of Stein-type LASSO estimators and evaluate the performance of the proposed estimators, compared to the LASSO, via a Monte Carlo simulation study. We further investigate the superiority of the proposed estimators compared to the LASSO using the prostate cancer data set.
Linear Model and Estimators
===========================
Consider the linear regression model $$\label{eq21}
Y_i=\beta_0+\beta_1x_{1i}+\ldots+\beta_px_{pi}+\epsilon_i=\beta_0+{\boldsymbol{x}}_i^{\rm \top}{\boldsymbol{\beta}}+\epsilon_i,\quad i=1,\ldots,n,$$ where $\epsilon_1,\ldots,\epsilon_n$ are i.i.d. random variables with mean $0$ and variance $\sigma^2$.
Without loss of generality, we will assume that the covariates are centered to have mean $0$ and take $\widehat\beta_0=n^{-1}\sum_{j=1}^n Y_i=\bar Y$ and replace $Y_i$ in by $Y_i-\bar Y$ to eliminate $\beta_0$. Then, we also assume $\bar Y=0$ to better concentrate on the estimation of ${\boldsymbol{\beta}}=(\beta_1,\ldots,\beta_p)^{\rm \top}$.
Following Knight and Fu (2000), we consider the bridge estimator of ${\boldsymbol{\beta}}$ by minimizing the penalized least squares criterion $$\label{eq22}
\sum_{i=1}^n\left(Y_i-{\boldsymbol{x}}_i^{\rm \top}{\boldsymbol{\beta}}\right)^2+\lambda_n\sum_{j=1}^p|\beta_j|^\gamma,$$ for a given $\lambda_n$ with $\gamma>0$.
In consequent study, we only focus on the special case $\gamma=1$, resulting the LASSO of Tibshirani (1996). We will provide some notes about the use of in conclusions.
Stein-type LASSO
----------------
Following Stein (1981), we define the following set of general shrinkage estimators emanating from the LASSO estimator as $$\label{general-estimator}
\widehat{\boldsymbol{\beta}}_n^{\rm S}=\widehat{\boldsymbol{\beta}}_n^{\rm L}+{\boldsymbol{g}}(\widehat{\boldsymbol{\beta}}_n^{\rm L}),$$ for some smooth and bounded function ${\boldsymbol{g}}:\mathbb{R}^p\to\mathbb{R}^p$.
Clearly, the shrinkage estimator $\widehat{\boldsymbol{\beta}}_n^{\rm S}$ has smaller ${\textnormal{L}}_2$-risk than LASSO, for all ${\boldsymbol{g}}(\cdot)$ satisfying the following inequality $$\label{condition1}
\|{\boldsymbol{g}}(\widehat{\boldsymbol{\beta}}_n^{\rm L})\|^2+2\nabla^{\rm \top}{\boldsymbol{g}}(\widehat{\boldsymbol{\beta}}_n^{\rm L})<0,\quad \textnormal{almost everywhere in}\;{\boldsymbol{g}}.$$
Let define $a=(n-p)(p-2)/(n-p+2)$, $\mathcal{W}_n= (\widehat{\boldsymbol{\beta}}_n^{\rm L})^{\rm \top}({\boldsymbol{X}}^{\rm \top}{\boldsymbol{X}})\widehat{\boldsymbol{\beta}}_n^{\rm L}/\widehat{\sigma}^2$ and $\widehat{\sigma}^2$ is a consistent estimator of $\sigma^2$ and ${\boldsymbol{X}}=\left( {\boldsymbol{x}}_{1},\ldots,{\boldsymbol{x}}_{n}\right)^{\top}$. A well-known function which satisfies the condition is ${\boldsymbol{g}}(\widehat{\boldsymbol{\beta}}_n^{\rm L})=-a\mathcal{W}_n^{-1}$, giving rise to the Stein-type estimator, for small enough $a$. However, incorporating such function in , gives an estimator with undesirable properties. Apparently as soon as $\mathcal{W}_n<a$, the proposed estimator changes the sign of LASSO. On the other hand, the new estimator does not scale LASSO component-wise. Hence, for $\widehat{\boldsymbol{\beta}}_n^{\rm L}=(\widehat\beta_{1n}^{\rm L},\ldots,\widehat\beta_{pn}^{\rm L})^{\rm \top}$, we define the Stein-type LASSO (SL) estimator with form
$$\widehat{\boldsymbol{\beta}}_n^{\rm SL}=\left(\left\{1-a\mathcal{W}_n^{-1}\right\}\widehat\beta_{jn}^{\rm L}|j=1,\ldots,p\right)^{\rm \top}.$$
Assume ${\boldsymbol{C}}_n=\frac1n\sum_{i=1}^n {\boldsymbol{x}}_i{\boldsymbol{x}}_i^{\rm \top}\to{\boldsymbol{C}}$, ${\boldsymbol{C}}$ is a non-negative definite matrix and $\frac1n\max_{1\leq i\leq n}{\boldsymbol{x}}_i^{\rm \top}{\boldsymbol{x}}_i\to0$. Clearly, if $\lambda_n$ is $\sqrt{n}$-consistent, i.e., $\lambda_n=O(\sqrt{n})$, then from Knight and Fu (2000) we have $\sqrt{n}(\widehat{\boldsymbol{\beta}}_n^{\rm L}-{\boldsymbol{\beta}})\overset{\mathcal{D}}{\to}{\mathcal{N}}_p({\boldsymbol{0}},\sigma^2{\boldsymbol{C}}^{-1})$ and the ${\textnormal{L}}_2$-risk of SL can be obtained using the Stein’s identity (1981).
To avoid negative values, the positive part of SL, namely positive rule Stein-type LASSO (PRSL) will be defined as $$\widehat{\boldsymbol{\beta}}_n^{\rm PRSL}=\left(\left\{1-a\mathcal{W}_n^{-1}\right\}^{+}\widehat\beta_{jn}^{\rm L}|j=1,\ldots,p\right)^{\rm \top},$$ where $b^+=\max(0,b)$.
Then, the ${\textnormal{L}}_2$-risk difference is given by $$\begin{aligned}
{\mathcal{D}}_1&=&{\textnormal{R}}({\boldsymbol{\beta}};\widehat{\boldsymbol{\beta}}_n^{\rm SL})-{\textnormal{R}}({\boldsymbol{\beta}};\widehat{\boldsymbol{\beta}}_n^{\rm PRSL})\cr
&=&-\sum_j{\textnormal{E}}\left[\left\{1-a\mathcal{W}_n^{-1}\right\}^2I\left(\mathcal{W}_n<a\right)\left(\widehat\beta_{jn}^{\rm L}\right)^2\right]\cr
&&+2\sum_j{\textnormal{E}}\left[\left\{1-a\mathcal{W}_n^{-1}\right\}I\left(\mathcal{W}_n<a\right)\left(\widehat\beta_{jn}^{\rm L}(\widehat\beta_{jn}^{\rm L}-\beta_j)\right)\right]\cr
&&<0 .\end{aligned}$$ Since for values $\mathcal{W}_n<a$, $1-a\mathcal{W}_n^{-1}<0$ and the expected value of a positive random variable is always positive. Hence the positive part of SL has uniformly smaller ${\textnormal{L}}_2$-risk compared to SL.
In forthcoming section, we investigate the performance of the PRSL estimator compared to the LASSO, via a Monte Carlo simulation.
Simulation
==========
In this section we conduct a Monte Carlo simulation study to evaluate the performance of the PRSL with respect to the LASSO of Tibshirani (1996).
We generate the vector of responses from following model: $$\label{sim.mod}
Y_i=\beta_1x_{1i}+\ldots+\beta_px_{pi}+\epsilon_i, i=1,\ldots,n,$$ where $E(\epsilon_{i}|{\boldsymbol{x}}_i)=0$ and $E(\epsilon_{i}^2)=1$. Furthermore, we generated the predictors $x_{ij}$ and errors $\epsilon_{i}$ from $\mathcal{N}\left(0,1\right)$. We consider the sample size $n \in \left \{50,100\right \}$ and the number of predictor variables $p \in \left \{10, 20, 30 \right \}$. We also consider the regression coefficients are set $\beta_j=c\sqrt{2\alpha}j^{-\alpha/2}$ with $\alpha=0.1,0.5,1$ for $j=1,\cdots,p$. The larger values of $\alpha$ indicates that the coefficients $\beta_j$ decline more quickly with $j$. Also, the value of $c$ controls the population ${\rm R}^2=c^2/(1+c^2)$, and is selected on a 20-point grid in $[0,{\rm R}^2]$.
The number of simulations is initially varied. Finally, each realization is repeated 1000 times to obtain stable results. For each realization, we calculated the MSE of suggested estimators. All computations were conducted using the software R.
The performance of an estimator $\widehat{\boldsymbol{\beta}}_n^{\rm \ast}$ was evaluated by using MSE criterion, scaled by the MSE of LASSO so that the values of relative MSE (RMSE), is given by $$\textnormal{RMSE}\left( \widehat{\boldsymbol{\beta}}_n^{\rm \ast}\right) =\frac{\textnormal{MSE}\left( \widehat{\boldsymbol{\beta}}_n^{\rm \ast}\right) }{\textnormal{MSE}\left( \widehat{\boldsymbol{\beta}}_n^{\rm L}\right) }.
\label{rmse}$$
If the RMSE is less than one, then it indicates performance superior to the LASSO.
![ The RMSEs of suggested estimator for different values of $\alpha$ when $\rm R^2\in [0,0.5]$ \[Fig:1\]](RMSEn50_100){width="15cm" height="10cm"}
![ The RMSEs of suggested estimator for different values of $\alpha$ when $\rm R^2\in [0,0.8]$ \[Fig:2\]](RMSE2n50_100){width="15cm" height="10cm"}
The results are reported graphically in Figures \[Fig:1\] and \[Fig:2\] for the ease of comparison. Each figure has six panel plots which correspond to three values of $\alpha$ for $n = 50, 100$ and $p=10,20,30$, and presents the RMSE values of the estimators in Equation \[rmse\] as a function of the population $\rm R^2$. According to these figures, we can see clear trends. For example, in Figure \[Fig:1\](b), if the $\rm R^2$ varies from 0 to 0.1, then the PRSL has the smallest RMSE when $\alpha=0.1$, which indicates that it performs better than LASSO, followed by the PRSL when $\alpha=0.5$ and $\alpha=1$. On the other hand, for the intermediate values of $\rm R^2$, the performance of PRSL is less efficient than the performance of LASSO. Also, the RMSE of PRSL when $\alpha=0.1,0.5$ is superior to LASSO when $\rm R^2$ is getting increased. If we take a closer look to Figure \[Fig:1\](e), which is the case $(n,p)=(100,20)$, then one can see a similar trend except that the RMSEs of the PRSL outshine the LASSO for each values of $\alpha$ when the population $\rm R^2$ is approaching to $0.5$. In Figure \[Fig:2\], as summary, the performance of PRSL is more efficient than LASSO for the small values of population $\rm R^2$, and it looses its efficiency when we increase in small amounts $\rm R^2$, and finally the relative performance of all estimators become almost similar when $\rm R^2$ is close to 0.8.
Prostate Data
=============
Prostate data came from the study of Stamey et al. (1989) about correlation between the level of prostate specific antigen (PSA), and a number of clinical measures in men who were about to receive radical prostatectomy. The data consist of 97 measurements on the following variables: log cancer volume (lcavol), log prostate weight (lweight), age (age), log of benign prostatic hyperplasia amount (lbph), log of capsular penetration (lcp), seminal vesicle invasion (svi), Gleason score (gleason), and percent of Gleason scores 4 or 5 (pgg45). The idea is to predict log of PSA (lpsa) from these measured variables.
A descriptions of the variables in this dataset is given in Table \[tab:1\].
Variables Description Remarks
----------- -------------------------------------------- ------------------
lpsa Log of prostate specific antigen (PSA) Response
lcavol Log cancer volume
lweight Log prostate weight
age Age Age in years
lbph Log of benign prostatic hyperplasia amount
svi Seminal vesicle invasion
lcp Log of capsular penetration
gleason Gleason score A numeric vector
pgg45 Percent of Gleason scores 4 or 5
: Discription of the variables of prostate data[]{data-label="tab:1"}
Playing around with the ${\boldsymbol{g}}$ function in (2.4) may give better candidates compared to LASSO. In this section, we further investigated the performance of the following alternatives $$\begin{aligned}
\widehat{\boldsymbol{\beta}}_n^{\rm SL2}&=&\left(\left\{1-\frac{a}{\mathcal{W}_n+1}\right\}\widehat\beta_{jn}^{\rm L}|j=1,\ldots,p\right)^{\rm \top}\\
\mbox{or}&&\cr
\widehat{\boldsymbol{\beta}}_n^{\rm SL3}&=&\left(\left\{1-\frac{ar\left(\mathcal{W}_n\right)}{\mathcal{W}_n}\right\}\widehat\beta_{jn}^{\rm L}|j=1,\ldots,p\right)^{\rm \top}\end{aligned}$$ where $r(x)$ is a concave function w.r.t to $x$, i.e., $r(x)=\sqrt{x}$ or $r(x)=\log |x|$. The latter can be viewed as a Baranchik-type estimator.
LASSO PRSL SL2 SL3($r(x)=\sqrt{x}$) SL3($r(x)=\log |x|$)
--------- ------- ------- ------- ---------------------- ----------------------
coef 2.478 2.294 2.303 0.852 1.691
lcavol 0.472 0.437 0.438 0.162 0.322
lweight 0.186 0.173 0.173 0.064 0.127
age 0.000 0.000 0.000 0.000 0.000
lbph 0.000 0.000 0.000 0.000 0.000
svi 0.368 0.340 0.342 0.126 0.251
lcp 0.000 0.000 0.000 0.000 0.000
gleason 0.000 0.000 0.000 0.000 0.000
pgg45 0.000 0.000 0.000 0.000 0.000
RPE 1.000 0.764 0.766 0.705 0.335
: Estimation coeffecients of the variables of prostate data[]{data-label="Tab:2"}
Our results are based on $1000$ case resampled bootstrap samples. Since there is no noticeable variation for larger number of replications, we did not consider further values. The performance of an estimator is evaluated by its prediction error (PE) via 10-fold cross validation (CV) for each bootstrap replicate. In order to easily compare, we also calculated the relative prediction error (RPE) of an estimator with respect to the prediction error of the LASSO. If the RPE of an estimator is larger than one, then its performance is superior to the LASSO. In Table \[Tab:2\], we report both the estimation coefficient and the APEs of the five methods. According to these results, all suggested estimators outperform the LASSO.
![ The estimation of coefficients versus $s$ tuning parameter of each methods. Here $s$ is selected via 10-fold CV. The vertical line $\widehat s=0.44$ is selected by “one standard error" rule. \[Fig:coef:paths\]](prostate_coef_paths.eps){width="14cm" height="10cm"}
![ Box plots of 1000 bootstrap values of the listed mothods coefficient estimates for the eight predictors in the prostate cancer example \[Fig:boxplot:prostate\]](prostate_boxplot){width="14cm" height="10cm"}
Figure \[Fig:coef:paths\] shows each estimates as a function of standardized bound $s=|{\boldsymbol{\beta}}|/max|{\boldsymbol{\beta}}|$. The vertical line represents the model for $\widehat s = 0.44$, the optimal value selected “one standard error" rule with 10-fold CV, in which we choose the most parsimonious model whose error is no more than one standard error above the error of the best model. So, all methods gave non-zero coefficients to lcavol, lweight and svi. Also, Figure \[Fig:boxplot:prostate\] shows box plots of 1000 bootstrap replications of each methods with $\widehat s=0.44$. And, the results are consistent with Tibshirani (1996).
Conclusions
===========
In this paper, we employed the shrinkage idea of Stein (1981) to shrink the LASSO of Tibshirani (1996) more. Hence, under the concept of double shrinking, we proposed a double shrinkage estimator namely Stein-type LASSO. Some other similar double shrinkage estimators including the positive part of Stein-type LASSO also proposed as alternative options. Performance analysis of the proposed estimators investigated through a Monte-Carlo simulation as well as a real data analysis. The new set of estimators propose smaller $L_2$-risk compared to the LASSO. Moreover, the prostate cancer data analysis illustrated that the Stein-type LASSO estimators have smaller prediction error compared to the LASSO.
Regarding the function ${\boldsymbol{g}}(\cdot)$ in , numerical analysis illustrated that convex and differentiable functions behave superiorly. All our candidates for ${\boldsymbol{g}}(\cdot)$ satisfied the regularity condition . Further, our proposal will also work for the minimizer of for all values $\gamma>0$, including the ridge regression estimator and subset selector. Hence, the proposed methodology can be applied for other estimators. Apart from this, there are many competitors to the LASSO in the context of variable selection, where we only focused on LASSO for the purpose of defining double shrinking idea. For further research, one can use this method to define double shrunken estimator other than the Stein-type LASSO. As such one can define the Stein-type SCAD estimator.
References {#references .unnumbered}
==========
\[\]
\[J\]ames, W. and Stein, C. (1961). Estimation of quadratic loss, Proc. of the Fourth Berkeley Symp. on Math. Statist. Prob., 1, 361–379.
\[K\]night K, Fu W. (2000). Asymptotics for LASSO-type estimators. [*Ann. Statist.*]{}, 10;28(5):1356-1378.
\[S\]aleh. A. K. Md. Ehsanes. (2006). Theory of Preliminary Test and Stein-Type Estimation with Applications, Wiley; United Stated of America.
\[S\]aleh, A. K. Md. Ehsanes and Raheem, E. (2015). Improved LASSO, arXiv:1503.05160v1, 1-46.
\[S\]tamey, T.A., Kabalin, J.N., McNeal, J.E., Johnstone, I.M., Freiha, F., Redwine, E.A. and Yang, N. (1989). Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate: II. radical prostatectomy treated patients, Journal of Urology 141(5), 1076–1083.
\[S\]tein, C. (1981). Estimation of the mean of a multivariate normal distribution, [*Ann. Statist.*]{} 9, 1135-1151.
\[T\]ibshirani, R. (1996). Regression shrinkage and selection via the LASSO, [*J. Royal. Statist. Soc. B.*]{}, [**58**]{}(1), 267-288.
[^1]: Corresponding author. Email:
|
---
abstract: 'Ever increasing number of cellular users and their high data requirements, necessitates need for improvement in the present heterogeneous cellular networks (HetNet). Carrier sensing prevents base stations within a certain range of the transmitter from transmitting and hence aids in reducing the interference. Non-orthogonal multiple access (NOMA) has proven its superiority for the 5th generation (5G) networks. This work proposes a mathematical model for an improved HetNet with macro base station (MBS) and femto base station (FBS) tier. The FBS tier is equipped to support NOMA and carrier sensing for its transmissions. Offloading is performed for load balancing in HetNet where the macro users (MU) from congested MBS tier are offloaded to the FBS tier. The FBS tier pairs the offloaded MU (OMU) with an appropriate pairing user (PU) to perform NOMA. The performance of the OMU is studied under different channel conditions with respect to the available PU at the FBS and some useful observations are drawn. A decrease in outage probability by $74.04\%$ for cell center user (CCU) and $48.65\%$ for cell edge user (CEU) is observed for low density FBS. The outage probability decreases by $99.60\%$, for both the CCU and CEU, for high density FBS using the proposed carrier sensing in NOMA. The results are validated using simulations.'
author:
- |
Pragya Swami, Vimal Bhatia, *Senior Member, IEEE*, Satyanarayana Vuppala, *Member, IEEE*,\
and Tharmalingam Ratnarajah, *Senior Member, IEEE* [^1] [^2]
bibliography:
- 'refJ2.bib'
title: 'Offloading of Users in NOMA-HetNet Using Repulsive Point Process'
---
Non-orthogonal multiple access, stochastic geometry, repulsive point process, heterogeneous cellular network, outage probability.
Introduction
============
Femto base station (FBS) deployment in the existing cellular network is one of the most viable solution to meet the intense consumer demands for mobile data while catering to ever increasing number of cellular users. The resulting network of macro base station (MBS) and FBS, termed as heterogeneous cellular networks (HetNet), provides a cost-effective expansion to existing cellular wireless networks. To increase this capacity further, non-orthogonal multiple access (NOMA), now included in the Release 15 of 3rd generation partnership project (3GPP), has gained wide interest recently as an enabling technique for 5th generation (5G) mobile networks and beyond. NOMA has proven to provide better spectral efficiency [@random], [@saito] as compared to orthogonal multiple access (OMA) adopted by 4G mobile communication systems standardized by 3GPP such as Long Term Evolution (LTE) [@LTE] and LTE-Advanced [@LTEA].
In a HetNet, the FBS acts as an offloading spot and helps in load balancing by serving some of the macro users (MU), called as offloaded user (OU) at the FBS, from the congested MBS tier [@offloadingsarabjit], [@VTC]. The FBSs are deployed opportunistically or randomly making conventional frequency planning strategies very difficult (and redundant) in a two-tier network [@pastpresentfuture], [@interferencesystem]. The lack of coordination between FBSs leads to randomized co-tier interference which results in performance degradation. This interference can be mitigated by using a medium access channel (MAC) protocol, as one of the possible solution, involving carrier sensing in the FBS tier for interference management such that the transmitting FBS does not end up using the channel that is already occupied by other FBS. Carrier sensing forbids the FBS contending for the same channel to transmit simultaneously.
To do this, each FBS senses the channel and transmits only if the channel is not occupied by any other contender. The distance within which carrier sensing is performed is called as contention radius (CR) and the FBSs within CR are called as contenders. Clearly, one of the contender wins and accesses the spectrum. Hence, we can say that this carrier sensing creates an exclusion region (equals to CR) around a FBS within which no other FBSs are allowed to transmit. The exclusion region around a FBS can be visualized as an existence of a minimum distance, equal to CR, between the FBSs. This makes the FBS’s positions correlated with other FBSs, since, it is required to maintain a minimum distance between FBSs. The formation of exclusion region around FBSs can be modeled spatially using repulsive point processes (RPP) for e.g. hard core point process (HCPP) with a hard core parameter (HCP). While modeling the base stations using RPP, the HCP physically equals the CR within which the base stations contend for spectrum access. The Poisson point process (PPP) model assumes no correlation amongst the nodes’ position thereby rendering PPP assumptions inaccurate for modeling the active transmitters that coordinates for spectrum access using carrier sensing [@stochasticsurvey]. The inaccuracy of PPP to model location of base stations (BS) for different tiers of HetNet is demonstrated in [@PPP1; @PPP2; @pinto; @andrewsPPPproblem]. To capture the characteristics of cellular networks using carrier sensing or MAC protocol, point processes for e.g., RPP where distances among BS are fixed have proved to be more accurate than the PPP assumptions [@stochasticsurvey], [@elsawy]. Thus, in this work we consider and analyze RPP for modeling the FBS tier.
Bertil Mat[é]{}rn proposed three approaches to construct a RPP from parent PPP leading to the formation of Type I, Type II, and Type III Mat[é]{}rn Hard core point process (MHCPP). Here, primary points are used to refer to the points in the parent PPP while secondary points are used to refer to the points of the constructed MHCPP. Type I MHCPP deletes all the primary points from the parent PPP that coexists within a distance less than the HCP. The construction of Type II MHCPP requires every point to be associated with a time mark and deletes the primary points coexisting within a distance smaller than the HCP, provided it also has a lowest time mark. However, this method leads to underestimating the intensity of simultaneously active transmitters [@elsawy], [@INRIA]. Type III MHCPP removes this flaw by following similar procedure as that for Type II, however, the primary point is deleted only if it coexist within a distance less than the HCP from another secondary point with a lower time mark. The aggregate interference for cognitive radio network under MHCPP is characterized in [@M1], [@M2], which is later approximated as a PPP model assuming fading and shadowing effects.
The MBS are generally studied as serving users with similar requirements, hence it distributes its power equally amongst them. On the other hand, for the deployed FBSs the range of user requirements varies from ultra high definition video transmissions to low power sensors in an Internet of Things setup [@IoT], [@NOMAIoT]. It may also be required to fulfill such varied requirements simultaneously. Hence, in this work, we use power splitting amongst the users appropriately using NOMA in FBSs to support the offloaded users. NOMA uses superimposition of users’ signal with different channel conditions in power domain unlike OMA [@LTE] which uses orthogonality in frequency, time or code to serve multiple users. In the literature, most system models that employ NOMA generally do not account for practical system characteristics such as the minimum distance constraint between the base stations. For instance, [@random], [@underlay] consider PPP distributed base stations (BS) for their analysis without employing any MAC protocol. Therefore, advanced system models with more realistic approach should be analyzed. Since, power domain NOMA involves superimposing signals of users with different channel conditions in power domain; to identify and differentiate the femto users (FU) with different channel conditions we assume two types of users namely cell center user (CCU) and cell edge user (CEU). CCU being close to the FBS as compared to the CEU has better channel condition than CEU. The difference in channel conditions leads to different performance at the two types of users, hence, user fairness needs to be maintained which is studied in the context of NOMA in [@userfairnessMIMONOMA]. CEU performance and user fairness is also studied in [@VTC; @CEU2; @fairnesssystem; @fairnesssystem2].
Motivation and Contribution
---------------------------
Motivated by the need of a carrier sensing in the current HetNet to model the randomized FBS tier, for interference management, the usefulness of the concept of offloading for load balancing in the congested HetNet, and the advances of NOMA to meet the requirements of 5G and beyond services, we propose a framework of an offloading model in HetNet using NOMA at FBS tier, referred as NOMA-HetNet. The FBS tier also uses carrier sensing to manage the interference caused by their dense and random deployments. The carrier sensing in FBS tier with NOMA is modeled using an RPP[^3], as explained in Section \[sec:retaining\] since, the PPP assumptions renders inaccurate results for modeling correlated points that occur as a result of carrier sensing. Towards this end, to the best of our knowledge, there exists no literature which studies and analyzes the impact of RPP with NOMA in the offloading environment of HetNets.
The key contributions of this work are listed below
- An analytical framework is designed for a HetNet where the FBS tier is equipped with carrier sensing for interference management and NOMA for power splitting.
- To model the carrier sensing amongst the FBSs we use RPP modeling. A retaining model for the RPP is explained in Section \[sec:retaining\] to decide the density (or number) of active FBSs based on the carrier sensing used.
- Offloading is performed for load balancing by handing some users from the congested MBS to the FBS tier. Since, FBS uses NOMA to support the offloaded users, it pairs the incoming offloaded macro user (OMU) with an available pairing user (PU). Also with NOMA, it becomes important to know whether the OMU is a CCU or CEU with respect to the available OMU. Hence, the concept of NOMA compatibility is discussed and the impact of offloading is analyzed.
- A comparative study between the HetNet using PPP and RPP for modeling FBS tier with NOMA is performed. To make the study broader we also include the comparison of the proposed model with that of HetNet using OMA technique and modeling based on PPP.
Rest of the paper is organized as follows. The general system model is given in Section \[sec:GSM\] and the retaining model is discussed in Section \[sec:retaining\]. Section \[sec:analytical\] derives some useful expressions of outage probabilities for the proposed model using RPP. Numerical results are discussed in Section \[sec:results\]. Finally, the work is concluded in Section \[sec:conclusion\].
General System Model {#sec:GSM}
====================
A HetNet comprising of MBS underlaid with FBS is considered in the analysis where the FBS tier supports offloaded users from congested MBS tier for load balancing. FBS tier employs NOMA (hence also referred as FBS-NOMA) and carrier sensing for transmission. We assume that $\Omega_{m}$ denotes the PPP distributed nodes for MBS tier with intensity $\lambda_{m}$ and $\Omega_{u}$ denotes the PPP distributed nodes for users with intensity $\lambda_{u}$. $\Omega_{f}$ denotes the marked PPP (refer Fig. \[fig:system\]) with intensity $\lambda_{f}$, i.e, $\Omega_{f} \in {(x_i^f, p_i^f); i = 1, 2, 3, \ldots }$, where $x_i^f$ denotes the position of $i^{th}$ FBS with associated time mark, $p_i^f$, distributed uniformly in the range $[0, 1]$. The marked PPP distribution of FBS tier underlaid with MBS tier is shown in Fig. \[fig:system\]. The FBS that are retained using carrier sensing is explained in detail in Section \[sec:retaining\]. $\Omega_{f}^R$ denotes the set constructed using carrier sensing at FBS tier. It should be noted that the MBS tier does not perform carrier sensing. Assuming $t \in \{m,f\}$ denoting MBS, FBS tier, respectively, the transmit power of $t^{th}$ tier is denoted by $P_t$ and the target rate of a typical user for both the tiers is represented by $R$. $\mathcal{Y}_t$ denotes the coverage range of the BS of $t^{th}$ tier. Bounded path loss model is considered as $L(r_t)=\frac{1}{1+r_t^{\alpha_t}}$ which ensures that path loss is always smaller than one even for small distances [@shotnoise], where $r_t$ is the distance between the typical user and the associated BS of $t^{th}$ tier and $\alpha_t$ is the path loss exponent of $t^{th}$ tier. Hence, the total channel gain is given by $\vert h \vert^2=\vert \hat{h} \vert^2 L(r_t)$, where $\vert \hat{h}\vert^2$ is Rayleigh distributed. The overall system transmission bandwidth is assumed to be 1 Hz. We assume a guard zone of radius $r_g >1$ around a receiver as shown in Fig. \[fig:system\]. The interference at a receiver is calculated beyond this guard zone.
![ System Model []{data-label="fig:system"}](J2.eps "fig:"){height="5.5cm" width="9cm"}\
[**[*Note*:]{}**]{} Throughout the paper $\vert \hat{h} \vert^2$ implies Rayleigh distribution, $\vert \tilde{h} \vert^2$ will denote unordered channel gains and $\vert {h} \vert^2$ will imply ordered channel gain.
Retaining Model for FBS Tier {#sec:retaining}
============================
In this section, we derive the density of active FBS retained under the applied carrier sensing. The process of finding the active FBSs using carrier sensing is termed as thinning process. The marked PPP model provides the baseline model (or parent model) for the distribution of all FBSs while the subset of FBSs that succeed to access the spectrum will be modeled using RPP. The parent model distribution consists of uniformly marked PPP as shown in Fig. \[fig:system\] and the contention radius is denoted by $r_c$. To carry out the contention, we first find the neighborhood set of a generic FBS, $x_i^f$, contending for the channel. The neighborhood set of generic FBS is denoted by $\mathcal{N}_{x_i^f}$. The notion of received signal strength is used to decide the neighborhood set of a generic FBS, mathematically written as $N_{x_i^f} = \{(x_i^f, p_i^f) \in \Omega_{f} | \gamma (x_i^f, x_j^f) > T_B\}_{i\ne j}$ where $\gamma (a,b)$ denotes the received SNR at node $a$ from node $b$. This implies that $N_{x_i^f}$ is the set of neighbor FBSs such that the received signal-to-noise ratio (SNR) at the generic FBS at $x_i^f$ is greater than the BS-sensing threshold, $T_B$. The criterion for selecting the FBS amongst all the FBS in the neighborhood is based on their time marks. The FBS that qualifies to transmit (or alternatively we may refer to the FBS which is retained) is determined by the lowest time mark amongst its neighborhood set as can be seen from Fig \[fig:system\]. For the three neighborhood set shown in Fig \[fig:system\], namely $N_{x_1^f}, N_{x_2^f}$, and $ N_{x_3^f}$, the FBS that wins the contention carries the lowest marks amongst all the other neighbors. This method is similar to the general carrier sensing multiple access (CSMA) protocol [@CSMAfollowsRPP1], [@CSMAfollowsRPP2]. Following this procedure, we find the retaining probability of FBS under the above conditions.
In the first step, the contention radius, $r_c$, is calculated. By bounding the observation to the region $B_{x_i^f}(r_c)$, gives us all the FBS inside the radius of $r_c$ centered at $x_i^f$. The contention radius is taken to be sufficiently large such that the probability of an FBS in the neighborhood of $x_i^f$ lying beyond $r_c$ is negligible. Mathematically, we write it as $$\label{eq:contentioncondition}
\mathcal{P}\left\{\rho_f \vert \hat{h}_(i,j) \vert^{2} L(r_j)> T_B\vert r_j > r_{c}\right\}\leq\epsilon,$$ where $\rho_f=P_f/\sigma_f^2$ denotes the transmit SNR of FBS tier, $\sigma_f^2$ is the noise variance, $r_j$ is the distance of $j^{th}$ BS in disc $B_{x_i^f}(r_c)$ to BS $x_i^f$. By rearranging (\[eq:contentioncondition\]), we calculate the contention radius as, $$\label{eq:contentionradius}
r_{c}=\left(\frac{\rho_f \vert \hat{h}_j \vert^2 L(r_j)}{T_B} {F}_{X}^{-1}(\epsilon)\right)^{1/\alpha_f},$$ where $X=\vert \hat{h}_j \vert^2$, ${F}_{X}^{-1}(\epsilon)$ represents the inverse of the cumulative distribution function (CDF) of the fading distribution evaluated at infinitesimal $\epsilon$. The neighborhood success probability (NSP) is defined as the probability that an FBS $x_j^f$ qualifies the minimum signal strength of $T_B$ at $x_i^f$ and become its neighbor. Amongst the neighboring FBS in $N_{x_i^f}$, the FBS with the lowest time mark wins the contention and is allowed to access the channel. The NSP is calculated using (\[eq:contentionradius\]) as $$\label{eq:NSP1}
\mathcal{P}_{s}=\mathcal{P}\left\{\rho_f X L(r_j)\geq T_B \vert x_{j}^f\in \mathcal{B}_{x_{i}^f}(r_{c})\right\}.$$ With the assumption of Rayleigh faded channel and bounded path loss model we write (\[eq:NSP1\]) as $$\begin{gathered}
\label{eq:NSP2}
\mathcal{P}_{s}=\int_{0}^{1} f(r_j) \left(1-F_{X}\left(\frac{T_B r_j^{\alpha_f} }{\rho_f}\right)\right)\mathrm{d}r_j + \\ \int_{1}^{r_c} f(r_j)\left(1-F_{X}\left(\frac{T_B r_j^{\alpha_f} }{\rho_f}\right)\right)\mathrm{d}r_j, \end{gathered}$$ where $f(r_j)=2 r_j/ r_c^2$. Solving (\[eq:NSP2\]) we get the NSP as $$\begin{gathered}
\label{eq:NSP}
\mathcal{P}_{s}=\frac{1}{r_c^2} e^{-\frac{T_B}{\rho_f}}+ \frac{ 2 ({T_B}/{\rho_f})^{-2/\alpha_f } \Gamma \left(\frac{2}{\alpha_f },{T_B}/{\rho_f}\right)}{\alpha_f r_c^2}-\\
\frac{2 \left(({T_B}/{\rho_f}) r_c^{\alpha_f }\right)^{-2/\alpha_f } \Gamma \left(\frac{2}{\alpha_f },{T_B}/{\rho_f} r_c^{\alpha_f
}\right)}{\alpha_f},\end{gathered}$$ where $\Gamma(a, x)= \int_{x}^{\infty} e^{-t} t^{a-1} dt$ is the incomplete gamma function. From [@elsawy], we can directly write the retaining probability for CSMA protocol as $\mathcal{P}_{R} = \frac {1-e^{N_e P_{s}}}{N_e P_{s}} $, with the expected number of FBSs in the disc of radius $r_c$ around $x_i^f$, i.e. in $B_{x_i^f}(r_c)$, as $N_e = \pi \lambda_f r_c^2 $.
**Remark 1:** The NSP in (\[eq:NSP\]) shows a vital dependence on the selection of $r_c$. As we increase the $r_c$, the probability of a FBS lying in the neighborhood of generic FBS decreases since the received SNR decreases with increase in distance between the base stations. Also, the $r_c$ needs to be selected sufficiently large such that NSP beyond $r_c$ is negligible. Hence, $r_c$ needs to be selected appropriately. Then, the intensity of active number of transmitting FBS using carrier sensing is given by $\lambda_{f}^R = \lambda_{f} \mathcal{P}_{R}$.
Signal to Interference and Noise Ratio at Typical Macro User {#sec:SINRMU}
------------------------------------------------------------
Given the signal intended for a typical macro user (MU) as $x_m$, the signal transmitted by the MBS can be written as $X_{m,tx}=\sqrt{P_m} x_m$ and the received signal can be written as $X_{m,rx}=\sqrt{P_m} x_m \tilde{h}_m +n_m$, where $n_m$ denotes channel noise at MBS tier. The useful signal power, noise and/or interference power for a given signal $X$ can be easily calculated using $P=\mathbb{E}\left[XX^{*}\right]$, where $\mathbb{E}[.]$ denotes the statistical expectation. Hence, the signal to interference and noise ratio (SINR) at a typical MU, normalized by noise variance, can be written as $$\label{eq:SINRMBS}
\text{SINR}_m=\frac {P_m \rho_m^r \vert \tilde{h}_m \vert^2}{\rho_{f} \mathcal{I}_{f}+1},$$ where $\rho_m^r=\mathbb{E}\left[x_m^2\right]/ \sigma_m^2$ and $\rho_{f}=P_{f}/ \sigma_f^2$ denotes the receiving transmit SNR of MBS and the transmit SNR of FBS tier, respectively, and $\sigma_m^2$ and $\sigma_f^2$ represents the noise variance of MBS and FBS tier, respectively. $\mathcal{I}_{f}= \sum _{v\in \Omega_{m}^R/\{0\}} \vert \tilde{h}_v \vert ^2$, where $\vert \tilde{h}_{v}\vert^2$ denotes the total channel gain from $v^{th}$ FBS (except at origin) to typical MU at the origin. It should be noted that we have assumed orthogonality in the MBS tier hence, the co-tier interference is neglected for the analysis of MBS tier. However, for the FBS tier, we consider both the cross-tier interference as well as the co-tier interference.
SINR at Typical Femto User (with NOMA)
--------------------------------------
Let us assume that M femto users (FUs) are being served by an FBS. The channel gains of the FUs are ordered as $\vert{h_1}{\vert^2} \leq \cdots \leq \vert{h_{M_f}}{\vert^2}$ and their respective power allocation factors are ordered as as ${a_1} \geq \cdots \geq {a_{M_f}}$. Given $x_i$ as the intended signal for $i^{th}$ FU such that $\mathbb{E}[x_i^2]$ is assumed to be equal $\forall{i} \in{(1,2,\cdots,{M_f})}$. The signal transmitted by the FBS is given by $X_{f}=\sum_{i=1}^{M_f} x_i \sqrt{a_i P_f}$. Hence, the signal received at $k^{th}$ typical FU (which can be either CCU or CEU as discussed later in the paper) is given by $X_{f_{rx}}=h_k (\sum_{i=1}^{M_f} x_i \sqrt{a_i P_f}) +n_k$, where $n_k$ denotes the channel noise at $k^{th}$ typical FU and $\vert h_k \vert^2$ denotes the total channel gain at $k^{th}$ typical FU.
SINR at $k^{th}$ typical FU to decode message of $j^{th}$ FU ($j<k$) is given by $${\label{eq:SINRNOMAdecode}}
\text{SINR}_{k\to j}^f=\frac{ \rho_f^r P_f a_j \vert h_k^f\vert^2}{ \rho_f^r P_f \vert h_k^f\vert^2 \sum _{l=j+1}^{M_f} a_l +\rho_f \mathcal{I}_f+\rho_m \mathcal{I}_m+1},$$ where $\rho_f^r=\mathbb{E}[x_i^2]/\sigma_f^2$ denotes the receiving transmit SNR at FU. $a_n$ denotes power allocation factor for FU with index $n={k,j,l}$. $\mathcal{I}_{m}$ denotes the cross-tier interference at typical FU and is given by $\mathcal{I}_{m}=\sum _{x\in \Omega_{m}/\{0\}} \vert h_x \vert ^2$, where $\vert h_{x}\vert^2$ denotes the total channel gain from $x^{th}$ MBS to typical FU assumed to be at origin using Slivnyak’s theorem [@Slivnyak]. $\mathcal{I}_{f}$ denotes the co-tier interference such that $\mathcal{I}_{f}= \sum _{y\in \Omega_{f}^R/\{0\}} \vert h_y \vert ^2$, where $\vert h_{y}\vert^2$ denotes the total channel gain from $y^{th}$ FBS to typical FU. SINR at $k^{th}$ typical FU to decode its own message is given by $${\label{eq:SINRNOMA}}
\text{SINR}_{k}^f=\frac{\rho_f^r P_f a_k \vert h_k^f\vert^2}{ \rho_f^r P_f \vert h_k^f\vert^2
\sum _{l=k+1}^{M_f} a_l +\rho_f \mathcal{I}_f+\rho_m \mathcal{I}_m+1}.$$
Performance Analysis {#sec:analytical}
====================
This section derives the outage probability of MBS tier and FBS tier with NOMA using carrier sensing. Outage probability of the FBS tier with NOMA includes the outage probability of both type of users i.e., the CCU and the CEU.
### SINR Outage Probability Analysis for MBS Tier {#sec:outageMU}
The outage probability of a typical MU is given as follows.
\[prop:OUTMU\] Conditioned on the fact that MU connects to the nearest MBS, the outage probability of a typical MU is given as $$\mathcal{P}_{O}^m=\pi \lambda_m \mathcal{Y}_m^2 \sum\limits_{n = 0}^{N} {b_{n}^m}e^{-c_{n}^m \frac{\phi} {{\rho_m P_m}} } e^{\mu_m^f},$$ where $N$ is a parameter to ensure a complexity-accuracy trade-off, ${b_{n}^m} = - {w_{N}}\sqrt {1 - \theta _{n}^2} \left({1 \over 2}\left({\theta _{n}+1} \right) \right) e^{-\pi \lambda_m \left({1 \over 2}\left({\theta _{n}+1} \right)\mathcal{Y}_m \right)^2 }$, ${b_0} = - \sum\nolimits_{{n} = 1}^N {b_{n}^m}$, ${c_{n}^m} = 1 + {\left({{{{{ \mathcal{Y}_m}}} \over 2}{\theta _{n}} + {{{{ \mathcal{Y}_m}}} \over 2}} \right)^\nu }$, ${c_0} = 0$, ${w_{N}} = {\pi \over N}$, ${\theta _{n}} = \cos \left({{{2{n} - 1} \over {2{N}}}\pi } \right)$ [@random], and $\phi=2^{2 R}-1$ denotes SINR threshold. $$\mu_m^f=-\lambda_m^R \frac{r_g^{-\alpha' } \left(\alpha_f s_m^f F(r_g,\alpha_f)-(\alpha') K \right)}{\alpha'},$$ where $s_m^f=\frac{ c_{n}^m \phi \rho_f} {{ \rho_m P_m }}$, $K=r_g^{\alpha_f} \text{ln} \left(s_m^f r_g^{-\alpha_f
}+1\right)$, $F(r_g,\alpha_f) ={}_2F_1\left(1,\frac{\alpha'}{\alpha_f };2-\frac{1}{\alpha_f};-s_m^f r_g^{-\alpha_f }\right)$ is the hypergeometric function and $\alpha'=\alpha_f-1$.
*Proof*: Please see Appendix \[appendix:P1\].\
### SINR Outage Analysis for FBS Tier (with NOMA) {#sec:outageFUNOMA}
The outage probability at the $k^{th}$ typical FU is expressed as
\[prop:OUTFUNOMA\] Conditioned on the uniform distance of a typical FU from FBS and ordered channel gain of the users, the outage probability at the $k^{th}$ typical FU is given as
$$\begin{gathered}
\label{eq:outageNOMA}
P_{k}^f=\psi _k \sum _{z=0}^{{M_f}-k}\left(\begin{array}{c} {M_f}-k \\ z \\ \end{array} \right) \frac{(-1)^z}{k+z} \sum_{T_k^z} \left(\begin{array}{c} k+z \\ q_0 \ldots q_{N} \\ \end{array} \right) \\
\left(\prod _{{n}=0}^{N} {b_{n}^f}^{q_{n}}\right) e^{-\sum _{{n}=0}^N q_{n} c_{n}^f \frac{\epsilon_{max}} {\rho_f P_f}} \mathcal{L}_{I_m}\left(s_f^m I_m\right) e^{\mu_f^m},
%\prod_{t_f} \mathcal{L}_{\mathcal{I}_{{t_f}}}(s_f \rho_{t_f}^I),
\end{gathered}$$
where $\epsilon_{max}=\text{max} \left(\epsilon_1, \epsilon_2, \dots, \epsilon_k \right)$ such that $\epsilon_j$ is evaluated as $$\epsilon_{j}=\frac{\phi_{j}}{\left({{a_{j}} - {\phi _{j}}\sum\limits_{i = {j} + 1}^{M_f} {a_i}} \right)},$$ where ${\phi _{j}= 2^{R_{j}} - 1}$ and $R_j$ denotes the target data rate of $j^{th}$ user such that $R_j=R ~\forall{j} \in{(1,2,\ldots,{{M_f}})}$. $s_f=\frac{ \epsilon_{max} \sum _{{n}=0}^N q_{n} c_{n}^f}{\rho_f P_f}$, ${b_{n}^f} = - {w_{N}}\sqrt {1 - \theta _{n}^2} \left({{{\mathcal{Y}_f} \over 2}{\theta _{n}} + {{\mathcal{Y}_f} \over 2}} \right)$, ${c_{n}^f} = 1 + {\left({{{\mathcal{Y}_f} \over 2}{\theta _{n}} + {{\mathcal{Y}_f} \over 2}} \right)^\nu }$, ${T_k^z}=\left({{q_0}, \ldots, {q_{N}}}~|~\sum_{i=0}^{N_f} q_i=k+z\right)$, $\psi_k={{{M_f}!} \over {(k - 1)!({M_f} - k)!}}$, $\left(\begin{array}{c} k+z \\ q_0 \ldots q_{N} \\ \end{array} \right) = {{{M_f}!} \over {{q_0}! \ldots {q_{N}}!}}$.
**Remark 2:** It can be noted that (\[eq:outageNOMA\]) contains two expectations terms that contribute to the role of cross-tier and co-tier interference in the outage probability of typical FU. The user with $k=1$ does not perform SIC, hence the term $\epsilon_{max}$ equals $\epsilon_1$. Also the outage probability is dependent on the transmit SNR of both MBS and FBS, and on the user’s target rate. The dependence is directly proportional to the target rate and the transmit SNR of MBS tier, while it is inversely proportional to the the transmit SNR of FBS tier as observed in Fig. \[fig:2\].
*Proof*: Please see Appendix \[appendix:P2\].\
Offloading and NOMA Compatibility (NC) Probability {#sec:moreprobabilities}
--------------------------------------------------
This section discusses the offloading (OF) probability and the NC probability for the proposed model. OF probability is conditioned on the long term power averaged biased-received-power (BRP) received from the FBS and MBS. The NC probability describes whether the OMU is a CEU or CCU with respect to the available PU at FBS. PU is the FU with which the incoming OMU is paired by the FBS and served using NOMA. Section \[sec:offloading\] decides whether MU will be offloaded to FBS tier, and Section \[sec:NC\] decides whether OMU is a CCU or CEU with respect to the available PU.
### OF Probability {#sec:offloading}
OF probability from MBS tier to FBS tier can be calculated as follows.
\[prop:OFFLOADING\] Offloading is based on maximum BRP [@cellassociationandrews], where a user is associated with the strongest BS in terms of long-term averaged BRP at the user. The closed form expression for the OF probability for $\nu_m=3$ and $\nu_f=4$ is given as
$$\begin{gathered}
\mathcal{P}_{}^{m\to f}=-\frac{3}{4} \text{E}\left(\frac{1}{4},\pi \lambda_m \left(\frac{B_m P_m}{B_f P_f}\right)^{\frac{1}{2}} \mathcal{Y}_{f}^{8/3}\right)\\
+\frac{3 \Gamma \left(\frac{3}{4}\right)}{4 ( \pi )^{3/4} \mathcal{Y}_{f}^2 \left(\lambda_m \left(\frac{B_m P_m}{B_f P_f}\right)^{\frac{1}{2}}\right)^{3/4}}- e^{-\pi \lambda_m \mathcal{Y}_{m}^2},
\end{gathered}$$
where $B_m$ and $B_f$ are the bias factor for MBS and FBS tier respectively. $\text{E} (n,x)$ evaluates the exponential integral as $\text{E}(n,x)= \int_1^{\infty } {e^{-xt}}/{t^n} \, dt$ and $\Gamma(x)=\int_0^{\infty } e^{-t} t^{x-1} dt$ is the complete gamma function.
*Proof*: Please see Appendix \[appendix:P3\].\
### NOMA Compatibility (NC) Probability {#sec:NC}
When the offloaded MU is served using NOMA, it becomes necessary to find out how the OMU will be treated by the FBS, i,e, whether the OMU will be accommodated by the FBS as a CCU or CEU with respect to the available PU. The probability of whether FBS can apply NOMA to the OMU or not is decided on whether the OMU satisfies the sufficiently different channel condition criterion and whether it will be accommodated as a CCU or a CEU. This condition for the OMU is checked with respect to the available PU. Assuming that index $k$ refers to the OMU and $n$ for the available PU, the probability of OMU to be offloaded as a CCU with respect to the PU can be calculated as $$\label{eq:pairingNC}
\mathcal{P}_{NC}=\mathcal{P}\left(\frac{\vert{h_n}{\vert^2} }{\vert{h_k}{\vert^2}} < p\right),$$ where $p$ (satisfying $0<p<1$ ) represents the ratio of channel gains PU and OMU. The probability density function (PDF) of the ratio of two order statistics [@orderstatpdfratio],[@globe] is given as $$\begin{gathered}
\label{eq:ratio}
f_{\frac{h_n^2}{h_k^2}} (z)=\frac{{M_f}!}{{(n-1)! (-n+k-1)! \ ({M_f}-k)!}}\\{ \sum_{j_1=0}^{(n-1)} \sum_{j_2=0}^{(-n+k-1)} \
\frac{(-1)^{j_1+j_2} \left(
\begin{array}{c}
n-1 \\
j_1 \\
\end{array}
\right) \left(
\begin{array}{c}
-n+k-1 \\
j_2 \\
\end{array}
\right)}{\left(z \
t_1+t_2\right){}^2}},\end{gathered}$$ where $t_1=j_1-j_2+k-n$, and $t_2={M_f}-k+1+j_2$. Hence, the probability can be calculated using $\mathcal{P}_{NC}=\int_{0}^p f_{({h_n^2}/{h_k^2})} (z) dz$. The value of $p$ signifies the amount of difference in the channel gains between the OMU and PU. Hence, we may say that $p$ is a measure of the channel condition of OMU with respect to the available PU. Results for different values of $p$ are discussed in Section \[sec:results\]. A lower value of $p$ signifies a large difference in the users’ channel gain, while a large value of $p$ signifies smaller difference in the users’ channel gain.
**Remark 3:** A tractable analysis is done with ${M_f}=2$, $k=2$ (OMU), and $n=1$ (PU). Hence, we get the NC probability as $\mathcal{P}_{NC}={2 p}/({p+1})$. NC probability helps us differentiate whether the OMU is a CCU or CEU with respect to the available PU.
Total Outage Probability After Offloading
-----------------------------------------
Combining outage probability, OF probability and NC probability, the total outage probability when a PU is assumed to be available with FBS for the incoming OMU can be written in three cases, depending on the relative channel condition of OMU with respect to the available PU as follows
- Case I: When MU is offloaded to FBS (without NOMA) $$\label{eq:Case1}
\mathcal{P}_{T}=(1-\mathcal{P}_{}^{m\to f})\mathcal{P}_{O}^m+\mathcal{P}_{}^{m\to f} \mathcal{P}_{O}^f.$$
- Case II: When MU is offloaded as a CCU with respect to available PU at FBS (with NOMA) $$\begin{gathered}
\label{eq:Case2}
\mathcal{P}_{{T}}^{C}=(1-\mathcal{P}_{}^{m\to f})\mathcal{P}_{O}^m+\mathcal{P}_{}^{m\to f} \mathcal{P}_{NC} \mathcal{P}_{k}^f.
\end{gathered}$$
- Case III: When MU is offloaded as a CEU with respect to available PU at FBS (with NOMA) $$\begin{gathered}
\label{eq:Case3}
\mathcal{P}_{{T}}^{E}=(1-\mathcal{P}_{}^{m\to f})\mathcal{P}_{O}^m+\mathcal{P}_{}^{m\to f} (1-\mathcal{P}_{NC}) \mathcal{P}_{k}^f.
\end{gathered}$$
**Remark 4:** The above equations in (\[eq:Case1\]), (\[eq:Case2\]), and (\[eq:Case3\]) combine two situations, one where no offloading takes place (denoted by the first terms), and second when offloading occurs (denoted by the second terms). Case II and Case III also includes the NC probability in their second terms as a check for whether the incoming OMU is a CEU or CCU with respect to the available PU. As can be seen from Fig. \[fig:3\] and Fig. \[fig:4\], the NC probability effects the performance of the OMU depending on its relative channel condition with respect to the PU.
Symbols Value
-------------------------------- -------------------------------------
$ P_m $, $P_f$ 40 W, 1 W
$\lambda_m $ $10^{-4}m^{-2}$
$\lambda_f $ $10^{-3}m^{-2}$ and $10^{-1}m^{-2}$
$a_k$ $0.2, 0.8$
$T_B$ $0$dB
$\mathcal{Y}_m, \mathcal{Y}_f$ $1 \text{km}, 5 \text{m}$
: Network Parameters[]{data-label="tab:data"}
Results and Discussions {#sec:results}
=======================
![ Variation of outage probability with transmit SNR for different FBS density ($\lambda_m=10^{-4}$). []{data-label="fig:2"}](Res1.eps "fig:"){height="7cm" width="9cm"}\
![Comparison of outage probability for PPP and RPP modeling at different FBS density ($\lambda_m=10^{-4}$). []{data-label="fig:3"}](Res2.eps "fig:"){height="7cm" width="9cm"}\
In this section, outage probability of the NOMA-HetNet with carrier sensing is studied based on the analytical expression derived in Section \[sec:analytical\] for the two-tier HetNet, where FBS tier uses NOMA and carrier sensing for its transmissions. The transmit SNR is varied from 0 to 30 dB for both the tiers and $N=10$. Transmit SNR at MBS tier and FBS tier is considered to be fixed at $\rho_m=16dB$ and $\rho_f=0dB$ [@reference], while analyzing the FBS tier and MBS tier performance, respectively. The graphs shows analytical (Anal.) curves verified using Monte Carlo simulation (Sim.) curves.
Fig. \[fig:2\] shows the variation of outage probability with transmit SNR of the FBS tier for different FBS densities. Also, for comparative study, outage probability of FBS tier using OMA (referred as FBS-OMA) modeled with the same carrier sensing, as used for FBS-NOMA, has been plotted. From simulations it is observed that for low transmit SNRs the performance of both FBS-NOMA and FBS-OMA using carrier sensing are nearly same however at higher transmit SNR, FBS-NOMA surpasses the performance of FBS-OMA. The performance of FBS-OMA degrades due to increase in co-tier interference at high transmit SNR from interferers in the vicinity. The improvement shown by FBS-NOMA using RPP results in decrease in the outage probability by $78.57 \%$ as compared to FBS-OMA with RPP. Construction of RPP (to enable the proposed carrier sensing in FBS tier) from parent PPP removes the FBS that do not fulfill the hard core parameter criterion (or minimum distance criterion between FBSs). This leads to the removal of nearby interferers, that have a large contribution in the total interference at typical FU. Clearly, for a dense FBS network the number of such removals will be higher as compared to a sparse FBS network. Hence, for a higher density the number of FBSs removed will be more as compared to when the FBS density is assumed to be low. This renders a major impact on the net interference at typical FU and hence also on the outage performance of FBS-NOMA. Since, the dominant interferers are removed, the outage probability of a FU (both CEU and CCU) decreases. It is worth pointing that the increase in FBS density (from $10^{-3}$ to $10^{-1}$) has a higher impact on CEU ($90.30 \%$ decrease in outage probability) as compared to CCU ($52.10 \%$ decrease in outage probability) as can also be observed from Fig. \[fig:2\]. Since a CCU is already present near to an FBS, increasing FBS density does not impact the CCU much. However, as mentioned earlier, CEU is farther away from FBS and has poorer channel condition as compared to the CCU. One way to improve the quality of service of CEU is to increase the density of FBS such that chances of an FBS lying close to CEU increases. However, higher density also implies higher co-tier interference hence, increasing the density does not always imply an improved performance. Employing carrier sensing on FBS-NOMA and hence using RPP to model the FBS-NOMA network with higher density, instead of using PPP, guarantees an increased chances of an FBS lying closer to CEU in addition to managed co-tier interference due to the thinning process from carrier sensing. Hence, increasing the density of FBS tier leads to larger decrease in outage probability for a CEU as compared to CCU. This improvement is suppressed in high density FBS network modeled using PPP due to increased co-tier interference from large number of FBS as shown in Fig. \[fig:3\]. Hence, we may infer that RPP also caters to the well known issue of performance enhancement in terms of decreased outage probability for CEUs by compensating the drawback of increased co-tier interference of PPP modeled FBS tier at higher densities.
Fig. \[fig:3\] shows the comparison of outage probability of FBS-NOMA tier for the two cases, when the network is modeled using PPP and using RPP. The current literature shows the performance enhancement of NOMA over OMA, however uses PPP assumptions for modeling the BSs. When the NOMA network is modeled using RPP, it gives even better results as compared to when the network is modeled using PPP as observed from Fig. \[fig:3\]. The reason, as discussed earlier, is the reduced co-tier interference due to the removal of dominant interferers, due to the thinning process of RPP, that otherwise hinders the performance of high density networks. As the density of FBS tier is increased for both PPP and RPP model, it is observed that higher FBS density increases the outage probability of FBS-NOMA modeled using PPP while decreases the outage probability of FBS-NOMA modeled using RPP. Carrier sensing manages the interference and the increasing density has a positive impact on the FBS tier performance instead of a negative effect as seen for PPP. As an observation it can also be noticed that the performance improvement (between PPP and RPP) for CCU ($74.04 \%$ decrease in outage probability) is higher for low density as compared to CEU ($ 48.65 \%$ decrease in outage probability). However, for higher density this performance enhancement becomes nearly the same ($ 99.6 \%$ decrease in outage probability) for both CCU and CEU. Hence, we may conclude that FBS-NOMA network modeled using RPP gives better performance, especially for CEU, as compared to PPP modeling.
[{width="5.8" height="5cm"}]{} \[fig:3zeq0pt1\]
[{width="5.8" height="5cm"}]{} \[fig:3zeq0pt5\]
[{width="5.8" height="5cm"}]{} \[fig:3zeq0pt8\]
Fig. \[fig:4\] shows the total outage probability of a typical MU after offloading to FBS tier for an FBS density of $\lambda_f=10^{-3}$. The figures are plotted for three different values of $p$, signifying the different channel conditions of the MU at the time of offloading. A comparison for the two cases, i.e., one where no carrier sensing on FBS-NOMA tier is used and other where carrier sensing is incorporated in the FBS-NOMA tier, is done. Fig. \[fig:4\] and Fig. \[fig:5\] also shows the outage probability of typical MU when offloading is not performed and is compared with the offloading scenario. As can be seen from Fig. \[fig:4\] and Fig. \[fig:5\], offloading to the FBS-NOMA tier modeled using RPP yields a better outage probability, in all the three cases, and for both CCU and CEU, when compared to the offloading to FBS-NOMA tier modeled using PPP assumptions. Also, when compared with the outage probability of typical MU without offloading, we may observe that offloading to FBS-NOMA modeled using RPP yields better outage probability which is not always the case for offloading to FBS-NOMA modeled using PPP.
From Fig. \[fig:4\] it can be observed that for the PPP modeling, when $p=0.1$, i.e, when the difference in OMU and PU is large, offloading as a CCU yields a decreased outage probability while offloading as a CEU does not give any visible improvement in the outage probability. This situation is reversed when $p=0.8$, i.e., when the difference in channel gain between OMU and PU decreases. For $p=0.8$, CEU performance is enhanced while the offloaded CCU does not show any improvement as compared to no offloading. For $p=0.5$, the difference in the channel gain between OMU and PU is larger than that of $p=0.8$ and smaller than $p=0.1$. Hence, for $p=0.5$, the OMU offloaded either as CCU or CEU yields nearly same outage performance.
[{width="5.8" height="5cm"}]{} \[fig:4zeq0pt1\]
[{width="5.8" height="5cm"}]{} \[fig:4zeq0pt5\]
[{width="5.8" height="5cm"}]{} \[fig:4zeq0pt8\]
Next, we increase the density of FBS tier from $\lambda_f=10^{-3}$ to $\lambda_f=10^{-1}$. Fig. \[fig:5\] shows the total outage probability of the OMU as a CCU or CEU to FBS-NOMA tier for an FBS density of $\lambda_f=10^{-1}$. Again, similar to Fig. \[fig:4\], the graphs are plotted for different channel condition of the OMU. It is observed that with the increased FBS density, the impact of carrier sensing can be seen more adequately. It can be noted from Fig. \[fig:5\] that for higher FBS density the offloading to FBS-NOMA tier without carrier sensing degrades the performance of the OMU for some cases. However, with carrier sensing at FBS-NOMA a gain in outage performance is seen for all the three cases of offloading. This is because without carrier sensing at FBS-NOMA, increased density of FBS also increases the aggregate interference at the OMU, while the increased interference is managed by using carrier sensing. Fig. \[fig:5\] (a) is plotted for a value of $p=0.1$ which implies that the difference in channel condition between the OMU and PU is large. This implies that for $p=0.1$ the offloaded CEU will have a much poorer channel condition and offloaded CCU will have a much better channel condition as compared to its corresponding PU. As can be seen from the curves, when modeling of FBS-NOMA is done using PPP assumption, due to lack of interference management the offloaded CEU’s performance is degraded as compared to when no offloading is done. However, a good channel condition for the offloaded CCU decreases the outage probability for the OMU. Similarly, Fig. \[fig:5\] (c) is plotted for $p=0.8$ which implies that there is not much difference in channel condition between OMU and PU. This indicates that the channel condition of OMU as CCU is not as good as compared to when $p=0.1$. This leads to degradation of OMU’s performance when offloaded as a CCU. This is because the power allocation factors are fixed for CEU and CCU to be $0.2$ and $0.8$, respectively, and therefore even though CCU’s channel condition is not as good for $p=0.8$ as it was for $p=0.1$, it is served by the same power as for $p=0.1$. This leads to the increase in outage probability of OMU as CCU. The interference management at the FBS-NOMA tier using RPP compensates any such degradation seen at offloaded CEU or CCU. Thus, we may conclude, no carrier sensing at FBS-NOMA tier leads to the degradation in outage performance of either the offloaded CEU or offloaded CCU, depending on their channel condition during offloading. However, the use of carrier sensing (or modeling using RPP), positions the active FBS such that the interference at the OMU is managed, and thus unlike PPP, none of the three cases of offloading results in degradation at OMU. Hence, carrier sensing in FBS-NOMA tier plays a crucial role in the interference management and hence in the performance gain at OMU from offloading.
Conclusion {#sec:conclusion}
==========
This work presents a mathematical framework of HetNet comprising MBS tier and FBS tier. The FBS tier uses NOMA and carrier sensing for transmission. The carrier sensing is modeled using an RPP. Offloading of MU from MBS tier to FBS tier helps in load balancing in HetNets. The offloading is studied under different channel conditions of OMU with respect to available PU at FBS tier and some useful observations are drawn. The comparison of the proposed carrier sensing model in FBS-NOMA tier is done with two existing techniques namely, FBS-NOMA without carrier sensing and FBS-OMA. Both the comparisons supports the superiority of the proposed model. It is also observed that the use of carrier sensing in high density FBS-NOMA tier (modeled using RPP) provides decreased outage probability for OMU in all the channel conditions during offloading unlike when the FBS-NOMA tier did not performed carrier sensing (modeled using PPP). Thus, the RPP model and its analysis of NOMA-HetNet is vital for 5G and beyond communication systems.
Proof of Proposition 1 {#appendix:P1}
======================
Assume that a typical MU connects to the nearest MBS, small scale fading is Rayleigh distributed, and users follows homogeneous PPP distribution, then by applying the polar coordinates, the cumulative density function (CDF) of the unordered channel gain of MBS tier can be written as [@random] $$\label{eq:CDFMBS}
{F_{\vert\tilde{h}_m{\vert^2}}}(y) = 2 \pi \lambda_m \int_0^{{{ \mathcal{Y}_m}}} \left({1 - {e^{- (1 + {r_m^{\nu_m} })y}}} \right) e^{-2 \pi \lambda_m r_m^2} r_m dr_m.$$ Using G-C quadrature [@GCQuad], (\[eq:CDFMBS\]) can be approximated as $$\label{eq:CDFMBS2}
{F_{\vert\tilde{h}_m{\vert^2}}}(y) \approx \pi \lambda_m \mathcal{Y}_m^2 \sum\limits_{n = 0}^N {b_{n}^m}e^{-c_{n}^m y},$$ where $N$ is a parameter to ensure a complexity-accuracy trade-off, ${b_{n}^m} = - {w_{N}}\sqrt {1 - \theta _{n}^2} \left({1 \over 2}\left({\theta _{n}+1} \right) \right) e^{-\pi \lambda_m \left({1 \over 2}\left({\theta _{n}+1} \right)\mathcal{Y}_m \right)^2 }$, ${b_0} = - \sum\nolimits_{{n} = 1}^N {b_{n}^m}$, ${c_{n}^m} = 1 + {\left({{{{{ \mathcal{Y}_m}}} \over 2}{\theta _{n}} + {{{{ \mathcal{Y}_m}}} \over 2}} \right)^{\nu_m} }$, $
{c_0} = 0$, ${w_{N}} = {\pi \over N}$, ${\theta _{n}} = \cos \left({{{2{n} - 1} \over {2{N}}}\pi } \right)$. The outage probability at a typical MU is given as following $$\begin{aligned}
\label{eq:MUoutagesteps}
\mathcal{P}_{O}^{m}&=\mathcal{P} \left(\alpha_m\times \text{log}(1+\text{SINR}_{{m}})<R \right),\\
%&= \mathcal{P} \left(\frac {P_m \rho_m \vert \tilde{h}_m \vert^2}{\sum_{t_m} \rho_{t_m}^I \mathcal{I}_{t_m}+1}<\epsilon\right)\nonumber\\
&= \mathcal{P} \left({ \vert \tilde{h}_m \vert^2 }<\frac{\phi} {{\rho_m P_m}} \left({1+\rho_f I_f}\right)\right),\nonumber\\
&={F_{\vert\tilde{h}_m{\vert^2}}}\left(\frac{\phi} {{\rho_m P_m}} \left({1 +\rho_f I_f}\right)\right), \nonumber\\
&\stackrel{\text{(a)}}=\pi \lambda_m \mathcal{Y}_m^2 \sum\limits_{n = 0}^N {b_n^m} e^{-c_n^m \frac{\phi} {{\rho_m P_m}} \left({1 +\rho_f I_f}\right)},\nonumber\\
&=\pi \lambda_m \mathcal{Y}_m^2 \sum\limits_{n = 0}^N {b_n^m} e^{- \frac{c_n^m \phi} {{\rho_m P_m}}}\mathbb{E}_{I_f}\left[e^{- \frac{ c_n^m \phi \rho_f I_f} {{\rho_m P_m}}}\right],\nonumber\\
&=\pi \lambda_m \mathcal{Y}_m^2 \sum\limits_{n = 0}^N {b_n^m} e^{- \frac{c_n^m \phi} {{\rho_m P_m}}} \times \mathcal{F}_I,\nonumber\\
%&=\pi \lambda_m \mathcal{Y}_m^2 \sum\limits_{n = 0}^N {b_n^m}e^{-c_n^m \frac{\phi} {{\rho_m P_m}} }\prod_{t_m} \mathcal{L}_{\mathcal{I}_{{t_m}}}(s_m \rho_{t_m}),\nonumber\end{aligned}$$ where (a) follows from (\[eq:CDFMBS2\]) and $\alpha_m$ is the fraction of bandwidth allocated to typical MU, $\mathcal{F}_I=\mathbb{E}_{I_f}\left[e^{- s_m^f I_f}\right]$, $s_m^f=\frac{ c_n^m \phi \rho_f} {{\rho_m P_m}}$ and $\phi=2^{2 R}-1$ denotes SINR threshold. Now, we calculate the cross-tier interference at typical MU from FBS tier as follows similar to [@basemaths1]. $$\begin{aligned}
\mathbb{E}_{\mathcal{I}_{{f}}}\left[e^{-s_m^f \mathcal{I}_{{f}} }\right]&=\mathbb{E}_{\mathcal{I}_{{f}}}\left[e^{-s_m^f \sum _{v\in \Omega_{f}^R/\{0\}} \vert h_v \vert ^2 }\right],\\\nonumber
&\stackrel{\text{(a)}}=\mathbb{E}_{\mathcal{I}_{{f}}}\left[e^{-s_m^f \sum _{v\in \Omega_{f}^R/\{0\}} \vert \hat{h}_v \vert ^2 r_v^{-\alpha_f} }\right],\\\nonumber
&=\mathbb{E}_{\Omega_{f}^R/\{0\}}\left[\prod_{v \in \Omega_{f}^R/\{0\}} \mathbb{E}_{\hat{h}_v} \left[ e^{-s_m^f \vert \hat{h}_v \vert ^2 r_v^{-\alpha_f} }\right]\right],\\\nonumber
&=\mathbb{E}_{\Omega_{f}^R/\{0\}}\left[ e^{-\sum_{\Omega_{f}^R/\{0\}} \text{ln}\left( 1+s_m^f r_v^{-\alpha_f}\right) } \right],\\\nonumber
&\stackrel{\text{(b)}}\ge e^{\mathbb{E}_{\Omega_{f}^R/\{0\}}\left[ {-\sum_{\Omega_{f}^R/\{0\}} \text{ln}\left( 1+s_m^f r_v^{-\alpha_f}\right) } \right]}, \\\nonumber\end{aligned}$$ where (a) follows from the assumption of a guard zone around receivers $r_g>1$, hence, the bounded path loss model is reduced to simply $r_v^{-\alpha_f}$ for the calculation of interference, where $r_v$ is the distance between $v^{th}$ FBS to typical MU and (b) follows from Jensen’s inequality. Let, $\mu_m^f={\mathbb{E}}_{\Omega_{f}^R/\{0\}}\left[-\sum_{v\in\Omega_{f}^R/\{0\}}\Delta_{v}\right],$ where $\Delta_{v}=\text{ln}\left( 1+s_m^f r_v^{-\alpha_f}\right) $. Using Campbell’s theorem [@dhillon], we can write as $$\begin{aligned}
\label{eq:mu} \mu_m^f&={\mathbb{E}}_{\Omega_{f}^R}^{!o}\left[-\sum_{v\in\Omega_{f}^R}\Delta_{v}\right]=\int_{r_g}^{\infty} \lambda_f^R \Delta_{v}(r_v) d r_v,\\\label{eq:mu}\nonumber
&=\int_{r_g}^{\infty} - \lambda_f^R \text{ln}\left( 1+s_m^f r_v^{-\alpha_f}\right) d r_v,\\ \nonumber
&=-\lambda_f^R \frac{r_g^{-\alpha' } \left(\alpha_f s_m^f F(r_g,\alpha_f)-(\alpha') r_g^{\alpha_f} \text{ln} \left(s_m^R r_g^{-\alpha_f
}+1\right)\right)}{\alpha'},\end{aligned}$$ where $F(r_g,\alpha_f) ={}_2F_1\left(1,\frac{\alpha'}{\alpha_f };2-\frac{1}{\alpha_f};-s_m r_g^{-\alpha_f }\right)$ is the hypergeometric function and $\alpha'=\alpha_f-1$.
Proof of Proposition 2 {#appendix:P2}
======================
Using the assumption of homogeneous PPP, the CDF of unordered channel gain of FU can be expressed as [@random], $$\label{eq:CDFFBS}
{F_{\vert\tilde{h}_f{\vert^2}}}(y) = {2 \over {\mathcal{Y}_f^2}}\int_0^{\mathcal{Y}_f} \left({1 - {e^{- (1 + {z^{\alpha_f} })y}}} \right)z\,dz.$$ By applying the G-C quadrature [@GCQuad] to (\[eq:CDFFBS\]), we get $$\label{eq:CDFFBS2}
{F_{\vert\tilde{h}_f{\vert^2}}}(y) \approx {1 \over {\mathcal{Y}_f}}\sum\limits_{n = 0}^{N} {b_{n}^f}e^{-c_{n}^f y},$$ where ${b_{n}^f} =-{w_{N}}\sqrt {1 - \theta _{n}^2} \left({{{\mathcal{Y}_f} \over 2}{\theta _{n}} + {{\mathcal{Y}_f} \over 2}} \right)$, ${c_{n}^f} = 1 + {\left({{{\mathcal{Y}_f} \over 2}{\theta _{n}} + {{\mathcal{Y}_f} \over 2}} \right)^{\alpha_f} }$.
The ordered channel gain of FBS tier is related with the unordered channel gain of FBS tier $F_{\vert{\tilde{h}_f}{\vert^2}}(y)$ [@underlay] as $$\label{eq:CDFFBS3}
{F_{\vert{h_k^f}{\vert^2}}}(y)=\psi _k \sum _{z=0}^{{M_f}-k}\left(\begin{array}{c} {M_f}-k \\ z \\ \end{array} \right) \frac{(-1)^z}{k+z} \left( {F_{\vert{\tilde{h}_f}{\vert^2}}}(y)\right)^{z+k},$$ where $\psi_k={{{M_f}!} \over {(k - 1)!({M_f} - k)!}}$. Substituting (\[eq:CDFFBS2\]) in (\[eq:CDFFBS3\]) and applying multinomial theorem we get the CDF of ordered channel gain as $$\begin{gathered}
\label{eq:CDFFBS4}
{F_{\vert{h_k^f}{\vert^2}}}(y)=\psi _k \sum _{z=0}^{{M_f}-k}\left(\begin{array}{c} {M_f}-k \\ z \\ \end{array} \right) \frac{(-1)^z}{k+z} \sum_{T_k^z} \left(\begin{array}{c} k+z \\ q_0 \ldots q_{N} \\ \end{array} \right) \\
\left(\prod _{{n}=0}^{N} {b_{n}^f}^{q_{n}}\right) e^{-\sum _{{n}=0}^N q_{n} c_{n}^f y},\end{gathered}$$ where ${T_k^z}=\left({{q_0}! \ldots {q_{N}}!}~|~\sum_{i=0}^{N} q_i=k+z\right)$, $\left(\begin{array}{c} k+z \\ q_0 \ldots q_{N} \\ \end{array} \right) = {{{M_f}!} \over {{q_0}! \ldots {q_{N}}!}}$.
Assuming the channel gains of ${M_f}$ users to be ordered as $\vert{h_1^f}{\vert^2} \leq \ldots \leq \vert{h_{M_f}^f}{\vert^2}$, and hence the corresponding power allocation factors as ${a_1} \geq \ldots \geq {a_{M_f}}$ we derive the outage probability at $k^{th}$ FU as
$$P_{k}^f = \mathcal{P}\left(\text{SINR}_{k \to j}^f < {\phi _j}, \text{SINR}_{k}^f < {\phi _k}\right),$$
where $\phi_n=2^{R} - 1$ such that $n$ denotes user index, $\text{SINR}_{k \to j}$ and $\text{SINR}_{k}$ are given in (\[eq:SINRNOMAdecode\]) and (\[eq:SINRNOMA\]), respectively.
We observe that the first user (i.e., $k=1$), according to the ordered channel gains, does not perform any SIC. All users after it (i.e., $k>1$) decodes the information of users preceding them (i.e., $j<k$), and then decode their own message. Since the outage probability is decided based on successful SIC followed by successful decoding of self message, we can write outage probability of $k^{th}$ user as $$P_{k}^f = \mathcal{P}\left({\vert{h_k^f}{\vert^2} < \frac{\epsilon_{max} (1+\rho_f \mathcal{I}_f+\rho_m \mathcal{I}_m)}{\rho_f P_f }}\right),$$ where $\epsilon_{max}=\text{max} \left(\epsilon_1, \epsilon_2, \dots, \epsilon_k \right)$ such that $\epsilon_j$ is calculated as $$\epsilon_{j}=\frac{\phi_{j}}{\left({{a_{j}} - {\phi _{j}}\sum\limits_{i = {j} + 1}^{M_f} {a_i}} \right)},$$ where ${\phi _{j}= 2^{R_{j}} - 1}$ and $R_j$ denotes the target data rate of $j^{th}$ user such that $R_j=R ~\forall{j} \in{(1,2,\ldots,{{M_f}})}$. This gives the outage probability as $$\label{eq:out}
P_k^f=F_{\vert h_k^f \vert^2}(y),$$ where $y=\frac{\epsilon_{max} (1+\rho_f \mathcal{I}_f+\rho_m \mathcal{I}_m)}{\rho_f P_f }$. Hence, the outage probability of $k^{th}$ user can be calculated using (\[eq:out\]) and (\[eq:CDFFBS4\]) as
$$\begin{gathered}
P_{k}^f=\psi _k \sum _{z=0}^{{M_f}-k}\left(\begin{array}{c} {{M_f}}-k \\ z \\ \end{array} \right) \frac{(-1)^z}{k+z} \sum_{T_k^z} \left(\begin{array}{c} k+z \\ q_0 \ldots q_{N} \\ \end{array} \right) \\
\left(\prod _{{n}=0}^{N} {b_{n}^f}^{q_{n}}\right) e^{-\sum _{{n}=0}^N q_{n} c_{n}^f \frac{\epsilon_{max}} {\rho_f P_f}} \mathbb{E}_{\mathcal{I}_{{m}}}\left[e^{-s_m \mathcal{I}_{{m}} }\right] \times \\ \mathbb{E}_{\mathcal{I}_{{f}}}\left[e^{-s_f \mathcal{I}_{{f}}}\right],\end{gathered}$$
where $s_f^m= \frac{ \rho_{m} \epsilon_{max} \sum _{s=0}^N q_n c_n^f}{\rho_f P_f}$ and $s_f^f= \sum _{{n}=0}^N q_{n} c_{n}^f \frac{\epsilon_{max}} { P_f} $. Hence, we write the outage probability as $$\begin{gathered}
P_{k}^f=\psi _k \sum _{z=0}^{{M_f}-k}\left(\begin{array}{c} {{M_f}}-k \\ z \\ \end{array} \right) \frac{(-1)^z}{k+z} \sum_{T_k^z} \left(\begin{array}{c} k+z \\ q_0 \ldots q_{N} \\ \end{array} \right) \\
\left(\prod _{{n}=0}^{N} {b_{n}^f}^{q_{n}}\right) e^{-\sum _{{n}=0}^N q_{n} c_{n}^f \frac{\epsilon_{max}} {\rho_f P_f}} \mathcal{L}_{I_m}(s_f^m) \times e^{\mu_f^f},\end{gathered}$$
Laplace transform of cross tier interference from MBS tier and is calculated as $$\mathcal{L}_{\mathcal{I}_{{m}}}(s)=e^{\pi \lambda_m \left(s^{\delta_m } \Gamma(1-\delta_m , s) -s^{\delta_m } \Gamma(1-\delta_m) \right)},$$ where $\delta_m=2/\nu_m$, $\Gamma(a,x)=\int_{x}^{\infty} t^{a-1} e^{-t}$ and $\Gamma(z)=\int_{0}^{\infty} x^{z-1} e^{-x}$. For co-tier interference, the interference is considered beyond the tagged BS. Hence, we replace $r_g$ by $\mathcal{Y}_f$ in (\[eq:mu\]) to calculate the co-tier interference.
Proof of Proposition 3 {#appendix:P3}
======================
Offloading is based on maximum BRP [@cellassociationandrews] and a user is associated with the strongest BS in terms of long-term averaged BRP at the user. Hence, the offloading probability can be calculated as follows $$\begin{aligned}
\mathcal{P}_{}^{m\to f}&=\mathbb{E}_{r_f}\left[\mathcal{P} \left(B_m P_m r_m^{-\nu_m }<B_f P_f r_f^{-\nu_f }\right)\right],\\
&=\mathbb{E}_{r_f}\left[\left(e^{- \pi \lambda_m r_f^{\frac{2 \nu_f}{\nu_m}}
\left(\frac{B_m P_m}{B_f P_f}\right)^{\frac{2}{\nu_m}} }-e^{- \pi \lambda_m
\mathcal{Y}_m^2}\right)\right], \nonumber\end{aligned}$$where $B_m$ and $B_f$ are the bias factor for MBS and FBS tier respectively. The probability distribution of $r_f$ can be expressed as $f(r_f)=2 r_f/\mathcal{Y}_{f}^2$ assuming uniform distribution of FU around FBS within radius $\mathcal{Y}_f$ and $r_m$ follows $f(r_m)=2 \pi r_m \lambda_{m} \times e^{-\pi r_m^2 \lambda_{m}}$ owing to NN policy. The path loss exponent is taken as $\nu_m=3$ for MBS tier and $\nu_f=4$ for FBS tier. Using these values we get a closed form expression for the offloading probability as
$$\begin{gathered}
\mathcal{P}_{}^{m\to f}=-\frac{3}{4} \text{E}\left(\frac{1}{4},\pi \lambda_m \left(\frac{B_m P_m}{B_f P_f}\right)^{\frac{1}{2}} \mathcal{Y}_{f}^{8/3}\right)\\
+\frac{3 \Gamma \left(\frac{3}{4}\right)}{4 ( \pi )^{3/4} \mathcal{Y}_{f}^2 \left(\lambda_m \left(\frac{B_m P_m}{B_f P_f}\right)^{\frac{1}{2}}\right)^{3/4}}- e^{-\pi \lambda_m \mathcal{Y}_{m}^2},\end{gathered}$$
where $\text{E} (n,x)$ evaluates the exponential integral as $\text{E}(n,x)= \int_1^{\infty } {e^{-xt}}/{t^n} \, dt$ and $\Gamma(x)=\int_0^{\infty } e^{-t} t^{x-1} dt$ is the complete gamma function.
[0.92]{}
[^1]: This work is supported by the MeitY, Indian Institute of Technology Indore, Indore, India, and University of Edinburgh, Edinburgh, UK.
[^2]: P. Swami and V. Bhatia are with Indian Institute of Technology Indore, Indore, India; S. Vuppala is with University of Luxembourg, Luxembourg; and T. Ratnarajah is with the Institute for Digital Communications, University of Edinburgh, Edinburgh, UK.
[^3]: The terms RPP and carrier sensing are used interchangeably in the paper
|
---
abstract: 'We present scanning tunneling spectroscopic and high-field thermodynamic studies of hole- and electron-doped (p- and n-type) cuprate superconductors. Our experimental results are consistent with the notion that the ground state of cuprates is in proximity to a quantum critical point (QCP) that separates a pure superconducting (SC) phase from a phase comprised of coexisting SC and a competing order, and the competing order is likely a spin-density wave (SDW). The effect of applied magnetic field, tunneling current, and disorder on the revelation of competing orders and on the low-energy excitations of the cuprates is discussed.'
address:
- |
Department of Physics, California Institute of Technology\
Pasadena, CA 91125, USA
- 'Pohang University of Science and Technology, Pohang 790-784, Republic of Korea'
author:
- 'N.-C. YEH[^1], C.-T. CHEN, V. S. ZAPF, A. D. BEYER, and C. R. HUGHES'
- 'M.-S. PARK, K.-H. KIM, and S.-I. LEE'
title: |
QUASIPARTICLE SPECTROSCOPY AND HIGH-FIELD\
PHASE DIAGRAMS OF CUPRATE SUPERCONDUCTORS\
– AN INVESTIGATION OF COMPETING ORDERS\
AND QUANTUM CRITICALITY
---
Introduction
============
There has been emerging consensus that the existence of competing orders[@Zhang97; @Sachdev03; @Kivelson03] in the ground state of cuprate superconductors is likely responsible for a variety of non-universal phenomena such as the pairing symmetry, pseudogap, commensuration of the low-energy spin excitations, and the spectral homogeneity of quasiparticle spectra.[@Yeh02a; @Yeh01; @Yeh02b; @Chen02] Among probable competing orders such as the spin-density waves (SDW),[@Demler01; @ChenY02] charge-density waves (CDW),[@LeeDH02] stripes,[@Kivelson03] and the staggered-flux phase,[@Kishine01] the dominant competing order and its interplay with superconductivity (SC) remain not well understood. In this work, we report experimental investigation of these issues via quasiparticle spectroscopic and high-field thermodynamic studies. We compare our results with a conjecture that cuprate superconductivity occurs near a quantum critical point (QCP)[@Sachdev03; @Vojta00] that separates a pure SC phase from a phase with coexisting SC and SDW.[@Demler01; @ChenY02] The scenario of SDW as the relevant competing order can be rationalized by the proximity of cuprate SC to the Mott antiferromagnetism, and also by experimental evidence for spin fluctuations in the SC state of various cuprates.[@Wells97; @Lake01; @Mook02; @Yamada03] Possible relevance of SDW in the cuprates to the occurrence of strong quantum fluctuations and the pseudogap phenomenon will be discussed.
Competing Orders and Quantum Criticality
========================================
We consider a conjecture of competing SDW and SC near a non-universal QCP at $\alpha = \alpha _c$,[@Demler01] where $\alpha$ is a material-dependent parameter that may represent the doping level, the electronic anisotropy, spin correlation, orbital ordering, or the degree of disorder for a given family of cuprates. As schematically illustrated in Fig. 1(a), in the absence of magnetic field $H$, the ground state consists of a pure SC phase if $\alpha _c < \alpha < \alpha _2$, a pure SDW phase if $\alpha < \alpha _1$, and a SDW/SC coexisting state if $\alpha _1 < \alpha < \alpha _c$. Upon applying magnetic field, delocalized spin fluctuations can be induced due to magnetic scattering from excitons around vortex cores, eventually leading to the occurrence of SDW coexisting with SC for fields satisfying $H^{\ast} (\alpha) < H < H_{c2} (\alpha)$ if the cuprate is sufficiently close to the QCP,[@Demler01] and $H_{c2}$ is the upper critical field. In general we expect stronger quantum fluctuations of the SC order parameter in the SDW/SC phase than in the pure SC phase because of excess low-energy excitations associated with the competing SDW. In principle such a difference between coexisting SDW/SC and pure SC phases can be manifested in the field dependence of the thermodynamic properties of the cuprates at $T \to 0$. That is, the proximity of a cuprate to the QCP at $\alpha _c$ can be estimated by determining a characteristic field $H^{\ast}$ using thermodynamic measurements at $T \to 0$, and a smaller magnitude of $(H^{\ast}/H_{c2}^0)$ would indicate a closer proximity to $\alpha _c$ if $\alpha > \alpha _c$, where $H_{c2}^0$ denotes the upper critical field of a given sample at $T = 0$. In contrast, for a cuprate superconductor with $\alpha _1 < \alpha < \alpha _c$, we find $H^{\ast} = 0$ so that SDW coexists with SC even in the absence of external fields, implying gapless SDW excitations (i.e. $\Delta _{SDW} = 0$) and strong excess fluctuations in the SC state. Thus, the thermodynamic quantity $h^{\ast} \equiv (H^{\ast}/H_{c2}^0)$ for a given cuprate is expected to reflect its susceptibility to low-energy excitations and its SC stiffness, which can be confirmed via studies of the quasiparticle spectra taken with a low-temperature scanning tunneling microscope (STM).
Experimental Approach and Results
=================================
To investigate the conjecture outlined above, we employ in this work measurements of the penetration depth $\lambda (T,H)$, magnetization $M(T,H)$, and third-harmonic susceptibility $\chi _3 (T,H)$ on different cuprates to determine the irreversibility field $H_{irr}(T)$ and the upper critical field $H_{c2}(T)$. The degree of quantum fluctuations in each sample is then estimated by the ratio $h^{\ast} \equiv (H^{\ast}/H_{c2}^0)$, where the characteristic field $H^{\ast}$ is defined as $H^{\ast} \equiv H_{irr}(T \to 0)$. The magnitude of $h^{\ast}$ for different cuprates is determined and compared with the corresponding quasiparticle spectra.
Specifically, the experiments results reported in this work consist of studies on the n-type optimally doped infinite-layer cuprate $\rm La_{0.1}Sr_{0.9}CuO_2$ (La-112, $T_c = 43$ K) and one-layer $\rm Nd_{1.85}Ce_{0.15}CuO_{4-\delta}$ (NCCO, $T_c = 21$ K); and the p-type optimally doped $\rm HgBa_2Ca_3Cu_4O_x$ (Hg-1234, $T_c = 125$ K) and $\rm YBa_2Cu_3O_{7-\delta}$ (Y-123, $T_c = 93$ K). Results obtained by other groups on p-type underdoped Y-123 ($T_c = 87$ K),[@O'Brien00] over- and optimally doped $\rm Bi_2Sr_2CaCu_2O_{8+x}$ (Bi-2212, $T_c = 60$ K and 93 K)[@Krusin-Elbaum04; @Krasnov00]; and n-type optimally doped $\rm Pr_{1.85}Ce_{0.15}CuO_{4-\delta}$ (PCCO, $T_c = 21$ K),[@Kleefisch01] are also included for comparison. The optimally doped samples of La-112 and Hg-1234 were prepared under high pressures, with details of the synthesis and characterizations described elsewhere.[@Jung02a; @KimMS98; @KimMS01] Detailed physical properties of the optimally doped Y-123 single crystal[@Yeh93] and NCCO epitaxial thin-film[@Yeh92] have also been given elsewhere.
The $M(T,H)$ measurements were conducted in lower DC fields using a Quantum Design SQUID magnetometer at Caltech, and in high magnetic fields (up to 50 Tesla in a $^3$He refrigerator) using a compensated coil in the pulsed-field facilities at the National High Magnetic Field Laboratory (NHMFL) in Los Alamos. The irreversibility field $H_{irr}(T)$ was identified from the onset of reversibility in the $M(T,H)$ loops, as exemplified in the inset of Fig. 2(a) for La-112 and in the main panel of Fig. 2(b) for Hg-1234. The penetration depths of La-112 and Hg-1234 were determined in pulsed fields up to 65 Tesla by measuring the frequency shift $\Delta f$ of a tunnel diode oscillator (TDO) resonant tank circuit with the sample contained in one of the component inductors.[@Mielke01] Small changes in the resonant frequency can be related to changes in the penetration depth $\Delta \lambda$ by $\Delta \lambda = -\frac{R^2}{r_s}\frac{\Delta f}{f_0}$, where $R$ is the radius of the coil and $r_s$ is the radius of the sample.[@Mielke01] In our case, $R \sim r_s = 0.7$ mm and the reference frequency $f_0 \sim 60$ MHz such that $\Delta f \sim$ (0.16 MHz/$\mu$m)$\Delta \lambda$. Further details for the pulsed-field measurements of La-112 can be found in Ref.[@Zapf04]. Third-harmonic magnetic susceptibility $\chi _3 (T,H)$ measurements were also performed on Hg-1234 sample using a 9-Tesla DC magnet and Hall probe techniques.[@Reed95] The $\chi _3 (T,H)$ data measured the non-linear response of the sample and were therefore sensitive to the occurrence of phase transformation.[@Reed95]
Selected data of these thermodynamic measurements of La-112 and Hg-1234 are shown in Figs. 2(a)-(b), and a collection of measured $H_{irr}^{ab}(T)$ and $H_{c2}^{ab}(T)$ curves for various cuprates are summarized in Fig. 1(b), with the reduced characteristic fields $(h^{\ast})$ of several representative cuprates explicitly given in Fig.1(a). We note that the Hg-1234 sample, while having the highest $T_c$ and upper critical field (estimated at $H_p \sim H_{c2}^{ab} \sim 500$ Tesla) among all cuprates shown here, has the lowest reduced irreversibility line $(H_{irr}^{ab}(T)/H_p)$, where $H_p \equiv \Delta _{SC}^0/(\sqrt{2} \mu _B)$ is the paramagnetic field, and $\Delta _{SC}^0$ denotes the superconducting gap at $T = 0$. The low irreversibility field is not only due to the extreme two-dimensionality (2D) of Hg-1234[@KimMS01] that leads to strong thermal fluctuations at high temperatures, but also due to its close proximity to the QCP, yielding strong field-induced quantum fluctuations at low temperatures.
To further evaluate our conjecture that cuprates with smaller $h^{\ast}$ are in closer proximity to a QCP at $\alpha _c$ and are therefore associated with a smaller SDW gap $\Delta _{SDW}$ and stronger SC fluctuations, we examine the SC energy gap $\Delta _{SC} (T)$ and the quasiparticle low-energy excitations of different cuprates. In Fig. 3(a), we compare the $\Delta _{SC}(T)$ data of La-112, taken with our low-temperature scanning tunneling microscope (STM), with those of Bi-2212 and PCCO obtained from intrinsic tunnel junctions[@Krasnov00] and grain-boundary junctions,[@Kleefisch01] respectively. Noting that the $h^{\ast}$ values are $\sim 0.53$ for PCCO (NCCO), $\sim 0.45$ for Bi-2212, and $\sim 0.24$ for La-112, we find that the rate of decrease in $\Delta _{SC}$ with $T$ also follows the same trend. These differences in $\Delta _{SC}(T)$ cannot be attributed to different pairing symmetries, because La-112 is consistent with $s$-wave pairing symmetry,[@Chen02] NCCO (PCCO) can exhibit $s$-wave[@Alff99] or $d$-wave pairing,[@Tsuei00] depending on both the cation and oxygen doping levels,[@Skinta02; @Biswas02] and Bi-2212 is $d$-wave pairing.[@Renner98; @Pan00; @Hudson01] Therefore, the sharp contrast in the $\Delta _{SC}(T)$ data between La-112 and NCCO (PCCO) suggests that the proximity to the QCP at $\alpha _c$ plays an important role in determining the low-energy excitations of the cuprates. These experimental findings have been further corroborated by recent $t$-$t^{\prime}$-$U$-$V$ model calculations[@ChenHY04] for optimally doped p-type cuprates, which demonstrate that the competing order SDW can appear with increasing temperature even though $\alpha > \alpha _c$ at $T = 0$.
Another experimental confirmation for our conjecture can be found in the quasiparticle tunneling spectra exemplified in Fig. 3(b). We find that excess sub-gap spectral weight exists in La-112, although the quasiparticle spectra of La-112 are momentum-independent and its response to quantum impurities is consistent with $s$-wave pairing.[@Yeh02b; @Chen02; @Yeh03a] The excess sub-gap spectral weight relative to the BCS prediction implies excess low-energy excitations that cannot be reconciled with simple quasiparticle excitations from a pure SC state, and is therefore strongly suggestive of the presence of a competing order with an additional channel of low-energy excitations. In contrast, substantial details of the quasiparticle spectrum on optimally doped Y-123 (for quasiparticle energies $|E|$ up to $>\sim \Delta _{SC}$) can be explained by the generalized BTK theory,[@Yeh01; @Wei98] implying that the spectral contribution due to the competing order is insignificant up to $|E| \sim \Delta _{SC}$. This finding is in agreement with a much larger $h^{\ast}$ value for Y-123 than for La-112, and therefore much weaker quantum fluctuations in the former.
A further verification for the closer proximity of La-112 to the QCP than Y-123 is manifested in Figs. 4(a)-(b), where the quasiparticle tunneling spectra of La-112 are found to be dependent on the tunneling current $I$. For lower tunneling currents, we find that the coherence peaks at $E = \pm \Delta _{SC}$ are systematically suppressed by increasing $I$ without changing the $\Delta _{SC}$ value. The coherence peaks eventually vanish while additional features at higher energies begin to emerge. Finally, in the large current limit, pseudogap-like features appear at $|E| \equiv \Delta _{PG} > \Delta _{SC}$, as illustrated in Fig. 4(a). More detailed evolution of the superconducting and pseudogap features with $I$ is summarized in Fig. 4(b). We also notice that the magnitude of $\Delta _{SC}$ determined under smaller tunneling currents is highly spatially homogeneous, whereas the $\Delta _{PG}$ value appears to vary significantly from one location to another. We suggest two effects associated with increasing tunneling current $I$: First, the localized high current density under the STM tip can effectively suppress the SC order parameter and therefore reduce $\alpha$. Second, the magnetic field induced by the localized high current density can also assist the evolution from an initial SC phase with $\alpha >\sim \alpha _c$ into the coexisting SDW/SC state with increasing $I$. The current-induced SDW can be manifested in the DC tunneling spectrum through coupling of the SDW order parameter to disorder.[@Polkovnikov02] Hence, the resulting quasiparticle spectrum under large tunneling currents contains convoluted spectral information of SDW, SC, and the disorder potential, thus it can be spatially inhomogeneous. In contrast, for optimally doped Y-123, no noticeable spectral variation was found with tunneling currents in the same range as that used for studying La-112. This finding is again consistent with our conjecture that Y-123 is farther from the QCP than La-112, so that it is more difficult to induce SDW in Y-123.
Disorder Effect on Quasiparticle Spectra and Pseudogap
======================================================
Next, we investigate the effect of disorder on the scenario depicted in Fig. 1(a). Generally speaking, disorder reduces the SC stiffness and tends to shift $\alpha$ closer to $\alpha _c$ if initially $\alpha > \alpha _c$.[@Vojta00] Therefore one can envision spatially varying $\alpha$ values in a sample if the disorder potential is spatially inhomogeneous. In particular, for strongly 2D cuprates like Bi-2212 and Hg-1234, disorder can help pin the fluctuating SDW locally,[@Yeh03b] so that regions with the disorder-pinned SDW can coexist with SC, as schematically illustrated in Fig. 5(a). These randomly distributed regions of pinned SDW are scattering sites for quasiparticles at $T < T_c$ and for normal carriers at $T > T_c$, provided that the SDW persists above $T_c$. We have performed numerical calculations for the quasiparticle local density of states (LDOS) of a 2D $d$-wave superconductor to examine these related issues.[@Chen03] We have considered two scenarios: one assumes that the ground state of the 2D $d$-wave cuprate is a pure SC with random point defects, and the other assumes that randomly pinned SDW regions coexist with SC at $T \ll T_c$. Using the Green’s function techniques detailed in Ref.[@Chen03], we find that the quasiparticle interference spectra for SC coexisting with a disorder-pinned SDW differ fundamentally from those due to pure SC with random point disorder. A representative real-space map of the quasiparticle LDOS, for a 2D $d_{x^2-y^2}$-wave superconductor with 24 randomly distributed pinned SDW regions in an area of $(400 \times 400)$ unit cells, is shown in Fig. 5(b). The Fourier transformation (FT) of the LDOS is illustrated in Fig. 5(c) for a pure SC with random point disorder at $T = 0$, in Fig. 5(d) for the FT-LDOS of pinned SDW at $T = 0$, and in Fig. 5(e) for the FT-LDOS of pinned SDW at $T = T_c$. We note that the superposition of the FT-LDOS in Figs. 5(c) and 5(d) is consistent with experimental observation on a slightly underdoped Bi-2212 at $T \ll T_c$,[@Hoffman02; @McElroy03] whereas the FT-LDOS in Fig. 5(e) is consistent with experimental observation on a similar sample at $T > \sim T_c$.[@Vershinin04] These findings suggest that a disorder-pinned SDW coexists with SC in Bi-2212 at low temperatures, and that only the SDW persists above $T_c$.[@Chen03] These studies therefore demonstrate the significant role of competing orders in determining the physical properties of cuprate superconductors. Moreover, disorder-pinned collective modes such as SDW can naturally account for the strong spatial inhomogeneity observed in the quasiparticle spectra of Bi-2212,[@Pan01; @Lang02] and are also likely responsible for the pseudogap phenomenon[@Timusk99] above $T_c$ in under- and optimally doped p-type cuprates.
Discussion
==========
The experimental results presented in the previous sections are generally consistent with the notion that significant quantum fluctuations can be induced by a magnetic field in all cuprate superconductors, and that the degree of excess low-energy excitations can be correlated with the proximity of the cuprates to a QCP. However, to further our understanding, systematic studies of more cuprates will be necessary. In particular, neutron scattering studies will be important for determining the SDW gaps of different cuprates. It is also imperative to examine the correlation between the low-temperature high-field phase diagram and the low-energy excitations of different cuprates through quasiparticle spectroscopic studies. For instance, quasiparticle tunneling spectroscopic studies of the highly 2D Hg-1234 can substantiate our conjecture if the following can be verified: (1) $\Delta _{SC}(T)$ decreases more rapidly with $T$ than other p-type cuprates with larger values of $h^{\ast}$; (2) the quasiparticle DOS exhibits strong spatial variation below $T_c$, similar to Bi-2212; (3) excess sub-gap DOS than the BTK prediction exists because of the relatively smaller $\Delta _{SDW}$ that provides an additional channel of low-energy excitations; and (4) strong quasiparticle spectral dependence on the tunneling current. Similarly, determination of the characteristic field $h^{\ast}$ as a function of impurity concentration can provide further verification for our conjecture that $h^{\ast}$ varies with increasing disorder.
Despite consensus for the existence of competing orders in the ground state of the cuprates, whether SDW is the dominant competing order is yet to be further verified. For instance, we note that certain experimental consequences (such as the quasiparticle spectra) due to a disorder-pinned SDW cannot be trivially distinguished from a disorder-pinned CDW in tetragonal cuprates. Therefore neutron scattering studies will be necessary to distinguish between these two competing orders. As for the staggered flux phase, unit-cell doubling features for the quasiparticle LDOS along the CuO$_2$ bonds must be demonstrated around a vortex core if the staggered flux phase is a relevant competing order.[@Kishine01] Finally, the most important issue remaining to be addressed in the studies of competing orders and quantum criticality is to investigate possible correlation between the existence of competing orders and the occurrence of cuprates superconductivity.
Summary
=======
In summary, we have investigated the quasiparticle tunneling spectra and the thermodynamic high-field phase diagrams of various electron- and hole-doped cuprate superconductors. The experimental data reveal significant magnetic field-induced quantum fluctuations in all cuprate superconductors, and the degree of quantum fluctuations appears to correlate well with the magnitude of excess low-energy excitations as the result of competing orders in the ground state. Moreover, our experimental results support the notion that the ground state of cuprates is in proximity to a QCP separating pure SC from coexisting SC/SDW, and our theoretical analysis further suggests that disorder-pinned fluctuating SDW can have significant effect on the LDOS of highly 2D cuprates. Additionally, the persistence of disorder-pinned SDW above $T_c$ may be accountable for the pseudogap phenomenon observed in the spectroscopic studies of highly 2D hole-doped cuprates.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work at Caltech is supported by the National Science Foundation through Grants \#DMR-0405088 and \#DMR-0103045, and at the Pohang University by the Ministry of Science and Technology of Korea through the Creative Research Initiative Program. The pulsed-field experiments were performed at the National High Magnetic Field Laboratory facilities in the Los Alamos National Laboratory under the support of the National Science Foundation.
[0]{} S.-C. Zhang, [*Science*]{} [**275**]{}, 1089 (1997).
S. Sachdev, [*Rev. Mod. Phys.*]{} [**75**]{}, 913 (2003).
S. A. Kivelson [*et al.*]{}, [*Rev. Mod. Phys.*]{} [**75**]{}, 1201 (2003).
N.-C. Yeh, [*Bulletin of Assoc. Asia Pacific Phys. Soc.*]{} [**12**]{}, 2 (2002); cond-mat/0210656.
N.-C. Yeh [*et al.*]{}, [*Phys. Rev. Lett.*]{}, [**87**]{}, 087003 (2001).
N.-C. Yeh [*et al.*]{}, [*Physica C*]{} [**367**]{}, 174 (2002).
C.-T. Chen [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**88**]{}, 227002 (2002).
E. Demler, S. Sachdev and Y. Zhang, [*Phys. Rev. Lett.*]{} [**87**]{}, 067202 (2001).
Y. Chen, H.-Y. Chen and C. S. Ting, [*Phys. Rev. B*]{} [**66**]{}, 104501 (2002).
D.-H. Lee, [*Phys. Rev. Lett.*]{} [**88**]{}, 227003 (2002).
J. Kishine, P. A. Lee and X.-G. Wen, [*Phys. Rev. Lett.*]{} [**86**]{}, 5365 (2001).
M. Vojta, Y. Zhang and S. Sachdev [*Phys. Rev. B*]{} [**62**]{}, 6721 (2000).
B. O. Wells [*et al.*]{}, [*Science*]{} [**277**]{}, 1067 (1997).
B. Lake [*et al.*]{}, [*Science*]{} [**291**]{}, 1759 (2001).
H. A. Mook [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**88**]{}, 097004 (2002).
K. Yamada [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**90**]{}, 137004 (2003).
N.-C. Yeh [*et al.*]{}, [*Phys. Rev. B*]{} [**47**]{}, 6146 (1993).
J. L. O’Brien [*et al.*]{}, [*Phys. Rev. B*]{} [**61**]{}, 1584 (2000).
N.-C. Yeh [*et al.*]{}, [*Phys. Rev. B*]{} [**45**]{}, 5710 (1992).
S. Kleefisch [*et al.*]{}, [*Phys. Rev. B*]{} [**63**]{}, 100507 (2001).
L. Krusin-Elbaum [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**92**]{}, 097004 (2004).
V. S. Zapf [*et al.*]{}, submitted to [*Phys. Rev. B*]{} (2004); cond-mat/0405072.
V. M. Krasnov [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**84**]{}, 5860 (2000).
C. U. Jung [*et al.*]{}, [*Physica C*]{} [**366**]{}, 299 (2002).
M.-S. Kim [*et al.*]{}, [*Phys. Rev. B*]{} [**57**]{}, 6121 (1998).
M.-S. Kim [*et al.*]{}, [*Phys. Rev. B*]{} [**63**]{}, 134513 (2001).
C. Mielke [*et al.*]{}, [*J. Phys.: Condens. Matter*]{} [**13**]{}, 8325 (2001).
D. S. Reed [*et al.*]{}, [*Phys. Rev. B*]{} [**51**]{}, 16448 (1995).
L. Alff [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**83**]{}, 2644 (1999).
C. C. Tsuei and J. Kirtley, [*Phys. Rev. Lett.*]{} [**85**]{}, 182 (2000).
J. A. Skinta [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**88**]{}, 207005 (2002).
A. Biswas [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**88**]{}, 207004 (2002).
Ch. Renner [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**80**]{}, 149 (1998).
S. H. Pan [*et al.*]{}, [*Nature*]{} [**403**]{}, 746 (2000).
E. W. Hudson [*et al.*]{}, [*Nature*]{} [**411**]{}, 920 (2001).
H.-Y. Chen and C. S. Ting, cond-mat/0405524 (2004).
N.-C. Yeh [*et al.*]{}, [*J. Low Temp. Phys.*]{} [**131**]{} 435 (2003).
J. Y. T. Wei [*et al.*]{}, [*Phys. Rev. Lett.*]{} [**81**]{}, 2542 (1998).
A. Polkovnikov, M. Vojta, and S. Sachdev, [*Phys. Rev. B*]{} [**65**]{}, 220509 (R) (2002).
N.-C. Yeh and C.-T. Chen, [*Int. J. Mod. Phys. B*]{} [**17**]{} 3575 (2003).
C.-T. Chen and N.-C. Yeh, [*Phys. Rev. B*]{} [**68**]{}, 220505(R)(2003).
J. E. Hoffman [*et al.*]{}, [*Science*]{} [**297**]{}, 1148 (2002).
K. McElroy [*et al.*]{}, [*Nature*]{} [**422**]{}, 592 (2003).
M. Vershinin [*et al.*]{}, [*Science*]{} [**303**]{}, 1995 (2004).
S. H. Pan [*et al.*]{}, [*Nature*]{} [**413**]{}, 282 (2001).
K. M. Lang [*et al.*]{}, [*Nature*]{} [**415**]{}, 412 (2002).
T. Timusk and B. Statt, [*Rep. Prog. Phys.*]{} [**62**]{}, 61 (1999).
[^1]: Corresponding author. E-mail: ncyeh@caltech.edu
|
---
abstract: 'We show that unitarity does not allow cloning of any two points in a ray. This has implication for cloning of the geometric phase information in a quantum state. In particular, the quantum history which is encoded in the geometric phase during cyclic evolution of a quantum system cannot be copied. We also prove that the generalized geometric phase information cannot be copied by a unitary operation. We argue that our result also holds in the consistent history formulation of quantum mechanics.'
address: 'Institute of Physics, Bhubaneswar-751005, Orissa, India '
author:
- 'Arun K. Pati [^1]'
title: Quantum History cannot be Copied
---
=10000
.5cm
In quantum theory state of a single quantum is represented by not just a vector $|\psi\ra$ in a separable Hilbert space ${\cal H}$, but by a ray in the ray space ${\cal R}$. A ray is a set of equivalence classes of states that differ from each other by complex numbers of unit modulus. Thus the ray space ${\cal R}$ is defined as ${\cal R} =
\{ |\psi'\ra : |\psi \ra \sim |\psi'\ra = c|\psi \ra \}$, where $c \in {\bf C} $ is a group of non-zero complex numbers and $|c|=1$. Given a quantum state $|\psi\ra$ we can generate a ray by the application of the ‘ray operator’ $R(c) = \exp[ i {\rm Arg}~( c ) |\psi\ra \la \psi| ]
= I + (c -1) |\psi\ra \la \psi|$ such that $R(c)|\psi \ra = |\psi' \ra$. Geometrically, we can represent all these equivalent classes of states as points in a ray and all of them represent the same physical state. The set of rays of the Hilbert space ${\cal H}$ is called the projective Hilbert space ${\cal P} = {\cal H}/U(1)$. If we have a continuous unitary time evolution of a quantum system $|\psi\ra \rightarrow U(t)|\psi \ra$ then the evolution can be represented as an open curve $\Gamma:
t \rightarrow |\psi(t)\ra$ in ${\cal R}$ whose projection in ${\cal P}$ is also an open curve ${\widehat \Gamma}$. The quantum state at different times can belong to different rays. If the quantum state at two different times belongs to the same ray, then it may trace an open curve $C$ in ${\cal R}$, but its projection in ${\cal P}$ is a closed curve ${\widehat C}$. Such an evolution is called a cyclic evolution.
In quantum information theory we view a quantum state $|\psi\ra$ as the carrier of both classical and quantum information. The fundamental unit of classical information is a bit and that of quantum information is a qubit. Classical bit can be copied but quantum bit or qubit cannot be copied. It is the linearity of quantum theory that does not allow us to produce a copy of an arbitrary quantum state [@wz; @dd]. Using unitarity one can prove that two non-orthogonal states cannot be copied either [@hpy]. However, orthogonal quantum states like $|0\ra$ and $|1\ra$ can be copied unitarily. Also, if we know a quantum state, we can make as many copies as we wish. The subject of quantum cloning is an active area of research. Recent studies have shown that one could make approximate copies by deterministic transformations [@bh] and exact copies by probabilistic operations [@dg; @akp0]. Another important limitation on quantum information is that it is impossible to delete an unknown quantum state [@pb]. We know that classical bit or quantum bit in orthogonal states can be deleted against a copy. However, linearity of quantum theory prohibits deletion of a qubit against a copy.
Here, we ask the following question: If we can make a copy of $|\psi\ra$ can we make copy of an equivalent state $|\psi'\ra$ by the same machine? That is whether $|\psi \ra \otimes |\Sigma\ra
\rightarrow |\psi\ra \otimes |\psi \ra$ and $|\psi' \ra \otimes |\Sigma\ra
\rightarrow |\psi' \ra \otimes |\psi' \ra$ is possible by a single unitary operator, where $|\Sigma\ra$ is the blank state and $|\psi'\ra = c|\psi\ra$. Intuitively, one would say that since $|\psi \ra$ and $ c|\psi\ra$ represent the same physical state from informational point of view it should be possible to make copies of $|\psi \ra$ and $ c|\psi\ra$ by the same cloning machine. But surprisingly, this intuition is not correct. The proof is simple but nevertheless it is important. This has important implication in cloning of relative phase information. For example, we prove that the geometric phase information during cyclic evolution of a quantum system cannot be copied by unitary machine or any physical operation (a completely positive trace preserving map). Furthermore, we prove that the non-cyclic geometric phase cannot be copied during an arbitrary evolution of a quantum system. The important implication of our theorems is that history of a quantum system which is encoded in the geometric phase cannot be copied. Interestingly, we argue that our result also holds in the consistent histories formulation of quantum theory where the geometric phase appears naturally in the histories.\
[**Theorem 1:**]{} [*In general, two points in a ray cannot be cloned by a single unitary machine.*]{}\
[*Proof:*]{} A ray is an equivalence classes of states up to global phases . Thus, rather than the usual cloning machine having the form $|\psi \rangle \otimes |\Sigma \ra \rightarrow |\psi \ra \otimes
|\psi \ra$ we should really allow for $$|\psi\ra \otimes |\Sigma \ra \rightarrow
e^{i \theta(\psi) } |\psi \ra \otimes |\psi \ra,$$ where $\theta(\psi)$ can be an arbitrary function of $|\psi \ra$. Now, for two equivalent states $|\psi \ra$ and $|\psi' \ra$ if we take the transformation $$\begin{aligned}
|\psi \rangle \otimes |\Sigma \ra &\rightarrow &
|\psi \ra \otimes |\psi \ra\nonumber\\
|\psi' \rangle \otimes |\Sigma \ra &\rightarrow & |\psi'\ra \otimes
|\psi' \ra, \end{aligned}$$ then this says that which specific member of the equivalence class we are in that needs to be preserved. One might argue that this would be against the spirit of having an equivalence class in the first place where any member of the equivalence class may be substituted with any other at any time. So, the general cloning transformation may be given by (1). Note that this ambiguity does not arise in usual cloning literature that describe approximate clones because there one asks only about the reduced state of each subsystem. It should be mentioned that we can also use the transformation (2) to prove our theorem but (1) is more general way of stating the action of cloning map on a ket. As we know for a fixed state vector, an overall phase is not important, since physical states are density matrices and not vectors. Indeed the cloning maps given by (1) and (2) are equivalent. Unless there is confusion, in this paper, we use sometime transformations (1) or (2).
Let us consider two equivalent classes of states $|\psi \ra$ and $|\psi' \ra$. The cloning transformation for two points in a ray is now given by $$\begin{aligned}
|\psi \ra \otimes |\Sigma \ra &\rightarrow&
e^{i \theta(\psi) } |\psi \ra \otimes |\psi \ra \nonumber\\
|\psi' \ra \otimes |\Sigma \ra &\rightarrow&
e^{i \theta(\psi') } |\psi \ra \otimes |\psi \ra\end{aligned}$$ Since unitarity preserves the inner product we must have $c = \exp[ i \theta(\psi) - \theta(\psi')]$. However, this cannot hold for arbitrary values of $c, \theta(\psi)$ and $\theta(\psi')$. This proves that two equivalent states cannot be copied by the same machine even if we know the state. The physical meaning of this theorem is that the relative phase between two points in a ray cannot be copied by a unitary machine.
The ‘no-cloning theorem for a ray’ will have implication for cloning of relative phase information in quantum systems. In recent years, there have been considerable interest in the study of the relative phases and in particular the Berry phases [@berry] in quantum systems. It is also hoped that the Berry phase which is of geometric origin may be used in design of robust logic gates in quantum computation. The original Berry phase was discovered in the context of quantum adiabatic theorem [@lis; @mes]. It is basically an extra phase that the system acquires when the Hamiltonian is slowly changed cyclically over one time period. The Berry phase is independent of the detailed dynamics of the system and is of purely geometric in origin. This arises due to the non-trivial curvature of the parameter space in which the state vector is transported around a closed loop. An early discovery of the geometric phase was made by Pancharatnam in the context of interference of light [@panch]. The Berry phase was then generalized to non-adiabatic but cyclic evolutions of quantum system by Aharonov and Anandan [@aa]. In fact, now we know that the geometric phase appears in much more general context than it was thought before [@samu; @ms; @akp].
Consider the unitary time evolution of a quantum system where the state vector evolves as $|\psi(0)\ra
\rightarrow |\psi(t) \ra = U(t)|\psi(0) \ra$ such that at $t=T$, $|\psi(T) \ra = e^{i\Phi}
|\psi(0) \ra$, $\Phi$ being the total phase. That is to say that the quantum system at $t=T$ comes back to its original state apart from a phase factor. As we know such an evolution is called the cyclic evolution. Even though $|\psi(0) \ra$ and $|\psi(T) \ra$ are equivalent it is the relative phase $\Phi$ between them that is observable. The total phase $\Phi$ that the system acquires during a cyclic evolution is composed of two phases, one is the dynamical phase $\delta$ and the other is the geometric phase $\beta({\widehat C})$. This $\beta({\widehat C})$ is also known as the Aharonov-Anandan (AA) phase [@aa]. Thus the total phase is given by $$\Phi = \delta + \beta({\widehat C}),$$ where $\delta$ is the dynamical phase and $\beta({\widehat C})$ is the geometric phase. The dynamical phase is given by $$\delta = - \frac{1}{\hbar} \int_0^T \la \psi(t)|H|\psi(t)\ra ~dt.$$ It represents, in a sense, an ‘internal clock’ of the quantum system. On the other hand the geometric phase is given by $$\beta({\widehat C}) = i\oint \la {\tilde \psi}(t)|{\dot {\tilde \psi}}(t)
\ra ~dt = i\oint \la {\tilde \psi}|d {\tilde \psi} \ra,$$ where $|{\tilde \psi}(t) \ra = \exp(-if(t)) |\psi(t)\ra$ with $f(t)$ being any smooth function that satisfies $f(T)- f(0)= \Phi$. Here, $ i \la {\tilde \psi}|d {\tilde \psi} \ra$ is the differential connection-form that gives rise to geometric phase. It is gauge invariant, reparameterization invariant, and depends only on the closed curve ${\widehat C}$ in the projective Hilbert space ${\cal P}$ of the quantum system. Unlike the dynamical phase, the geometric phase indeed depends on the path of the evolution [@pati91] and is a non-integrable quantity.
In this context, if we are able to clone $|\psi(0) \ra$, one may think that we can also clone $|\psi(T) \ra$ as they really belong to the same ray. This is because during a cyclic evolution the system starts from a ray and after a time period $T$ comes back to the same ray but at a different point. Now, application of our theorem tells us that we cannot clone $|\psi(0) \ra$ and $|\psi(T) \ra$ by a single unitary machine.\
[**Proposition:**]{} [*Quantum history which is encoded in the geometric phase during cyclic evolution of a quantum system cannot be copied by a unitary transformation.*]{}\
[*Proof:*]{} Suppose we could copy $|\psi(0) \ra$ and $|\psi(T) \ra$ by a unitary machine. That is $$\begin{aligned}
|\psi(0) \ra \otimes |\Sigma\ra & \rightarrow & |\psi(0) \ra \otimes
|\psi(0) \ra \nonumber\\
|\psi(T) \ra \otimes |\Sigma\ra & \rightarrow & |\psi(T) \ra \otimes
|\psi(T) \ra.\end{aligned}$$ By unitarity, Eq(7) implies that we must have $\la \psi(0)|\psi(T)\ra
\rightarrow \la \psi(0)|\psi(T)\ra^2$, i.e., $\exp(i\Phi) \rightarrow
\exp(2 i\Phi)$. However, this is not possible by a unitary machine. If it is possible, then that would mean $\exp(i\Phi) =1$ which is not true in general. Moreover, if the system undergoes parallel transportation then it acquires a pure geometric phase and we will have $\Phi = \beta({\widehat C})$ [@aa]. Here we would like to mention that an [*arbitrary quantum state cannot be parallel transported*]{} either. (This is another no-go theorem. See the notes in the end). Then the no-cloning theorem for a ray tells us that we cannot clone the geometric phase information of a quantum system during a cyclic evolution. Since the geometric phase attributes memory to a quantum system it remembers the history of the evolution. This then implies that the [*quantum history cannot be copied.*]{} Therefore, even if we can make copy of a known quantum states we cannot copy its history. The only way to copy the history is to first make a copy of the state and then pass the copied quantum system through the same cycle again. Physical reason for this impossibility is the following. We are able to make copy of $|\psi(0) \ra$ because we have complete knowledge of it. But the geometric phase not only depends on $|\psi(0) \ra$ but also on the path of the evolution that the system has undergone in the past. Unless we have knowledge of the past history we cannot copy the geometric phase.
In fact, we can prove that quantum history cannot be copied by any physical operation. A physical operation in quantum theory we mean a completely positive (CP) trace preserving mappin g.\
[**Theorem 2:**]{} [*In general, geometric phase during a cyclic evolution cannot be copied by a completely positive map.*]{}\
[*Proof:*]{} We know that by including ancilla, any CP map can be realized as a unitary evolution in an enlarged Hilbert space. Consider the cloning of $|\psi(0) \ra$ and $|\psi(T) \ra$ including ancilla. This may be given by $$\begin{aligned}
|\psi(0) \ra \otimes |\Sigma \ra \otimes | A \ra & \rightarrow &
|\psi(0) \ra \otimes |\psi(0) \ra \otimes | A(0) \ra \nonumber\\
|\psi(T) \ra \otimes |\Sigma \ra \otimes |A \ra & \rightarrow &
|\psi(T) \ra \otimes |\psi(T) \ra \otimes |A(T) \ra,\end{aligned}$$ where $|A \ra$ is the initial state and $|A(0)\ra$, $|A(T)\ra$ are the final states of the ancilla. By unitarity in the enlarged Hilbert space, we have $$\begin{aligned}
e^{i\Phi} = e^{2i\Phi} \la A(0)|A(T)\ra.\end{aligned}$$ This cannot hold in general, hence it is impossible to copy quantum history by any physical operation. However, if it so happens that environment also undergoes a cyclic evolution and acquires equal and opposite relative phase as that of the quantum system, i.e., $|A(T)\ra = \exp(-i \Phi) |A(0) \ra$, then possibly quantum history can be copied by a completely positive map.
Next, we prove that the geometric phase information during a general quantum evolution cannot be copied. When a quantum system evolves in time it traces an open path $\Gamma: t \rightarrow |\psi(t)\ra$ in the ray space ${\cal R}$ and the quantum state at different times belong to different rays. The projection of the evolution path $\Gamma$ in the projective Hilbert space ${\cal P}$ is also an open path ${\widehat \Gamma}$. For example, consider the time evolution $|\psi(0)\ra \rightarrow |\psi(t)\ra$. The evolution under consideration [*need not be adiabatic, cyclic, and even unitary*]{}. All that is required is that there is a linear map and evolution curve should be smooth with a inner product defined over the Hilbert space. In this case the initial and final states are not equivalent. For any two non-orthogonal states the relative phase (or total phase difference) between them is given by the Pancharatnam phase $\Phi_P =
{\rm Arg} \la \psi(0)|\psi(t) \ra$ [@panch; @samu]. It is always possible to write the total phase $\Phi_P$ as sum of the dynamical phase $\Phi_D$ and the geometric phase $\Phi_G$. The dynamical phase is given by $$[\Phi_D]_0^t = - i \int \la \psi(t)|{\dot \psi}(t)\ra ~dt.$$ It depends on the detailed dynamics that the system is undergoing. The geometric phase is given by $$\begin{aligned}
[\Phi_G]_0^t & = &
{\rm Arg} \la \psi(0)|\psi(t) \ra +
i \int \la \psi(t)|{\dot \psi}(t)\ra ~dt\nonumber\\
& = & i\int \la \chi(t)|{\dot \chi}(t) \ra ~dt
= i\int \la \chi|d \chi \ra,\end{aligned}$$ where $|\chi(t) \ra = \frac{ \la \psi(t)| \psi(0)\ra}
{ |\la \psi(t)| \psi(0)\ra| } |\psi(t) \ra$ is a reference-section introduced in [@akp; @akp1]. Here, $ i \la \chi|d \chi \ra$ is the differential connection-form that gives rise to the most general geometric phase. It is again $U(1)$ gauge invariant, reparameterization invariant, and depends only on the geometry of the open curve ${\widehat \Gamma}$ in the projective Hilbert space ${\cal P}$ of the quantum system. It can be shown that it is a [*non-additive quantity*]{} which in turn implies that the system remembers along which path it has been transported. Thus, the most general geometric phase remembers the history of quantum system and attributes a memory to the quantum system. Next we prove the following.\
[**Theorem 3:**]{} [*Quantum history which is encoded in the generalized geometric phase during arbitrary evolution of a quantum system cannot be copied unitarily.*]{}\
[*Proof:*]{} Consider a sequence of time evolution of a quantum system from $t=0$ to $t_1$ and then from $t_1$ to $t_2$. Thus, we have the time evolution $|\psi(0)\ra \rightarrow |\psi(t_1)\ra \rightarrow |\psi(t_2)\ra$. Suppose, we want to clone the quantum states $|\psi(0)\ra$, $|\psi(t_1)\ra$ and $|\psi(t_2)\ra$ by a unitary machine. Then we will have the following cloning transformation $$\begin{aligned}
|\psi(0) \ra \otimes |\Sigma\ra & \rightarrow & |\psi(0) \ra \otimes
|\psi(0) \ra \nonumber\\
|\psi(t_1) \ra \otimes |\Sigma\ra & \rightarrow & |\psi(t_1) \ra \otimes
|\psi(t_1) \ra \nonumber\\
|\psi(t_2) \ra \otimes |\Sigma\ra & \rightarrow & |\psi(t_2) \ra \otimes
|\psi(t_2) \ra.\end{aligned}$$ Now, by unitarity, taking all the inter inner products we will have $$\begin{aligned}
\la \psi(0)|\psi(t_1) \ra \la \psi(t_1)|\psi(t_2) \ra \la \psi(t_2)
|\psi(0) \ra =
[\la \psi(0)|\psi(t_1) \ra \la \psi(t_1)|\psi(t_2) \ra \la \psi(t_2)
|\psi(0) \ra]^2.\end{aligned}$$ Let us define a complex quantity called as the three point Bargmann invariant $\Delta^{(3)}$ as $$\Delta^{(3)} = \la \psi(0)|\psi(t_1) \ra \la \psi(t_1)|\psi(t_2) \ra
\la \psi(t_2)|\psi(0) \ra.$$ The three point Bargmann invariant remains the same under unitary and antiunitary transformation and plays an important role in the kinematic approach to the theory of geometric phases developed by Mukunda and Simon [@ms]. Now taking the argument of both the sides of Eq(13), we will have ${\rm Arg} \Delta^{(3)} \rightarrow 2 {\rm Arg} \Delta^{(3)}$ which implies that ${\rm Arg} \Delta^{(3)} = 0$. But ${\rm Arg} \Delta^{(3)}$ is nothing but the excess geometric phase that the system may acquire in going from $t_0 = 0$ to $t_1$ and then from $t_1$ to $t_2$ instead of going from $t_0 = 0$ to $t_2$ directly [@akp]. More precisely, let $[\Phi_G]_{0}^{t_1}$ is the geometric phase that the system acquires during the evolution from time $t_0=0$ to $t_1$, $[\Phi_G]_{t_1}^{t_2}$ is the geometric phase that the system acquires during the evolution from time $t_1$ to $t_2$, and $[\Phi_G]_{0}^{t_2}$ is the geometric phase that the system acquires during the evolution directly from time $t_0=0$ to $t_2$. An important property of the geometric phase is that it is [*non-additive*]{}. We will indeed have $ [\Phi_G]_{0}^{t_1} + [\Phi_G]_{t_1}^{t_2} \not= [\Phi_G]_{0}^{t_2}$. Thus, the excess geometric phase given by $$[\Phi_G]_{0}^{t_1} + [\Phi_G]_{t_1}^{t_2} - [\Phi_G]_{0}^{t_2} =
{\rm Arg} [ \la \psi(0)|\psi(t_1) \ra \la \psi(t_1)|\psi(t_2) \ra
\la \psi(t_2)|\psi(0) \ra] = {\rm Arg} \Delta^{(3)}.$$ In the above the dynamical phase has disappeared because that is simply an additive quantity. Thus, ${\rm Arg} \Delta^{(3)} \not=0$. Hence this shows that the quantum history encoded in the generalized geometric phase during an arbitrary quantum evolution cannot be copied.
Our result has implication in the context of consistent history formulation of quantum mechanics developed by Griffiths [@grif], Omne [@omn] and Gell-Mann and Hartle [@hart]. Recently, it has been shown by Anastopoulos and Savvidou that the geometric phase is manifested in the probabilistic structure of histories. Specifically, they have shown that the geometric phase is the basic building block of the interference phase between pair of histories [@ana].
In consistent history approach, typically a history can be thought of as a sequence of properties or events which correspond to a time-ordered sequence of propositions about the quantum system. These events are represented by projectors $\Pi_1, \Pi_2,
\ldots \Pi_n$ on the Hilbert space ${\cal H}$ at a succession of times $t_1 < t_2 < \cdots t_n$. These projectors at different times need not commute. One defines the space of all histories by a history Hilbert space ${\cal H}_h = {\cal H}_{t_i}^{\otimes t_i}$ which consists of tensor product of copies of the Hilbert space ${\cal H}_{t_i}$ at $i$th instant of time. On the history Hilbert space one represents history as a projector $P= \Pi_1 \otimes \Pi_2 \cdots
\Pi_n$. The meaning of such a history is that events $\Pi_i$ occurs in the system at time $t_i$, respectively. One assigns a realistic interpretation to such a history provided certain consistency conditions are satisfied.
By introducing Heisenberg projector $ \Pi_i(t_i) = U(t_i)^{\dagger}
\Pi_i U(t_i)$ one can define a weight operator $C_{P}$ to each history defined by $$C_{P} = U(t_n)^{\dagger} \Pi_n U(t_n) \cdots U(t_1)^{\dagger}
\Pi_1 U(t_1) = \Pi_n(t_n) \cdots \Pi_1(t_1).$$ Given a pair of weight operators $C_P$ and $C_{P'}$ for two histories one defines an important quantity –that is the decoherence functional which is a complex valued function $d(P, P')$ of history propositions $P, P'$ that measures the quantum mechanical interference between them. More precisely, the decoherence functional is a mapping $d: {\cal H}_h \times {\cal H}_h \rightarrow {\bf C}$ that satisfies the following conditions [@isham]:\
1. $d(P, P') = d(P', P)^*$ for all $P, P'$. This is hermiticity.\
2. $d(P, P) \ge 0$ for all $P$. This is positivity.\
3. If $P$ and $P'$ are orthogonal, then for all $P''$, $d(P \oplus P', P'') = d(P, P'') + d(P', P'')$. This is additivity.\
4. $d(I, I) = 1$. This is normalization.
One way of defining the decoherence functional is $$d(P, P') = {\rm Tr} \big( C_P \rho_0 C_{P'}^{\dagger} \big)$$ for an initial density matrix $\rho_0$. This can be interpreted as a probability in standard quantum theory under certain conditions. When $d(P, P') = 0$ for $P\not= P'$ in a set of histories that satisfies $\sum_i \Pi_i = 1$ and $P P' = P \delta_{PP'}$, then under this condition $d(P, P)$ can be regarded as the probability that the history proposition $P$ is true. The decoherence functional $d(P, P')$ can be thought of as the degree of interference between the histories $P$ and $P'$.
To see how the geometric phase appears in the consistent history formulation, consider the time ordered events with projectors at different times as $\Pi_0, \Pi_1, \ldots \Pi_n$. If one assumes that the projectors are fine grained, they can be represented as elements of the projective Hilbert space ${\cal P}$, i.e., $\Pi_i= \Pi_{t_i} = |\psi(t_i)\ra \la \psi(t_i)|$. If one neglects the dynamical evolution, i.e., set the Hamiltonian equal to zero, then trace of the weight operator is given by $${\rm Tr} C_P = \la \psi(0)|\psi(t_n) \ra \la \psi(t_n)|\psi(t_{n-1}) \ra
\cdots \la \psi(t_1)|\psi(0) \ra.$$ By assuming the number of time steps $n$ very large and each time steps differs by $\delta t$ with $\delta t \sim O(1/n)$ one can approximate this by a continuous time history and we have $${\rm Arg}[ {\rm Tr} C_P] = \Phi_G.$$ That is to each history one can assign a geometric phase [@ana]. Actually, one can understand the origin of the geometric phase in history formulation as follows. Note that $${\rm Tr}C_P = {\Delta^{(n+1)}}^*,$$ where $\Delta^{(n+1)}$ is the $(n+1)$-point Bargmann invariant defined as $$\Delta^{(n+1)}= \la \psi(0)|\psi(t_1) \ra \la \psi(t_1)|\psi(t_2) \ra
\cdots \la \psi(t_{n-1})|\psi(t_n) \ra.$$ We already know that in the standard quantum theory the Bargmann invariant represents the excess geometric phase if system undergoes sequence of times evolution between $t_0 < t_1 < \cdots t_n $ and a direct evolution from $t_0$ to $t_n$. This is true even if we do not assume that the Hamiltonian is zero. So in the consistency history formulation the appearance of geometric phase is natural. It has been also shown that the information about the geometric phase for a set of histories is sufficient to reconstruct the decoherence functional [@ana]. Now, from our theorem we know that geometric phase cannot be copied, then this also holds for the decoherence functional in the consistent history formulation.
In conclusion, we have proved that two equivalence classes of states representing the same physical state cannot be cloned by a unitary operator. It is the relative phase between two points in a ray that is impossible to clone. Even though the proof is really simple its implications may be important. During cyclic evolution of a quantum system the initial and final states are equivalent and can be represented as two points in a ray. One implication of our theorem is that the geometric phase information during a cyclic evolution cannot be copied. Since the geometric phase attributes a memory to a quantum system and remembers the history, this suggests that even though a state can be copied its quantum history cannot be copied. We have also shown that the geometric phase information during a cyclic evolution cannot be copied by a physical operation. In addition, we have proved that the geometric phase during arbitrary quantum evolution cannot be copied by a unitary machine. Interestingly, we have argued that our result also holds in the consistent history formulation of quantum theory. We hope that the impossibility of copying quantum history may have some application is quantum cosmology.
[*Notes:*]{} Here we prove that in general an arbitrary quantum state cannot be parallel transported. The parallel transport condition for a pure quantum state is that it never rotates locally (so it does not acquire any phase infinitesimally) but can under go a net rotation globally. This reflects the curvature of the quantum state space, i.e., the projective Hilbert space ${\cal P}$ in which the vector is parallel transported. Thus, during a parallel transportation a state can acquire a phase if brought back to its original position along a closed path ${\widehat C}$. This phase is essentially the holonomy angle or the geometric phase $\beta({\widehat C})$. Mathematically, the parallel transport condition for a vector $|\psi(t) \ra$ can be expressed as $\la \psi(t)|{\dot \psi}(t)\ra =0$. If $|\psi(t) \ra$ satisfies this then $|\psi(T) \ra =
\exp(i\beta({\widehat C}) )
|\psi(0) \ra$ [@aa] during a cyclic evolution and $ \la \psi(0)|\psi(t) \ra = |\la \psi(0)|\psi(t) \ra| \exp[\Phi_G]$ during arbitrary non-cyclic evolution [@akp; @akp1].
Suppose we have a set of known orthonormal bases $\{ |\psi_n(t)\ra \}$ with $\la \psi_n(t)|\psi_m(t)\ra = \delta_{nm}$ for all $t$. If these bases undergo parallel transportation then they satisfy $\la \psi_n(t)|{\dot \psi_n}(t)\ra =0$. Now let $|\psi(t) \ra$ be an arbitrary state: $|\psi(t) \ra = \sum_{n=1}^N c_n(t) |\psi_n(t) \ra \in
{\cal H}^N $. The question is if $\la \psi_n(t)|{\dot \psi_n}(t)\ra =0$ does that mean $\la \psi(t)|{\dot \psi}(t)\ra =0$? The answer is no. To be explicit we have $$\la \psi(t)|{\dot \psi}(t)\ra = \sum_n c_n^* {\dot c}_n +
\sum_{nm, n\not=m} c_m^* c_n \la \psi_m(t)|{\dot \psi_n}(t)\ra \not= 0.$$ Thus, in general an arbitrary quantum state cannot be parallel transported. In other words, [*universal parallel transportation machine cannot exist*]{}. However, there can be some special cases where it can be. When $c_n(t)$’s are time-independent and $\la \psi_m(t)|{\dot \psi_n}(t)\ra =0$ for $m\not=n$ then an arbitrary state can also undergo parallel transportation. This is a both necessary and sufficient condition. It would be an interesting problem by itself to find what kind of Hamiltonian would satisfy the parallel transport condition for an arbitrary quantum state.
[99]{}
W. Wootters and W. H. Zurek, Nature (London), [**299**]{}, 802 (1982)
D. Dieks, Phys. Lett. A [**92**]{}, 271 (1982).
H. P. Yuen, Phys. Lett. A [**113**]{}, 405 (1986).
V. Buzek and M. Hillery, Phys. Rev. A [**54**]{}, 1844 (1996).
L. M. Duan and G. C. Guo, Phys. Rev. Lett. [**80**]{}, 4999 (1998).
A. K. Pati, Phys. Rev. Lett. [**83**]{}, 2849 (1999).
A. K. Pati and S. L. Braunstein, Nature, [**404**]{}, 164 (2000).
M. V. Berry, Proc. Roy. Soc. (Lond.) [**392**]{}, 45 (1984).
S. Pancharatnam, Proc. Indian Acad. Sci. A [**44**]{}, 247 (1956).
L. I. Schiff, [*Quantum Mechanics*]{} (McGraw-Hill, 1968).
A. Messiah, [*Quantum Mechanics*]{} (North-Holland, Amsterdam, 1962).
Y. Aharonov and J. Anandan, Phys. Rev. Lett.[**58**]{}, 1539 (1987).
J. Samuel and R. Bhandari, Phys. Rev. Lett. [**60**]{}, 2339 (1988).
N. Mukunda and R. Simon, Ann. Phys. [**228**]{}, 205 (1993).
A. K. Pati, Phys. Rev. A [**52**]{}, 2576 (1995)
A. K. Pati, J. of Phys. A [**28**]{}, 2087 (1995).
A. K. Pati, Phys. Lett. A [**159**]{}, 105 (1991)
R. B. Griffiths, J. Stat. Phys. [**36**]{}, 219 (1984).
R. Omnes, J. Stat. Phys. [**53**]{}, 893 (1988).
M. Gell-Mann and J. B. Hartle, Phys. Rev. D [**47**]{}, 3345 (1993).
C. Anastopoulos and N. Savvidou, Int. J. Theo. Phys. [**41**]{}, 1572 (2002).
C. J. Isham, Int. J. Theo. Phys. [**36**]{}, 785 (1997).
[^1]: Email: akpati@iopb.res.in
|
---
abstract: 'This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e. involving time derivatives) area integrals associated to a non-negative self-adjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e. involving space derivatives) area integrals associated to a non-negative self-adjoint operator satisfying in addition a pointwise upper bounds for the gradient of the heat kernel. As applications, we obtain sharp estimates for the operator norm of the area integrals on $L^p(\RN)$ as $p$ becomes large, and the growth of the $A_p$ constant on estimates of the area integrals on the weighted $L^p$ spaces.'
address:
- ' Ruming Gong, Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou, 510275, P.R. China'
- ' Lixin Yan, Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou, 510275, P.R. China'
author:
- Ruming Gong and Lixin Yan
title: |
Weighted $L^p$ estimates for the area integral\
\[2pt\] associated to self-adjoint operators
---
\[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Definition]{} \[theorem\][Corollary]{} \[theorem\][Example]{} \[theorem\][Remark]{}
Introduction
=============
[**1.1. Background.**]{}Let $\varphi\in C_0^{\infty}({\RN})$ with $\int \varphi =0.$ Let $\varphi_t(x)=t^{-n}\varphi(x/t), t>0$, and define the Lusin area integral by
$$\begin{aligned}
\label{e1.1}
S_{\varphi}(f)(x)=\bigg(\int_{|x-y|<t}
\big|f\ast \varphi_t(y)\big|^2 {dy \, dt\over t^{n+1}}\bigg)^{1/2}.\end{aligned}$$
A celebrated result of Chang-Wilson-Wolff ([@CWW]) says that for all $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and all $f\in \mathcal{S}(\RN)$, there is a constant $C=C(n, \varphi)$ independent of $w$ and $f$ such that
$$\begin{aligned}
\label{e1.2}
\int_{\RN}S^2_{\varphi}(f)w\, dx\leq C\int_{\RN}|f|^2Mw\, dx,\end{aligned}$$
where $Mw$ denotes the Hardy-Littlewood maximal operator of $w$.
The fact that $\varphi$ has compact support is crucial in the proof of Chang, Wilson and Wolff. In [@CW], Chanillo and Wheeden overcame this difficulty, and they obtained weighted $L^p$ inequalities for $1<p<\infty$ of the area integral, even when $\varphi$ does not have compact support, including the classical area function defined by means of the Poisson kernel.
From the theorem of Chang, Wilson and Wolff, it was already observed in [@FP] that R. Fefferman and Pipher obtained sharp estimates for the operator norm of a classical Calderón-Zygmund singular integral, or the classical area integral for $p$ tending to infinity, e.g.,
$$\begin{aligned}
\label{e1.3}
\big\|S_{\varphi}(f)\big\|_{L^p(\RN)}\leq C p^{1/2} \big\|f\big\|_{L^p(\RN)}\end{aligned}$$
as $p\rightarrow \infty.$\
[**1.2. Assumptions, notation and definitions.**]{} In this article, our main goal is to provide an extension of the result of Chang-Wilson-Wolff to study some weighted norm inequalities for the area integrals associated to non-negative self-adjoint operators, whose kernels are not smooth enough to fall under the scope of [@CWW; @CW; @W]. The relevant classes of operators is determined by the following condition:
[**Assumption $(H_1)$.**]{} Assume that $L$ is a non-negative self-adjoint operator on $L^2({\mathbb R}^{n}),$ the semigroup $e^{-tL}$, generated by $-L$ on $L^2(\RN)$, has the kernel $p_t(x,y)$ which satisfies the following Gaussian upper bound if there exist $C$ and $c$ such that for all $x,y\in {\mathbb R}^{n}, t>0,$
$$|p_{t}(x,y)| \leq \frac{C}{t^{n/2} } \exp\Big(-{|x-y|^2\over
c\,t}\Big).
\leqno{(GE)}$$
Such estimates are typical for elliptic or sub-elliptic differential operators of second order (see for instance, [@Da] and [@DOS]).\
For $f\in {\mathcal S}(\RN)$, define the (so called vertical) area functions $S_{P}$ and $S_{H}$ by
$$\begin{aligned}
\label{e1.4}
S_{P}f(x)&=&\bigg(\int_{|x-y|<t}
|t\nabla_y e^{-t\sqrt{L}} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2},\\
S_{H}f(x)&=&\bigg(\int_{|x-y|<t}
|t\nabla_y e^{-t^2L} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2}, \label{e1.5}\end{aligned}$$
as well as the (so-called horizontal) area functions $s_{p}$ and $s_{h}$ by
$$\begin{aligned}
\label{e1.6}
s_{p}f(x)&=&\bigg(\int_{|x-y|<t}
|t\sqrt{L} e^{-t\sqrt{L}} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2},\\
s_{h}f(x)&=&\bigg(\int_{|x-y|<t}
|t^2L e^{-t^2L} f(y)|^2 {dy dt\over t^{n+1}}\bigg)^{1/2}.\label{e1.7}\end{aligned}$$
It is well known (cf. e.g. [@St; @G]) that when $L=\Delta$ is the Laplacian on $\RN$, the classical area functions $S_{P}, S_{H}, s_{p}$ and $s_{h}$ are all bounded on $L^p(\RN), 1<p<\infty.$ For a general non-negative self-adjoint operator $L$, $L^p$-boundedness of the area functions $S_{P}, S_{H}, s_{p}$ and $s_{h}$ associated to $L$ has been studied extensively – see for examples [@A], [@ACDH], [@ADM], [@CDL], [@St1] and [@Y], and the references therein.
[**1.3. Statement of the main results.**]{} Firstly, we have the following weighted $L^p$ estimates for the area functions $s_{p}$ and $s_{h}.$
\[th1.1\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy Gaussian bounds $(GE)$. If $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and $f\in \mathcal{S}(\RN)$, then
$$\begin{aligned}
\hspace{-2.5cm}
&{\rm (a)}& \hspace{0.1cm}
\int_{\RN}s_h(f)^pw\, dx\leq C(n,p)\int_{\RN}|f|^pMw\, dx,\ \ \ 1<p\leq 2,\\
\hspace{-2.5cm}&{\rm (b)}& \hspace{0.1cm}\int_{\{s_h(f)>\lambda\}} wdx\leq {C(n)\over \lambda}\int_{\RN}|f| Mwdx,\ \ \ \lambda>0,
\\
\hspace{-2.5cm}&{\rm (c)}& \hspace{0.1cm} \int_{\RN} s_h(f)^pwdx\leq C(n,p)\int_{\RN}|f|^p(Mw)^{p/2}w^{-(p/2-1)}dx, \ \ \ 2<p<\infty.\end{aligned}$$
Also, estimates (a), (b) and (c) hold for the operator $s_p.$
To study weighted $L^p$-boundedness of the (so-called vertical) area integrals $S_{P}$ and $S_{H}$, one assumes in addition the following condition:
[**Assumption $(H_2)$.**]{} Assume that the semigroup $e^{-tL}$, generated by $-L$ on $L^2(\RN)$, has the kernel $p_t(x,y)$ which satisfies a pointwise upper bound for the gradient of the heat kernel. That is, there exist $C$ and $c$ such that for all $x,y\in{\RN}, t>0$,
$$\big|\nabla_x p_t(x,y)\big|\leq {C\over t^{(n+1)/2}} \exp\Big(-{|x-y|^2\over
c\,t}\Big).
\leqno{(G)}$$
Then the following result holds.
\[th1.2\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy conditions $(GE)$ and $(G)$. If $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and $f\in \mathcal{S}(\RN)$, then $(a), (b)$ and $(c)$ of Theorem \[th1.1\] hold for the area functions $S_{P}$ and $S_{H}$.
Let us now recall a definition. We say that a weight $w$ is in the the Muckenhoupt class $A_p, 1<p<\infty,$ if
$$\begin{aligned}
\|w\|_{A_p}\equiv \sup_Q \bigg({1\over |Q|}\int_Q w(x)dx \bigg)\bigg({1\over |Q|}\int_Q w(x)^{-1/(p-1)}dx \bigg)^{p-1}<\infty.\end{aligned}$$
$\|w\|_{A_p}$ is usually called the $A_p$ constant (or characterization or norm) of the weight. The case $p=1$ is understand by replacing the right hand side by $(\inf_Q w)^{-1}$ which is equivalent to the one defined above. Observe the duality relation:
$$\|w\|_{A_p}=\|w^{1-p'}\|^{p-1}_{A_{p'}}.$$
Following the R. Fefferman-Pipher’s method, we can use Theorems \[th1.1\] and \[th1.2\] to establish the $L^p$ estimates of the area integrals as $p$ becomes large.
\[th1.3\] Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$ Under assumptions of Theorems \[th1.1\] and \[th1.2\], there exists a constant $C$ such that for all $w\in A_1$, the following estimate holds:
$$\begin{aligned}
\label{e1.8}
\|Tf\|_{L^2_w(\RN)}\leq C\|w\|_{A_1}^{1/2}\|f\|_{L^2_w(\RN)}.\end{aligned}$$
This inequality implies that as $p\rightarrow \infty$
$$\begin{aligned}
\label{e1.9}
\|Tf\|_{L^p(\RN)}\leq Cp^{1/2}\|f\|_{L^p(\RN)}.\end{aligned}$$
The next result we will prove is the following.
\[th1.4\] Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$ Under assumptions of Theorems \[th1.1\] and \[th1.2\], there exists a constant $C$ such that for all $w\in A_p$, the following estimate holds for all $f\in L^p_w(\RN), 1<p<\infty$:
$$\begin{aligned}
\label{e1.10}
\|T f\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{\beta_p+1/(p-1)} \|f\|_{L^p_w(\RN)} \ \ (1<p<\infty),\end{aligned}$$
where $\beta_p=\max\{1/2,1/(p-1)\}.$
We should mention that Theorems \[th1.1\] and \[th1.2\] are of some independent of interest, and they provide an immediate proof of weighted $L^p$ estimates of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}$ on $L^p_w(\RN), 1<p<\infty $ and $ w\in A_p $ (see Lemma \[le5.1\] below). In the proofs of Theorems \[th1.1\] and \[th1.2\], the main tool is that each area integral is controlled by $\gL$ pointwise:
$$\begin{aligned}
\label{e3.2}
Tf(x) \leq C\gL(f)(x),\ \ \ x\in\RN,\end{aligned}$$
where $T$ is of $S_P, S_H, s_p$ and $s_H$, and $\gL$ is defined by
$$\begin{aligned}
\label{e3.1} \hspace{1cm}
\gL(f)(x)=\Bigg(\iint_{\RR_{+}^{n+1}}
\bigg({t\over t+|x-y|}\bigg)^{n\mu}
|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}}\Bigg)^{1/2}, \ \ \mu>1\end{aligned}$$
with some $\Psi\in{\mathcal S}(\RN)$. The idea of using $\gL$ to control the area integrals is due to Calderón and Torchinsky [@CT] (see also [@CW] and [@W]). Note that the singular integral $\gL$ does not satisfy the standard regularity condition of a so-called Calderón-Zygmund operator, thus standard techniques of Calderón-Zugmund theory ([@CW; @W]) are not applicable. The lacking of smoothness of the kernel was indeed the main obstacle and it was overcome by using the method developed in [@CD; @DM], together with some estimates on heat kernel bounds, finite propagation speed of solutions to the wave equations and spectral theory of non-negative self-adjoint operators.
The layout of the paper is as follows. In Section 2 we recall some basic properties of heat kernels and finite propagation speed for the wave equation, and build the necessary kernel estimates for functions of an operator, which is useful in the proof of weak-type $(1,1)$ estimate for the area integrals. In Section 3 we will prove that the area integral is controlled by $\gL$ pointwise, which implies Theorems \[th1.1\] and \[th1.2\] for $p=2$, and then we employ the R. Fefferman-Pipher’s method to obtain sharp estimates for the operator norm of the area integrals on $L^p(\RN)$ as $p$ becomes large. In Section 4, we will give the proofs of Theorems \[th1.1\] and \[th1.2\]. Finally, in Section 5 we will prove our Theorem \[th1.4\], which gives the growth of the $A_p$ constant on estimates on the weighted $L^p$ spaces.
Throughout, the letter “$c$" and “$C$" will denote (possibly different) constants that are independent of the essential variables.
Notation and preliminaries
==========================
Let us recall that, if $L$ is a self-adjoint positive definite operator acting on $L^2({\mathbb R}^n)$, then it admits a spectral resolution
$$\begin{aligned}
L=\int_0^{\infty} \lambda dE(\lambda).\end{aligned}$$
For every bounded Borel function $F:[0,\infty)\to{\mathbb{C}}$, by using the spectral theorem we can define the operator
$$\begin{aligned}
\label{e2.1}
F(L):=\int_0^{\infty}F(\lambda)\,dE_{L}(\lambda).\end{aligned}$$
This is of course, bounded on $L^2({\mathbb R}^n)$. In particular, the operator $\cos(t\sqrt{L})$ is then well-defined and bounded on $L^2({\mathbb R}^{n})$. Moreover, it follows from Theorem 3 of [@CS] that if the corresponding heat kernels $p_{t}(x,y)$ of $e^{-tL}$ satisfy Gaussian bounds $(GE)$, then there exists a finite, positive constant $c_0$ with the property that the Schwartz kernel $K_{\cos(t\sqrt{L})}$ of $\cos(t\sqrt{L})$ satisfies $$\begin{aligned}
\label{e2.2} \hspace{1cm}
{\rm supp}K_{\cos(t\sqrt{L})}\subseteq
\big\{(x,y)\in {\mathbb R}^{n}\times {\mathbb R}^{n}: |x-y|\leq c_0 t\big\}.\end{aligned}$$ See also [@CGT] and [@Si]. The precise value of $c_0$ is inessential and throughout the article we will choose $c_0=1$.
By the Fourier inversion formula, whenever $F$ is an even, bounded, Borel function with its Fourier transform $\hat{F}\in L^1(\mathbb{R})$, we can write $F(\sqrt{L})$ in terms of $\cos(t\sqrt{L})$. More specifically, we have $$\begin{aligned}
\label{e2.3}
F(\sqrt{L})=(2\pi)^{-1}\int_{-\infty}^{\infty}{\hat F}(t)\cos(t\sqrt{L})\,dt,\end{aligned}$$ which, when combined with (\[e2.2\]), gives $$\begin{aligned}
\label{e2.4} \hspace{1cm}
K_{F(\sqrt{L})}(x,y)=(2\pi)^{-1}\int_{|t|\geq |x-y|}{\hat F}(t)
K_{\cos(t\sqrt{L})}(x,y)\,dt,\qquad \forall\,x,y\in{\mathbb R}^{n}.\end{aligned}$$
The following result is useful for certain estimates later.
\[le2.1\] Let $\varphi\in C^{\infty}_0(\mathbb R)$ be even, $\mbox{supp}\,\varphi \subset (-1, 1)$. Let $\Phi$ denote the Fourier transform of $\varphi$. Then for every $\kappa=0,1,2,\dots$, and for every $t>0$, the kernel $K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)$ of the operator $(t^2L)^{\kappa}\Phi(t\sqrt{L})$ which was defined by the spectral theory, satisfies
$$\begin{aligned}
\label{e2.5}
{\rm supp}\ \! K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}
\subseteq \big\{(x,y)\in \RN\times \RN: |x-y|\leq t\big\}\end{aligned}$$
and
$$\begin{aligned}
\label{e2.6}
\big|K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)\big|
\leq C \, t^{-n}\end{aligned}$$
for all $t>0$ and $x,y\in \RN.$
The proof of this lemma is standard (see [@SW] and [@HLMMY]). We give a brief argument of this proof for completeness and convenience for the reader.
For every $\kappa=0,1,2,\dots$, we set $\Psi_{\kappa, t}(\zeta):=(t\zeta)^{2\kappa}\Phi(t\zeta)$. Using the definition of the Fourier transform, it can be verified that $$\widehat{\Psi_{\kappa,t}}(s)=(-1)^{\kappa}
{1\over t}\psi_{\kappa}({s\over t}),$$ where we have set $\psi_{\kappa} (s)={d^{2\kappa}\over ds^{2\kappa}}\varphi(s)$. Observe that for every $\kappa=0,1,2,\dots$, the function $\Psi_{\kappa,t}\in{\mathcal S}(\mathbb R)$ is an even function. It follows from formula (\[e2.4\]) that
$$\label{e2.7}
K_{(t^2L)^{\kappa}\Phi(t\sqrt{L})}(x,y)
=(-1)^{\kappa}{1\over 2\pi}\int_{|st|\geq |x-y|}
{d^{2\kappa }\over ds^{2\kappa}}\varphi({s})K_{\cos(st\sqrt{L})}(x,y)\,ds.$$
Since $\varphi\in C^{\infty}_0(\mathbb R)$ and $\mbox{supp}\,\varphi \subset(-1, 1)$, (\[e2.5\]) follows readily from this.
Note that for any $m\in {\Bbb N}$ and $t>0$, we have the relationship
$$(I+tL)^{-m}={1\over (m-1)!} \int\limits_{0}^{\infty}e^{-tsL}e^{-s} s^{m-1} ds$$
and so when $m>n/4$,
$$\begin{aligned}
\big\| (I+tL)^{-m} \big\|_{L^2\rightarrow L^{\infty}}\leq {1\over (m-1)!} \int\limits_{0}^{\infty}
\big\| e^{-tsL}\big\|_{L^2\rightarrow L^{\infty}} e^{-s} s^{m-1} ds\leq C t^{-n/4}
\end{aligned}$$
for all $t>0.$ Now $ \big\| (I+tL)^{-m} \big\|_{L^1\rightarrow L^{2}}=\big\| (I+tL)^{-m} \big\|_{L^2\rightarrow L^{\infty}}\leq
C t^{-n/4}$, and so
$$\begin{aligned}
\big\|(t^2L)^{\kappa}\Phi(t\sqrt{L})\big\|_{L^1\rightarrow L^{\infty}} \leq
\big\| (I+t^2L)^{2m}(t^2L)^{\kappa}\Phi(t\sqrt{L}) \big\|_{L^2\rightarrow L^2} \big\| (I+t^2L)^{-m} \big\|^2_{L^2\rightarrow L^{\infty}}.
\end{aligned}$$
The $L^2$ operator norm of the last term is equal to the $L^{\infty}$ norm of the function $(1+t^2|s|)^{2m} (t^2|s|)^{\kappa}\Phi(t\sqrt{|s|})$ which is uniformly bounded in $t>0$. This implies that (\[e2.6\]) holds. The proof of this lemma is concluded.
\[le2.2\] Let $\varphi\in C^{\infty}_0(\mathbb R)$ be even function with $\int \varphi =1$, $\mbox{supp}\,\varphi \subset (-1/10, 1/10)$. Let $\Phi$ denote the Fourier transform of $\varphi$ and let $\Psi(s)=s^{2n+2}\Phi^3(s)$. Then there exists a positive constant $C=C({n,\Phi})$ such that the kernel $K_{\Psi(t\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)$ of $ \Psi(t\sqrt{L})(1-\Phi(r\sqrt{L}))$ satisfies
$$\begin{aligned}
\label{e2.8}
\big|K_{\Psi(t\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)\big|
\leq C\, { r\over t^{n+1}}\Big(1+{|x-y|^2\over t^2}\Big)^{-(n+1)/2}\end{aligned}$$
for all $t>0, r>0$ and $x, y\in \RN$.
By rescaling, it is enough to show that
$$\begin{aligned}
\label{e2.9}
|K_{\Psi(\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)|
\leq C{r}\big(1+{|x-y|^2}\big)^{-(n+1)/2}.\end{aligned}$$
Let us prove (\[e2.9\]). One writes $\Psi(s)=\Psi_1(s)\Phi^2(s)$, where $\Psi_1(s)=s^{2n+2}\Phi(s)$. Then we have $\Psi(\sqrt{L})=\Psi_1(\sqrt{L})\Phi^2(\sqrt{L})$. It follows from Lemma 2.1 that $|K_{\Phi(\sqrt{L}) }(z,y)|\leq C$ and $K_{\Phi(\sqrt{L}) }(z,y)=0$ when $|z-y|\geq 1.$ Note that if $|z-y|\leq 1$, then $\big(1+|x-y|\big) \leq 2(1+|x-z|)$. Hence,
$$\begin{aligned}
&&\hspace{-1cm}\Big|\big(1+|x-y|\big)^{n+1}K_{\Psi(\sqrt{L})(1-\Phi(r\sqrt{L}))}(x,y)\Big|\\
&&= \big(1+|x-y|\big)^{n+1}\Big|\int_{\RN}
K_{\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})}(x,z) K_{\Phi(\sqrt{L}) }(z,y) dz\Big|\\
&&\leq C\int_{\RN}\big|K_{\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})}(x,z)\big|\big(1+|x-z| \big)^{n+1}dz.\end{aligned}$$
By symmetry, we will be done if we show that
$$\begin{aligned}
\label{e2.10}
\int_{\RN}\big|K_{\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})}(x,z)\big|\big(1+|x-z| \big)^{n+1}dx\leq Cr.\end{aligned}$$
Let $G_r(s)=\Psi_1(s)(1-\Phi(rs))$. Since $G_r(s)$ is an even function, apart from a $(2\pi)^{-1}$ factor we can write
$$G_r(s)=\int^{+\infty}_{-\infty}\widehat{G_r}(\xi){\rm cos}(s\xi)d\xi,$$
and by (\[e2.3\]),
$$\begin{aligned}
\label{e2.11}\Psi_1(\sqrt{L})(1-\Phi(r\sqrt{L}))\Phi(\sqrt{L})
=\int^{+\infty}_{-\infty}\widehat{G_r}(\xi){\rm cos}(\xi\sqrt{L})\Phi(\sqrt{L})d\xi.\end{aligned}$$
By Lemma 2.1 again, it can be seen that $K_{{\rm cos}(\xi\sqrt{L})\Phi(\sqrt{L})}(x,z)=0$ if $|x-z|\geq 1+|\xi|.$ Using the unitarity of $\cos(\xi\sqrt{L})$, estimates (\[e2.5\]) and (\[e2.6\]), we have
$$\begin{aligned}
\int_{\RN} \big|K_{{\rm cos}(\xi\sqrt{L})\Phi(\sqrt{L})}(x,z)\big|dx &=&
\int_{\RN} \big|\cos(\xi\sqrt{L}) \big(K_{\Phi(\sqrt{L})}(\cdot\ , z)\big)(x)\big|dx\nonumber\\
&\leq& (1+|\xi|)^{n/2} \big\| \cos(\xi\sqrt{L}) \big(K_{\Phi(\sqrt{L})}(\cdot\ , z)\big)\big\|_{L^2(\RN)} \nonumber\\
&\leq& (1+|\xi|)^{n/2} \big\| K_{\Phi(\sqrt{L})}(\cdot\ , z) \big\|_{L^2(\RN)} \nonumber\\
&\leq& (1+|\xi|)^{n/2}.\end{aligned}$$
This, in combination with (\[e2.11\]), gives
$$\begin{aligned}
\label{e2.12}
{\rm LHS\ \ of \ \ } (\ref{e2.10}) \ &\leq& C\int^{+\infty}_{-\infty} |\widehat{G_r}(\xi)| \, (1+|\xi|)^{2n+1}\, d\xi\nonumber\\
&\leq& C\Big(\int^{+\infty}_{-\infty} |\widehat{G_r}(\xi)|^2 \, (1+|\xi|)^{4n+4}\, d\xi\Big)^{1/2}\nonumber\\
&\leq& C\big\|G_r\|_{W^{2n+2, \,2}(\RN)}.\end{aligned}$$
Next we estimate the term $\big\|G_r\|_{W^{2n+2, \,2}(\RN)}$. Note that $G_r(s)=\Psi_1(s)(1-\Phi(rs)), \Phi(0)=\widehat{\varphi}(0)=\int \varphi =1$ and $\Phi=\widehat{\varphi}\in \mathcal{S}(\RR),$ also $\Psi_1(s)=s^{2n+2}\Phi(s).$ We have
$$\begin{aligned}
\label{e2.13} \hspace{1cm}
\|G_r\|_{L^2}^2= \int_{\RR} |\Psi_1(s)|^2|1-\Phi(rs)|^2ds
\leq C \|\Phi'\|^2_{L^\infty} \int_{\RR} |\Psi_1(s)|^2\, (rs)^2\,ds \leq C r^2.\end{aligned}$$
Moreover, observe that for any $k\in{\mathbb N}$, $\big|{d^k\over ds^k}\big(1-\Phi(rs)\big)\big|=r^k|\Phi^{(k)}(rs)|\leq Crs^{1-k}.$ By Leibniz’s rule, we obtain
$$\begin{aligned}
\label{e2.14}
\Big\|{d^{2n+2}\over ds^{2n+2}}G_r(s)\Big\|_{L^2}&=& \Big\|{d^{2n+2}\over ds^{2n+2}}\Big(\Psi_1(s)\big(1-\Phi(rs)\big)\Big)\Big\|_{L^2(\RN)}\nonumber\\
&\leq&\sum_{m+k=2n+2}\Big\|{d^{m}\over ds^{m}}\Big(s^{2n+2}\Phi\Big) {d^{k}\over ds^{k}}\Big(1-\Phi(rs)\Big)\Big\|_{L^2(\RN)}\nonumber\\
&\leq&Cr \sum_{m=0}^{2n+2}\Big\|s^{m-(2n+1)}{d^{m}\over ds^{m}}\Big(s^{2n+2}\Phi\Big) \Big\|_{L^2(\RN)}\nonumber\\
&\leq&Cr.\end{aligned}$$
From estimates (\[e2.13\]) and (\[e2.14\]), it follows that $\big\|G_r\|_{W^{2n+2, \,2}(\RN)}\leq Cr$. This, in combination with (\[e2.12\]), shows that the desired estimate (\[e2.10\]) holds, and concludes the proof of Lemma 2.2.
Finally, for $s>0$, we define $${\Bbb F}(s):=\Big\{\psi:{\Bbb C}\to{\Bbb C}\ {\rm measurable}: \ \
|\psi(z)|\leq C {|z|^s\over ({1+|z|^{2s}})}\Big\}.$$ Then for any non-zero function $\psi\in {\Bbb F}(s)$, we have that $\{\int_0^{\infty}|{\psi}(t)|^2\frac{dt}{t}\}^{1/2}<\infty$. Denote by $\psi_t(z)=\psi(tz)$. It follows from the spectral theory in [@Yo] that for any $f\in L^2(\RN)$,
$$\begin{aligned}
\Big\{\int_0^{\infty}\|\psi(t\sqrt{L})f\|_{L^2(\RN)}^2{dt\over t}\Big\}^{1/2}
&=&\Big\{\int_0^{\infty}\big\langle\,\overline{ \psi}(t\sqrt{L})\,
\psi(t\sqrt{L})f, f\big\rangle {dt\over t}\Big\}^{1/2}\nonumber\\
&=&\Big\{\big\langle \int_0^{\infty}|\psi|^2(t\sqrt{L}) {dt\over t}f,
f\big\rangle\Big\}^{1/2}\nonumber\\
&=& \kappa \|f\|_{L^2(\RN)},
\label{e2.15}\end{aligned}$$
where $\kappa=\big\{\int_0^{\infty}|{\psi}(t)|^2 {dt/t}\big\}^{1/2},$ an estimate which will be often used in the sequel.
An auxiliary $\gL$ function
===========================
The $\gL$ function
------------------
Let $\varphi\in C^{\infty}_0(\mathbb R)$ be even function with $\int \varphi =1$, $\mbox{supp}\,\varphi \subset (-1/10, 1/10)$. Let $\Phi$ denote the Fourier transform of $\varphi$ and let $\Psi(s)=s^{2n+2}\Phi^3(s)$ (see Lemma 2.2 above). We define the $\gL$ function by
$$\begin{aligned}
\label{e3.1} \hspace{1cm}
\gL(f)(x)=\Bigg(\iint_{\RR_{+}^{n+1}}
\bigg({t\over t+|x-y|}\bigg)^{n\mu}
|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}}\Bigg)^{1/2}, \ \ \mu>1.\end{aligned}$$
In this section, we will show that the area integrals $s_p$, $s_h $, $s_H$ and $ S_H$ are all controlled by $\gL$ pointwise. To achieve this, we need some results on the kernel estimates of the semigroup. Firstly, we note that the Gaussian upper bounds for $p_t(x,y)$ are further inherited by the time derivatives of $p_{t}(x,y)$. That is, for each $k\in{\mathbb N}$, there exist two positive constants $c_k$ and $C_k$ such that
$$\begin{aligned}
\label{e3.0}
\Big|{\partial^k \over\partial t^k} p_{t}(x,y) \Big|\leq
\frac{C_k}{ t^{n /2+k} } \exp\Big(-{|x-y|^2\over
c_k\,t}\Big)\end{aligned}$$
for all $t>0$, and $x, y\in {\mathbb R}^{n}$. For the proof of (\[e3.0\]), see [@Da] and [@Ou], Theorem 6.17.
Note that in the absence of regularity on space variables of $p_t(x,y)$, estimate (\[e3.0\]) plays an important role in our theory.
\[le3.1\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels $p_t(x,y)$ of the semigroup $e^{-tL}$ satisfy Gaussian bounds $(GE)$. Then for every $\kappa=0,1, ..., $ the operator $ (t\sqrt{L})^{2\kappa} e^{-t\sqrt{L}}$ satisfies
$$\begin{aligned}
\label{e3.00}\hspace{1cm}
\big|K_{(t\sqrt{L})^{2\kappa} e^{-t\sqrt{L}}}(x,y)\big|\leq C_\kappa t^{-n}\Big(1+{ |x-y|\over t}\Big)^{-(n+2\kappa+1)}, \ \ \forall t>0\end{aligned}$$
for almost every $x,y\in \RN.$
The proof of (\[e3.00\]) is simple. Indeed, the subordination formula $$e^{-t\sqrt{L}}={1\over \sqrt{\pi}}\int^\infty_0e^{-u}u^{-1/2}e^{-{t^2\over 4u}L}du$$ allows us to estimate
$$\begin{aligned}
\big|K_{(t\sqrt{L})^{2\kappa} e^{-t\sqrt{L}}}(x,y)\big|&\leq&
C_\kappa\int_0^{\infty}{e^{-u}\over\sqrt{u}}\Big({t^2\over u}\Big)^{-n/2}\exp \Big(-{u|x-y|^2\over c t^2}\Big)u^\kappa du\\
&\leq&C_\kappa t^{-n}\int_0^{\infty}e^{-u}u^{n/2+\kappa-1/2} \exp \Big(-{u|x-y|^2\over c t^2}\Big)\, du\\
&\leq&C_\kappa t^{-n}\Big(1+{|x-y| \over t }\Big)^{-(n+2\kappa+1)}\end{aligned}$$
for every $t>0$ and almost every $x,y\in \RN$.
\[le3.2\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels $p_t(x,y)$ of the semigroup $e^{-tL}$ satisfy $(GE)$ and ($G$). Then for every $\kappa=0,1, ..., $ the operator $ t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )$ satisfies
$$\begin{aligned}
\Big|K_{t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )}(x,y)\Big|
&\leq& C t^{-n} \exp\Big(-{|x-y|^2\over
c \,t^2}\Big), \ \ \ \forall t>0 \end{aligned}$$
for almost every $x,y\in \RN.$
Note that $ t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )= t\nabla e^{-{t^2\over 2}L} \circ (t^2L)^\kappa e^{-{t^2\over 2}L}.$ Using (\[e3.0\]) and the pointwise gradient estimate ($G$) of heat kernel $p_t(x,y)$, we have
$$\begin{aligned}
\Big|K_{t^{2\kappa+1}\nabla (L^\kappa e^{-t^2L} )}(x,y)\Big|&=& \Big| \int_{\RN} K_{t\nabla e^{-{t^2\over 2}L}}(x,z) K_{(t^2L)^\kappa e^{-{t^2\over 2}L}} (z,y)dz\Big|\nonumber\\
&\leq& C t^{-2n}\int_{\RN} \exp\Big(-{|x-z|^2\over
c \,t^2}\Big) \exp\Big(-{|z-y|^2\over
c \,t^2}\Big) dz\nonumber\\
&\leq& C t^{-n} \exp\Big(-{|x-y|^2\over
c \,t^2}\Big) \end{aligned}$$
for every $t>0$ and almost every $x,y\in \RN$.
Now we start to prove the following Propositions \[prop3.3\] and \[prop3.4\].
\[prop3.3\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy condition $(GE)$. Then for $f\in {\mathcal S}(\RN),$ there exists a constant $C=C_{n,\mu,\Psi}$ such that the area integral $s_p$ satisfies the pointwise estimate:
$$\begin{aligned}
\label{e3.4}
s_pf(x) \leq C\gL(f)(x).\end{aligned}$$
Estimate (\[e3.4\]) also holds for the area integral $s_h$.
\[prop3.4\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy conditions $(GE)$ and $(G)$. Then for $f\in {\mathcal S}(\RN),$ there exists a constant $C=C_{n,\mu,\Psi}$ such that the area integral $S_P$ satisfies the pointwise estimate:
$$\begin{aligned}
\label{e3.2}
S_Pf(x) \leq C\gL(f)(x).\end{aligned}$$
Estimate (\[e3.2\]) also holds for the area integral $S_H$.
\[Proofs of Propositions \[prop3.3\] and \[prop3.4\]\] Let us begin to prove (\[e3.2\]). By the spectral theory ([@Yo]), for every $f\in {\mathcal S}(\RN)$ and every $\kappa \in{\mathbb N}$,
$$\begin{aligned}
f =C_\Psi\int^\infty_0
(t^2L)^{\kappa}e^{-t^2{L}}\Psi(t\sqrt{L}) f {dt\over t}\end{aligned}$$
with $C^{-1}_{\Psi}=\int^\infty_0 t^{2\kappa}
e^{-t^2} \Psi(t ){dt/t}$, and the integral converges in $L^2(\RN)$. Recall the subordination formula:
$$e^{-t\sqrt{L}} ={1\over \sqrt{\pi}}\int^\infty_0 e^{-u} {u}^{-1/2} e^{-{t^2\over 4u}L} du.$$
One writes
$$\begin{aligned}
\label{e3.666}
\hspace{-1cm} s\nabla e^{-s\SL}f(y)
&=& {1\over \sqrt{\pi}}\int^\infty_0e^{-u} {u}^{-1/2}s\nabla e^{-{s^2\over 4u}L}f(y)du \nonumber\\
&=&{ C_{\Psi} \over \sqrt{\pi}}\int^{\infty}_{0}\int^\infty_0e^{-u} {u}^{-1/2} st^{2\kappa}\nabla\big(L^\kappa
e^{-({s^2\over 4u}+t^2)L}\big)\Psi(t\sqrt{L})f (y){dt\, du\over t}.\end{aligned}$$
Fix $\kappa=[{n(\mu-1)\over 2}]+1$. Using Lemma \[le3.2\] and the Hölder inequality, we can estimate (\[e3.666\]) as follows:
$$\begin{aligned}
&&\hspace{-1cm}|s\nabla e^{-s\SL}f(y)|\\
&\leq&C \int^{\infty}_0\int^{\infty}_0\int_{\RN}e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
|\Psi(t\SL)f(z)|{dzdtdu\over t}\\
&\leq&C A\cdot B,\end{aligned}$$
where $$\begin{aligned}
A^2 &= & \int^{\infty}_0\int^{\infty}_0\int_{\RN}|\Psi(t\SL)f(z)|^2e^{-u} {u}^{-1/2}s t^{2\kappa}
\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dzdtdu\over t} \end{aligned}$$
and $$\begin{aligned}
B^2
&=& \int^{\infty}_0\int^{\infty}_0\int_{\RN}e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dzdtdu\over t}\\
&=&C \int^{\infty}_0\int^{\infty}_0\int_{0}^{\infty}e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa}e^{-r^2 }
r^{n-1}{drdtdu\over t}\\
&\leq& C \int^{\infty}_0\int^{\infty}_0 e^{-u} \, v^{2\kappa}\Big(1+{v^2 }\Big)^{-1/2-\kappa} { du dv\over v} \\
&\leq& C.\end{aligned}$$
Note that in the first equality of the above term $B$, we have changed variables $|y-z|\rightarrow r({s^2\over 4u}+{t^2 })^{1/2}$ and $t\rightarrow v ({s^2/u})^{1/2}$. Hence,
$$\begin{aligned}
&&\hspace{-0.3cm}|s\nabla e^{-s\SL}f(y)|^2\\
&&\leq C
\int^{\infty}_0\int^{\infty}_0\int_{\RN}|\Psi(t\SL)f(z)|^2e^{-u}u^{-1/2}st^{2\kappa}\Big({s^2\over 4u}
+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dzdtdu\over t}.\end{aligned}$$
Therefore, we put it into the definition of $S_p$ to obtain
$$\begin{aligned}
&&\hspace{-0.6cm}S^2_P(f)(x) = \int^{\infty}_0\int_{|x-y|<s}|s\nabla e^{-s\SL}f(y)|^2{dyds\over s^{n+1}}\\
&&\leq C\iint_{\RR^{n+1}_{+}}|\Psi(t\SL)f(z)|^2\\
&&\hspace{0.2cm}\times\Bigg(\int^{\infty}_0\int^{\infty}_0
\int_{|x-y|<s}e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y-z|^2/c({s^2\over 4u}+{t^2 })}
{dydsdu\over s^{n+1}}\Bigg)
{dzdt\over t^{n+1}}.\end{aligned}$$
We will be done if we show that
$$\begin{aligned}
\label{e3.3}
&&
\hspace{-1.2cm}
\int^{\infty}_0\int^{\infty}_0\int_{|v-y|<s}e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|y|^2/c({s^2\over 4u}+{t^2 })}
{dydsdu\over s^{n+1}}\\
&\leq& C\Big({t\over t+|v|}\Big)^{n\mu},\nonumber\end{aligned}$$
where we set $x-z=v,$ and we will prove estimate (\[e3.3\]) by considering the following two cases.
[*Case 1.*]{} $|v|\leq t.$ In this case, it is easy to show that
$$\begin{aligned}
{\rm LHS\ of \ (\ref{e3.3})}
&&\leq C \int^{\infty}_0\int^{\infty}_0 e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}
{ dsdu\over s }\\
&&\leq C \int^{\infty}_0\int^{\infty}_0 e^{-u}u^{-1/2}\sqrt{u}s\big({s^2}+1\big)^{-1/2-\kappa-n/2}
{ dsdu\over s }\\
&&\leq C.\end{aligned}$$
But $|v|\leq t$, so
$$\Big({t\over t+|v|}\Big)^{n\mu}\geq C_{n,\mu}.$$
This implies that (\[e3.3\]) holds when $|v|\leq t.$
[*Case 2.*]{} $|v|> t$. In this case, we break the integral into two pieces:
$$\begin{aligned}
\int^\infty_0\int^{|v|/2}_0\int_{|v-y|<s}\cdots +\int^\infty_0\int^{\infty}_{|v|/2}\int_{|v-y|<s}\cdots =:I+{\it II}.\end{aligned}$$
For the first term, note that $|y|\geq |v|-|v-y|>|v|/2.$ This yields
$$\begin{aligned}
I&\leq&\int^{\infty}_0\int^{|v|/2}_0\int_{|v-y|<s}e^{-u}u^{-1/2}st^{2\kappa+n}
\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2}e^{-|v|^2/c({s^2\over 4u}+{t^2 })}
{dydsdu\over s^{n+1}}\\
&\leq&C\int^{\infty}_0\int^{\infty}_0e^{-u}u^{-1/2}st^{2\kappa+n}
\Big({s^2\over 4u}+{t^2}\Big)^{-1/2-\kappa-n/2} \Big({s^2\over 4u}+{t^2 }\Big)^{n\mu/2}/|v|^{n\mu}
{ dsdu\over s }\\
&\leq&C\Big({t\over |v|}\Big)^{n\mu}\int^{\infty}_0\int^{\infty}_0
e^{-u}u^{-1/2}st^{2\kappa+n-n\mu}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa-n/2+n\mu/2}
{ dsdu\over s }\\
&\leq&C\Big({t\over |v|}\Big)^{n\mu}\int^{\infty}_0\int^{\infty}_0e^{-u} s \big({s^2+1}\big)^{-1/2-\kappa-n/2+n\mu/2}
{ dsdu\over s }\\
&\leq&C\Big({t\over |v|}\Big)^{n\mu},\end{aligned}$$
where we used condition $\kappa=[{n(\mu-1)\over2}]+1$ in the last inequality. Since $|v|>t$, so $I\leq C_{n,\mu}\Big({t \over t+|v|}\Big)^{n\mu}.$
For the term $\it{II}$, we have
$$\begin{aligned}
\it{II}
&\leq&C\int^{\infty}_0\int_{|v|/2}^{\infty}\int_{0}^{\infty}e^{-u}
u^{-1/2}st^{2\kappa+n}\big({s^2\over 4u}+{t^2 }\big)^{-1/2-\kappa-n/2}e^{-r^2/ ({s^2\over 4u}+{t^2 })}r^{n-1}
{drdsdu\over s^{n+1}}\\
&\leq&C\int^{\infty}_0\int_{|v|/2}^{\infty} e^{-u}u^{-1/2}st^{2\kappa+n}\Big({s^2\over 4u}+{t^2 }\Big)^{-1/2-\kappa}
{ dsdu\over s^{n+1}}\\
&\leq&C\int^{\infty}_0\int_{|v|/(4\sqrt{u}t)}^{\infty} e^{-u}u^{-1/2}(\sqrt{u})^{1-n}
\Big({s^2+1}\Big)^{-1/2-\kappa}
{ dsdu\over s^{n}}\\
&\leq&C\int^{\infty}_0 e^{-u}u^{-1/2}(\sqrt{u})^{1-n}
\Big({\sqrt{u}\, t\over |v|}\Big)^{2\kappa+n}
du\\&\leq&C\Big({t\over |v|}\Big)^{2\kappa+n}\\
&\leq&
C \Big({t\over t+|v|}\Big)^{n\mu},\end{aligned}$$
since $\kappa=[{n(\mu-1)\over2}]+1$ and $|v|>t$.
From the above [*Cases 1*]{} and [*2*]{}, we have obtained estimate (\[e3.3\]), and then the proof of estimate (\[e3.2\]) is complete. The similar argument as above gives estimate (\[e3.2\]) since for the area integral $S_H$, and this completes the proof of Proposition \[prop3.4\].
For the area functions $s_P$ and $s_h$, we can use a similar argument to show Proposition \[prop3.3\] by using either estimate (\[e3.0\]) or Lemma \[le3.1\] instead of Lemma \[le3.2\] in the proof of estimate (\[e3.2\]), and we skip it here.
Weighted $L^2$ estimate of $\gL$.
---------------------------------
\[th3.5\] Let $\mu>1$. Then there exists a constant $C=C_{n,\mu,\Phi}$ such that for all $w\geq 0$ in $L_{loc}^{1}(\RN)$ and all $f\in {\mathcal S}(\RN)$, we have $$\begin{aligned}
\label{e3.6}
\int_{\RN}\gL (f)^2wdx\leq C\int_{\RN}|f|^2Mwdx.\end{aligned}$$
The proof essentially follows from [@CWW] and [@CW] for the classical area function. Note that by Lemma \[le2.1\], the kernel $K_{\Psi(t\sqrt{L})}$ of the operator $\Psi(t\sqrt{L})$ satisfies supp $K_{\Psi(t\sqrt{L})}\subseteq \big\{ (x,y)\in \RN\times \RN: |x-y|\leq t\big\}.$ By (\[e3.1\]), one writes
$$\begin{aligned}
\label{e3.7}\hspace{1cm}
\int_{\RN}\gL (f)^2wdx
&=& \int_{\RN} \int_{\RR_{+}^{n+1}}
\bigg({t\over t+|x-y|}\bigg)^{n\mu}
|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}} dx\nonumber\\
&=&
\int_{\RR^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2\bigg({1\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-y|}\Big)^{n\mu}dx\bigg){dydt\over t}.\end{aligned}$$
For $k$ an integer, set
$$A_k=\Big\{(y,t):\ 2^{k-1}<{1\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-y|}\Big)^{n\mu}dx\leq 2^k\Big\}.$$
Then
$$\begin{aligned}
\label{e3.8}
{\rm RHS \ of \ (\ref{e3.7})} \ \leq \sum_{k\in \mathbb{Z}}2^k
\int_{\RR^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2\chi_{A_k}(y,t){dydt\over t}.\end{aligned}$$
We note that if $(y,t)\in A_k$, then since $\mu>1$,
$$2^{k-1}\leq {1\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-y|}\Big)^{n\mu}dx\leq CMw(y).$$
Now if $|y-z|<t$, then $t+|x-y|\approx t+|x-z|$. Thus if $|y-z|<t$ and $(y,t)\in A_k,$
$$2^{k-1}\leq {C\over t^n}\int_{\RN}w(x)
\Big({t\over t+|x-z|}\Big)^{n\mu}dx\leq CMw(z).$$
In particular, if $(y,t)\in A_k$ and $|y-z|<t$, then $z\in E_k=\{z:\ Mw(z)\geq C2^k\}$. Now since ${\rm supp}\ \! K_{ \Psi(t\sqrt{L})}(y,z)
\subseteq \big\{(y,z)\in \RN\times \RN: |y-z|\leq t\big\}$, for $(y,t)\in A_k$,
$$\Psi(t\sqrt{L})f(y)=\int_{|y-z|<t}K_{ \Psi(t\sqrt{L})}(y,z)f(z)dz
=\int_{\RN}K_{ \Psi(t\sqrt{L})}(y,z)f(z)\chi_{E_k}(z)dz.$$
Therefore,
$$\begin{aligned}
{\rm RHS \ of \ (\ref{e3.7})} \ &\leq& \sum_{k\in \mathbb{Z}}2^k
\int_{\RR^{n+1}_{+}}|\Psi(t\sqrt{L})(f\chi_{E_k})(y)|^2\chi_{A_k}(y,t){dydt\over t}\\
&\leq& \sum_{k\in \mathbb{Z}}2^k
\int_{\RR^{n+1}_{+}}|\Psi(t\sqrt{L})(f\chi_{E_k})(y)|^2{dydt\over t}\\
&=& \sum_{k\in \mathbb{Z}}2^k
\int_0^{\infty}\|\Psi(t\sqrt{L})(f\chi_{E_k})\|_{L^2(\RN)}^2{ dt\over t}\\
&=& C_{\Psi} \sum_{k\in \mathbb{Z}}2^k \| f\chi_{E_k} \|_{L^2(\RN)}^{2}\end{aligned}$$
with $C_{\Psi}=\int^\infty_0|\Psi(t)|^2{dt/t}<\infty,$ and the last inequality follows from the spectral theory (see [@Yo]). By interchanging the order of summation and integration, we have
$$\begin{aligned}
\int_{\RN}\gL (f)^2wdx&\leq& C \sum_{k\in \mathbb{Z}}2^k\int_{\RN}|f|^2\chi_{E_k}dx\\
&\leq& C\int_{\RN}|f|^2\Big(\sum_{k\in \mathbb{Z}}2^k\chi_{E_k}\Big)dx\\
&\leq& C \int_{\RN}|f|^2Mwdx.\end{aligned}$$
This concludes the proof of the theorem.
As a consequence of Propositions \[prop3.3\], \[prop3.4\] and Theorem \[th3.5\], we have the following analogy for the area function of the result of Chang, Wilson and Wolff.
\[c3.3\] Let $T$ be of the area integrals $s_h$, $s_p$, $S_{P}$ and $S_{H}.$ Under assumptions of Theorems \[th1.1\] and \[th1.2\], there exists a constant $C$ such that for all $w\geq 0$ in $L_{loc}^{1}(\RN)$ and all $f\in {\mathcal S}(\RN)$,
$$\begin{aligned}
\int_{\RN}|Tf|^2wdx\leq C\int_{\RN}|f|^2Mwdx.\end{aligned}$$
Proof of Theorem 1.3
--------------------
Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$ For $w\in A_1$, we have $Mw(x)\leq \|w\|_{A_1}w(x)$ for a.e. $x\in \RN$. According to Corollary \[c3.3\],
$$\begin{aligned}
\int_{\RN}T(f)^2wdx\leq C\int_{\RN}|f|^2Mwdx\leq C\|w\|_{A_1}\int_{\RN}|f|^2wdx.\end{aligned}$$
This implies (\[e1.8\]) holds.
For (\[e1.9\]), we follow the method of Cordoba and Rubio de Francia (see pages 356-357, [@FP]). Let $p>2$ and take $f\in L^p(\RN).$ Then from duality, we know that there exist some $\varphi\in L^{(p/2)'}(\RN),$ with $\varphi\geq0,$ $\|\varphi\|_{L^{(p/2)'}(\RN)}=1$, such that
$$\|Tf\|^2_{L^p(\RN)}\leq \int_{\RN}|Tf|^2\varphi dx.$$
Set
$$v=\varphi +{M\varphi\over 2\|M\|_{L^{(p/2)'}(\RN)}}+{M^2\varphi\over (2\|M\|_{L^{(p/2)'}(\RN)})^2}+\cdots$$
following Rubio de Francia’s familiar method (Here $\|M\|_{L^{(p/2)'}(\RN)}$ denotes the operator norm of the Hardy-Littlewood maximal operator on $L^{(p/2)'}(\RN)$). Then $\|v\|_{L^{(p/2)'}(\RN)}\leq 2$ and $\|v\|_{A_1}\leq 2\|M\|_{L^{(p/2)'}(\RN)}\equiv O(p)$ as $p\to \infty.$ Therefore
$$\begin{aligned}
\|Tf\|^2_{L^p(\RN)}&\leq& \int_{\RN}|Tf|^2\varphi dx\\
&\leq&\int_{\RN}|Tf|^2v dx\\
&\leq&C\|v\|_{A_1}\int_{\RN}|f|^2v dx\\
&\leq&Cp\|f\|^2_{L^p(\RN)}.\end{aligned}$$
This proves (\[e1.9\]), and then the proof of this theorem is complete. $\Box$
Note that in Theorem \[e1.3\], when $L=-\Delta$ is the Laplacian on $\RN$, it is well known that estimate (\[e1.9\]) of the classical area integral on $L^p(\RN)$ is sharp, in general (see, e.g., [@FP]).
Proofs of Theorems 1.1 and 1.2
==============================
Note that from Propositions \[prop3.3\] and \[prop3.4\], the area functions $S_H, S_P, s_H$ and $s_p$ are all controlled by the $\gL$ function. In order to prove Theorems 1.1 and 1.2, it suffices to show the following result.
\[th4.1\] Let $L$ be a non-negative self-adjoint operator such that the corresponding heat kernels satisfy Gaussian bounds $(GE)$. Let $\mu>3$. If $w\geq 0$, $w\in L_{\rm loc}^{1}(\RN)$ and $f\in {\mathcal S}(\RN)$, then
$$\begin{aligned}
\hspace{-2.5cm}&{\rm (a)}& \hspace{0.1cm}\int_{\{\gL (f)>\lambda\}} wdx\leq {c(n)\over \lambda}\int_{\RN}|f| Mwdx,\ \ \ \lambda>0,\\
\hspace{-2.5cm}
&{\rm (b)}& \hspace{0.1cm}
\int_{\RN}\gL (f)^pw\, dx\leq c(n,p)\int_{\RN}|f|^pMw\, dx,\ \ \ 1<p\leq 2,
\\
\hspace{-2.5cm}&{\rm (c)}& \hspace{0.1cm} \int_{\RN} \gL (f)^pwdx\leq c(n,p)\int_{\RN}|f|^p(Mw)^{p/2}w^{-(p/2-1)}dx, \ \ \ 2<p<\infty.\end{aligned}$$
Weak-type $(1,1)$ estimate
--------------------------
We first state a Whitney decomposition. For its proof, we refer to Chapter 6, [@St].
\[le4.2\] Let $F$ be a non-empty closed set in $\RN$. Then its complement $\Omega$ is the union of a sequence of cubes $Q_k$, whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances from $F$. More explicitly:
\(i) $\Omega=\RN\setminus F=\bigcup\limits_{k=1}^{\infty}Q_k.$
\(ii) $Q_j\bigcap Q_k=\varnothing$ if $j\neq k$.
\(iii) There exist two constants $c_1, c_2 > 0$, (we can take $c_1 = 1$, and $c_2 = 4$), so that
$$c_1 {\rm diam}(Q_k)\leq {\rm dist}(Q_k,\ F)\leq c_2 {\rm diam}(Q_k).$$
Note that if $\Omega$ is an open set with $\Omega=\bigcup\limits_{k=1}^{\infty}Q_k$ a Whitney decomposition, then for every $\varepsilon:\ 0<\varepsilon<1/4,$ there exists $N\in \mathbb{N}$ such that no point in $\Omega$ belongs to more than $N$ of the cubes $Q_{k}^{\ast}$, where $Q_{k}^{\ast}=(1+\varepsilon)Q_k.$
Since $g^{\ast}_{\mu',\Psi}(f)\leq \gL(f)$ whenever $\mu'\geq \mu$, it is enough to prove ($a$) of Theorem \[th4.1\] for $3<\mu<4.$ Since $\gL$ is subadditive, we may assume that $f\geq0$ in the proof (if not we only need to consider the positive part and the negative part of $f$).
For $\lambda>0,$ we set $\Omega=\{x\in\RN:\ Mf(x)>\lambda\}$. By [@FS] it follows that
$$\begin{aligned}
\label{e4.2}
\int_{\Omega}wdx\leq {C\over \lambda}\int_{\RN}|f|Mwdx.\end{aligned}$$
Let $\Omega=\cup Q_j$ be a Whitney decomposition, and define
$$\begin{aligned}
h(x)&=&\left\{
\begin{array}{ll}
f(x), & x\notin \Omega \\ [12pt]
{1\over |Q_j|}\int_{Q_j}f(x)dx, & x\in Q_j
\end{array}
\right.\\[12pt]
b_j(x)&=&\left\{
\begin{array}{ll}
f(x)-{1\over |Q_j|}\int_{Q_j}f(x)dx, & x\in Q_j \\ [12pt]
0, & x\notin Q_j.
\end{array}
\right.\end{aligned}$$
Then $f=h+\sum_jb_j$, and we set $b=\sum_jb_j.$ As in [@St], we have $|h|\leq C\lambda$ a.e. By (\[e4.2\]), it suffices to show
$$\begin{aligned}
\label{e4.3}
w\{x\notin\Omega:\ \gL(f)(x)>\lambda\}\leq {C\over \lambda}\int_{\RN}|f|Mwdx.\end{aligned}$$
By Chebychev’s inequality and Theorem \[th3.5\],
$$\begin{aligned}
w\{x\notin\Omega:\ \gL(h)(x)>\lambda\}&\leq& {1\over \lambda^2}
\int_{\RN}\gL(h)^2(w\chi_{\RN\setminus\Omega})dx\\
&\leq& {C\over \lambda^2}
\int_{\RN}|h|^2 M(w\chi_{\RN\setminus\Omega})dx\\
&\leq&{C\over \lambda }
\int_{\RN}|h| M(w\chi_{\RN\setminus\Omega})dx\end{aligned}$$
since $|h|\leq C\lambda$ a.e. By definition of $h$, the last expression is at most
$$\begin{aligned}
\label{e4.4}
{C\over \lambda }
\int_{\RN}|f| Mwdx +\sum_j{C\over \lambda }
\int_{Q_j}\Big({1\over |Q_j|}\int_{Q_j}|f(z)|dz\Big) M(w\chi_{\RN\setminus\Omega})(x)dx.\end{aligned}$$
From the property (iii) of Lemma \[le4.2\], we know that for $x,z\in Q_j$ there is a constant $C$ depending only on $n$ so that $M(w\chi_{\RN\setminus\Omega})(x)\leq CM(w\chi_{\RN\setminus\Omega})(z)$. Thus (\[e4.4\]) is less than
$${C\over \lambda }
\int_{\RN}|f| Mwdx +\sum_j{C\over \lambda }
\int_{Q_j}\Big({1\over |Q_j|}\int_{Q_j}|f(z)|Mw(z)dz\Big) dx\leq {C\over \lambda }
\int_{\RN}|f| Mwdx.$$
This gives
$$w\{x\notin\Omega:\ \gL(h)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx.$$
Therefore, estimate (\[e4.3\]) will follow if we show that
$$\begin{aligned}
\label{eb}
w\{x\notin\Omega:\ \gL(b)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx.\end{aligned}$$
To prove (\[eb\]), we follow an idea of [@DM] to decompose $b=\sum_jb_j=\sum_j\Phi_j(\sqrt{L})b_j + \sum_j\big(1-\Phi_j(\sqrt{L})\big)b_j,$ where $\Phi_j(\sqrt{L})=\Phi\Big({\ell(Q_j)\over32}\sqrt{L}\Big),$ $\Phi$ is the function as in Lemma \[le2.2\] and $\ell(Q_j)$ is the side length of the cube $Q_j.$ See also [@CD]. So, it reduces to show that
$$\begin{aligned}
\label{ei1}
w\{x\notin\Omega:\ \gL\Big(\sum_j\Phi_j(\sqrt{L})b_j\Big)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx\end{aligned}$$
and
$$\begin{aligned}
\label{ei2}
w\{x\notin\Omega:\ \gL\Big(\sum_j\big(1-\Phi_j(\sqrt{L})\big)b_j\Big)(x)>\lambda\}\leq{C\over \lambda }
\int_{\RN}|f| Mwdx.\end{aligned}$$
By Chebychev’s inequality and Theorem \[th3.5\] again, we have
$$\begin{aligned}
{\rm LHS\ \ of \ \ (\ref{ei1})}\ &\leq&{C\over \lambda^2}\int_{\RN}\Big|\gL\Big(\sum_j\Phi_j(\sqrt{L})b_j\Big)\Big|^2
(w\chi_{\RN\setminus\Omega})dx\\
&\leq&{C\over \lambda^2}\int_{\RN}\Big| \sum_j\Phi_j(\sqrt{L})b_j \Big|^2
M(w\chi_{\RN\setminus\Omega})dx.\end{aligned}$$
Note that $\Phi_j(\sqrt{L})=\Phi\Big({\ell(Q_j)\over32}\sqrt{L}\Big),$ it follows from Lemma \[le2.1\] that ${\rm supp}\ \Phi_j(\sqrt{L})b_j\subset {{17 }Q_j/16} $ and $\big|K_{\Phi_j(\sqrt{L})}(x,y)\big|\leq C/\ell(Q_j)$. Hence, the above inequality is at most
$$\begin{aligned}
{C\over \lambda^2}\sum_j \int_{\RN}\Big| \Phi_j(\sqrt{L})b_j \Big|^2
M(w\chi_{\RN\setminus\Omega})dx.\end{aligned}$$
This, together with Lemma \[le2.1\] and the definition of $b$, yields
$$\begin{aligned}
{\rm LHS\ \ of \ \ (\ref{ei1})}\
&\leq& {C\over \lambda^2}\sum_j\int_{{17 }Q_j/16}\Big({\ell(Q_j)^{-n}}\int_{Q_j} |b(y)|dy \Big)^2
M(w\chi_{\RN\setminus\Omega})(x)dx\\
&\leq& {C\over \lambda^2}\sum_j\int_{{17 }Q_j/16}\Big( {1\over|Q_j|}\int_{Q_j}|f(y)| dy \Big)^2
M(w\chi_{\RN\setminus\Omega})(x)dx\\
&\leq& {C\over \lambda }\sum_j\int_{{17 }Q_j/16}\Big( {1\over|Q_j|}\int_{ Q_j}|f(y)| dy \Big)
M(w\chi_{\RN\setminus\Omega})(x)dx\\
&\leq& {C\over \lambda }\sum_j{1\over|Q_j|}\int_{{17 }Q_j/16} \int_{ Q_j}|f(y)|M(w\chi_{\RN\setminus\Omega})(y) dy
dx\\
&\leq&{C\over \lambda }\int_{\RN}|f |Mwdy.\end{aligned}$$
This proves the desired estimate (\[ei1\]).
Next we turn to estimate (\[ei2\]). It suffices to show that
$$\sum_j\int_{\RN\setminus\Omega}\gL\Big( \big(1-\Phi_j(\sqrt{L})\big)b_j\Big)wdx
\leq C\int_{\RN}|f|Mwdx.$$
Further, the above inequality reduces to prove the following result:
$$\begin{aligned}
\label{ei22}
\int_{\RN\setminus\Omega}\gL\Big( \big(1-\Phi_j(\sqrt{L})\big)b_j\Big)wdx
\leq C\int_{Q_j}|f|Mwdx.\end{aligned}$$
Let $x_j$ denote the center of $Q_j$. Let us estimate $\Psi(t\sqrt{L})\big(1-\Phi_j(\sqrt{L})\big)b_j(y)=:\Psi_{jt}(\sqrt{L})b_j(y)$ by considering two cases: $t\leq \ell(Q_j)/4$ and $t>\ell(Q_j)/4$.
[*Case 1. $t\leq \ell(Q_j)/4$*]{}. In this case, we use Lemma \[le2.1\] to obtain
$$\begin{aligned}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|&\leq& |\Psi(t\sqrt{L})b_j(y)|+\Big|\Psi(t\sqrt{L})\Phi_j(\sqrt{L})b_j(y)\Big|\\
&\leq& \Big|\int_{Q_j}K_{\Psi(t\sqrt{L})}(y,z)b(z)dz\Big|\\
&&+\Big|\int_{{17\over16}Q_j}
K_{\Psi(t\sqrt{L})}(y,z)\Big(\int_{Q_j}K_{\Phi_j(\sqrt{L})}(z,x)b(x)dx\Big)dz\Big|\\
&\leq& C\|b_j\|_{ 1}t^{-n}.\end{aligned}$$
[*Case 2. $t> \ell(Q_j)/4$*]{}. Using Lemma \[le2.2\], we have
$$\begin{aligned}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|
& \leq&\int_{\RN}\Big|K_{\Psi(t\sqrt{L})\big(1-\Phi_j(\sqrt{L})\big)}(y,z)\Big||b_j(z)|dz\\
& \leq& C \|b_j\|_{ 1}\ell(Q_j)t^{-n-1}.\end{aligned}$$
From the property (iii) of Lemma \[le4.2\], we know that if $x\notin\Omega,$ then $|x-x_j|> (\sqrt{n}+1/2)\ell(Q_j)$. By Lemma \[le2.1\], we have $\Psi(t\sqrt{L})\big(1-\Phi_j(\sqrt{L})\big)b_j(y)=0$ unless $|y-x_j|\leq t+(1/32+\sqrt{n}/2)\ell(Q_j).$ Note that for $x\notin\Omega$, $0<t\leq \ell(Q_j)/4$ and $|y-x_j|\leq t+(1/32+\sqrt{n}/2)\ell(Q_j)$, $|x-y|\geq|x-x_j|-|y-x_j|>{\sqrt{n}/2+7/32\over \sqrt{n}+1/2}|x-x_j|$. Denote $F_j=:\{y:\ |y-x_j|<(9/32+\sqrt{n}/2)\ell(Q_j)\}$. Then for $x\notin\Omega$ and $\mu>3$, we have
$$\begin{aligned}
&&\hspace{-1cm}\bigg(\int^{\ell(Q_j)/4}_{0}\int_{F_j}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{n/2}\over |x-x_j|^{n\mu/2}}
\bigg(\int^{\ell(Q_j)/4}_{0}
t^{n\mu-2n-n-1}dt\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{n/2}\over |x-x_j|^{n\mu/2}}\ell(Q_j)^{(n\mu-3n)/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{-n}\over (1+|x-x_j|/\ell(Q_j))^{n\mu/2}} \\
&&\leq C |f\chi_{Q_j}|\ast \tau_{\ell(Q_j)}(x),\end{aligned}$$
where $\tau_{\ell(Q_j)}(x)=1/(1+|x|)^{n\mu/2}\in L^1(\RN)$.
For the next part of the integral we consider two cases: $n=1$ and $n>1$. Note that for $x\notin\Omega$, $\ell(Q_j)/4<t\leq |x-x_j|/4$ and $y\in E_{jt}=:\{y:\ |y-x_j|\leq t+(1/32+\sqrt{n}/2)\ell(Q_j)\}$, $|x-y|\geq|x-x_j|-|y-x_j|>{\sqrt{n}/4+11/32\over \sqrt{n}+1/2}|x-x_j|$. Thus for $3<\mu<4$, if $n=1$,
$$\begin{aligned}
&&\hspace{-1cm}\bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}\int_{E_{jt}}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\[3pt]
&&\leq C {\|b_j\|_1\ell(Q_j)\over |x-x_j|^{ \mu/2}}
\bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}
t^{-4+1+\mu-2}dt\bigg)^{1/2}\\[3pt]
&&\leq C {\|b_j\|_1\ell(Q_j) \over |x-x_j|^{ \mu/2}}\ell(Q_j)^{( \mu-4)/2}\\[3pt]
&&\leq C {\|b_j\|_1\ell(Q_j)^{-1}\over (1+|x-x_j|/\ell(Q_j))^{ \mu/2}} \\[3pt]
&&\leq C |f\chi_{Q_j}|\ast \sigma_{\ell(Q_j)}(x),\end{aligned}$$
where $\sigma_{\ell(Q_j)}(x)=1/(1+|x|)^{ \mu/2} $. On the other hand, for $3<\mu<4$, if $n>1$,
$$\begin{aligned}
&&\hspace{-1cm} \bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}\int_{E_{jt}}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)\over |x-x_j|^{ n\mu/2}}
\bigg(\int_{\ell(Q_j)/4}^{|x-x_j|/4}
t^{-2n-2+n\mu+n-n-1}dt\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j) \over |x-x_j|^{ n\mu/2}}\ell(Q_j)^{( n\mu-2n-2)/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j)^{-n}\over (1+|x-x_j|/\ell(Q_j))^{ n+1}} \\
&&\leq C |f\chi_{Q_j}|\ast P_{\ell(Q_j)}(x),\end{aligned}$$
where $P_{\ell(Q_j)}(x)=1/(1+|x|)^{ n+1} $.
Finally, since $t/(t+|x-y|)\leq 1$, so
$$\begin{aligned}
&&\hspace{-1cm}\bigg(\int^{\infty}_{|x-x_j|/4}\int_{E_{jt}}
\big|\Psi_{jt}(\sqrt{L})b_j(y)\big|^2
\Big({t\over t+|x-y|}\Big)^{n\mu}{dydt\over t^{n+1}}\bigg)^{1/2}\\
&&\leq C {\|b_j\|_1\ell(Q_j) }
\bigg(\int^{\infty}_{|x-x_j|/4}
t^{-2n-3}dt\bigg)^{1/2}\\
&&\leq C |f\chi_{Q_j}|\ast P_{\ell(Q_j)}(x).\end{aligned}$$
Therefore, if $x\notin\Omega$, and $n>1$, then $\gL\Big(\big(1-\Phi_j(\sqrt{L})\big)b_j\Big)(x)\leq C |f\chi_{Q_j}|\ast P_{\ell(Q_j)}(x).$ And
$$\begin{aligned}
\int_{\RN\setminus\Omega}\gL\Big( \big(1-\Phi_j(\sqrt{L})\big)b_j\Big)wdx
&\leq& C \int_{\RN\setminus\Omega}|f\chi_{Q_j}|\ast P_{\ell(Q_j)}wdx\\
&\leq& C \int_{Q_j}|f|( P_{\ell(Q_j)}\ast w)dx\\
&\leq& C \int_{Q_j}|f|Mwdx.\end{aligned}$$
If $n=1$ we get the same thing, but with $P$ replaced by $\sigma.$ This concludes the proof of (\[e4.3\]). And the proof of this theorem is complete.
Estimate for $2<p<\infty$
-------------------------
We proceed by duality. If $h(x)\geq0$ and $h\in L^{(p/2)'}(wdx)$, then
$$\begin{aligned}
\int_{\RN}\gL(f)^2hwdx=\int_{\RR^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2{1\over t}
\bigg({1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}dx\bigg)dydt.\end{aligned}$$
Set
$$E_k=\Big\{(y,t):\ {1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}dx\sim2^k\Big\}.$$
Note that if $|y-z|<t$, then $t+|x-y|\sim t+|x-z|$. Thus, if $(y,t)\in E_k$ and $|y-z|<t$, then
$$\begin{aligned}
2^k<{1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}dx\sim
{1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-z|}\Big)^{n\mu}dx.\end{aligned}$$
The last expression is at most
$$\begin{aligned}
&&\hspace{-1.2cm}C\sum_{j=0}^{\infty}{1\over2^{jn\mu}}{1\over t^n}\int_{B(z,2^jt)}hwdx\\
&=&C\sum_{j=0}^{\infty}{1\over2^{jn(\mu-1)}}
{w(B(z,2^jt))\over (2^jt)^n}{1\over w(B(z,2^jt))}\int_{B(z,2^jt)}hwdx\\
&\leq&C\sum_{j=0}^{\infty}{1\over2^{jn(\mu-1)}}Mw(z)M_w(h)(z)\\
&\leq&C Mw(z)M_w(h)(z),\end{aligned}$$
where
$$M_w(h)(z)=\sup_{t>0}\Big({1\over w(B(z,t))}\int_{B(z,t)}hwdx\Big).$$ Recall that ${\rm supp}\ K_{\Psi(t\sqrt{L})}(y,\cdot)\subset B(y,t)$. Since for $(y,t)\in E_k$ and $|y-z|<t$ we have $z\in A_k=\{z:\ Mw(z)M_w(h)(z)\geq C_{n,\mu}2^k\},$ it follows that for $(y,t)\in E_k$, $\Psi(t\sqrt{L})f(y)=\Psi(t\sqrt{L})(f\chi_{A_k})(y).$ Thus,
$$\begin{aligned}
&&\hspace{-1.2cm}\int_{\RR^{n+1}_{+}}|\Psi(t\sqrt{L})f(y)|^2{1\over t}
\bigg({1\over t^n}\int_{\RN}h(x)w(x)\Big({t\over t+|x-y|}\Big)^{n\mu}\bigg)dydt\\
&&\leq \sum_k 2^{k+1}\int_{E_k}|\Psi(t\sqrt{L})f(y)|^2{dydt\over t}\\
&&= \sum_k 2^{k+1}\int_{E_k}|\Psi(t\sqrt{L})(f\chi_{A_k})(y)|^2{dydt\over t}\\
&&\leq C \sum_k 2^{k+1}\int_{\RN}| f|^2\chi_{A_k}{dy }\\
&&\leq C \int_{\RN}| f|^2 {MwM_w(h) }{dy }.\end{aligned}$$
Applying the Hölder inequality with exponents $p/2$ and $(p/2)'$, we obtain the bound
$$C \bigg(\int_{\RN}| f|^p (Mw)^{p/2}w^{-(p/2-1)}{dy }\Big)^{2/p}
\Big(\int_{\RN}M_w(h)^{(p/2)'}w{dy }\bigg)^{(p-2)/p}.$$ However, since $M_w$ is the centered maximal function, we have
$$\int_{\RN}M_w(h)^{(p/2)'}wdx\leq C_{n,p}\int_{\RN}h^{(p/2)'}wdx,$$ by a standard argument based on the Besicovitch covering lemma. Since $h$ is arbitrary, we obtain our result.
Proof of Theorem \[th1.4\]
===========================
In order to prove Theorem \[th1.4\], we first prove the following result.
\[le5.1\] Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$, $S_{H}$ and $\gL$ with $\mu>3$. Under assumptions of Theorems \[th1.1\], \[th1.2\] and \[th4.1\], for $w\in A_p,\
1<p<\infty$, we have
$$\begin{aligned}
\label{e5.1}\|Tf\|_{L^p_w(\RN)}\leq C \|f\|_{L^p_w(\RN)}\end{aligned}$$
where constant $C$ depends only on $p$, $n$ and $w$.
Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$, $S_{H}$ and $\gL$ with $\mu>3$. Note that if $w\in A_1$, then $Mw\leq Cw$ a.e. By Theorems \[th1.1\], \[th1.2\] and \[th4.1\], $T$ is bounded on $L^p_w(\RN), 1<p<\infty,$ for any $w\in A_1,$ i.e.,
$$\|Tf\|_{L^p_w(\RN)}\leq C \|f\|_{L^p_w(\RN)}.$$
By extrapolation theorem, these operators are all bounded on $L^p_w(\RN), 1<p<\infty,$ for any $w\in A_p,$ and estimate (\[e5.1\]) holds. For the detail, we refer the reader to pages 141-142, Theorem 7.8, [@D].
Going further, we introduce some definitions. Given a weight $w$, set $w(E)=\int_E w(x)dx$. The non-increasing rearrangement of a measurable function $f$ with respect to a weight $w$ is defined by (cf. [@CR])
$$\begin{aligned}
f^\ast_w(t)=\sup_{w(E)=t} \inf_{x\in E}|f(x)|\ \ \ \ (0<t<w(\RN)).\end{aligned}$$
If $w\equiv1$, we use the notation $f^\ast(t)$.
Given a measurable function $f$, the local sharp maximal function $M^\sharp_{\lambda}f$ is defined by
$$\begin{aligned}
M^\sharp_{\lambda}f(x)=\sup_{Q\ni x}\inf_{c}\big((f-c)\chi_Q\big)^\ast(\lambda|Q|)\ \ \ (0<\lambda<1).\end{aligned}$$
This function was introduced by Strömberg [@S], and motivated by an alternate characterization of the space $BMO$ given by John [@J].
\[le5.4\] For any $w\in A_p$ and for any locally integrable function $f$ with $f^\ast_w(+\infty)=0$ we have
$$\begin{aligned}
\label{e5.2}
\|Mf\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{\gamma_{p,q}}\cdot\|M^\sharp_{\lambda_n}
(|f|^q)\|^{1/q}_{L^{p/q}_w(\RN )}\ \ \ (1<p<\infty,1\leq q<\infty),\end{aligned}$$
where $\gamma_{p,q}=\max\{1/q,1/(p-1)\}$, $C$ depends only on $p,q$ and on the underlying dimension $n$, and $\lambda_n$ depends only on $n$.
For the proof of this lemma, see Theorem 3.1 in [@L1].
\[pro5.5\] Let $\gL$ be a function with $\mu>3$ in (\[e3.1\]). Then for any $f\in C^\infty_0(\RN)$ and for all $x\in \RN,$
$$M^\sharp_{\lambda}\big(\gL(f)^2\big)(x)\leq CMf(x)^2,$$ where $C$ depends on $\lambda,\mu,\Psi$ and $n$.
Given a cube $Q$, let $T(Q)=\{(y,t):\ y\in Q,0<t<l(Q)\}$, where $\ell(Q)$ denotes the side length of $Q$. For $(y,t)\in T(Q)$, using (\[e2.5\]) of Lemma \[le2.1\] we have
$$\begin{aligned}
\label{e5.3}
\Psi(t\sqrt{L})f(y)=\Psi(t\sqrt{L})(f\chi_{3Q})(y).\end{aligned}$$
Now, fix a cube $Q$ containing $x$. For any $z\in Q$ we decompose $\gL(f)^2$ into the sum of
$$I_1(z)=\iint_{T(2Q)}|\Psi(t\sqrt{L})f(y)|^2\Big({t\over t+|z-y|}\Big)^{n\mu}{dydt\over t^{n+1}}$$ and
$$I_2(z)=\iint_{\RR^{n+1}_{+} \setminus T(2Q)}|\Psi(t\sqrt{L})f(y)|^2\Big({t\over t+|z-y|}\Big)^{n\mu}{dydt\over t^{n+1}}.$$
From Theorem \[th4.1\], we know that for $\mu>3$, $\gL(f)$ is of weak type $(1, 1)$. Then using (\[e5.3\]), we have
$$\begin{aligned}
\label{e5.4}
(I_1)^\ast(\lambda|Q|)&\leq& \Big(\gL(f\chi_{6Q})\Big)^{\ast}(\lambda|Q|)^2\\
&\leq&\Big({C\over \lambda|Q|}\int_{6Q}|f|\Big)^2\leq CMf(x)^2.\nonumber\end{aligned}$$
Further, for any $z_0 \in Q$ and $(y, t)\notin T (2Q)$, by the Mean Value Theorem,
$$(t+|z-y|)^{-n\mu}-(t+|z_0-y|)^{-n\mu}\leq C\ell(Q)(t+|z-y|)^{-n\mu-1}.$$ From this and (\[e5.3\]), using Lemma \[le2.1\] again and $\mu>3$, we have
$$\begin{aligned}
&&\hspace{-1.2cm}|I_2(z)-I_2(z_0)|\\
&&\leq C\ell(Q)\iint_{\RR^{n+1}_{+}\setminus
T(2Q)}t^{n\mu}|\Psi(t\sqrt{L})f(y)|^2\Big({1\over
t+|z-y|}\Big)^{n\mu+1}{dydt\over t^{n+1}}\\
&&\leq C\sum^\infty_{k=1}{1\over 2^k}{1\over (2^k\ell(Q))^{n\mu}}
\iint_{T(2^{k+1}Q)\setminus
T(2^{k}Q)}t^{n\mu}|\Psi(t\sqrt{L})f(y)|^2{dydt\over t^{n+1}}\\
&&\leq C\sum^\infty_{k=1}{1\over 2^k}{|2^{k+1}Q|\over
(2^k\ell(Q))^{n\mu}}\Big(\int^{2^{k+1}\ell(Q)}_0t^{n\mu-3n-1}dt\Big)
\Big(\int_{6\cdot2^kQ}|f|\Big)^2\\
&&\leq C\sum^\infty_{k=1}{1\over 2^k}\Big({1\over
|2^{k+1}Q|}\int_{6\cdot2^kQ}|f|\Big)^2\leq CMf(x)^2.\end{aligned}$$
Combining this estimate with (\[e5.4\]) yields
$$\begin{aligned}
\inf_{c}\Big((\gL(f)^2-c)\chi_Q\Big)^\ast(\lambda|Q|)&\leq&
\big((I_1+I_2-I_2(z_0))\chi_Q\big)^\ast(\lambda|Q|)\\
&\leq&(I_1)^\ast(\lambda|Q|)+CMf(x)^2\\
&\leq& CMf(x)^2,
\end{aligned}$$
which proves the desired result.
Then we have the following result.
\[th5.4\] Let $T$ be of the area functions $s_h$, $s_p$, $S_{P}$,$S_{H}$ and $\gL$ with $ \mu>3.$ Under assumptions of Theorems \[th1.1\], \[th1.2\] and \[th4.1\], for $w\in A_p,\
1<p<\infty$, if $\|f\|_{L^p_w(\RN )}<\infty$, then
$$\begin{aligned}
\label{e5.5}
\Big(\int_{\RN}\big(M(Tf)\big)^pwdx\Big)^{1/p}\leq C\|w\|_{A_p}^{\beta_p}\Big(\int_{\RN}
\big(M( f)\big)^pwdx\Big)^{1/p},\end{aligned}$$
where $\beta_p=\max\{1/2,1/(p-1)\}$, and a constant $C$ depends only on $p$ and $n$.
Suppose $T=\gL.$ From Lemma \[le5.1\], we know that $\gL$ is bounded on $L^p_w(\RN ) $ when $w\in A_p$. Therefore, assuming that $\|f\|_{L^p_w(\RN)}$ is finite, we clearly obtain that $(\gL)^\ast_w(+\infty)=0 $. Letting $\gL(f)$ instead of $f$ in (\[e5.2\]) with $q=2$ and applying Proposition \[pro5.5\], we get
$$\Big(\int_{\RN}\big(M(\gL(f))\big)^pwdx\Big)^{1/p}\leq C\|w\|_{A_p}^{\beta_p}\Big(\int_{\RN}
\big(M( f)\big)^pwdx\Big)^{1/p}.$$
Under assumptions of Theorems \[th1.1\] and \[th1.2\], it follows that the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}$ are all controlled by $\gL$ pointwise. So we have the estimate (\[e5.5\]) for $s_h$, $s_p$, $S_{P}$ and $S_{H}$. Then the proof of this theorem is complete.
In [@B], Buckley proved that for the Hardy-Littlewood maximal operator,
$$\begin{aligned}
\label{m}
\|M\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{1/(p-1)}\ \ (1<p<\infty),\end{aligned}$$
and this result is sharp.
From (\[m\]) and Theorem \[th5.4\], there exists a constant $C=C(T, n, p)$ such that for all $w\in A_p$,
$$\begin{aligned}
\label{5.7}
\|T\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{ {1\over p-1}+ \, \max\big\{{1\over 2},\, {1\over p-1}\big\}} \ \ \ \ \ \ (1<p<\infty),\end{aligned}$$
where $T$ is of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H}.$ This proves Theorem \[th1.4\].
[**Remarks.**]{}
\(i) Note that when $L=-\Delta$ is the Laplacian on $\RN$, it is well known that the exponents $\beta_p$ of (\[e5.5\]) in Theorem \[th5.4\] is best possible, in general (see, e.g., Theorem 1.5, [@L1]).
\(ii) For the classical area function $S_{\varphi}$ in (\[e1.1\]), the result of Theorem \[th1.4\] was recently improved by A. Lerner in [@L2], i.e., there exists a constant $C=C(S_{\varphi}, n, p)$ such that for all $w\in A_p, 1<p<\infty$,
$$\begin{aligned}
\label{e5.8}
\|S_{\varphi}\|_{L^p_w(\RN)}\leq C\|w\|_{A_p}^{ \, \max\big\{{1\over 2},\, {1\over p-1}\big\}},\end{aligned}$$
and the estimate (\[e5.8\]) is the best possible for all $1<p<\infty.$ However, we do not know whether one can deduce the same bounds (\[e5.8\]) for the $L^p_w$ operator norms of the area functions $s_h$, $s_p$, $S_{P}$ and $S_{H},$ and they are of interest in their own right.
Note that sharp weighted optimal bounds for singular integrals has been studied extensively, see for examples, [@CMP; @HLRSUV; @LOP1; @LOP2; @P] and the references therein.
\(iii) Finally, for $f\in {\mathcal S}(\RN)$, we define the (so called vertical) Littlewood-Paley-Stein functions ${\mathcal G}_P $ and $ {\mathcal G}_H$ by
$$\begin{aligned}
{\mathcal G}_P(f)(x)&=&\bigg(\int_0^{\infty}
|t\nabla_x e^{-t\sqrt{L}} f(x)|^2{ dt\over t }\bigg)^{1/2},\\
{\mathcal G}_H(f)(x)&=&\bigg(\int_0^{\infty}
|t\nabla_x e^{-t^2L} f(x)|^2 { dt\over t }\bigg)^{1/2}, \end{aligned}$$
as well as the (so-called horizontal) Littlewood-Paley-Stein functions $ g_p$ and $g_h$ by
$$\begin{aligned}
g_p(f)(x)&=&\bigg(\int_0^{\infty}
|t\sqrt{L} e^{-t\sqrt{L}} f(x)|^2 { dt\over t }\bigg)^{1/2},\\
g_h(f)(x)&=&\bigg(\int_0^{\infty}
|t^2L e^{-t^2L} f(x)|^2 { dt\over t }\bigg)^{1/2}. \end{aligned}$$
One then has the analogous statement as in Theorems 1.1, 1.2, 1.3 and 1.4 replacing $s_p, s_h, S_P, S_H $ by $g_p, g_h, {\mathcal G}_P,
{\mathcal G}_H$, respectively.
[**Acknowledgment**]{}: The research of Lixin Yan is supported by NNSF of China (Grant No. 10771221) and National Science Foundation for Distinguished Young Scholars of China (Grant No. 10925106).
[99999]{}
0.12cm P.Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\RR$ and related estimates. [*Memoirs of the Amer. Math. Soc.*]{} 186, no. 871 (2007).
0.12cm
P. Auscher, T. Coulhon, X.T. Duong and S. Hofmann, Riesz transform on manifolds and heat kernel regularity. [*Ann. Sci. École Norm. Sup.*]{}, [**37**]{}, (2004), 911-957.
0.12cm P. Auscher, X.T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces. Unpublished preprint (2005).
0.12cm
P. Auscher, J.M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights, [*Adv. in Math.*]{}, [**212**]{}(2007), 225-276.
0.12cm
S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities. [*Trans. Amer. Math. Soc.,*]{} [**340**]{} (1993), 253-272.
0.12cm A.P. Calderón and A. Torchinsky, Parabolic maximal function associated with a distribution. [*Adv. Math.*]{}, [**16**]{} (1975), 1-64.
0.12cm
S.-Y.A. Chang, J.M. Wilson and T. Wolff, Some weighted norm inequalities concerning the Schrödinger operator. [*Comm. Math. Helv.,*]{} [**60**]{} (1985), 217-246.
0.12cm
S. Chanillo and R. L. Wheeden, Some weighted norm inequalities for the area integral. [ *Indiana Univ. Math. J.*]{}, [**36**]{} (1987), 277-294.
0.12cm
J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplacian and the geometry of complete Riemannian manifolds. [*J. Differential Geom.*]{}, [**17**]{} (1982), 15-53.
0.12cm
K.M. Chong, N.M. Rice, Equimeasurable rearrangements of functions. Queen¡¯s Papers in Pure and Applied Mathematics, vol. 28, Queen¡¯s University, Kingston, Ont., 1971.
0.12cm T. Coulhon and X.T. Duong, Riesz transforms for $1\leq p\leq 2$. [*Trans. Amer. Math. Soc.,*]{} [**351**]{} (1999), 1151-1169.
0.12cm T. Coulhon, X.T. Duong and X.D. Li, Littlewood-Paley-Stein functions on complete Riemannian manifolds for $1\leq p\leq 2.$ [*Studia Math.,*]{} [**154**]{} (2003), 37-57.
0.12cm
T. Coulhon and A. Sikora, Gaussian heat kernel upper bounds via Phragmén-Lindelöf theorem. [*Proc. Lond. Math.,*]{} [**96**]{} (2008), 507-544.
0.12cm
D. Cruz-Uribe, J.M. Martell and C. Perez, Sharp weighted estimates for classical operators (2010), available at http://arxiv.org/abs/1001-4254.
0.12cm
E.B. Davies, [*Heat kernels and spectral theory*]{}, Cambridge Univ. Press, 1989.
0.12cm J. Duoandikoetxea, [*Fourier analysis.*]{} Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29. American Mathematical Society, Providence, RI, 2001.
0.12cm X.T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains. [*Rev. Mat. Iberoamericana*]{}, [**15**]{} (1999), 233-265.
0.12cm
X.T. Duong, E.M. Ouhabaz and A. Sikora, Plancherel-type estimates and sharp spectral multipliers. [*J. Funct. Anal.*]{}, [**196**]{} (2002), 443–485.
0.12cm
R. Fefferman and J. Pipher, Multiparameter operators and sharp weighted inequalities. [ *Amer. J. Math.*]{}, [**119**]{} (1997), 337-369.
0.12cm
C. Fefferman and E.M. Stein, Some maximal inequalities. [ *Amer. J. Math.*]{}, [**92**]{} (1971), 107–115.
0.12cm
J. Garcá-Cuerva and J.L. Rubio de Francia, [*Weighted Norm Inequalities and Related Topics*]{}, North Holland math., Studies [**116**]{}, North Holland, Amsterdam, 1985.
0.12cm
L. Grafakos, [*Modern Fourier Analysis*]{}, Springer-Verlag, Graduate Texts in Mathematics [**250**]{}, Second Edition, 2008.
0.12cm
S. Hofmann, G.Z. Lu, D. Mitrea, M. Mitrea and L.X. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. To appear in [*Memoris of Amer. Math. Soc.*]{}, 2010.
0.12cm
T. Hytönen, M. Lacey, M. Reguera, E. Sawyer, I. Uriarte-Tuero and A. Vagharshakyan, Weak and strong type $A_p$ estimates for Calderón-Zygmund operators (2010), available at http://arxiv.org/abs/1006.2530.
0.12cm
F. John, Quasi-isometric mappings. Seminari 1962¨C1963 di Analisi, Algebra, Geometria e Topologia, Rome, 1965.
0.12cm
A.K. Lerner, On some sharp weighted norm inequalities. [ *J. Funct. Anal.*]{}, [**32**]{} (2006), 477-494.
0.12cm
A.K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals (2010), available at http://arXiv.org/abs/1005.1422.
0.12cm
A.K. Lerner, S. Ombrosi and C. Perez, $A_1$ bounds for Calderón-Zygmund operators related a problem of Muckenhoupt and Wheeden, [*Math. Res. Lett.*]{}, [**16**]{} (2009), 149-156.
0.12cm
0.12cm
A.K. Lerner, S. Ombrosi and C. Perez, Sharp $A_1$ bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, [*Int. Math. Res. Not. IMRN*]{}, [**6**]{} (2008), Art. ID rnm 161, 11.
0.12cm
E.M. Ouhabaz, Analysis of heat equations on domains. [*London Math. Soc. Monographs*]{}, Vol. [**31**]{}, Princeton Univ. Press (2004).
0.12cm
C. Perez, A course on singular integrals and weights. Preprint (2010).
0.12cm
J.-O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. [ *Indiana Univ. Math. J.*]{}, [**28**]{} (1979), 511-544.
0.12cm
A. Sikora, Riesz transform, Gaussian bounds and the method of wave equation. [*Math. Z.*]{}, [**247**]{} (2004), 643-662.
0.12cm
A. Sikora and J.Wright, Imaginary powers of Laplace operators. [*Proc. Amer. Math. Soc.*]{}, [**129**]{} (2001), 1745-1754.
0.12cm
E.M. Stein, [*Singular integral and differentiability properties of functions*]{}. Princeton Univ. Press, [**30**]{}, (1970).
0.12cm
E.M. Stein, [*Topics in harmonic analysis related to the Littlewood-Paley theory*]{}. Princeton Univ. Press, (1970).
0.12cm
J.M. Wilson, Weighted norm inequalities for the continuous square functions. [ *Trans. Amer. Math. Soc.*]{}, [**314**]{} (1989), 661-692.
0.12cm
L.X. Yan, Littlewood-Paley functions associated to second order operators. [ *Math. Z.*]{}, [**246**]{} (2004), 655-666.
0.12cm K. Yosida, [*Functional Analysis*]{} (Fifth edition). Spring-Verlag, Berlin, 1978.
0.05cm
|
---
abstract: 'We study the pulsar timing, focusing on the time delay induced by the gravitational field of the binary systems. In particular, we study the gravito-magnetic correction to the Shapiro time delay in terms of Keplerian and post-Keplerian parameters, and we introduce a new post-Keplerian parameter which is related to the intrinsic angular momentum of the stars. Furthermore, we evaluate the magnitude of these effects for the binary pulsar systems known so far. The expected magnitude is indeed small, but the effect is important per se.'
author:
- 'Matteo Luca Ruggiero$^{1,2 }$ and Angelo Tartaglia$^{1,2 }$'
title: 'A post-Keplerian parameter to test gravito-magnetic effects in binary pulsar systems'
---
Introduction {#sec:intro}
============
The first binary radio pulsar (PSR B 1913+16) was discovered some 30 years ago by Taylor and Hulse[@hulse75]. Since those years, a great amount of work has been done, and pulsars in binary systems have proved to be celestial laboratories for testing the relativistic theories of gravity (see [@stairs03] and references therein). Indeed, up to the present day, Einstein’s theory of gravity, General Relativity (GR), has passed all observational tests with excellent results. However, even if the aim of the experimental relativists is always to achieve a greater precision, we must not forget that most of the tests of GR come from solar system experiments, where the gravitational field is in the “weak” regime. On the other hand, it is expected that deviations from GR can occur for the first time in the “strong” field regime: hence, the solar system experiments are inadequate to this end. On the contrary, the strong gravitational field is best tested by means of pulsars. Pulsars, which are highly magnetized rotating neutron stars, are important both for testing relativistic theories of gravity and for studying the interstellar medium, stars, binary systems and their evolution, plasma physics in extreme conditions. As for the tests of gravity, the recent discovery of the first double pulsar PSR J0737-3039 [burgay03]{},[@lyne04] provided an astonishing quantity of data, which make this system a rare relativistic laboratory [@kramer05].
As we pointed out elsewhere [@tartaglia05], this system in particular, and binary pulsars systems in general, could be useful for testing the so called gravito-magnetic effects. These effects are originated by the rotation of the sources of the gravitational field, which gives rise to the presence of off-diagonal $g_{0i}$ terms in the metric tensor. The gravitational coupling with the angular momentum of the source is indeed much weaker than the coupling with mass alone, the so called gravito-electric interaction. In fact, the ratio between the former and the latter can be estimated to be of the order of $$\varepsilon =\frac{g_{0\phi }}{g_{00}} \simeq \frac{J R_{S}}{r^{2} Mc}\;,
\label{rapporto}$$ where $R_{S}=2GM/c^2$ is the Schwarzschild radius of the source, $M$ being its mass, and $J$ (the absolute value of) its angular momentum. At the surface of the Sun, which is the most favorable place in the solar system, evaluation of eq. (\[rapporto\]) gives $\varepsilon \sim 10^{-12}$, thus evidencing the weakness of the gravito-magnetic versus the gravito-electric interaction. The smallness of $\varepsilon $ is the reason why, though having been suggested from the very beginning of the relativistic age [lense]{}, the experimental verification of the existence of gravito-magnetic effects has been very difficult until today (see [@ruggiero02] and references therein). The relevance of pulsar systems for the detection of the gravito-magnetic effects lays in the fact that the ratio (\[rapporto\]) can be less unfavorable whenever $r$ is approaching the Schwarzschild radius of the source: this can be the case of a source of electromagnetic (e.m.) signals orbiting around a compact, collapsed object.
As for the rotation effects in pulsars binary systems, in general, the coupling of the intrinsic angular momentum of the stars with the orbital angular momentum, and the coupling of the intrinsic angular momenta of the two stars themselves were studied, together with the corresponding precession effects [@barker75],[@barker76],[@oconnell04],[weisberg02]{},[@stairs04]. The gravitomagnetic effects on the propagation of light in a binary system were studied by Kopeikin and Mashhoon (see [@kopeikin02] and references therein).
In a previous paper [@tartaglia05] we studied the effects of the gravitational field on e.m. pulses propagating in a binary system, and we emphasized the gravito-magnetic contribution. In doing that, we used a simplified model which considered circular orbits only. Here we generalize those results, by taking into account orbits with arbitrary eccentricity, which is a more realistic approach on the basis of the knowledge of binary systems discovered until today. In particular, we study the gravito-magnetic correction to the Shapiro contribution to the time of flight of the signals, focusing on its effect on the arrival times perceived by the experimenter on the Earth, and we introduce a new Post-Keplerian parameter which is related to the intrinsic angular momentum of the stars. Finally, we evaluate the magnitude of these effects for the binary pulsar systems known so far.
The Context: Pulsars Timing {#sec:timing}
===========================
When studying pulsars, what is measured are the pulse arrival times at the (radio) telescope over a suitably long period of time. In fact, even though individual pulses are generally weak and have an irregular profile, a regular mean profile is obtained by averaging the received pulses over a long time. Pulses traveling to the Earth are delayed because of the dispersion caused by the interstellar medium. Besides this delay, other gravitational factors influence the arrival times of the signals emitted by a pulsar in a binary system: the strong field in the vicinity of the pulsar, the relatively weak field between the two compact objects forming the binary system and, finally, the weak field of the Solar System. Consequently, the arrival time $T_{N}$ of the $N$th pulse, as measured on the Earth, depends on a set of parameters $\alpha _{1},\alpha _{2}...,\alpha _{K}$, which include a description of the orbit of the binary system: $$T_{N}=F(N,\alpha _{1},\alpha _{2},...,\alpha _{K}) \label{eq:toa1}$$In particular, the set $\alpha _{1},\alpha _{2}...,\alpha _{K}$ includes the Keplerian parameters, together with the so called “post-Keplerian” (PK) parameters which describe the relativistic corrections to the Keplerian orbit of the system. Since seven parameters are needed to completely describe the dynamics of the binary system (see [@taylor89] and references therein), the measurement of any two PK parameters, besides the five Keplerian ones, allows to predict the remaining PK parameters. For instance, if the two masses are the only free parameters, the measurement of three or more PK parameter[s]{} over-constrains the system and introduces theory-dependent lines in a mass-mass diagram that should intersect, in principle, in a single point [@damour91]. This is of course true as far as the intrinsic angular momenta are not taken into account.
It is possible to obtain a relation which links the time of arrival of a pulse on the Earth to its time of emission. More in detail, the following timing formula holds, which relates the reception (topocentric) time $%
T_{Earth}$ on the Earth with the emission time $T_{pulsar}$ in the comoving pulsar frame [@straumann04]: $$\begin{aligned}
T_{pulsar}=&T_{Earth}-t_0-\frac{D}{f^2}+\Delta_{R_\odot}+\Delta_{E_\odot}-%
\Delta_{S_\odot} \notag \\
&- \left(\Delta_R+\Delta_E+\Delta_S+\Delta_A \right) \label{eq:timing1}\end{aligned}$$ where $t_0$ is a reference epoch, $D/f^2$ is the dispersive delay (as a function of the frequency of the pulses, $f$), $\Delta_R,\Delta_E,\Delta_S$ are, respectively the Roemer delay, the Einstein delay and the Shapiro delay due to the gravitational field of the binary system (while $%
\Delta_{R_\odot},\Delta_{E_\odot},\Delta_{S_\odot}$ are the corresponding terms due to the Solar System field) and $\Delta_A$ is the delay due to aberration.
Here we are concerned with the gravito-magnetic corrections due to the intrinsic angular momentum of the stars to the Shapiro delay $\Delta_{S}$, which we analyze in the following section.
The Shapiro Time Delay {#sec:shapiro}
======================
![[Notation used for describing the pulsar orbit.]{}[]{data-label="fig:fig1"}](fig1.eps){width="9cm" height="9cm"}
We want to calculate the relation between the coordinate emission time $t_e$ and the coordinate arrival time $t_a$ (as measured at the Solar System center of mass).
To this end, we consider a reference frame at rest in the center of mass of the binary system. In this reference frame, the vector pointing to the pulsar emitting e.m. signals is $\vec{\bm{x}}_1$, while the one pointing to its companion star is $\vec{\bm{x}}_2$; in the following, the suffix “1” will always refer to the pulsar, and “2” to its companion. Furthermore, $\vec{\bm{x}}_b$, is the position of the center of mass of the solar system.
We use the notation of Figure \[fig:fig1\] for the description of the pulsar orbit. We choose a first set of Cartesian coordinates $\{x,y,z\}$, with origin in the center of mass of the binary system, and such that the line of sight is parallel to the $z$ axis. Then, we introduce another set of Cartesian coordinates $\{X,Y,Z\}$, with the same origin: the $X$ axis is directed along the ascending node, the $Z$ axis is perpendicular to the orbital plane. The angle between the $x$ and $X$ axes is $\Omega$, the longitude of the ascending node, while the angle between the $z$ and $Z$ axes is $i$, the inclination of the orbital plane. Let $\vec{%
\bm{x}}_1=r_1 \hat{\bm{x}}_1$ be the orbit of the pulsar: it is described by $$\hat{\bm{x}}_1 = \cos(\omega+\varphi) \hat{\bm{X}}+\sin(\omega+\varphi) \hat{%
\bm{Y}}, \label{eq:orbita1}$$ in terms of the argument of the periastron, $\omega$, and the true anomaly, $%
\varphi$. Let us pose $$\theta \doteq \omega+\varphi, \label{eq:deftheta}$$ for the sake of simplicity. Then, we use the notation $$\vec{\bm{r}} \doteq \vec{\bm{x}}_1-\vec{\bm{x}}_2 \label{eq:defr1}$$ to describe the position of the pulsar with respect to its companion, and we remember that we have, for the Keplerian problem $$r = \frac{a(1-e^2)}{1+e\cos\varphi}, \label{eq:kep1}$$ where $a$ is the semi-major axis of the relative motion and $e$ is the eccentricity. The astronomical elements $\Omega$, $i$, $\omega$, $a$, $e$ represent the Keplerian parameters. In what follows, we will use also the definitions $r_b=|\vec{\bm{x}}_b|$, $r=|\vec{\bm{r}}|$ and $\hat{\bm{n}}
\doteq \frac{\vec{\bm{x}}_b}{r_b}$.
That being said, let us focus on the physical situation we are dealing with. If the gravito-magnetic effects are neglected, the metric describing the gravitational field of the binary system is given by (see, for instance, [@will]) $$ds^2= \left(1+2\phi \right)dt^2-\left(1-2\phi \right)|d\vec{\bm{x}}|^2,
\label{eq:ds1}$$ where $$\phi (x,y,z) = -\frac{M_1}{|\vec{\bm{x}}-\vec{\bm{x}}_1|}-\frac{M_2}{|\vec{%
\bm{x}}-\vec{\bm{x}}_2|}. \label{eq:ds2}$$ We see that the total gravitational potential is the sum of the two contributions, due to the pulsar, whose mass is $M_1$, and to its companion, whose mass is $M_2$[^1]. Starting from ([eq:ds1]{}), the mass contribution to the time delay can be evaluated following the standard approach, described, for instance, in [straumann04]{}. However, we want to generalize this approach in order to take into account the effects of the rotation of the sources of the gravitational field, i.e. the gravito-magnetic effects.
To this end, we may guess that the metric, in the coordinates $%
\{X,Y,Z\}$, assumes the form $$\begin{aligned}
ds^2= &\left(1+2\phi \right)dt^2-\left(1-2\phi \right)\left(dX^2+dY^2+dZ^2
\right) \notag \\
&+4 \vec{\bm{A}} \cdot d\vec{\bm{X}}dt, \label{eq:ds3}\end{aligned}$$ where $\vec{\bm{A}}$ is the (total) gravito-magnetic vector potential of the system. Eq. (\[eq:ds3\]) generalizes the weak field metric around a rotating source (see [@mashh1],[@mashhoon03]). We suppose that the dominant contribution to the total gravito-magnetic potential is due to the intrinsic angular momenta $\vec{\bm{J}}_1,\vec{\bm{J}}_2$ of the stars; we suppose also that $\vec{\bm{J}}_1,\vec{\bm{J}}_2$ are aligned with the total orbital angular momentum $\vec{\bm{L}}$ , i.e. perpendicular to the orbital plane. Furthermore, since we assume that the signals emitted by pulsar 1 propagate along a straight line, and because the gravito-magnetic coupling depends on the impact parameter, we may conclude that the only relevant gravito-magnetic contribution comes from the companion star 2. Consequently, the metric (\[eq:ds3\]) becomes $$\begin{aligned}
ds^2 = & \left(1+2\phi \right)dt^2-\left(1-2\phi \right)\left(dX^2+dY^2+dZ^2
\right) \notag \\
& +4J_2\frac{\left(\vec{\bm{x}}-\vec{\bm{x}}_2 \right)\cdot \hat{\bm{X}}}{|%
\vec{\bm{x}}-\vec{\bm{x}}_2|^3} dY dt-4J_2\frac{\left(\vec{\bm{x}}-\vec{%
\bm{x}}_2 \right)\cdot \hat{\bm{Y}}}{|\vec{\bm{x}}-\vec{\bm{x}}_2|^3} dX dt.
\label{eq:ds4}\end{aligned}$$
In (\[eq:ds4\]) $|\vec{\bm{x}}|$ varies along the straight line defined by $$\vec{\bm{x}}(t)=\vec{\bm{x}}_1(t_e)+\frac{t-t_e}{t_a-t_e}\left(\vec{\bm{x}}%
_b(t_a)-\vec{\bm{x}}_1(t_e) \right). \label{eq:time21}$$ If we set $$\alpha \doteq \frac{t-t_e}{t_a-t_e}, \label{eq:defalfa1}$$ then $\alpha=0$ when $t=t_e$, and $\alpha=1$ when $t=t_a$. Consequently, we may write $$\vec{\bm{x}}(t)=\vec{\bm{x}}_1(t_e)+\alpha\left(\vec{\bm{x}}_b(t_a)-\vec{%
\bm{x}}_1(t_e) \right). \label{eq:time2}$$ After some straightforward manipulations, the line element (\[eq:ds4\]) can be written in the form $$ds^2=g_{tt}dt^2+g_{\alpha\alpha}d\alpha^2+2g_{t\alpha}dtd\alpha,
\label{eq:ds41}$$ where $$\begin{aligned}
g_{tt} = & 1-\frac{2M_1}{|\vec{\bm{x}}-\vec{\bm{x}}_1|}-\frac{2M_2}{|\vec{%
\bm{x}}-\vec{\bm{x}}_2|}, \notag \\
g_{\alpha\alpha} = & -\left(1+\frac{2M_1}{|\vec{\bm{x}}-\vec{\bm{x}}_1|}+%
\frac{2M_2}{|\vec{\bm{x}}-\vec{\bm{x}}_2|} \right)\left(r_b^2+r_1^2+2r_1 r_b
\sin i \sin \theta \right), \label{eq:ds42} \\
g_{t\alpha} = & -2J_2\frac{r r_b \sin i \cos \theta}{|\vec{\bm{x}}-\vec{%
\bm{x}}_2|^3}. \notag\end{aligned}$$
By setting $ds^2=0$, we easily see that the propagation time is made of three contributions, up to first order in the masses of the stars, and in the spin angular momentum: $$\Delta t= \Delta t_0 + \Delta t_M + \Delta t_J, \label{eq:deltatitot}$$ where $$\Delta t_0 = \int_0^1 \sqrt{r_b^2+r_1^2+2r_br_1\sin i \sin \theta}d\alpha,
\label{eq:tzero1}$$ $$\Delta t_M = \int_0^1 \left(\frac{2M_2}{|\vec{\bm{x}}-\vec{\bm{x}}_2|}
\right)\sqrt{r_b^2+r_1^2+2r_1 r_b \sin i \sin \theta}d\alpha, \label{eq:tM1}$$ $$\Delta t_J = \int_0^1 2J_2\frac{r r_b \sin i \cos \theta}{|\vec{\bm{x}}-\vec{%
\bm{x}}_2|^3} d\alpha. \label{eq:tJ1}$$\
Let us comment on (\[eq:tzero1\])-(\[eq:tJ1\]): the first contribution is a purely geometric one, and is due to the propagation, in flat space, of e.m. signals from the pulsar toward the Solar System. The second contribution represents the time depending part of the total Shapiro delay. As such it is entirely due to the mass of the companion star; in fact it is easy to verify that the term containing the mass of the pulsar emitting e.m. signals gives a contribution which remains constant during the orbital motion. Finally, the third contribution is due to the gravito-magnetic field of the companion star. To lowest order, we obtain $$\Delta t_0 \simeq r_b+r_1\sin i \sin \theta \label{eq:tzero2}$$ $$\Delta t_M \simeq 2M_2 \ln \left[\frac{2r_b}{\hat{\bm{n}}\cdot \vec{\bm{r}}+r%
} \right] \label{eq:tM2}$$ $$\Delta t_J \simeq \frac{2J_2 \sin i \cos \theta}{r}\frac{1-\hat{\bm{n}}\cdot
\hat{\bm{r}}}{\sin^2 i \sin^2 \theta -1} \label{eq:tJ2}$$ The contributions (\[eq:tzero2\]) and (\[eq:tM2\]) to the time delay are in agreement with the standard results (see [@straumann04]), while equation (\[eq:tJ2\]) gives the gravito-magnetic corrections.
From the time delay (\[eq:tM2\]) we may extract the contribution which varies during the orbital motion; if we use the equation of the orbit (\[eq:orbita1\]) it can be written in the form $$\Delta t_M^*=-2M_2\ln \left[\frac{1-\sin i \sin(\omega+\varphi)}{%
1+e\cos\varphi} \right]. \label{eq:dsm7}$$
If we introduce the PK Shapiro parameters $$\begin{aligned}
\mathcal{R} & \doteq & M_2, \label{eq:shapr} \\
\mathcal{S} & \doteq & \sin i, \label{eq:shaps}\end{aligned}$$ the mass (or, gravito-electric) contribution to the time delay becomes $$\Delta t_M^*=-2\mathcal{R}\ln \left[\frac{1-\mathcal{S} \sin(\omega+\varphi)%
}{1+e\cos\varphi} \right]. \label{eq:dsm8}$$
On the other hand, the gravito-magnetic contribution ([eq:tJ2]{}) changes continuously during the orbital motion, so that, similarly, we may write $$\Delta t_J^* = -\frac{J_2 \sin i }{a (1-e^2)}\left[\frac{\left(\cos(\omega+%
\varphi)\right)\left(1+e\cos \varphi\right)}{1-\sin i \sin (\omega+\varphi)} %
\right]. \label{eq:dsj3}$$ If we introduce the PK parameter $\mathcal{S}$, and define a new PK parameter $$\mathcal{J} \doteq J_2 \label{eq:j2p}$$ the gravito-magnetic contribution to the Shapiro time delay can be written in the form $$\Delta t_J^* = -\frac{\mathcal{J} \mathcal{S} }{a (1-e^2)}\left[\frac{%
\left(\cos(\omega+\varphi)\right)\left(1+e\cos \varphi\right)}{1-\mathcal{S}
\sin (\omega+\varphi)} \right]. \label{eq:dsj4}$$ We notice that the new PK parameter $\mathcal{J}$ coincides with the intrinsic angular momentum of the source of the gravito-magnetic field. In particular, if we knew the rotation frequency of the source of the gravito-magnetic field (which is possible, if the latter is a visible pulsar, for instance), the new PK parameter could give information on its moment of inertia.
The gravito-electric and gravito-magnetic contributions to the time delay (\[eq:dsm8\]) and (\[eq:dsj4\]) can be written in the form $$\begin{aligned}
\Delta t_{M}^{\ast } &=&A_{M}F_{M}(\varphi ), \label{eq:afm1} \\
\Delta t_{J}^{\ast } &=&A_{J}F_{J}(\varphi ), \label{eq:afj1}\end{aligned}$$where we have introduced the following constant “amplitudes” $$\begin{aligned}
A_{M} &\doteq &-2\mathcal{R}, \label{eq:afm2} \\
A_{J} &\doteq &-\frac{\mathcal{J}\mathcal{S}}{a(1-e^{2})}, \label{eq:afj2}\end{aligned}$$and the varying “phase” terms $$\begin{aligned}
F_{M}(\varphi ) &\doteq &\ln \left[ \frac{1-\mathcal{S}\sin (\omega +\varphi
)}{1+e\cos \varphi }\right] , \label{eq:afm3} \\
F_{J}(\varphi ) &\doteq &\frac{\left( \cos (\omega +\varphi )\right) \left(
1+e\cos \varphi \right) }{1-\mathcal{S}\sin (\omega +\varphi )}.
\label{eq:afj3}\end{aligned}$$In particular we see that when $\mathcal{S}=\sin i=1$, $F_{J}(\varphi )$ tends to diverge as far as $\omega +\varphi
\rightarrow \pi /2$ (i.e. close to the conjunction position, when the impact parameter goes to zero). $F_{M}$ too diverges in the same configuration and we see that the divergences have different “strength”, the former being an inverse power, the latter logarithmic. The reason for that difference is easily referable to the gravito-magnetic potential affecting $F_{J}$, which has a dipolar structure, and to the monopolar gravito-electric one affecting $F_{M}$.
Even though the divergences have no physical meaning because the actual compact objects have finite dimensions and the e.m. signals cannot pass through the center of the companion star, we see that the gravito-magnetic contribution is bigger for those systems that are seen nearly edge-on from the Earth, which are the ideal candidates for revealing the gravito-magnetic effects. This is the case, for instance, of the binary system PSR J0737-3039, where, however, unfortunately the presence of a large magnetosheath zone makes the effective impact parameter much bigger than the actual linear dimension of a neutron star [tartaglia05]{}. That being said, in the following section we give numerical estimates for the constant amplitudes $A_{M}$ and $A_{J}$ for the known binary pulsar systems.
Numerical Estimates {#sec:numerical}
===================
The PK parameters related to the Shapiro time delay $\mathcal{R}%
,\mathcal{S}$ have been successfully measured with great accuracy in some binary pulsar systems, such as PSR B 1913+16 (see [@weisberg02],[taylor89]{}), and the recently discovered system PSR J0737-3039 (see [burgay03]{},[@lyne04],[@kramer05]). Indeed, the analysis of these systems is very accurate, because of their favorable geometrical properties and, through the measurements of several PK parameters, they provided very accurate tests of GR as confronted to alternative theories of gravity. In [@tartaglia05], we studied the gravito-magnetic corrections to pulsar timing in a simplified situation, taking into account circular orbits only. Here, since we have generalized those results to arbitrary elliptic orbits, we may apply the formalism developed so far to all the binary pulsar systems known up to this moment, in order to estimate the magnitude of the gravito-magnetic corrections to the time delay. In Table \[tab:table1\] $%
A_{M}$ and $A_{J}$ are evaluated for the binary systems known until today[^2]. A few comments on how the table has been obtained: for those systems where the available data are not complete, we have chosen for the missing data the most favorable value (see the caption of the table). Furthermore, we have estimated the intrinsic angular momentum of the sources of the gravitational field supposing that the progenitor star was only a little bigger than the Sun, and that most of the angular momentum was preserved during the collapse, so that $J_{2}\simeq J_{\odot }$.
From Table \[tab:table1\] it is clear that the gravito-magnetic contribution is much smaller than the mass contribution, as expected. However it is possible, at least in principle, to distinguish the former from the latter, on the basis of their different dependance from the geometric parameters. In fact, from (\[eq:tzero2\]),(\[eq:dsm8\]) and (\[eq:dsj4\]) it is clear that the geometric and the gravito-electric contribution are symmetric with respect to the conjunction and opposition points, while the gravito-magnetic contribution is anti-symmetric. As we pointed out in the previous paper [@tartaglia05], if it were possible to identify conjunction and opposition points in the sequence of arriving pulses, this fact could be exploited for extracting the gravito-magnetic effect.
Nowadays, the uncertainties in pulsar data timing are of the order of $10^{-6}\sim 10^{-7} s$ (see for instance [@kramer05]). However, as we pointed out above, gravito-magnetic effects can become larger if the geometry of the system is favorable: in particular, when $\sin i=1$, $F_{J}$ tends to diverge close to the conjunction. On the other hand, we can give an estimate of the value of the geometrical parameters needed to make $A_{J}$ of the order of magnitude of the present day uncertainties. So, if we assume $J_{2}\simeq J_{\odot }$, in order to have $A_{J}\geq 10^{-7}\ s$, we must have $$a(1-e^{2})\leq \frac{a(1-e^{2})}{\sin i}\leq 5\times 10^{4}\ m
\label{eq:estim1}$$Hence, small orbits with great eccentricity allow, at least in principle, the measurement of the gravito-magnetic effects. It might be useful, also, to estimate the rate of decay of the orbit because of the emission of gravitational waves. If we assume, for the sake of simplicity, $e=0,M_{1}=M_{2}=1.44M_{\odot }$ and that $a$ fulfills (\[eq:estim1\]), we get (see [@straumann04]) $$\left\vert \frac{a}{\dot{a}}\right\vert =\frac{5}{64}\frac{a^{4}}{%
M_{1}M_{2}(M_{1}+M_{2})}\leq 2\times 10^{7}\ s \label{eq:estim2}$$Consequently, we may argue that these effects may become larger in the final phase of the evolution of the binary systems, i.e. during their coalescence, even though, in that phase, the weak field approach that we used in this paper would probably be rather poordemanding for different analysis techniques.
The smallness of the gravito-magnetic delay would also require long data taking times so posing the problem of the stability of the pulsar frequency. However a peculiarity which is not blurred by any drift or noise is the physical antisymmetry of the gravito-magnetic effect, which should emerge in the long period over all other phenomena.
System $A_M$ $[\mu s]$ $A_J$ $[p s]$
--------------------------------------------- ----------------- ---------------
[PSR B1913+16$^{a}$]{} 6.9 4.2
PSR J0737-3039$^{b}$ 6.2 11.8
PSR B1534+12$^{c}$ 6.7 2.3
PSR J1756-2251$^{d}$ 5.9 2.8
PSR J1829+2456$^{e}$ 6.1 1.1
PSR J1518+4904$^{f}$ 7.2 0.5
PSR J1811-1736$^{g}$ 3.5 0.4
PSR B2127+11C$^{h}$ 6.8 6.1
$^a$: [@weisberg02],[@taylor89]
$^b$: [@kramer05]
$^c$: [@stairs02],[@stairs04],[@thorsett04]
$^d$: [@faulkner04]
$^e$: [@champion04]
$^f$: [@nice96]
$^g$: [@lyne99]
$^h$: [@deich96]
: Evaluation of the gravito-electric and gravito-magnetic contributions to the time delay. For the systems PSR J1756-2251,PSR J1829+2456, PSR J1518+4904, PSR J1811-1736, PSR B2127+11C, since the present data do not provide the inclination of the orbit, we chose the most favorable value for the calculations of $A_J$, i.e. $\sin i=1$. Similarly, for the calculations of $A_M$, we chose the best estimate for the mass of the companion star for the systems PSR J1829+2456, PSR J1518+4904, PSR J1811-1736, since the available data do not constrain it completely.[]{data-label="tab:table1"}
\
Conclusions {#sec:conc}
============
In this paper, we have studied the effects of the gravitational interaction on the time delay of electromagnetic signals coming from a binary system composed of a radio-pulsar and another compact object. In particular, we have focused our attention on the gravito-magnetic corrections to the time delay due to the gravitational field of the binary system (Shapiro time delay).In doing so, we have generalized the results obtained in a previous paper, where we considered a simplified situation, taking into account circular orbits only. Here arbitrary elliptic orbits are allowed. Furthermore, by following a standard approach, we have expressed the time delay and its gravito-magnetic component in terms of Keplerian and post-Keplerian parameters.In particular, a new post-Keplerian parameter has been introduced, which coincides with the intrinsic angular momentum of the source of the gravitational field, and could give, in some cases, information on its moment of inertia.We have given numerical estimates of the amount of the gravito-magnetic corrections for all binary pulsar systems known until today, and we have seen that, even though they are usually much smaller than the corresponding gravito-electric ones, they can become larger for those systems that are seen nearly edge-on from the Earth, which are the ideal candidates for revealing the gravito-magnetic effects in this context.Among the known binary pulsar systems, the one having the most favorable geometrical properties for the detection of the gravito-magnetic effect is PSR J0737-3039, which, unfortunately, has a large magnetosheath that keeps the magnitude of the gravito-magnetic correction below the detectability threshold. However, we cannot exclude, at the present discovery rate of new binary pulsars, that other binary systems with favorable geometrical configurations can be found. This fact, together with the expected improvement of the sensitivity and precision of the timing of pulses, makes us cherish the hope that, in the future, it will be possible to measure the gravito-magnetic corrections to the time delay, and in particular the newly introduced post-Keplerian parameter.
[99]{}
Hulse R.A., Taylor J.H., *Astrophys. J.* **195**, L51 (1975) Stairs I.H., *Living Rev. Relativ.* **5** (2003), `http://www.livingreviews.org/lrr-2003-5` Burgay M. *et al.*, *Nature* **426**, 531 (2003) Lyne A. *et al.*, *Science* **303**, 1153 (2004) Kramer M. *et al.*, in *Proceedings of The 22nd Texas Symposium on Relativistic Astrophysics, Stanford Universisty, December 2004* (2004), astro-ph/0503386 Tartaglia A., Ruggiero M.L., Nagar A., *Phys. Rev. D* **71**, 023003 (2005), gr-qc/0501059 Lense J. and Thirring H., *Phys. Z.* **19**, 156 (1918); the English translation available in Mashhoon B., Hehl F.W. and Theiss D.S., *Gen. Rel. Grav.* **16**, 711 (1984). Ruggiero M.L., Tartaglia A., *Il Nuovo Cimento B* **117**, 743 (2002), gr-qc/0207065 Barker B.M., O’Connell R.F. *Phys. Rev. D* **12**, 329 (1975) Barker B.M., O’Connell R.F. *Phys. Rev. D* **14**, 861 (1976) O’ Connell R.F., *Phys. Rev. Lett.* **93**, 081103 (2004), gr-qc/0409101 Weisberg J.M., Taylor J.H., in *Proceedings of Binary Pulsars, Chania, Crete 2002 ASP Conference Sries* (2002), astro-ph/0211217 Stairs I.H., Thorsett S.E., Arzoumanian Z., *Phys. Rev. Lett* (2004), astro-ph/0408457 Kopeikin S., Mashhoon B., *Phys. Rev. D* **65**, 064025 (2002), gr-qc/0110101 Taylor J.H., Weisberg J.M., *Astrophys. J.* **345**, 434 (1989) Damour T., Taylor J.H., *Phys. Rev. D* **45**, 1840 (1992) Straumann N., *General Relativity*, Springer, Berlin (2004) Will C.M., *Theory and Experiment in Gravitational Physics*, Cambridge University Press, Cambridge (1993) Mashhoon B., Gronwald F., Lichtenegger H.I.M., *Lect. Notes Phys.* **562**, 83 (2001), gr-qc/9912027 Mashhoon B., (2003), gr-qc/0311030 Stairs I.H, Thorsett S.E., Taylor J.E., Wolszczan A., *Astrophys. J.* **581**, 501 (2002), astro-ph/0208357 Thorsett S.E., Dewey R.J., Stairs I.H., *Astrophys. J.* **619**, 1036 (2005), astro-ph/0408458 Faulkner A.J. *et al.*, *Astrophys. J.* **618**, L119 (2005), astro-ph/0411796 Champion D.J. *et al.*, *Mon. Not. Roy. Astron. Soc.* **350**, L61 (2004), astro-ph/0403553 Nice D.J., Sayer R.W., Taylor J.H., *Astrophys. J.* **466**, L87 (1996) Lyne A.G. *et al.*, *Mon. Not. Roy. Astron. Soc.* **312**, 698 (2000), astro-ph/9911313 Deich W.T.S., Kulkarni S.R., in *Compact Stars in Binaries*, van Paradijs J., van den Heuvel E.P.J., Kuulkers E. eds, Kluwer Academic Publishers, Dordrecht (1996)
[^1]: We use units such that c=G=1; the signature of the space-time metric is $(1,-1,-1,-1)$.
[^2]: Notice that, in physical units we have $A_{M}=\frac{%
GM_{2}}{c^{3}}$, $A_{J}=\frac{GJ_{2}\sin i}{c^{4}a(1-e^{2})}$.
|
---
abstract: 'Idioms pose problems to almost all Machine Translation systems. This type of language is very frequent in day-to-day language use and cannot be simply ignored. The recent interest in memory augmented models in the field of Language Modelling has aided the systems to achieve good results by bridging long-distance dependencies. In this paper we explore the use of such techniques into a Neural Machine Translation system to help in translation of idiomatic language.'
author:
- 'Giancarlo D. Salton'
- 'Robert J. Ross'
- |
John D. Kelleher\
Applied Intelligence Research Centre\
School of Computing\
Dublin Institute of Technology\
Ireland\
[{giancarlo.salton,robert.ross,john.d.kelleher}@dit.ie]{}\
bibliography:
- 'coling2018.bib'
title: Exploring the Use of Attention within an Neural Machine Translation Decoder States to Translate Idioms
---
Introduction {#sec:intro}
============
Neural Machine Translation {#sec:related}
==========================
{#sec:model}
NMT Experiments {#sec:experiments}
===============
Analysis of the Models {#sec:analysis}
======================
Conclusions {#sec:conclusions}
===========
Acknowledgements {#acknowledgements .unnumbered}
================
The acknowledgements should go immediately before the references. Do not number the acknowledgements section. Do not include this section when submitting your paper for review.
|
---
abstract: 'In this survey, we present five different proofs for the transcendence of Kempner’s number, defined by the infinite series $\sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$. We take the opportunity to mention some interesting ideas and methods that are used for proving deeper results. We outline proofs for some of these results and also point out references where the reader can find all the details.'
---
\[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{}
\[thm\][Definition]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Question]{}
\[thm\][Remark]{}
Boris Adamczewski\
CNRS, Université de Lyon, Université Lyon 1\
Institut Camille Jordan\
43 boulevard du 11 novembre 1918\
69622 Villeurbanne Cedex\
France\
<Boris.Adamczewski@math.univ-lyon1.fr>\
.2 in
[*À Jean-Paul Allouche, pour son soixantième anniversaire.*]{}
[*Le seul véritable voyage, le seul bain de Jouvence,\
ce ne serait pas d’aller vers de nouveaux paysages,\
mais d’avoir d’autres yeux...*]{}
[Marcel Proust, [*À la recherche du temps perdu*]{}]{}
Introduction
============
Proving that a given real number is transcendental is usually an extremely difficult task. Even for classical constants like $e$ and $\pi$, the proofs are by no means easy, and most mathematicians would be happy with a single proof of the transcendence of $e+\pi$ or $\zeta(3)$. In contrast, this survey will focus on the simple series $$\kappa := \sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$$ that can be easily proved to be transcendental. The first proof is due to Kempner [@Kempner] in 1916 and, in honor of this result, we refer to $\kappa$ as the [*Kempner number*]{}[^1]. If the transcendence of $\kappa$ is not a real issue, our aim is instead to look at the many faces of $\kappa$, which will lead us to give five different proofs of this fact. This must be (at least for the author) some kind of record, even if we do not claim this list of proofs to be exhaustive. In particular, we will not discuss Kempner’s original proof. Beyond the transcendence of $\kappa$, the different proofs we give all offer the opportunity to mention some interesting ideas and methods that are used for proving deeper results. We outline proofs for some of these results and also point out references where the reader can find all the details.
The outline of the paper is as follows. In Section \[section: knight\], we start this survey with a totally elementary proof of the transcendence of the Kempner number, based on a digital approach. Quite surprisingly, a digital approach very much in the same spirit has a more striking consequence concerning the problem of finding good lower bounds for the number of non-zero digits among the first $N$ digits of the binary expansion of algebraic irrational numbers. In Section \[section: mahler\], we give a second proof that relies on Mahler’s method. We also take the opportunity to discuss a little-known application of this method to transcendence in positive characteristic. Our third proof is a consequence of a $p$-adic version of Roth’s theorem due to Ridout. It is given in Section \[section: roth\]. More advanced consequences of the Thue–Siegel–Roth–Schmidt method are then outlined. In Section \[section: folding\], we give a description of the continued fraction expansion of $\kappa$ which turns out to have interesting consequences. We present two of them, one concerning a question of Mahler about the Cantor set and the other the failure of Roth’s theorem in positive characteristic. Our last two proofs rely on such a description and the Schmidt subspace theorem. They are given, respectively, in Sections \[sec: cf1\] and \[sec: cf2\]. The first one uses the fact that $\kappa$ can be well approximated by a familly of quadratic numbers of a special type, while the second one uses the fact that $\kappa$ and $\kappa^2$ have very good rational approximations with the same denominators. Both proofs give rise to deeper results that are described briefly.
Throughout this paper, $\lfloor x\rfloor$ and $\lceil x\rceil$ denote, respectively, the floor and the ceiling of the real number $x$. We also use the classical notation $f(n)\ll g(n)$ (or equivalently $g(n)\gg f(n)$), which means that there exists a positive real number $c$, independent of $n$, such that $f(n)< cg(n)$ for all sufficiently large integers $n$.
An ocean of zeros {#section: knight}
=================
We start this survey with a totally elementary proof of the transcendence of the Kempner number, due to Knight [@Kn91]. This proof, which is based on a digital approach, is also reproduced in the book of Allouche and Shallit [@AS Chap. 13].
Set $$f(x) := \sum_{n=0}^{\infty} x^{2^n} ,$$ so that $\kappa = f(1/2)$. For every integer $i\geq 0$, we let $a(n,i)$ denote the coefficient of $x^n$ in the formal power series expansion of $f(x)^i$. Thus $a(n,i)$ is equal to the number of ways that $n$ can be written as a sum of $i$ powers of $2$, where different orderings are counted as distinct. For instance, $a(5,3)=3$ since $$5 = 1+2+2=2+1+2=2+2+1 .$$ Note that for positive integers $n$ and $i$, we clearly have $$\label{eq: ank}
a(n,i) \leq (1+\log_2 n)^{i} .$$ The expression $$\kappa^{i} = f(1/2)^{i} = \sum_{n=0}^{\infty} \frac{a(n,i)}{2^n}$$ can be though of as a “fake binary expansion" of $\kappa^{i}$ in which carries have not been yet performed.
Let us assume, to get a contradiction, that $\kappa$ is an algebraic number. Then there exist integers $a_0,\ldots,a_d$, with $a_d>0$, such that $$a_0 + a_1 \kappa+\cdots + a_d\kappa^d =0 .$$ Moving all the negative coefficients to the right-hand side, we obtain an equation of the form $$\label{eq: fake}
a_{i_1}\kappa^{i_1} + \cdots + a_{i_r}\kappa^{i_r} + a_d\kappa^{d} = b_{j_1}\kappa^{j_1} + \cdots + b_{j_s}\kappa^{j_s} ,$$ where $r+s=d$, $0\leq i_1<\cdots <i_r<d$, $0\leq j_1<\cdots <j_s$, and coefficients on both sides are nonnegative.
Let $m$ be a positive integer and set $N := (2^{d}-1)2^{m}$, so that the binary expansion of $N$ is given by $$(N)_2 = \underbrace{1\cdots 1}_{d}\; \underbrace{0\cdots 0}_{m} .$$ Then for every integer $n$ in the interval $I := [N - (2^{m-1}-1) , N + 2^{m}-1]$ and every integer $i$, $0\leq i\leq d$, we have $$a(n,i) = \left\{
\begin{array}{ll}
d!, & \mbox{if } n=N \mbox{ and } i=d; \\
0,& \mbox{otherwise.}
\end{array}
\right.$$ Indeed, every $n\not=N\in I$ has more than $d$ nonzero digits in its binary expansion, while $N$ has exactly $d$ nonzero digits.
Now looking at Equality (\[eq: fake\]) as an equality between two fake binary numbers, we observe that
- on the right-hand side, all fake digits with position in $I$ are zero (an ocean of zeros),
- on the left-hand side, all fake digits with position in $I$ are zero except for the one in position $N$ that is equal to $a_dd!$ (an island).
Note that $d$ is fixed, but we can choose $m$ as large as we want. Performing the carries on the left-hand side of (\[eq: fake\]) for sufficiently large $m$, we see that the fake digit $a_dd!$ will produce some nonzero binary digits in a small (independent of $m$) neighborhood of the position $N$. On the other hand, the upper bound (\[eq: ank\]) ensures that, for sufficiently large $m$, carries on the right-hand side of (\[eq: fake\]) will never reach this neighborhood of the position $N$. By uniqueness of the binary expansion, Equality (\[eq: fake\]) is thus impossible. This provides a contradiction.
Beyond Knight’s proof
---------------------
Unlike $\kappa$, which is a number whose binary expansion contains absolute oceans of zeros, it is expected that all algebraic irrational real numbers have essentially random binary expansions (see the discussion in Section \[section: roth\]). As a consequence, if $\xi$ is an algebraic irrational number and if $\mathcal P(\xi,2,N)$ denotes the number of $1$’s among the first $N$ digits of the binary expansion of $\xi$, we should have $$\mathcal P(\xi,2,N) \sim \frac{N}{2} \cdot$$ Such a result seems to be out of reach of current approaches, and to find good lower bounds for $\mathcal P(\xi,2,N)$ remains a challenging problem.
A natural (and naive) approach to study this question can be roughly described as follows: if the binary expansion of $\xi$ contains too many zeros among its first digits, then some partial sums of its binary expansion should provide very good rational approximations to $\xi$; but on the other hand, we know that algebraic irrationals cannot be too well approximated by rationals. More concretely, we can argue as follows. Let $\xi:= \sum_{i\geq 0}1/2^{n_i}$ be a binary algebraic number. Then there are integers $p_k$ such that $$\displaystyle\sum_{i=0}^k \frac{1}{2^{n_i}}= \frac{p_k}{2^{n_k}} \;\;\mbox{ and }\;\;
\left \vert \xi - \frac{p_k}{2^{n_k}} \right\vert < \frac{2}{2^{n_{k+1}}} \cdot$$ On the other hand, since $\xi$ is algebraic, given a positive $\varepsilon$, Ridout’s theorem (see Section \[section: roth\]) implies that $$\left\vert \xi - \frac{p_k}{2^{n_k}} \right\vert > \frac{1}{2^{(1+\varepsilon)n_k}} ,$$ for every sufficiently large integer $k$. This gives that $n_{k+1} < (1 + \varepsilon) n_k+1$ for such $k$. Hence, for any positive number $c$, we have $$\label{ri}
\mathcal P(\xi,2,N) > c \log N ,$$ for every sufficiently large $N$.
Quite surprisingly, a digital approach very much in the same spirit as Knight’s proof of the transcendence of $\kappa$ led Bailey, J. M. Borwein, Crandall, and Pomerance [@BBCP04] to obtain the following significant improvement of (\[ri\]).
Let $\xi$ be an algebraic real number of degree $d\geq 2$. Then there exists an explicit positive number $c$ such that $$\mathcal P(\xi,2,N) > cN^{1/d} ,$$ for every sufficiently large $N$.
We do not give all the details, for which we refer the reader to [@BBCP04]. Let $\xi$ be an algebraic number of degree $d\geq 2$, for which we assume that $$\label{eq: H}
\mathcal P(\xi,2,N) < c N^{1/d} ,$$ for some positive number $c$. Let $a_0,\ldots,a_d$, $a_d>0$ such that $$a_0+a_1\xi +\cdots +a_d\xi^d=0 .$$ Let $\sum_{i\geq 0} 1/2^{n_i}$ denote the binary expansion of $\xi$ and set $f(x) := \sum_{i\geq 0} x^{n_i}$. We also let $a(n,i)$ denote the coefficient of $x^n$ in the power series expansion of $f(x)^i$. Without loss of generality we can assume that $n_0=0$. This assumption is important, in fact, for it ensures that $$a(n,d-1)=0 \implies a(n,i)=0, \mbox{ for every }i, 0\leq i\leq d-1 .$$ Set $T_i(R):=\sum_{m\geq 1} a(R+m,i)/2^m$ and $T(R):=\sum_{i=0}^d a_iT_i(R)$. A fundamental remark is that $T(R)\in\mathbb Z$. Let $N$ be a positive integer and set $K:=\lceil 2d\log N\rceil$. Our aim is now to estimate the quantity $$\sum_{R=0}^{N-K} \vert T(R)\vert .$$
[*Upper bound.*]{} We first note that $$\label{eq: ineq}
a(n,i)\leq {n+i-1\choose i-1} \mbox{ and }
\sum_{R=0}^Na(R,i) \leq \mathcal P(\xi,2,N)^{i} .$$ Using these inequalities, it is possible to show that $$\begin{aligned}
\
\sum_{R=0}^{N-K} T_i(R) & = & \sum_{m=1}^{\infty} 2^{-m} \sum_{R=0}^{N-K} a(R+m,i)\nonumber \\
&<& \sum_{R=0}^{N} a(R,i) + 2^{-K} \sum_{R=K}^N T_i(R) \nonumber \\
& \leq & \mathcal P(\xi,2,N)^i+1 ,\nonumber \end{aligned}$$ for $N$ sufficiently large. We thus obtain that $$\begin{aligned}
\label{eq: maj}
\sum_{R=0}^{N-K} \vert T(R)\vert & \leq & \sum_{i=1}^d \vert a_k\vert \left(\mathcal P(\xi,2,N)^{i}+1\right) \nonumber \\
&\leq & a_dc^dN + O(N^{1-1/d}) .\end{aligned}$$
[*Lower bound.*]{} We first infer from (\[eq: H\]) and (\[eq: ineq\]) that $$\mbox{Card} \left\{ R \in [0,N] \mid a(R,d-1)>0\right\} < c^{d-1}N^{1-1/d} .$$ Let $0=R_1<R_2<\cdots <R_M$ denote the elements of this set, so that $M<c^{d-1}N^{1-1/d}$. Set also $R_{M+1}:=N$. Then $$\sum_{i=1}^M(R_{i+1}-R_i) = N .$$ Let $\delta >0$ and set $$\mathcal I := \left\{ i \in [0,M] \mid R_{i+1}-R_i \geq \frac{\delta}{3}c^{1-d}N^{1/d} \right\} .$$ Then we have $$\label{eq: min1}
\sum_{i\in \mathcal I} \left( R_{i+1} - R_i\right)\geq \left(1-\frac{\delta}{3}\right) N .$$ Now let $i\in \mathcal I$. Note that Roth’s theorem (see Section \[section: roth\]) allows us to control the size of blocks of consecutive zeros that may occur in the binary expansion of $\xi$. Concretely, it ensures the existence of an integer $$j_i\in\left( \frac{1}{2+\delta/2}(R_{i+1}-R_i-d\log N), (R_{i+1}-R_i-d\log N)\right)$$ such that $a(j_i,1)>0$. Thus $a(R_i+j_i,d)>0$ since by assumption $n_0=0$, and then a short computation gives that $T(R_i+j_i-1)>0$.
By definition, $a(R,d-1)=0$ for every $R\in (R_{i},R_{i+1})$ and thus $a(R,i)=0$ for every $R\in (R_i,R_{i+1})$ and every $i\in[0,d-1]$. For such integers $R$, a simple computation gives $$T(R-1) = \frac{1}{2}T(R) +\frac{1}{2} a_da(R,d)$$ and thus $T(R)>0$ implies $T(R-1)>0$. Applying this argument successively to $R$ equal to $R_i+j_i-1,R_i+j_i-2,\ldots, R_i+1$, we finally obtain that $T(R)>0$ for every integer $R\in [R_i,R_i +j_i)$. The number of integers $R\in [0,N]$ such that $T(R)>0$ is thus at least equal to $$\sum_{i\in\mathcal I} \frac{1}{2+\delta/2}\left( R_{i+1}-R_i-d\log N\right) ,$$ which, by (\[eq: min1\]), is at least equal to $(1/2 -\delta/3)N$ for sufficiently large $N$. Since $T(R)\in \mathbb Z$, we get that $$\label{eq: min2}
\sum_{R=0}^{N-K} \vert T(R)\vert \geq \left(\frac{1}{2}-\frac{\delta}{3}\right)N ,$$ for sufficiently large $N$.
[*Conclusion.*]{} For sufficiently large $N$, Inequalities (\[eq: maj\]) and (\[eq: min2\]) are incompatible as soon as $c\leq ((2+\delta)a_d)^{-1/d}$. Thus, choosing $\delta$ sufficiently small, this proves the theorem for any choice of $c$ such that $c<(2a_d)^{-1/d}$.
We end this section with a few comments on Theorem BBCP.
- It is amusing to note that replacing Roth’s theorem by Ridout’s theorem in this proof only produces a minor improvement: the constant $c$ can be replaced by a slightly larger one (namely by any $c< a_d^{-1/d}$).
- A deficiency of Theorem BBCP is that it is not effective: it does not give an explicit integer $N$ above which the lower bound holds. This comes from the well-known fact that Roth’s theorem is itself ineffective. The authors of [@AdFa] show that one can replace Roth’s theorem by the much weaker Liouville inequality to derive an effective version of Theorem BBCP. This version is actually slightly weaker, because the constant $c$ is replaced by a smaller constant, but the proof becomes both totally elementary and effective.
- Last but not least: Theorem BBCP immediately implies the transcendence of the number $$\sum_{n=0}^{\infty} \frac{1}{2^{\left\lfloor n^{\log\log n}\right\rfloor}} ,$$ for which no other proof seems to be known!
Functional equations {#section: mahler}
====================
Our second proof of the transcendence of $\kappa$ follows a classical approach due to Mahler. In a series of three papers [@Mah29; @Mah30A; @Mah30B] published in 1929 and 1930, Mahler initiated a totally new direction in transcendence theory. [*Mahler’s method*]{} aims to prove transcendence and algebraic independence of values at algebraic points of locally analytic functions satisfying certain type of functional equations. In its original form, it concerns equations of the form $$f(x^k) = R(x,f(x)) ,$$ where $R(x,y)$ denotes a bivariate rational function with coefficients in a number field. In our case, we consider the function $$f(x) := \sum_{n=0}^{\infty} x^{2^n} ,$$ and we will use the fact that it is analytic in the open unit disc and satisfies the following basic functional equation: $$\label{eq: fe}
f(x^2) = f(x) - x .$$
Note that we will in fact prove much more than the transcendence of $\kappa=f(1/2)$, for we will obtain the transcendence of $f(\alpha)$ for every nonzero algebraic number $\alpha$ in the open unit disc. This is a typical advantage when using Mahler’s method. Before proceeding with the proof we need to recall a few preliminary results.
[*Preliminary step 1.*]{} The very first step of Mahler’s method consists in showing that the function $f(x)$ is transcendental over the field of rational function $\mathbb C(x)$. There are actually several ways to do that. Instead of giving an elementary but [*ad hoc*]{} proof, we prefer to give the following general statement that turns out to be useful in this area.
Let $(a_n)_{n\geq 0}$ be an aperiodic sequence with values in a finite subset of $\mathbb Z$. Then $f(x)=\sum_{n\geq 0}a_nx^n$ is transcendental over $\mathbb C(x)$.
Note that $f(x)\in\mathbb Z[[x]]$ has radius of convergence one and the classical theorem of Pólya–Carlson[^2] thus implies that $f(x)$ is either rational or transcendental. Furthermore, since the coefficients of $f(x)$ take only finitely many distinct values and form an aperiodic sequence, we see that $f(x)$ cannot be a rational function.
[*Preliminary step 2.*]{} We will also need to use Liouville’s inequality as well as basic estimates about [*height functions*]{}. There are, of course, several notions of heights. The most convenient works with the absolute logarithmic Weil height that will be denoted by $h$. We refer the reader to the monograph of Waldschmidt [@Wa_book Chap. 3] for an excellent introduction to heights and in particular for a definition of $h$. Here we just recall a few basic properties of $h$ that will be used in the sequel. All are proved in [@Wa_book Chap. 3]. For every integer $n$ and every pair of algebraic numbers $\alpha$ and $\beta$, we have $$\label{eq: height1}
h(\alpha^n)=\vert n\vert h(\alpha)$$ and $$\label{eq: height1bis}
h(\alpha+\beta) \leq h(\alpha)+h(\beta) +\log 2 .$$ More generally, if $P(X,Y)\in\mathbb Z[X,Y]\setminus\{0\}$ then $$\label{eq: height2}
h(P(\alpha,\beta)) \leq \log L(P) + (\deg_XP)h(\alpha) + (\deg_Y P)h(\beta) ,$$ where $L(P)$ denote the length of $P$, which is classically defined as the sum of the absolute values of the coefficients of $P$. We also recall Liouville’s inequality: $$\label{eq: liouville}
\log\vert \alpha\vert \geq - d h(\alpha) ,$$ for every nonzero algebraic number $\alpha$ of degree at most $d$.
We are now ready to give our second proof of transcendence for $\kappa$.
Given a positive integer $N$, we choose a nonzero bivariate polynomial $P_N\in \mathbb Z[X,Y]$ whose degree in both $X$ and $Y$ is at most $N$, and such that the order of vanishing at $x=0$ of the formal power series $$A_N(x) := P_N(x,f(x))$$ is at least equal to $N^2$. Note that looking for such a polynomial amounts to solving a homogeneous linear system over $\mathbb Q$ with $N^2$ equations and $(N+1)^2$ unknowns, which is of course always possible. The fact that $A_N(x)$ has a large order of vanishing at $x=0$ ensures that $A_N$ takes very small values around the origin. More concretely, for every complex number $z$, $0\leq \vert z\vert <1/2$, we have $$\label{eq: anmaj}
\vert A_n(z) \vert \leq c(N) \vert z\vert^{N^2} ,$$ for some positive $c(N)$ that only depends on $N$.
Now we pick an algebraic number $\alpha$, $0< \vert \alpha\vert <1$, and we assume that $f(\alpha)$ is also algebraic. Let $L$ denote a number field that contains both $\alpha$ and $f(\alpha)$ and let $d:=[L:\mathbb Q]$ be the degree of this extension. The functional equation (\[eq: fe\]) implies the following for every positive integer $n$: $$A_N(\alpha^{2^{n}}) = P_N(\alpha^{2^{n}},f(\alpha^{2^{n}})) = P_N\left(\alpha^{2^{n}},f(\alpha)- \sum_{k=0}^{n-1} \alpha^{2^k}\right) \in L .$$ Thus $A_N(\alpha^{2^{n}})$ is always an algebraic number of degree at most $d$. Furthermore, we claim that $A_N(\alpha^{2^{n}})\not=0$ for all sufficiently large $n$. Indeed, the function $A_N(x)$ is analytic in the open unit disc and it is nonzero because $f(x)$ is transcendental over $\mathbb C(x)$, hence the identity theorem applies.
Now, using (\[eq: height1\]), (\[eq: height1bis\]) and (\[eq: height2\]), we obtain the following upper bound for the height of $A_N(\alpha^{2^n})$: $$\begin{array}{lll}
h(A_N(\alpha^{2^n})) & = &h(P_N(\alpha^{2^{n}},f(\alpha^{2^{n}})))\\ \\
& \leq & \log L(P_N) + Nh(\alpha^{2^n}) + Nh(f(\alpha^{2^{n}})) \\ \\
& = & \log L(P_N) + 2^nNh(\alpha) + Nh(f(\alpha) - \sum_{k=0}^{n-1} \alpha^{2^k}) \\ \\
& \leq & \log L(P_N) + 2^{n+1}Nh(\alpha) + Nh(f(\alpha)) + n\log 2 .\\ \\
\end{array}$$ From now on, we assume that $n$ is sufficiently large to ensure that $A_N(\alpha^{2^{n}})$ is nonzero and that $\vert \alpha^{2^n}\vert < 1/2$. Since $A_N(\alpha^{2^n})$ is a nonzero algebraic number of degree at most $d$, Liouville’s inequality (\[eq: liouville\]) implies that $$\log \vert A_N(\alpha^{2^n})\vert \geq - d \left(\log L(P_N) + 2^{n+1}Nh(\alpha) + Nh(f(\alpha)) + n\log 2\right) .$$ On the other hand, since $\vert \alpha^{2^n}\vert < 1/2$, Inequality (\[eq: anmaj\]) gives that $$\log \vert A_N(\alpha^{2^n})\vert \leq \log c(N) + 2^nN^2\log \vert \alpha\vert .$$ We thus deduce that $$\log c(N) + 2^nN^2\log \vert \alpha\vert \geq - d \left(\log L(P_N) + 2^{n+1}Nh(\alpha) + Nh(f(\alpha)) + n\log 2\right) .$$ Dividing both sides by $2^n$ and letting $n$ tend to infinity, we obtain $$N \leq \frac{2dh(\alpha)}{\vert \log \vert\alpha\vert \vert} \cdot$$ Since $N$ can be chosen arbitrarily large independently of the choice of $\alpha$, this provides a contradiction and concludes the proof.
Beyond Mahler’s proof {#allouche}
---------------------
Mahler’s method has, by now, become a classical chapter in transcendence theory. As observed by Mahler himself, his approach allows one to deal with functions of several variables and systems of functional equations as well. It also leads to algebraic independence results, transcendence measures, measures of algebraic independence, and so forth. Mahler’s method was later developed by various authors, including Becker, Kubota, Loxton and van der Poorten, Masser, Nishioka, and Töpfer, among others. It is now known to apply to a variety of numbers defined by their decimal expansion, their continued fraction expansion, or as infinite products. For these classical aspects of Mahler’s theory, we refer the reader to the monograph of Ku. Nishioka [@Ni_liv] and the references therein.
We end this section by pointing out another feature of Mahler’s method that is unfortunately less well known. A major deficiency of Mahler’s method is that, in contrast with the Siegel $E$- and $G$-functions, there is not a single classical transcendental constant that is known to be the value at an algebraic point of an analytic function solution to a Mahler-type functional equation. Roughly, this means that the most interesting complex numbers for number theorists seemingly remain beyond the scope of Mahler’s method. However, a remarkable discovery of Denis is that Mahler’s method can be applied to prove transcendence and algebraic independence results involving [*periods of $t$-modules*]{}, which are variants of the more classical periods of abelian varieties, in the framework of the arithmetic of function fields of positive characteristic. For a detailed discussion on this topic, we refer the reader to the recent survey by Pellarin [@Pel2], and also [@Pel1]. Unfortunately, we cannot begin to do justice here to this interesting topic. We must be content to give only a hint about the proof of the transcendence of an analogue of $\pi$ using Mahler’s method, and we hope that the interested reader will look for more in [@Pel1; @Pel2].
Let $p$ be a prime number and $q=p^{e}$ be an integer power of $p$ with $e$ positive. We let $\mathbb F_q$ denote the finite field of $q$ elements, $\mathbb F_q[t]$ the ring of polynomials with coefficients in $\mathbb F_q$, and $\mathbb F_q(t)$ the field of rational functions. We define an absolute value on $\mathbb F_q[t]$ by $\vert P\vert = q^{\deg_t P}$ so that $\vert t\vert =q$. This absolute value naturally extends to $\mathbb F_q(t)$. We let $\mathbb F_q((1/t))$ denote the completion of $\mathbb F_q(t)$ for this absolute value and let $C$ denote the completion of the algebraic closure of $\mathbb F_q((1/t))$ for the unique extension of our absolute value to the algebraic closure of $\mathbb F_q((1/t))$. Roughly, this allows to replace the natural inclusions $$\mathbb Z\subset \mathbb Q\subset \mathbb R\subset \mathbb C$$ by the following ones $$F_q[t]\subset F_q(t)\subset F_q((1/t))\subset C .$$ The field $C$ is a good analogue for $\mathbb C$ and allows one to use some tools from complex analysis such as the identity theorem. In this setting, the formal power series $$\Pi := \prod_{n=1}^{\infty} \frac{1}{1- t^{1-q^n}} \in \mathbb F_q((1/t))\subset C$$ can be thought of as an analogue of the number $\pi$. To be more precise, the Puiseux series $$\widetilde{\Pi} = t(-t)^{1/(q-1)} \prod_{n=1}^{\infty} \frac{1}{1- t^{1-q^n}} \in C$$ is a fundamental period of Carlitz’s module and, in this respect, it appears to be a reasonable analogue for $2i\pi$. Of course, proving the transcendence of either $\Pi$ or $\widetilde{\Pi}$ over $\mathbb F_p(t)$ remains the same. As discovered by Denis [@Denis], it is possible to deform the infinite product given in our definition of $\Pi$, in order to obtain the following “analytic function" $$f_{\Pi}(x) := \prod_{n=1}^{\infty} \frac{1}{1- tx^{q^n}}$$ which converges for all $x\in C$ such that $\vert x\vert <1$. A remarkable property is that the function $f_{\Pi}(x)$ satisfies the following Mahler-type functional equation: $$f_{\Pi}(x^q) = \frac{f_{\Pi}(x)}{(1-tx^q)} \cdot$$ As the principle of Mahler’s method also applies in this framework, one can prove along the same lines as in the proof we just gave for the transcendence of $\kappa$ that $f_{\Pi}$ takes transcendental values at every nonzero algebraic point in the open unit disc of $C$. Considering the rational point $1/t$, we obtain the transcendence of $\Pi = f_{\Pi}(1/t)$.
Note that there are many other proofs of the transcendence of $\Pi$. The first is due to Wade [@Wa] in 1941. Other proofs were then given by Yu [@Yu] using the theory of Drinfeld modules, by Allouche [@All90] using automata theory and Christol’s theorem, and by De Mathan [@DM] using tools from Diophantine approximation.
$p$-adic rational approximation {#section: roth}
===============================
The first transcendence proof that graduate students in mathematics usually meet concerns the so–called Liouville number $${\mathcal L}:= \sum_{n=1}^{\infty} \frac{1}{b^{n!}} \cdot$$ This series is converging so quickly that partial sums $$\frac{p_n}{q_n} := \sum_{k=1}^{n}\frac{1}{b^{k!}}$$ provide infinitely many extremely good rational approximations to $\mathcal L$, namely $$\left\vert {\mathcal L} - \frac{p_n}{q_n} \right\vert
< \frac{2}{q_n^{n+1}} \cdot$$ In view of the classical Liouville inequality [@Li], these approximations prevent $\mathcal L$ from being algebraic. Since $\kappa$ is also defined by a lacunary series that converges very fast, it is tempting to try to use a similar approach. However, we will see that this requires much more sophisticated tools.
Liouville’s inequality is actually enough to prove the transcendence for series such as $\sum_{i=0}^{\infty} 1/2^{n_i}$, where $\limsup (n_{i+1}/n_i)=+\infty$, but it does not apply if $n_i$ has only an exponential growth like $n_i=2^{i}$, $n_i=3^{i}$ or $n_i=F_i$ (the $i$th Fibonacci number). In the case where $\limsup (n_{i+1}/n_i)>2$, we can use Roth’s theorem [@Ro55].
Let $\xi$ be a real algebraic number and $\varepsilon$ be a positive real number. Then the inequality $$\left\vert \xi - \frac{p}{ q} \right\vert < \frac{1}{q^{2 + \varepsilon}}$$ has only a finite number of rational solutions $p/q$.
For instance, the transcendence of the real number $\xi := \sum_{i=0}^{\infty} 1/2^{3^{i}}$ is now a direct consequence of the inequality $$0<\left\vert \xi - \frac{p_n}{q_n} \right\vert
< \frac{2}{q_n^3} ,$$ where $p_n/q_n := \sum_{i=0}^n 1/2^{3^{i}}$. However, the same trick does not apply to $\kappa$, for we get that $$\left\vert \kappa - \frac{p_n}{q_n} \right\vert
\gg \frac{1}{q_n^2} ,$$ if $p_n/q_n := \sum_{i=0}^n 1/2^{2^{i}}$.
The transcendence of $\kappa$ actually requires the following $p$-adic extension of Roth’s theorem due to Ridout [@Ri57]. For every prime number $\ell$, we let $\vert \cdot \vert_\ell$ denote the $\ell$-adic absolute value normalized such that $\vert \ell \vert_\ell = \ell^{-1}$.
Let $\xi$ be an algebraic number and $\varepsilon$ be a positive real number. Let $S$ be a finite set of distinct prime numbers. Then the inequality $$\left( \prod_{\ell \in S} \vert p \vert_\ell \cdot \vert q\vert_\ell \right) \cdot
\left\vert \xi - \frac{p}{q} \right\vert < \frac{1}{q^{2+\varepsilon}}$$ has only a finite number of rational solutions $p/q$.
With Ridout’s theorem in hand, the transcendence of $\kappa$ can be easily deduced: we just have to take into account that the denominators of our rational approximations are powers of $2$.
Let $n$ be a positive integer and set $$\rho_n := \sum_{i=1}^{n}\frac{1}{2^{2^{i}}}.$$ Then there exists an integer $p_n$ such that $\rho_n = p_n/q_n$ with $q_n = 2^{2^n}$. Observe that $$\left\vert \kappa - \frac{p_n}{q_n} \right\vert < \frac{2}{2^{2^{n+1}}}
= \frac{2}{(q_n)^{2}},$$ and let $S=\{2\}$. Then, an easy computation gives that $$\vert q_n \vert_2 \cdot \vert p_n\vert_2 \cdot
\left\vert \kappa - \frac{p_n}{q_n}\right\vert < \frac{2}{(q_n)^{3}} \cdot$$ Applying Ridout’s theorem, we get that $\kappa$ is transcendental.
Of course there is no mystery, the difficulty in this proof is hidden in the proof of Ridout’s theorem.
Beyond Roth’s theorem
---------------------
The Schmidt subspace theorem [@Schmidt80] provides a formidable multidimensional generalization of Roth’s theorem. We state below a simplified version of the $p$-adic subspace theorem due to Schlickewei [@Sch77], which turns out to be very useful for proving transcendence of numbers defined by their base-$b$ expansion or by their continued fraction expansion. Note that our last two proofs of the transcendence of $\kappa$, given in Sections \[sec: cf1\] and \[sec: cf2\], both rely on the subspace theorem. Several recent applications of this theorem can also be found in [@Bilu].
We recall that a [*linear form*]{} (in $m$ variables) is a homogeneous polynomial (in $m$ variables) of degree $1$.
Let $m\ge 2$ be an integer and $\varepsilon$ be a positive real number. Let $S$ be a finite set of distinct prime numbers. Let $L_1, \ldots , L_m$ be $m$ linearly independent linear forms in $m$ variables with real algebraic coefficients. Then the set of solutions ${\bf x} = (x_1, \ldots, x_m)$ in $\mathbb Z^m$ to the inequality $$\left(\prod_{i=1}^m\prod_{\ell \in S} \vert x_i\vert_\ell \right) \cdot \prod_{i=1}^m
{\vert L_{i} ({\bf x}) \vert } \leq
(\max\{|x_1|, \ldots , |x_m|\})^{-\varepsilon}$$ lies in finitely many proper subspaces of $\mathbb Q^m$.
Let us first see how the subspace theorem implies Roth’s theorem. Let $\xi$ be a real algebraic number and $\varepsilon$ be a positive real number. Consider the two independent linear forms $\xi X - Y$ and $X$. The subspace theorem implies that all the integer solutions $(p, q)$ to $$\label{Ch7:equation:rothlin}
\vert q \vert \cdot \vert q \xi - p\vert < \vert q\vert^{-\varepsilon}$$ are contained in a finite union of proper subspaces of $\mathbb Q^2$. There thus is a finite set of lines $x_1 X + y_1 Y = 0,\ \ldots ,\
x_t X + y_t Y = 0$ such that, every solution $(p, q)\in\mathbb Z^2$ to (\[Ch7:equation:rothlin\]), belongs to one of these lines. This means that the set of rational solutions $p/q$ to $\left\vert \xi - p/q \right\vert < q^{-2-\varepsilon}$ is finite, which is Roth’s theorem.
### A theorem of Corvaja and Zannier
Let us return to the transcendence of $\kappa$. Given an integer $b\geq 2$ and letting $S$ denote the set of prime divisors of $b$, it is clear that the same proof also gives the transcendence of $\sum_{n=0}^{\infty}1/b^{2^n}$. However, if we try to replace $b$ by a rational or an algebraic number, we may encounter new difficulties. As a good exercise, the reader can convince himself that the proof will still work with $b={5 \over 2}$ or $b={{17} \over 4}$, but not with $b={3 \over 2}$ or $b={5 \over 4}$. Corvaja and Zannier [@CZ] make clever use of the subspace theorem that allows them to overcome the problem in all cases. Among other results, they proved the following nice theorem.
Let $(n_i)_{i\geq 0}$ be a sequence of positive integers such that $\liminf n_{i+1}/n_i>1$ and let $\alpha$, $0<\vert \alpha\vert<1$, be an algebraic number. Then the number $$\sum_{i=0}^{\infty} \alpha^{n_i}$$ is transcendental.
Of course, we recover the fact, already proved in Section \[section: mahler\] by Mahler’s method, that the function $f(x)= \sum_{n=0}^{\infty} x^{2^n}$ takes transcendental values at every nonzero algebraic point in the open unit disc. The proof of Theorem CZ actually requires an extension of the $p$-adic subspace theorem to number fields (the version we gave is sufficient for rational points). We also take the opportunity to mention that the main result of [@Ad04] is actually a consequence of Theorem 4 in [@CZ].
In order to explain the idea of Corvaja and Zannier we somewhat oversimplify the situation by considering only the example of $f({4 \over 5})$. We refer the reader to [@CZ] for a complete proof. We assume that $f({4 \over 5})$ is algebraic and we aim at deriving a contradiction. A simple computation gives that $$\left\vert f\left({4 \over 5}\right) - \sum_{k=0}^n \left(\frac{4}{5}\right)^{2^k} \right\vert < 2 \left(\frac{4}{5}\right)^{2^{n+1}} ,$$ for every nonnegative integer $n$. This inequality can obviously be rephrased as $$\left\vert f\left({4 \over 5}\right) - \sum_{k=0}^n \left(\frac{4}{5}\right)^{2^k} - \left(\frac{4}{5}\right)^{2^{n+1}} - \left(\frac{4}{5}\right)^{2^{n+2} }\right\vert
< 2 \left(\frac{4}{5}\right)^{2^{n+3}} ,$$ but the subspace theorem will now take care of the fact that the last two terms on the left-hand side are $S$-units (for $S=\{2,5\}$). Multiplying by $5^{2^{n+2}}$, we obtain that $$\left\vert 5^{2^{n+2}}f\left({4 \over 5}\right) - 5^{2^{n+2}-2^n}p_n - 4^{2^{n+1}}5^{2^{n+1}} - 4^{2^{n+2}}\right\vert < 2 \left(\frac{4}{5}\right)^{2^{n+3}}5^{2^{n+2}} ,$$ for some integer $p_n$. Consider the following four linearly independent linear forms with real algebraic coefficients: $$\begin{array}{ll}
L_1 (X_1,X_2,X_3,X_4) = & f({4 \over 5})X_1-X_2-X_3-X_4, \\
L_2 (X_1,X_2,X_3,X_4) = & X_1, \\
L_3 (X_1,X_2,X_3,X_4) = & X_3 , \\
L_4(X_1,X_2,X_3,X_4)=& X_4.
\end{array}$$ For every integer $n \geq 1$, consider the integer quadruple $${\bf x}_n = (x_1^{(n)},x_2^{(n)},x_3^{(n)},x_4^{(n)}):=\left( 5^{2^{n+2}} , 5^{2^{n+2}-2^n}p_n , 4^{2^{n+1}}5^{2^{n+1}} , 4^{2^{n+2}} \right) .$$ Note that $\Vert {\bf x}_n\Vert_{\infty}\leq 5\cdot 5^{2^{n+2}}$. Set also $S=\{2,5\}$. Then a simple computation shows that $$\left(\prod_{i=1}^4\prod_{\ell \in S} \vert x_i^{(n)}\vert_\ell \right) \cdot \prod_{i=1}^4
{\vert L_{i} ({\bf x}_n) \vert } \leq 2 \left(\frac{4^8}{5^7}\right)^{2^{n}} <
\Vert {\bf x}_n\Vert_{\infty}^{-\varepsilon} ,$$ for some $\varepsilon>0$. We then infer from the subspace theorem that all points ${\bf x}_n$ lie in a finite number of proper subspaces of $\mathbb Q^4$. Thus, there exist a nonzero integer quadruple $(x,y,z,t)$ and an infinite set of distinct positive integers ${\mathcal N}$ such that $$\label{eq: plan}
5^{2^{n+2}} x + 5^{2^{n+2}-2^n}p_n y + 4^{2^{n+1}}5^{2^{n+1}} z + 4^{2^{n+2}} t = 0,$$ for every $n$ in ${\mathcal N}$. Dividing (\[eq: plan\]) by $ 5^{2^{n+2}}$ and letting $n$ tend to infinity along ${\mathcal N}$, we get that $$x + \kappa y=0 .$$ Since $\kappa$ is clearly irrational, this implies that $x=y=0$. But then Equality (\[eq: plan\]) becomes $$4^{2^{n+1}}5^{2^{n+1}} z = -4^{2^{n+2}} t ,$$ which is impossible for large $n\in\mathcal N$ unless $z=t=0$ (look at, for instance, the $5$-adic absolute value). This proves that $x=y=z=t=0$, a contradiction.
Note that the proof of the transcendence of $f(\alpha)$, for every algebraic number $\alpha$ with $0<\vert \alpha\vert <1$, actually requires the use of the subspace theorem with an arbitrary large number of variables (depending on $\alpha$). For instance, we need $14$ variables to prove the transcendence of $f(2012/2013)$.
### The decimal expansion of algebraic numbers
The decimal expansion of real numbers such as $\sqrt 2$, $\pi$, and $e$ appears to be quite mysterious and, for a long time, has baffled mathematicians. After the pioneering work of É. Borel [@Borel1; @Borel2], most mathematicians expect that all algebraic irrational numbers are [*normal numbers*]{}, even if this conjecture currently seems to be out of reach. Recall that a real number is normal if for every integer $b\geq 2$ and every positive integer $n$, each one of the $b^n$ blocks of digits of length $n$ occurs in its base-$b$ expansion with the same frequency. We end this section by pointing out an application of the $p$-adic subspace theorem related to this problem.
Let $\xi$ be a real number and $b\geq 2$ be a positive integer. Let $(a_n)_{n\geq -k}$ denote the base-$b$ expansion of $\xi$, that is, $$\xi= \displaystyle\sum_{n\geq -k} \frac{a_n}{b^n} = a_{-k}\cdots a_{-1}a_0{\scriptscriptstyle \bullet} a_1a_2\cdots .$$ Following Morse and Hedlund [@HM], we define the [*complexity function*]{} of $\xi$ with respect to the base $b$ as the function that associates with each positive integer $n$ the positive integer $$p(\xi,b,n) := \mbox{Card} \{(a_j,a_{j+1},\ldots, a_{j+n-1}), \; j \geq 1\}.$$ A normal number thus has the maximum possible complexity in every integer base, that is, $p(\xi,b,n) = b^n$ for every positive integer $n$ and every integer $b\geq 2$. One usually expects such a high complexity for numbers like $\sqrt 2$, $\pi$, and $e$. Ferenczi and Mauduit [@FM] gave the first lower bound for the complexity of all algebraic irrational numbers by means of Ridout’s theorem. More recently, Adamczewski and Bugeaud [@AdBuAnnals] use the subspace theorem to obtain the following significant improvement of their result.
Let $b \geq 2$ be an integer and $\xi$ be an algebraic irrational number. Then $$\label{bfm}
\lim_{n\to \infty} \frac{p(\xi,b,n)}{ n} = + \infty .$$
Note that Adamczewski [@Ad10] obtains a weaker lower bound for some transcendental numbers involving the exponential function. For a more complete discussion concerning the complexity of the base-$b$ expansion of algebraic numbers, we refer the reader to [@Ad10; @AdBuAnnals; @AdBuCant; @Miw09].
We only outline the main idea for proving Theorem AB1 and refer the reader to [@AdBuAnnals] or [@AdBuCant] for more details. Let $\xi$ be an algebraic number and let us assume that $$\label{eq: comp}
\liminf_{n\to \infty} \frac{p(\xi,b,n)}{ n} < +\infty,$$ for some integer $b\geq 2$. Our goal is thus to prove that $\xi$ is rational. Without loss of generality, we can assume that $0<\xi<1$.
Our assumption implies that the number of distinct blocks of digits of length $n$ in the base-$b$ expansion of $\xi$ is quite small (at least for infinitely many integers $n$). Thus, at least some of these blocks of digits have to reoccur frequently, which forces the early occurrence of some repetitive patterns in the base-$b$ expansion of $\xi$. This rough idea can be formalized as follows. We first recall some notation from combinatorics on words. Let $V=v_1\cdots v_r$ be a finite word. We let $\vert V\vert=r$ denote the length of $V$. For any positive integer $k$, we write $V^k$ for the word $$\overbrace{V\cdots V}_{\mbox{$k$ times}} .$$ More generally, for any positive real number $w$, $V^w$ denotes the word $V^{\lfloor w \rfloor}V'$, where $V'$ is the prefix of $V$ of length $\left\lceil(w-\lfloor w\rfloor)\vert V\vert\right\rceil$. With this notation, one can show that the assumption (\[eq: comp\]) ensures the existence of a real number $w>1$ and of two infinite sequences of finite words $(U_n)_{n\geq 1}$ and $(V_n)_{n\geq 1}$ such that the base-$b$ expansion of $\xi$ begins with the block of digits $0{\scriptscriptstyle \bullet} U_nV_n^w$ for every positive integer $n$. Furthermore, if we set $r_n := \vert U_n \vert$ and $s_n := \vert V_n\vert$, we have that $s_n$ tends to infinity with $n$ and there exists a positive number $c$ such that $r_n/s_n <c$ for every $n\geq 1$.
This combinatorial property has the following Diophantine translation. For every positive integer $n \ge 1$, $\xi$ has to be close to the rational number with ultimately periodic base-$b$ expansion $$0{\scriptscriptstyle \bullet} U_nV_nV_nV_n\cdots .$$ Precisely, one can show the existence of an integer $p_n$ such that $$\label{Ch7:equation:pj}
\left\vert \xi - \frac{p_n}{ b^{r_n} (b^{s_n} - 1)} \right\vert
\ll \frac{1 }{b^{r_n + w s_n}} \cdot$$
Consider the following three linearly independent linear forms with real algebraic coefficients: $$\begin{array}{ll}
L_1 (X_1, X_2, X_3) = & \xi X_1 - \xi X_2 - X_3, \\
L_2 (X_1, X_2, X_3) = & X_1, \\
L_3 (X_1, X_2, X_3) = & X_2.
\end{array}$$ Evaluating them at the integer points $${\bf x}_n =(x_1^{(n)},x_2^{(n)},x_3^{(n)}):= (b^{r_n+s_n}, b^{r_n}, p_n) ,$$ we easily obtain that $$\left(\prod_{i=1}^3\prod_{p \in S} \vert x_i^{(n)}\vert_p \right) \cdot
\prod_{i=1}^3 {\vert L_i ({\bf x}_n) \vert } \ll
\left( \max\{b^{r_n + s_n}, b^{r_n}, p_n\} \right)^{-\varepsilon},$$ where $\varepsilon := (\omega-1)/2(c+1) >0$ and $S$ denotes the set of prime divisors of $b$. We then infer from the subspace theorem that all points ${\bf x}_n$ belong to a finite number of proper subspaces of $\mathbb Q^3$. There thus exist a nonzero integer triple $(x,y,z)$ and an infinite set of distinct positive integers ${\mathcal N}$ such that $$\label{Ch7:equation:plan}
x b^{r_n+s_n} + y b^{r_n} + z p_n = 0 ,$$ for every $n$ in ${\mathcal N}$. Dividing (\[Ch7:equation:plan\]) by $b^{r_n+s_n}$ and letting $n$ tend to infinity along ${\mathcal N}$, we get that $$x+\xi z = 0 ,$$ as $s_n$ tends to infinity. Since $(x,y,z)$ is a nonzero vector, this implies that $\xi$ is a rational number. This ends the proof.
Interlude: from base-$b$ expansions to continued fractions {#section: folding}
==========================================================
It is usually very difficult to extract any information about the continued fraction expansion of a given irrational real number from its decimal or binary expansion and vice versa. For instance, $\sqrt 2$, $e$, and $\tan 1$ all have a very simple continued fraction expansion, while they are expected to be normal and thus should have essentially random expansions in all integer bases. In this section, we shall give an exception to this rule: our favorite binary number $\kappa$ has a predictable continued fraction expansion that enjoys remarkable properties involving both repetitive and symmetric patterns (see Theorem Sh1 below). Our last two proofs of transcendence for $\kappa$, given in Sections \[sec: cf1\] and \[sec: cf2\], both rely on Theorem Sh1.
For an introduction to continued fractions, the reader is referred to standard books such as Perron[@Perron], Khintchine [@Kh], or Hardy and Wright [@HaWr]. We will use the classical notation for finite or infinite continued fractions $$\frac{p}{q}
= a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}
= [a_0, a_1,\cdots, a_n]$$ resp., $$\xi =
a_0 + \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\ddots+\cfrac{1}
{a_n + \cfrac{1}{\ddots}}}}} = [a_0, a_1,\cdots, a_n, \cdots]$$ where $p/q$ is a positive rational number (resp. $\xi$ is a positive irrational real number), $n$ is a nonnegative integer, $a_0$ is a nonnegative integer, and the $a_i$’s are positive integers for $i \geq 1$. Note that we allow $a_n=1$ in the first equality. If $A=a_1a_2\cdots$ denotes a finite or an infinite word whose letters $a_i$ are positive integers, then the expression $[0,A]$ stands for the finite or infinite continued fraction $[0,a_1,a_2,\ldots]$. Also, if $A=a_1a_2\cdots a_n$ is a finite word, we let $A^R:=a_na_{n-1}\cdots a_1$ denote the reversal of $A$. As in the previous section, we use $\vert A\vert$ to denote the length of the finite word $A$.
The following elementary result was first discovered by Mendès France [@MF][^3].
Let $c, a_0, a_1, \ldots,a_n$ be positive integers. Let $p_n/q_n := [a_0, a_1, \cdots, a_n]$. Then $$\label{fold}
\frac{p_n}{q_n} + \frac{(-1)^n}{c q_n^2}
= [a_0, a_1, a_2, \cdots, a_n, c, -a_n, -a_{n-1}, \cdots, -a_1].$$
For a proof of the folding lemma, see, for instance, [@AS p. 183]. In Equality (\[fold\]) negative partial quotients occur. However, we have two simple rules that permit to get rid of these forbidden partial quotients: $$\label{eq: rule1}
[\ldots,a,0,b,\ldots] = [\ldots,a+b,\ldots]$$ and $$\label{eq: rule2}
[\ldots,a, -b_1, \cdots, -b_r] = [,\ldots,a-1,1,b_1-1,b_2,\ldots,b_r] .$$
As first discovered independently by Shallit [@Sh1; @Sh2] and Kmošek [@Km], the folding lemma can be used to describe the continued fraction expansion of some numbers having a lacunary expansion in an integer base, such as $\kappa$. Following Theorem 11 in [@Sh1], we give now a complete description of the continued fraction expansion of $2\kappa$. The choice of $2\kappa$ instead of $\kappa$ is justified by obtaining a nicer formula.
Let $A_1:=1112111111$ and $B_1:=11121111$. For every positive integer $n$, let us define the finite words $A_{n+1}$ and $B_{n+1}$ as follows: $$A_{n+1} = A_n12 (B_n)^R$$ and $$B_{n+1} \mbox{ is the prefix of } A_{n+1} \mbox{ with length }\vert A_{n+1}\vert -2 .$$ Then the sequence of words $A_n$ converges to an infinite word $$A_{\infty} = 111 2 111111 12 1111 2 111 12 \cdots$$ and $$2\kappa = [1,A_{\infty}] =[1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,\ldots] .$$
In particular, the partial quotients of $2\kappa$ take only the values $1$ and $2$. This shows that $\kappa$ is badly approximable by rationals and [*a fortiori*]{} that the transcendence of $\kappa$ is beyond the scope of Roth’s theorem (the $p$-adic version of Roth’s theorem was thus really needed in Section \[section: roth\]).
First note that by the definition of $A_{n+1}$, the word $A_n$ is a prefix of $A_{n+1}$, for every nonnegative integer $n$, which implies that the sequence of finite words converges (for the usual topology on words) to an infinite word $A_{\infty}$.
For every integer $n\geq 0$, we set $$\frac{P_n}{Q_n} := \sum_{k=0}^{n} \frac{2}{2^{2^k}} \cdot$$ We argue by induction to prove that $$P_n/Q_n = [1,A_{n-2}] ,$$ for every integer $n\geq 3$. We first note that $P_3/Q_3=[1,1,1,1,2,1,1,1,1,1,1]=[1,A_1]$. Let $n\geq 3$ be an integer and let us assume that $P_n/Q_n=[1,A_{n-2}]$. By the definition of $P_n/Q_n$, we have that $$\label{eq: pnqn}
\frac{P_{n+1}}{Q_{n+1}} = \frac{P_n}{Q_n} + \frac{1}{2Q_n^2} \cdot$$ Furthermore, an easy induction shows that for every integer $k\geq 1$, $\vert A_k\vert$ is even and $A_k$ ends with $11$ so that $$\label{eq: 11}
(A_k)^R = 11 (B_k)^R .$$ Since $\vert A_{n-2}\vert$ is even, we can apply the folding Lemma and we infer from Equalities (\[eq: pnqn\]) and (\[eq: 11\]), and from the transformation rules (\[eq: rule1\]) and (\[eq: rule2\]) that $$\begin{array}{lll}
P_{n+1}/Q_{n+1} &=& [1,A_{n-2},2,- (A_{n-2})^R] \\ \\
&=& [1,A_{n-2},1,1, 0,1,(B_{n-2})^R]\\ \\
&=& [1,A_{n-2},1,2, (B_{n-2})^R] \\ \\
&=& [1,A_{n-1}] .
\end{array}$$ This proves that $P_n/Q_n=[1,A_{n-2}]$ for every $n\geq 3$. Since the sequence $(P_n/Q_n)_{n\geq 0}$ converges to $2\kappa$ and $A_n$ is always a prefix of $A_{n+1}$, we obtain that $2\kappa=[1,A_{\infty}]$, as desired.
Two applications
----------------
In the second part of the paper [@Sh1], Shallit [@Sh2] extends his construction and obtained the following general result[^4].
Let $b\geq 2$ and $n_0\geq 0$ be integers and let $(c_n)_{n\geq 0}$ be a sequence of positive integers such that $c_{n+1}\geq 2c_n$, for every integer $n\geq n_0$. Set $d_n:=c_{n+1}-2c_n$ and $$S_b(n) := \sum_{k=0}^n \frac{1}{b^{c_k}} \cdot$$ If $n\geq n_0$ and $S_b(n)=[a_0,a_1,\ldots,a_r]$, with $r$ even, then $$S_b(n+1) = [a_0,a_1,\ldots,a_r,b^{d_n}-1,1,a_r-1,a_{r-1},\ldots,a_1] .$$
This result turns out to have interesting consequences, two of which are recalled below.
### A question of Mahler about the Cantor set
Mahler [@Mah84] asked the following question: how close can irrational numbers in the Cantor set be approximated by rational numbers? We recall that the [*irrationality exponent*]{} of an irrational real number $\xi$, denoted by $\mu(\xi)$, is defined as the supremum of the real numbers $\mu$ for which the inequality $$\left\vert \xi - \frac{p}{ q} \right\vert < \frac{1}{q^{\mu}}$$ has infinitely many rational solutions $p/q$. Mahler’s question may thus be interpreted as follows: are there elements in the Cantor set with any prescribed irrationality exponent?
This question was first answered positively by Levesley, Salp and Velani [@LSV] by means of tools from metric number theory. A direct consequence of Shallit’s result is that one can also simply answer Mahler’s question by providing explicit example of numbers in the Cantor set with any prescribed irrationality exponent. We briefly outline how to prove this result and refer the reader to [@Bu08] for more details. Some refinements along the same lines can also be found in [@Bu08].
Let $\tau\geq 2$ be a real number. Note first that the number $$\xi_{\tau} := 2 \sum_{n= 1}^{\infty} \frac{1}{3^{\lfloor \tau^n\rfloor}}$$ clearly belongs to the Cantor set. Furthermore, the partial sums of $\xi_{\tau}$ provide infinitely many good rational approximations which ensure that $\mu(\xi_{\tau})\geq \tau$. When $\tau\geq (3+\sqrt 5)/2$, a classical approach based on triangles inequalities allows to show that $\mu(\xi_{\tau})\leq \tau$. However, the method fails when $\tau$ satisfies $2\leq \tau<(3+\sqrt 5)/2$.
In order to overcome this difficulty, we can use repeatedly Theorem Sh2 with $b=3$ and $c_n=\lfloor \tau^n\rfloor$ to obtain the continued fraction expansion of $\xi_{\tau}/2$. Set $\xi_{\tau}/2:=[0,b_1,b_2,\ldots]$ and let $s_n$ denote the denominator of the $n$th convergent to $\xi_{\tau}/2$. If $\tau=2$, we see that the partial quotients $b_n$ are bounded, which implies $\mu(\xi_2)=\mu(\xi_2/2)=2$, as desired. We can thus assume that $\tau >2$. Let us recall that once we know the continued fraction of an irrational number $\xi$, it becomes easy to deduce its irrationality exponent. Indeed, if $\xi=[a_0,a_1,\ldots]$, it is well-known that $$\label{eq: mes}
\mu(\xi) = 2 + \limsup_{n\to\infty} \frac{\ln a_{n+1}}{\ln q_n} ,$$ where $p_n/q_n$ denotes the $n$th convergent to $\xi$. Equality (\[eq: mes\]) is actually a direct consequence of the inequality $$\frac{1}{(2+a_{n+1})q_n^2}<\left\vert \xi - \frac{p_n}{q_n} \right\vert < \frac{1}{a_{n+1}q_n^2 }$$ and the fact that the convergents provide the best rational approximations (see, for instance, [@Kh Chapter 6]). When $\tau >2$, the formula given in Theorem Sh2 shows that the large partial quotients[^5] of $\xi_{\tau}/2$ are precisely those equal to $3^{d_n}-1$ which occur first at some positions, say $r_n+1$. But then Theorem Sh2 implies that $s_{r_n}$ is the denominator of $\sum_{k=1}^n1/3^{\lfloor \tau^k\rfloor}$, that is $s_{r_n}= 3^{\lfloor \tau^n\rfloor}$. A simple computation thus shows that $$\limsup_{n\to\infty} \frac{\ln b_{n+1}}{\ln s_n} = \limsup_{n\to\infty} \frac{\ln (3^{d_n}-1)}{\ln 3^{\lfloor \tau^n\rfloor}}= \tau-2 ,$$ since $d_n=\lfloor 3^{\tau^{n+1}}\rfloor - 2 \lfloor 3^{\tau^n}\rfloor$. Then we infer from Equality (\[eq: mes\]) that $\mu(\xi_{\tau})= \mu(\xi_{\tau}/2)= \tau$, as desired.
### The failure of Roth’s theorem in positive characteristic
We consider now Diophantine approximation in positive characteristic. Let $\mathbb F_p((1/t))$ be the field of Laurent series with coefficients in the finite field $\mathbb F_p$, endowed with the natural absolute value $\vert \cdot \vert$ defined at the end of Section \[section: mahler\]. In this setting, the approximation of real numbers by rationals is naturally replaced by the approximation of Laurent series by rational functions. In analogy with the real case, we define the irrationality exponent of $f(t) \in\mathbb F_p((1/t))$, denoted by $\mu(f)$, as the supremum of the real number $\mu$ for which the inequality $$\left\vert f(t) - \frac{P(t)}{Q(t)} \right\vert < \frac{1}{\deg Q^{\mu}}$$ has infinitely many rational solutions $P(t)/Q(t)$.
It is well-known that Roth’s theorem fails in this framework. Indeed, Mahler [@Mah49] remarked that it is even not possible to improve Liouville’s bound for the power series $$f(t): = \sum_{n=0}^{\infty}t^{-p^n} \in\mathbb F_p[[1/t]]$$ is algebraic over $\mathbb F_p(t)$ with degree $p$, while $\mu(f)=p$. Osgood [@Os] and then Lasjaunias and de Mathan [@LaDeM] obtained an improvement of the Liouville bound (namely the Thue bound) for a large class of algebraic functions. However not much is known about the irrationality exponent of algebraic functions in $\mathbb F_p((1/t))$. For instance, it seems that we do not know whether $\mu(f)=2$ for almost every[^6] algebraic Laurent series in $\mathbb F_p((1/t))$. We also do not know what the set $\mathcal E$ of possible values taken by $\mu(f)$ is precisely when $f$ runs over the algebraic Laurent series.
In this direction, we mention that it is possible to use an analogue of Theorem Sh2 for power series with coefficients in a finite field (the proof of which is identical). Thakur [@Th99] uses such a result in order to exhibit explicit power series $f(t)\in\mathbb F_p[[1/t]]$ with any prescribed irrationality measure $\nu\geq 2$, with $\nu$ rational. In other words, this proves that $\mathbb Q_{\geq 2}\subset \mathcal E$, where $\mathbb Q_{\geq 2}:=\mathbb Q\cap [2,+\infty)$. These power series are defined as linear combinations of Mahler-type series which shows that they are algebraic, while the analogue of Theorem Sh2 allows us to describe their continued fraction expansion and thus to easily compute the value of $\mu(f)$, as previously. Note that this result can also be obtained by considering only continued fractions, as shown independently by Thakur [@Th99] and Schmidt [@Schmidt00]. It is expected, but not yet proved, that $\mathcal E=\mathbb Q_{\geq 2}$. For a recent survey about these questions, we refer the reader to [@Th09].
Approximation by quadratic numbers {#sec: cf1}
==================================
A famous consequence of the subspace theorem provides a natural analogue of Roth’s theorem in which rational approximations are replaced by quadratic ones. More precisely, if $\xi$ is an algebraic number of degree at least $3$ and $\varepsilon$ is a positive real number, then the inequality $$\label{eq: sch}
\left\vert \xi - \alpha\right\vert < \frac{1}{H(\alpha)^{3+\varepsilon}} ,$$ has only finitely many quadratic solutions $\alpha$. Here $H(\alpha)$ denotes the (naive) height of $\alpha$, that is, the maximum of the modulus of the coefficients of its minimal polynomial.
In this section, we give our fourth proof of transcendence for $\kappa$ which is obtained as a consequence of Theorem Sh1 (see Section \[section: folding\]) and Theorem AB2 stated below. We observe that some repetitive patterns occur in the continued fraction expansion of $2\kappa$ and then we use them to find infinitely many good quadratic approximations $\alpha_n$ to $2\kappa$. However, a more careful analysis would show that $$\left\vert 2\kappa - \alpha_n \right\vert \gg \frac{1}{H(\alpha_n)^3} ,$$ so that we cannot directly apply (\[eq: sch\]). Fortunately, the subspace theorem offers a lot of freedom and adding some information about the minimal polynomial of our approximations finally allows us to conclude.
We keep the notation from Sections \[section: roth\] and \[section: folding\]. Let ${\bf a}=a_1a_2\cdots$ be an infinite word and $w\geq 1$ be a real number. We say that ${\bf a}$ satisfies Condition $(*)_w$ if there exists a sequence of finite words $(V_n)_{n \ge 1}$ such that the following hold.
- For any $n \ge 1$, the word $V_n^w$ is a prefix of the word ${\bf a}$.
- The sequence $(\vert V_n\vert)_{n \ge 1}$ is increasing.
The following result is a special instance of Theorem 1 in [@AdBuActa].
Let $(a_n)_{n \ge 1}$ be a bounded sequence of positive integers such that ${\bf a}=a_1a_2\cdots$ satisfies Condition $(*)_w$ for some real number $w>1$. Then the real number $$\xi:= [0, a_1, a_2, \ldots]$$ is either quadratic or transcendental.
We only outline the main idea of the proof and refer the reader to [@AdBuActa] for more details.
Assume that the parameter $w > 1$ is fixed, as well as the sequence $(V_n)_{n \ge 1}$ occurring in the definition of Condition $(*)_w$. Set also $s_n:=\vert V_n\vert$. We want to prove that the real number $$\xi:= [0, a_1, a_2, \ldots]$$ is either quadratic or transcendental. We assume that $\alpha$ is algebraic of degree at least three and we aim at deriving a contradiction.
Let $p_n/q_n$ denote the $n$th convergent to $\xi$. The key fact for the proof is the observation that $\xi$ has infinitely many good quadratic approximations obtained by truncating its continued fraction expansion and completing by periodicity. Let $n$ be a positive integer and let us define the quadratic number $\alpha_n$ as having the following purely periodic continued fraction expansion: $$\alpha_n:= [0, V_nV_nV_n \cdots ] .$$ Then $$\left\vert \xi -\alpha_n\right\vert < \frac{1}{q_{\lfloor ws_n\rfloor^2}} ,$$ since by assumption the first $\lfloor ws_n\rfloor$ partial quotients of $\xi$ and $\alpha_n$ are the same. Now observe that $\alpha_n$ is a root of the quadratic polynomial $$P_n (X) := q_{s_n-1} X^2 + (q_{s_n} - p_{s_n-1}) X - p_{s_n} .$$ By Rolle’s theorem, we have $$\vert P_n (\xi)\vert = \vert P_n (\xi) - P_n (\alpha_n)\vert
\ll q_{s_n} |\xi - \alpha_n| \ll \frac{q_{s_n} }{ q_{\lfloor w s_n\rfloor}^2} \cdot$$ Furthermore, by the theory of continued fractions we also have $$\vert q_{s_n} \xi - p_{s_n}\vert \le \frac{1}{q_{s_n}} \cdot$$ Consider the four linearly independent linear forms with real algebraic coefficients: $$\begin{array}{lll}
L_1(X_1, X_2, X_3, X_4)& = & \xi^2 X_2 + \xi (X_1 - X_4) - X_3, \\
L_2(X_1, X_2, X_3, X_4) &= & \xi X_1 - X_3, \\
L_3(X_1, X_2, X_3, X_4) &= & X_1, \\
L_4(X_1, X_2, X_3, X_4) &= & X_2.
\end{array}$$ Evaluating them on the integer quadruple $(q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})$, a simple computation using continuants shows that $$\prod_{1 \le j \le 4} |L_j (q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})|
\ll q_{s_n}^2 q_{\lfloor w s_n\rfloor}^{-2} < \frac{1}{q_{s_n}^{\varepsilon}} ,$$ for some positive number $\varepsilon$, when $n$ is large enough. We can thus apply the subspace theorem. We obtain that all the integer points $(q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})$, $n\in\mathbb N$, belong to a finite number of proper subspaces of $\mathbb Q^4$. After some work, it can be shown that this is possible only if $\xi$ is quadratic, a contradiction.
We are now ready to give a new proof of transcendence for $\kappa$.
Shallit [@Sh1] proves that the continued fraction of $\kappa$ is not ultimately periodic. Here we include, for the sake of completeness, a similar proof that the word $A_{\infty}$ is not ultimately periodic. Set $A_{\infty}:=a_1a_2\cdots$. We argue by contradiction assuming that $A_{\infty}$ is ultimately periodic. There thus exist two positive integers $r$ and $n_0$ such that $$\label{eq: per}
a_{n+jr} = a_{n} ,$$ for every $n\geq n_0$ and $j\geq 1$.
For every positive integer $i$, set $k_i:= \vert A_i\vert = 10\cdot 2^{i-1}$. Let us fix a positive integer $n$ such that $k_n \geq r+ n_0$. Theorem Sh1 implies that $A_{\infty}$ begins with $$A_{n+1} = a_1\cdots a_{k_n-2}a_{k_n-1}a_{k_n} 12 a_{k_n-2}\cdots a_1$$ and since $A_n$ always ends with $11$ (see (\[eq: 11\])), we obtain that $$\label{eq: 1112}
A_{n+1}=a_1\cdots a_{k_n-2} 11 12 a_{k_n-2}\cdots a_1 .$$ We thus have $$\label{eq: sym}
a_{k_n+x+1} = a_{k_n-x} ,$$ for every integer $x$ with $2\leq x \leq k_n-1$. Since the word $1112$ occurs infinitely often in $A_{\infty}$, we see that $r\geq 4$. We can thus choose $x=r-2$ in (\[eq: sym\]), which gives that $$a_{k_n+r-1} = a_{k_n-r+2} .$$ Then (\[eq: per\]) implies that $$a_{k_n+r-1} = a_{k_n-1}$$ and $$a_{k_n-r+2} =a_{k_n +2} .$$ Finally, we get that $$a_{k_n-1} = a_{k_n +2} ,$$ that is $$1=2 ,$$ a contradiction. Hence, $A_{\infty}$ is not ultimately periodic.
Now, set $$\xi := \frac{4\kappa-3}{2-2\kappa} \cdot$$ Clearly it is enough to prove that $\xi$ is transcendental. By Theorem Sh1, we have that $$2\kappa =[1,A_{\infty}] = [1,a_1,a_2,\ldots] $$ and a simple computation shows that $$\xi = [0,a_3,a_4,\cdots] .$$ Let us show that the infinite word $a_3a_3\cdots$ satisfies Condition $(*)_{1+3/10}$. Let $n$ be a positive integer. Using the definition of $A_{n+2}$, we infer from (\[eq: 1112\]) that $$A_{n+2} = a_1\cdots a_{k_n-2} 1112 a_{k_n-2}\cdots a_1 12 a_3a_4\cdots a_{k_n-2} 2 111a_{k_n-2}\cdots a_1$$ Setting $V_n := a_3a_4\cdots a_{k_n-2} 1112 a_{k_n-2}\cdots a_112$, we obtain that $A_{n+2}$ begins with $$a_1a_2 V_n^{1+ (k_n-4)/2k_n} .$$ Since $k_n\geq 10$, we thus deduce that $A_{\infty}$ begins with $a_1a_2V_n^{1+3/10}$, for every integer $n\geq 1$. This shows that the infinite word $a_3a_4\cdots$ satisfies Condition $(*)_{1+3/10}$. Applying Theorem AB2, we obtain that $\xi$ is either quadratic or transcendental. However, we infer from Lagrange’s theorem that $\xi$ cannot be quadratic, for we just have shown that the word $a_3a_4\cdots$ is not ultimately periodic. Thus $\xi$ is transcendental, concluding the proof.
Beyond this proof
-----------------
Very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. From numerical evidence and a belief that these numbers behave like most of the numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. Apparently, Khintchine [@Kh] was the first to consider this question (see [@All00; @Sha; @MIW] for surveys including a discussion of this problem). Although almost nothing has been proved yet in this direction, Lang [@LangSMF; @Lang] made some more general speculations, including the fact that algebraic numbers of degree at least three should behave like most of the numbers with respect to the Gauss–Khintchine–Kuzmin–Lévy laws. This conjectural picture is thus very similar to the one encountered in Section \[section: roth\] regarding the expansion of algebraic irrational numbers in integer bases.
As a first step, it is worth proving that real numbers with a “too simple" continued fraction expansion are either quadratic or transcendental. Of course, the term “simple" can lead to many interpretations. It may, for instance, denote real numbers whose continued fraction expansion can be produced in a simple algorithmical, dynamical, combinatorial or arithmetical way. In any case, it is reasonable to expect that such expansions should be close to be periodic. In this direction, there is a long tradition of using an excess of periodicity to prove the transcendence of some continued fractions (see, for instance, [@ADQZ; @Baker62; @Bax; @Dav1; @Dav2; @Dav3; @Maillet; @Quef1; @Quef2]). Adamczewski and Bugeaud [@AdBuActa] use the freedom offered by the subspace theorem to prove some combinatorial transcendence criteria, which provide significant improvements of those previously obtained in [@ADQZ; @Dav1; @Dav2; @Quef1; @Quef2]. Some transcendence measures for such continued fractions are also recently given in [@AdBuJEMS; @Bu3], following the general approach developed in [@AdBuPLMS] (see also [@AdBuCrelle] for similar results related to integer base expansions).
Theorem AB2 gives the transcendence of non-quadratic real numbers $\xi$ whose continued fraction expansion begins with arbitrarily long blocks of partial quotients of the form $V_n^{1+\varepsilon}$, for some positive $\varepsilon$. The key fact in the proof to apply the subspace theorem is to see that the linear form $$\xi^2 X_2 + \xi (X_1 - X_4) - X_3$$ takes small values at the integer quadruples $(q_{s_n}, q_{s_n-1}, p_{s_n}, p_{s_n-1})$, where $s_n=\vert V_n\vert$ and where $p_n/q_n$ denotes the $n$th convergent to $\xi$. This idea can be naturally generalized to the important case where the repetitive patterns do not occur at the very beginning of the expansion. Indeed, if the continued fraction expansion of $\xi$ begins with a block of partial quotients of the form $U_nV_n^{1+\varepsilon}$, then $\xi$ is well approximated by the quadratic number $\alpha'_n:=[0,U_nV_nV_nV_n\cdots]$. As shown in [@AdBuActa], when dealing with these more general patterns, we can argue similarly, but we have now to estimate the linear form $$L(X_1,X_2,X_3,X_4):= \xi^2X_1-\xi(X_2+X_3)+X_4$$ at the integer quadruples $$\begin{array}{c}
{\bf x}_n:= (q_{r_n-1}q_{r_n+s_n}-q_{r_n}q_{r_n+s_n}, q_{r_n-1}p_{r_n+s_n}-q_{r_n}p_{r_n+s_n-1}, \hspace{3cm}\\
\hspace{2cm}p_{r_n-1}q_{r_n+s_n}-p_{r_n}q_{r_n+s_n-1}, p_{r_n-1}p_{r_n+s_n}-p_{r_n}p_{r_n+s_n-1}) ,
\end{array}$$ where $r_n:=\vert U_n\vert$, $s_n:=\vert V_n\vert$, and $p_n/q_n$ denotes the $n$th convergent to $\xi$. In a recent paper, Bugeaud [@Bu12] remarks that the quantity $\vert L({\bf x}_n)\vert$ was overestimated in the proof given in [@AdBuActa][^7]. Taking into account this observation, the method developed in [@AdBuActa] has much striking consequences than those initially announced. As an illustration, we now have that the continued fraction expansion of an algebraic number of degree at least $3$ cannot be generated by a finite automaton.
Simultaneous approximation by rational numbers {#sec: cf2}
==============================================
Another classical feature of the subspace theorem is that it can also deal with simultaneous approximation of several real numbers by rationals. Our last proof of the transcendence of $\kappa$ relies on this principle. We use the occurrences of some symmetric patterns in the continued fraction expansion of $2\kappa-1$ to find good simultaneous rational approximations of $2\kappa-1$ and $(2\kappa-1)^2$.
We keep the notation of Sections \[section: folding\] and \[sec: cf1\]. Set $\xi:=(2\kappa-1)$. Clearly it is enough to prove that $\xi$ is transcendental. We argue by contradiction assuming that $\xi$ is algebraic.
By Theorem Sh1, we have that $\xi=[0,A_{\infty}]= [0,a_1,a_2,\ldots]$. We let $p_n/q_n$ denote the $n$th convergent to $\xi$. We also set as previously $k_n:=\vert A_n\vert$. By the theory of continued fraction, we have $$\left\vert \xi - \frac{ p_{k_n-1} }{ q_{k_n-1} } \right\vert < \frac{1}{q_{k_n-1}^2}
\mbox{ and }
\left\vert \xi - \frac{ p_{k_n} }{ q_{k_n} } \right\vert < \frac{1}{q_{k_n}^2} \cdot$$ Let us also recall the so-called mirror formula (see, for instance, [@AdAl]): $$\label{eq: mirror}
\frac{q_{n-1}}{q_{n}} =[0,a_n,\ldots,a_1] .$$ Since $A_{n} = B_{n-1} 1112 (B_{n-1})^R$, we have $$\frac{q_{k_n-1}}{q_{k_n}} =[0,B_{n-1},2,1,1,1, (B_{n-1})^R]$$ and a simple computation using continuants shows that[^8] $$\left\vert \xi - \frac{q_{k_n-1}}{q_{k_n}} \right\vert \ll \frac{1}{q_{k_n}}
\mbox{ and }
\left\vert \frac{p_{k_n}}{q_{k_n}} - \frac{q_{k_n-1}}{q_{k_n}} \right\vert \ll \frac{1}{q_{k_n}} \cdot$$ Then we have $$\begin{array}{lll}
\left\vert \xi^2 - \frac{ p_{k_n-1} }{ q_{k_n} } \right\vert &=&
\left\vert \xi^2 - \frac{ p_{k_n-1} }{ q_{k_n-1} } \cdot \frac{ q_{k_n-1} }{ q_{k_n} } \right\vert \\ \\
&=& \left\vert \left (\xi-\frac{ p_{k_n-1} }{ q_{k_n-1} }\right)\left(\xi+\frac{ q_{k_n-1} }{ q_{k_n} } \right)
+ \xi\left(\frac{ p_{k_n-1} }{ q_{k_n-1} }- \frac{ q_{k_n-1} }{ q_{k_n} } \right)\right\vert \\ \\
& \ll& q_{k_n}^{-1} \cdot
\end{array}$$
Consider the four linearly independent linear forms with real algebraic coefficients: $$\begin{array}{lll}
L_1(X_1, X_2, X_3, X_4)& = & \xi^2 X_1 - X_4, \\
L_2(X_1, X_2, X_3, X_4) &= & \xi X_1 - X_3, \\
L_3(X_1, X_2, X_3, X_4) &= & \xi X_2-X_4, \\
L_4(X_1, X_2, X_3, X_4) &= & X_2.
\end{array}$$ Evaluating them on the integer quadruple $(q_{k_n}, q_{k_n-1}, p_{k_n}, p_{k_n-1})$, our previous estimates implies that $$\prod_{1 \le j \le 4} \left\vert L_j (q_{k_n}, q_{k_n-1}, p_{k_n}, p_{k_n-1})\right\vert
\ll \frac{1}{q_{k_n}} \cdot$$ The subspace theorem thus implies that all the integer points $(q_{k_n}, q_{k_n-1}, p_{k_n}, p_{k_n-1})$, $n\geq 1$, belong to a finite number of proper subspaces of $\mathbb Q^4$. Thus, there exists a nonzero integer quadruple $(x,y,z,t)$ such that $$q_{k_n}x+ q_{k_n-1}y+ p_{k_n}z+ p_{k_n-1}t = 0 ,$$ for all $n$ in an infinite set of positive integers $\mathcal N$. Dividing by $q_{k_n}$ and letting $n$ tend to infinity along $\mathcal N$, we obtain that $$x+ \xi(y+z) + \xi^2 t =0 .$$ Since $A_{\infty}$ is not ultimately periodic (see Section \[sec: cf1\]), it follows from Lagrange’s theorem that $\xi$ is not quadratic and thus $x=t=(y+z)=0$. This gives that $q_{k_n-1} = p_{k_n}$ for every $n\in\mathcal N$. But $$\frac{q_{k_n-1} }{q_{k_n}} = [0,B_n,2,1,1,1, (B_n)^R]
\not= [0,B_n,1,1,1,2, (B_n)^R] = \frac{p_{k_n} }{q_{k_n}} ,$$ a contradiction.
Beyond this proof
-----------------
As already mentioned in Section \[sec: cf1\], there is a long tradition in using an excess of periodicity to prove the transcendence of some continued fractions. The fact that occurrences of symmetric patterns can actually give rise to transcendence statements is more surprising. The connection between palindromes[^9] in continued fractions and simultaneous approximation of a number and its square is reminiscent of works of Roy about extremal numbers [@Roy03bis; @Roy03; @Roy04] (see also [@Fi]). Inspired by this original discovery of Roy, Adamczewski and Bugeaud [@AdBuFourier] use the subspace theorem and the mirror formula (\[eq: mirror\]) to establish several combinatorial transcendence criteria for continued fractions involving symmetric patterns (see also [@Bu12] for a recent improvement of one of these criteria). The same authors [@AdBuJEMS] also give some transcendence measures for such continued fractions. As a simple illustration, they prove in [@AdBuFourier] that if the continued fraction of a real number begins with arbitrarily large palindromes, then this number is either quadratic or transcendental.
It is amusing to mention that in contrast there are only very partial results about the transcendence of numbers whose decimal expansion involves symmetric patterns (see [@AdBuIMRN]). In particular, it is not known whether a real number whose decimal expansion begins with arbitrarily large palindromes is either rational or transcendental.
Beyond the study of extremal numbers and transcendence results, Adamczewski and Bugeaud [@AdBuJLMS] use continued fractions with symmetric patterns to provide explicit examples for the famous Littlewood conjecture on simultaneous Diophantine approximation.
Acknowledgments
===============
This work was supported by the project Hamot, ANR 2010 BLAN-0115-01. The idea to make this survey came up with a talk I gave for the conference *Diophantine Analysis and Related Fields* that was held in Tokyo in March 2009. I would like to take the opportunity to thank the organizers of this conference and especially Noriko Hirata-Kohno. I would also like to thank the anonymous referees for their useful comments. My interest for transcendence grew quickly after I followed some lectures by Jean-Paul Allouche when I was a Ph.D. student. I am deeply indebted to Jean-Paul for giving such stimulating lectures!
[99]{}
B. Adamczewski, Transcendance “à la Liouville" de certains nombres réels, [*C. R. Acad. Sci. Paris*]{} [**338**]{} (2004), 511–514.
B. Adamczewski, On the expansion of some exponential periods in an integer base, [*Math. Ann.*]{} [**346**]{} (2010), 107–116.
B. Adamczewski and J.-P. Allouche, Reversals and palindromes in continued fractions, [*Theoret. Comput. Sci.*]{} [**320**]{} (2007), 220–237.
B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers, II. continued fractions, [*Acta Math.*]{} [**195**]{} (2005), 1–20.
B. Adamczewski and Y. Bugeaud, On the Littlewood conjecture in simultaneous Diophantine approximation, [*J. London Math. Soc.*]{} [**73**]{} (2006), 355–366.
B. Adamczewski and Y. Bugeaud, Real and $p$-adic expansions involving symmetric patterns, [*Int. Math. Res. Notes*]{}, Volume 2006 (2006), Article ID 75968, 17 pages.
B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers I. Expansions in integer bases, [*Ann. of Math. (2)*]{} [**165**]{} (2007), 547–565.
B. Adamczewski and Y. Bugeaud, On the Maillet-Baker continued fractions, [*J. Reine Angew. Math.*]{} [**606**]{} (2007), 105–121.
B. Adamczewski and Y. Bugeaud, Palindromic continued fractions, [*Ann. Inst. Fourier*]{} [**57**]{} (2007), 1557–1574.
B. Adamczewski and Y. Bugeaud, Diophantine approximation and transcendence, in [*Combinatorics, Automata and Number Theory*]{}, Encyclopedia Math. Appl., Vol. 135, Cambridge, 2010, pp. 424–465.
B. Adamczewski et Y. Bugeaud, Mesures de transcendance et aspects quantitatifs de la méthode de Thue-Siegel-Roth-Schmidt, [*Proc. London Math. Soc.*]{} [**101**]{} (2010), 1–31.
B. Adamczewski and Y. Bugeaud, Trancendence measure for continued fractions involving repetitive or symmetric patterns, [*J. Eur. Math. Soc.*]{} [**12**]{} (2010), 883–914.
B. Adamczewski et Y. Bugeaud, Nombres réels de complexité sous-linéaire : mesures d’irrationalité et de transcendance, [*J. Reine Angew. Math.*]{} [**658**]{} (2011), 65–98.
B. Adamczewski et C. Faverjon, Chiffres non nuls dans le développement en base entière des nombres algébriques irrationnels, [*C. R. Acad. Sci. Paris*]{} [**350**]{} (2012), 1–4.
J.-P. Allouche, Sur la transcendance de la série formelle $\Pi$, [*J. Théor. Nombres Bordeaux*]{} [**2**]{} (1990), 103–117.
J.-P. Allouche, Nouveaux résultats de transcendance de réels à développement non aléatoire, [*Gaz. Math.*]{} [**84**]{} (2000) 19–34.
J.-P. Allouche, J. L. Davison, M. Queffélec, and L. Q. Zamboni, Transcendence of Sturmian or morphic continued fractions, [*J. Number Theory*]{} [**91**]{} (2001) 39–66.
J.-P. Allouche and J. Shallit, [*Automatic Sequences: Theory, Applications, Generalizations*]{}, Cambridge, 2003.
D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance, On the binary expansions of algebraic numbers, [*J. Théor. Nombres Bordeaux*]{} [**16**]{} (2004), 487–518.
A. Baker, Continued fractions of transcendental numbers, [*Mathematika*]{} [**9**]{} (1962) 1–8.
C. Baxa, Extremal values of continuants and transcendence of certain continued fractions, [*Adv. in Appl. Math.*]{} [**32**]{} (2004), 754–790.
Yu. Bilu, The many faces of the Subspace Theorem \[after Adamczewski, Bugeaud, Corvaja, Zannier$\ldots$\], [*Astérisque*]{} [**317**]{} (2008), 1–38. Séminaire Bourbaki. Vol. 2006/2007, Exp. No. 967.
É. Borel, Les probabilités dénombrables et leurs applications arithmétiques, [*Rend. Circ. Mat. Palermo*]{} [**27**]{} (1909), 247–271.
É. Borel, Sur les chiffres décimaux de $\sqrt{2}$ et divers problèmes de probabilités en chaîne, [*C. R. Acad. Sci. Paris*]{} [**230**]{} (1950), 591–593.
Y. Bugeaud, Diophantine approximation and Cantor sets, [*Math. Ann.*]{} [**341**]{} (2008), 677–684.
Y. Bugeaud, Continued fractions with low complexity: Transcendence measures and quadratic approximation, [*Compos. Math.* ]{} [**148**]{} (2012), 718–750.
Y. Bugeaud, Automatic continued fractions are transcendental or quadratic, [*Ann. Sci. Éc. Norm. Supér.*]{}, to appear.
P. Corvaja and U. Zannier, Some new applications of the subspace theorem, [*Compos. Math.*]{} [**131**]{} (2002), 319–340.
J. L. Davison, A class of transcendental numbers with bounded partial quotients, in [*Number theory and applications (Banff, AB, 1988)*]{}: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 265, Kluwer Acad. Publ., 1989, pp. 365–371.
J. L. Davison, Continued fractions with bounded partial quotients, [*Proc. Edinburgh Math. Soc.*]{} [**45**]{} (2002), 653–671.
J. L. Davison, Quasi-periodic continued fractions, [*J. Number Theory*]{} [**127**]{} (2007), 272–282.
L. Denis, Indépendance algébrique de différents $\pi$, [*C. R. Acad. Sci. Paris*]{} [**327**]{} (1998), 711–714.
S. Ferenczi and Ch. Mauduit, Transcendence of numbers with a low complexity expansion, [*J. Number Theory*]{} [**67**]{} (1997), 146–161.
S. Fischler, Palindromic prefixes and episturmian words, [*J. Combin. Theory Ser. A*]{} [**113**]{} (2006), 1281–1304.
G. H. Hardy and E. M. Wright, [*An introduction to the theory of numbers*]{}, Oxford, sixth edition, 2008, Revised by D. R. Heath-Brown and J. H. Silverman.
A. J. Kempner, On Transcendental Numbers, [*Trans. Amer. Math. Soc.*]{} [**17**]{} (1916), 476–482.
A. Y. Khintchine, [*Continued fractions*]{}, 2nd edition, Moscow–Leningrad, 1949 (Russian); English translation: Chicago, 1964.
M. Kmošek, Rozwiniecie niektórych liczb niewymiernych na u[ł]{}amki [ł]{}ańcuchowe, Master thesis, Warsaw, 1979.
M. J. Knight, An “oceans of zeros” proof that a certain non-Liouville number is transcendental, [*Amer. Math. Monthly*]{} [**98**]{} (1991), 947–949.
G. Köhler, Some more predictable continued fractions, [*Monatsh. Math.*]{} [**89**]{} (1980), 95–100.
S. Lang, Report on Diophantine approximations, [*Bull. Soc. Math. France*]{} [**93**]{} (1965), 177–192.
S. Lang, [*Introduction to Diophantine Approximations*]{}, 2nd edition, Springer, 1995.
A. Lasjaunias and B. de Mathan, Thue’s theorem in positive characteristic, [*J. Reine Angew. Math.*]{} [**473**]{} (1996), 195–206.
J. Levesley, C. Salp, and S. Velani, On a problem of K. Mahler: Diophantine approximation and Cantor sets, [*Math. Ann.*]{} [**338**]{} (2007), 97–118.
J. Liouville, Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationelles algébriques, [*C. R. Acad. Sci. Paris*]{} [**18**]{} (1844), 883–885; 910-911.
K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, [*Math. Ann.*]{} [**101**]{} (1929), 342–367.
K. Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendente Funktionen, [*Math. Z.*]{} [**32**]{} (1930), 545–585.
K. Mahler, Über das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen, [*Math. Ann.*]{} [**103**]{} (1930), 573–587.
K. Mahler, On a theorem of Liouville in fields of positive characteristic, [*Can. J. Math.*]{} [**1**]{} (1949), 397–400.
K. Mahler, Some suggestions for further research, [*Bull. Austral. Math. Soc.*]{} [**29**]{} (1984), 101–108.
E. Maillet, [*Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions*]{}, Gauthier-Villars, 1906.
B. de Mathan, Irrationality measures and transcendence in positive characteristic, [*J. Number Theory* ]{} [**54**]{} (1995), 93–112.
M. Mendès France, Sur les fractions continues limitées, [*Acta Arith.*]{} [**23**]{} (1973), 207–215.
M. Morse and G. A. Hedlund, Symbolic dynamics, [*Amer. J. Math.*]{} [**60**]{} (1938), 815–866.
Ku. Nishioka, [*Mahler Functions and Transcendence*]{}, Lect. Notes in Mathematics, Vol. 1631, Springer, 1997.
C. Osgood, Effective bounds on the “diophantine approximation” of algebraic functions over fields of arbitrary characteristic and applications to differential equations, [*Indag. Math.*]{} [**37**]{} (1975), 105–119.
F. Pellarin, Aspects de l’indépendance algébrique en caractéristique non nulle, [*Astérisque*]{} [**317**]{} (2008) 205–242 (2008). Séminaire Bourbaki. Vol. 2006/2007, Exp. No. 973.
F. Pellarin, An introduction to Mahler’s method for transcendence and algebraic independence, to appear in [*EMS proceedings of the conference ‘Hodge structures, transcendence and other motivic aspects"*]{}.
O. Perron, [*Die Lehre von den Kettenbrüchen*]{}, Teubner, 1929.
M. Queffélec, Transcendance des fractions continues de Thue–Morse, [*J. Number Theory*]{} [**73**]{} (1998), 201–211.
M. Queffélec, Irrational number with automaton-generated continued fraction expansion, in J.-M. Gambaudo, P. Hubert, P. Tisseur, and S. Vaienti, eds., [*Dynamical Systems: From Crystal to Chaos*]{}, World Scientific, 2000, pp. 190–198.
D. Ridout, Rational approximations to algebraic numbers, [*Mathematika*]{} [**4**]{} (1957), 125–131.
, Rational approximations to algebraic numbers, [*Mathematika*]{} [**2**]{} (1955), 1–20; corrigendum, 169.
D. Roy, Approximation simultannée d’un nombre et de son carré, [*C. R. Acad. Sci. Paris*]{} [**336**]{} (2003), 1–6.
D. Roy, Approximation to real numbers by cubic algebraic integers, II, [*Ann. of Math. (2)*]{} [**158**]{} (2003), 1081–1087.
D. Roy, Approximation to real numbers by cubic algebraic integers, I, [*Proc. London Math. Soc.*]{} [**88**]{} (2004), 42–62.
H. P. Schlickewei, The ${\mathfrak p}$-adic Thue-Siegel-Roth-Schmidt theorem, [*Arch. Math. (Basel)*]{} [**29**]{} (1977), 267–270.
W. M. Schmidt, [*Diophantine Approximation*]{}, Lect. Notes in Math., Vol. 785, Springer, 1980.
W. M. Schmidt, On continued fractions and diophantine approximation in power series fields, [*Acta Arith.*]{} [**95**]{} (2000), 139–166.
W. T. Scott and H. S. Wall, Continued fraction expansions for arbitrary power series [*Ann. of Math. (2)*]{} [**41**]{} (1940), 328–349.
J. Shallit, Simple continued fractions for some irrational numbers, [*J. Number Theory*]{} [**11**]{} (1979) 209–217.
J. Shallit, Simple continued fractions for some irrational numbers, II, [*J. Number Theory*]{} [**14**]{} (1982) 228–231.
J. Shallit, Real numbers with bounded partial quotients: a survey, [*Enseign. Math.*]{} [**38**]{} (1992) 151–187.
D. Thakur, Diophantine approximation exponents and continued fractions for algebraic power series, [*J. Number Theory*]{} [**79**]{} (1999), 284–291.
D. Thakur, Approximation exponents in function fields, in W. Chen, T. Gowers, H. Halberstam, W. Schmidt, and R. Vaughn, eds., [*Analytic Number Theory, Essays in Honor of Klaus Roth*]{}, Cambridge, 2009, pp. 421–435.
L. I. Wade, Certain quantities transcendental over $GF(p^n,x)$, [*Duke Math. J.*]{} [**8**]{} (1941), 701–720.
M. Waldschmidt, Un demi-siècle de transcendance, in [*Development of Mathematics 1950–2000*]{}, Birkhäuser, 2000, pp. 1121–1186.
M. Waldschmidt, [*Diophantine Approximation on Linear Algebraic Groups*]{}, Grundlehren der Mathematischen Wissenschaften, Vol. 326, Springer, 2000.
M. Waldschmidt, Words and transcendence, in W. Chen, T. Gowers, H. Halberstam, W. Schmidt, and R. Vaughn, eds., [*Analytic Number Theory, Essays in Honor of Klaus Roth*]{}, Cambridge, 2009, pp. 449–470.
J. Yu, Transcendence and special zeta values in characteristic $p$, [*Ann. of Math. (2)*]{} [**134**]{} (1991), 1–23.
[^1]: The number $\kappa$ is sometimes erroneously called the Fredholm number (see, for instance, the discussion in [@Sha]).
[^2]: Note that this argument could also be replaced by the use of two important results from automata theory: the Cobham and Christol theorems (see [@AS] and also Section \[allouche\] for another use of Christol’s theorem).
[^3]: The folding lemma is an avatar of the so–called mirror formula, another very useful elementary identity for continued fractions, which is the object of the survey [@AdAl]. Many references to work related to these two identities can be found in [@AdAl].
[^4]: Köhler [@Ko] also obtains independently almost the same result after he studied [@Sh1]. It is also worth mentioning that this result was somewhat anticipated, although written in a rather different form, by Scott and Wall in 1940 [@SW].
[^5]: That is those which are larger than all previous ones.
[^6]: This could mean something like, among algebraic Laurent series $f$ of degree and height at most $M$, the proportion of those with $\mu(f)=2$ tends to one as $M$ tends to infinity.
[^7]: The authors of [@AdBuActa] use that $\vert L({\bf x}_n)\vert \ll q_{r_n} q_{r_n+s_n} q_{ r_n \lfloor (1+\varepsilon) s_n \rfloor }^{-2}$ while it is actually elementary to see that $\vert L({\bf x}_n)\vert \ll q_{r_n}^{-1} q_{r_n+s_n} q_{ r_n \lfloor (1+\varepsilon) s_n \rfloor }^{-2}$.
[^8]: Indeed, the theory of continuants implies that the denominator of the rational number $[0,B_{n-1}]$, say $r_n$, grows roughly $\sqrt{q_{k_n}}$ and thus $r_n^{-2}$ essentially behaves like $q_{k_n}^{-1}$. For more details about continuants, see, for instance, [@AdBuCant Sec.8.2], [@AdBuCrelle1 Sec. 5], or [@AdBuJLMS Sec.3].
[^9]: Recall that a palindrome is a word $W=a_1\cdots a_n$ that is equal to its reversal $W^R=a_n\cdots a_1$. Thus a palindrome may be considered as a perfect symmetric pattern.
|
---
abstract: 'The field of the extended TeV source HESS J1804$-$216 was serendipitously observed with the [*Chandra*]{} ACIS detector on 2005 May 4. The data reveal several X-ray sources within the bright part of HESS J1804$-$216. The brightest of these objects, CXOU J180432.4$-$214009, which has been also detected with [*Swift*]{} (2005 November 3) and [*Suzaku*]{} (2006 April 6), is consistent with being a point-like source, with the 0.3–7 keV flux $F_{X}=(1.7\pm0.2)\times10^{-13}$ ergs s$^{-1}$ cm$^{-2}$. Its hard and strongly absorbed spectrum can be fitted by the absorbed power-law model with the best-fit photon index $\Gamma\approx0.45$ and hydrogen column density $n_{\rm H}\approx4\times10^{22}$ cm$^{-2}$, both with large uncertainties due to the strong correlation between these parameters. A search for pulsations resulted in a 106 s period candidate, which however has a low significance of 97.9%. We found no infrared-optical counterparts for this source. The second brightest source, CXOU J180441.9$-$214224, which has been detected with [*Suzaku*]{}, is either extended or multiple, with the flux $F_{X}\sim 1\times 10^{-13}$ ergs cm$^{-2}$ s$^{-1}$. We found a nearby M dwarf within the X-ray source extension, which could contribute a fraction of the observed X-ray flux. The remaining sources are very faint ($F_X <3\times 10^{-14}$ ergs cm$^{-2}$ s$^{-1}$), and at least some of them are likely associated with nearby stars. Although one or both of the two brighter X-ray sources could be faint accreting binaries or remote pulsars with pulsar wind nebulae (hence possible TeV sources), their relation to HESS J1804$-$216 remains elusive. The possibility that HESS J1804$-$216 is powered by the relativistic wind from the young pulsar B1800–21, located at a distance of $\sim 10$ pc from the TeV source, still remains a more plausible option.'
author:
- 'O. Kargaltsev, G. G. Pavlov, and G. P. Garmire'
title: ' The TeV source HESS J1804$-$216 in X-rays and other wavelengths'
---
Introduction
============
Recent observations with the High Energy Stereoscopic System (HESS) and other modern very high energy (VHE) telescopes have revealed a rich population of TeV $\gamma$-ray sources (Aharonian et al. 2005). A significant fraction of these sources are associated with various types of known astrophysical phenomena (see Ong 2006 for a review). The list of Galactic TeV sources with firm associations includes high mass X-ray binaries (HMXBs), supernova remnants (SNRs), and pulsar wind nebulae (PWNe). Extragalactic TeV sources are so far represented only by AGNs (mostly blazars). Many of the newly discovered TeV sources are extended and resolved in the HESS images. Most of the identified extended sources are PWNe and SNRs, although there is an indication that some HMXBs could also produce extended TeV emission (e.g., HESS J1632–478; Aharonian et al. 2006a, hereafter Ah06). Among the known Galactic TeV sources, only HMXBs are variable in TeV, some of them showing variations with the binary orbital period (e.g., the microquasar LS 5039, Aharonian et al. 2006b). The extragalactic AGN sources appear to be point-like at TeV energies and can also be variable.
A quarter of the $\approx 50$ VHE sources known to date[^1] do not have firm identifications, although possible counterparts/associations have been suggested for some of them. HESS J1804–216 (hereafter HESS J1804), the brightest among such sources, has been recently discovered during the HESS Galactic plane scan in 2004 May–October (Aharonian et al. 2005). The “best-fit position” of the source (which is close to, but may be different from, the peak in the TeV brightness distribution; see Ah06 for definition) is R.A.=$18^{\rm h}04^{\rm m}31^{\rm s}$, Decl.=$-21^{\circ}42\farcm0$, with a 13 uncertainty in each of the coordinates. The distribution of the TeV brightness shows an extended source with elongated morphology (see the contours in Fig. 17 of Ah06). The size of the source, $\gtrsim 20'\times 10'$, substantially exceeds the size of the HESS point spread function (PSF), $\approx6'$ for this observation (Ah06). The large extent of the TeV emission rules out its association with an AGN, which means that HESS J1804 is a Galactic source.
Ah06 point out that the TeV emission does not perfectly line up with any known sources in the field. Among possible counterparts, Ah06 mention the young Vela-like pulsar B1800–21 and the SNR G8.7–0.1, both of which have been detected in X-rays (Kargaltsev, Pavlov, & Garmire 2006a and Finley & Ögelman 1994, respectively). Ah06 also do not dismiss the possibility that HESS J1804 and other unidentified TeV sources belong to a new class of objects sometimes dubbed “dark particle accelerators” (Aharonian et al. 2005a) because of the apparent lack of counterparts outside the TeV band.
Following the discovery of HESS J1804, the field was observed in X-rays by the [*Swift*]{} X-ray Telescope (XRT) instrument on 2005 November 3 (Landi et al. 2006) and [*Suzaku*]{} X-ray Imaging Spectrometers (XIS) on 2006 April 6 (Bamba et al. 2006). Landi et al. (2006) detected three X-ray sources in the $23\farcm6\times23\farcm6$ [*Swift*]{} XRT detector field-of-view (FOV), at distances of $13\farcm3$, $7\farcm4$, and $2\farcm0$ (positional uncertainty $\sim5''$–$6''$) from the best-fit HESS position (we will call them Sw1, Sw2, and Sw3 hereafter). Sw1 and Sw2 had been previously detected with [*ROSAT*]{}. Sw1 (= 1RXS J180404.6–215325), the brightest of the 3 sources, shows a very soft thermal-like spectrum ($kT\approx 0.3$ keV for an optically thin thermal bremsstrahlung model), and it is positionally coincident with a bright star outside the extension of the TeV source. The spectra of Sw2 (= 1WGA J1804.0-2142) and Sw3 could not be measured because of the small numbers of counts detected ($22\pm 7$ and $26\pm 6$ counts, respectively, in the 11.6 ks exposure). Sw2 could also be associated with a star close to the boundary of the XRT error circle, while Sw3, closest to the center of HESS J1804, did not show obvious counterparts at other wavelengths.
The subsequent deeper (40 ks) [*Suzaku*]{} XIS observation revealed two distinct X-ray sources (Suzaku J1804–2142 and Suzaku J1804–2140; Su42 and Su40 hereafter) in the $18'\times 18'$ XIS FOV. Su40 is positionally coincident with Sw3 within the large ($\sim 1'$) positional uncertainty of [*Suzaku*]{} XIS. Bamba et al. (2006) found that Su40 is extended (or multiple) while Su42 is unresolved at the [*Suzaku*]{} resolution (half power PSF diameter $\approx 2'$). Spectral fits with a power-law (PL) model show markedly different spectral parameters for the two sources. Su42 was found to be unusually hard (photon index $\Gamma = -0.3^{+0.5}_{-0.5}$, the errors are at the 90% confidence for one interesting parameter) with a moderate (albeit rather uncertain) hydrogen column density, $n_{\rm H,22}\equiv n_{\rm H}/(10^{22}\,{\rm cm}^{-2})
=0.2^{+2.0}_{-0.2}$. Su40 showed a softer ($\Gamma=1.7^{+1.4}_{-1.0}$), strongly absorbed ($n_{\rm H,22}=11^{+10}_{-6}$) spectrum. The sources have comparable fluxes, $\sim2.5$ and $4.3\times 10^{-13}$ ergs cm$^{-2}$ s$^{-1}$ in 2–10 keV, for Su42 and Su40, respectively. Despite an appreciable probability of chance coincidence (obvious from the [*Chandra*]{} images in §2), Bamba et al. (2006) conclude that both [*Suzaku*]{} sources are physically associated with HESS J1804. They mention that the harder Su42 could be an HMXB while the softer Su40 could be either a PWN or, more likely, it could be associated with SNR G8.7–0.1. The authors also point out that the ratios of the $\gamma$-ray flux of HESS J1804 to the X-ray fluxes of Su42 and Su40 are surprisingly high compared to those seen in TeV sources with known associations, including SNRs and PWNe. Thus, the [*Swift*]{} and [*Suzaku*]{} data do not provide a conclusive result on the nature of HESS J1804, and its association with the found X-ray sources remains unclear.
In the course of our observation of PSR B1800–21 and its PWN with the [*Chandra X-ray Observatory*]{}, the most interesting part of the HESS J1804 field happened to be within the FOV. Detailed results of the PWN/PSR B1800–21 study have been presented by Kargaltsev et al. (2006a). In this paper we present the analysis of X-ray sources in the vicinity of HESS J1804, including the two sources detected with [*Suzaku*]{}[^2]. The details of the [*Chandra*]{} observation and the data analysis, supplemented with the analysis of optical-infrared-radio data, are presented in §2. We discuss the nature of the [*Chandra*]{} sources and the likelihood of their association with HESS J1804 in §3, and summarize our findings in §4.
Observations and Data Analysis
==============================
We serendipitously observed the field of HESS J1804 with the Advanced CCD Imaging Spectrometer (ACIS) on board [*Chandra*]{} on 2005 May 4. The total useful scientific exposure time was 30,236 s. The observation was carried out in Faint mode. The aim point was chosen on S3 chip, near the PSR B1800–21 position (see Kargaltsev et al. 2006a). In addition to S3, the S0, S1, S2, I2, and I3 chips were turned on. The detector was operated in Full Frame mode which provided time resolution of 3.24 seconds. The data were reduced using the Chandra Interactive Analysis of Observations (CIAO) software (ver. 3.2.1; CALDB ver. 3.0.3).
Chandra images
--------------
Figure 1 shows the ACIS image of the HESS J1804 field with overlaid TeV contours, extracted from Figure 17 of Ah06. The brightest portion of HESS J1804 falls onto the I3 and I2 chips, its best-fit position is offset by $\approx 11\farcm2$ from the aim point. We searched for possible X-ray counterparts within the HESS J1804 extension and found a relatively bright source, which we designate CXOU J180432.4$-$214009 (hereafter Ch1), located at ${\rm R.A.}=18^{\rm h}04^{\rm
m}32\fs462$, ${\rm decl.}=-21^{\circ}40' 09\farcs91$ (the $1\sigma$ centroid uncertainty is 038 in R.A. and 032 in decl.; the $1\sigma$ error in absolute [*Chandra*]{} astrometry is $\approx 0\farcs4$ for each of the coordinates), well within the brightest portion of HESS J1804 and just $1\farcm9$ north of the best-fit position (Ah06). Although Ch1 appears to be slightly extended in the ACIS image, a PSF simulation shows that this is likely the result of the off-axis location (off-axis angle $\theta=10\farcm3$), which is also responsible for the relatively large centroiding uncertainty quoted above. The position of Ch1 is consistent (within the uncertainties) with that of Sw3 and Su40 (see §1). Therefore, we conclude that Ch1, Sw3, and Su40 represent the same source, although we found no evidence of the $\sim 2'$–$3'$ extension reported by Bamba et al. (2006) for Su40.
![ [*Chandra*]{} ACIS image (0.5–8 keV; smoothed with a $r=6''$ gaussian kernel) of the central part of HESS J1804 with the TeV contours overlayed. The best-fit position of HESS J1804 and its uncertainty are marked by the cross. The arrows show the four brightest X-ray sources, Ch1 (CXOU J180432.4$-$214009 = Sw3 = Su40), Ch2 (CXOU 180441.9$-$214224 = Su42), Ch3 (CXOU J180421.5$-$214233), and Ch4 (CXOU J180423.1$-$213932), detected in the brighter part of the TeV image, the pulsar B1800–21, and the [*ROSAT*]{} source 1WGA 1804.0$-$2142 (= Sw2). (Sw1, the brightest of the sources detected with [*Swift*]{}, is out of the ACIS FOV: it is shown in the [*ROSAT*]{} image in Fig. 8.) ](f1.eps){width="3.2in"}
![ [*Chandra*]{} ACIS-I3 image (in the 0.5–8 keV band; binned by a factor of 8) of the HESS J1804 central region. The best-fit position of HESS J1804 from Ah06 is shown by the cross. The position of the M-type dwarf (see §2.4) is shown by the box. Two larger circles ($r=1'$) are centered at the positions of Su40 and Su42 as reported by Bamba et al. (2006). The smaller circle ($r=44''$) shows the region used to estimate the count rates from Ch2 while the small ellipse shows the region used for the Ch1 spectral extraction. An offset of about $15''$ between the positions of the [*Chandra*]{} sources and [*Suzaku*]{} sources is apparently due to inaccuracy in [*Suzaku*]{} aspect solution. The fainter Ch3 and Ch4 sources (see text) are also marked. ](f2.eps){width="3.2in"}
We barely see some excess counts within the Su42 error circle in the original ACIS image, scattered over an area exceeding the PSF even with account for the large off-axis angle, $\theta \approx 14'$. However, when we filter out photons with energies $>8$ keV (which effectively reduces the background by a factor of 2.7) and bin by a factor of 8 (i.e., the new pixel size is 39), an extended (or multiple) source becomes visible, with a size of $\simeq1\farcm5-2'$ (see Fig. 2). The best-fit centroid of the source (obtained with the CIAO [*wavdetect*]{} tool) is ${\rm R.A.}=18^{\rm h}04^{\rm m}41\fs924$, Decl.$=-21^{\circ}42'24\farcs09$; we designate the source as CXOU J180441.9$-$214224 (hereafter Ch2).
In addition to Ch1 and Ch2, we found a dozen fainter sources on the I3 and I2 chips, of which CXOU J180421.5$-$214233 and CXOU J180423.1$-$213932 (hereafter Ch3 and Ch4, respectively) are the brightest and the closest to the best-fit position of HESS J1809 (see Figs. 1 and 2). Ch3 is consistent with being point-like, while Ch4 is either extended or, more likely, multiple.
We also attempted to search for signatures of diffuse emission (e.g., an SNR) on the I3 chip. A direct visual inspection of the ACIS image did not show clear signatures of large-scale diffuse emission. We applied the exposure map correction and smoothed the image with various scales, but failed to find statistically significant deviations from a uniform brightness distribution. To estimate an upper limit on the SNR emission, we measured the count rate from the entire I3 chip (with all identifiable point sources removed). The count rate, $0.266\pm0.003$ counts s$^{-1}$ in the 0.5–7 keV band, exceeds the nominal I3 background of 0.17 counts s$^{-1}$ ([*Chandra*]{} Proposers’ Observatory Guide[^3], v.8, §6.15.2), which could be caused by an elevated particle background, diffuse X-ray background, or SNR emission. Since we see no clear evidence of an SNR, we consider the difference, $\approx0.09$ counts s$^{-1}$, as an upper limit on the SNR count rate in the 70 arcmin$^2$ of the chip area, which corresponds the average surface brightness limit of 1.3 counts ks$^{-1}$ arcmin$^{-2}$.
![ Hard (2–7 keV; [*left*]{}) and soft (0.5–2 keV; [*right*]{}) band [*Chandra*]{} images of the HESS J1804 central region. The best-fit position of HESS J1804 (Ah06) is shown by the cross. The circles ($r=1'$) are centered at the positions of the two X-ray sources seen by [*Suzaku*]{} XIS (Bamba et al. 2006). The images in the [*top*]{} panels are binned by a factor of 8 (pixel size $3\farcs9$) while the same images in the [*middle*]{} panels are binned by a factor of 20 (pixel size $9\farcs8$). The [*bottom*]{} panels show the same images binned by a factor of 20 and smoothed with a $30''$ gaussian kernel. ](f5.eps){width="3.4in"}
Spectral analysis of the [*Chandra*]{} sources
----------------------------------------------
We extracted the Ch1 spectrum from the elliptical region (with the minor and major axes of $4\farcs9$ and $10\farcs8$; see Fig. 2), which accounts for the elongated shape of the off-axis PSF and contains $\approx83\%$ of the source counts. The background was measured from a larger circular annulus; it contributes about 15% to the total of 127 counts within the source extraction region. We group the spectra into 10 spectral bins between 0.3 and 7 keV. The spectrum of the source (shown in Fig. 3) is strongly absorbed, with only five counts below 2 keV (the lowest photon energy is 0.35 keV). The absorbed PL model fits the spectrum well, $\chi_{\nu}^{2}=0.99$ for 7 degrees of freedom, with $n_{\rm H,22}
\approx 3.8$, $\Gamma\approx 0.45$, and the absorbed and unabsorbed fluxes of $(1.7\pm0.2)$ and $(2.5^{+0.9}_{-0.4})$ $\times10^{-13}$ erg cm$^{-2}$ s$^{-1}$ in 0.3–7 and 0.3–8 keV, respectively. (Here and below the [*Chandra*]{} fluxes, luminosities and PL normalizations are corrected for vignetting and for the finite extraction aperture.) The uncertainties of the fits are listed in Table 1 and illustrated by confidence contours in Figures 3 and 4. As one can see, fixing the absorption at the best-fit value substantially reduces the uncertainties of the remaining parameters since $\Gamma$ and $n_{H}$ are strongly correlated. At a fiducial distance of 8 kpc, the observed PL flux corresponds to the unabsorbed luminosity of $\sim 2\times10^{33}$ ergs s$^{-1}$. Even with account for the large uncertainties, the spectral parameters are in poor agreement with those obtained by Bamba et al. (2006) for Su40, although an accurate comparison is difficult because those authors do not provide confidence contours. The [*Chandra*]{} and [*Suzaku*]{} unabsorbed fluxes, which are more accurately measured than the spectral parameters, are consistent within their uncertainties: $3.3^{+1.2}_{-0.5}\times 10^{-13}$ versus $4.3_{-1.1}^{+4.0}\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ in the 2–10 keV band, respectively. The ACIS spectrum of Ch1 also fits an absorbed black-body (BB) model with the temperature of 2.3 keV and emitting region radius of $\sim
30 (d/8\,{\rm kpc})$ m. The uncertainties of the BB fit are even larger than those of the PL fit.
For Ch2, the total number of background-subtracted counts within the $r=44''$ aperture centered at the source position (see Fig. 2) is $73\pm19$ in 0.3–8 keV (the total number of counts is 307, of which 234 counts are estimated to come from the background). Restricting the photon energies to the hard, 2–7 keV, band results in a similar $S/N=3.1$ ($55\pm12$ net source counts), while $S/N=2.1$ ($28\pm13$ net source counts) in the soft, 0.5–2 keV, band. These numbers indicate a relatively hard spectrum of the source, in qualitative agreement with the Su42 spectrum as reported by Bamba et al. (2006). The hard and soft band images are shown in Figure 5. The low $S/N$ values preclude a reliable spectral fitting. The measured count rates correspond to the observed 0.3–8 keV flux of $(1.0\pm 0.3)\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ in the $r=44''$ aperture, and the unabsorbed flux of $\approx1.5\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ in 2–10 keV, using the best-fit spectral parameters reported by Bamba et al. (2006) for Su42. The estimated unabsorbed flux of Ch2 is a factor of $\approx 1.7$ smaller than the flux of Su42, $(2.5\pm0.4)\times 10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ reported by Bamba et al. (2006); however, the difference may be due to unaccounted systematic errors.
For Ch3 and Ch4, the background-subtracted numbers of counts in the 0.5–8 keV band are $19\pm5$ and $44\pm10$, in $r=7\farcs4$ and $21''$ apertures, respectively (we chose the larger aperture for Ch4 because it looks extended or multiple). Their observed fluxes can be crudely estimated as $\sim1.2$ and $\sim2.5\times 10^{-14}$ ergs cm$^{-2}$ s$^{-1}$, respectively. The low S/N does not allow a meaningful spectral analysis of these sources. Figure 5 shows, however, that both Ch3 and Ch4 are better seen in the soft band, which means that they are less absorbed (hence less distant) than Ch1 and Ch2. Since the other off-axis sources on I3 and I2 chips are even fainter, their flux estimates are not reliable.
Timing of Ch1
-------------
We searched for pulsations of Ch1, using the arrival times of the 127 photons (of which $\approx85$% on are expected to come from the source) recalculated to the solar system barycenter using the CIAO [axBary]{} tool. The ACIS time resolution of 3.24 s and the total time span of 30 ks allow a search for pulsations in a $3\times 10^{-5}$–0.1 Hz range. We calculated the $Z_{1}^{2}$ statistic (e.g., Zavlin et al. 2000) at $10^{5}$ equally spaced frequencies $\nu$ in the $3\times 10^{-5}$–0.1 Hz range. This corresponds to oversampling by a factor of about 33, compared to the expected width of $T_{\rm span}^{-1}\approx 33$ $\mu$Hz of the $Z_{1}^{2}(\nu)$ peaks, and guarantees that we miss no peaks. The most significant peak, $Z_{\rm 1,max}^2=23.70$, was found at $\nu=0.009423\, {\rm Hz}\pm 5\, \mu{\rm Hz}$ ($P\approx
106.12\pm0.05$ s). The maximum value of $Z_{1}^{2}$ corresponds to the 97.9% ($\approx 2.3\sigma$) significance level, for the number of independent trials $\mathcal{N}=\nu_{\rm max}
T_{\rm{span}}\approx 3\times 10^{3}$. The pulse profile folded with the above frequency is shown in Figure 6 ([*top*]{}). The corresponding observed pulsed fraction is $58\%\pm 13\%$ (intrinsic source pulsed fraction $\approx 67\%\pm 15\%$) The significance of the period candidate is rather low, so the periodicity should be tested in a longer observation.
We also produced the unfolded light curve of Ch1 (Fig. 8, [*bottom*]{}) using a 2 ks binning. The light curve indicates that the source may experience some non-periodic variability on a few ks scale.
[ccccccc]{} Model & $n_{\rm H,22}$ & $\mathcal{N}$ or Area & $\Gamma$ or kT & $\chi^{2}$/dof & $L_{\rm X}$ or $L_{\rm bol}$\
PL & $3.8_{-2.5}^{+4.2}$ & $10.2_{-5.7}^{+200}$ & $0.45_{-1.45}^{+2.05}$ & $6.9/6$ & $1.9_{-0.3}^{+0.7}$\
PL & 3.8 & $10.2_{-2.7}^{+3.6}$ & $0.45_{-0.39}^{+0.34}$ & $6.9/7$ & $1.9\pm0.2$\
BB & $2.6_{-1.2}^{+1.8}$ & $\sim2.9$ & $\sim2.3$ & $6.8/6$ & $\sim7.9$\
![ [*Top:*]{} 8 $\mu$m [*Spitzer*]{} IRAC image of the HESS J1804 field with TeV contours overlayed. The best-fit position of HESS J1804 is shown by the cross. The positions of Ch1 and Ch2 are marked with the star and box respectively. The position of PSR B1800–21 is marked with a diamond, and the possible new SNR G8.31–0.09 is shown by the arrow. [*Bottom:*]{} Blow-up of the central part of the image. The two $r=10''$ circles, the larger $28''$ circle, and the $14''\times20''$ ellipse are centered at the positions of Ch1, Ch3, Ch2, and Ch4, respectively. The M dwarf near the Ch2 position is shown by the box. ](f7.eps){width="3.2in"}
![ [*Top:*]{} 10 ks [*ROSAT*]{} PSPC image of the HESS J1804 field. The white circle ($r=12'$) is centered at the best-fit position of HESS J1804. The diameter of the circle roughly corresponds to the extent of the $\gamma$-ray emission (see Ah06) The positions of the other sources discussed in the text are also marked. The brightest source Sw1 (=1RXS J180404.6– 215325) is most likely a nearby star DENIS J180403.2–215336 with magnitudes $R=12.0$, $J=10.1$, and $K_s=9.1$. Sw2 (= 1WGA 1804.0$-$2142) corresponds to the bright source at the very bottom of the ACIS-S3 chip in Fig. 1; it positionally coincides with the other bright star, DENIS J180400.6–214252 ($R=13.9$, $J=11.3$, $K=10.15$ ). [*Bottom:*]{} NVSS $\lambda=20$ cm image of the same size (beam FWHM=$45''$). The only bright source within the circle is the G8.31–0.09 SNR candidate. The much fainter NVSS J180434–214025 (see §2.4) is not discernible in this image. ](f8.eps){width="3.4in"}
Optical-IR-radio data
---------------------
We found no counterparts to Ch1 within $9''$ from its position in the Two Micron All Sky Survey (2MASS; Cutri et al. 2003) or Digital Sky Survey (DSS2)[^4] catalogs up to the limiting magnitudes $K_s=15.4$, $H=16$, $J=17.5$, $R=19$, and $B=21$. However, since the interstellar extinction towards the inner Galactic buldge is very large ($A_{V}\simeq18$ in the direction of Ch1 \[$l=8\fdg 429$, $b=-0\fdg018$\]; Schultheis et al. 1999), the limits are not very restrictive. We also examined the publically available data from the [*Spitzer*]{} GLIMPSE-II survey[^5] covering the vicinity of HESS J1804 (see the 8 $\mu$m IRAC image in Fig. 7, [*top*]{}) but found no IR sources within $10''$ from the Ch1 position, down to limiting fluxes of 5 and 6 $\mu$Jy at 4.5 and 8 $~\mu$m, respectively.
The closest match to Ch1 in radio catalogs was found in the NRAO VLA Sky Survey (NVSS) catalog (Condon et al. 1998). The catalog position of the relatively faint ($27.6\pm3.8$ mJy) radio source, NVSS J180434–214025, is offset by $32''$ from the Ch1 position, less than the NVSS beam size ($45''$ FWHM). However, the apparently extended NVSS source (linear size $\sim 1\farcm5$) looks like a part of a larger ($\sim4'$ in diameter) diffuse structure, barely discernible in the NVSS image. Since the image of NVSS J180434$-$214025 shows some artificial structures, we cannot consider it as a true radio counterpart of Ch1 until it is confirmed by deeper observations.
The optical/NIR source nearest to Ch2 is located $\sim28''$ away from the best-fit X-ray centroid (see Fig. 7). Having the magnitudes $B=14.54$, $V=13.30$, $R=12.19$, $J=8.64$, $H=8.05$, and $K=7.67$, and the proper motion of $\approx10.6$ mas yr$^{-1}$ (NOMAD1 0682$-$0650954; Zacharias et al. 2005), it is likely a late-type M dwarf located at $d\approx10$ pc. Such a dwarf could provide an X-ray flux of $\sim 10^{-13}$–$10^{-12}$ ergs cm$^{-2}$ s$^{-1}$ (see, e.g., Hünsch et al. 1999; Preibisch et al. 2005), similar to those observed from Ch2/Su42. However, given its large offset from the brightest part of Ch2 (Fig. 2), the dwarf cannot account for the entire extended X-ray emission, although its flare might be responsible for the possible difference between the fluxes measured with [*Suzaku*]{} and [*Chandra*]{} (§2.2).
Ch3 is positionally coincident with the optical-NIR source DENIS J180421.4$-$214233, with magnitudes $K=11.9$, $H=12.3$, $J=13.3$, $I=15.5$, $R=16.8$, $V=17.5$, and $B=19.0$, which is also seen in the $4.5$ and $8~\mu$m IRAC images. Within the X-ray extent of Ch4, there are five relatively bright 2MASS and DENIS sources (H magnitudes ranging from 10 to 14). Two IR sources within the X-ray extent of Ch4 are clearly seen in the IRAC images. One of them (northeast of the X-ray centroid of Ch4) is positionally coincident with the DENIS source J180423.6$-$213928 ($J=12.7$, $H=10.0$, and $K=8.8$). The other IR source has a NIR counterpart NOMAD1 0683–0642056 ($V=17.2$, $J=15.4$, $H=11.7$, and $K=10.0$), with the proper motion of 208 mas yr$^{-1}$.
All the stars we found within the X-ray extents of Ch3 and Ch4 exhibit extremely red colors. Explaining such colors solely by extinction would require a very large absorbing column that would absorb any soft X-rays ($\lesssim2$ keV) from this direction, in contradiction with the fact that we do see such X-rays from Ch3 and Ch4. The extremely red colors can be naturally explained if the NIR/IR objects are young pre-main-sequence (T Tauri) stars surrounded by dusty disks or infalling envelopes (e.g., Hartman et al. 2005). Indeed, the IRAC images show that the Ch3 and Ch4 regions are immersed in the extended diffuse IR emission (see Fig. 7) that may be associated with a nearby starforming region. The large proper motion of NOMAD1 0683–0642056 suggests a small distance to this star, $d\approx 100\, (v_\perp/100\,{\rm km\,s}^{-1}$) pc. Since the colors of this star are similar to those of the other stars around, it is likely that most of them belong to the same group, which is, perhaps, one of the nearest regions of star formation. Although the nearby T Tauri stars can easily account for the observed X-ray fluxes from Ch3 and Ch4 (e.g., Preibisch et al. 2005), such stars cannot produce TeV emission and, therefore, Ch3 and Ch4 are not associated with HESS J1804.
The IRAC images of the field (e.g., Fig. 7, [*top*]{}) reveal a large-scale diffuse emission with complex morphology. However, the IR brightness distribution does not correlate with the TeV brightness (shown by the contours in the same figure), nor with the large-scale X-ray brightness distribution seen in the archival [*ROSAT*]{} PSPC image (Fig. 8, [*top*]{}). The recently discovered radio source G8.31–0.09 (see the NVSS image in Fig. 8, [*bottom*]{}), classified as a possible SNR (Brogan et al. 2006), coincides well with the shell-like structure seen in the IRAC images (Fig. 7, [*top*]{}), thus confirming that the source is indeed a new SNR with an interesting IR morphology.
Discussion.
===========
We see from the [*Chandra*]{} ACIS image (Fig. 1) that the X-ray sky in the region of HESS J1804 is rich with point sources with fluxes of $\sim 10^{-14}-10^{-13}$ erg cm$^{-2}$ s$^{-1}$, most of which are possibly stars. Therefore, it is not surprising to find a few sources located relatively close to each other in this region of the sky. However, Ch1 does not have a known IR/optical counterpart while Ch2 appears to be extended, and both of them are located within the brightest part of HESS J1804 (19 and 25 from the best-fit TeV position). This raises a possibility that at least one them is associated with the TeV source. Below we discuss whether Ch1 or Ch2 could be X-ray counterparts of HESS 1804, for several possible interpretations of the TeV source. Since the large extent of the TeV emission rules out association with extragalactic sources, we limit our consideration to the Galactic sources only.
A High Mass X-ray Binary?
--------------------------
As there are several HMXBs among the identified TeV sources (see examples in Table 2), we can consider the possibility that HESS J1804 is an HMXB and therefore may have a compact X-ray counterpart, such as Ch1 or Ch2. It is believed that in HMXBs particles can be accelerated up to $\sim 10$ TeV or even higher energies either in jets produced as the result of accretion onto a compact object (e.g., Bosch-Ramon 2006 and references therein) or in the pulsar wind, if the compact object is an active pulsar (e.g., Dubus 2006). Examples of such systems are the famous HMXB with the young PSR B1259$-$63 and the microquasars LS 5039 and LSI$+61^\circ303$, for which the nature of the central engine (NS or BH) is still under debate. So far these are the only HMXB firmly detected in both the TeV and GeV bands. The ultra-relativistic particles can produce TeV emission via the inverse Compton scattering (ICS) of the optical-UV photons emitted by the non-degenerate companion or through the synchrotron self-Compton (SSC) process.
HMXBs produce X-rays either in the course of accretion of the matter from the secondary companion onto the compact object or via the synchrotron radiation in the shocked pulsar wind. We see from Table 2 that the TeV-to-X-ray (1–10 TeV to 1–10 keV) flux ratio, $f_{\gamma}/f_{\rm X}$, is $\lesssim 1$ for all the four HMXBs with more or less secure TeV associations, much smaller than $f_{\gamma}/f_{\rm X}\sim 30$ and $50$ for Ch1 and Ch2, respectively. However, given the small size of the HMXB sample in Table 2 and the fact that $f_{\gamma}/f_{\rm X}$ varies by at least a factor of 10 within the sample, it is possible that some HMXBs have a higher $f_{\gamma}/f_{\rm X}$. Indeed, most of accreting binaries are strongly variable X-ray sources, some of them being X-ray transients. For instance, IGR J16358–4726, which is likely associated with HESS 1634–472 (Ah06), is a strongly variable X-ray source, with the 2$-$10 keV flux varying by a factor of $\gtrsim 4000$ (Patel et al. 2004; Mereghetti et al. 2006). This example demonstrates that the $f_{\gamma}/f_{\rm X}$ ratio in HMXBs may vary dramatically, especially in the cases when the TeV and X-ray fluxes are not measured simultaneously. Thus, the rather modest X-ray luminosities of Ch1 and Ch2 could be explained assuming that either of them is an HMXB in the low/hard state.
The hard ($\Gamma\sim 0.5$) X-ray spectrum of Ch1 is strongly absorbed; the hydrogen column density, $n_{\rm H,22}\simeq 4$, is a factor of 2–3 larger than the total Galactic HI column ($\simeq 1.5\times 10^{22}$ cm$^{-2}$; Dickey & Lockman 1990) and a factor of 2–4 larger than the $n_{\rm H,22}\sim1.4$ inferred from the X-ray spectrum of PSR B1800–21 (and its PWN) located at the distance of $\approx 4$ kpc (Kargaltsev et al. 2006a). Taking into account that the $n_{\rm H}$ value deduced from an X-ray spectrum under the assumption of standard element abundances generally exceeds the $n_{\rm HI}$ measured from 21 cm observations by a factor of 1.5–3 (e.g., Baumgartner & Mushotzky 2005), the large $n_{\rm H}$ (consistent with $A_{V}\sim20$; e.g., Predehl & Schmitt 1995) suggests that Ch1 is either located within (or even beyond) the Galactic Buldge or it shows intrinsic absorption, often seen in X-ray spectra of HMXBs (e.g., Walter et al. 2006).
As the spin periods of NSs in HMXBs range from a fraction of second to thousands of seconds, the HMXB interpretation provides a plausible explanation for the putative 106 s periodicity in Ch1, which would be difficult to interpret otherwise. On the other hand, the lack of an IR/NIR counterpart is somewhat surprising, although the upper limits on the unabsorbed IR/NIR fluxes (see §2.4 and Fig. 9) still cannot rule out a B-giant at a distance of $\gtrsim 8$ kpc.
We found no [*CGRO*]{} EGRET counterparts for Ch1 and other sources in the HESS J1804 field. The nearest EGRET source (Hartman et al. 1999) is located $\simeq2\fdg2$ from the Ch1 position, too far to be associated with HESS J1804 or the [*Chandra*]{} sources. However, only three HMXBs (PSR B1259–63, LS 5039 and LSI$+61^\circ303$) have been identified with EGRET sources so far. The upper limit on GeV flux, obtained from the EGRET upper limits map (Fig. 3 from Hartman et al. 1999), is not deep enough to test the connection between the X-ray spectrum of Ch1 and the TeV spectrum of HESS J1804 (see Fig. 9). The [*Intergal*]{} ISGRI upper limit (A. Bykov 2006, priv. comm.), shown in the same figure, appears to be even less restrictive. From Figure 9 we can only conclude that the TeV spectrum of HESS J1804 breaks somewhere between the EGRET and HESS energy ranges, as observed for many TeV sources of different kinds.
The spectral parameters of the Ch2 source are very uncertain. Although the flux measured with [*Chandra*]{} is somewhat lower than that measured with [*Suzaku*]{} a year later (see §2.2), the difference is only marginal because of the large uncertainties of the measurements. However, if confirmed, the variability would be an argument supporting an HMXB interpretation of Ch2. On the contrary, the rather large X-ray extent of Ch2 \[$\sim 1'= 2 (d/7\, {\rm kpc})$ pc\] argues against the X-ray binary interpretation[^6]. Although possible X-ray emission from a nearby M-dwarf (§2.4) may contribute to Ch2, it cannot account for the entire emission from this extended or multiple source.
Even if either of the X-ray sources is an HMXB, a major problem with its association with HESSJ1804 is the extended morphology of the latter. Although, there is no an [*a priori*]{} reason to believe that HMXBs cannot produce extended TeV emission, the observational evidence for that is currently rather weak. So far, among the TeV sources possibly associated with HMXBs, only two, HESS J1632–478 and HESS J1634–472, might show extended TeV emission (Ah06), and the evidence for the extension is marginal in both cases.
![ Unabsorbed spectra of Ch1 and HESS J1804 (Ah06), together with the [*CGRO*]{} EGRET and [*INTEGRAL*]{} IBIS/ISGRI upper limits. The open triangles show the upper limits on the dereddened NIR fluxes in the K$_s$, H, and J bands (see §2.4). ](f9.eps){width="2.7in"}
Thus, although an HMXB at a distance of $\sim 8$–15 kpc remains a plausible interpretation for Ch1[^7] and somewhat less plausible for Ch2, the association between them and the TeV source is very questionable. An HMXB origin of Ch1 or Ch2 would be firmly established if the periodic (and/or non-periodic) variability is confirmed for Ch1 (or found for Ch2) in a deeper X-ray observation, or a companion star is detected in the IR-optical. At the same time, a deeper on-axis observation with [*Chandra*]{} can measure the true extent and spatial structure of Ch2.
A Pulsar Wind Nebula?
---------------------
Among other types of Galactic X-ray sources, only SNR shocks and PWNe are believed to be able to produce extended TeV emission. In fact, the second highest (persistent) TeV-to-X-ray flux ratio, $f_{\gamma}/f_{\rm
X}=3.4$, in Table 2 belongs to the PWN G18.0–0.7 around the Vela-like pulsar B1823–13 ($\dot{E}\approx3\times10^{36}$ erg s$^{-1}$; $d\approx4$ kpc), likely associated with HESS J1825–137 (Ah06). Although no SNR has been associated with this pulsar, it powers a luminous extended X-ray PWN ($L_{X}\sim 3 \times 10^{33}$ erg s$^{-1}$, angular size $\gtrsim5'$; Gaensler et al. 2003). In addition to the extended low-surface-brightness component, G18.0–0.7 has a much more compact ($5''$–$10''$) brighter core, resolved by [*Chandra*]{} (Teter et al. 2004; Kargaltsev et al. 2006b). The TeV emission detected with HESS covers a much larger area than the X-ray emission from G18.0–0.7, extending up to $1^{\circ}$ southward from the pulsar (Aharonian et al. 2006c). However, both the TeV and the low-surface-brightness X-ray emission have similarly asymmetric shapes, and they are offset in the same direction with respect to the pulsar position. A similar picture is observed around the Vela pulsar ($\dot{E}\approx7\times10^{36}$ erg s$^{-1}$; $d\approx300$ pc). An X-ray bright, compact ($\sim40''$ in diameter) PWN centered on the pulsar is accompanied by a much larger ($\sim50'$) but dimmer asymmetric diffuse X-ray component (sometimes referred to as “Vela X”), which also has a TeV counterpart (Aharonian et al. 2006d).
The asymmetry in the extended PWN components can be caused by the reverse SNR shock that had propagated through the inhomogeneous SNR interior towards the SNR center and reached one side of the PWN sooner than the other side (e.g., Blondin, Chevalier, & Frierson 2001). The wind, produced by the pulsar over a substantial period of time (up to a few kyrs) and therefore occupying a substantial volume, could be swept up by the reverse shock wave into a smaller volume on one side of the PWN. The swept-up wind confined within the formed “sack” emits synchrotron radiation in X-rays. At the same time, the wind can produce TeV radiation via the ICS of the cosmic microwave background (CMB) and synchrotron photons off the relativistic electrons[^8]. The Lorentz factor of the electron that upscatters the CMB photon to the energy $\mathcal{E}_{\gamma}$ is $\gamma\approx10^8
(\mathcal{E}_{\gamma}/9~{\rm TeV})^{1/2}$. Electrons with such Lorentz factors emit synchrotron photons with energies $\mathcal{E}_{\rm syn}
\sim0.5\gamma_{8}^{2}(B/10~\mu{\rm G})\,{\rm keV}
\sim 0.5(\mathcal{E}_{\gamma}/9~{\rm TeV})(B/10~\mu{\rm G})$ keV. Therefore, the observed TeV spectrum of HESS J1804, spanning from 0.2 to 10 TeV (Ah06), would correspond to the $\approx0.01$–0.6 keV range of the synchrotron photon energies in $B=10~\mu$G. These EUV and soft X-ray synchrotron photons are heavily absorbed at $n_{\rm H}\gtrsim10^{22}$ cm$^{-2}$ and hence are difficult to detect. Thus, if the swept-up wind is cold enough \[e.g., $\gamma\lesssim10^{8} (B/10\,\mu{\rm G})^{-1/2}$\], the sack may be bright in TeV but faint in the [*Chandra*]{} band. Furthermore, the magnetic field inside the sack is lower than that in the compact PWN, leading to a lower synchrotron brightness since the latter depends on the magnetic field strengths as $B^{(p+1)/2}$ for the PL distribution of electrons, $dn_{e}=K\gamma^{-p}d\gamma$. This could explain why the TeV emitting region is dimmer in X-rays than the compact PWN populated with more energetic electrons, but it does not explain why the compact PWN shows lower surface brightness in TeV than the extended asymmetric PWN. The brightness of the TeV emission produced via the ICS on CMB photons does not depend on the magnetic field; therefore, the simplest explanation could be that the sack contains a larger number (and perhaps a higher column density) of the swept-up TeV-emitting electrons compared to those within the compact PWN.
One could try to apply the above interpretation to HESS J1804, assuming that Ch1 (or Ch2) is a pulsar with a PWN. The off-axis position may not allow one to resolve a compact PWN. Furthermore, Bamba et al. (2006) report Su40 (=Ch1) as an extended source, which could mean that the more sensitive (on large angular scales) [*Suzaku*]{} XIS observation has detected a fainter extended PWN component (similar to the [*XMM-Newton*]{} observation of B1823–13; Gaensler et al. 2003). The faintness of a possible extended PWN component could be at least partly attributed to the strong X-ray absorption in this direction. On the other hand, Ch2 is resolved by [*Chandra*]{} into an extended X-ray source, which might be a PWN. However, the low S/N and the off-axis location hamper the assessment of the spatial structure and the spectrum of Ch2.
The 3.24 s time resolution of the ACIS observation also precludes a search for sub-second pulsations expected from a young pulsar (the putative 106 s period of Ch1 is certainly too long for a young isolated pulsar and hence should be attributed to a statistical fluctuation in this interpretation). Keeping in mind the above examples of Vela X and G18.0–0.7, the large extent of HESS J1804 should not be alarming. A lack of strong asymmetry with respect to the pulsar, which is the distinctive feature of all the other extended TeV PWNe (Table 2; de Jager 2006), could be attributed to the low sensitivity of the [*Chandra*]{} observation to extended emission of low surface brightness or to the projection effect (i.e., the TeV PWN could be displaced from the pulsar along the line of sight). The large $f_{\gamma}/f_{\rm X}$ values cast additional doubts on the PWN interpretation; however, even a luminous extended X-ray component of low surface brightness could remain undetected in the relatively shallow off-axis ACIS exposure. A deeper on-axis observation with [*Chandra*]{} would test the nature of Ch1 and Ch2 and the PWN interpretation. Overall, although not excluded, the possibility that Ch1 or Ch2 are the pulsars powering the TeV PWN does not look very compelling at this point.
On the other hand, the association of HESS J1804 with the Vela-like pulsar B1800–21 remains a plausible option. To date, young Vela-like pulsars have been found in the vicinity of $\sim10$ extended TeV sources (e.g., de Jager 2006; Gallant 2006). Since both pulsars and TeV sources are concentrated in the Galactic plane, and the extended TeV sources have typical sizes of $\sim5'-15'$, one could attempt to explain this by a chance coincidence. However, the probability of chance coincidence is low. For instance, within the $\simeq 300$ square degrees area of the Galactic plane, surveyed by HESS (Ah06) the surface density of young ($\leq100$ kyrs) pulsars is $\approx0.13$ deg$^{-2}$ (based on the ATNF Pulsar Catalog data; Manchester et al. 2005). On the other hand, the same area includes four extended TeV sources (HESS J1825–137, HESS J1809–193, HESS J1804–216, and HESS J1616–508) located within $15'$ from one of the young pulsars. Since the probability of finding a young pulsar within an arbitrary placed R$=15'$ circle is only $2.6$%, the probability of accidentally having all the four TeV sources within the $15'$ distances from the young pulsars is negligible, $0.026^{4}\approx5\times10^{-7}$. This strongly suggests a physical connection between the two phenomena (e.g., de Jager 2006). Furthermore, there are several pairs, such as PSRB0833–45/HESSJ0835–455, PSRB1509–58/HESSJ1514–591, and PSRB1823–13/HESSJ1825–137, for which the connection is supported by the correlation between the TeV and X-ray brightness distributions. Note, that in these pairs the pulsars are offset by $10'$–$15'$ from the peaks of the TeV brightness.
From the theoretical perspective, the “crushed PWN” hypothesis (Blondin et al. 2001), briefly discussed above, provides a possible explanation for the observed offsets. From the observational point of view, the associations are supported by the detection of large, asymmetric X-ray structures correlated with the TeV brightness distributions and apparently connected to the pulsars. However, in several possible associations the existing X-ray images are not deep enough to reveal an extended PWN component. In particular, the X-ray images of the PWN around B1800–21 (Kargaltsev et al. 2006a) show a hint of a dim, asymmetric PWN component extended toward HESS J1804, but the sensitivity of the [*Chandra*]{} observation was possibly insufficient to detect the PWN beyond $15''$–$20''$ from the pulsar. This is similar to PSR B1823–13, where the arcminute-scale PWN was well seen only in a long [*XMM-Newton*]{} observation, and only [*a posteriori*]{} a hint of it was found in the [*Chandra*]{} data (Kargaltsev et al. 2006b). Hence, there is a good chance that PSR B1800–21 also has a dim, asymmetric PWN. It could be detected in a deep [*XMM-Newton*]{} exposure, thereby establishing the association between HESS J1804 and PSR B1800–21.
An SNR shock?
-------------
While discussing the Su40 (=Ch1) and Su42 (=Ch2) association with HESS J1804, Bamba et al. (2006) suggest that the X-ray and TeV emission could come from an SNR shock (possibly in G8.7$-$0.1). In our opinion, the fact that the angular extent of the two X-ray sources is much smaller than the extent of the TeV emission (see Fig. 1) is a strong argument against such an interpretation[^9]. Nevertheless, a possibility that an SNR (so far undetected in X-rays) could produce the observed TeV emission in HESSJ1804 (see Fatuzzo, Melia, & Crocker 2006) cannot be ruled out if the TeV source is not associated with Ch1, Ch2, or PSR B1800–21.
Indeed, contrary to the conclusion by Bamba et al. (2006)[^10], the close match in the sky positions of Ch1 (or Ch2) and HESS J1804 can merely be a chance coincidence, and HESS J1804 may have no point-like X-ray counterparts down to the $3\sigma$ limiting flux of $\lesssim1\times10^{-14}$ ergs s$^{-1}$ cm$^{-2}$ within the TeV bright region. However, one cannot exclude the presence of faint diffuse X-ray emission, e.g. from an SNR whose image size exceeds the chip size. Since it is difficult to estimate which fraction of the observed diffuse count rate (1.3 counts ks$^{-1}$ arcmin$^{-2}$ in the I3 chip; see §2.1) comes from the background and what is the nature of the remaining flux (e.g., thermal emission from an SNR or nonthermal emission from an extended PWN), we can only put an upper limit of $2.5\times10^{-12}$ ergs s$^{-1}$ cm$^{-2}$ on the 2–10 keV flux in the I3 chip area, corresponding to $f_{\gamma}/f_{\rm X}\gtrsim 4$ (this estimate assumes $n_{\rm H,22}=1$ and a PL model with $\Gamma=1.5$). However, we do not see any significant large-scale (in comparison with the off-axis PSF size) X-ray brightness variations in the ACIS image. (see Figs. 1 and 2). Although such uniformity is somewhat unusual for an SNR, we note that the interior of the shell-type SNR RX J1713.7–3946 (resolved into a $\approx1^{\circ}$ shell in TeV; Aharonian et al. 2004) is relatively faint and homogeneous in X-rays (Hiraga et al. 2005). Furthermore, following Kargaltsev et al. (2006a), we conclude that if the X-ray spectrum and luminosity of the undetected SNR are similar to those of the Vela SNR, the expected off-axis ACIS-I3 surface brightness is $<0.3$ counts ks$^{-1}$ arcmin$^{-2}$ in the 0.5–7 keV band (for the Raymond-Smith thermal plasma emission models with $T<3$ MK and $n_{\rm H,22}=1$), i.e. at least a factor of 4 below the observed upper limit (see §2.1.1).
On the other hand, the TeV brightness distribution in HESSJ1804 poorly correlates with the radio brightness distribution. Although located within the boundaries of G8.7$-$0.1, the region around HESS J1804 in the radio image is much dimmer than the northeast part of G8.7$-$0.1 that also emits X-rays observed with [*ROSAT*]{} (see Fig. 8). This, in our view, argues against the HESS J1804 and G8.7$-$0.1 association (see, however, Fatuzzo et al.2006, who argue that the TeV emission can be produces by a shock in the G8.7–0.1 interacting with a molecular cloud).
A possibility that HESS J1804 is associated with the recently discovered faint radio (and IR) source G8.31–0.09, likely an SNR (Brogan et al. 2006), is not attractive either. G8.31–0.09 is outside the ACIS FOV, and it is not seen in the archival [*ROSAT*]{} PSPC image (Fig. 8; [*top*]{}). However, G8.31–0.09 is far from the peak of the TeV brightness distribution (see the [*Spitzer*]{} image in Fig. 7). Furthermore, the size of the shell-like G8.31–0.09 in the [*Spitzer*]{} image is much smaller than the TeV extent of HESS J1804 and hence, even if G8.31–0.09 is indeed an SNR, it is unlikely to be related to HESS J1804.
Summary
=======
We serendipitously detected several X-ray sources, whose positions are close to the maximum of the TeV brightness distribution of the extended VHE source HESSJ1804. Among these sources, only Ch1 and Ch2 might be related to HESSJ1804. The fact that HESS J1804 is an extended source rules out an extragalactic (i.e. AGN) origin, and it also argues against an HMXB interpretation.
On the other hand, the marginal detection of 106 s pulsations in Ch1 suggests that Ch1 might be an HMXB unrelated to HESSJ1804. There also remains a possibility that Ch1 is a new obscured pulsar/PWN couple, possibly associated with G8.7$-$0.1. In this case no variability is expected on time scales $\gtrsim 1$ s, but one could expect to see an X-ray PWN, which has not been detected in the [*Chandra*]{} observation possibly because of the off-axis placement on the ACIS detector.
A possible variability of Ch2 on a year timescale might also suggest that Ch2 is an accreting binary, which makes the association with HESS J1804 unlikely. On the other hand, the extended appearance of Ch2 argues in favor of a PWN or a remote SNR. In the former case, there remains a possibility of association of Ch2 with HESS J1804. Further on-axis observations with [*Chandra*]{} ACIS are needed to firmly establish the nature of the two sources.
It is possible that neither Ch1 nor Ch2 are associated with HESS J1804. In this case the most plausible interpretation of HESS J1804 is that the TeV emission comes from an X-ray dim part of the asymmetric PWN created by PSR B1800–21. A longer observation with [*XMM-Newton*]{} or [*Chandra*]{}, combined with deep high-resolution imaging in the radio and IR, will finally differentiate between these possibilities and establish the nature the two [*Chandra*]{} sources as well as the origin of the TeV emission.
Our thanks are due to Andrey Bykov for providing the [*Integral*]{} IBIS/ISGRI upper limit for the HESS J1804 flux. We are also grateful to Konstantin Getman for the useful discussions about multiwavelength emission from young stars. This work was partially supported by by NASA grants NAG5-10865 and NAS8-01128 and [*Chandra*]{} awards AR5-606X and SV4-74018.
Aharonian, F. et al. 2004, Nature, 432, 75
Aharonian, F. et al. 2005a, Science, 307, 1938
Aharonian, F. et al. 2005b, A&A, 442, 1
Aharonian, F. et al. 2005c, A&A, 437, 95
Aharonian, F. et al. 2005d, A&A, 437, 7
Aharonian, F. et al. 2005e, A&A, 435, 17
Aharonian, F. et al. 2005f, A&A, 432, 25
Aharonian, F., et al. 2006a, ApJ, 636, 777 (Ah06)
Aharonian, F., et al. 2006b, A&A, accepted (astro-ph/0607192)
Aharonian, F., et al. 2006c, A&A, accepted (astro-ph/0607548)
Aharonian, F., et al. 2006d, A&A, 448, 43
Aharonian, F., et al. 2006e, A&A, 460, 743
Aharonian, F., et al. 2006f, A&A, 449, 223
Aharonian, F., et al. 2006g, A&A, 457, 899
Albert, J., et al. 2006a, Science, 312, 1771
Albert, J., et al. 2006b, ApJ, 642, 119
Arons, J. 2004, Adv. Space Res. 33, 466
Bamba, A., et al. 2006, PASJ, in press (astro-ph/0608310)
Baumgartner, W. H., & Mushotzki, R. F. 2006, ApJ, 639, 929
Blondin. J., Chevalier, R., & Frierson, D. 2001, ApJ, 563, 806
Bosch-Ramon, V., et al. 2005, ApJ, 628, 388
Bosch-Ramon, V. 2006, Astrophys. Space Sci, in press (astro-ph/0612318)
Brogan, C. L., Gelfand, J. D., Gaensler, B. M., Kassim, N. E., & Lazio, T. J. 2006, ApJ, 639, L25
Chernyakova, M., Neronov, A., Lutovinov, A., Rodriguez, J., & Johnston, S. 2006, MNRAS, 367, 1201
Condon, J. J., Cotton, W. D., Greisen, E. W., Yin, Q. F., Perley, R. A., Taylor, G. B., & Broderick, J. J. 1998, AJ, 115, 1693
Corbel, S., Fender, R. P., Tzioumis, A. K., Tomsick, J. A., Orosz, J. A., Miller, J. M., Wijnands, R., & Kaaret, P. 2002, Science, 298, 196
Cui, W., & Konopelko, A. 2006, ApJ, 625, L109
Cutri, R. M., et al. 2003, 2MASS All-Sky Catalog of Point Sources, VizieR On-line Data Catalog: II/246
de Jager, O. 2006, On the Present and Future of Pulsar Astronomy, 26-th meeting of IAU, Joint Discussion 2, 16-17 August, 2006, Prague, Czech Republic, JD02, \#53
Dickey, J. M., & Lockman. F. J. 1990, ARA&A, 28, 215
Dubus, G. 2006, A&A, 456, 801
Finley, J. P., & Ögelman, H. 1994, ApJ, 434, L25
Fatuzzo, M., Melia, F., & Crocker, R. M. 2006, ApJ, accepted (astro-ph/0602330)
Gaensler, B. M., Pivovaroff, M. J., & Garmire, G. P. 2001, ApJ, 556, 107
Gaensler, B. M., et al. 2002, ApJ, 569, 878
Gaensler, B. M., et al. 2003, ApJ, 588, 441
Gallant, Y. 2006, to appear in Proc. “The Multi-Messenger Approach to High-Energy Gamma-Ray Sources”, Barcelona, July 2006 (astro-ph/0611720)
Hartman, R. C., et al. 1999, ApJS, 123, 79
Hartmann, L., et al. 2005, ApJ, 629, 881
Harrison, F. A., et al. 2000, ApJ, 528, 454
Heindl, W. A., Tomsick, J. A., Wijnands, R., & Smith, D. M. 2003, ApJ, 588, L97
Hiraga, J. S. 2005, A&A, 431, 953
Horns, D., et al. 2006, to appear in Proc. “The Multi-Messenger Approach to High-Energy Gamma-Ray Sources”, Barcelona, July 2006 (astro-ph/0609386)
Hünsch, M., Schmitt, J. H. M. M., Sterzik, M. F., & Voges, W. 1999, A&A Supp., 135, 319
Iyudin, A. F., Aschenbach, B., Becker, W., Dennerl, K., & Haberl, F. 2005, A&A, 429, 225
Kargaltsev, O., Pavlov, G. G., & Garmire, G. P. 2006a, ApJ, submitted (astro-ph/0611599)
Kargaltsev, O., Pavlov, G. G., Sanwal, D., Wong, J., & Garmire G. P. 2006b, AAS HEAD meeting 9, \#7.57
Landi, R., et al. 2006, ApJ, 651, 190
Lutovinov, A., et al. 2005, A&A, 444, 821
Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129,1993
Markwardt, C. B., & Ögelman, H. B. 1997, ApJ, 480, 13
Mereghetti, S., et al. 2006, A&A, 450, 759
Migliari, S., Fender, R. P., & Méndez, M. 2002, Science, 297, 1673
Ong, R. A. 2006, Rapporteur Talk at 29th International Cosmic Ray Conference in Pune, India, 2005 (astro-ph/0605191)
Patel, S. K., et al. 2004, ApJ, 602, 45
Predehl, P., & Schmitt, J. H. M. M. 1995, A&A, 293, 889
Preibisch, T., et al. 2005, ApJS, 160, 401
Teter, M. A., Sanwal, D., Pavlov, G. G., & Tsuruta, S. 2003, BAAS, 35, 706
Schultheis, M., et al. 1999, 349, L69
Walter, R., et al. 2006, A&A, 453, 133
Zacharias, N., et al. 2005, VizieR On-line Data Catalog: I/297
Zavlin, V. E., Pavlov, G. G., Sanwal, D., & Trümper, J. 2000, ApJ, 540, 25
[ccccccccccc]{} Name & Type & $f_{X}$ & $\Gamma_{X}$ & $f_{\gamma}$ & $\Gamma_{\gamma}$ & $f_{\gamma}$/$f_{X}$ & Extended in TeV? & Variability & X-ray counterpart & Ref.\
LSI +61 303 & HMXB/$\mu$-quasar & 0.64 & 1.8 & 0.54 & 2.6 & 0.8 & no & $P_{\rm orb}=26$ d & yes & 1,2\
LS 5039 & HMXB/$\mu$-quasar & 0.90 & 1.6 & 0.62 & 2.1 & 0.7 & no & $P_{\rm orb}=4.4$ d & yes & 3,4\
PSR B1259–63 & HMXB/pulsar & 0.6 & 1.4 & 0.24 & 2.7 & 0.4 & no & $P_{\rm spin}=48$ ms, $P_{\rm orb}=3.4$ yrs & yes & 5,6\
HESS J1634–472 & HMXB/NS? & 0.004–16.5 & 0.5 & 0.50 & 2.4 & 0.03–125 & yes? & $P=5890$ s , X-ray transient & IGR J16358–4726? & 7,8\
HESS J1632–478 & HMXB? & 8.8 & 0.7 & 1.7 & 2.1 & 0.2 & yes? & $P_{\rm spin }=1300$ s, $P_{\rm orb}=9$ days & IGR J16320–4751 & 7,9,10\
1ES 1218+30.4 & BL Lac & 2.4 & 1.4 & 0.73 & 3.0 & 0.3 & no & yes & yes & 11\
Mkn 421 & BL Lac & 24.9 & 1.5 & 49.3 & 2.1 & 2.0 & no & yes & yes & 12\
RX J1713.7–3946& SNR & 80 & 2.3 & 6 & 2.2 & 0.075 & yes & no & G347.3–0.5 & 13,14\
G266.6–1.2 & SNR & 11.8 & 2.6 & 7 & 2.1 & 0.6 & yes & no & Vela Junior & 15, 16\
Crab & PWN & 868 & 2.1 & 6.7 & 2.6 & 0.008 & no & no & Crab PWN & 17\
HESS J1825–137 & PWN & 0.14 & 2.3 & 0.48 & 2.4 & 3.4 & yes & no & B1823–13 PWN & 18, 19\
MSH 15–52 & PWN & 5.5 & 1.9 & 1.5 & 2.3 & 0.27 & yes & no & yes & 20, 21\
Vela X & PWN & 7.4 & 2.1 & 4.6 & 1.45 & 0.6 & yes & no & yes & 22, 23\
G0.9+0.1 & PWN? & 1.6 & 2.3 & 0.19 & 2.4 & 0.1 & no? & no & yes & 24 25\
HESS J1804/Ch1 & ? & 0.03 & 0.45 & 0.91 & 2.7 & 30 & yes & $P_{\rm spin}=106$ s ? & yes & –\
HESS J1804/Ch2 & ? & 0.02 & ? & 0.91 & 2.7 & 50 & yes & ? & yes & –\
HESS J1804/Diff. & ? & $\lesssim0.25$ & 1.5 & 0.91 & 2.7 & $\gtrsim 4$& yes & no & yes & –\
[^1]: See the catalogs at http://www.icrr.u-tokyo.ac.jp/morim/TeV-catalog.htm and http://www.mpi-hd.mpg.de/hfm/HESS/public/HESS\_catalog.htm
[^2]: It should be noted that after this paper had been generally completed, Cui & Konopelko (2006) published an ApJ letter using the same [*Chandra*]{} data. Using wrong coordinates of PSR B1800–21, they could not identify the pulsar in the [*Chandra*]{} image, failed to notice one of the two [*Suzaku*]{} sources, and did not provide a thorough analysis of the other Suzaku source. We correct the shortcomings of that work in our paper.
[^3]: See http://asc.harvard.edu/proposer/POG/index.html
[^4]: see http://archive.eso.org/dss/dss
[^5]: http://www.astro.wisc.edu/sirtf/glimpsedata.html
[^6]: To our knowledge, extended X-ray emission has been reported only from three HMXBs: SS 433 (Migliari, Fender, & Méndez 2002), Cyg X-3 (Heindl et al. 2003 ), and XTE J1550–564 (Corbel et al. 2002). This emission is often attributed to jets. In these systems the angular extent of the resolved X-ray emission ranges from $3''$ to $30''$ corresponding to physical lengths of 0.1 to 0.8 pc at the nominal distances to these systems. No TeV emission has been reported from these HMXBs yet.
[^7]: We should mention that, based on the strongly absorbed, hard X-ray spectrum, this source can also be a background AGN.
[^8]: An alternative TeV production mechanism is $\pi^{0}\rightarrow\gamma + \gamma$ decay, with $\pi^{0}$ being produced when the relativistic protons of the pulsar wind interact with the ambient matter (Horns 2006). Although the presence of the hadronic component in the pulsar wind has not yet been established observationally, it is expected to be present according to some pulsar wind acceleration models (e.g., Arons 2005).
[^9]: For instance, the RX J1713.7–3946 and G266.6-1.2 SNRs have comparable sizes in X-rays and TeV.
[^10]: These authors state that the expected number of sources within the area defined by the error bars of the HESS J1804 best-fit position should be very small, $(4-9)\times10^{-3}$. First, one should not use the uncertainty of the best-fit TeV position for such an estimate when the TeV source is clearly extended and asymmetric. Second, as we see from Fig. 1, the probability of finding an X-ray source with a flux of $10^{-14}$–$10^{-13}$ ergs s$^{-1}$ cm$^{-2}$ within an arbitrary placed $r=1'$ circle is quite high.
|
---
abstract: |
We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times. Consider incremental connectivity, letting $t_u$ be the time to insert an edge and $t_q$ be the query time. For $t_u = \Omega(t_q)$, the problem is equivalent to the well-understood *union–find* problem: ${\textnormal{\scshapeInsertEdge}}(s,t)$ can be implemented by ${\textnormal{\scshapeUnion}}({\textnormal{\scshapeFind}}(s), {\textnormal{\scshapeFind}}(t))$. This gives worst-case time $t_u = t_q = O(\lg n / \lg\lg n)$ and amortized $t_u = t_q =
O(\alpha(n))$.
By contrast, we show that if $t_u = o(\lg n / \lg\lg n)$, the query time explodes to $t_q \ge n^{1-o(1)}$. In other words, if the data structure doesn’t have time to find the roots of each disjoint set (tree) during edge insertion, there is no effective way to organize the information!
For amortized complexity, we demonstrate a new inverse-Ackermann type trade-off in the regime $t_u = o(t_q)$.
A similar lower bound is given for fully dynamic connectivity, where an update time of $o(\lg n)$ forces the query time to be $n^{1-o(1)}$. This lower bound allows for amortization and Las Vegas randomization, and comes close to the known $O(\lg n \cdot (\lg\lg
n)^{O(1)})$ upper bound.
author:
- |
Mihai Pǎtraşcu\
AT&T Labs
- |
Mikkel Thorup\
AT&T Labs
bibliography:
- '../../general.bib'
title: 'Don’t Rush into a Union: Take Time to Find Your Roots'
---
Introduction
============
We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times. Previous trade-offs where smooth and much weaker. The new explosive lower bounds are found hidden in some very well-studied problems: incremental and fully-dynamic connectivity.
Our Results
-----------
The union–find problem is to support the following operations on a collection of disjoint sets, starting from $n$ singleton sets $\{1\},
\dots, \{n\}$:
Return an element in the same set as $v$ that uniquely identifies the set. (This is called the root, or the representative of the set.)
Join the sets identified by $u$ and $v$, *assuming* these are roots of their own sets.
The terminology for this problem stems from the usual implementation as a forest, in which each tree represents a set. [[]{.nodecor}]{} involves walking to the root of $v$’s tree, potentially doing useful work (path compression). [[]{.nodecor}]{} simply involves adding an edge between the roots (whose direction is usually determined by the size of each subtree, cf. union by rank).
The union–find problem has been studied into excruciating detail and is now essentially understood. From an amortized perspective, Tarjan [@tarjan75uf] showed that a sequence of $n-1$ unions and $m$ finds can be supported in time $O(n + m
\alpha(m,n))$. See [@tarjan84uf; @lapoutre90uf] for different analyses and trade-offs between amortized running times. From a worst-case perspective, the classic union-by-rank gives union in constant time and find in $O(\log n)$ time. Trade-offs were addressed by Blum [@blum86uf], with an improvement by Smid [@smid90uf]. They show that, if the time for union is bounded by $t_{{\textnormal{\scshapeUnion}}}$, [[]{.nodecor}]{} can be supported in worst-case $O(\lg
n / \lg t_{{\textnormal{\scshapeUnion}}})$. Finally, Alstrup et al. [@alstrup99uf] showed that the amortized and worst-case trade-offs can be achieved *simultaneously*. These bounds are known to be optimal in the powerful cell-probe model (see below for a review of the lower bounds).
Here we consider an obvious strengthening of the problem, where we allow:
Join the sets containing $u$ and $v$ if these sets are different.
The link–find problem is a natural way to solve one of the most basic graph problems: *incremental connectivity*. This is the problem of maintaining an undirected graph under edge insertions and connectivity queries. New edges may link arbitrary nodes, possibly introducing cycles. Two nodes are connected if they find the same identifier.
We now consider the worst-case trade-offs for link-find and incremental connectivity. Since link-find solves incremental connectivity, we want upper-bounds are for link-find and lower-bounds for incremental connectivity.
Let ${t_{{\textnormal{\scshapeLink}}}}$ be the link time and ${t_{{\textnormal{\scshapeFind}}}}$ be the find time. In the regime ${t_{{\textnormal{\scshapeLink}}}}\ge {t_{{\textnormal{\scshapeFind}}}}$, the problem can be solved by union–find, since we have time to find the roots of $u$ and $v$ and union them if they are different. Using the upper bounds for union–find, we obtain ${t_{{\textnormal{\scshapeFind}}}}= O(\lg n / \lg t_u)$, and in particular the balanced point ${t_{{\textnormal{\scshapeLink}}}}= {t_{{\textnormal{\scshapeFind}}}}= O(\lg n / \lg\lg n)$.
If we insist on ${t_{{\textnormal{\scshapeLink}}}}= o({t_{{\textnormal{\scshapeFind}}}})$, union–find no longer suffices. In fact, we show a surprisingly abrupt trade-off that essentially signifies the “end of data structuring” even for incremental connectivity:
\[thm:inc\] Any data structure for incremental connectivity over $n$ vertices that supports edge insertions in worst-case time ${t_{{\textnormal{\scshapeLink}}}}= o(\frac{\lg
n}{\lg\lg n})$ must have worst-case query time ${t_{{\textnormal{\scshapeFind}}}}\ge n^{1-o(1)}$ in the cell-probe model with cells of $O(\lg n)$ bits.
By reduction, we get the same trade-off for link–find. This can be contrasted with the very smooth trade-off for union–find, $t_{{\textnormal{\scshapeFind}}} = O(\lg n / \lg t_{{\textnormal{\scshapeUnion}}})$, e.g., the standard union-by-rank with $O(1)$-time [[]{.nodecor}]{} and $O(\lg n)$-time [[]{.nodecor}]{}. Our result shows a remarkable dependence of edge insertion on the [[]{.nodecor}]{} operation. As soon as a new link doesn’t have time to locate the roots, the query degenerates into almost linear time.
We will also analyze the amortized bounds for link–find, which are again weaker than those for union–find when $t_q\geq t_u$, but the bounds are less striking.
We show a similar computational phenomenon for fully dynamic connectivity where both edge insertions and deletions. In this fully-dynamic case, we hit the wall even when we amortize.
\[thm:full\] Any data structure for fully dynamic connectivity in a graph of $n$ vertices with update time $t_u = o(\lg n)$ must have query time $t_q
\ge n^{1-o(1)}$. This bound allows amortization and Las Vegas randomization (expected running times), and holds in the cell-probe model with cells of $O(\lg n)$ bits.
Thorup [@thorup00connect] has an almost matching upper bound of $t_u = O(\lg n \cdot (\lg\lg n)^3)$ and $t_q = o(\lg n)$. This data structure uses both Las Vegas randomization and amortization.
#### Supporting both ${\textnormal{\scshapelink}}$ and ${\textnormal{\scshapeunion}}$.
To fully appreciate the difficulty of finding roots, consider a data structure that supports both a traditional ${\textnormal{\scshapeunion}}$ update between roots and ${\textnormal{\scshapelink}}$ between arbitrary nodes. We know from previous works that if ${\textnormal{\scshapeunion}}$ takes ${t_{{\textnormal{\scshapeUnion}}}}$ time, then the best worst-case query time is $\Theta(\log n/\log {t_{{\textnormal{\scshapeUnion}}}})$. This holds both for find-root and connecitivity queries. We can always implement links with find-root and union in $O(\log n/\log {t_{{\textnormal{\scshapeUnion}}}}~+~ {t_{{\textnormal{\scshapeUnion}}}})$ time, and this preserves the query time. However, in the case where the query time dominates the union time, that is, ${t_{{\textnormal{\scshapeUnion}}}}=o(\log
n/\log {t_{{\textnormal{\scshapeUnion}}}})$, we would like to avoid finding the roots, and get a query time closer to ${t_{{\textnormal{\scshapeUnion}}}}$.
A similar phenomenon appeared in connection with union-find with deletions. Kaplan et al. [@kaplan02union] considered this problem but wanted bounds where $n$ represented the size of the actual tree(s) involved in an operation. All worst-case bounds are trivially local, and [@kaplan02union] proved refined the standard amortized analysis to work locally, though the bound becomes a bit weird with the standard notation: $\alpha(n)$ is OK, but otherwise, it becomes $\alpha(n\cdot \lceil M/N\rceil,n)$ amortized time per find where $M$ and $N$ are the global number of finds and unions, respectively. With the notation from [@alstrup05union], the local amortized find bound is $O(\alpha_{\lceil M/N\rceil}(n))$. They showed how to augment union-find with a delete operation if we when deleting an element $x$, first find the root and then perform a local rebuilding step in the tree that $x$ is deleted from. For $t_u=O(1)$, this gave them both find-root and delete in $O(\log n)$ time. Similar to our case, they asked if the deletion time could be made better than this find time. For the deletions, the answer was yes. Alstrup et al. [@alstrup05union] proved that deletions could be supported locally in constant time without affecting the $O(\log n)$ bound on the query time.
Back to our union-find with links problem, as in the deletions case, we would like to support ${\textnormal{\scshapelink}}$ better than ${\textnormal{\scshapefind}}$ without affecting the ${\textnormal{\scshapefind}}$ bound. Here we show that any such positive result is totally impossible. If we try to beat the query time, supporting links in ${t_{{\textnormal{\scshapeLink}}}}= o(\log n/\log {t_{{\textnormal{\scshapeUnion}}}})$ time, then the query time explodes from ${t_{{\textnormal{\scshapeFind}}}}= O(\log n/\log t_u)$ to ${t_{{\textnormal{\scshapeFind}}}}\ge n^{1-o(1)}$ time.
Lower Bounds
------------
Many of the early lower bounds for union–find were in (restricted versions of) the pointer machine model [@tarjan79uf; @banachowski80uf; @lapoutre96uf; @blum86uf].
In STOC’89, Fredman and Saks [@fredman89cellprobe] were the first to show dynamic lower bounds in the cell-probe model. They studied the partial sums problem and the union–find problem. The partial sums problem asks to maintain an array $A[1{\mathinner{\ldotp\ldotp}}n]$ under pointwise updates and queries for a prefix sum: $\sum_{i \le k} A[i]$. For partial sums and for worst-case union–find, Fredman and Saks showed a lower bound of $t_q = \Omega(\lg n / \lg(t_u \lg n))$. For amortized union–find, they gave an optimal inverse-Ackermann lower bound. A different proof of the same bounds was given by Ben-Amram and Galil in FOCS’91 [@benamram01sums].
In STOC’99, Alstrup, Ben-Amram and Rauhe [@alstrup99uf] improved the trade-off for union–find to $t_q = \Omega(\lg n / \lg t_u)$, which was also the highest known trade-off for any problem. In STOC’02, Kaplan, Shafrir and Tarjan [@kaplan05meldable] showed that the optimal worst-case and amortized trade-offs for union–find also hold for a weaker Boolean version where the user specifies set identifiers and where we only have membership queries. From a lower bound perspective, the tricky part is that the query output is a single bit. Identifiers can always be viewed as special elements of sets. Thus they get the same lower bound trade-off for incremental connectivity: edges are only added between current set identifiers, and connectivity queries are between arbitrary nodes and current set identifiers. This lower-bound trade-off for incremental connectivity is tight when $t_u=\Omega(t_q)$, matching the previously mentioned upper-bounds for link–find. However, by our Theorem \[thm:inc\], the incremental connectivity queries hit a wall when the update time becomes lower.
The work of Pǎtraşcu and Demaine from STOC’04 [@patrascu06loglb] gives the best trade-offs known today, for any explicit problem. They considered partial sums and fully dynamic connectivity, and showed that, if $\max \{ t_u, t_q \} = O(B \lg n)$, then $\min \{ t_u, t_q \} = \Omega(\log_B n)$. In particular, their bounds implied $\max \{ t_u, t_q \} = \Omega(\lg n)$, whereas previous results implied $\max \{ t_u, t_q \} = \Omega(\lg n / \lg\lg n)$.
These bounds are easily seen to be optimal for the partial sums problem. The standard solution is to create an ordered binary tree with leaf set $[n]$; each internal node maintains the sum of its children. Updates and queries are trivially supported in $\Theta(\log
n)$ time. To get a trade-offs, we can instead use a $B$-tree with degree $B$. The time of an update is the height of the tree, which is $O(\log_B n)$. However, to answer a query, we need to add up all left siblings from the path to the root, so the query time is $O(B \log_B
n)$.
Our results significantly improve the known trade-offs in the regime of fast query times. Note that the previous strongest bounds from [@patrascu06loglb] could at most imply $t_q = \Omega(n^{\varepsilon})$ even for constant update time. Here ${\varepsilon}$ depends on the constant in the update time. For example, allowing only 4 cell probes for the updates, [@patrascu06loglb careful inspection] gets a query lower bound of $\Omega(n^{\frac1{16}})$. Our Theorem \[thm:full\] says for another problem that we with $o(\log n)$ probes get a query lower bound $\geq n^{1-o(1)}$ queries.
The trade-offs of [@patrascu06loglb] are optimal in the full range for the partial sums problem. For incremental and fully dynamic connectivity, the previous mild trade-offs [@kaplan05meldable; @patrascu06loglb] are optimal in the regime $t_u \gg t_q$; it is only the regime of fast updates that causes the abrupt transitions in Theorems \[thm:inc\] and \[thm:full\].
#### Lower bounds beyond the balanced tree.
The previous lower-bounds we discussed are essentially all showing that the we cannot do much better than maintaining information in a balanced tree. All operations follow well-understood paths to the roots. Trade-offs were obtained by increasing the degree, decreasing the height: the faster of updates and queries would just follow the path to the root while the slower would have to consider siblings on the way. The lower bounds from [@patrascu06loglb] are best possible in this regard.
Our stronger trade-offs for incremental and fully-dynamic connectivity shows that there is no such simple way of organizing information; that the links between arbitrary vertices changes the structure too much if the update times is not long enough, we cannot maintain the balanced information tree.
Simulation by Communication Games Results
=========================================
Generally, for the data structure problems considered, we are going to find an input distribution that will make any deterministic algorithm perform badly on the average. This also implies expected lower bounds for randomized algorithms.
Consider an abstract dynamic problem with operations ${\textnormal{\scshapeUpdate}}(u_i)$ and ${\textnormal{\scshapeQuery}}(q_i)$. Assume the sequence of operations is of fixed length, and that the type of each operation (query versus update) is fixed a priori. The “input” $u_i$ or $q_i$ of the operation is not fixed yet. Let $I_A$ and $I_B$ be two *adjacent* intervals of operations, and assume that every input $u_i$ or $q_i$ outside of $I_A \cup I_B$ has been fixed. What remains free are the inputs $X_A$ during interval $I_A$ and $X_B$ during interval $I_B$. These inputs $(X_A,X_B)$ follow a given distribution ${\mathcal{D}}$.
It is natural to convert this setting into a communication game between two players: Alice receives $X_A$, Bob receives $X_B$, and their goal is to answer the queries in $X_B$ (which depend on the updates in $X_A$). In our applications below, the queries will be Boolean, and it will even be hard for the players to compute the *and* of all queries in the $I_B$ interval. Each player is deterministic, and the two players can exchange bits of information. The last bit communicated should be the final answer of the game, which here is the and of the queries in $I_B$. The complexity of the game is defined as the total communication (in bits) between the players, in expectation over ${\mathcal{D}}$.
We will work in the cell-probe model with $w$-bit cells; in the applications below, $w = \Theta(\lg n)$. For notational convenience, we assume the data structure must read a cell immediately before writing it (but it may choose to read a cell without rewriting it). Let $W_A$ be the set of cells written during time interval $I_A$, and $R_B$ be the set of cells read during interval $I_B$.
\[lem:bloom\] For any $p \ge 0$, the communication game can be solved by a zero-error protocol with complexity ${\mathbf{E}}_{\mathcal{D}}\big[ |W_A|\cdot O(\lg
\frac{1}{p}) + O(w) \cdot \big( |W_A \cap R_B| + p |R_B| \big)
\big]$.
Alice first simulates the data structure on the interval $I_A$. The memory state at the beginning of $I_A$ is fixed. After this simulation Alice constructs a Bloom filter [@bloom70filter] with error (false positive) probability $p$ for the cells $W_A$. The hash functions needed by the Bloom filter can be chosen by public coins, which can later be fixed since we are working under a distribution. Alice’s first message is the Bloom filter, which requires $|W_A| \cdot O(\lg
\frac{1}{p})$ bits.
Bob will now attempt to simulate the data structure on $I_B$. The algorithm may try to read a cell of the following types:
a cell previously written during $I_B$: Bob already knows its contents.
a cell that is positive in the Bloom filter: Bob sends the address of the cell to Alice, who replies with its contents; this exchange takes $O(w)$ bits.
a cell that is negative in the Bloom filter: Bob knows for sure that the cell was not written during $I_A$. Thus, he knows its contents, since it comes from the old fixed memory snapshot before the beginning of $I_A$.
With this simulation, Bob knows all the his answers and can transmit the final bit telling if they are all true. The number of messages from Bob is $|W_A \cap R_B|$ (true positives) plus an expected number of false positives of at most $p |R_B|$.
We will use the simulation to obtain lower bounds for $|W_A \cap
R_B|$, comparing the complexity of the protocol with a communication lower bound. This simulation works well when $|W_A \cap R_B| \approx
|W_A \cup R_B| / \frac{\lg n}{\lg\lg n}$, since we can use $p \approx
\frac{1}{\lg n}$, and make the term $|W_A \cap R_B|$ dominate. Unfortunately, it does not work in the regime $|W_A \cap
R_B| \approx |W_A \cup R_B| / \lg n$, since one of the terms proportional to $|W_A|$ or $|R_B|$ will dominate, for any $p$.
To give a tighter simulation, we use a stronger communication model: nondeterministic complexity. In this model, a prover sends a public proof $Z$ to both Alice and Bob. Alice and Bob independently decide whether to accept the message, and they can only accept if the output of the communication game is “true” (i.e. all queries in $I_B$ return true). In this model Alice and Bob do not communicate with each other. Alice’s answer is a deterministic function $f_A(X_A,Z)$ of her own input and the public proof. Similarly, we have Bob’s answer $f_B(X_B,Z)$. For the protocol to be correct, $f_A(X_A,Z)$ and $f_B(X_B,Z)$ may only both be true if this is the answer to the game.
Our goal for the prover is to define a short public proof $Z(X_A,X_B)$ that will lead Alice and Bob to the desired answer $f_A(X_A,Z(X_A,X_B))\wedge f_B(X_B,Z(X_A,X_B))$. The complexity of the protocol is the of the game should be the and of all queries in $I_B$. Since we are working under a distribution, the bit length of the prover’s message $Z(X_A,X_B)$ is a random variable, and we define the complexity of the protocol as its expectation.
\[lem:bloomier\] The communication game can be solved by a nondeterministic protocol with complexity ${\mathbf{E}}_{\mathcal{D}}\big[ O(w) \cdot |W_A \cap R_B| + O(|W_A
\cup R_B|) \big]$.
We will use a retrieval dictionary (a.k.a. a Bloomier filter, or a dictionary without membership). Such a dictionary must store a set $S$ from universe $U$ with $k$ bits of associated data per element of $S$. When queried for some $x\in S$, the dictionary must retrieve $x$’s associated data. When queried about $x \notin S$, it may return anything. One can construct retrieval dictionaries with space $O(k|S|
+ \lg\lg |U|)$; see e.g. [@dietzfel08retrieval].
The message $Z(X_A,X_B)$ of the prover will consist of the addresses and contents of the cells $X = |W_A\cap R_B|$, taking $O(w)$ bits each. In addition, he will provide a retrieval dictionary for the symmetric difference $W_A
\Delta R_B = (W_A \setminus R_B) \cup (R_B \setminus W_A)$. In this dictionary, every element has one associated bit of data: zero if the cell is from $W_A \setminus R_B$ and one if from $R_B \setminus W_A$. The dictionary takes $O(\lg w + |W_A \cup R_B|)$ bits.
Alice first simulates the data structure on $I_A$. Then she verifies that all cells $X$ were actually written ($X \subseteq W_A$), and their content is correct. Furthermore, she verifies that for all cells from $W_A \setminus X$, the retrieval dictionary returns zero. If some of this fails, she rejects with a false.
Bob simulates the data structure on $I_B$. The algorithm may read cells of the following types:
cells previously written during $I_B$: Bob knows their contents.
cells from $X$: Bob uses the contents from public proof (Alice verified these contents).
cells for which the retrieval dictionary returns *one*: Bob uses the contents from the fixed memory snapshot before the beginning of $I_A$ (Alice verified she didn’t write such cells).
cells for which the retrieval dictionary return *zero*: Bob rejects. The prover is trying to cheat, since in a correct simulation all cells of $R_B \setminus X$ has a one bit in the dictionary.
If neither player rejects, we know that $R_B \setminus X$ is disjoint from $W_A \setminus X$, so the simulation of Bob is correct. Finally Bob rejects if any of his answers are false.
Lower Bound for Incremental Connectivity
========================================
Any data structure for incremental connectivity over $n$ vertices that supports edge insertions between roots in worst-case time ${t_{{\textnormal{\scshapeUnion}}}}=
o(\frac{\lg n}{\lg\lg n})$ and arbitrary edge insertions in worst-case time ${t_{{\textnormal{\scshapeLink}}}}= o(\frac{\lg n}{\lg\lg n})$ must have query time ${t_{{\textnormal{\scshapeFind}}}}\ge n^{1-o(1)}$.
Let ${\varepsilon}=o(1)$ be such that ${t_{{\textnormal{\scshapeUnion}}}}= o({\varepsilon}^2 \lg n / \lg\lg n)$. Define $B = \lg^2 n$, $C = n^{\varepsilon}$, and $M=n^{1-{\varepsilon}}$. The starting point of our hard instance is essentially taken from Fredman and Saks’ seminal paper [@fredman89cellprobe]. The hard instance will randomly construct a forest of $M$ trees. Each tree will be a perfect tree of degree $B$ and height $\log_B (n/M)$. On layer $0$ of the forest we have the $M$ roots. On layer $i$, we have exactly $M \cdot B^i$ vertices with $B^i$ vertices from each tree.
We can describe the edges between level $i$ and $i-1$ as a function $f_i: [M\cdot B^i] \to [M\cdot B^{i-1}]$ that is balanced: for each $x
\in [MB^{i-1}]$, $|(f_i)^{-1}(x)| = B$. We will use the following convenient notation for composition: $f_{\ge i} = f_i \circ f_{i+1}
\circ \cdots$. For example, the ancestor on level $i-1$ of leaf $x$ is $f_{\ge i}(x)$.
Our hard instance will insert the edges describing $f_i$’s in bottom-up fashion (i.e. by decreasing $i$, from the largest level up to the roots). We call “epoch $i$” the period of time when the edges $f_i$ are inserted. Let $W_i$ (respectively $R_i$) be the cells written (respectively, read) in epoch $i$. Observe that $|W_i| + |R_i|
\le M\cdot B^i {t_{{\textnormal{\scshapeUnion}}}}$. We will use the following convenient notation for set union: $W_{\le i} = \bigcup_{j \le i} W_j$. The cells $W_i
\setminus W_{<i}$ are those *last* written in epoch $i$.
All the above edges where added in union-find style from roots of current trees, and indeed the above constitutes the hard case for union-find from [@fredman89cellprobe]. At this point [@fredman89cellprobe] shows that finding a root from a random leaf would entail reading cells from most epochs in $\Omega(\log n/\log B)$ expected time.
Our goal is to show that linking arbitrary vertices may lead to much more expensive queries. We will describe some very powerful metaqueries that combines links to roots and leaves with a few connectivity to reveal far more information than if we only had the regular connectivity queries. The metaqueries will be provably hard to answer, so if the links are done too quickly, the queries must be very slow.
Our graph contains $C$ additional special vertices, conceptually colored with the colors $1 {\mathinner{\ldotp\ldotp}}C$. Each colored vertex is connected to $M/C$ nodes on level 0 (the final roots of our trees). This is done in a fixed pattern: colored vertex $1$ is connected to roots $1, \dots, M/C$; colored vertex $2$ to the next $M/C$ roots; etc. These edges can be inserted at the very beginning of the execution, prior to any interesting updates.
At the end of epoch 1 all trees are complete. In this state, we say the [*root color*]{} of a vertex is the color that its root is connected to. Conceptually, the hard distribution colors a random set $Q$ of exactly $M$ leaves and verifies that these are the root colors.
To implement this test by incremental connectivity operations (${\textnormal{\scshapeLink}}$), we first link each query leaf to the proposed colored vertex. Then, for $i=2 {\mathinner{\ldotp\ldotp}}C$, we query whether colored vertex $i$ is connected to colored vertex $i-1$, and then insert an edge between these two color nodes. The metaquery returns “true” iff all connectivity queries are negative.
We claim that if the metaquery answers true, the coloring of $Q$ must be consistent with the coloring of the roots. Indeed, if some leaf is colored $i$ and its root is colored $j\ne i$, this inconsistency is caught at step $\max\{i,j\}$. At this step, everything with color $\le
\max\{i,j\}-1$ has been connected into a tree, so the connectivity query will return true.
Let ${\chi}(Q)$ be the coloring of leaves in $Q$ that matches their root colors. In the hard distribution, the metaquery always receives proposed colors from ${\chi}(Q)$, so it should answer true. Nevertheless, the data structure will need to do a lot of work to verify this. Let $R^Q$ be the cells read during the metaquery. We have $|R^Q| \le C\cdot t_q
+ 2M\cdot {t_{{\textnormal{\scshapeUnion}}}}$. The main claim of our proof is:
\[lem:inc-main\] For any $i \in \{1, \dots, \log_B (n/M) \}$, we have ${\mathbf{E}}[|R^Q \cap (W_i
\setminus W_{<i})|] = \Omega({\varepsilon}M)$.
Before we prove the lemma, we show that it implies our lower bound. The sets $W_i \setminus W_{<i}$ are disjoint by construction, so $\sum_i {\mathbf{E}}[|R^Q \cap (W_i \setminus W_{<i})|] \le {\mathbf{E}}[|R^Q|]$. Remember that we have $\log_B(n/M) = O(\log(n^{\varepsilon}) / \lg\lg n) = O({\varepsilon}\lg n
/ \lg\lg n)$ epochs. Thus ${\mathbf{E}}[|R^Q|] = \Omega(M \cdot {\varepsilon}^2 \lg n /
\lg\lg n)$. But we always have $|R^Q| \le C \cdot t_q + 2M \cdot t_u = C t_q
+ o(M \frac{{\varepsilon}^2 \lg n}{\lg\lg n} )$, by choice of ${\varepsilon}$. It follows that $Ct_q$ is the dominant term in ${\mathbf{E}}[|R^Q|]$, so $t_q = \Omega(M{\varepsilon}^2 (\lg n /
\lg\lg n)/C)\ge n^{1-2{\varepsilon}}$.
#### Proof of Lemma \[lem:inc-main\].
Fix $i$. We will prove the stronger statement that the lower bound holds no matter how we fix the edges outside epoch $i$ (all $f_j$’s for $j\ne
i$).
To dominate the work of later epochs $i-1,\dots,1$, we consider $B^i$ i.i.d. metaqueries. Choose sets $Q^1, Q^2, \dots,Q^{B^i}$ independently, each containing $M$ uniformly chosen leaves. Starting from the memory state where all trees are completely built and the roots have been colored, we simulate each metaquery $(Q^j, {\chi}(Q^j))$ in isolation. We do not need to write any cells in this simulation, for the cell-probe model has unbounded state to remember intermediate results and in our hard distribution there is no operation after the metaquery. Thus the simulations of the different metaqueries do not influence each other. Let $R^\star$ be the cells read by all $B^i$ metaqueries. By linearity of expectation, ${\mathbf{E}}[|R^\star \cap (W_i \setminus W_{<i})|] \le B^i \cdot {\mathbf{E}}[|R^Q \cap
(W_i \setminus W_{<i})|]$.
Let $Q^\star = \bigcup_j Q^j$. Since we have fixed all $f_{>i}$, asking about the root color of a leaf $q\in Q^\star$ is equivalent to asking about the root color of node $f_{>i}(q)$ on level $i$.
We have ${\mathbf{E}}[|f_{>i}(Q^\star)|] \ge (1- \frac{1}{e}) MB^i$.
Each leaf $x$ in some $Q^j$ is chosen uniformly, so its ancestor $f_{<i}(x)$ is also uniform. The $M\cdot B^i$ trials are independent (for different $Q^j, Q^k$), or positively correlated (inside the same $Q^j$, since the leaves must be distinct). Thus, we expect to collect $(1-1/e) MB^i$ distinct ancestors.
By the Markov bound $|f_{>i}(Q^\star)| \ge \frac{1}{2} MB^i$ with probability at least $1 - 2/e$. Thus we may fix the sequence $(Q^1, Q^2, \dots,Q^{B^i})$ to a value that achieves $|f_{>i}(Q^\star)| \ge \frac{1}{2} MB^i$ while increasing ${\mathbf{E}}[|R^\star \cap (W_i \setminus W_{<i})|]$ by at most $(1-2/e)^{-1} = O(1)$.
The only remaining randomness in our instance are the edges $f_i$ from epoch $i$ and the proposed colorings ${\chi}(Q^j)$ given to each metaquery $Q^j$. To be valid, these colorings are functions of $f_i$, for as soon as we know $f_i$, we know the whole forest including the root colors of all the leaves in the different $Q^j$. The metaquery colors have to agree on common leaves, so they provide us a coloring ${\chi}(Q^*)$. With $f_i$ yet unknown, we claim that ${\chi}(Q^\star)$ has a lot of entropy:
\[cl:balance-color\] ${\mathrm{H}}({\chi}(Q^\star))=\Omega(MB^i \lg C)$.
Let $X$ be the unknown coloring of all vertices on level $i$. We claim it has entropy ${\mathrm{H}}(X) = MB^i \cdot \log_2 C - O(C\lg
n)$. We have not fixed anything impacting this coloring so $X$ is a random balanced vector from $[C]^{MB^i}$. Indeed, any balanced coloring is equiprobable, because the coloring of the roots is balanced, all trees have the same sizes, and $f_i$ is a random balanced function. We claim that it has entropy ${\mathrm{H}}(X) = MB^i \cdot \log_2 C - O(C\lg
n)$. The number of balanced colorings is given by the multinomial coefficient $\binom{MB^i}{MB^i/C, ~MB^i/C,
~\dots}$. This is the central multinomial coefficient, so it is the largest. It must therefore be at least a fraction $(MB^i)^{-C} \ge
n^{-C}$ of the sum of all multinomial coefficients. This sum is $C^{MB^i}$ (the total number of possible colorings), so ${\mathrm{H}}(X) \ge
\log_2 (C^{MB^i} / n^C)=MB^i\log_2 C-C\log_2 n$.
We argue that ${\mathrm{H}}({\chi}(Q^\star)) =\Omega(MB^i \lg C)$. Indeed, ${\chi}(Q^\star)$ reveals the coloring of vertices $f_{<i}(Q^\star)$ on level $i$, which number at least $\frac{1}{2}
MB^i$. Given ${\chi}(Q^\star)$, to encoding $X$, we just write all other colors explicitly using $\frac{1}{2} MB^i \log_2 C$ bits. Therefore ${\mathrm{H}}({\chi}(Q^\star)) \geq {\mathrm{H}}(X)-\frac{1}{2} MB^i \log_2 C\geq
MB^i \log_2 C -C\lg_2 n -\frac{1}{2}MB^i \log_2 C =\Omega(MB^i \lg C)$.
We consider the communication game in which Alice represents the time of epoch $i$ (her private input is $X_A=f_i$), and Bob represents the time of epochs $i-1, \dots, 1$ and the metaqueries (his private input is $X_B={\chi}(Q^\star)$). Their goal is to determine whether all the metaqueries return true.
\[cl:high-complex\] Any zero-error protocol must have average case bit complexity $\Omega(MB^i \lg C)$.
We turn our attention to the communication game. The set of inputs of Alice and Bob that lead to a fixed transcript of the communication protocol forms a combinatorial rectangle. More precisely, a transcript $t$ represents a sequence of transmissions between Alice and Bob. On Alice’s side, there will be a certain set ${\cal X}^t_A$ of inputs making her follow $t$ provided that Bob follows $t$, and we have a corresponding input set ${\cal X}^t_B$ from Bob. Inputs $X_A$ and $X_B$ will lead to $t$ if and only if $(X_A,X_B)\in {\cal X}^t_A\times {\cal X}^t_B$. Since the players must verify $X_B={\chi}(Q^\star)$ and the protocol has zero error, the rectangle cannot contain two inputs of Bob with different ${\chi}(Q^\star)$, that is, $|{\cal X}^t_B|=1$ for all valid $t$. Thus the transcript for a coloring ${\chi}(Q^\star)$ is unique with no smaller entropy.
We will use Lemma \[lem:bloom\] to obtain a communication protocol, setting the rate of false positives in the Bloom filter to $p= 1/\lg n$. The cells written in Alice’s interval are precisely $W_i$; the cells read in Bob’s interval are $R_{<i} \cup R^\star$ where $R^\star$ is the union of the cells read by all the metaqueries. By Lemma \[lem:bloom\], the communication complexity is: $$\begin{aligned}
& & {\mathbf{E}}\big[
|(R_{<i} \cup R^\star) \cap W_i| \cdot O(\lg n)
~+~ W_i \cdot O(\lg\lg n)
~+~ \tfrac{1}{\lg n} |R_{<i} \cup R^\star|\cdot O(\lg n) \big] \\
&\le&
{\mathbf{E}}[|R^\star \cap W_i|]\cdot O(\lg n)
~+~ O(M B^i t_u \cdot \lg\lg n) ~+~ O(MB^{i-1} t_u \cdot \lg n) ~+~ O(|R^\star|)\end{aligned}$$ We compare this to the lower bound of $\Omega(MB^i \lg C) =
\Omega(MB^i \cdot {\varepsilon}\lg n)$ from Claim \[cl:high-complex\]. Remember that $t_u = o({\varepsilon}^2 \lg n /
\lg\lg n)$, so the second term is $o(MB^i {\varepsilon}^2 \lg n)$, which is asymptotically lower than the lower bound. Also, we set $B = \lg^2 n$, so the third term is $o(MB^i)$. Finally, we have $|R^\star| = O(B^i
Mt_u)$. To see this, recall that $|R^\star|\leq B^i (Mt_u+C t_q)$, so if the statement was false, we would have $B^i C t_q=\omega(B^iM)$ and $t_q=\omega(M/C)=\omega(n^{1-2{\varepsilon}})$. Since $O(B^i Mt_u)$ is also low order term, the first term must dominate, which means ${\mathbf{E}}[|R^\star \cap (W_i \setminus W_{<i})|] = \Omega(MB^i
{\varepsilon})$. Therefore, ${\mathbf{E}}[|R^\star \cap (W_i \setminus W_{<i})|] =
\Omega({\varepsilon}M)$. This completes the proof of Lemma \[lem:inc-main\] from which we got our lower bound for incremental connectivity.
Lower Bound for Dynamic Connectivity
====================================
Any data structure for dynamic connectivity in graphs of $n$ vertices that has (amortized) update time $t_u = o(\lg n)$ must have (amortized) query time $t_q \ge
n^{1-o(1)}$.
Let ${\varepsilon}$ be such that $t_u = o({\varepsilon}^2 \lg n)$, and define $M =
n^{1-{\varepsilon}}$ and $C = n^{\varepsilon}$. The shape of our graphs is depicted in Figure \[fig:graphs\]. The vertices are points of a grid $[M] \times
[n/M]$. The edges of our graph are matchings between consecutive columns. Let $\pi_1, \dots, \pi_{n/M -1}$ be the permutations that describe these matchings. We let $\pi_{\le j} = \pi_j\circ \pi_{j-1}
\circ \dots \circ \pi_1$. Node $i$ in the first column is connected in column $j+1$ to $\pi_{\le j}(i)$.
The graph also contains $C$ special vertices, which we imagine are colored with the colors $1, \dots, C$. At all times, a colored vertex is connected to a fixed set of $M/C$ vertices in the first column. (For concreteness, colored vertex $1$ is connected to vertices $1, \dots, M/C$; colored vertex $2$ to the next $M/C$ vertices; etc.)
We will allow two meta-operations on this graph: ${\textnormal{\scshapeUpdate}}$ and ${\textnormal{\scshapeQuery}}$. Initially, all permutations are the identity (i.e. all edges are horizontal). ${\textnormal{\scshapeUpdate}}(j, \pi_{new})$ reconfigures the edges between columns $j$ and $j+1$: it sets $\pi_j$ to the permutation $\pi_{new}$. This entails deleting $M$ edges and inserting $M$ edges, so ${\textnormal{\scshapeUpdate}}$ takes time $2M\cdot t_u$.
${\textnormal{\scshapeQuery}}(j, x)$ receives a vector $\chi \in [C]^M$, which it treats as a proposed coloring for vertices on column $j$. The goal of the query is to test whether this coloring is consistent with the coloring of the vertices in the first column. More specifically, a node $i$ of color $a$ in the first column must have ${\chi}[\pi_{<j}(i)] = a$. A [[]{.nodecor}]{} can be implemented efficiently by connectivity operations. First each vertex $i$ in column $j$ is connected to the colored vertex ${\chi}[i]$. Then, for $i = 2 {\mathinner{\ldotp\ldotp}}M$, we run a connectivity query to test whether colored vertex $i$ is connected to colored vertex $i-1$. If so, [[]{.nodecor}]{} return false. Otherwise, it inserts an edge between colored vertices $i$ and $i-1$ and moves to the next $i$. At the end, [[]{.nodecor}]{} deletes all vertices it had inserted. The total cell-probe complexity of ${\textnormal{\scshapeQuery}}$ is $O(M)\cdot t_u + C \cdot t_q$. It is easy to observe that this procedure correctly tells whether the colorings are consistent (as in our instance of incremental connectivity).
We will now describe the hard distribution over problem instances. We assume $\frac{n}{M}-1$ is a power of two. Let $\sigma$ be the bit-reversal permutation on $\{0, \dots, \frac{n}{M} - 2\}$: $\sigma(i)$ is the reversal of $i$, treated as a vector of $\log_2
(\frac{n}{M} - 1)$ bits. For $i= 0, \dots, \frac{n}{M}-1$, we execute an ${\textnormal{\scshapeUpdate}}$ to position $j=\sigma(i)+1$, and a ${\textnormal{\scshapeQuery}}$ to the same position $j$. The update sets $\pi_j$ to a new random permutation. The query always receives the consistent coloring, and should answer true. The total running time is $$T \leq n/M(2Mt_u+O(M)t_u + C t_q)=O(n t_u + (n/M)C t_q).$$ If we can prove a lower bound $T=\omega(nt_u)$, then this will yield a high lower bound for $t_q$.
For the lower bound proof, we consider a perfect ordered binary tree with $n/M - 1$. The leaves are associated with the pairs of ${\textnormal{\scshapeUpdate}}$ and ${\textnormal{\scshapeQuery}}$ operations in time order. Let $W(v)$ (respectively $R(v)$) be the set of cells written (respectively, read) while executing the operations in the subtree of $v$. Note that $W(v)
\subseteq R(v)$, since we have assumed a cell must be read before it is written. Our main claim is:
\[lem:full-main\] Let $v$ be a node with $2k$ leaves in its subtree, and let $v_L, v_R$ be its left and right children. Then ${\mathbf{E}}[|W(v_L) \cap R(v_R)| +
\frac{1}{\lg n} |W(v_L) \cup R(v_R)|] = \Omega(k\cdot {\varepsilon}M)$.
Before we prove the lemma, we use it to derive the desired lower bound. We claim that the total expected running time is $T \ge \sum_v
{\mathbf{E}}[|W(v_L) \cap R(v_R)|]$, where the sum is over all nodes in our lower bound tree. Consider how a fixed instance is executed by the data structure. We will charge each read operation to a node in the tree: the lowest common ancestor of the time when the instruction executes, and the time when the cell was last written. Thus, each $W(v_L) \cap R(v_R)$ corresponds to (at least) one read instruction, so there is no double-counting in the sum.
We now sum the lower bound of Lemma \[lem:full-main\] over all nodes; observe that $\sum_v k_v = \Theta(\frac{n}{M} \lg
\frac{n}{M})$, since the tree has $n/M - 1$ leaves. We obtain $\sum_v
{\mathbf{E}}[|W(v_L) \cap R(v_R)|] + \frac{1}{\lg n} \sum_v {\mathbf{E}}[|W(v_L) \cup
R(v_R)|] = \Omega(\frac{n}{M} \lg \frac{n}{M} \cdot {\varepsilon}M)$. The first term is at most $T$, as explained above. In the second term is also bounded by $T$. This is because $\sum_v {\mathbf{E}}[|W(v_L) \cup R(v_R)|] \le T \lg \frac{n}{M}$ since every cell probe is counted once for every ancestor of the time it executes. Thus $2T \geq \Omega(\frac{n}{M} \lg \frac{n}{M} \cdot {\varepsilon}M) =
\Omega({\varepsilon}^2 n \lg n)$. In our construction, the total running time was $T = O(n t_u + \frac{n}{M} C t_q)$. Since $t_u = o({\varepsilon}^2 \lg n)$, the second term must dominate: $\frac{nC}{M} t_q = \Omega({\varepsilon}^2 n \lg
n)$, so $t_q > M/C = n^{1-2{\varepsilon}} = n^{1-o(1)}$.
#### Proof of Lemma \[lem:full-main\].
We will prove the stronger statement that the lower bound holds no matter how we fix the updates outside node $v_L$.
We transform the problem into the natural communication game: Alice receives the update permutations in the subtree $v_L$ and Bob receives the colorings of the queries in the subtree $v_R$ (the updates are fixed). They have to check whether all queries are positive in the sequence of [[]{.nodecor}]{} and [[]{.nodecor}]{} operations defined by their joint input.
We apply Lemma \[lem:bloomier\] to construct a nondeterministic communication protocol for this problem, with complexity ${\mathbf{E}}[ |W(v_L)
\cap R(v_R)| \cdot O(\lg n) + O(|W(v_L) \cup R(v_R)|)]$. The conclusion of Lemma \[lem:full-main\] follows by comparing this protocol to the following communication lower bound:
The game above has nondeterministic (average-case) communication complexity $\Omega(k M \lg C)$.
Let $X_A$ and $X_B$ be the inputs of the two players. For any choice of $X_A$, there is a unique sequence of colorings $X_B$ that Bob should accept. As in the proof of Lemma \[cl:high-complex\], we conclude that the public proof is an encoding of $X_B$ so we can lower bound the complexity via ${\mathrm{H}}(X_B)$.
Let $J_A$ and $J_B$ be the columns touched (updated and queried) in Alice’s input and in Bob’s input. Bob’s input consists of the coloring of column $j$, for each $j\in J_B$. This is $\pi_{<j}$ applied to the fixed coloring in the first column.
Since $J_A$ and $J_B$ are defined by the bit-reversal permutation, we know that they interleave perfectly: between every two values in the sorted order of $J_B$, there is a unique value in $J_A$. Thus, the coloring for different $j\in J_B$ are independent random variables, since an independent uniform permutation from $J_A$ is composed into $\pi_{<j}$ compared to all indices from $J_B$ below $j$. Each coloring is uniformly distributed among balanced colorings, so it has entropy $M\lg C -
O(C\lg M)$ (c.f. proof of Claim \[cl:balance-color\]). We conclude that ${\mathrm{H}}(X_B) =
\Omega(kM\lg C)$.
Amortized link-find bounds
==========================
In this section we consider the amortized complexity of the link-find problem which is like the union-find problem except that we can link arbitrary nodes, not just roots. In link-find, we may not necessarily have an obvious notion of a root that we can find. The fundamental requirement to a component is that if we call find from any vertex in it, we get the same root as long as the component is not linked with other components.
Let $u$ be the number of updates and $q$ the number of queries. With union-find, the complexity over the whole sequence is $\Theta(\alpha(q,u)q)$ if $q\geq u$, and $\Theta(\alpha(q,q)q+u)$ if $q\leq u$. With link-find, we get the same complexity when $q\geq u$, but a higher complexity of $\Theta(\alpha(q,u)u)$ when $q\leq u$. Thus, with link-find, we get a symmetric formula in $q$ and $u$ of $$\label{eq:link-find}
\Theta(\alpha(\max\{q,u\},\min\{q,u\})\max\{q,u\}).$$ We get the upper-bound in (\[eq:link-find\]) via a very simple reduction to union-find.
The link-find data structure
----------------------------
Nodes have three types: free, leaf, and union nodes. A leaf node has a pointer to a neighboring union node, and the union nodes will participate in a standard union-find data structure. The parent of a leaf is the union node it points to. The parent of a union node is as in the union-find structure and the parent of a root is the root itself.
All nodes start as free nodes. We preserve the invariant that if a component has a free node, then all nodes in the component are free.
To perform a find on a free node $v$, we scan the component of $v$. If it is a singleton, we just return it. Otherwise, assuming some initial tie-breaking order, we make the smallest node in the component a union node and all other nodes leaf nodes pointing to is. The union node which is its own root is returned. All this is paid for by the nodes that lost their freedom.
To perform a find on a non-free node, we perform it on the parent which is in the union-find data structure.
We now consider the different types of links. When we perform link between two free nodes, nothing happens except that an edge is added in constant time.
If we link a free node $v$ with a non-free node $w$, we make all nodes in the components of $v$ leaves pointing to the parent of $w$. This is paid for by the new leaves.
If we link two non-free nodes, we first perform a find from their parents which are union nodes. If they have different roots we unite them.
This completes the description of our link-find data structure which spends linear time reducing to a union-find data structure. A union node requires a find on a non-singleton node, so the number of union nodes is at most $\min\{q,u\}$. Concerning finds in the union-find data structure, we get one for each original find on a non-free node. In addition, we get two finds for each link of two non-free nodes, adding up to at most $q+2u$ finds. Our total complexity is therefore $$O(u+q+\alpha(q+2u,\min\{q,u\})(q+2u))=O(\alpha(\max\{q,u\},\min\{q,u\})
\max\{q,u\}).$$ We are going to present a matching lower bound.
The link-find data structure for a forest
-----------------------------------------
We will now show that it is the links between nodes in the same components that makes link-find harder than union-find in the sense that if no such links appear, we get the same $O$-bound as with union-find.
The modification to the above link-find reduction is simple. Using standard doubling ideas, we can assume that $u$ and $q$ are known in advance. If $q\geq u$, we are already matching the union-find bound, so assume $q\leq u$.
To do a find on a free node, we again scan its component. However, if it has less than $\alpha(q,q)$ nodes, we just return the smallest but leaving the component free. Otherwise, as before, we make the smallest node a union node and all other nodes leaf nodes pointing to it. This is the only change to our link-find algorithm.
In the case where the component has $\alpha(q,q)$ nodes, we clearly pay only $O(\alpha(q,q))$ for a find. The advantage is that we now create at most $u/\alpha(q,q)$ union nodes. Links involving a free node have linear total cost, and now, when we perform a link of non-free nodes, we know they are from different components to be united, so this will reduce the number of union roots by one. Hence we get at most $2u/\alpha(q,q)$ finds resulting from these links. Thus, in the union-find data structure, we end up with $q+2u/\alpha(q,q)$ finds and $u/\alpha(q,q)$ unions. The total cost is $$O(u+q+\alpha(q+2u/\alpha(q,q),u/\alpha(q,q))(q+2u/\alpha(q,q))=
O(\alpha(q,q)q+n)$$ time. The simplification uses that $\alpha$ is increasing in its first and decreasing in its second argument, and that the whole time bound is linear if $q\leq u/\alpha(q,q)$.
Appendix $\alpha$.Lower Bounds for Amortized Link–Find {#appendix-alpha.lower-bounds-for-amortized-linkfind .unnumbered}
======================================================
We will now sketch a proof for the lower-bound in (\[eq:link-find\]) with $u$ link updates and $q$ find queries. When $q\geq u$, we get this from the union-find lower bound of $\Omega(\alpha(q,u)q)$ from [@fredman89cellprobe]. However, for $q\ll u$, we need to prove a higher lower-bound than that for union-find. The lower bound we want in this case is $\Omega(\alpha(u,q)u)$.
We would get the desired lower bound if we could code a union-find problem with $\Omega(q)$ updates and $\Omega(u)$ queries. We cannot make such a black-box reduction, but we can do it inside the proof construction from [@fredman89cellprobe]. We will only present the idea in the “reduction”. For a real proof one has to carefully examine the whole proof from [@fredman89cellprobe] to verify that nothing really breaks.
The lower bound construction from [@fredman89cellprobe] proceeds in rounds. We start with singleton roots. In a union round, we pair all current roots randomly, thus halving the number of roots. In a find round, we perform a number of finds on random leaves. The number of finds are adjusted depending on the actions of the data structure. From [@kaplan05meldable] we know that the lower bound also holds if the finds just have to verify the current root of a node.
In our case, we will start with $n$ roots. In a union-round, we just link roots as in union-find. However, in a find round, instead of calling find from a leaf $v$, we link $v$ to its current root $r$. We want to turn this leaf-root link into a verification. We will not do that for the individual links, but we will do it for the find-round as a whole (one needs to verify that this batching preserves the lower-bound). At the end of the find-round, we simply perform a find on each root. All these finds should return the root itself. If one of the links $(v,r)$ had gone to the wrong root and $r'$ was the correct root, then $r$ and $r'$ would be connected in the same tree, which means that they cannot both be roots. One of the finds would therefore return a different root. If the union-find problem we code used $f$ finds, then our link-find solution ends up with $u=n-1+f$ link updates and $q=n-1$ find verifications, hence with the desired lower bound of $$\Omega(\alpha(f,n)f)=\Omega(\alpha(u,q)u).$$
|
---
abstract: 'We present a probabilistic model for Sketch-Based Image Retrieval (SBIR) where, at retrieval time, we are given sketches from novel classes, that were not present at training time. Existing SBIR methods, most of which rely on learning class-wise correspondences between sketches and images, typically work well only for previously seen sketch classes, and result in poor retrieval performance on novel classes. To address this, we propose a generative model that learns to generate images, conditioned on a given novel class sketch. This enables us to reduce the SBIR problem to a standard image-to-image search problem. Our model is based on an inverse auto-regressive flow based variational autoencoder, with a feedback mechanism to ensure robust image generation. We evaluate our model on two very challenging datasets, Sketchy, and TU Berlin, with novel train-test split. The proposed approach significantly outperforms various baselines on both the datasets.'
author:
- |
Vinay Kumar Verma\*, Aakansha Mishra$^{\ddag}$, Ashish Mishra$^{\dag}$ and Piyush Rai\*\
$^*$IIT-Kanpur, $^{\ddag}$IIT-Guwahati, $^{\dag}$IIT-Madras\
[vkverma@cse.iitk.ac.in, ak.kkb@iitg.ac.in, mishra@iitm.ac.in,piyush@cse.iitk.ac.in]{}
bibliography:
- 'egbib.bib'
title: 'Generative Model for Zero-Shot Sketch-Based Image Retrieval'
---
Introduction
============
The commonly used approaches to search for an image from a database of images are: (1) Text-based image retrieval, in which we search for an image using a text-based query and (2) Content-based image retrieval (CBIR), in which a related image is used as a query image. Image as the query has a much richer content as compared to text-based query. CBIR gives excellent search results but requires giving a *real* image as the query, which may not always be possible. Often it is more convenient to draw an outline sketch of the image and use that as a query to search for the desired image(s). The retrieval of images by giving the sketch as a query is termed as sketch-based image retrieval (SBIR) [@conf/iccv/SBIR1; @conf/cvpr/SBIR2; @conf/cvpr/SBIR15; @Siamese2]. The topic has drawn considerable attention recently. However, existing SBIR systems assume that the class represented by the input sketch at query time was also present in the image-sketch pairs used to train the SBIR model, and consequently, these systems suffer when the input sketch is from a previously *unseen/novel* class.
![Illustration of Zero-Shot Sketch-Based Image Retrieval (ZS-SBIR)[]{data-label="fig:general"}](zssbir2.jpg){height="5cm" width="8.5cm"}
In this work, we present a method to handle the SBIR task for the unseen/novel class at test time. These novel classes are either absent at the training time or not used in training. This type of setup, to handle the previously unseen classes at test time is called Zero-Shot Learning (ZSL), and has been extensively investigated recently for problems, such as image classification [@vinaycvpr; @ConSE; @verma2017simple; @akata2015evaluation], action classification [@liu2011recognizing; @mishrawacv], image tagging [@zsl_tagging], and visual question answering [@journals/corr/RamakrishnanPSM17] etc. To the best of our knowledge, the only works that have investigated SBIR in the zero-shot setting include [@imagehashing; @ashisheccv2018]. Among these, [@imagehashing] used a hashing approach for the ZS-SBIR. This approach is motivated by other ZSL approaches where some side information about unseen classes is present, e.g., their textual description, word2vec or attribute based vectors are used for the knowledge transfer. Recently [@ashisheccv2018] proposed a vanilla conditional variational autoencoder (CVAE) architecture and adversarial autoencoder for the ZS-SBIR task.
In this paper, we address the drawbacks of existing approaches for SBIR to handle the retrieval of *novel/unseen* class examples. We propose a *conditional* generative model that can generate image features conditioned on the attributes (raw sketch or word2vec[@word2vec]) of a given class. Like [@ashisheccv2018] we also have a generative model, but our approach is significantly different from their model which uses a standard conditional VAE. In contrast, our proposed approach is built upon the *Inverse autoregressive flow* (IAF) based variational autoencoder [@kingma2016improved], with a *feedback* based mechanism [@controlabletext]. The IAF helps to learn the complex latent-space distribution of the images while the feedback mechanism further helps in making the generated image distribution follow the original distribution more closely. The other recently proposed ZS-SBIR approach [@imagehashing] requires side information in the form of description of the sketch, which may not always be available. In contrast, our proposed approach requires no side information and still performs significantly better than [@imagehashing]. We also use a residual decoder that helps to learn a complex model with a deeper network. Notably, since we are able to generate images from any specified class, we are able to transform the zero-shot problem into a typical supervise learning problem. The main contributions of this paper can be summarized as follows:
- We propose a sketch-conditioned image generation scheme to solve the ZS-SBIR problem, using a generative model consisting of an inverse autoregressive flow based encoder.
- We leverage a feedback mechanism [@controlabletext; @vinaycvpr] to encourage the synthesized distribution to be not too far from the original distribution of the observed unlabeled images.
- Unlike the other recently proposed approaches for ZS-SBIR [@imagehashing], even without any side information (e.g., word2vec based attributes of the classes), our method yields significantly better results as compared to [@imagehashing].
{height="6.0cm" width="17.5cm"}
ZS-SBIR Setting
===============
In the zero-shot setting, we partition image dataset into two parts based on sketch classes. One part is the training set which has paired seen-class (S) sketches and image. The second part is test set which has unseen-class (U) sketches only (and no images). Note that the training set is essentially labelled. The training and testing set are mutually exclusive in terms of the sketch classes. In zero-shot setting, we train our model in such a way that it can generalize to unseen class sketches. The mathematical formulation of the zero-shot problem for SBIR is given below:
Let $A=\{(\mathbf{x_i}^{skt},\mathbf{x_i}^{img},y_i)|y_i \in \mathcal{Y}\}$ be the triplet consisting of sketch, image and the class label, where $\mathcal{Y}$ is the set of all class labels. We partition the class labels into two disjoint set $\mathcal{Y}_{tr}$ and $\mathcal{Y}_{te}$ for train and test set respectively. Let $A_{tr}=\{\mathbf{x_i}^{skt},\mathbf{x_i}^{img},y_i|y_i \in Y_{tr}\}$ and $A_{te}=\{\mathbf{x_i}^{skt},\mathbf{x_i}^{img},y_i|y_i \in Y_{te}\}$ be the partition of $A$ into train and test set, respectively. Another assumption for the ZS-SBIR is that $A_{tr}\cap A_{te}=\emptyset$ i.e. train and test classes are disjoint. For simplicity, we will represent $\mathbf{x}^{skt}$ as ”$\mathbf{a}$” and $\mathbf{x}^{img}$ as ”$\mathbf{x}$” throughout this exposition.
Background
==========
As discussed earlier, our approach is based on turning the sketch-to-image search problem into an image-to-image search problem. To this end, we need a model that can generate high-quality images, given a sketch of the class representing that image. This, essentially is a conditional image generation problem. To model the complex distribution of real-world images, we leverage the inverse auto-regressive flow (IAF) based variational autoencoder [@kingma2016improved], and adapt it using a feedback mechanism to integrate the information provided by the sketch attribute. Before describing our architecture, we first provide a background of the components we build upon.
Variational Inference and Learning
----------------------------------
Suppose ${\boldsymbol{x}}=\{{\boldsymbol{x}}^1,\cdot\cdot,{\boldsymbol{x}}^N \}$ be a set of $N$ i.i.d. observations (e.g., $N$ images). Let us denote each sample by ${\boldsymbol{x}}$ and assume ${\boldsymbol{z}}$ be the latent variable associated with ${\boldsymbol{x}}$. For a given dataset ${{{\bf X}}}$, the marginal likelihood of observations is denoted as $\log p({\boldsymbol{x}})=\sum_{i=1}^{N}\log p({\boldsymbol{x}}^i)$. The posterior over the latent variable is denoted by $q({\boldsymbol{z}}|{\boldsymbol{x}})$. We can define a variational lower bound on the marginal log-likelihood $$\small
\log p({\boldsymbol{x}})\geq E_{q({\boldsymbol{z}}|{\boldsymbol{x}})}[\log p({\boldsymbol{x}},{\boldsymbol{z}})-\log q({\boldsymbol{z}}|{\boldsymbol{x}})] = L({\boldsymbol{x}};\theta)$$ where $p$ and $q$ are distributions whose parameters are collectively denoted by $\theta$, and $L$ is the Evidence Lower Bound (ELBO), defined as $$L({\boldsymbol{x}};\theta) = \log p({\boldsymbol{x}})-D_{kl}(q({\boldsymbol{z}}|{\boldsymbol{x}})||p({\boldsymbol{z}}|{\boldsymbol{x}}))
\label{eq:elbo}$$ Maximizing the lower bound $L({\boldsymbol{x}};\theta)$ w.r.t. $\theta$ also maximizes $\log p({\boldsymbol{x}})$ and minimizes $D_{kl}(q({\boldsymbol{z}}|{\boldsymbol{x}})||p({\boldsymbol{z}}|{\boldsymbol{x}})$, where $p({\boldsymbol{z}}|{\boldsymbol{x}})$ is the true posterior over the latent variables and $q({\boldsymbol{z}}|{\boldsymbol{x}})$ is the approximate posterior (often also called the inference network). In order to infer complex true posterior $p({\boldsymbol{z}}|{\boldsymbol{x}})$, we need to have a sufficiently expressive approximation $q({\boldsymbol{z}}|{\boldsymbol{x}})$. Normalizing Flows [@germain2015made] is an idea that helps accomplish this be defining a series of transformations for a latent variable that enable learning sufficiently rich distribution for that variable.
Normalizing Flow
----------------
For the inference network $q({\boldsymbol{z}}|{\boldsymbol{x}})$, we need a highly flexible method that captures the complex nature of the true posterior distribution. Normalizing flow is a popular approach used for the variational inference of posterior over latent space. Normalizing flow [@germain2015made] depends on sequence of invertible mappings for transforming the initial probability density. Suppose $z_0$ be the initial random variable with a simple probability density function $q({\boldsymbol{z}}_0|{\boldsymbol{x}})$ and ${\boldsymbol{z}}_t$ be the final output of a sequence of invertible transformations $f_t$ on ${\boldsymbol{z}}_0$. ${\boldsymbol{z}}_t$ can be computed as: ${\boldsymbol{z}}_t=f_t({\boldsymbol{z}}_{t-1},{\boldsymbol{x}})$ $\forall t=1,\cdots,T$. If Jacobian determinant of each $f_t$ can be computed, then the final probability density function can be computed as: $$\small
\log q({\boldsymbol{z}}_T|{\boldsymbol{x}})=\log({\boldsymbol{z}}_0|x)-\sum_{t=1}^T \log\det\left|\frac{\partial {\boldsymbol{z}}_t}{\partial {\boldsymbol{z}}_{t-1}}\right|$$
Inverse Autoregressive Transformations (IAF)
--------------------------------------------
Let ${\boldsymbol{v}}$ be a variable which is modeled by the Gaussian version of the autoregressive model. Suppose $[\mu({\boldsymbol{v}}),\sigma({\boldsymbol{v}})]$ be the representation of function that maps ${\boldsymbol{v}}$ to the mean $\mu$ and variance $\sigma$. Due to the autoregressive structure, the Jacobian is lower triangular matrix with zeros on the diagonal. Mean and standard deviation of $i^{th}$ element of ${\boldsymbol{v}}$ are computed from ${\boldsymbol{v}}_{1:i-1}$ i.e., previous elements of ${\boldsymbol{v}}$. To sample from such a model, we use a sequence of transformations from a noise vector $\epsilon \sim N(0,I)$ to the corresponding vector ${\boldsymbol{v}}$ as: ${\boldsymbol{v}}_0=\mathbf{\mu}_0+\mathbf{\sigma}_0 \odot \mathbf{\epsilon}_0$ and for $i>0$ ${\boldsymbol{v}}_i=\mu_i({\boldsymbol{v}}_{1:i-1})+\sigma_i({\boldsymbol{v}}_{1:i-1})\epsilon_i$. Variational inference makes sampling from posterior, such models are not interesting to be directly used for the normalizing flow. Although, the inverse transformation is interesting for normalizing flows, as long as we have $\sigma_i>0$ the transformation is one-to-one and it can be inverted as : $\mathbf{\epsilon}_i=\frac{{\boldsymbol{v}}_i-\mathbf{\mu}_i({\boldsymbol{v}}_{1:i-1})}{\mathbf{\sigma}_i(y_{i:i-1})}$. Two key observation for IAF as follows:
- As computation of every element $\mathbf{\epsilon}_i$ does not depend on one another, inverse transformation can be parallelized $\mathbf{\epsilon}=\frac{{\boldsymbol{v}}-\mathbf{\mu}({\boldsymbol{v}})}{\mathbf{\sigma_y}}$ (subtraction and division are element-wise).
- Inverse autoregressive operation has a simple Jacobian determinant. It is lower triangular matrix. As an outcome, the log-determinant of Jacobian of transformation is simple to compute: $\log\det|\frac{\partial \epsilon}{\partial {\boldsymbol{v}}}|=\sum_{i=1}^D -\log \mathbf{\sigma}_i({\boldsymbol{v}})$
IAF step
--------
As shown in Fig. \[fig:model\], the output of initial encoder network is $\mathbf{\mu}_0$, $\mathbf{\sigma}_0$ and one extra output ${{\boldsymbol{h}}\xspace}$ which is consider as one extra input to each subsequent step in the flow. In other word, the parameters of encoder are refined iteratively based on output of previous step $\mathbf{\mu}_0$, $\mathbf{\sigma}_0$ and ${{\boldsymbol{h}}\xspace}$. The sampled vector from latent space of initial encoder is defined as : ${\boldsymbol{z}}_0=\mathbf{\mu}_0 + \mathbf{\sigma}_0 \odot \mathbf{\epsilon}$. Where $\mathbf{\epsilon} \sim N(0,I)$. After $t$ steps the refinement of sample ${\boldsymbol{z}}_0$ is recursively defined as : ${\boldsymbol{z}}_t=\mathbf{\mu}_t + \mathbf{\sigma}_t \odot {\boldsymbol{z}}_{t-1}$. In this sequential step the predicted posterior fits more closely to the true posterior.
Finding an appropriate latent space for sampling is a crucial part of generative models as in variational autoencoder (VAE). VAE based generative models compute latent space in one step which may not be sufficient to capture a complex distribution. So the distribution of the predicted posterior and true posterior could be different with adequate margin. Whereas in IAF based variational autoencoder, predicted posterior are transformed to the true posterior using some simple sequential transformation. This sequence of simple transformation can be reduced to any complex distribution. Therefore using an auto-regressive method we can reduce the difference between the distribution of the estimated posterior and true posterior as compare to standard VAE.
Zero-Shot Sketch-Based Image Retrieval
======================================
In this section, we describe the various components of our proposed model. Again, note that the goal is to learn to generate high-quality images, given the sketch and optionally other side information (e.g., word2vec description of the class).
Inverse Autoregressive Flow-Based Encoder
-----------------------------------------
Learning the complex distribution of ${\boldsymbol{z}}$ in the high dimensional latent space is not feasible by a single step transformation. Therefore, in the plain VAE, the approximate posterior can be far away from the true posterior of ${\boldsymbol{z}}$. IAF provides a way to learn the complex distribution by using the chain of simple transformation. The final latent variable $z$ can be given as: $$\mathbf{z}_T=f_T(...f_2(f_1(f_0(\mathbf{z_0})))...)
\vspace{-5pt}$$ Here each $f_i$ is simple transformation function and are invertible in nature. Figure \[fig:model\] shows the pipeline of IAF architecture.
VAE with feedback mechanism
---------------------------
In our model, the encoder consists of standard encoder coupled with an IAF module. The output of IAF based encoder is refinement of the latent code which is initialized by standard encoder, denoted as $p_E(z_t|x)$ with parameters $\theta_E$. The regressor output distribution is denoted as $p_R(\mathbf{a}|x)$, and the VAE loss function is given by (assuming the regressor to be fixed): $$\small
\begin{aligned}
\mathcal{L}_{VAE}(\theta_E,\theta_G) &= -\mathbb{E}_{p_{E}(\mathbf{z_t}|\mathbf{x}),p(\mathbf{a}|\mathbf{x})} [\log p_{{G}}(\mathbf{x}|\mathbf{z_t},\mathbf{a})]\\ &+ \text{KL}(p_{E}(\mathbf{z_t}|\mathbf{x})||p(\mathbf{z_t}))
\end{aligned}
\label{eq:VAE}$$ where the first term on the R.H.S. is generator’s reconstruction error and the second term promotes the estimated posterior to be close to the prior.
### Regressor/Cyclic-consistency Loss
In our proposed model, the regressor, defined by a probabilistic model $p_R({{\boldsymbol{a}}}|{\boldsymbol{x}})$ with parameters $\theta_R$, is a feed-forward neural network that learns to project the example ${\boldsymbol{x}}\in \mathbb{R}^D$ to its corresponding class-attribute vector ${{\boldsymbol{a}}}\in \mathbb{R}^L$. The objective of the regressor is to minimize the cyclic-consistency loss. The regressor is learned using two sources of data:
- Labeled examples $\{{\boldsymbol{x}}_n,{{\boldsymbol{a}}}_n\}_{n=1}^{N_S}$ from the seen classes, on which we can define a supervised loss, given by $$\mathcal{L}_{Sup} (\theta_R) = -\mathbb{E}_{{\boldsymbol{x}}_n}[p_R({{\boldsymbol{a}}}_n|{\boldsymbol{x}}_n)]$$
- Synthesized examples $\hat{{\boldsymbol{x}}}$ from the generator, for which we can define an unsupervised loss, given by $$\small
\mathcal{L}_{Unsup}(\theta_R) = -\mathbb{E}_{p_{\theta_G}}(\mathbf{\hat{x}}|\mathbf{z_t})p(\mathbf{z_t})p(\mathbf{a})[p_R(\mathbf{a}|\mathbf{\hat{x}})]$$ The weighted combination of supervised and unsupervised loss is defined as the overall objective to minimize the cyclic-consistency/regressor loss: $$\vspace{-5pt}
\min_{\theta_{R}} \mathcal{L}_R = \mathcal{L}_{Sup} + \lambda_R \cdot \mathcal{L}_{Unsup}
\label{eq:disc_total}
\vspace{-5pt}$$
### Regressor-Driven Learning
Regressor-Driven learning helps to minimize the cyclic-consistency loss and guide the generator to generate high-quality samples. The cyclic loss encourages the decoder/generator to generates example $\hat{{\boldsymbol{x}}}$ coherent with its sketch feature vector ${{\boldsymbol{a}}}$. This is done using a loss function described below.
In the first case, suppose the generator generates low-quality samples. Then the regressor will incur a high cyclic loss for these samples. In this case, the regressor assumes that it has optimal parameters and will not regress to the correct value. This loss occurs because of the bad quality samples generated by the generator. Minimizing this loss w.r.t $\theta_G$ helps generator to improve the samples quality. The objective function is given by $$\mathcal{L}_{c}(\theta_{G}) = -\mathbb{E}_{p_G(\mathbf{\hat{x}}|\mathbf{z_t},\mathbf{a})p(\mathbf{z_t})p(\mathbf{a})}[\log p_{R}(\mathbf{a}|\hat{{\boldsymbol{x}}})]
\vspace{-5pt}$$ The other loss which acts as a regularizer that encourages the generator to generate a good class-specific sample even from a random $\mathbf{z_t}$ drawn from the prior distribution $p(\mathbf{z_t})$ and combined with the sketch from $p({{\boldsymbol{a}}})$ is $$\mathcal{L}_{Reg}(\theta_G) = -\mathbb{E}_{p(\mathbf{z_t})p(\mathbf{a})}[\log p_G(\mathbf{\hat{x}}|\mathbf{z_t},\mathbf{a})]
\vspace{-5pt}$$ The above two loss functions help us increase the coherence of $\mathbf{\hat{x}} \sim p_G(\mathbf{\hat{x}}|\mathbf{z},\mathbf{a})$ with class-attribute $\mathbf{a}$. A third loss function is used to ensure that the sampling distribution $p(\mathbf{z_t})$ and the distribution obtained from the generated examples $p_{E}(\mathbf{z_t}|\mathbf{\hat{x}})$ follow the same distribution. $$\begin{aligned}
\mathcal{L}_{E}(\theta_G) = -\mathbb{E}_{\mathbf{\hat{x}}\sim p_G(\mathbf{\hat{x}}|\mathbf{z_t},\mathbf{a})}\text{KL}[(p_{E}(\mathbf{z_t}|\mathbf{\hat{x}})||q(\mathbf{z_t}))]
\end{aligned}
\vspace{-5pt}$$ Hence the complete learning objective for the generator and encoder is given by, $$\begin{aligned}
\min_{\theta_G,\theta_E} \mathcal{L}_{VAE} + \lambda_c \cdot \mathcal{L}_{c} + \lambda_{reg} \cdot \mathcal{L}_{Reg} + \lambda_E \cdot \mathcal{L}_E \\
\end{aligned}
\label{eq:EG_total}
\vspace{-5pt}$$
Residual Decoder
----------------
The proposed decoder is a combination of the deep and shallow network. The deep network is responsible for the better reconstruction of visual space while shallow network reduces over-fitting. This architecture is motivated by ResNet [@resnet] where the network has skip connections. These skip connections provide more paths to the network for information propagation. While some paths are deeper, others are shallow [@he2016identity]. If in the deeper path the gradient vanishing or explosion problem occurs, the shallow paths still work, and proper gradient flows in the backward direction. In the residual network, the output of a neural network layer is given by $f_o(x)=f_{in}({\boldsymbol{x}})+{\boldsymbol{x}}$, (here $f_{in}({\boldsymbol{x}})$, is the direct output), i.e., the output does not only depend on the current layer neural network, but it depends on input as well.
Related Work
============
Images have rich and vibrant content, while a sketch only provides rough information like shape and size. It is easy for a human to match the sketch from the image, but for machines, this is a very complex task. Since for an algorithm, it is very difficult to learn the features that are invariant to color, shape, size, pose, etc. The common pipeline for SBIR is to project the images and sketches in common subspace such that the same class images and sketches are close to each other on some metric space. Then any similarity metric can be used for the retrieval task. Most of the traditional approaches for SBIR have used hand-crafted features such as gradient field HOG descriptor [@HOG], SIFT [@sift] and SURF [@surf] etc. [@sarthak] proposed a dynamic programming based method for SBIR which is effective in translation, rotation, and scale (similarity). Recent advancement of deep learning provides an automatic feature extraction technique which learns the pose and color invariant feature. Recently [@journals/ijcv/CNNinSKETCH; @sketchy; @imagehashing; @ashisheccv2018] have used deep feature for SBIR task. Instead of finding the common subspace other approach projects the sketch space to image space or vice versa such that the information gap between the sketches and the real images are minimum [@hu2013performance; @sarthak].
Recently zero-shot learning drew more attention due to its capability of classifying a novel class object during the test phase. In the ZSL each class is associated with side information like description of the class, attribute or unsupervised word embedding (Word2vec [@word2vec], Glove [@glove], etc.). This side information of the class is called the semantic features/attributes. In ZSL, the core concept is to learn projection between class feature and side information, using labeled seen class data only. We can categories all proposed models for ZSL in three types based on projection.The most popular work learns the projection between visual space to semantic space and vice-versa [@xu2017transductive; @akata2015evaluation; @ConSE; @vinayaaai; @verma2017simple; @mishracvpr]. Another popular approach projects the visual and semantic features in a shared subspace such that same class visual features and semantic attributes map closer, whereas different class visual features and semantic attributes are well-separated [@wang2017zero].
Recently generative models are emerging as the most popular approach for zero-shot image classification. This type of approach gaining popularity because of its ability to synthesize the unseen class sample and can reduce the ZSL problem to a supervised learning problem. These approach learns the data distribution based on the given conditions [@guo2017synthesizing; @verma2017simple; @vinaycvpr; @xian2018feature]. Most of the previous methods for zero-shot learning are focused on image classification. However, a few models are used for zero-shot action classification, zero-shot image tagging and zero-shot multi-label learning as well [@lampert2014attribute; @xu2015semantic; @mishrawacv; @xu2017transductive; @zsl_tagging; @uai_gaure].
Recently [@imagehashing; @ashisheccv2018] have proposed a model for the ZS-SBIR. [@imagehashing] proposed a hashing based approach for the ZS-SBIR. The hashing architecture is based on the multi-model deep network. [@ashisheccv2018] proposed a generative model for the ZS-SBIR based on the CVAE architecture. The proposed approach is also a generative in nature based on IAF to get the improved variational inference [@kingma2016improved]. Here our encoder is based on the IAF architecture that learns the complex latent encoding of the input into the latent space. It can learn the complex distribution with the simple sequential transformation. Also, we are using the $\beta$-VAE [@betavae] architecture for the disentangled representation. The residual decoder is used that gives the better generation of the sample because it can flow the gradient with the deeper layers. In the proposed approach the external feedback mechanism provides the feedback to the encoder about the generation quality. Hence the generator has better guidance for generating the robust sample.
Experiments and Results
=======================
To show the effectiveness of our proposed model we ause two challenging datasets: Sketchy [@sketchy] and TU-Berlin [@berlin]. Originally, Sketchy dataset [@sketchy] contains 75471 hand-drawn sketches and 12500 corresponding images from 125 classes. [@liu] have provided 60502 more real images from all 125 classes, which extends the original dataset. TU-Berlin extended [@berlin] is a large scale dataset having 20000 sketches and 204489 images from 250 different categories provided by [@liu; @sketchnet].
The visual features for images and sketches are extracted using ResNet-152 [@resnet], pretrained on ImageNet [@imagenet2015] dataset. The sketches and image features are extracted from the last fully connected layer. It gives 2048-dimensional feature vectors. We believe that further finetuning on this dataset on ResNet-152 architecture will give better performance. The visual features of the sketches are used as a class attributes in our proposed generative model.
Sketchy Dataset (Extended) {#sketch}
--------------------------
For fair comparison with the recent work [@imagehashing; @ashisheccv2018], we have two splits of the dataset. [@imagehashing] randomly selected 25 classes of sketches as the test set ($A_{te}$) and the remaining labeled 100 classes are used as the training set ($A_{tr}$). Here $A_{tr}\cap A_{te}=\phi$, i.e. train and test class are disjoint. We have another split of the dataset similar to [@ashisheccv2018], this contains 104 classes in training, and used 21 classes images and sketches as a test set.
Random split proposed by [@imagehashing] is not the realistic Zero-Shot setting [@ZSL-GBU]. Since in random split test set may have some classes that are present in the ImageNet class. Since we are using the ImageNet pre-trained model for the feature extraction and this training is done in a supervised manner. This violates the assumption that $A_{te}$ are the unseen classes. Therefore it is not the exact Zero-Shot setting. The split proposed by [@ashisheccv2018] is the realistic setup for ZS-SBIR, where the split is done in such a way that any of the $A_{te}$ classes are not present in the ImageNet dataset. In our setup for training, we need a paired image and sketch set. To make the paired set, we selected a random image and sketch from the same training class and paired them. This process repeated 1000 time, i.e., each class in the training set has 1000 pair of data point. We are comparing our model with the previous approach in their original setup; therefore, we are using both the split.
TU Berlin Dataset (Extended)
----------------------------
Similar to [@imagehashing] for the fair comparison randomly 30 classes are selected for the $A_{te}$ and remaining 220 classes are used for the $A_{tr}$. This dataset is highly biased; few classes have large examples while few have only limited samples. In the Zero-shot setup, learning with biased data is a very hard problem. Therefore form training we removed the biases. For doing so, we are sampling the equal number of image and sketch sample pairs from each class. In the testing, we selected the class that has more than 400 samples. For making image and sketch pair, we follow the same pattern mentioned in the previous section. Here again, each class has a 1500 pair of image and sketch.
Implementation details
----------------------
In our model, we have three components, Encoder (E), Generator (G) and Regressor (R). The encoder is based on the IAF architecture, refer to figure-\[fig:model\]. The encoder contains the two fully connected layers of size 4096 followed by one layer that gives the $\mu$ and $\sigma$ this passed to 3 layers IAF architecture. The generator has five layers of the fully connected neural network (NN) with the residual connection. It is a combination of the deep and shallow network. Here sigmoid activation is used. All layers are of the same size of 6144. Regressor takes the reconstructed samples $\hat{x}$ and regresses the sketch. It uses the two-layer fully connected NN of size 4096. The learning rate ($\eta$) is set as a stepwise decreasing rate. Initially for the 5 epoch $\eta=0.001$ then after each 10 epoch it changed to \[0.0005,0.0001,0.00001\]. Here instead of $\mathcal{N}(0,I)$ prior, we found from the validation data $p\sim\mathcal{N}(0,0.005)$ gives the better performance. Also, for the ablation, we experiment with the plain autoencoder. The autoencoder used contains the same architecture as the IAF encoder with feedback connection; only the difference is that the dimension of $z$ is zero. Therefore the generated sample is deterministic and depends only on the given sketch feature ${{\boldsymbol{a}}}$.
Training and Testing
--------------------
There are two modules in the model, IAF-VAE and regressor. We are alternately optimizing the IAF-VAE and regressor. These two module helps each other to learn the robust generator. In the VAE training, we are minimizing the loss w.r.t. $E$ and $G$’s parameters, and for regressor training, we are minimizing the regressor’s loss w.r.t $R$’s parameter only. The alternate optimization is done Until convergence. The complete setup is for the zero-shot learning; therefore the testing is performed from the unseen class sketch to unseen class image. In the testing phase each $x^{skt}$ is concatenated with the $z\sim\mathcal{N}(0,0.005)$ and generate the $c$ samples using generator $G$. Now from these $c$, samples find the image that gives maximum cosine similarity. The similarity of the sample $x_i$ from the query sketch $x^{skt}$ can be given as: $$\label{eq:test}
\small
s({\boldsymbol{x}}^{skt},{\boldsymbol{x}}_i)=\max_{t=1:c} cosine\left(G(\theta({\boldsymbol{x}}^{skt}_t)),\theta({\boldsymbol{x}}_i)\right)$$ Here, ${\boldsymbol{x}}_i$ is the image in the query database. Equation-\[eq:test\] is repeated for each image and find the $K$ samples with maximum similarity scores for the $top@K$ retrieval. $\theta$ is the ResNet-152 model and gives the feature vector for each image, and $G$ is the generator.
![Top-6 retrieved images for randomly chosen five sketches for ZS-SBIR.[]{data-label="fig:retrieved"}](visual_results_f.jpg)
Result Analysis with existing methods {#result-analysis-with-existing-methods .unnumbered}
-------------------------------------
Since best of our knowledge, only two very recent works ZSIH [@imagehashing] and CVAE [@ashisheccv2018] have been proposed for ZS-SBIR. Therefore for supporting the performance of the proposed model, we compare the performance of our model with several other state-of-the-art. We have analyzed two types of baselines methods, 1- Sketch-Based Image Retrieval (SBIR) Baselines and 2- Zero-Shot Learning (ZSL) baselines.
### Sketch base image retrieval baselines (SBIR) {#sketch-base-image-retrieval-baselines-sbir .unnumbered}
Several approaches have been proposed for SBIR. We compare our model with Siamese-1 [@conf/cvpr/Siamese1], Siamese-2 [@Siamese2], Coarse-grained triplet [@cgt], Fine-grained triplet [@sketchy], DSH [@journals/corr/LiuSSLS17], SaN [@journals/ijcv/CNNinSKETCH], GN Triplet [@sketchy], 3D Shape [@journals/corr/SBIR13], Siamese CNN [@conf/icip/SBIR16]. Since these baselines are not originally proposed for zero-shot setting, [@imagehashing] provides these baseline for the zero-shot setting. We have taken these baseline result directly from the paper [@imagehashing].
### Zero-Shot baselines (ZSL) {#zero-shot-baselines-zsl .unnumbered}
The most of the existing approaches for zero-shot learning are proposed for the zero-shot image classification. We select a set of zero-shot learning approaches as baseline to compare with our proposed model. These ZSL baseline approaches are CMT [@conf/nips/CMT], DeViSE [@conf/nips/DeviSE], SAE [@SAE2017], SSE [@zhang2015zero], ESZSL [@romera2015embarrassingly], CAAE [@AAE], JLSE [@zhang2016zero], DAP [@lampert2014attribute]. The baseline results are borrowed from the [@ashisheccv2018; @imagehashing]. We again reproduce the the baseline results reported in the table-\[tab:newsplit\].
The recent work on the ZS-SBIR is ZSIH [@imagehashing] and CVAE [@ashisheccv2018]. ZSIH [@imagehashing] shows the experiment on the TU-Berlin and Sketchy dataset. They reported the result of precision@100 and mAP@all for all datasets. [@imagehashing] using the word2vec[@word2vec] as side information in their model. As mentioned earlier we are not using any side information but have significantly better result compare to ZSIH. The comparison result with the baseline and ZSIH are shown in the table-\[tab:oldsplit\]. We can see without any side information our approach performs significantly better than all the previous approach that used the side information. Also, we experimented the proposed approach with autoencoder only and found that the proposed VAE model significantly outperforms the autoencoder model. Please refer to table-\[tab:oldsplit\] for the more details.
Another approach CVAE [@ashisheccv2018] has proposed a generative model for ZS-SBIR; they showed the result on the Sketchy dataset. CVAE suggested the realistic train-test split similar to [@ZSL-GBU] for the ZS-SBIR. [@ashisheccv2018] evaluated the performance of the model over precision@200 and mAP@200 metric. We are following the same setup to compare our result with CVAE. Our result on the Sketchy-dataset shows that the proposed approach is significantly better compare to CVAE. CVAE not using any side information and without using any side information (e.g., word2vec), our method shows the $3.0\%$ and $5.8\%$ relative improvement over precision@200 and mAP@200 metric.
In the Figure-\[fig:retrieved\] we have illustrated the top-6 retrieved result using the unseen class sketches from the image database. Retrieved images are closely matched to the outline of the sketches. Since our model learns the mapping between sketches and images based on components. So it may retrieve some other class images which are significantly similarity with the sketch. In Figure-\[fig:retrieved\] we can see that for helicopter sketch our model retrieve the fish because the outline of sketches of the helicopter and fish are very similar. Also, we have shown the t-SNE [@tsne] plot of the original data and the reconstructed data for the Sketchy [@sketchy] dataset. In the t-SNE plot, we can observe that the generated samples for the novel classes are not as good as the original one. But the generated sample nearly follow the same distribution as the original one. The few class samples are as good as the original samples. Please refer to figure-\[fig:tsne\] for the t-SNE plot.
![t-SNE plot for the original and the reconstructed sample for the sketchy dataset without using any side information.[]{data-label="fig:tsne"}](tsne.png)
![Ablation of our proposed VAE with the plain autoencoder without any side information. ***AutoW/O-w2v***: Autoencoder without word2vec, ***W/O-w2v***: Proposed approach without word2vec.[]{data-label="fig:ablation_autoencoder"}](ablation_autoencoder.png)
Ablation Study {#ablation-study .unnumbered}
==============
We now show the significance of the different components of the model as compared to the basic VAE model. We have found that the proposed approach outperforms by a significant margin across all the dataset. Even though without using any side information we are performing better than the previous approach [@imagehashing] that used the side information. In the section-\[\[withvae\]\] we are showing the ablation with and without VAE. Also in section-\[\[withiaf\]\] we are showing the significance of the IAF component.
With/Without VAE {#withvae}
----------------
We also perform the ablation analysis over the different component of the proposed approach. In the first experiment, we compare the performance of autoencoder with the proposed VAE architecture. We found that the proposed model with the VAE component always outperforms compare to autoencoder architecture. Using the feedback-VAE architecture on the Sketchy-dataset the model shows the $20\%$ and $16\%$ relative improvement on the precision@100 and mAP@all metric compare to plain autoencoder architecture. The similar pattern we observe for the Tu-Berlin dataset also. The feedback-VAE shows the $17.8\%$ and $26.3\%$ relative improvement over the plain autoencoder architecture on the precision@100 and mAP@all metric. Please refer to figure-\[fig:ablation\_autoencoder\] for comparison details.
With/Without IAF {#withiaf}
----------------
We present the ablation study with IAF and without IAF component, without using any side information. We have found that if we remove the IAF component, the performance drop is significant as compared to with-IAF. For the Sketchy dataset, we reported in Table-\[tab:ablation\] that with-IAF component, the precision@100 and mAP@all are 0.358 and 0.289, respectively. If we remove the IAF component, the performance drop is significant, and precision@100 and mAP@all are 0.313 and 0.261, respectively. Therefore we have 12.6% and 9.7% relative drop in the performance without-IAF. We also observed a similar pattern on the TU-Berlin dataset. Here in Table-\[tab:ablation\] with-IAF we have 0.334 and 0.238, precision@100 and mAP@all, respectively. But if we drop the IAF component, our precision@100 and mAP@all are 0.294 and 0.198, respectively. Therefore we have 12.0% and 16.8% performance drop, respectively. Please refer to Table-\[tab:ablation\] for the more details.
Conclusion
==========
In this paper, we addressed the Zero-Shot Sketch-Based Image Retrieval problem, which is a challenging and more realistic setting as compared to the conventional SBIR. The proposed generative approach can solve the SBIR problem when the classes are growing with time, and does not require all classes to be present at the training time. We have found that the proposed approach, based on the IAF architecture with the feedback mechanism, generates high-quality samples of the novel classes. Moreover, without using any side information, our proposed generative model can retrieve novel class examples and gives state-of-art results on benchmark datasets. In this work, we assume that the test query comes from the unseen classes only. In future, it will be interesting to explore the *Generalized* ZS-SBIR problem where test query can come from the seen as well as unseen classes. Also, the domain shift is a critical problem in the ZSL. It will be an interesting direction of future work to handle the domain shift for zero-shot SBIR. The recent model shows a significant improvement inj ZS-SBIR using the side information. In future, it will also be exciting to explore the model with the help of side information.
|
---
abstract: 'We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7.'
address:
- |
Kevin Kordek\
School of Mathematics\
Georgia Institute of Technology\
686 Cherry St.\
Atlanta, GA 30332
- |
Dan Margalit\
School of Mathematics\
Georgia Institute of Technology\
686 Cherry St.\
Atlanta, GA 30332
author:
- Kevin Kordek
- Dan Margalit
bibliography:
- 'bnprime.bib'
title: Homomorphisms of commutator subgroups of braid groups
---
Introduction
============
Let ${B}_n$ denote the braid group on $n$ strands and let ${B}_n'$ denote its commutator subgroup. We say that two homomorphisms $\rho_1 : {B}_n' \to {B}_n$ and $\rho_2 : {B}_n' \to {B}_n$ are *equivalent* if there is an automorphism $\alpha$ of ${B}_n$ such that $\alpha\circ \rho_1 = \rho_2$. The following is our main result.
\[thm:main\] Let $n\geq 7$, and let $\rho : {B}_n'\rightarrow {B}_n$ be a nontrivial homomorphism. Then $\rho$ is equivalent to the inclusion map.
In 2017, Orevkov [@orevkov] showed for $n \geq 4$ that $\operatorname{Aut}({B}_n') \cong \operatorname{Aut}({B}_n)$. Theorem \[thm:main\] implies Orevkov’s result for $n \geq 7$. Another proof of Orevkov’s result for $n \geq 7$ was given previously by McLeay [@mcleay].
In his 2004 paper, Lin [@linbp Theorem A] proved there are no nontrivial homomorphisms from ${B}_n'$ to ${B}_m$ when $n \geq 5$ and $m < n$. Theorem \[thm:main\] implies Lin’s result for $n \geq 7$.
In 1981, Dyer–Grossman [@dg] proved for $n \geq 3$ that $\operatorname{Aut}({B}_n) \cong {B}_n/Z({B}_n) \rtimes {\mathbb{Z}}/2$, solving a problem of Artin [@artin] from 1947. Then in 2016 Castel [@castel] classified for $n \geq 6$ all homomorphisms ${B}_n \to {B}_{n+1}$ (recently Chen and the authors [@chenkordekmargalit] generalized Castel’s result by classifying all homomorphisms ${B}_n \to {B}_{2n}$ for $n \geq 5$). Castel’s theorem implies the theorem of Dyer–Grossman [@dg]. In Section \[sec:castel\] we explain how to derive a special case of Castel’s theorem from Theorem \[thm:main\]; specifically, we obtain the classification of homomorphisms ${B}_n \to {B}_n$ for $n \geq 7$. Thus our work gives a new proof of the Dyer–Grossman for $n \geq 7$.
The main new tool we use to prove Theorem \[thm:main\] is the notion of a totally symmetric set, which we define in Section \[sec:tss\]. Briefly, a totally symmetric subset of a group $G$ is a subset $X$ of commuting elements with the property that any permutation of $X$ can be achieved by conjugation in $G$. Recently, totally symmetric sets have also been used by Chudnovsky, Li, Partin, and the first author to give a lower bound for the cardinality of a finite quotient of the braid group [@reu].
*Outline of the paper.* In Section \[sec:tss\], we introduce totally symmetric sets. We prove the following fundamental lemma: the image of a totally symmetric set under a homomorphism is either a totally symmetric set of the same size or a singleton (Lemma \[lem:tss\]). The section culminates with a classification of totally symmetric subsets of ${B}_n$ (Lemma \[lem:bnclass\]). In Section \[sec:proof\] we prove Theorem \[thm:main\] using the classification of totally symmetric sets and the fundamental lemma. Finally, in Section \[sec:castel\], we apply Theorem \[thm:main\] to prove the aforementioned special case of Castel’s theorem.
Totally symmetric sets {#sec:tss}
======================
In this section we introduce the main new technical tool in the paper, namely, totally symmetric sets. After giving some examples, we derive some basic properties of totally symmetric sets, in particular developing the relationship with canonical reduction systems. In the next section we explain how these ideas are used in the proof of the main theorem.
[*Totally symmetric subsets of groups*.]{} Let $X$ be a subset of a group $G$. We may conjugate $X$ by an element $g$ of $G$, meaning that we conjugate each element of $X$ by $g$. We say that $X$ is *totally symmetric* if
- the elements of $X$ commute pairwise and
- every permutation of $X$ can be achieved via conjugation by an element of $G$.
As a first example, any singleton $\{x\}$ is totally symmetric.
We further say that $X \subseteq G$ is totally symmetric with respect to a subgroup $H$ of $G$ if $X$ satisfies the above definition, with the additional constraint that the conjugating elements $g$ can be chosen to lie in $H$. We observe that if $X \subseteq G$ is totally symmetric with respect to $H \leqslant G$, and $X \subseteq H$, then $X$ is a totally symmetric subset of $H$.
The definition of a totally symmetric set is inspired by the work of Aramayona–Souto, who studied one particular example of a totally symmetric set in their work on homomorphisms between mapping class groups; see [@AS Section 5].
[*New totally symmetric sets from old*.]{} Let $X =\{x_1,\dots,x_m\}$ be a totally symmetric subset of $G$. There are several ways of obtaining new totally symmetric sets from $X$. Let $k,\ell \in {\mathbb{Z}}$ with $k$ and $\ell$ not both zero, and let $z$ be an element of $G$ with the property that each permutation of $X$ can be achieved by an element of $G$ that commutes with $z$ (for example $z$ can lie in $Z(G)$). Also, for each $i$ let $x_i^*$ denote the product of the $x_j \in X$ with $j \neq i$. Starting from $X$, we may create the following totally symmetric sets: $$\begin{aligned}
X^k &= \{ x_1^k ,\dots, x_m^k \} \\
X^* &= \{x_1^*,\dots,x_m^*\} \\
X^{k,\ell} &= \{x_1^k(x_1^*)^\ell,\dots,x_m^k(x_m^*)^\ell\} \\
X' &= \{x_1x_2^{-1},\dots,x_1x_m^{-1}\} \\
X^z &= \{x_1z,\dots,x_mz \}.\end{aligned}$$ We can combine these constructions, for instance $(X^k)^z$ and $(X^*)^*$ is totally symmetric. Also, if all permutations of $X$ are achievable by elements of a subgroup $H$ of $G$, then the same is true for $X^k$, $X^*$, $X'$, and $X^z$.
[*A fundamental lemma*.]{} We have the following fundamental fact about totally symmetric sets. It is an analog of Schur’s lemma from representation theory.
\[lem:tss\] Let $X$ be a totally symmetric subset of a group $G$ and let $\rho : G \to H$ be a homomorphism of groups. Then $\rho(X)$ is either a singleton or a totally symmetric set of cardinality $|X|$.
It is clear from the definition of a totally symmetric set that $\rho(X)$ is totally symmetric and that its cardinality is at most $|X|$. Suppose that the restriction of $\rho$ to $X$ is not injective, so that there are $x_1,x_2 \in X$ with $\rho(x_1)=\rho(x_2)$. For any $x_i \in X$ there is (by the definition of a totally symmetric set) a $g \in G$ so that $(gx_1g^{-1},gx_2g^{-1}) = (x_1,x_i)$. Thus $$\rho(x_1x_i^{-1}) = \rho((gx_1g^{-1})(gx_2^{-1}g^{-1})) = \rho(g)\rho(x_1x_2^{-1})\rho(g)^{-1} = 1.$$ The lemma follows.
[*Totally symmetric sets in braid groups*.]{} In the braid group ${B}_n$ the most basic example of a totally symmetric set is $$X_n = \{\sigma_1,\sigma_3,\sigma_5,\dots,\sigma_{N}\}$$ where $N$ is the largest odd integer less than $n$. As above, the sets $X_n^k$ and $X_n^*$, $X_n'$, and $X^t$ are totally symmetric. In the following lemma, let $z \in {B}_n$ be a generator for the center $Z(B_n)$; the signed word length of $z$ is $n(n-1)$. We also define $$\begin{aligned}
Y_n &= X_n', \text{ and} \\
Z_n &= \left(X_n^{n(n-1)}\right)^{z^{-1}}.\end{aligned}$$
Let $n \geq 2$. The set $X_n \subseteq {B}_n$ is totally symmetric with respect to ${B}_n'$. In particular, $Y_n$ and $Z_n$ are totally symmetric subsets of ${B}_n'$.
Suppose some permutation $\tau$ of $X_n$ is achieved by $g \in {B}_n$. Since $\sigma_1$ commutes with each element of $X_n$, the permutation $\tau$ is also achieved by $g\sigma_1^k$ for all $k \in {\mathbb{Z}}$. If we take $k$ to be the negative of the signed word length of $g$, then $g\sigma_1^k$ lies in ${B}_n'$. The first statement follows. The second statement also follows, once we observe that each element of $Y_n$ and $Z_n$ lies in ${B}_n'$.
The goal of the remainder of the section is to classify totally symmetric subsets of size $\lfloor n/2 \rfloor$ in ${B}_n$. The end result is Lemma \[lem:bnclass\] below. The proof requires two auxiliary tools, Lemmas \[lem:multiclass\] and \[lem:matrix\].
[*Totally symmetric multicurves*.]{} The first tool is a topological version of total symmetry. Let $X$ be a set and let $S$ be a surface. We say that a multicurve $M$ is *$X$-labeled* if each component of $M$ is labeled by a subset of $X$. The symmetric group $\Sigma_X$ acts on the set of $X$-labeled multicurves by acting on the labels. The mapping class group $\operatorname{Mod}(S)$—the group of homotopy classes of orientation-preserving homeomorphisms of $S$ fixing the boundary of $S$—also acts on the set of $X$-labeled multicurves via its action on the set of unlabeled multicurves.
Let $M$ be a $X$-labeled multicurve in $S$. We say that $M$ is *totally symmetric* if for every $\sigma \in \Sigma_X$ there is an $f \in \operatorname{Mod}(S)$ so that $\sigma \cdot M = f \cdot M$. As in the case of totally symmetric sets, we say that $M$ is totally symmetric with respect a subgroup $H$ of $\operatorname{Mod}(S)$ if the elements $f$ from the definition can all be chosen to lie in $H$.
We say that a $X$-labeled multicurve has the *trivial labeling* if each component of the multicurve has the label $X$. Every such multicurve is totally symmetric (with respect to any subgroup $H$ of $\operatorname{Mod}(S)$).
We can describe an $X$-labeled multicurve in a surface $S$ as a set of pairs $\{(c_i,A_i)\}$ where each $c_i$ is a curve in $S$, where each $A_i$ is a subset of $X$, and where $\{c_i\}$ is a multicurve in $S$.
Let $X$ be a subset of a group $G$. If $M=\{(c_1,A_1),\dots,(c_m,A_m)\}$ is a totally symmetric $X$-labeled multicurve in a surface $S$, then we may create new totally symmetric multicurves from $X$ as follows. For a subset $A$ of $X$, we denote by $A^c$ the complement $X \setminus A$. The new totally symmetric multicurve is
$$\begin{aligned}
M^* &= \{(c_1,A_1^c),\dots,(c_m,A_m^c)\}\end{aligned}$$
For $1 \leq i \leq n-1$ let $c_i$ be the curve in ${D}_n$ with the property that the half-twist about $c_i$ is $\sigma_i$. The $X_n$-labeled multicurve $$M_n = \{(c_1,\sigma_1),\dots,(c_N,\sigma_N)\}$$ is totally symmetric.
For the statement of the next lemma, we require several further definitions. For a natural number $k$, let $[k]$ denote the set $\{1,\dots,k\}$. Also, for $H$ a subgroup of $\operatorname{Mod}(S)$, we say that two labeled multicurves in $S$ are $H$-equivalent if they lie in the same orbit under $H$.
Let $c_0$ denote the standard curve in ${D}_n$ that surrounds the first $n-1$ marked points (so that $c_0$ is disjoint from $c_i$ for $i=1,3,\dots,N$). The multicurve in ${D}_n$ (with $n$ odd) whose components are $c_0, c_1, c_3,\ldots, c_N$ is depicted in Figure \[fig:mc\]. The labeled multicurves $M_n\cup\{c_0,[k]\}$ and $M_n^*\cup\{c_0,[k]\}$ for $n$ odd are depicted in Figure \[fig:lmc\].
$c_1$ at 57 110 $c_3$ at 114 110 $c_{N}$ at 224 110 $c_0$ at 140 245 $c_1$ at 410 110 $c_3$ at 460 110 $c_{N}$ at 570 110 ![The curves $c_1,c_3,\ldots, c_N$ and the curve $c_0$ on the disk ${D}_n$ with $n$ odd (*left*) and $n$ even (*right*).[]{data-label="fig:mc"}](Mnc0 "fig:")![The curves $c_1,c_3,\ldots, c_N$ and the curve $c_0$ on the disk ${D}_n$ with $n$ odd (*left*) and $n$ even (*right*).[]{data-label="fig:mc"}](Mneven "fig:")
\[lem:multiclass\] Let $n \geq 1$ and let $k = \lfloor n/2 \rfloor$.
1. If $n$ is even, then every totally symmetric $[k]$-labeled multicurve in ${D}_n$ with nontrivial labeling is ${B}_n$-equivalent to $M_n$ or $M_n^*$.
2. If $n$ is odd, then every totally symmetric $[k]$-labeled multicurve in ${D}_n$ with nontrivial labeling is ${B}_n$-equivalent to $M_n$, $M_n^*$, $M_n \cup \{(c_0,[k])\}$, or $M_n^* \cup \{(c_0,[k])\}$.
Say that $M$ is a totally symmetric $[k]$-labeled multicurve with nontrivial label. Let $c$ be a curve in $M$ with nontrivial label $A \subsetneq [k]$. The symmetric group $\Sigma_k$ acts on the power set of $[k]$ and the orbit of $A$ under this action has at least $k$ elements. Since $M$ is totally symmetric, this means that there are curves $d_1,\dots,d_k \in M$ with the property that every permutation of $\{d_1,\dots,d_k\}$ is realized by an element of ${B}_n$. In particular the curves $d_1,\dots,d_k$ all surround the same number of marked points. It follows that each surrounds exactly two marked points. We can further conclude from all of this that there are no other curves in $M$ with nontrivial label besides $d_1,\dots,d_k$. Up to ${B}_n$-equivalence we may also assume that the curves $d_1,\dots,d_k$ are exactly the components of $M_n$.
Suppose there are curves $b_1,\dots,b_m$ of $M$ with trivial label. The curves $b_1,\dots,b_m$ induce a partition of the set $\{d_1,\dots,d_k\}$: the curve $d_i$ and $d_j$ are in the same subset of the partition if and only if they are not separated by an element of $\{b_1,\dots,b_m\}$. Since the $b_i$ are essential and distinct from the $c_i$, and since $k = \lfloor n/2 \rfloor$, it must be that either (1) $m=1$ and $b_1=c_0$ or (2) the partition is nontrivial, meaning that it contains more than one subset. The first possibility can only arise when $n$ is odd. The second possibility violates the assumption that $M$ is totally symmetric. Indeed, if we choose a permutation $\tau$ of $[k]$ with the property that the corresponding permutation of $\{d_1,\dots,d_k\}$ does not respect the partition then there is no mapping class $f$ with $f \cdot M = \tau \cdot M$ (any $f$ with $f \cdot M = \tau \cdot M$ must preserve the multicurve $\{b_1,\dots,b_m\}$ and hence preserve the partition of $\{d_1,\dots,d_k\}$). The lemma follows.
[*From totally symmetric sets to totally symmetric multicurves*.]{} Associated to each element $f$ of $\operatorname{Mod}(S)$ is its canonical reduction system ${\Gamma}(f)$. We will make use of several basic facts about canonical reduction systems. First, we have ${\Gamma}(f) = \emptyset$ if and only if $f$ is periodic or pseudo-Anosov. Next, if $f$ and $g$ commute then ${\Gamma}(f) \cap {\Gamma}(g) = \emptyset$. Also, for any $f$ and $g$ we have ${\Gamma}(gfg^{-1}) = g{\Gamma}(f)$. See the paper by Birman–Lubotzky–McCarthy for background on canonical reduction systems [@blm].
Given a totally symmetric subset $X = \{x_1,\dots,x_m\}$ of $\operatorname{Mod}(S)$ we obtain an $X$-labeled multicurve as follows: the underlying multicurve $M$ is obtained from the disjoint union of the ${\Gamma}(x_i)$ by identifying homotopic curves, and the label of a curve $c \in M$ is the set of $x_i$ with $c$ a component of ${\Gamma}(x_i)$. We denote this $X$-labeled multicurve by ${\Gamma}(X)$. We have the following lemma, which follows immediately from the definitions and the stated facts about canonical reduction systems.
\[lem:symcrs\] If $X$ is a totally symmetric subset of $\operatorname{Mod}(S)$ then ${\Gamma}(X)$ is totally symmetric.
The totally symmetric multicurves associated to $X_n$, $Y_n$, and $Z_n$ are $$\begin{aligned}
{\Gamma}(X_n) &= \{(c_1,\sigma_1),\dots,(c_N,\sigma_N)\} \\
{\Gamma}(Y_n) &= \{(c_1,Y_n)\} \cup \{(c_3,\sigma_1\sigma_3^{-1}),\dots,(c_N,\sigma_1\sigma_N^{-1})\}, \text{ and}\\
{\Gamma}(Z_n) &= \{(c_1,\sigma_1^{(n(n-1)}z^{-1}),\dots,(c_N,\sigma_N^{n(n-1)}z^{-1}\}.\end{aligned}$$
$\{1\}$ at 48 105 $\{2\}$ at 105 105 $\{k\}$ at 215 105 $\{1,2,\dots,k\}$ at 139 230 $\{1\}^c$ at 387 105 $\{2\}^c$ at 444 105 $\{k\}^c$ at 553 105 $\{1,2,\dots,k\}$ at 478 230 ![*Left:* The labeled multicurve $M_n\cup\{c_0,[k]\}$; *Right:* The labeled multicurve $M_n\cup\{c_0,[k]\}$[]{data-label="fig:lmc"}](Mnc0 "fig:")![*Left:* The labeled multicurve $M_n\cup\{c_0,[k]\}$; *Right:* The labeled multicurve $M_n\cup\{c_0,[k]\}$[]{data-label="fig:lmc"}](Mnc0 "fig:")
[*Classification of derived totally symmetric subsets*.]{} Let $X = \{x_1,\dots,x_m\}$ be a totally symmetric subset of a group $G$. We say that a totally symmetric set $Y$ in $G$ is *derived* from $X$ if $Y$ lies in the abelian subgroup $\langle X \rangle$ of $G$. We have already seen examples of derived totally symmetric sets, such as $X^k$, $X^*$, $X^{k,\ell}$, and $X'$.
Let $X$ be a totally symmetric subset of a group $G$. We consider the action of $G$ on itself by conjugation and write $\operatorname{Stab}_G(X)$ for the stabilizer of the set $X$. We say that $X$ is *rigid* if $$\operatorname{Stab}_G(Y) \subseteq \operatorname{Stab}_G(X).$$ It follows from Nielsen–Thurston theory that the totally symmetric sets $X_n$, $Y_n$, and $Z_n$ are rigid totally symmetric subsets of ${B}_n$.
\[lem:matrix\] Let $X = \{x_1,\dots,x_m\}$ be a rigid totally symmetric subset of a group $G$ with $m \geq 2$, and let $Y$ be a derived totally symmetric set with $m$ elements. Then $Y$ is equal to some $X^{k,\ell}$.
Say that the elements of $Y$ are $\{y_1,\dots,y_m\}$ are given as $$y_i = x_1^{a_{i,1}} \cdots x_m^{a_{i,m}}.$$ Let $A$ be the $m\times m$ matrix $\left(a_{i,j}\right)$. As such, the $i$th row of $A$ records the exponents on the $x_j$ in the expression for $y_i$.
The statement of the lemma is equivalent to the statement that there exist integers $k$ and $\ell$ such that, up to relabeling the rows of $A$, we have
$$A =
\begin{pmatrix}
\ell & k & \cdots & k \\
k & \ell & \cdots & k\\
\vdots & \vdots & \ddots &\vdots\\
k & k& \cdots & \ell
\end{pmatrix}$$
We claim that any permutation of the rows of $A$ is achieved by a permutation of the columns of $A$. Let $\sigma\in S_m$ and let $g\in G$ be such that $gy_ig^{-1} = y_{\sigma(i)}$. This determines a permutation of the rows of $A$. By the total symmetry, every permutation of the rows arises in this way. On the other hand, since $Y$ is rigid, the conjugating element $g$ also permutes $X$, and hence determines a permutation of the columns of $A$. As both permutations have the same effect on the set of $y_i$, the claim follows.
We next claim that if $v$ is a column of $A$, and $w$ is an element of ${\mathbb{Z}}^m$ obtained by permuting the entries of $v$, then $w$ is also a column of $A$. The claim now follows from the previous claim and the fact that any permutation of the entires of $v$ can be achieved by a permutation of the rows of $A$.
Let $v_1$ be the first column of $A$. It must be that, up to reordering the rows of $A$, we have that $v_1 = (\ell,k,\dots,k)$ for some integers $k$ and $\ell$. Indeed, otherwise, there would be more than $m$ distinct permutations of the entries of $v_1$, violating the previous claim. It further follows from the previous claim that the $m$ columns of $A$ are the $m$ distinct permutations of the entries of $v_1$. After reordering the rows, $A$ has the desired form.
[*Classification of totally symmetric subsets of the braid group*.]{} For the following lemma, recall that when $n$ is odd, $c_0$ is the standard curve in ${D}_n$ that surrounds the first $n-1$ marked points. Also, we say that two totally symmetric subsets of a group $G$ are *$G$-equivalent* if there is an automorphism of $G$ taking one to the other. In this lemma and throughout the paper, we use the usual identification of ${B}_n$ with the mapping class group $\operatorname{Mod}({D}_n)$.
\[lem:bnclass\] Let $n \geq 5$ and let $X=\{x_1,x_3,\dots,x_N\}$ be a totally symmetric subset of ${B}_n$ with $|X| = \lfloor n/2 \rfloor$. Assume that ${\Gamma}(X)$ has a nontrivial labeling.
1. If $n$ is even, $X$ is ${B}_n$-equivalent to $(X_n^k)^{z^s}$ or $((X_n^*)^k)^{z^s}$ for some $k,s \in {\mathbb{Z}}$ with $k \neq 0$.
2. If $n$ is odd, $X$ is ${B}_n$-equivalent to $(X_n^k)^{T_{c_0}^rz^s}$ or $((X_n^*)^k)^{T_{c_0}^rz^s}$ for some $k,r,s \in {\mathbb{Z}}$ with $k \neq 0$.
We treat the case where $n$ is odd. The other case is similar (and simpler). By Lemma \[lem:symcrs\], the multicurve ${\Gamma}(X)$ is totally symmetric. It then follows from Lemma \[lem:multiclass\] that ${\Gamma}(X)$ is ${B}_n$-equivalent to $M_n$ or $M_n^*$. We may assume then that ${\Gamma}(X)$ is equal to $M_n$ or $M_n^*$. We treat these cases in turn.
Suppose first that ${\Gamma}(X)$ is equal to $M_n$. This means that ${\Gamma}(x_1)$ is equal to either $\{c_1\}$ or $\{c_0,c_1\}$. We assume we are in the latter case, the other case being similar (and again simpler). The multicurve ${\Gamma}(x_1)$ divides into three regions, each corresponding to a periodic or pseudo-Anosov Nielsen–Thurston component of $x_1$. To prove the lemma it suffices to show that the outer two Nielsen–Thurston components of $x_1$ are trivial.
The only region with enough marked points to support a pseudo-Anosov mapping class is the one lying between $c_0$ and $c_1$. However, since $x_1$ commutes with $x_3$ and ${\Gamma}(x_3) = \{c_3\}$ it follows that $x_1$ has no pseudo-Anosov components. Thus, all of the Nielsen–Thurston components of $x_1$ are periodic.
The outermost Nielsen–Thurston component of $x_1$ (exterior to $c_0$) is necessarily trivial since this outermost region is a pair of pants (after collapsing the boundary components to points), and since $x_1$ fixes two of the three marked points (the ones corresponding to $c_0$ and the boundary of ${D}_n$).
The Nielsen–Thurston component of $x_1$ corresponding to the region between $c_0$ and $c_1$ must also be trivial. Indeed, if we collapse $c_0$ and $c_1$ to marked points, we obtain a sphere with $n-1$ marked points. A periodic mapping class of this sphere is a rotation fixing the marked points coming from $c_0$ and $c_1$. Since, as above, $x_1$ fixes $c_3$ (which surrounds two marked points) it follows that the rotation is trivial, and so this Nielsen–Thurston component fo $x_1$ is trivial, as desired.
Proof of Theorem \[thm:main\] {#sec:proof}
=============================
In this section, we prove Theorem \[thm:main\]. For the proof, we require two definitions: for a multicurve $M$ on ${D}_n$, we let $\operatorname{Stab}_{{B}_n}(M)$ denote the subgroup of ${B}_n$ whose elements preserve $M$ and we let $\operatorname{Fix}_{{B}_n}(M)$ denote the subgroup of ${B}_n$ whose elements identically fix each component of $M$.
As in the statement, assume $n \geq 7$ and let $\rho : {B}_n' \to {B}_n$ be a homomorphism. We will prove the theorem inductively. The inductive hypothesis is that if $2 \leq m \leq n$ and $\rho : {B}_n' \to {B}_m$ is a nontrivial homomorphism, then it is equivalent to the identity (in particular $m=n$). The base case $m=2$ holds because ${B}_2$ is abelian and ${B}_n'$ is perfect.
Let $Z_n$ be the totally symmetric subset of ${B}_n'$ defined in Section \[sec:tss\]. By Lemma \[lem:tss\], $\rho(Z_n)$ is either a singleton or a totally symmetric subset of ${B}_n$ of cardinality $\lfloor n/2 \rfloor$. Let $p = n(n-1)$. If $\rho(Z_n)$ is a singleton then it follows that both of the elements $\sigma_1^{p}z^{-1}$ and $\sigma_3^{p}z^{-1}$ have the same image under $\rho$, and hence that $$(\sigma_1^{p}z^{-1})(\sigma_3^{p}z^{-1})^{-1} = (\sigma_1\sigma_3^{-1})^{p}$$ lies in the kernel of $\rho$. It follows that $\rho(\sigma_1\sigma_3^{-1})^{p} = 1$. Since ${B}_n$ is torsion-free, we therefore have that $\rho(\sigma_1\sigma_3^{-1}) = 1$, and since the normal closure of $\sigma_1\sigma_3^{-1}$ in ${B}_n'$ is equal to ${B}_n'$ (see [@chenkordekmargalit Lemma 8.4]) it follows that $\rho$ is trivial.
By the previous paragraph, we may henceforth assume that $\rho(Z_n)$ is a totally symmetric subset of ${B}_n$ of cardinality $\lfloor n/2 \rfloor$. For $1\leq i \leq n-1$, we define $$x_i = \rho(\sigma_i^{p}z^{-1})$$ and so $\rho(Z_n) = \{x_1, x_3,\ldots, x_N\}$, where $N$ is the largest odd number less than $n$.
Let $M$ denote the labeled multicurve ${\Gamma}(\rho(Z_n))$. There are three cases to consider:
1. $M$ is empty,
2. $M$ is nonempty and is trivially labeled, and
3. $M$ is non-empty and is not trivially labeled.
*Case 1.* In this case we will prove that $\rho$ is trivial. To say that $M$ is empty is to say that either all of the $x_i$ are periodic or all of the $x_i$ are pseudo-Anosov. Since the image of $\rho$ is contained in ${B}_n'$ and since the only periodic element of ${B}_n'$ is the identity, we must have that $x_i = 1$ for all $i$. This implies that $\rho(Z_n)$ is a singleton, a contradiction. It therefore suffices to deal with the case in which the $x_i$ are all pseudo-Anosov.
Let $\bar {B}_n$ denote the quotient ${B}_n/Z({B}_n)$ and let $\bar x_i$ denote the image of $x_i$ in this quotient. Since the $x_i$ are pseudo-Anosov, so too are the $\bar x_i$. By a theorem of McCarthy [@mccarthy], there is a short exact sequence $$1 \to F \to C_{\bar {B}_n}(\bar x_1) \to {\mathbb{Z}}\to 1,$$ where $C_{\bar {B}_n}(\bar x_1)$ is the centralizer of $x_1$ in $\bar {B}_n$ and $F$ is a finite subgroup of $\bar {B}_n$. Since $\bar x_3$ commutes with $\bar x_1$ and is conjugate, it follows that $(\bar x_1 \bar x_3^{-1})^p$ lies in $F$ for some $p$. In particular $\bar x_1 \bar x_3^{-1}$, hence $x_1x_3^{-1}$, is periodic. We again use the fact that ${B}_n'$ contains no nontrivial periodic elements to conclude that $x_1 x_3^{-1} = 1$. This contradicts the assumption that the $x_i$ are distinct.
*Case 2.* We will again prove that $\rho$ is trivial. Let $M_0$ denote the multicurve given by the set of “largest” components of $M$, that is, the set of components surrounding the largest number of marked points.
We claim that $\rho({B}_n')$ lies in $\operatorname{Stab}_{{B}_n}(M_0)$. Since $\operatorname{Stab}_{{B}_n}(M) \subseteq \operatorname{Stab}_{{B}_n}(M_0)$, it suffices to show that $\rho({B}_n')$ lies $\operatorname{Stab}_{{B}_n}(M)$. The commutator subgroup ${B}_n'$ is generated by the set of elements of the form $\sigma_1\sigma_i^{-1}$ where $2\leq i \leq n-1$; see [@linbp p. 7] or [@kordek Prop. 3.1]. Let $p = n(n-1)$. For each $i$, there exists an odd $j$ such that $\sigma_j^pz^{-1}$ commutes with $\sigma_1\sigma_i^{-1}$. This implies that each $\sigma_1\sigma_i^{-1}$ preserves the canonical reduction system of some element $\sigma_j^pz^{-1}$ with $j$ odd. It follows that each $\sigma_1\sigma_i^{-1}$ preserves $M$, which is the canonical reduction system of each $\sigma_j^pz^{-1}$. This completes the proof of the claim.
We next claim that $\rho({B}_n')$ lies in $\operatorname{Fix}_{{B}_n}(M_0)$. Say that $M_0$ has $q$ components, each surrounding $p$ marked points, and let $\ell = m-(p-1)q$. There is a homomorphism $$\Pi_e : \operatorname{Stab}_{{B}_n}(M_0) \to B_\ell$$ obtained by crushing to a marked point each disk bounded by component of $M_0$ (the ‘$e$’ in the subscript stands for ‘exterior’). By the previous claim, $\Pi_e \circ \rho$ is well defined. Then, by our inductive hypothesis, $\Pi_e \circ \rho$ is trivial. The claim follows.
We are now ready to complete the proof. There is a direct product decomposition $$\operatorname{Fix}_{{B}_n}(M_0) \cong ({B}_p \times \cdots \times {B}_p) \times {B}_\ell$$ where there are $q$ copies of ${B}_p$ in the first factor; see [@chenkordekmargalit Section 6]. Post-composing $\rho$ with each of the $q+1$ projection maps associated to the direct product decomposition, we see by induction that each such composition is trivial. It follows that $\rho$ is trivial.
*Case 3.* In this case we will prove that $\rho$ is equivalent to the identity. The proof has five steps. In the fourth step, we say that a sequence of curves $e_1,\dots,e_k$ in ${D}_n$ forms a chain if each $e_i$ surrounds two marked points, if $i(e_i,e_{i+1})=2$ for $1 \leq i \leq k-1$, and if the $e_i$ are disjoint otherwise.
- Up to equivalence, we have $\rho(\sigma_1\sigma_j^{-1}) = \sigma_1\sigma_j^{-1}$ for all odd $j$.
- For each $i\geq 5$ there exists a curve $d_i$ such that $\rho(\sigma_1\sigma_i^{-1}) = \sigma_1H_{d_i}^{-1}$.
- For $i=2,3,4$ there exists a curve $d_i$ such that $\rho(\sigma_1\sigma_i^{-1}) = \sigma_1H_{d_i}^{-1}$.
- The curves $c_1,d_2,\dots,d_{n-1}$ form a chain.
- The homomorphism $\rho$ is equivalent to the inclusion map.
*Step 1.* When $n$ is even, Lemma \[lem:bnclass\] implies that there is a nonzero $\ell$ so that $\rho(Z_n)$ is ${B}_n$-equivalent to one of the following totally symmetric sets: $$(X_n^{0,\ell})^{T_{c_0}^rz^s}\quad \text{or}\quad (X_n^{\ell, 0})^{T_{c_0}^rz^s}.$$ When $n$ is odd, Lemma \[lem:bnclass\] implies that there is a nonzero $\ell$ so that $\rho(Z_n)$ is ${B}_n$-equivalent to one of the following totally symmetric sets: $$(X_n^{0,\ell})^{z^s}\quad (X_n^{\ell, 0})^{z^s}\quad (X_n^{0,\ell})^{T_{c_0}^rz^s}\quad \text{or} \quad (X_n^{\ell, 0})^{T_{c_0}^rz^s}.$$ Therefore, up to replacing $\rho$ by an equivalent homomorphism, we may assume that $\rho(Z_n)$ is equal to one of these sets.
We claim that there exists $q\in {\mathbb{Z}}$, depending only on $\rho$, such that $$\rho(\sigma_{1}\sigma_{j}^{-1}) = (\sigma_{1}\sigma_{j}^{-1})^{q}.$$ for all odd $j$ with $1 < j \leq n-1$. Let $p = n(n-1)$. Regardless of which of the above sets $\rho(Z_n)$ is equal to, we have $$\rho(\sigma_{1}\sigma_{j}^{-1})^p =\rho(\sigma_{1}^pz^{-1})\rho(\sigma_{j}^pz^{-1})^{-1} = (\sigma_{1}\sigma_{j}^{-1})^{\pm\ell}$$ (in all cases, the $T_{c_0}^r$ and $z^s$ terms cancel each other; the sign of the exponent in the last term depends on whether we have $X_n^{\ell,0}$ or $X_n^{0,\ell}$). In our earlier paper [@chenkordekmargalit Lemma 8.7], we proved that $(\sigma_{1}\sigma_{j}^{-1})^{\ell}$ has a $k$th root if and only if $k$ divides $\ell$ and in this case there is a unique root, namely, $(\sigma_{1}\sigma_{j}^{-1})^{\ell/p}$. Setting $q = \pm \ell/p$ then gives the claim.
We next claim that $q=\pm 1$. Let $g$ be an element of ${B}_n'$ such that $$g\sigma_1 g^{-1} = \sigma_2$$ and such that $g$ commutes with each $\sigma_5$ (for example $g = \sigma_2^{-2}\sigma_1\sigma_2$). Then $g$ conjugates $\sigma_1\sigma_5^{-1}$ to $\sigma_2\sigma_5^{-1}$. Define $d$ to be the curve with $$H_{d} = \rho(g)\sigma_1\rho(g)^{-1}.$$ We then have that $$\begin{aligned}
\rho(\sigma_2\sigma_5^{-1}) = \rho\left(g(\sigma_1\sigma_5^{-1})g^{-1}\right) = \rho(g)\rho(\sigma_1\sigma_5^{-1})\rho(g)^{-1} = \rho(g)\left(\sigma_1\sigma_5^{-1}\right)^\ell\rho(g)^{-1} =
H_{d}^{q}\sigma_5^{-q}.\end{aligned}$$ The element $\sigma_1\sigma_5^{-1}$ satisfies a braid relation with $\sigma_2\sigma_5^{-1}$, and so $\rho(\sigma_1\sigma_5^{-1})$ satisfies a braid relation with $\rho(\sigma_2\sigma_5^{-1})$. It follows that $(\sigma_1\sigma_5^{-1})^{q}$ satisfies a braid relation with $H_{c_2}^{q}\sigma_5^{-q}$. Since $\sigma_5$ commutes with both $\sigma_1$ and $H_{d}$, we further have that $\sigma_1^q$ satisfies a braid relation with $H_{d}^q$. A result of Bell and second author [@bellmargalit Lemma 4.9] states that if two half-twists $H^r_a$ and $H^{s}_b$ satisfy a braid relation, then $r=s = \pm 1$. The claim follows.
If $q = 1$, then $\rho(\sigma_1\sigma_j^{-1}) = \sigma_1\sigma_j^{-1}$ for $j$ odd, as desired. If $q = -1$, then we may further postcompose $\rho$ with the inversion automorphism of ${B}_n$ to obtain again that $\rho(\sigma_1\sigma_j^{-1}) = \sigma_1\sigma_j^{-1}$. This completes the first step.
*Step 2.* Fix some $i\geq 5$. There exists $g_i\in {B}_n'$ such that $$g_i\sigma_5g_i^{-1} = \sigma_i$$ and such that $g_i$ commutes with each of $\sigma_1$ and $\sigma_3$; for instance we may take $$g_i = \sigma_i^{9-2i} (\sigma_{i-1} \cdots \sigma_5)(\sigma_i \cdots \sigma_5).$$ Let $d_i$ be the curve such that $$H_{d_i} = \rho(g_i)\sigma_5\rho(g_i)^{-1}.$$ Since $g_i$ commutes with $\sigma_1$ and $\sigma_3$, and hence with $\sigma_1\sigma_3^{-1}$, it follows that $\rho(g_i)$ commutes with $\rho(\sigma_1\sigma_3^{-1}) = \sigma_1\sigma_3^{-1}$. It follows further that $\rho(g_i)$ commutes with each of $\sigma_1$ and $\sigma_3$ (it cannot be that $\rho(g_i)$ interchanges $c_1$ and $c_3$ because the signs of the half-twists differ). Hence $H_{d_i}$ commutes with $\sigma_1$.
Using the above properties of $g_i$ and the fact (from Step 1) that $\rho(\sigma_1\sigma_5^{-1}) = \sigma_1\sigma_5^{-1}$, we have $$\begin{aligned}
\rho(\sigma_1\sigma_i^{-1}) =\rho(g_i(\sigma_1\sigma_5^{-1})g_i^{-1}) &= \rho(g_i)\rho(\sigma_1\sigma_5^{-1})\rho(g_i)^{-1}
= \rho(g_i)\sigma_1\sigma_5^{-1}\rho(g_i)^{-1}
= \sigma_1H_{d_i}^{-1}.
\end{aligned}$$ This completes the second step.
*Step 3.* We begin with the case $i=2$. By Step 1, there is a half-twist $H_{d}$ that commutes with $\sigma_5$ and such that $\rho(\sigma_2\sigma_5^{-1}) = H_{d}\sigma_5^{-1}$. Let $d_2=d$. Since $\sigma_5$ commutes with both $\sigma_1$ and $\sigma_2$, we have that $$\sigma_1\sigma_2^{-1} = \sigma_1\sigma_5^{-1}\left(\sigma_5\sigma_2^{-1}\right) = \sigma_1\sigma_5^{-1}\left(\sigma_2\sigma_5^{-1}\right)^{-1}$$ and hence that $$\begin{aligned}
\rho(\sigma_1\sigma_2^{-1}) = \rho\left(\sigma_1\sigma_5^{-1}\left(\sigma_2\sigma_5^{-1}\right)^{-1}\right) = (\sigma_1\sigma_5^{-1})(H_{d_2}\sigma_5^{-1})^{-1}
= \sigma_1H_{d_2}^{-1}.\end{aligned}$$
Now let $i=3$. Let $c_3$ denote the simple closed curve such that $H_{c_3} = \sigma_3$. By Step 1, we have that $\rho(\sigma_1\sigma_3^{-1}) = \sigma_1\sigma_3^{-1}$, and hence that $\rho(\sigma_1\sigma_3^{-1}) = \sigma_1H_{c_3}^{-1}$.
Finally we treat the case $i=4$. Choose $g_4\in B_n'$ such that $g_4\sigma_3g_4^{-1} = \sigma_4$ and such that $g_4$ commutes with $\sigma_1$ and each $\sigma_i$ with $i\geq 6$ (for instance $g_4 = \sigma_4^{-2}\sigma_3 \sigma_4$). The second condition implies that $g_4$ commutes with $\sigma_1\sigma_6^{-1}$ and hence that $\rho(g_4)$ commutes with $\rho(\sigma_1\sigma_6^{-1}) = \sigma_1H_{d_6}^{-1}$. Equivalently, $\rho(g_4)$ commutes with $\sigma_1$ and $H_{d_6}$. Define $d_4$ to be the curve such that $$H_{d_4} = \rho(g_4)\sigma_3\rho(g_4)^{-1}$$ We then have that $$\rho(\sigma_1\sigma_4^{-1}) = \rho(g_4(\sigma_1\sigma_3^{-1})g_4^{-1}) = \rho(g_4)(\sigma_1\sigma_3^{-1})\rho(g_4)^{-1} = \sigma_1H_{d_4}^{-1}.$$ This completes the third step.
*Step 4.* Let $d_1$ be the standard curve $c_1$. We will show first that $i(d_i, d_j) = 0$ if $j-i \geq 2$ and then that $i(d_i,d_{i+1}) = 2$ for each $1 \leq i \leq n-2$.
We begin by showing that $i(d_i, d_j) = 0$ if $j-i \geq 2$. We treat separately the cases of $i > 2$, $i=2$, and $i=1$.
For the first case we fix some $i > 2$. Suppose first that $j-i > 1$. In this case $\sigma_i$ commutes with $\sigma_j$ and $\sigma_1\sigma_i^{-1}$ commutes with $\sigma_1\sigma_j^{-1}$. This implies that $\rho(\sigma_1\sigma_i^{-1})=\sigma_1H_{d_i}^{-1}$ commutes with $\rho(\sigma_1\sigma_j^{-1})=\sigma_1H_{d_j}^{-1}$. Since both $H_{d_i}$ and $H_{d_j}$ commute with $\sigma_1$, this implies that $H_{d_i}$ commutes with $H_{d_j}$. Equivalently, $i(d_i,d_j) = 0$, as desired.
Next we show that $i(d_2,d_j)=0$ for $j \geq 4$. We have that $$\sigma_2\sigma_j^{-1} = (\sigma_1\sigma_2^{-1})^{-1}(\sigma_1\sigma_j^{-1}).$$ Applying Steps 1 and 3 we have that $$\rho(\sigma_2\sigma_j^{-1}) = (\sigma_1H_{d_2}^{-1})^{-1} (\sigma_1H_{d_j}^{-1}) = H_{d_2}H_{d_j}^{-1}.$$ Since $\sigma_2\sigma_j^{-1}$ is conjugate to $\sigma_1\sigma_3^{-1}$ in ${B}_n'$, it follows that $H_{d_2}H_{d_j}^{-1}$ is conjugate to $\rho(\sigma_1\sigma_3^{-1})=\sigma_1\sigma_3^{-1}$ (by Step 1) and hence that $H_{d_2}$ commutes with $H_{d_j}$. Thus $i(d_2,d_j) = 0$.
Finally, we show that $i(d_1,d_j)=0$ for $j \geq 3$. We have already shown in Steps 2 and 3 that $\sigma_1$ commutes with each $H_{d_j}$ with $j\geq 4$. Since $H_{d_3} = \sigma_3$, it follows that $\sigma_1$ commutes with $H_{d_3}$, as desired.
We now proceed to show that $i(d_i,d_{i+1}) = 2$ for each $1 \leq i \leq n-2$. We again treat separately the cases $i > 2$, $i=2$, and $i=1$.
Fix some $i \geq 3$. Since $\sigma_i$ satisfies a braid relation with $\sigma_{i+1}$, the element $\sigma_1\sigma_i^{-1}$ satisfies a braid relation with $\sigma_1\sigma_{i+1}^{-1}$. It follows that $\rho(\sigma_1\sigma_i^{-1})=\sigma_1H_{d_i}^{-1}$ satisfies a braid relation with $\rho(\sigma_1\sigma_{i+1}^{-1})=\sigma_1H_{d_{i+1}}^{-1}$. Since both $H_{d_i}$ and $H_{d_{i+1}}$ commute with $\sigma_1$, this implies that $H_{d_i}$ satisfies a braid relation with $H_{d_{i+1}}$. As in Step 1, it follows that $i(d_i,d_{i+1}) = 2$.
Next we show that $i(d_2,d_3)=2$. Similar to the previous paragraph, $\rho(\sigma_2\sigma_5^{-1})$ satisfies a braid relation with $\rho(\sigma_3\sigma_5^{-1})$. Since $\rho(\sigma_2\sigma_5^{-1}) = H_{d_2}\sigma_5^{-1}$ and $$\rho(\sigma_3\sigma_5^{-1}) = \rho\left((\sigma_1\sigma_3^{-1})^{-1}(\sigma_1\sigma_5^{-1})\right) = \rho(\sigma_1\sigma_3^{-1})^{-1}\rho(\sigma_1\sigma_5^{-1}) = \sigma_3\sigma_5^{-1} = H_{d_3}\sigma_5^{-1}.$$ As in the previous paragraph, it follows that $i(d_2,d_3) = 2$.
Finally, we already showed in Step 1 that $\sigma_1$ satisfies a braid relation with $H_{d_2}$. Again, it follows that $i(d_1,d_2) = 2$. The completes the fourth step.
*Step 5.* Since the curves $c_1,d_2,\dots,d_{n-1}$ form a chain, there is an element $\alpha$ of ${B}_n$ such that the curves $\alpha(c_1), \alpha(d_2),\ldots, \alpha(d_{n-1})$ are equal to $c_1,\dots,c_{n-1}$, respectively (this is an instance of the change of coordinates principle for mapping class groups [@primer Section 1.3]).
After replacing $\rho$ by its post-composition with the inner automorphism of ${B}_n$ induced by $\alpha$, we have that $$\rho(\sigma_1\sigma_i^{-1}) = \sigma_1\sigma_i^{-1}$$ Since the elements $\sigma_1\sigma_i^{-1}$ generate ${B}_n'$, it follows that $\rho$ is equal to the standard inclusion. This completes Step 5, and thus the theorem is proven.
Homomorphisms between braid groups {#sec:castel}
==================================
In this section we classify homomorphisms ${B}_n \to {B}_n$ for $n \geq 7$. As discussed in the introduction, this is a special case of a theorem of Castel.
Let $\rho : {B}_n\rightarrow {B}_n$ be a homomorphism, and let $k$ be an integer. The *transvection* of $\rho$ by $z^k$ is the homomorphism given by $$\rho^{z^k}(\sigma_i) = \rho(\sigma_i)z^k$$ for all $1 \leq i \leq n-1$. There is an equivalence relation on the set of homomorphisms ${B}_n \to {B}_n$ whereby $\rho_1 \sim \rho_2$ if $\rho_2 = \alpha \circ \rho_1^{z^k}$ for some automorphism $\alpha$ of ${B}_n$ and some $k \in {\mathbb{Z}}$. This notion of equivalence is more complicated than the one we defined for homomorphisms ${B}_n' \to {B}_n$ in the introduction, in that it involves the transvections. There are no analogous transvections of homomorphisms ${B}_n' \to {B}_n$, since the image of ${B}_n'$ must lie in ${B}_n'$, and the only power of $z^k$ in ${B}_n'$ is the identity.
The following theorem represents the special case of Castel’s theorem that we will prove.
Let $n \geq 7$, and let $\rho : {B}_n \to {B}_n$ be a homomorphism whose image is not cyclic. Then $\rho$ is equivalent to the identity.
Assume that $\rho: {B}_n\rightarrow {B}_n$ is a homomorphism with non-cyclic image. This is equivalent to the assumption that restriction $\rho'$ of $\rho$ to ${B}_n'$ is nontrivial. Theorem \[thm:main\] then implies that there is an automorphism $\alpha$ of ${B}_n$ such that $\alpha\circ \rho'$ is the identity. Thus replacing $\rho$ by $\alpha\circ \rho$, we may assume that $\rho'$ is the inclusion map.
We claim that $\rho(z)$ is equal to $z^k$ for some integer $k$. As in Section \[sec:proof\] let $p = n(n-1)$. Since $z\in {B}_n$ is central, we have that $\rho(z)$ commutes with each $\rho(\sigma_i^pz^{-1})$. Since $$\rho(\sigma_i^pz^{-1}) = \rho'(\sigma_i^pz^{-1}) = \sigma_i^pz^{-1},$$ it follows that $\rho(z)$ commutes with each $\sigma_i^p$, hence with each $\sigma_i$. The claim follows.
We next claim that $\rho(\sigma_i)^p = \sigma_i^pz^{k-1}$. By the previous claim we have $$\sigma_i^pz^{-1} = \rho(\sigma_i^pz^{-1}) = \rho(\sigma_i^p)\rho(z)^{-1} = \rho(\sigma_i^p)z^{-k} = \rho(\sigma_i)^pz^{-k},$$ as desired.
We now claim that there exist integers $r$ and $s$ such that for all $1\leq i\leq n-1$ we have $$\rho(\sigma_i) = \sigma_i^rz^s.$$ By the previous claim, the canonical reduction system of $\rho(\sigma_i)$ is equal to $c_i$. Since $\sigma_1$ commutes with $\sigma_j$ for $j \geq 3$, we have that $\rho(\sigma_1)$ fixes each $c_j$ with $j \geq 3$. Thus the Nielsen–Thurston component of $\rho(\sigma_1)$ corresponding to the region between $c_1$ and the boundary of ${D}_n$ cannot be pseudo-Anosov or a nontrival periodic element. It follows that $\rho(\sigma_1) = \sigma_1^rz^s$ for some $r$ and $s$. Since $\rho(\sigma_i)$ is conjugate to $\rho(\sigma_1)$ for $1 \leq i \leq n-1$, the claim follows.
Next we claim that $r=1$. We have $$\sigma_1\sigma_3^{-1} = \rho(\sigma_1\sigma_3^{-1}) = \sigma_1^rz^s\sigma_3^{-r}z^{-s} = \sigma_1^r\sigma_3^{-r},$$ whence the claim.
We now have that $\rho(\sigma_i) = \sigma_iz^s$ for $1 \leq i \leq n-1$. The transvection of $\rho$ by $z^{-t}$ is then equal to the identity. This completes the proof of the theorem.
|
---
abstract: 'The profiles of the chromo-electric field generated by static quark-antiquark, $Q{\bar Q}$ and three-quark, $QQQ$ sources are calculated in Coulomb gauge. Using a variational ansatz for the ground state, we show that a flux tube-like structure emerges and combines to the “Y”-shape field profile for three static quarks. The properties of the chromo-electric field are, however, not expected to be the same as those of the full action density or the Wilson line and the differences are discussed.'
author:
- 'Patrick O. Bowman and Adam P. Szczepaniak'
title: ' Chromo-Electric flux tubes '
---
Introduction
============
An intuitive picture of quark-gluon dynamics emerges in the Coulomb gauge, $\na\cdot \A^a = 0$ [@clee; @zw1; @zw2]. In this case QCD is represented as a many-body system of strongly interacting physical quarks, antiquarks and gluons. In particular the gluon degrees of freedom have only the two transverse polarizations and in the non-interacting limit reduce to the physical massless plane wave states. In the interacting theory gluonic states, just like any other colored objects, are expected to be non-propagating, [*i.e.*]{} confined on the hadronic scale. The non-propagating nature of colored states follows from the infrared enhanced dispersion relations which can be set up in the Coulomb gauge [@zw2; @as1; @as2; @as3].
In the Coulomb gauge the $A^0$ component of the 4-vector potential results in an instantaneous interaction (potential) between color charges. Unlike QED, where the corresponding potential is a function only of the relative distance between the electric charges, in QCD it is a functional of the transverse gluon components, ${\bf A}$ [@clee]. Thus the numerical value of the potential cannot be obtained without knowing the correct wave functional of the state and its dependence on the gluon coordinates. So in QCD the chromo-electric field is expected to be non-local and to depend on the global distribution of charges, which set up the gluon wave functional.
Even though the exact solution to the general many-body problem is unavailable it is often possible to obtain good approximations if the dominant correlations can be identified. In Coulomb gauge QCD (in the Schrödinger field representation) the domain of the transverse gluon field, $\A$ is bounded and non-flat, and is referred to as the Gribov region. It is expected that the strong interaction between static charges originates from the long-range modes near the boundary of the Gribov region, the so called Gribov horizon. For example it has been recently shown that center vortices, when transformed to the Coulomb gauge, indeed reside on the Gribov horizon [@goz].
The curvature of the Gribov region contributes to matrix elements via the functional measure determined by the determinant of the Faddeev-Popov operator. This determinant prevents analytical calculations of functional integrals, however it has been shown that its effect can be approximated by imposing appropriate boundary conditions on the gluon wave functional [@as4; @hr]. This wave functional is in turn constrained by minimizing the expectation value of the energy density which leads to a set of coupled self-consistent Dyson equations [@as1; @sw]. Once the wave functional is determined it is possible to calculate the distribution of the chromo-electric field in the system. This is the main subject of this paper.
In the following we study the chromo-electric field in the presence of the static quark-antiquark and three-quark systems, prototypes for a meson and a baryon respectively. Recent lattice computations indicate that the gluonic field near the static $Q-{\bar Q}$ state forms flux tubes. There are also indications that for the $QQQ$ state the fields arrange in the so called “Y”-shape [@tak; @latY], although some work supports the “$\Delta$”-shape [@latD]. String-like behavior has been observed in the chromo-electric field in Ref. [@sho] and the “Y”-shape interaction advocated in Ref. [@kuzsim]. A recent reevaluation of the center-vortex model also supports the “Y”-shape [@corn].
In the following section we summarize the relevant elements of the Coulomb gauge formalism and discuss the approximations used. This is followed by numerical results and outlook of future studies. There is a fundamental difference between lattice gauge flux tubes corresponding to the distribution for the action density and the chromo-electric field profiles. In the context of the potential energy of the sources, this difference was emphasized in Zwanziger, Greensite and Olejnik [@goz; @zwancon]. We discuss those in Section. IV.
Chromo-electric Coulomb field in the presence of static charges
===============================================================
The Coulomb gauge Hamiltonian
-----------------------------
The Yang-Mills Coulomb gauge Hamiltonian in the Schrödinger representation, $H=H(\Pi, A)$ is given by $$H = \frac{1}{2} \int d\x \left[ \bm{\Pi}^a(\x) \cdot \bm{\Pi}^a(\x)
+ \B^a(\x)\cdot \B^a(\x) \right] + {\hat V}_C. \label{h}$$ The gluon field satisfies the Coulomb gauge condition, $\na \cdot \A^a(\x)=0$, for all color components $a=1\cdots N_c^2-1$. The conjugate momenta, $\bm{\Pi}^a(\x)=
-i\partial/\partial {\A^a(\x)}$ obey the canonical commutation relation, $[\Pi^{i,a}(\x), A^{j,b}(\y)] = -i\delta_{ab}
\delta^{ij}_T(\na_\x)\delta(\x-\y)$, with $\delta^{ij}_T(\na) =
\delta_{ij} - \nabla_i \nabla_i/\na^2$. The canonical momenta also correspond to the negative of the transverse component of the chromo-electric field, $\bm{\Pi}^a(\x) = -
\E^a_T(\x)$, $\na \cdot \E^a_T = 0$. The chromo-magnetic field, $\B$ contains linear and quadratic terms in $\A$. It will also be convenient to transform to the momentum space components of the fields by $$\A^a(\k) = \int d\x\A^a(\x) e^{-i\k\cdot x},$$ and similarly for $\bm{\Pi}^a(\k)$. The Coulomb potential ${\hat V}_C$ may be expressed in terms of the longitudinal component of the chromo-electric field, $${\hat V}_C = \frac{1}{2}\int d\x {\bf E}^a(\x) {\bf E}^a(\x),$$ with $${\bf E}^a(\x) = \int d\y d\z { \frac{\x - \y}{4\pi|\x - \y|^3}}
\left[ \frac{g}{1 - \lambda}\right]^{ab}_{\y,\z} \rho^b(\z). \label{e}$$ Here $(1-\lambda)$ is the Faddeev-Popov (FP) operator which in the configuration-color space is determined by, $$[\lambda]^{ab}_{\x,\y} = \int {\frac{d\p}{(2\pi)^3}} {\frac{d\q}{(2\pi)^3}}
e^{i\p\cdot\x} e^{-i\q\cdot\y}
\lambda_{ab}(\p,\q),$$ where $$\lambda_{ab}(\p,\q) = ig f_{acb} {\frac{\A^c(\p-\q)\cdot \q}{\q^2}},$$ $f$ are the $SU(N_c)$ structure constants, and $g$ is the bare coupling. In [Eq. (\[e\])]{}, $\rho$ is the color charge density given by $$\rho^a(\x) = \psi^{\dag}(\x)T^a \psi(\x) + f_{abc} \A^b(\x)\cdot
\bm{\Pi}^c(\x),$$ with the two terms representing the quark and the gluon contribution, respectively; the former is replaced by a $c$-number for static quarks. Without light flavors there is no other dependence on the quark degrees of freedom. The energy of the static $Q{\bar Q}$ or $QQQ$ systems measured with respect to the state with no sources is thus given by the Coulomb term and is determined by the expectation value of the longitudinal component of the chromo-electric field.
It is the dependence of the chromo-electric field and the Coulomb interaction on the static vector potential (through $\lambda$) that produces the differences between QCD and QED. In QED the kernel in the bracket in [Eq. (\[e\])]{} reduces to $[\cdots] \to
\delta(\y-\z)$ and the Abelian expression for the electric field emerges. In QCD the chromo-electric field and the Coulomb potential are enhanced due to long-wavelength transverse gluon modes on the Gribov horizon where the FP operator vanishes. The combination of two effects on the Gribov horizon: enhancement of $(1 - \lambda)^{-1}$ in the longitudinal electric field and vanishing of the functional norm, which is proportional to $\det(1-\lambda)$, leads to finite, albeit large, expectation values of the static interaction between color charges. In [Eq. (\[h\])]{} we have omitted the FP measure since, as mentioned earlier in Ref. [@as4], its effect can be approximately accounted for by imposing specific boundary conditions on the ground state wave functional.
Since the chromo-electric field depends on the distribution of the transverse vector potential it is necessary to know the wave functional of the system. A self-consistent variational ansatz can be chosen in a Gaussian form,
$$\Psi[A] = \exp\left( - \frac{1}{2}
\int {\frac{d\p}{(2\pi)^3}} \omega(p)
\A^a(\p)\cdot \A^a(-\p) \right). \label{varia}$$
The parameter $\omega(p)$ ($p\equiv |\p|$) is determined by minimizing the expectation value of the energy density of the vacuum ([*i.e.*]{} without sources). The boundary condition referred to above corresponds to setting $\omega(0) \equiv \mu$ to be finite, which plays the role of $\Lambda_{QCD}$, [*i.e*]{} it controls the position of the Landau pole. Minimizing the energy density of the vacuum leads to a set of coupled self-consistent integral equations: one for $\omega$, one for the expectation value of the inverse of the FP operator, $d(p)$, $$\begin{gathered}
(2\pi)^3 \delta(\p-\q)\delta_{ab} d(p)
\equiv \\
\int d\x d\y e^{-i\p\cdot\x} e^{i\q\cdot\y}
\langle \Psi| \left[ \frac{g}{1 - \lambda}\right]^{ab}_{\x,\y}
|\Psi \rangle / {\langle \Psi|\Psi \rangle },
\label{d}\end{gathered}$$ and one for the expectation value of the square of the inverse of the FP operator, which appears in the matrix elements of $V_C$, $$\begin{gathered}
(2\pi)^3 \delta(\k-\q)\delta_{ab} f(p)d^2(p)
\equiv \\
\int d\x d\y e^{-i\p\cdot\x} e^{i\q\cdot\y}
\langle \Psi| \left[ \left(\frac{g}{1 - \lambda}\right)^2
\right]^{ab}_{\x,\y}
|\Psi \rangle / {\langle \Psi|\Psi \rangle }.
\label{ff}\end{gathered}$$ The approximation $f=1$ ignores the dispersion in the expectation value of the inverse of the FP operator, $$\Bigg\langle \left[ \frac{g}{1 - \lambda} \right]^2 \Bigg\rangle
\to \Bigg\langle \frac{g}{1 - \lambda} \Bigg\rangle^2. \label{disp}$$ This approximation has been extensively used, [*e.g.*]{} in Refs. [@zw1; @zwan]. The three Dyson equations were analyzed in Ref. [@as1] where it was found that the solution of $\omega$ can be well approximated by the simple function $\omega(p) = \theta( \mu - p) \mu + \theta( p -\mu)
p$. The renormalization scale $\mu$, being the only parameter in the theory, can constrained by the long range part of the Coulomb kernel $\langle V_C \rangle \propto fd^2$. We will discuss this more in the subsection below. The low momentum, $p<\mu$ dependence of $d(p)$ and of the Coulomb potential $V_C(p) = f(p)d(p)^2$ is well approximated by a power-law, $$d(p) = d(\mu) \left( \frac{\mu}{p}\right)^\alpha, f(p) = f(\mu)
\left( \frac{\mu}{p}\right)^\beta \label{df}$$ with $\alpha \sim 0.5$ and $\beta \sim 1$. The exponents are bounded by $ 2\alpha + \beta \le 2$ and the upper limit corresponds to the linearly rising confining potential. At large momentum, $p >> \mu$, as expected from asymptotic freedom, both $d$ and $f$ are proportional to $1/\log^\gamma(p)$, with $\gamma = O(1)$. Adding static sources does not modify the parameters of the vacuum gluon distribution, [*e.g.*]{} $\omega(p)$. This is because the vacuum energy is an extensive quantity while sources contribute a finite amount to the total energy. Thus we can use the three functions $\omega$, $d$ and $f$ calculated in the absence of the sources to compute the expectation value of the chromo-electric field in the presence of static sources. The ansantz state obtained by applying quark sources to the variational vacuum of Eq. (\[varia\]) does not, however optimize the state with sources.
The field lines in the $Q{\bar Q}$ and $QQQ$ systems
------------------------------------------------------
For a quark and an antiquark at positions $\x_q \equiv \R/2 =
R{\hat \z}/2$ and $\x_{\bar q} = -\R/2 = -R{\hat \z}/2$, respectively, and the gluon field distributed according to $\Psi[\A]$, the expectation value of the square of the magnitude of the chromo-electric field measured at position $\x$ is given by
$$\begin{gathered}
\langle \E^2(\x,\R) \rangle = { \frac{C_F}{(4\pi)^2}}
\sum_{\z_1=\pm \R/2} \sum_{\z_2 =\pm \R/2} \pm
\int d\y_1 d\y_2 \\
\times{ \frac{ (\x - \y_1)\cdot (\x-\y_2)}{|\x - \y_1|^3
|\x - \y_2|^3}} E(\z_1,\y_1;\z_2,\y_2), \label{qq}\end{gathered}$$
where the $+ (-)$ sign is for the $\z_1 =(\ne) \z_2$ contributions, and $$E(\z_1,\y_1;\z_2,\y_2)
\equiv \frac{\langle \Psi |
\left[ \frac{g}{1 - \lambda}\right]_{\z_1,\y_1}
\left[ \frac{g}{1 - \lambda}\right]_{\y_2,\z_2}|\Psi \rangle }
{ \langle \Psi|\Psi \rangle }. \label{ee}$$ The color factors leading to $C_F$ can be extracted from the expectation value in [Eq. (\[ee\])]{} (the ground state expectation value of the inverse of two FP operators is an identity in the adjoint representation). In the Abelian limit, $E(\z_1 \cdots \y_2) \to \delta(\y_1 -
\z_1)\delta(\y_2 - \z_2)$ and Eq. (\[qq\]) gives the dipole field distribution, $\langle \E^2 \rangle_{QED}$. One should note that [Eq. (\[qq\])]{} contains the two self energies. These self energies are necessary to produce the correct asymptotic behavior at $x >>
R$ for charge-neutral systems, (in QED and QCD) [*i.e*]{} $\E^2$ has to fall-off at least as $1/\x^4$ at large distances from the sources.
The infrared, $|\x| \sim |\R| >> 1/\mu$ enhancement in QCD arises from the expectation value of the inverse of the FP operator. If $\langle \E^2(\x,\R) \rangle$ is integrated over $\x$ one obtains the expectation value of the Coulomb energy of the $Q{\bar Q}$ source. The mutual interaction energy is given by, $$\begin{aligned}
V_C(\R) &= {1\over 2}\int d\x \langle \E^2(\x,\R) \rangle \nonumber \\
&\hspace{-5mm}= -C_F {{\langle \Psi |
\left[ {g\over {1 - \lambda}} \left(-{1\over \na^2}\right)
{g\over {1 - \lambda}}\right]_{{\R\over 2},-{\R\over 2}}
|\Psi \rangle }
/{ \langle \Psi|\Psi \rangle }
}, \nonumber \\
&\hspace{-5mm} = -C_F \int {{d\p}\over {(2\pi)^3}} {{d^2(p) f(p)}\over {p^2}}
e^{i\p\cdot \R},
\label{vc} \end{aligned}$$ and the net self-energy contribution is, $$\begin{aligned}
\Sigma &= C_F {{\langle \Psi |
\left[ {g\over {1 - \lambda}} \left(-{1\over \na^2}\right)
{g\over {1 - \lambda}}\right]_{\pm {\R\over 2},\pm {\R\over 2}}
|\Psi \rangle }
/{ \langle \Psi|\Psi \rangle } }, \nonumber \\
&= C_F \int {{d\p} \over {(2\pi)^3}} {{d^2(p) f(p)}\over {p^2}}.
\label{sigma} \end{aligned}$$ In lattice simulations it has been shown [@Greensite] that the Coulomb energy and the phenomenological static $Q\bar Q$ potential obtained from the Wilson loop are different. In particular it was found that the Coulomb potential string tension is about three times larger than the phenomenological string tension. This is in agreement with the “no confinement without Coulomb Confinement” scenario discussed by Zwanziger [@zwancon]. It is simple to understand the origin of the difference. Even if $|\Psi[\A]\rangle $ were the true vacuum state (without sources) of the Coulomb gauge QCD Hamiltonian (here we approximate it by a variational ansatz) the state $|Q{\bar Q},R\rangle \equiv Q(\R/2){\bar Q}(-\R/2) |\Psi[A]\rangle$ is no longer an eigenstate. For example ${\hat V}_C$ acting on $|Q{\bar
Q},R\rangle$ excites any number of gluons and couples them to the quark sources. The Coulomb energy was defined as the expectation value, $V_C$ in $|Q{\bar Q},R\rangle$ minus the vacuum energy and it is therefore different from the phenomenological static potential energy which corresponds to the total energy (measured with respect to the vacuum) of the true eigenstate of the Hamiltonian with a $Q{\bar Q}$ pair. If one defines [@goz] $$\begin{aligned}
G(R,T) &\equiv \langle Q{\bar Q},R|e^{-(H-E_0)T}|Q{\bar Q},R\rangle
\nonumber \\
& = \sum_n |\langle Q {\bar Q},R,n|Q{\bar Q},R\rangle|^2 e^{-(E_n -
E_0) T},\end{aligned}$$ then the Coulomb potential on the lattice can be calculated from $$V_C(R) = \lim_{T=0} -{d\over {dT}} \log(G(R,T)),$$ and the phenomenological potential from $$V(R) = \lim_{T=\infty} -{d\over {dT}} \log(G(R,T)).$$ Thus one should be comparing $V_C(R)$ in [Eq. (\[vc\])]{} to the lattice Coulomb potential and not to the phenomenological potential obtained from the Wilson loop. Finally, one could try to optimize the state with sources, [*e.g.*]{} by adding gluonic components. In this case terms in the Hamiltonian beyond the Coulomb term would contribute to the energy of the system and one could compare with the true (Wilson loop) static energy. In our previous studies, where we extracted numerical values for $\mu$ and the critical exponents $\alpha,\beta,\gamma$ ([*cf.*]{} [Eq. (\[df\])]{}) we have instead compared $V_C$ to the phenomenological, Wilson potential [@as1]. In what follows we will use the larger value of the string tension, to be in agreement with Ref. [@goz].
If the two exponents $\alpha$ and $\beta$, which determine the infrared behavior of $d(p)$ and $f(p)$ respectively, satisfy $2\alpha + \beta > 2$, then the self energy in [Eq. (\[sigma\])]{} is divergent and so is the [*rhs*]{} of [Eq. (\[vc\])]{}. This reflects the long-range behavior of the effective confining potential generated by self-interactions between the gluons that make up the Coulomb operator. For the colorless $Q\bar Q$ system the total energy which is the sum of $V_C$ and $\Sigma$, is finite as it should be. For a colored system, [*e.g.*]{} a quark-quark source, the sign of $V_C$ changes, there is no cancellation between the infrared singularities, and in the confined phase the system would be un-physical with infinite energy. The integral determining the self energy also becomes divergent in the UV, since for $p\to \infty$ the product $d^2(p)f(p)$ only falls-off logarithmically. Modulo these logarithmic corrections this UV divergence is the same as in the Abelian theory and can be removed by renormalizing the quark charge.
It follows from translational invariance of the matrix element in [Eq. (\[ee\])]{}, that $E$ depends only on the relative coordinates, $\z_1 - \y_1$ and $\z_2 - \y_2$. We therefore introduce the momentum space representation, $$\begin{aligned}
E(\z_1,\y_1;\z_2,\y_2) & = & \int {{d\p} \over {(2\pi)^3}}
{{d\q} \over {(2\pi)^3}} e^{i\p \cdot (\z_1 - \y_1)
-i \q\cdot (\z_2 - \y_2) } \nonumber \\
& & \qquad\times d(p) d(q) E(\p;\q), \label{ft}\end{aligned}$$ and define $F_\L(\l) \equiv E(\l+\L/2; \l-\L/2)$ with $\l \equiv (\p + \q)/2$ and $\L \equiv \p - \q$. The Dyson equation for $F$ can be derived in the rainbow-ladder approximation which, as shown in Ref. [@as1; @as2], sums up the dominant infrared and ultra-violet contributions to the expectation value of the inverse of two FP operators,
$$F_\L(\l)
= 1 + N_c \int {{d\k}\over {(2\pi)^3 }}
{{\left[ (\k-\L/2)\delta_T(\k + \l)(\k+\L/2) \right] }
\over {2\omega(\k + \l)}}
{{d(\k - \L/2)}\over {(\k - \L/2)^2}}
{{d(\k + \L/2)}\over {(\k + \L/2)^2}} F_\L(\k), \label{F}$$
It follows from [Eq. (\[ft\])]{} that $\L$ and $\l$ are conjugate to the [*center of mass*]{}, $\R \equiv [ (\z_1 - \y_1) + (\z_2 - \y_2)
]/2$ and the [*relative*]{}, $\r \equiv [ (\z_1 - \y_1) - (\z_2 - \y_2)
]$ coordinate respectively. The Dyson equation for $F_L$ is UV divergent if for $p/\mu >> 1$, and $d(p) \ge \log^{1/2}(p^2)$. This divergence can be removed by the Coulomb operator renormalization constant. The renormalized equation is obtained from the once-subtracted equation $F_\L(\l) - F_{\L_0}(\l_0)$. For example, if the subtraction is chosen at $|\l_0| = \mu$ and $\L_0 = {\bf 0}$, the renormalized coupling $F_0(\mu)$ can be fixed from the Coulomb potential. After integrating [Eq. (\[qq\])]{} (over $\x$) one obtains $\delta(\y_1 -
\y_2)$ multiplying $E(\z_1, \cdots, \y_2)$. Therefore, it follows from [Eq. (\[ft\])]{} that $V_C(\R)$ is determined by $F_0(\l)$ and $F_0(\l) = F_0(l) = f(l)$ with $f$ defined in [Eq. (\[ff\])]{}.
In [Eq. (\[F\])]{} $\L$ is a parameter, [*i.e.*]{} the Dyson equation does not involve self-consistency in $\L$. We have just shown that as $\L \to {\bf 0}$, $F_\L(\l)$ has a finite limit: it is given by $f$. For large $L=|\L|$ ($L/\mu >> 1$), due to asymptotic freedom, $F_\L$ is expected to vanish logarithmically, $F_\L \to d^2(L) \propto 1/\log(L^2)$. We do not attempt here to solve [Eq. (\[F\])]{}, instead we use a simple interpolation formula between the $L=0$ and $L\to \infty$ limits,
$$F_\L(\l) =
f(\l) \theta(\mu - |\l|) \theta(\mu -|{\L\over 2}|) + \left[ 1 -
\theta(\mu - |\l|)\theta(\mu - |{\L\over 2}|) \right]
\sim f\left( {\p + \q}\over 2 \right) \theta(\mu - p) \theta( \mu - q)
+ \left[ 1 - \theta(p) \theta(q) \right], \label{f}$$
[*i.e.*]{} in the term in the bracket we ignore the short distance logarithmic corrections. It is easy to show that if logarithmic corrections are ignored then the short-range, $p,q > \mu$ contribution to the energy density is the same as in the Abelian case. Since we are mainly interested in the long range behavior of the chromo-electric field, in the following we shall ignore contributions from the region $p,q>\mu$ all together. In the long-range approximation, $x, R >> 1/\mu$ the expectation value of $\E^2$ is then given by,
$$\langle \E^2(\x,\R) \rangle =
{{C_F}\over {(4\pi)^2}}
\sum_{ij=1}^2 \xi^{Q\bar Q}_{ij} \int d\r f_L(r)
{{\z_i - \x - \r/2} \over {|\z_i - \x - \r/2|}}
\cdot {{\z_j - \x + \r/2} \over {|\z_j - \x + \r/2|}}
d'_L(\z_i - \x - \r/2) d'_L(\z_j - \x + \r/2).
\label{el}$$
$\xi^{Q\bar Q}_{ij} = 1$ for $i=j$ and $-1$ for $i\ne j$, $\z_{1,(2)} = (-)\R/2$, $$d'_L(r) \equiv {2\over {\pi}}\int^\mu_0 p dp j_1(rp) d(p), \label{dpl}$$ is the derivative of $d_L$ w.r.t. $r$, $$f_L(r) = {1\over {2\pi^2}} \int^\mu_0 dp p^2 f(p) j_0(pr), \label{fl}$$ and $j_0, j_1$ are Bessel’s functions. We note that the expression in [Eq. (\[el\])]{} is not necessarily positive. In the limit $f(p)=1$, the matrix element of the square of the inverse of the FP operator is approximated by the square of matrix elements (cf. [Eq. (\[disp\])]{}) and $\langle \E^2\rangle$ becomes positive.
The expression for $\langle \E^2\rangle$ for the three quark system is derived by taking the expectation value of the Coulomb operator, ${\hat V}_C$ in a color-singlet state $\epsilon_{ijk}Q_i(\z_1) Q_j(\z_2)
Q_k(\z_3) |\Psi[\A]\rangle$, which gives
$$\langle \E^2(\x,\R_i) \rangle = {{C_F}\over {(4\pi)^2}}
\sum_{ij=1}^3 \xi^{QQQ}_{ij} \int d\r f_L(r) {{ \z_i - \x - \r/2 } \over
{ |\z_i - \x - \r/2| } }
\cdot
{{ \z_j - \x + \r/2 } \over
{ |\z_j - \x + \r/2| }} d'_L(\z_i - \x - \r/2)
d'_L(\z_j - \x + \r/2)$$
where $\xi^{QQQ}_{ij} = 1$ if $i = j$ and $\xi^{QQQ}_{ij} = -1/2$ if $i \ne j$. We note that the energy density for the $QQQ$ system comes from two-body correlations between the $QQ$ pairs.
Numerical results
===================
We first consider the simple approximation to the expectation value of the Coulomb kernel of [Eq. (\[disp\])]{} in which $f(p)=1$. If one wishes to have the confining potential grow linearly at large distances then it is necessary to set $\alpha = 1$, [*i.e.*]{} $d(p) \propto \mu/p$ for $p/\mu < 1$. In this case, assuming that the long-range behavior of the potential is of the form $V_C(r) = b_C r$, we obtain from [Eq. (\[vc\])]{}, $$b_C = C_F d^2(\mu)\mu^2/(8\pi). \label{bc}$$ We use the Coulomb string tension $b_C = 0.6 \mbox{ GeV}^2$. For the $Q\bar Q$ system the long-range contribution to the electric fields is then given by, $$\begin{gathered}
\langle \E^2(\x,\R) \rangle =
{{2b_C}\over {\pi^3}}
\biggl[
{{(\R/2 - \x)} \over {|\R/2 -
\x|^2}}\left(1-j_0(\mu|\R/2-\x|)\right) \\
+ (\x \to -\x) \biggr]^2. \label{elnof}\end{gathered}$$
![\[fig1\] $R^2 \langle \E^2(x) \rangle$ in units of $2b_C/\pi^3(\hbar c)^2 $ as a function of the distance $x$ along the $Q{\bar Q}$ axis. We employ the $f(p) = 1$ approximation. The quark and the antiquark are located at $R/2=5\mbox{ fm}$ and $-R/2 = -5\mbox{ fm}$ respectively. The renormalization scale $\mu=1.1\mbox{ GeV}$ is calculated from [Eq. (\[bc\])]{} using $d(\mu) = 3.5$ from Ref. [@as1]. The dashed line is the contribution from the two self energies, the dash-dotted line represents mutual interactions and the solid line is the total.](Fig1a.eps){width="3.in"}
In Fig. 1 we show the Coulomb energy density as a function of position on the $Q{\bar Q}$ axis, $\x = x\hat{\R}$, for $R=|\R|=10\mbox{ fm}$. The small oscillations come from the sharp-cutoff introduced by the $\theta$-functions in [Eq. (\[f\])]{} which produces the Bessel’s functions in [Eq. (\[elnof\])]{}. For a smooth cutoff, [*e.g.*]{} with $\theta(\mu - p) \to \exp(-p/\mu)$ in [Eq. (\[elnof\])]{} one should replace $1 - j_0(\mu|\R/2-\x|)$ by $1 - \arctan(\mu|\R/2-\x|)/\mu|\R/2-\x|$. The cut-off is also responsible for the rapid variations near the quark positions, $\x = \pm R/2$.
We note that for large separations between the quarks, $R >> 1/\mu$ and $x << R$, the Coulomb energy density behaves as expected from dimensional analysis, $$\langle \E^2(\x,R\mu\to \infty) \rangle \to {{32 b_C}\over {\pi^3R^2}}.
\label{short}$$ which is consistent with linear confinement, [*i.e.*]{} if $\langle \E^2(\x,R\mu\to \infty) \rangle$ is integrated over $\x$ in the region $|\x|<R$ on obtains $V_C(R) \propto R$.
At large distances $x >> R >> 1/\mu$ we obtain $$\langle \E^2(|\x|/R \to \infty,R\mu\to \infty) \rangle \to
{{2 b_C R^2}\over {\pi^3\x^4}}. \label{inf}$$ If there were a finite correlation length one would expect $\langle \E^2(|\x|/R \to \infty,R\mu\to \infty)\rangle$ to fall-off exponentially with $|\x|$ [@DDS] and not as a power-law. The power-law behavior obtained in Eq. (\[inf\]) is again related to the difference between the $|Q{\bar Q}, R\rangle$ state used here, which is built by adding quark sources to the vacuum and the true ground state of the $Q{\bar Q}$ system as discussed in Sec. IIA. In other words the profile of the chromo-electric field distribution for such a state is not expected to agree with the profile of the flux-tube or action density. To illustrate this difference, in Fig. 2 we plot the energy density as a function of the magnitude of the distance transverse to the $Q\bar Q$ axis, $x_\perp = |\x_\perp|$, $\R\cdot \x = \R\cdot \x_\perp = 0$.
![\[fig1b\] $R^2 \langle \E^2(x) \rangle$ in units of $2b_C/\pi^3(\hbar c)^2 $ as a function of the distance $x$ transverse to the $Q{\bar Q}$ axis. The units and the setting are as in Fig. 1.](Fig1b.eps){width="3.in"}
Finally, in Fig. 3, we show the contour plot of the energy density as a function of the position in the $xz$ plane with quark and antiquark on the $z$ axis at $R/2$ and $-R/2$ respectively.
![\[fig2\] $R^2 \langle \E^2(x) \rangle$ as a function of position in the $xz$ plane. The units and the same setting as in Fig. 1. ](Fig2.eps){width="3.in"}
It is clear from Figs. 2 and 3 that a flux tube like structure emerges and from [Eq. (\[short\])]{} that it has the correct scaling as a function of the $Q{\bar Q}$ separation but, as discussed above it does not have a finite correlation length (large $x$ behavior).
The field distribution for the $QQQ$ system in the $f_L(p) =1 $ approximation is equal to the sum of three terms each representing a contribution from a $QQ$ pair. We place each of the three quarks in a corner of an equilateral triangle, $\z_i$, $i=1,\cdots3$ $$\begin{gathered}
\langle \E^2(\x,\R_i) \rangle = \frac{C_F}{32\pi^2}
\bigl[ (\D_1 - \D_2)^2 \\
+ (\D_1 - \D_3)^2 + (\D_2 - \D_3)^2 \bigr],\end{gathered}$$ where $$\D_i = {{ \z_j - \x + \r/2 } \over
{ |\z_j - \x + \r/2| }} d'_L(\z_i - \x - \r/2).$$
The contour plot of of energy density in this case is shown in Fig. 4. Even though the field originates from the two-particle correlations the net field seems to form into a “Y”-shape structure. This structure has also recently been seen to emerge in Euclidean lattice simulations.
![\[fig3\] $R^2 \langle \E^2(x) \rangle$ as a function of position in the $xz$ plane. The units and the same setting as in Fig. 1. The upper panel show the total field distribution and the lower the distribution from mutual interaction (no self-energies) only. ](Fig3.eps){width="2.5in"}
Finally, to study the effects of $f_L(p)$, in Fig. 5 we show the predictions for the $Q{\bar Q}$ field distribution given by [Eq. (\[vc\])]{} where we use $d(p)$ and $f(p)$ in the form given by [Eq. (\[df\])]{} with $\alpha = 1/2$ and $\beta=1$ and normalized such that $V(R) \to bR$ at large distances. Furthermore, to remove the oscillations introduced by the momentum space cutoff, we now cut the small $x$ region in coordinate space, by i) extending the upper limits of integration in Eqs. (\[dpl\]) and (\[fl\]) to infinity and ii) cutting off the position space functions at short distances, $$d'_L(r) = {2\over {\pi\alpha}} \sin(\pi\alpha/2)
\Gamma(2 -\alpha) \theta(r\mu - 1)
{ {\mu^2} \over {(\mu r)^{2-\alpha}}}$$ $$f_L(r) = {1\over {2\pi^2}} \sin(\pi\beta/2)
\Gamma(2-\beta) \theta(r \mu - 1)
{ {\mu^3} \over {(\mu r)^{3-\beta}}}.$$
Comparing Fig. 3 and Fig. 5 we observe a narrowing of the flux tube. This is to be expected as the action of $f(p)$ is to introduce additional gluonic correlations. That said, there is no major qualitative change in the field distribution.
![\[fig4\] $R^2 \langle \E^2(x) \rangle$ for $Q{\bar Q}$ from [Eq. (\[el\])]{} with $\alpha = 1/2$ and $\beta=1$. The units and the same setting as in Fig. 1, except that the contribution from $f(p)$ has been included.](Fig4.eps){width="2.5in"}
Summary
=======
We have calculated the distribution of the longitudinal chromo-electric field in the presence of static $Q{\bar Q}$ and $QQQ$ sources using a variational model for the ground state wave functional. Despite this wave functional having no string-like correlations a flux tube like picture does emerge. In particular the on-axis energy density of the $Q{\bar Q}$ system behaves as $b_c/R^2$ for large inter-quark separation, $R$ and the field falls off like $b_c R^2/x^4$ at large distances from the center of mass of the $Q{\bar Q}$ system, $x$. This is weaker than in the Abelian case ($\sim R^2/x^6$) and implies that moments of the average transverse spread of the tube, defined as proportional to $\langle |x_\perp|^n \E^2(z,x_\perp) \rangle$, are finite for $n<2$ only. Thus there is no finite correlation length for the longitudinal component of the chromo-electric field, as expected for the state which does not take into account screening of the Coulomb line by the transverse gluons (flux tube). This also leads to large Van der Waals forces, which is bothersome, but it is consistent with the scenario of “no confinement without Coulomb confinement” of Zwanziger. The Coulomb potential leads to a variational (stronger) upper bound to the true confining interaction.
Similar behavior at large distances is also true for the three quark sources, except that here we find the emergence of the “Y”-shape junction. This is consistent with lattice simulations, but is remarkable in our case as it arises from two-body forces. It will be interesting to examine field distributions which include transverse field excitations. In that case the only lattice results available are for the potential, not for the field distributions. Finally we note that, since the mean field calculation provides a variational upper bound, the long range behavior of the field distribution falls-off more slowly than expected for the Van der Waals force. Certainly as the complete string develops this is expected to disappear and it would be interesting to build a string-like model for the ansatz ground state to verify this assertion.
The authors wish to thank J. Greensite, H. Reinhardt, Y. Simonov, F. Steffen, H. Suganuma and D. Zwanziger for helpful feedback. This work was supported in part by the US Department of Energy under contract DE-FG0287ER40365. The numerical computations were performed on the AVIDD Linux Clusters at Indiana University funded in part by the National Science Foundation under grant CDA-9601632.
[99]{}
N. H. Christ and T. D. Lee, Phys. Rev. D [**22**]{}, 939 (1980) \[Phys. Scripta [**23**]{}, 970 (1981)\].
D. Zwanziger, Nucl. Phys. B [**485**]{}, 185 (1997) \[arXiv:hep-th/9603203\]. A. Cucchieri and D. Zwanziger, Phys. Rev. Lett. [**78**]{}, 3814 (1997) \[arXiv:hep-th/9607224\].
A. P. Szczepaniak and E. S. Swanson, Phys. Rev. D [**65**]{}, 025012 (2002) \[arXiv:hep-ph/0107078\].
A. P. Szczepaniak and E. S. Swanson, Phys. Rev. D [**62**]{}, 094027 (2000) \[arXiv:hep-ph/0005083\]. A. P. Szczepaniak and E. S. Swanson, Phys. Rev. D [**55**]{}, 1578 (1997) \[arXiv:hep-ph/9609525\]. J. Greensite, S. Olejnik and D. Zwanziger, arXiv:hep-lat/0401003.
L. Del Debbio, A. Di Giacomo and Y. A. Simonov, Phys. Lett. B [**332**]{}, 111 (1994) \[arXiv:hep-lat/9403016\].
A. P. Szczepaniak, arXiv:hep-ph/0306030.
C. Feuchter and H. Reinhardt, arXiv:hep-th/0402106.
A. R. Swift, Phys. Rev. D [**38**]{}, 668 (1988).
T. T. Takahashi, H. Matsufuru, Y. Nemoto and H. Suganuma, Phys. Rev. Lett. [**86**]{}, 18 (2001) \[arXiv:hep-lat/0006005\]; Phys. Rev. D [**65**]{}, 114509 (2002) \[arXiv:hep-lat/0204011\]; T. T. Takahashi and H. Suganuma, Phys. Rev. Lett. [**90**]{}, 182001 (2003). V.G. Bornyakov [*et al.*]{} \[DIK Collaboration\], arXiv:hep-lat/0401026.
C. Alexandrou, P. De Forcrand and A. Tsapalis, Phys. Rev. D [**65**]{}, 054503 (2002) \[arXiv:hep-lat/0107006\].
A. I. Shoshi, F. D. Steffen, H. G. Dosch and H. J. Pirner, Phys. Rev. D [**68**]{}, 074004 (2003) \[arXiv:hep-ph/0211287\]. D. S. Kuzmenko and Y. A. Simonov, Phys. Atom. Nucl. [**66**]{}, 950 (2003) \[Yad. Fiz. [**66**]{}, 983 (2003)\] \[arXiv:hep-ph/0202277\]. J. M. Cornwall, Phys. Rev. D [**69**]{}, 065013 (2004) \[arXiv:hep-th/0305101\].
D. Zwanziger, Phys. Rev. Lett. [**90**]{}, 102001 (2003) \[arXiv:hep-lat/0209105\].
D. Zwanziger, arXiv:hep-ph/0312254. J. Greensite and S. Olejnik, Phys. Rev. D [**67**]{}, 094503 (2003) \[arXiv:hep-lat/0302018\].
|
---
abstract: 'Shape-constrained density estimation is an important topic in mathematical statistics. We focus on densities on $\mathbb{R}^d$ that are log-concave, and we study geometric properties of the maximum likelihood estimator (MLE) for weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the optimal log-concave density is piecewise linear and supported on a regular subdivision of the samples. This defines a map from the space of weights to the set of regular subdivisions of the samples, i.e. the face poset of their secondary polytope. We prove that this map is surjective. In fact, every regular subdivision arises in the MLE for some set of weights with positive probability, but coarser subdivisions appear to be more likely to arise than finer ones. To quantify these results, we introduce a continuous version of the secondary polytope, whose dual we name the Samworth body. This article establishes a new link between geometric combinatorics and nonparametric statistics, and it suggests numerous open problems.'
author:
- 'Elina Robeva, Bernd Sturmfels, and Caroline Uhler'
title: '**Geometry of Log-Concave Density Estimation**'
---
Introduction
============
Let $X = (x_1,x_2,\ldots,x_n)$ be a configuration of $n$ distinct labeled points in ${\mathbb{R}}^d$, and let $w = (w_1,w_2,\ldots,w_n)$ be a vector of positive weights that satisfy $w_1 + w_2 + \cdots + w_n =1$. The pair $(X,w)$ is our dataset. Think of experiments whose outcomes are measurements in ${\mathbb{R}}^d$. We interpret $w_i$ as the fraction among our experiments that led to the sample point $x_i$ in ${\mathbb{R}}^d$.
From this dataset one can compute the sample mean $\,\hat \mu = \sum_{i=1}^n w_i x_i \,$ and the sample covariance matrix $\,\hat \Sigma = \sum_{i=1}^n w_i (x-\hat \mu) (x - \hat \mu)^T $. Suppose that $\hat \Sigma$ has full rank $d$ and we wish to approximate the sample distribution by a Gaussian with density $f_{\mu,\Sigma}$ on ${\mathbb{R}}^d$. Then $(\hat \mu, \hat \Sigma) $ is the best solution in the likelihood sense, i.e. this pair maximizes the log-likelihood function $$\label{ex:loglikelihood} (\mu,\Sigma) \,\,\, \mapsto \,\,\, \sum_{i=1}^n w_i \cdot
{\rm log}(f_{\mu,\Sigma} (x_i)) .$$
In [*nonparametric statistics*]{} one abandons the assumption that the desired probability density belongs to a model with finitely many parameters. Instead one seeks to maximize $$\label{eq:loglikelihood2}
f \,\,\,\mapsto \,\,\, \sum_{i=1}^n w_i \cdot {\rm log}(f (x_i))$$ over all density functions $f$. However, since $f$ can be chosen arbitrarily close to the finitely supported measure $\sum_{i=1}^n w_i \delta_{x_i}$, it is necessary to put constraints on $f$. One approach to a meaningful maximum likelihood problem is to impose [*shape constraints*]{} on the graph of $f$. This line of research started with Grenander [@Grenander], who analyzed the case when the density is monotonically decreasing. Another popular shape constraint is convexity of the density [@Groeneboom].
In this paper, we consider maximum likelihood estimation, under the assumption that $f$ is [*log-concave*]{}, i.e. that ${\rm log}(f)$ is a concave function from ${\mathbb{R}}^d$ to ${\mathbb{R}}\cup \{-\infty\}$. Density estimation under log-concavity has been studied in depth in recent years; see e.g. [@CSS; @Duembgen; @Walther]. Note that Gaussian distributions $f_{\mu,\Sigma}$ are log-concave. Hence, the following optimization problem naturally generalizes the familiar task of maximizing (\[ex:loglikelihood\]) over all pairs of parameters $(\mu,\Sigma)$: $$\label{eq:ourproblem}
\begin{matrix}
\hbox{Maximize the log-likelihood (\ref{eq:loglikelihood2}) of the given sample $(X,w)$ over all} \\
\hbox{integrable functions $f: {\mathbb{R}}^d \rightarrow {\mathbb{R}}_{\geq 0} $ such that
${\rm log}(f)$ is concave and
$\,\int_{{\mathbb{R}}^d} f(x) dx = 1$.}
\end{matrix}$$
A solution to this optimization problem was given by Cule, Samworth and Stewart in [@CSS]. They showed that the logarithm of the optimal density $\hat f$ is a piecewise linear concave function, whose regions of linearity are the cells of a regular polyhedral subdivision of the configuration $X$. This reduces the infinite-dimensional optimization problem (\[eq:ourproblem\]) to a convex optimization problem in $n$ dimensions, since $\hat f$ is uniquely defined once its values at $x_1,\dots, x_n$ are known. An efficient algorithm for solving this problem is described in [@CSS]. It is implemented in the [R]{} package [LogConcDEAD]{} due to Cule, Gramacy and Samworth [@CGS].
$\quad\quad$
\[ex:octahedron\] Let $d=2$, $n=6$, $w = \frac{1}{6}(1,1,1,1,1,1)$, and fix the point configuration $$\label{eq:sixpoints}
X \,\,=\,\, \bigl( \,(0, 0)\,, \,(100, 0)\,,\, (0, 100)\,,\, (22, 37)\,, \,(43, 22)\,, \,(36, 41) \,\bigr).$$ The graphical output generated by [LogConcDEAD]{} is shown on the left in Figure \[fig:octahedron\]. This is the graph of the function ${\rm log}(\hat f)$ that solves (\[eq:ourproblem\]). This piecewise linear concave function has seven linear pieces, namely the triangles on the right in Figure \[fig:octahedron\], with vertices taken from $X$.
The purpose of this paper is to establish a link between nonparametric statistics and geometric combinatorics. We develop a generalization of the theory of regular triangulations arising in the context of maximum likelihood estimation for log-concave densities.
Our paper is organized as follows. In Section 2 we first review the relevant mathematical concepts, especially polyhedral subdivisions and secondary polytopes [@DRS; @GKZ]. We then generalize results in [@CSS] from the case of unit weights $w = \frac{1}{n}(1,1,\ldots,1)$ to arbitrary weights $w$. Theorem \[thm:samworth\] casts the problem (\[eq:ourproblem\]) as a linear optimization problem over a convex subset $\mathcal{S}(X)$ of ${\mathbb{R}}^n$, which we call the [*Samworth body*]{} of $X$. Theorem \[thm:IntegralFormula\] uses integrals as in [@Bar] to give an unconstrained formulation of this problem with an explicit objective function.
Cule, Samworth and Stewart [@CSS] discovered that log-concave density estimation leads to regular polyhedral subdivisions. In this paper we prove the following converse to their result:
\[thm:converse\] Let $\Delta$ be any regular polyhedral subdivision of the configuration $X$. There exists a non-empty open subset $\,\mathcal{U}_\Delta$ in ${\mathbb{R}}^n$ such that, for every $w \in \mathcal{U}_\Delta$, the optimal solution $\hat f$ to (\[eq:ourproblem\]) is a piecewise log-linear function whose regions of linearity are the cells of $\Delta$.
The proof of Theorem \[thm:converse\] appears in Section 3. We introduce a remarkable symmetric function $H$ that serves as a key technical tool. The theory behind $H$ seems interesting in its own right. In Theorem \[thm:normalcone\] we characterize the normal cone at any boundary point of the Samworth body. In other words, for a given concave piecewise log-linear function $f$, we determine the set of all weight vectors $w$ such that $f$ is the optimal solution in (\[eq:ourproblem\]).
In Section 4 we view (\[eq:ourproblem\]) as a parametric optimization problem, as either $w$ or $X$ vary. Variation of $w$ is explained by the geometry of the Samworth body. We explore empirically the probability that a given subdivision is optimal. We observe that triangulations are rare. Thus pictures like the triangulation in Figure \[fig:octahedron\] are exceptional and deserve special attention.
In Section 5 we focus our attention on the case of unit weights, and we examine the constraints this imposes on $\Delta$. Theorem \[thm:d+2points\] shows that triangulations never occur for $n=d+2$ points in ${\mathbb{R}}^d$ with unit weights. A converse to this result is established in Theorem \[thm:d+3points\].
Sections 4 and 5 conclude with several open problems. These suggest possible lines of inquiry for a future research theme that might be named [*Nonparametric Algebraic Statistics*]{}.
Geometric Combinatorics
=======================
We begin by reviewing concepts from geometric combinatorics, studied in detail in the books by De Loera, Rambau and Santos [@DRS] and Gel’fand, Kapranov and Zelevinsky [@GKZ]. See Thomas [@Tho §7-8] for a first introduction. Let $X = (x_1,\ldots,x_n)$ be a configuration as before and $P = {\rm conv}(X)$ its convex hull in ${\mathbb{R}}^d$. We assume that the polytope $P$ has dimension $d$.
Fix a real vector $y = (y_1,\ldots,y_n)$. We write $h_{X,y}$ for the smallest concave function $h$ on ${\mathbb{R}}^n$ such that $h(x_i) \geq y_i$ for $i=1,\ldots,n$. The graph of $h_{X,y}$ is the upper convex hull of $\{(x_1,y_1),\ldots,(x_n,y_n)\}$ in ${\mathbb{R}}^{n+1}$. Hence $h_{X,y}(t)$ is the largest real number $h^*$ such that $(t,h^*)$ is in the convex hull of $\{(x_1,y_1),\ldots,(x_n,y_n)\}$. In particular, $h_{X,y}(t) = - \infty$ for $t \not\in P$. Up to sign, the function $h_{X,y}$ is called the [*characteristic section*]{} in [@DRS Definition 5.2.12]. We also refer to $h_{X,y}$ as the [*tent function*]{}, with (some of) the points $(x_i,y_i)$ being the [*tent poles*]{}. The vector $y$ is called [*relevant*]{} if $h_{X,y}(x_i) = y_i$ for $i=1,\ldots,n$, i.e. if each $(x_i,y_i)$ is a tent pole. This fails, for example, if $x_i$ lies in the interior of $P$ and $y_i$ is small relative to the other $y_j$.
A [*regular subdivision*]{} $\Delta$ of $X$ is a collection of subsets of $X$ whose convex hulls are the regions of linearity of the function $h_{X, y}$ for some $y\in \mathbb R^n$. These regions are $d$-dimensional polytopes, and are called the [*cells*]{} of $\Delta$. A regular subdivision $\Delta$ is a [*regular triangulation*]{} of $X$ if each cell is a $d$-dimensional simplex. The [*secondary polytope*]{} $\Sigma(X)$ is a polytope of dimension $n-d-1$ in ${\mathbb{R}}^n$ whose faces are in bijection with the regular subdivisions of $X$. In particular, the vertices of $\Sigma(X)$ correspond to the regular triangulations of $X$; see [@DRS §5].
If $\Delta$ is a regular triangulation of $X$, then the $k$-th coordinate of the corresponding vertex $z^\Delta$ of $\Sigma(X) \subset {\mathbb{R}}^n$ is the sum of the volumes of all simplices in $\Delta$ that contain $x_k$. In symbols, $$\label{eq:GKZvector}
z^\Delta_k \,\,= \,\, \sum_{\sigma \in \Delta: \atop x_k \in \sigma} {\rm vol}(\sigma) .$$ We call $z^\Delta = (z^\Delta_1,\ldots,z^\Delta_n) $ the [*GKZ vector*]{} of the triangulation $\Delta$, in reference to [@GKZ].
The support function of the secondary polytope $\Sigma(X)$ is the piecewise linear function $${\mathbb{R}}^n \rightarrow {\mathbb{R}}, \quad y \,\mapsto \, \int_P h_{X,y}(t) dt.$$ This follows from the equation in [@DRS page 232]. The function is linear on each cone in the [*secondary fan*]{} of $X$. For every $y$ in the secondary cone of a given regular triangulation $\Delta$, $$\label{eq:intislinear}
\int_P h_{X,y}(t) dt \,\,= \,\, z^\Delta \cdot y \,\,=\,\, \sum_{i=1}^n z^\Delta_i y_i .$$ This means that the convex dual to the secondary polytope has the representation $$\Sigma(X)^* \quad = \quad \bigl\{
y \in {\mathbb{R}}^n \,:\, z^\Delta \cdot y \leq 1 \,\,\hbox{for all} \,\,\Delta \bigr\} \,\,\, = \,\,\,
\bigl\{ y \in {\mathbb{R}}^n \,:\,\int_P h_{X,y}(t) dt \leq 1 \,\bigr\}.$$ Note that $\Sigma(X)^*$ is an unbounded polyhedron in ${\mathbb{R}}^n$ since $\Sigma(X)$ has dimension $n-d-1$. Indeed, $\Sigma(X)^*$ is the product of an $(n-d-1)$-dimensional polytope and an orthant ${\mathbb{R}}_{\geq 0}^{d+1}$.
We now introduce an object that looks like a continuous analogue of $\Sigma(X)^*$. We define $$\label{eq:samdef}
\mathcal{S}(X) \quad = \quad \bigl\{\,
y \in {\mathbb{R}}^n \,:\,\int_P {\rm exp}(h_{X,y}(t)) dt \leq 1 \,\bigr\}.$$
Inspired by [@CGS; @CSS], we call $\mathcal{S}(X)$ the [*Samworth body*]{} of the point configuration $X$.
The Samworth body $\mathcal{S}(X)$ is a full-dimensional closed convex set in ${\mathbb{R}}^n$.
Let $y, y' \in \mathcal{S}(X)$ and consider a convex combination $y'' = \alpha y + (1-\alpha)y'$ where $0 \leq \alpha \leq 1$. For all $t \in P$, we have $h_{X,y''}(t) \leq \alpha h_{X,y}(t) + (1-\alpha) h_{X,y'}(t)$, and therefore $${\rm exp}(h_{X,y''}(t)) \,\leq \, {\rm exp} \bigl(\alpha h_{X,y}(t) + (1-\alpha) h_{X,y'}(t) \bigr) \,\leq \,
\alpha \cdot {\rm exp} ( h_{X,y}(t)) + (1-\alpha) \cdot {\rm exp}(h_{X,y'}(t) ) .$$ Now integrate both sides of this inequality over all $t \in P$. The right hand side is bounded above by $1$, and hence so is the left hand side. This means that $y'' \in \mathcal{S}(X)$. We conclude that $\mathcal{S}(X)$ is convex. It is closed because the defining function is continuous, and it is $n$-dimensional because all points $y$ whose $n$ coordinates are sufficiently negative lie in $\mathcal{S}(X)$.
Every boundary point $y$ of the Samworth body $\mathcal{S}(X)$ defines a log-concave probability density function $f_{X,y}$ on ${\mathbb{R}}^d$ that is supported on the polytope $P = {\rm conv}(X)$. This density is $$\label{eq:fydensity}
f_{X,y} \,\,: \,\,\,
t \,\,\mapsto\,\, \begin{cases} {\rm exp}(h_{X,y}(t)) & {\rm if} \,\,\, t \in P, \\
\qquad 0 & {\rm otherwise}. \end{cases}$$
We fix a positive real vector $w = (w_1,\ldots,w_n) \in {\mathbb{R}}^n_{\geq 0}$ that satisfies $\sum_{i=1}^n w_i =1$. The following result rephrases the key results of Cule, Samworth and Stewart [@CSS Theorems 2 and 3], who proved this, in a different language, for the unit weight case $w = \frac{1}{n} (1,1,\ldots,1)$.
\[thm:samworth\] The linear functional $ \,y \mapsto w \cdot y = \sum_{i=1}^n w_i y_i$ is bounded above on the Samworth body $\,\mathcal{S}(X)$. Its maximum over $\,\mathcal{S}(X)\,$ is attained at a unique point $y^*$. The corresponding log-concave density $f_{X,y^*}$ is the unique optimal solution to the estimation problem (\[eq:ourproblem\]).
We are claiming that $\mathcal{S}(X)$ is strictly convex and its recession cone is contained in the negative orthant ${\mathbb{R}}^n_{\leq 0}$. The point $y^*$ represents the solution to the optimization problem $$\label{eq:constrained}
\hbox{Maximize $\,w \cdot y\,$ subject to $\,y \in \mathcal{S}(X)$.}$$
The equivalence of (\[eq:ourproblem\]) and (\[eq:constrained\]) stems from the fact that the optimal solution $\hat f$ to the maximum likelihood problem (\[eq:ourproblem\]) has the form $ f = f_{X, y}$ for some choice of $ y \in {\mathbb{R}}^n$. This was proven in [@CSS] for unit weights $w = \frac{1}{n}(1,1,\ldots,1)$. The general case of positive rational weights $w_i$ can be reduced to the unit weight case by regarding $(X,w)$ as a multi-configuration. We extend this from rational weights to non-rational real weights by a continuity argument.
Let $N$ be the sample size, so that $N_i = Nw_i$ is a positive integer for $i=1,\ldots,n$. We think of $x_i$ as a sample point in ${\mathbb{R}}^d$ that has been observed $N_i$ times. If $f$ is any probability density function on ${\mathbb{R}}^d$, then the log-likelihood of the $N$ observations with respect to $f$ equals $$\label{eq:MLF}
N \cdot \sum_{i=1}^n w_i \cdot {\rm log}( f(x_i)).$$ Maximizing (\[eq:MLF\]) over log-concave densities is equivalent to maximizing (\[eq:loglikelihood2\]). We know from [@CSS Theorem 2] that the maximum is unique and is attained by $ f = f_{X,y^*}$ for some $y^* \in {\mathbb{R}}^n$. Here $y^*$ is the unique relevant point in $\, \mathcal{S}(X) = \bigl\{ y \in {\mathbb{R}}^n : \int_{{\mathbb{R}}^d} f_{X,y}(t)dt \leq 1 \bigr\}\,$ that maximizes the linear functional $w \cdot y$. Hence (\[eq:ourproblem\]) and (\[eq:constrained\]) are equivalent for all $w \in {\mathbb{R}}^n_{\geq 0}$.
The constrained optimization problem (\[eq:constrained\]) can be reformulated as an unconstrained optimization problem. For the unit weight case $w_1 = \cdots = w_n = 1/n$, this was done in [@CSS §3.1]. This result can easily be extended to general weights. In the language of convex analysis, Proposition \[prop:unconstrained\] says that the optimal value function of the convex optimization problem (\[eq:constrained\]) is the [*Legendre-Fenchel transform*]{} of the convex function $y \mapsto \int_P {\rm exp}(h_{X,y}(t)) dt$.
\[prop:unconstrained\] The constrained optimization problem [(\[eq:constrained\])]{} is equivalent to the unconstrained optimization problem $$\label{eq:unconstrained}
\hbox{{\rm Maximize} $\,\,w \cdot y - \int_{P} \exp(h_{X,y}(t)) dt\,\,$ over all $\,\,y \in {\mathbb{R}}^n$},$$ where, as before, $P$ denotes the convex hull of $x_1, \dots , x_n\in\mathbb{R}^d$ and $h_{X,y}$ is the tent function, i.e., $h_{X,y} : \mathbb{R}^d \to \mathbb{R}$ is the least concave function satisfying $h_{X,y}(x_i) \geq y_i$ for all $i = 1,\ldots , n$.
A proof for uniform weights is given in [@CSS]. We here present the proof for arbitrary weights $w_1,\ldots,w_n$. These are positive real numbers that sum to $1$. This ensures that the objective function in (\[eq:constrained\]) is bounded above, since the exponential term dominates when the coordinates of $y$ become large. Clearly, the optimum of (\[eq:constrained\]) is attained on the boundary $\partial \mathcal{S}(X)$ of the feasible set $\mathcal{S}(X)$, and we could equivalently optimize over that boundary.
Now suppose that $y^*$ is an optimal solution of (\[eq:unconstrained\]). This implies that $h_{X,y^*}(x_i) = y^*_i$, i.e. each tent pole touches the tent. Otherwise $\,w \cdot y$ in the objective function can be increased without changing $\int_{P} \exp(h_{X,y}(t)) dt$. Let $c:= \int_{P} \exp(h_{X,y^*}(t)) dt$. We claim that $c=1$.
Let $\hat{y}$ be a vector in ${\mathbb{R}}^n$, also satisfying $h_{X,\hat{y}}(x_i) = \hat{y}_i$ for all $i$, such that $\exp(h_{X,y^*}(t)) = c \exp(h_{X,\hat{y}}(t))$ and $\int_{P} \exp(h_{X,\hat{y}}(t)) dt = 1$. This means that $h_{X,y^*}(t) = \log(c) + h_{X,\hat{y}}(t)$ for all points $t$ in the polytope $ P$. In particular, we have $ y_i^* - \hat{y}_i = \log(c) $ for $i=1,2,\ldots,n$.
We now analyze the difference of the objective functions at the points $\hat{y}$ and $y^*$: $$w \cdot \hat{y} - \int_{P} \exp(h_{X,\hat{y}}(t)) dt - \left(w \cdot y^*
- \int_{P} \exp(h_{X,y^*}(t)) dt\right) = -\log(c) -1+c.$$
Note that the function $c \mapsto -\log(c) -1+c$ is nonnegative. Since $y^*$ maximizes $w \cdot y - \int_{P} \exp(h_{X,y}(t)) dt$, it follows that $-\log(c) -1+c = 0$, which implies that $c=1$. So, the claim holds. We have shown that the solution $y^*$ of (\[eq:unconstrained\]) also solves the following problem: $$\label{eq:constrained_2}
\hbox{Maximize $\,w \cdot y - \int_{P} \exp(h_{X,y}(t)) dt$ \; subject to\; $\int_{P} \exp(h_{X,y}(t)) dt = 1$.}$$ But this is equivalent to the constrained formulation (\[eq:constrained\]), and the proof is complete.
The objective function in (\[eq:unconstrained\]) looks complicated because of the integral and because $h_{X,y}(t)$ depends piecewise linearly on both $y$ and $t$. To solve our optimization problem, a more explicit form is needed. This was derived by Cule, Samworth and Stewart in [@CSS Section B.1]. The formula that follows writes the objective function locally as an exponential-rational function. This can also be derived from work on polyhedral residues due to Barvinok [@Bar].
\[lem:integral\] Fix a simplex $\sigma= {\rm conv}(x_0,x_1,\ldots,x_d)$ in ${\mathbb{R}}^d$ and an affine-linear function $\ell: {\mathbb{R}}^d \rightarrow {\mathbb{R}}$, and let $y_0=\ell(x_0), y_1 = \ell(x_1),\ldots,y_d = \ell(x_d)$ be its values at the vertices. Then $$\int_{\sigma} {\rm exp}\bigl( \ell (t)\bigr) dt \,\, \,= \,
\,\,{\rm vol}(\sigma) \cdot \sum_{i=0}^d {\rm exp}(y_i) \!
\!\prod_{j \in \{0,\ldots,d\}\backslash \{i\}} \!\! \!\!\! (y_i-y_j)^{-1}.$$
This follows directly from equation (B.1) in [@CSS Section B.1], and it can also easily be derived from Barvinok’s formula in [@Bar Theorem 2.6].
This lemma implies the following formula for integrating exponentials of piecewise-affine functions on a convex polytope. This can be regarded as an exponential variant of (\[eq:intislinear\]).
\[thm:IntegralFormula\] Let $\Delta$ be a triangulation of the configuration $X=(x_1,\ldots,x_n)$ and $h : P \rightarrow {\mathbb{R}}$ the piecewise-affine function on $\Delta$ that takes values $h(x_i) = y_i$ for $i = 1,2,\ldots,n$. Then $$\int_{P} {\rm exp}\bigl( h (t)\bigr) dt \,\, \,= \,\,\,
\sum_{i=1}^n
{\rm exp}(y_i) \sum_{\sigma \in \Delta: \atop i \in \sigma} \frac{{\rm vol}(\sigma)}
{\prod_{j \in \sigma \backslash i} (y_i-y_j)}$$
We add the expressions in Lemma \[lem:integral\] over all maximal simplices $\sigma$ of the triangulation $\Delta$, and we collect the rational function multipliers for each of the $n$ exponentials ${\rm exp}(y_i)$.
This formula underlies the efficient solution to the estimation problem (\[eq:ourproblem\]) that is implemented in the [R]{} package [LogConcDEAD]{} [@CGS]. We record the following algebraic reformulation, which will be used in our study in the subsequent sections. This follows from Theorem \[thm:IntegralFormula\].
\[cor:optsecondary\] The equivalent optimization problems (\[eq:ourproblem\]), (\[eq:constrained\]), (\[eq:unconstrained\]) are also equivalent to $$\label{eq:optsecondary}
{\rm Maximize} \,\,\,w \cdot y \,- \,\sum_{\sigma \in \Delta}
\sum_{i \in \sigma}
\frac{{\rm vol}(\sigma) \cdot {\rm exp}(y_i)}
{\prod_{j \in \sigma \backslash i} (y_i-y_j)},$$ where $y $ runs over $ {\mathbb{R}}^n$ and $\Delta$ is a regular triangulation of $X$ whose secondary cone contains $y$.
We close this section with an example that illustrates the various concepts seen so far.
\[ex:hexagon\] Let $d=2$ and $n=6$. Take $X$ to be six points in convex position in the plane, labeled cyclically in counterclockwise order. The normalized area of the triangle formed by any three of the vertices of the hexagon $P = {\rm conv}(X)$ is computed as a $3 \times 3$-determinant $$\label{eq:areas}
\quad v_{ijk} \,\, := \,\,
{\rm vol}\bigl( {\rm conv}(x_i, x_j, x_k) \bigr) \,\, = \,\,
{\rm det}\begin{pmatrix} 1 & 1 & 1 \\ x_i & x_j & x_k \end{pmatrix}
\,\,\quad {\rm for}\, \,\, \,1 \leq i < j < k \leq 6. \quad$$ The configuration $X$ has $14$ regular triangulations. These come in three symmetry classes: six triangulations like $\Delta = \{123,134,145,156\}$, six triangulations like $\,\Delta' = \{123,134,146,456\}$, and two triangulations like $\,\Delta'' = \{123,135, 156,345\} $. The corresponding GKZ vectors are $$\begin{matrix}
z^{\Delta} &=& \bigl(\,v_{123}+v_{134}+v_{145}+v_{156}\,,\,
v_{123}\,,\,v_{123}+v_{134}\,,\,v_{134}+v_{145}\,,\,v_{145}+v_{156}\,,v_{156}\, \bigr) ,\\
z^{\Delta'} &=&
\bigl(\,v_{123}+v_{134}+v_{146}\,,\, v_{123}\,,\, v_{123}+v_{134}\,,\, v_{134}+v_{146}+v_{456}\,,
\, v_{456}\,, \,v_{146}+v_{456}\, \bigr), \\
z^{\Delta'} &=&
\bigl(\, v_{123}+v_{135}+v_{156} \,,\, v_{123}\,, \, v_{123}+v_{135}+v_{345}\,,\,
v_{345}\,, \,v_{135}+v_{156}+v_{345}\,, \,v_{156}\, \bigr) ,
\end{matrix}$$ as defined in (\[eq:GKZvector\]). The secondary polytope $\Sigma(X)$ is the convex hull of these $14$ points in ${\mathbb{R}}^6$. This is a simple $3$-polytope with $14$ vertices, $21$ edges and $9$ facets, shown in Figure \[fig:associahedron\]. This polytope is known as the [*associahedron*]{}. It has $45 = 14+21+9+1$ faces in total, one for each of the $45$ polyhedral subdivisions of $X$. These are the supports of the functions $h_{X,y}$.
For example, the edge of $\Sigma(X)$ that connects $z^{\Delta}$ and $z^{\Delta'}$ represents the subdivision $\{123,134,1456\}$, with two triangles and one quadrangle. The smallest face containing $\{z^{\Delta},z^{\Delta'},z^{\Delta''}\}$ is two-dimensional. It is a pentagon, encoding the subdivision $\{123, 13456\}$.
The Samworth body $\mathcal{S}(X)$ is full-dimensional in ${\mathbb{R}}^6$. Its boundary is stratified into $45$ pieces, one for each subdivision of $X$. For any given $w \in {\mathbb{R}}^6$, the optimal solution $y^*$ to (\[eq:constrained\]) lies in precisely one of these $45$ strata, depending on the shape of the optimal density $f_{X,y^*}$.
Algebraically, we can find $y^*$ by computing the maximum among $14$ expressions like $$\label{eq:maximumamong14}
\begin{matrix} w_1 y_1 + w_2 y_2 + \cdots + w_6 y_6
& - & v_{123}\cdot \bigl( \frac{{\rm exp}(y_1)}{(y_1-y_2)(y_1-y_3)}+
\frac{{\rm exp}(y_2)}{(y_2 - y_1)(y_2 - y_3)}+
\frac{{\rm exp}(y_3)}{(y_3 - y_1)(y_3 - y_2)}\bigr) \smallskip \\
& - & v_{134}\cdot \bigl( \frac{{\rm exp}(y_1)}{(y_1-y_3)(y_1-y_4)}+
\frac{{\rm exp}(y_3)}{(y_3 - y_1)(y_3 - y_4)}+
\frac{{\rm exp}(y_4)}{(y_4 - y_1)(y_4 - y_3)}\bigr) \smallskip \\
& - & v_{145}\cdot \bigl( \frac{{\rm exp}(y_1)}{(y_1-y_4)(y_1-y_5)}+
\frac{{\rm exp}(y_4)}{(y_4 - y_1)(y_4 - y_5)}+
\frac{{\rm exp}(y_5)}{(y_5 - y_1)(y_5 - y_4)}\bigr) \smallskip \\
& - & v_{156}\cdot \bigl( \frac{{\rm exp}(y_1)}{(y_1-y_5)(y_1-y_6)}+
\frac{{\rm exp}(y_5)}{(y_5 - y_1)(y_5 - y_6)}+
\frac{{\rm exp}(y_6)}{(y_6 - y_1)(y_6 - y_5)}\bigr) .
\end{matrix}$$ This formula is the objective function in (\[eq:optsecondary\]) for the triangulation $\Delta = \{123,134,145,156\}$. The mathematical properties of this optimization process will be studied in the next sections.
Every Regular Subdivision Arises
================================
Our goal in this section is to prove Theorem \[thm:converse\]. We begin by examining the function $$\label{eq:defH}
H\,:\, {\mathbb{R}}^d \rightarrow {\mathbb{R}}\,,\,\,\,
(u_1,\ldots, u_d) \,\, \mapsto \,\, (-1)^d\frac{1 + u_1^{-1} + \cdots + u_d^{-1}}{u_1 u_2 \cdots u_d }
\,+\, \sum_{j=1}^d \frac{e^{u_j}}{u_j^2 \prod_{k \not= j} (u_j - u_k)}.$$
\[prop:Hformula\] The function $H$ is well-defined on ${\mathbb{R}}^d$. It admits the series expansion $$\label{eq:Hidentity} H(u_1,\dots, u_d) \quad = \quad \sum_{r=0}^\infty\frac{h_r(u_1,\ldots,u_d)}{(r+d+1)!},$$ where $h_r$ is the complete homogeneous symmetric polynomial of degree $r$ in $d$ unknowns.
We substitute the Taylor expansion of the exponential function in the sum on the right hand side of (\[eq:defH\]). This sum then becomes $$\sum_{j=1}^d \frac{e^{u_j}}{u_j^2 \prod_{k \not= j} (u_j - u_k)} \quad= \quad\sum_{\ell=0}^\infty \frac{1}{\ell !} \sum_{j=1}^d \frac{u_j^{\ell-2}}{\prod_{k \not= j} (u_j - u_k)}
\quad$$ $$= \quad
\sum_{\ell=0}^\infty \frac{1}{\ell !} \sum_{j=1}^d \frac{u_j^{\ell-d-1}}{\prod_{k \not= j} (1-u_k/u_j )}
\quad = \quad
\sum_{r=-d-1}^\infty \frac{1}{( r+d+1)!} \sum_{j=1}^d
\frac{u_j^r}{\prod_{k \not= j} (1-u_k/u_j )}$$ For nonnegative values of the summation index $r = \ell - d- 1$, the inner summand equals $h_r(u_1,\ldots,u_d)$, by Brion’s Theorem [@MS Theorem 12.13]. For negative values of $r$, we use Ehrhart Reciprocity, in the form of [@MS Lemma 12.15, eqn (12.7)], as seen in [@MS Example 12.14]. The two terms for $r \in \{-d-1,-d\}$ cancel with the left summand on the right hand side of (\[eq:defH\]). The terms for $r \in \{-d+1,\ldots,-2,-1\}$ are zero. This implies (\[eq:Hidentity\]).
We shall derive a useful integral representation of our function $H$. What follows is a Lebesgue integral over the standard simplex $\,\Sigma_d = \{(y_1,\dots, y_d) \in {\mathbb{R}}^d: \,y_i \geq 0 ,\, \sum_{i}y_i \leq 1\}$.
\[prop:magic\] The function $H$ can be expressed as the following integral: $$\begin{aligned}
\label{eqn:HIntegralExpression}
H(u_1,\ldots, u_d) \,\,=\, \int_{\Sigma_d}
\left(1 - \sum_{i=1}^d t_i\right)\exp\left(\sum_{i=1}^d u_i t_i\right)\text{d} t_1\dots\text{d} t_d.\end{aligned}$$
The complete homogeneous symmetric polynomial $h_r$ equals the Schur polynomial $s_{(r)}$ corresponding to the partition $\lambda = (r)$. By formula (2.11) in [@GR] we have $s_{(r)} = Z_{(r)}$, where $Z_\lambda(u_1,\dots, u_d)$ is the [*zonal polynomial*]{}, or [*spherical function*]{} [@GR]. Therefore, we conclude $$H(u_1,\ldots, u_d) \,=\,\sum_{r=0}^{\infty}\frac{Z_{(r)}(u_1,\ldots, u_d)}{(r+d+1)!}
\,=\, \frac1{(d+1)!}\sum_{r=0}^{\infty}
\frac{Z_{(r)}(u_1,\ldots, u_d)\cdot [1]_{(r)}}{[d+2]_{(r)} \cdot r!},$$ where $[a]_\lambda = \prod_{j=1}^{m}(a-j+1)_{\lambda_j}$ for a partition $\lambda = (\lambda_1,\dots, \lambda_m)$, and $(a)_s = a(a+1)\cdots(a+s-1)$. In particular, $[1]_{(r)} = r!$, and $[1]_{\lambda} = 0$ if $\lambda$ has more than one nonzero part. Therefore, $$H(u_1,\ldots, u_d) \,\,= \,\, \frac1{(d+1)!}\sum_{\text{all partitions } \lambda}
\frac{Z_\lambda(u_1,\ldots, u_d) \cdot [1]_\lambda}{[d+2]_\lambda \cdot |\lambda|!}.$$ By [@GR (4.14)], this can be written in terms of the confluent hypergeometric function ${}_1F_1$: $$H(u_1,\ldots, u_d) \,\,= \,\,\frac1{(d+1)!} \cdot \, {}_1F_1(1;d+2;u_1,\ldots, u_d) .$$ The right hand side has the desired integral representation (\[eqn:HIntegralExpression\]), by [@GR equation (5.14)].
\[cor:positive\] The function $H$ is positive, increasing in each variable, and convex.
The integrand in is nonnegative. Hence, $H(u_1,\ldots, u_d) > 0$ for all $(u_1,\ldots, u_d)\in\mathbb R^d$. After taking derivatives with respect to $u_i$, the integrand remains positive. Therefore, $H$ is increasing in $u_i$. Finally, the integrand is a convex function, and hence so is $H$.
We now embark towards the proof of Theorem \[thm:converse\]. Recall that a vector $y \in {\mathbb{R}}^n$ is relevant if $h_{X,y}(x_i) = y_i$ for all $i$, i.e. the regular subdivision of $X$ induced by $y$ uses each point $x_i$.
\[lem:weightsAnyDim\] Fix a configuration $X$ of $n$ points in $\mathbb R^d$. For any relevant $y^* \in {\mathbb{R}}^n$ that satisfies $\int_{{\mathbb{R}}^d} f_{X,y^*}(t)dt = 1$, there are weights $w \in {\mathbb{R}}^n_{> 0}$ such that $y^* $ is the optimal solution to -.
We use the formulation (\[eq:optsecondary\]) which is equivalent to (\[eq:ourproblem\]), (\[eq:constrained\]), and (\[eq:unconstrained\]). Let $\Delta$ be any regular triangulation that refines the regular subdivision given by $y$. In other words, we choose $\Delta$ so that (\[eq:intislinear\]) is maximized. The objective function in Corollary \[cor:optsecondary\] takes the form $$S(y_1,\dots, y_n) \,\,\,= \,\,\,w\cdot y - \sum_{i=1}^n\exp(y_i)\sum_{\sigma\in\Delta,\atop i\in\sigma}\frac{\text{vol}(\sigma)}{\prod_{j\in \sigma\setminus i }(y_i-y_j)}.$$
Consider the partial derivative of the objective function $S$ with respect to the unknown $y_k$: $$\begin{aligned}
\frac{\partial S}{\partial y_k} \,\,\,= \,\,\,w_k \,\, - \,
&\sum_{\sigma\in\Delta,\atop k\in\sigma}\text{vol}(\sigma)\exp(y_k)\frac1{\prod_{j\in \sigma\setminus k }(y_k-y_j)}\left(1 - \sum_{j\in \sigma\setminus k }\frac{1}{(y_k - y_j)}\right) \\
-&\, \sum_{\sigma\in\Delta,\atop k\in\sigma}\text{vol}(\sigma)\sum_{j\in \sigma\setminus k } \exp(y_j) \frac1{\prod_{i\in \sigma\setminus j } (y_j - y_i)} \frac1{(y_j - y_k)}.\end{aligned}$$ Using the formula (\[eq:defH\]) for the symmetric function $H(u_1,\ldots,u_d)$, this can be rewritten as $$\frac{\partial S}{\partial y_k} \,\,\,= \,\,\,w_k \,-\,
\sum_{\sigma\in\Delta,\atop k\in\sigma} \text{vol}(\sigma)\exp(y_k)H(\{ y_i - y_k :
i\in\sigma \backslash k\}).$$ We now consider the specific given vector $y^* \in {\mathbb{R}}^n$, and we use it to define $$\begin{aligned}
\label{weightsFormula}
w_k \,\,= \,\,\sum_{\sigma\in\Delta,\atop k\in\sigma} \text{vol}(\sigma)\exp(y^*_k)H(\{
y^*_i - y^*_k : i\in\sigma \backslash k \}).\end{aligned}$$ By Corollary \[cor:positive\], the vector $w=(w_1,\ldots,w_n)$ is well-defined and has positive coordinates. Consider now our estimation problem (\[eq:ourproblem\]) for that $w \in {\mathbb{R}}^n_{>0}$. By construction, the gradient vector of $S$ vanishes at $y^*$. Furthermore, recall that the choice of the triangulation $\Delta$ was arbitrary, provided $\Delta$ refines the subdivision of $y$. This ensures that all subgradients of the objective function in (\[eq:unconstrained\]) vanish. Since this function is strictly convex, as shown in [@CSS], we conclude that the given $y^*$ is the unique optimal solution for the choice of weights in (\[weightsFormula\]).
We note that the function $H$ and Lemma \[lem:weightsAnyDim\] are quite interesting even in dimension one.
\[ex:d=1\] Let $d=1$. So, we here examine log-concave density estimation for $n$ samples $x_1 < x_2 < \cdots < x _n$ on the real line. The function we defined in (\[eq:defH\]) has the representations $$H(u) \,\,=\,\, \frac{e^u - u - 1}{u^2} \,\,=\,\,\int_{0}^1 (1-y) e^{uy} dy \,\,=\,\,
\frac{1}{2} + \frac{1}{6} u + \frac{1}{24} u^2 + \frac{1}{120} u^3 + \cdots .$$ A vector $y^* \in {\mathbb{R}}^n$ is relevant if and only if $$\label{eq:relevant}
{\rm det}
\begin{pmatrix}
1 & 1 & 1 \\
x_{i-1} & x_i & x_{i+1} \\
y^*_{i-1} & y^*_i & y^*_{i+1}
\end{pmatrix} \,\leq \, 0 \,
\,\quad \hbox{for} \quad i=2,3,\ldots,n-1.$$ The desired vector $w \in {\mathbb{R}}^n_{>0}$ is defined by the formula in (\[weightsFormula\]). The $k$-th coordinate of $w$ is $$w_k = \begin{cases} (x_2 - x_1)e^{y^*_1}H(y^*_2 - y^*_1)& {\rm if}\,\, k = 1,\\
(x_k - x_{k-1})e^{y^*_k}H(y^*_{k-1} - y^*_k) + (x_{k+1} - x_k)e^{y^*_k}H(y^*_{k+1} - y^*_k)&
{\rm if}\,\, 2\leq k \leq n-1,\\
(x_n-x_{n-1})e^{y^*_n}H(y^*_{n-1} - y^*_n)& {\rm if}\,\, k = n.
\end{cases}$$ If we now further assume that $f_{X,y^*} = {\rm exp}(h_{X,y^*})$ is a density, i.e. $\int_{-\infty}^\infty f_{X,y^*}(t)dt = 1$, then $f_{X,y^*}$ is the unique log-concave density that maximizes the likelihood function for $(X,w)$.
\[ex:H2\] For $d=2$, our symmetric convex function $H$ has the form $$H(u, v) \,=\, \frac{1}{uv} + \frac{1}{u^2 v} + \frac{1}{u v^2}
+ \frac{e^u}{u^2(u{-}v)} + \frac{e^v}{v^2(v{-}u)} \,= \,
\frac{1}{6} + \frac{1}{24}(u+v) + \frac{1}{120}(u^2 + uv + v^2) + \cdots.$$ For planar configurations $X$, we use this function to map each point $y^*$ in the boundary of the Samworth body $\mathcal{S}(X)$ to a hyperplane $w \in \partial \mathcal{S}(X)^*$ that is tangent to $\partial \mathcal{S}(X)$ at $y^*$.
The set of all vectors $w \in {\mathbb{R}}^n$ that lead to a desired optimal solution $y^* \in \partial \mathcal{S}(X)$ is a convex polyhedral cone in ${\mathbb{R}}^n$. The following theorem characterizes that convex cone.
\[thm:normalcone\] Fix a vector $y^* \in {\mathbb{R}}^n$ that is relevant for $X$. Let $\Delta_1,\Delta_2,\ldots, \Delta_m$ be all the regular triangulations of $X$ that refine the subdivision of $X$ given by $y^*$, and let $w^{\Delta_i} \in {\mathbb{R}}^n_{>0}$ be the vector defined by (\[weightsFormula\]) for $\Delta_i$. Then, a vector $w \in {\mathbb{R}}^n_{>0} $ lies in the convex cone that is spanned by $\,w^{\Delta_1},w^{\Delta_2}, \ldots, w^{\Delta_m}\,$ if and only if $\,y^*$ is the optimal solution for (\[eq:ourproblem\]),(\[eq:constrained\]),(\[eq:unconstrained\]),(\[eq:optsecondary\]).
This follows from the fact that the cone of subgradients at each $y^*$ is convex, and, the gradients for each triangulation on which $h_{X,y^*}$ is linear are also subgradients at $y^*$; cf. [@CSS]. We can take any convex combination of these subgradients to obtain another subgradient.
\[ex\_4points\_plane\] Fix four points $x_1,x_2, x_3, x_4$ in counterclockwise convex position in ${\mathbb{R}}^2$. These admit two regular triangulations, $\Delta_1 = \{124,234\}$ and $\Delta_2 = \{123,134\}$. Consider any $y \in {\mathbb{R}}^4$ with $\int_{{\mathbb{R}}^2} f_{X,y}(t) dt = 1$. The vector $w^{\Delta_1} \in {\mathbb{R}}^4$ has coordinates $$\begin{aligned}
w_1^{\Delta_1} &\,\,=\,\, v_{124}e^{y_1}H(y_2-y_1,y_4-y_1)\\
w_2^{\Delta_1} &\,\,=\,\, v_{124}e^{y_2}H(y_1-y_2, y_4-y_2)
+ v_{234}e^{y_2}H(y_3-y_2, y_4-y_2)\\
w_3^{\Delta_1} &\,\,=\,\, v_{234}e^{y_3}H(y_2-y_3, y_4-y_3)\\
w_4^{\Delta_1} &\,\,=\,\, v_{124}e^{y_4}H(y_1-y_4, y_2-y_4)
+ v_{234}e^{y_4}H(y_2-y_4, y_3-y_4).\end{aligned}$$ Here $v_{ijk}$ denotes the triangle area in (\[eq:areas\]). Similarly, the vector $w^{\Delta_2}$ has coordinates $$\begin{aligned}
w_1^{\Delta_2} &\,\,=\,\, v_{123}e^{y_1}H(y_2-y_1, y_3-y_1)
+ v_{134}e^{y_1}H(y_3-y_1, y_4-y_1)\\
w_2^{\Delta_2} &\,\,=\,\, v_{123}e^{y_2}H(y_1-y_2, y_3-y_2)\\
w_3^{\Delta_2} &\,\,=\,\, v_{123}e^{y_3}H(y_1-y_3, y_2-y_3)
+ v_{134}e^{y_3}H(y_1-y_3, y_4-y_3)\\
w_4^{\Delta_2} &\,\,=\,\, v_{134}e^{y_4}H(y_1-y_4, y_3-y_4).\end{aligned}$$ In these formulas, the bivariate function $H$ can be evaluated as in Example \[ex:H2\].
We now distinguish three cases for $y$, depending on the sign of the $4 \times 4$-determinant $$\label{eq:tetrahedron}
{\rm det} \begin{pmatrix}
1 & 1 & 1 & 1 \\
x_1 & x_2 & x_3 & x_4 \\
y_1 & y_2 & y_3 & y_4
\end{pmatrix}.$$ If (\[eq:tetrahedron\]) is positive then $y$ induces the triangulation $\Delta_1$. In that case, $y$ is the unique solution to our optimization problem whenever $w$ is any positive multiple of $w^{\Delta_1}$. If (\[eq:tetrahedron\]) is negative then $y$ induces $\Delta_2$ and it is the unique solution whenever $w$ is a positive multiple of $w^{\Delta_2}$. Finally, suppose (\[eq:tetrahedron\]) is zero, so $y$ induces the trivial subdivision $1234$. If $w$ is any vector in the cone spanned by $w^{\Delta_1}$ and $w^{\Delta_2}$ in ${\mathbb{R}}^4$ then $y$ is the optimal solution for (\[eq:ourproblem\]),(\[eq:constrained\]),(\[eq:unconstrained\]),(\[eq:optsecondary\]).
We next observe what happens in Theorem \[thm:normalcone\] when all coordinates of $y^*$ are equal.
\[cor:cccc\] Fix the constant vector $y^* = (c,c,\ldots,c)$, where $c = - {\rm log}({\rm vol}(P))$, so as to ensure that $\int_{{\mathbb{R}}^d} f_{X,y^*}(t) dt = 1$. For any regular triangulation $\Delta_i$, the weight vector in (\[weightsFormula\]) is a constant multiple of the GKZ vector in (\[eq:GKZvector\]). More precisely, we have $w^{\Delta_i} = \frac{e^c}{(d+1)!} \cdot z^{\Delta_i}$. Hence $y^*$ is the optimal solution for any $w$ in the cone over the secondary polytope $\Sigma(X)$.
The constant term of the series expansion in Proposition \[prop:Hformula\] equals $$H(0,0,\ldots,0) \,\,= \,\, \frac{1}{(d+1)!} .$$ This implies that the sum in (\[weightsFormula\]) simplifies to $\frac{e^c}{(d+1)!}$ times the sum in (\[eq:GKZvector\]). The last statement follows from Theorem \[thm:normalcone\] because the cone over $\sigma(X)$ is spanned by all GKZ vectors $z^\Delta$.
We shall now prove the result that was stated in the Introduction.
Let $\Delta_1, \ldots,\Delta_m$ be all regular triangulations that refine a given subdivision $\Delta$. To underscore the dependence on $y$, we write $w^{\Delta_i}_y$ for the vector defined in (\[weightsFormula\]). Let $\mathcal{C}_\Delta$ denote the secondary cone of $\Delta$. This is the normal cone to $\Sigma(X)$ at the face with vertices $z^{\Delta_1},\ldots, z^{\Delta_m}$. In particular, we have $\,\dim (\text{span} (z^{\Delta_1},\dots, z^{\Delta_m})) = n - \dim (\mathcal{C}_\Delta)$.
For $y \in {\mathbb{R}}^n$ we abbreviate $N(y) =
\dim (\text{span} (w^{\Delta_1}_{y}, \dots, w^{\Delta_m}_{y}))$. The closure of the cone $\mathcal{C}_\Delta$ contains the constant vector $y^0 = (c,c,\ldots,c)$, where $c = - {\rm log}({\rm vol}(P))$. Corollary \[cor:cccc\] implies that $N(y_0) = n - \dim (\mathcal{C}_\Delta)$. The matrix $(w^{\Delta_1}_{y}, \dots, w^{\Delta_m}_{y})$ depends analytically on the parameter $y$. Its rank is an upper semicontinuous function of $y$. Thus, there exists an open ball $\hat{\mathcal{B}}$ in ${\mathbb{R}}^n$ that contains $y_0$ and such that $N(y)\geq n - \dim (\mathcal{C}_\Delta)$ for every $y\in\hat{\mathcal B}$. Now, let $\mathcal B = \mathcal{C}_\Delta \cap \hat{\mathcal B}$. The set $\mathcal B$ is full-dimensional in $\mathcal{C}_\Delta$, and $N(y)\geq n - \dim(\mathcal{C}_\Delta)$ for all $y \in \mathcal{B}$.
For each $ y \in \mathcal{B}$ we consider the convex cone in Theorem \[thm:normalcone\], which consists of all weight vectors $w$ for which the optimum occurs at $y$. We denote it by $\,{\rm cone}(w^{\Delta_1}_{y}, \ldots, w^{\Delta_m}_{y})$. These convex cones are pairwise disjoint as $y$ runs over $\mathcal{B}$, and they depend analytically on $y$. Since the dimension of each cone is at least $n-{\rm dim}(\mathcal{B}) $, it follows that the semi-analytic set $$\label{eq:fulldimset}
\bigcup_{y \in \mathcal{B}} {\rm cone}(w^{\Delta_1}_{y}, \dots, w^{\Delta_m}_{y}).$$ is full-dimensional in ${\mathbb{R}}^n$. By Theorem \[thm:normalcone\], for each $w$ in the set (\[eq:fulldimset\]), the optimal solution $\hat f$ to (\[eq:ourproblem\]) is a piecewise log-linear function whose regions of linearity are the cells of $\Delta$.
We believe that the rank of the matrix $(w^{\Delta_1}_{y}, \ldots, w^{\Delta_m}_{y})$ is the same for all vectors $y$ that induce the regular subdivision $\Delta$, namely $N(y) = n-{\rm dim}(\mathcal{C}_\Delta)$. At present we do not know how to prove this. For the proof of Theorem \[thm:converse\], it was sufficient to have this constant-dimension property for all $y$ in a relatively open subset $\mathcal{B}$ of the secondary cone $\mathcal{C}_\Delta$.
The Samworth Body
=================
The maximum likelihood problem studied in this paper is a linear optimization problem over a convex set. We named that convex set the Samworth body, in recognition of the contributions made by Richard Samworth and his collaborators [@CGS; @CSS]. In what follows we explore the geometry of the Samworth body. We begin with the following explicit formula:
The Samworth body of a given configuration $X$ of $\,n$ points in ${\mathbb{R}}^d$ equals $$\label{eq:samformula}
\! \mathcal{S}(X) \,\, = \,\,
\biggl\{ (y_1,\ldots,y_n) \in {\mathbb{R}}^n \,:\,
\sum_{\sigma \in \Delta}
\sum_{i \in \sigma}
\frac{{\rm vol}(\sigma) \cdot {\rm exp}(y_i)}
{\prod_{j \in \sigma \backslash i} (y_i-y_j)} \leq 1 \,\,\,
\hbox{for all $\Delta$ that refine $y\,$} \biggr\}.$$ This is a closed convex subset of $\,{\mathbb{R}}^n$. In the defining condition we mean that $\Delta$ runs over all regular triangulations that refine the regular polyhedral subdivision of $X$ specified by $y$.
This is a reformulation of the definition (\[eq:samdef\]) using the formulas in Theorem \[thm:IntegralFormula\] and Corollary \[cor:optsecondary\]. Closedness and strict convexity of $\mathcal{S}(X)$ were noted in Theorem \[thm:samworth\].
Maximization of a linear function $w$ over $\mathcal{S}(X)$ becomes an unconstrained problem via the Legendre-Fenchel transform as in (\[eq:optsecondary\]). By solving this problem for many instances of $w$, one can approximate the shape of $\mathcal{S}(X)$. Indeed, each regular subdivision of $X$ specifies a full-dimensional subset in the boundary of the dual body $\mathcal{S}(X)^*$, by Theorem \[thm:converse\]. If we choose a direction $w$ at random in ${\mathbb{R}}^n$, then a unique positive multiple $\lambda w$ lies in $\partial \mathcal{S}(X)^*$, in the stratum associated to the subdivision of $X$ specified by the optimal solution $y^* \in \partial \mathcal{S}(X)$. By evaluating the map $w \mapsto y^*$ many times, we thus obtain the empirical distribution on the subdivisions, indicating the proportion of volumes of the strata in $\partial \mathcal{S}(X)^*$. In the next example we compute this distribution when the double sum in (\[eq:samformula\]) looks like that in (\[eq:maximumamong14\]).
\[ex:associahedron2\] Let $d=2$, $n=6$, and take our configuration $X$ to be the six points $(0,0),(1,0),(2,1),(2,2),(1,2), (0,1)$. We sampled 100,000 vectors $w$ uniformly from the simplex $\{w \in {\mathbb{R}}^6_{\geq 0} : \sum_{i=1}^6 w_i = 1\}$. For each $w$, we computed the optimal $y^* \in {\mathbb{R}}^6$, and we recorded the subdivision of $X$ that is the support of $h_{X,y^*}$. We know from Example \[ex:hexagon\] that the secondary polytope $\Sigma(X)$ is an associahedron, which has $14+21+9+1 = 45$ faces. We here code each subdivision by a list of length $3,2,1$ or $0$ from among the diagonal segments $$13, \,14, \, 15, \,24, \,25,\,26, \,35, \,36,\,46.$$ For instance, the list $13 \,\,14 \,\,15$ encodes the triangulation $\Delta$ in Example \[ex:hexagon\]. The edge connecting the triangulations $\Delta$ and $\Delta'$ from Example \[ex:hexagon\] is denoted $13 \, 14 $. We write $\emptyset$ for the trivial flat subdivision. The following table of percentages shows the empirical distribution we observed for the $45$ outcomes of our experiment:
$$\begin{matrix}
\emptyset & 35 \,&\, 46 \,&\, 24 \,&\, 15 \,&\, 13 \,&\, 26 \,&\, 25 \,&\, 14 \,&\, 36 \\
30.5 \,\,&\,\, 5.95 \,\,&\,\, 5.85 \,\,&\,\, 5.84 \,\,&\,\, 5.83 \,\,&
\,\, 5.75 \,\, &\,\, 5.70 \,\,&\,\, 3.91 \,\,&\,\, 3.90 \,\,&\,\, 3.87 \\
\end{matrix}$$ $$\begin{matrix}
13 \,15 & 26 \, 46 & 15 \, 35 & 13 \, 35 & 24 \, 26 & 24 \, 46 & 13 \, 14 & 35\, 36 & 14 \, 24 & 26 \, 36 & 14 \, 46 & 25 \, 35 & 15 \, 25 \\
1.23 & 1.21 & 1.21 & 1.20 & 1.16 &
1.14 & 0.96 & 0.92 & 0.92 & 0.92 & 0.92 & 0.90 & 0.90
\end{matrix}$$ $$\begin{matrix}
25 \,26 &
14 \, 15 &
36 \, 46 &
24 \, 25 &
13 \, 36 &
13 \,46 &
26 \, 35 &
15\, 24 &
13 \, 14 \, 15 &
13 \,15 \,35 &
14 \,24 \,46 &
24 \,26 \, 46 \\
0.89 &
0.89 &
0.87 &
0.87 &
0.84 &
0.82 &
0.77 &
0.70 &
0.25 &
0.24 & 0.23 & 0.22
\end{matrix}$$ $$\begin{matrix}
15 \, 25 \, 35 &
26\, 36 \, 46 &
13 \, 35 \, 36 &
24 \, 25 \, 26 &
13 \, 36 \, 46 &
25 \, 26\, 35 &
15 \,24 \, 25 &
14 \, 15 \, 24 &
13 \,14 \, 46 &
26 \, 35\, 36 \\
0.22 &
0.21 &
0.20 &
0.18 &
0.18 &
0.16 &
0.15 &
0.15 & 0.15 & 0.14
\end{matrix}$$
The entry marked $\emptyset$ reveals that the trivial subdivision occurs with the highest frequency. This means that a large portion of the dual boundary $\partial \mathcal{S}(X)^*$ is flat. Equivalently, the Samworth body $\mathcal{S}(X)$ has a “very sharp edge” along the lineality space of the secondary fan.
To get a better understanding of the geometry of the Samworth body $\mathcal{S}(X)$, at least when $d$ or $n-d$ are small, we can also use the algebraic formula in (\[eq:samformula\]) for explicit computations.
\[ex:fancyteewurst\] Let $d=3$, $n=6$, and fix the configuration of vertices of a [*regular octahedron*]{}: $$X \, = \,(x_1,x_2,\ldots,x_6) \,=\, \bigl( \,+e_1\,,\,-e_1\,,\,\,+e_2,\,-e_2\,,\,\,+e_3\,,\,-e_3 \,\bigr).$$ Here $e_i$ denotes the $i$th unit vector in ${\mathbb{R}}^3$. The secondary polytope $\Sigma(X)$ is a triangle. Its edges correspond to the three subdivisions of the octahedron $X$ into two square-based pyramids, $\Delta_{1234} = \{12345, 12346\}$, $\Delta_{1256} = \{12356,12456\}$, and $\Delta_{3456} = \{13456, 23456\}$. Its vertices correspond to the three triangulations of $X$, namely $\Delta_{12} = \{1235, 1236, 1245, 1256\}$, $\Delta_{34} = \{ 1345, 1346, 2345, 2346 \}$, and $\Delta_{56} = \{1356, 1456, 2356, 2456\}$.
The normal fan of $\Sigma(X)$, which is the secondary fan of $X$, has three full-dimensional cones in ${\mathbb{R}}^6$. A vector $y$ in ${\mathbb{R}}^6$ selects the triangulation $\Delta_{ij}$ if $y_i+y_j$ is the uniquely attained minimum among $\{y_1+y_2,\,y_3+y_4,\,y_5+y_6\}$. It selects $\Delta_{1234}$ if $y_1+y_2 = y_3+y_4 < y_5+y_6$, and it leaves the octahedron unsubdivided when $y$ is in the lineality space $\,\{y \in {\mathbb{R}}^6: y_1+y_2 = y_3+y_4 = y_5+y_6\}$.
The Samworth body $\mathcal{S}(X)$ is defined in ${\mathbb{R}}^6$ by the following system of three inequalities. Use the $i$th inequality when the $i$th number in the list $(y_1{+}y_2, y_3 {+} y_4, y_5 {+} y_6)$ is the smallest: $$\begin{matrix}
\frac{ e^{y_1} (2 y_1-y_6-y_5) (2 y_1-y_4-y_3)}{(y_1{-}y_2)(y_1{-}y_3)(y_1{-}y_5)(y_1{-}y_6)(y_1{-}y_4) }
- \frac{ e^{y_2} (2 y_2-y_6-y_5) (2 y_2-y_4-y_3)}{(y_1{-}y_2)(y_2{-}y_3)(y_2{-}y_5)(y_2{-}y_6)(y_2{-}y_4) }
+ \frac{e^{y_3} (2 y_3-y_6-y_5) }{ (y_1{-}y_3)(y_2{-}y_3)(y_3{-}y_5)(y_3{-}y_6) } \smallskip \\ \,\,
+ \frac{e^{y_4} (2 y_4-y_6-y_5) }{ (y_1{-}y_4)(y_2{-}y_4)(y_4{-}y_5)(y_4{-}y_6) }
- \frac{e^{y_5} (y_4-2 y_5+y_3) }{ (y_1{-}y_5)(y_2{-}y_5)(y_3{-}y_5)(y_4{-}y_5) }
- \frac{e^{y_6} (y_4-2 y_6+y_3) }{ (y_1{-}y_6)(y_2{-}y_6)(y_3{-}y_6)(y_4{-}y_6) } \,\,\leq \,1 \smallskip
\end{matrix}$$ $$\begin{matrix}
\frac{ e^{y_1} (2 y_1-y_6-y_5) }{ (y_1-y_3)(y_1-y_4)(y_1-y_5)(y_1-y_6) }
+ \frac{ e^{y_2} (2 y_2-y_6-y_5) }{ (y_2-y_3)(y_2-y_4)(y_2-y_5)(y_2-y_6) }
- \frac{ e^{y_3} (2 y_3-y_6-y_5) (y_2-2 y_3+y_1)}{(y_1-y_3)(y_3-y_4)(y_3-y_5)(y_3-y_6)(y_2-y_3) }
\smallskip \\
+ \frac{ e^{y_4} (2 y_4-y_6-y_5) (-2 y_4+y_1+y_2)}{(y_1-y_4)(y_3-y_4)(y_4-y_5)(y_4-y_6)(y_2-y_4) }
- \frac{ e^{y_5} (y_2-2 y_5+y_1) }{ (y_1-y_5)(y_2-y_5)(y_3-y_5)(y_4-y_5) }
- \frac{ e^{y_6} (y_2-2 y_6+y_1) }{ (y_1-y_6)(y_2-y_6)(y_3-y_6)(y_4-y_6) }\,\,\leq \,1 \smallskip
\end{matrix}$$ $$\!
\begin{matrix}
\frac{ e^{y_1} (2 y_1-y_4-y_3) }{ (y_1-y_3)(y_1-y_4)(y_1-y_5)(y_1-y_6) }
+ \frac{ e^{y_2} (2 y_2-y_4-y_3) }{ (y_2-y_3)(y_2-y_4)(y_2-y_5)(y_2-y_6) }
- \frac{ e^{y_3} (y_2-2 y_3+y_1) }{ (y_1-y_3)(y_2-y_3)(y_3-y_5)(y_3-y_6) } - \smallskip \\
\frac{ e^{y_4} (-2 y_4+y_1+y_2) }{ (y_1-y_4)(y_2-y_4)(y_4-y_5)(y_4-y_6) }
{+} \frac{ e^{y_5} (y_4-2 y_5+y_3) (y_2-2 y_5+y_1)}{(y_1-y_5)(y_3-y_5)(y_5-y_6)(y_4-y_5)(y_2-y_5) }
{-} \frac{ e^{y_6} (y_4-2 y_6+y_3) (y_2-2 y_6+y_1)}{(y_1-y_6)(y_3-y_6)(y_5-y_6)(y_4-y_6)(y_2-y_6) }
\leq 1
\smallskip
\end{matrix}$$
The dual convex body $\mathcal{S}(X)^*$ has seven strata of faces in its boundary: a $3$-dimensional manifold of $2$-dimensional faces, corresponding to the trivial subdivision, three $4$-dimensional manifolds of edges corresponding to $\Delta_{1234}, \Delta_{1256}, \Delta_{3456}$, and three $5$-dimensional manifolds of extreme points, corresponding to $\Delta_{12},\Delta_{34},\Delta_{56}$. Each $2$-dimensional face of $\mathcal{S}(X)^*$ is a triangle, like the secondary polytope $\Sigma(X)$. The dual to this convex set is the Samworth body $\mathcal{S}(X)$, which is strictly convex. Its boundary is singular along three $4$-dimensional strata are formed when two of the three inequalities above are active. These meet in a highly singular $3$-dimensional stratum which is formed when all three inequalities are active. These singularities of $\partial \mathcal{S}(X)$ exhibit the secondary fan of $X$. It is instructive to draw a cartoon, in dimension two or three, to visualize the boundary features of $\mathcal{S}(X)$ and $\mathcal{S}(X)^*$.
Up until this point, the premise of this paper has been that the configuration $X$ is fixed but the weights $w$ vary. Example \[ex:fancyteewurst\] was meant to give an impression of the corresponding geometry, by describing in an intuitive language how a Samworth body $\mathcal{S}(X)$ can look like.
However, our premise is at odds with the perspective of statistics. For a statistician, the natural setting is to fix unit weights, $w = \frac{1}{n}(1,1,\ldots,1)$, and to assume that $X$ consists of $n$ points that have been sampled from some underlying distribution. Here, one cares about one distinguished point in $\partial \mathcal{S}(X)$ and less about the global geometry of the Samworth body. Specifically, we wish to know which face of $\mathcal{S}(X)^*$ is pierced by the ray $\bigl\{ (\lambda,\ldots,\lambda) \,:\, \lambda \geq 0 \bigr\}$.
-------- -------- ---------- --------- ---------- -------------------- --------- --------- ---------
Convex Gaussian Uniform Circular Circular
3-gons 4-gons 5-gons 6-gons hull $\mathcal{N}(0,1)$ $a=0.5$ $a=0.3$ $a=0.1$
1 0 0 0 3 948 533 257 34
0 1 0 0 4 8781 6719 4596 1507
0 0 1 0 5 8209 9743 10554 8504
0 0 0 1 6 1475 2805 4495 9887
2 0 0 0 4 8 3 6 7
1 1 0 0 5 1 2 1 2
3 0 0 0 3 6 2 2 1
2 1 0 0 4 39 16 4 7
2 0 1 0 5 1 1 0 1
1 2 0 0 5 1 0 1 6
4 0 0 0 4 1 0 0 0
3 1 0 0 3 114 38 10 1
3 0 1 0 4 39 20 9 2
2 2 0 0 4 59 19 16 9
5 0 0 0 3 3 0 0 0
4 1 0 0 4 1 0 0 0
4 0 1 0 3 90 27 8 1
3 2 0 0 3 120 32 11 0
5 1 0 0 3 50 11 3 0
7 0 0 0 3 2 1 0 0
-------- -------- ---------- --------- ---------- -------------------- --------- --------- ---------
: \[tab:caroline\] The optimal subdivisions for six random points in the plane
\[ex:Gaussian\] Let $d=2$ and $n=6$ as in Example \[ex:associahedron2\], but now with unit weights $w = \frac{1}{6}(1,1,1,1,1,1)$. We sample i.i.d. points $x_1,\ldots,x_6$ from various distributions $f$ on ${\mathbb{R}}^2$, some log-concave and others not, and we compare the resulting maximum likelihood densities $\hat f$.
In what follows, we analyze the case where $f$ is a standard Gaussian distribution or a uniform distribution on the unit disc, and we contrast this to distributions of the form $X=(U_1^a \cos(2\pi U_2), U_1^a \sin(2\pi U_2))$, where $U_1$ and $U_2$ are independent uniformly distributed on the interval $[0,1]$ and $a<0.5$. Such distributions have more mass towards the exterior of the unit disc and are hence not log-concave. For $a=0.5$ this is the uniform distribution on the unit disc. We drew 20,000 samples $X = (x_1,\ldots,x_6)$ from each of these four distributions.
For each experiment, we recorded the number of vertices of the convex hull of the sample, we computed the optimal subdivision using [LogConcDEAD]{}, and we recorded the shapes of its cells. Our results are reported in Table \[tab:caroline\]. Each of the four right-most columns shows the number of experiments out of 20,000 that resulted in a subdivision as described in the five left-most columns. These columns do not add up to 20,000, because we discarded all experiments for which the optimization procedure did not converge due to numerical instabilities.
In the vast majority of cases, reported in the first four rows, the optimal solution $\hat f$ is log-linear. Here the subdivision is trivial, with only one cell. For instance, the fourth row is the 30.5% case in Example \[ex:associahedron2\]. In the last row, ${\rm conv}(X)$ is a triangle and the subdivision is a triangulation that uses all three interior points. We saw such a triangulation in Example \[ex:octahedron\]. In fact, we constructed the data (\[eq:sixpoints\]) by modifying one of the examples with seven cells found by sampling from a Gaussian $\mathcal{N}(0,1)$ distribution. Note that the subdivisions resulting from Gaussian samples tend to have more cells than those from other distributions.
The examples in this section illustrate two different interpretations of the data set $(X,w)$: either the configuration $X$ is fixed and the weight vector $w$ varies, or $w$ is fixed and $X$ varies. These are two different parametric versions of our optimization problem (\[eq:ourproblem\]), (\[eq:constrained\]), (\[eq:unconstrained\]), (\[eq:optsecondary\]). This generalizes the interpretation of the secondary polytope $\Sigma(X)$ seen in [@DRS Section 1.2], namely as a geometric model for [*parametric linear programming*]{}. The vertices of $\Sigma(X)$ represent the various collections of optimal bases when the matrix $X$ is fixed and the cost function $w$ varies. See [@DRS Exercise 1.17] for the case $d=2, n=6$, as in Examples \[ex:hexagon\], \[ex:associahedron2\] and \[ex:Gaussian\]. Of course, it is very interesting to examine what happens when both $X$ and $w$ vary, and to study $\Sigma(X)$ as a function on the space of configurations $X$. This was done in [@universal]. The same problem is even more intriguing in the statistical setting introduced in this paper.
Study the Samworth body as a function $X \mapsto \mathcal{S}(X)$ on the space of configurations. Understand log-concave density estimation as a parametric optimization problem.
This problem has many angles, aspects and subproblems. Here is one of them:
For fixed $w$ and a fixed combinatorial type of subdivision $\Delta$, study the semi-analytic set of all configurations $X$ such that $\Delta$ is the optimal subdivision for the data $(X,w)$.
For instance, suppose we fix the triangulation $\Delta$ seen on the right of Figure \[fig:octahedron\]. How much can we perturb the configuration in (\[eq:sixpoints\]) and retain that $\Delta$ is optimal for unit weights? For $n{=}6,d{=}2$, give inequalities that characterize the space of all datasets $(X,w)$ that select $\Delta$.
An ultimate goal of our geometric approach is the design of new tools for nonparametric statistics. One aim is the development of test statistics for assessing whether a given sample comes from a log-concave distribution. Such tests are important, e.g. in economics [@An1; @An2].
Design a test statistic for log-concavity based on the optimal subdivision $\Delta$.
The idea is that $\Delta$ is likely to have more cells when $X$ is sampled from a log-concave distribution. Hence we might use the f-vector of $\Delta$ as a test statistic for log-concavity. The study of such tests seems related to the approximation theory of convex bodies developed by Adiprasito, Nevo and Samper [@ANS]. What does their “higher chordality” mean for statistics?
Unit Weights
============
In this section we offer a further analysis of the uniform weights case. Example \[ex:Gaussian\] suggests that the flat subdivision occurs with overwhelming probability when the sample size is small. Our main result in this section establishes this flatness for the small non-trivial case $n=d+2$:
\[thm:d+2points\] Let $X$ be a configuration of $n=d+2$ points that affinely span ${\mathbb{R}}^d$. For $w = \frac{1}{n}(1,\ldots,1)$, the optimal density $\hat f$ is log-linear, so the optimal subdivision of $X$ is trivial.
We shall use the following lemma, which can be derived by a direct computation.
\[lem:surprise\] The symmetric function $H$ in Section 4 satisfies the differential equation $$\frac{\partial H}{\partial x_1}
(x_1,\dots, x_d) \quad = \quad \frac{e^{x_1}H(-x_1, x_2-x_1,\dots, x_d-x_1) - H(x_1,\dots, x_d)}{x_1}.$$
Our $d+2$ points in $\mathbb R^d$ can be partitioned uniquely into two affinely independent subsets whose convex hulls intersect. This gives rise to a unique identity $$\sum_{i=1}^k \alpha_ix_i \,\,\,= \,\, \sum_{j=k+1}^{d+2}\beta_j x_j,$$ where $1\leq k \leq d+1, \,\,\alpha_1,\dots, \alpha_k, \beta_{k+1},\dots, \beta_{d+2} \geq 0$, and $\sum \alpha_i = \sum \beta_j = 1$. We abbreviate $\mathcal{D} = \{1,2,\ldots,d+2\}$. There are precisely three regular subdivisions of the configuration $X$:
1. the triangulation $\,\bigl\{ \mathcal{D} \backslash \{1\},
\mathcal{D} \backslash \{2\}, \ldots, \mathcal{D} \backslash \{k\} \bigr\}$,
2. the triangulation $\,\bigl\{ \mathcal{D} \backslash \{k{+}1\},
\mathcal{D} \backslash \{k{+}2\}, \ldots, \mathcal{D} \backslash \{d{+}2\} \bigr\}$,
3. the flat subdivision $\bigl\{ \mathcal{D} \bigr\}$.
The simplex volumes $\,\sigma_{\mathcal{D} \setminus i}
= {\rm vol}\bigl({\rm conv}(\,x_\ell : \ell \in \mathcal{D} \backslash \{i\})\bigr) \,$ satisfy the identity $$\label{eq:simplexvolumes}
\qquad \quad \sum_{i=1}^k \sigma_{\mathcal{D}\setminus i} \,\,\,= \,\,\sum_{j=k+1}^{d+2}
\! \sigma_{\mathcal{D}\setminus j}
\quad = \quad {\rm vol}({\rm conv}(X)).$$
Now let $w \in {\mathbb{R}}^{d+2}$ be a positive weight vector, and suppose that the optimal heights $y_1,\dots, y_{d+2}$ do not induce the flat subdivision (iii). This means that the optimal subdivision is one of the triangulations (i) and (ii). We will show that in that case $w \not= (\lambda, \lambda,\ldots,\lambda)$.
After relabeling we may assume that (ii) is the optimal triangulation for the given weights $w$. This is equivalent to the inequality $$\sum_{i=1}^k y_i\sigma_{\mathcal{D}\setminus i}
\,\,\,> \,\, \sum_{j=k+1}^{d+2}y_j \sigma_{\mathcal{D}\setminus j}. \quad \qquad$$ In light of (\[eq:simplexvolumes\]), at least one of $y_1,\dots, y_{k}$ has to be larger than at least one of $y_{k+1},\dots, y_{d+2}$. After relabeling once more, we may assume that $\,y_1 > y_{k+1}$.
Theorem \[thm:normalcone\] states that the weight vector $w$ is uniquely determined (up to scaling) by the optimal height vector $y$. Namely, the coordinates of $w$ are given by the formula for the optimal triangulation (ii). That formula gives $$\label{eq:w_1}
w_1 \,\,\,= \,\, \sum_{j=k+1}^{d+2} \sigma_{\mathcal{D}\backslash j}e^{y_1}
H \bigl(y_\ell-y_1 : \ell\in\mathcal{D}\backslash \{1, j\} \bigr), \quad$$ and $$\label{eq:w_{k+1}}
w_{k+1} \,\,\,=\,\, \sum_{j=k+2}^{d+2}\sigma_{\mathcal{D}\backslash j}e^{y_{k+1}}
H \bigl(y_\ell-y_{k+1}: \ell\in \mathcal{D}\backslash \{k{+}1,j\} \bigr).$$ For any index $j\in \{k{+}2,\dots, d{+}2\}$ we consider the expression $$\begin{aligned}
\label{eq:difference}
e^{y_{1}}H(y_\ell-y_1 : \ell\in\mathcal{D}\backslash j )\,\,-\,\,
e^{y_{k+1}}H(y_\ell-y_{k+1}: \ell\in \mathcal{D}\backslash j)\, \, \\
= \,\,\, \bigl(\,e^{y_1-y_{k+1}} H(y_{\ell} - y_{k+1} - (y_1-y_{k+1}) : \ell\in \mathcal{D}\backslash j)
\, - \,H(y_\ell-y_{k+1}: \ell\in \mathcal{D}\backslash j)
\,\bigr). \notag\end{aligned}$$ If we divide the parenthesized difference by $x_1 = y_1 - y_{k+1}$, then we obtain an expression as in the right hand side of Lemma \[lem:surprise\]. Then, by Lemma \[lem:surprise\], the expression in becomes $$e^{y_{k+1}} \cdot (y_1-y_{k+1}) \cdot \frac{\partial H}{\partial x_1}\bigl( \,
y_\ell-y_{k+1}: \ell\in \mathcal{D}\backslash j \,\bigr) .$$ By Corollary \[cor:positive\], all partial derivatives of $H$ are positive. Also, recall that $y_1>y_{k+1}$. Therefore, the expression in is positive. Hence, for any $j\in\{k{+}2,\dots, d{+}2\}$, we have $$e^{y_{1}}H(y_\ell-y_1 : \ell\in\mathcal{D}\backslash j)
\,\,\,> \,\,\, e^{y_{k+1}}H(y_\ell-y_{k+1}: \ell\in \mathcal{D}\backslash j).$$ In the left expression it suffices to take $ \,\ell\in\mathcal{D}\backslash \{1, j\} $, and in the right expression it suffices to take $\, \ell\in \mathcal{D}\backslash \{k{+}1,j\}$. Summing over all $j$, the identities (\[eq:simplexvolumes\]), (\[eq:w\_1\]) and (\[eq:w\_[k+1]{}\]) now imply $$w_1 \,\,> \,\, w_{k+1}.$$ This means that $w \not= (\lambda,\lambda,\ldots,\lambda)$ for all $\lambda > 0$. We conclude that it is impossible to get a nontrivial subdivision of $X$ as the optimal solution when all the weights are equal.
We now show that the result of Theorem \[thm:d+2points\] is the best possible in the following sense.
\[thm:d+3points\] For any integer $d \geq 2$, there exists a configuration of $n=d+3$ points in ${\mathbb{R}}^d$ for which the optimal subdivision with respect to unit weights is non-trivial.
The hypothesis $d \geq 2$ is essential in this theorem. Indeed, for $d=1$ it can be shown, using the formulas in Example \[ex:d=1\], that the flat subdivision is optimal for any configuration of $d+3=4$ points on the line $\mathbb R$ with unit weights. Here is an illustration of Theorem \[thm:d+3points\].
$\quad\qquad$
Fix unit weights on the following five points in the plane: $$\begin{aligned}
\label{eq:fivepoints}
X \,\, = \,\, \bigl(\, (0,0), \,(40, 0),\, (20, 40),\, (17, 10),\, (21, 15) \,\bigr).\end{aligned}$$ Using [LogConcDEAD]{} [@CGS], we find that the optimal subdivision equals $\{124, 245, 235, 1345\}$.
To derive Theorem \[thm:d+3points\], we first study the following configuration of $d+2$ points in ${\mathbb{R}}^d$:
$$\label{eq:specialconfig}
X\,\, =\,\, \biggl( e_1\,,\,e_2\,,\,\ldots\,,\, e_d\,,\, 0\,,\,\, \frac1{d+1} \sum_{i=1}^de_i \,\biggr).$$
Let $\alpha > 0$ and assign weights as follows to the configuration $X$ in (\[eq:specialconfig\]): $$\begin{aligned}
\label{eq:equalWeightsFormula}
w_1=w_2 =\cdots = w_{d+1} > 0,
\text{ and } \,\,w_{d+2} \,=\,
w_1\frac{(d+1)e^{\alpha} H(-\alpha, -\alpha,\dots, -\alpha)}{dH(\alpha, 0,\dots,0)}.
\end{aligned}$$ Then the optimal heights satisfy $\,y_1= y_2 = \cdots = y_{d+1}\,$ and $\,y_{d+2} = y_1+\alpha$.
Let $\mathcal D = \{1,\dots, d+2\}$ and fix $w$ as in (\[eq:equalWeightsFormula\]). The volumes $\text{vol}(\mathcal D\backslash\{i\})$ are equal for $i\in\{1,\dots, d+1\}$. Set $\sigma = \text{vol}(\mathcal D\backslash\{i\})$. We will show that the heights $y_1=\cdots=y_{d+1} = y$ and $y_{d+2} = y+\alpha$ solve the Lagrange multiplier equations for our optimization problem, assuming that $\Delta$ is the triangulation $\{\mathcal D\backslash \{1\},\ldots, \mathcal D\backslash \{d{+}1 \}\}$. Indeed, from we derive $$\begin{matrix}
& w_i &= &
d \cdot \sigma \cdot e^y \cdot H(\alpha, 0, \ldots, 0) \quad & \hbox{
for $i\leq d+1$} \\
\hbox{and} \qquad & w_{d+2}
& = & (d{+}1)\cdot \sigma \cdot e^{y+\alpha} \cdot H(-\alpha, \dots, -\alpha). &
\end{matrix}$$ By taking ratios, we now obtain . Of course, the weights must be scaled so that they sum to one. Since $\alpha > 0$, the subdivision induced by $y$ is indeed $\{\mathcal D\backslash 1, \ldots, \mathcal D\backslash \{d{+}1 \} \}$.
We now note that, by Lemma \[lem:surprise\], $$e^\alpha \cdot H(-\alpha, \dots, -\alpha) - H(\alpha, 0, \ldots, 0)
\,\,=\,\, \alpha \frac{\partial H}{\partial \alpha}(\alpha, 0, \ldots, 0).$$ This is positive for $\alpha > 0$, zero for $\alpha = 0$, and negative for $\alpha < 0$. The first case implies:
\[cor:setup\] Fix the configuration $X$ in (\[eq:specialconfig\]) and suppose that $w_1=\cdots=w_{d+1}$. Then $\frac{w_{d+2}}{w_1} > \frac{d+1}d$ if and only if the optimal subdivision is the triangulation $\{\mathcal D\backslash \{1\},
\ldots, \mathcal D \backslash \{ d{+}1 \}\}$.
We are now prepared to pass from $d+2$ to $d+3$ points, and to offer the missing proof.
We use Corollary \[cor:setup\] with $\frac{w_{d+2}}{w_1} = 2$. This is strictly bigger than $\frac{d+1}{d}$ whenever $d\geq 2$. We redefine $(X,w)$ by splitting the last point $x_{d+2}$ into two nearby points with equal weights. Then $n=d+3$ and the optimal subdivision is non-trivial. This holds because, for any fixed $w \in {\mathbb{R}}^n$, the set of $X$ whose optimal subdivision is trivial is described by the vanishing of continuous functions. It is hence closed in the space of configurations.
We conclude this paper with a pair of challenges for Nonparametric Algebraic Statistics.
What is the smallest size $n $ of a configuration $X$ in ${\mathbb{R}}^d$ such that the optimal subdivision of $\,X$ with unit weights has at least $c$ cells? This $n$ is a function of $c$ and $d$.\
We just saw that $n(2,d) = d+3$ for $d \geq 2$. Determine upper and lower bounds for $n(c,d)$.
We can also ask for a characterization of combinatorial types of triangulations that are realizable as in Figures \[fig:octahedron\] and \[fig:fivepoints\]. Such a triangulation in ${\mathbb{R}}^d$ is obtained by removing a facet from a $(d{+}1)$-dimensional simplicial polytope with $\leq n$ vertices. If we are allowed to vary $w \in {\mathbb{R}}^n$, then Theorem \[thm:converse\] tells us that all simplicial polytopes have such a realization. Hence, in the following question, we seek configurations $X$ in ${\mathbb{R}}^d$ with $w = \frac{1}{n}(1,\ldots,1)$.
Which simplicial polytopes can be realized by points in ${\mathbb{R}}^d$ with unit weights?
For example, the octahedron can be realized with unit weights, as was seen in Figure \[fig:octahedron\].
[**Acknowledgements.**]{} We thank Donald Richards for very helpful discussions regarding Proposition \[prop:magic\]. Bernd Sturmfels was partially supported by the Einstein Foundation Berlin and the NSF (DMS-1419018). Caroline Uhler was partially supported by DARPA (W911NF-16-1-0551), NSF (DMS-1651995) and ONR (N00014-17-1-2147).
[10]{}
K. Adiprasito, E. Nevo and J. Samper: [*A geometric lower bound theorem*]{}, Geom. Funct. Anal. [**26**]{} (2016) 359–378.
M.Y. An: [*Log-concave probability distributions: theory and statistical testing*]{}, Duke University, Department of Economics Working Paper No. 95-03.
M.Y. An: [*Log-concavity versus log-convexity: a complete characterization*]{}, Journal of Economic Theory [**80**]{} (1998) 350–369.
A. Barvinok: [*Computing the volume, counting integral points, and exponential sums*]{}, Discrete Comput. Geom. [**10**]{} (1993) 123–141.
M. Cule, R.B. Gramacy and R. Samworth: LogConcDEAD: an R package for maximum likelihood estimation of a multivariate log-concave density. J. Statist. Software [**29**]{} (2009) Issue 2.
M. Cule, R. Samworth and M. Stewart: [*Maximum likelihood estimation of a multi-dimensional log-concave density*]{}, J. R. Stat. Soc. Ser. B Stat. Methodol. [**72**]{} (2010) 545–607.
J. De Loera, S. Hoşten, F. Santos and B. Sturmfels: [*The polytope of all triangulations of a point configuration*]{}, Documenta Mathematica [**1**]{} (1996) 103–119.
J. De Loera, J. Rambau and F. Santos: [*Triangulations. Structures for Algorithms and Applications*]{}, Algorithms and Computation in Mathematics [**25**]{}, Springer-Verlag, Berlin, 2010.
L. Dümbgen and K. Rufibach: [*Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency*]{}, Bernoulli [**15**]{} (2009) 40–68.
I.M. Gel’fand, M.M. Kapranov and A.V. Zelevinsky: [*Discriminants, Resultants and Multidimensional Determinants*]{}, Birkhäuser, Boston, 1994.
U. Grenander: [*On the theory of mortality measurement II*]{}, Skandinavisk Aktuarietidskrift [**39**]{} (1956) 125–153.
P. Groeneboom, G. Jongbloed and J. A. Wellner: [*Estimation of a convex function: Characterizations and asymptotic theory*]{}, Annals of Statistics [**29**]{} (2001) 1653–1698.
K. Gross and D. Richards: [*Total positivity, spherical series, and hypergeometric functions of matrix argument*]{}, Journal of Approximation Theory [**59**]{} (1989) 224–246
E. Miller and B. Sturmfels: [*Combinatorial Commutative Algebra*]{}, Graduate Texts in Mathematics, Vol. 227, Springer Verlag, New York, 2004.
R. Thomas: [*Lectures in Geometric Combinatorics*]{}, Student Mathematical Library [**33**]{}, IAS/Park City Mathematical Subseries, American Mathematical Society, Providence, RI, 2006.
G. Walther: [*Inference and modeling with log-concave distributions*]{}, Statistical Science [**24**]{} (2009) 319–327.
Elina Robeva, Massachusetts Institute of Technology, Department of Mathematics, [erobeva@mit.edu]{}
Bernd Sturmfels, MPI-MiS Leipzig, [bernd@mis.mpg.de]{} and UC Berkeley, [bernd@berkeley.edu]{}
Caroline Uhler, Massachusetts Institute of Technology, IDSS and EECS Department, [cuhler@mit.edu]{}.
|
---
abstract: 'Light rays passing very close to a black hole may experience very strong deviations. Two geometries were separately considered in the recent literature: a source behind the black hole (standard gravitational lensing); a source in front of the black hole (retro-lensing). In this paper we start from the Strong Field Limit approach to recover both situations under the same formalism, describing not only the two geometries just mentioned but also any intermediate possible configurations of the system source-lens-observer, without any small-angle limitations. This is done for any spherically symmetric black holes and for the equatorial plane of Kerr black holes. In the light of this formalism we revisit the previous literature on retro-lensing, sensibly improving the observational estimates. In particular, for the case of the star S2, we give sharp predictions for the magnitude of the relativistic images and the time of their highest brightness, which should occur at the beginning of year 2018. The observation of such images would open fascinating perspectives on the measure of the physical parameters of the central black hole, including mass, spin and distance.'
author:
- 'V. Bozza$^{1,2,3}$ and L. Mancini$^{2,3,4}$'
date: 'Received / Accepted'
title: 'Gravitational Lensing by Black Holes: a comprehensive treatment and the case of the star S2'
---
Introduction
============
The fact that light rays can wind an arbitrary number of times around a black hole before emerging back towards spatial infinity is well known since early times of General Relativity [@Dar; @Atk; @Lum; @Oha; @Nem] (see e.g. the treatment in @Cha). In practice, a source behind a black hole not only produces the two classical weak field gravitational lensing images, but also an infinite number of strong field images corresponding to photons with winding numbers running from 1 to infinity. This phenomenon has been revived in a work by @VirEll, who showed that the supermassive black hole at the center of our Galaxy may be a suitable lens candidate. The resolution needed for such observation is very high but should be reached by next future Very Long Baseline Interferometry experiments, such as ARISE[^1] and MAXIM[^2]. In order to describe gravitational lensing in such extreme cases, we cannot use any weak field approximation. However, it is possible to take advantage of the opposite limit and finally get a very simple and efficient analytical approximation, called the Strong Field Limit (SFL) [@Boz1]. This method, firstly emerged in Schwarzschild black hole studies [@Dar; @Oha; @BCIS], was then applied to Reissner-Nordstrom black holes [@ERT], to charged black holes of heterotic string theory [@Bha] and finally generalized to an arbitrary spherically symmetric metric [@Boz1]. The microlensing situation was considered by @Pet, while the Kerr black hole was explored analytically for quasi-equatorial motion [@Boz2] and numerically for arbitrary motion [@VazEst]. It is also interesting that a time delay measurement would give a precise estimate of the distance of the lens [@BozMan]. For former and/or alternative formulations of strong field gravitational lensing see @FriNew [@FKN; @DabSch; @Per].
Already in the first studies of gravitational lensing in strong fields by @Dar, it was noticed that the whole sky is mapped in the vicinity of what it was then called a “compact sun”. In particular, it was known that a source in front of the lens could yield relativistic images and Einstein rings as bright as a source behind the lens (see e.g. @Lum). In more recent years retro-lensing was re-discovered quite independently from standard gravitational lensing. @HolWhe proposed that a black hole of a few solar masses passing within 1 pc from the solar system would redirect photons from the Sun backwards to the Earth. An observer would thus see a “star” lighting up and then switching off in the sky as the Earth in its motion enters and leaves the best alignment position. @DepS2 suggested the black hole in Sgr A\* at the center of our Galaxy as a suitable retro-lens, proposing the star S2 (the star with the smallest average distance from Sgr A\* discovered so far) as a candidate source. At the moment, this seems to be the best known candidate for gravitational lensing in strong fields, deserving a closest investigation. At the same time, @EirTor considered retro-lensing by Sgr A\*, using the analytical framework of the SFL method to give correct estimates for such a phenomenon; their investigation, however, is limited to small angles and cannot cover the case of S2. Finally, @DepRM extended Holz & Wheeler proposal to Kerr retro-lensing, using the SFL method to calculate the position of the images but using the Holz & Wheeler formula for the amplification (which is inadequate for spinning black holes), finding no significant deviation from the light curves of the Schwarzschild case.
In this paper, we give up any limitation due to small-angle approximation and treat the standard and retro-gravitational lensing on a unified ground, where they just come up as particular cases. Besides recovering these two situations, we also address the gravitational lensing problem for any intermediate geometries, where the strong field images are still present as always, but they are usually dimmer (see Sect. 2). This allows us to treat in a very accurate way the case of the star S2, giving sharp predictions for the light curves of its images in the next years. Moreover, we clarify several aspects which were not clearly stated in the former literature, constructing a unique analytical framework for the whole phenomenology.
In Sect. 2 we treat the spherically symmetric lens; we give the formulae for the position and the magnification of the images, and discuss the differences with the magnification formula by @HolWhe; we also comment on the importance of time delay measurements. Sect. 3 is devoted to the study of S2, the best candidate source for gravitational lensing in the SFL; we use our formalism to draw analytical curves of the images, accurately predicting the epoch of their luminosity peak. In Sect. 4, in the light of the results by @Boz2, we guess about the possible changes in S2 relativistic images if the black hole at the center of our galaxy has non-vanishing spin; we also compare our results in Sun retro-lensing by Kerr black holes with those by @DepRM. Sect. 5 contains a summary of the work.
Gravitational lensing by spherically symmetric black holes
==========================================================
According to the SFL method, the deflection angle of a light ray passing very close to a black hole can be expanded around the minimum impact angle $\theta_m$, which separates the light rays absorbed by the black hole ($\theta<\theta_m$) from the light rays which are simply deflected ($\theta>\theta_m$). As previously shown [@Boz1], the deflection angle always diverges logarithmically at $\theta \sim \theta_m$ for any class of spherically symmetric black holes. The logarithmic term and the constant term give a sufficient approximation to the deflection angle in order to explain the whole phenomenology. The fundamental formula reads
$$\alpha(\theta)=-\overline{a} \log \left(\frac{\theta}{\theta_m}-1
\right) +\overline{b}, \label{SFL}$$
up to higher order terms in $(\theta-\theta_m)$. The numerical coefficients $\overline{a}$ and $\overline{b}$ depend on the characteristics of the black hole (electric charge, coupling to a scalar field, the particular gravitation theory we are using, etc.). We refer the reader to [@Boz1] for its full derivation and for some examples (see also @Bha).
Using the formula (\[SFL\]), @EirTor have correctly calculated the position and the magnification of the retro-lensing images. However, their treatment (as well as the treatment in [@Boz1] for standard lensing) is limited to small separations of the source from the optical axis (defined as the line connecting the observer with the lens). In order to address the general case, we have to write the lens equation in a suitable way, without restricting to particular cases.
Lens equation and position of the images
----------------------------------------
The source, the lens and the observer define the plane where the whole motion of the photon takes place in the case of spherically symmetric black holes. We define $\gamma$ as the angle between the source-lens direction and the optical axis (see Fig. \[Fig Lens Eq\]). $\theta$ is the angular position of the image in the sky of the observer, with respect to the position of the black hole. It coincides with the impact angle of the light ray as seen by the observer. The impact angle as seen from the source is called $\overline{\theta}$. Then the lens equation is simply (see also @Boz2) $$\gamma=\alpha(\theta)- \theta -\overline{\theta}. \label{Lens Eq}$$ Now, we can give a unique treatment allowing $\gamma$ to run over the whole range $[0,+\infty)$. In fact, $\gamma \simeq 0$ would yield the weak field gravitational lensing; $\gamma \simeq \pi$ would be gravitational retro-lensing; $\gamma \simeq 2\pi$ would give standard gravitational lensing in the SFL and so on. The collection of strong field gravitational lensing images would be recovered solving the lens equation for $\gamma \simeq 2n \pi$ with $n \geq 1$, while the collection of retro-lensing images is recovered with $\gamma=(2n-1)\pi$ with $n\geq 1$.
Since we are only interested in strong field gravitational lensing of far sources, the impact angle $\theta=\arcsin (u/D_{OL})$ is always negligible compared to $\gamma$ and $\alpha$, since it is of the order of the Schwarzschild radius of the black hole divided by the distance of the lens which is usually much larger (unless we are falling into the black hole!). So we can safely drop $\theta$ from Eq. (\[Lens Eq\]). For $\overline{\theta}$ we can apply the same argument as long as the source is far enough from the black hole (see e.g. @CunBar or @Vie for the more complicated case where the source is orbiting very close to the black hole).
The general solution of the lens equation is then $$\theta=\theta_m \left(1+ e^{\frac{\overline{b}-
\gamma}{\overline{a}}} \right). \label{theta}$$ This formula is valid both for standard gravitational lensing (when $\gamma \simeq 2n\pi$, compare with @Boz1) and for retro-lensing (when $\gamma \simeq (2n-1) \pi $, compare with @EirTor). It is also valid for any intermediate situation. Its limits of validity are fixed by the accuracy of the SFL formula (\[SFL\]) for the deflection angle when we slip from strong field to weak field gravitational lensing. This point deserves a deeper discussion.
Estimates of the accuracy of the SFL approximation are easily done in the Schwarzschild case, where it is possible to calculate the deflection angle exactly [@BCIS]. So let us call $\alpha_{ex}(\theta)$ the exact deflection angle for Schwarzschild black hole. Let $\theta_{ex}$ be the image calculated using $\alpha_{ex}$ in the lens equation (\[Lens Eq\]). What is really interesting in the images is their difference from the minimum impact angle, i.e. $\theta-\theta_m$ and $\theta_{ex}-\theta_m$. So, in Fig. \[Fig acc\], we finally plot $(\theta_{ex}-\theta)/(\theta_{ex}-\theta_m)$ vs $\gamma$.
Of course when $\gamma \simeq 0$ we should rather use the weak field approximation and the error becomes very large (we should also take into account the fact that the additional terms in the lens equation are no longer negligible). What is more interesting is looking at the error in the determination of the retro-lensing images ($\gamma \simeq \pi$), which turns to be of the order of $6\%$. The accuracy grows exponentially with $\gamma$ so that the outermost relativistic image of standard strong field gravitational lensing ($\gamma=2\pi$) is determined with an accuracy of $0.3\%$. To improve these numbers, which is perhaps desirable for the outermost retro-lensing image, higher order terms in the formula for the deflection angle should be included, but this task is beyond the scope of this work. A $6\%$ accuracy in retro-lensing will be sufficient for most of the following discussions.
Magnification of the images
---------------------------
In the first retro-lensing paper [@HolWhe] a rather unusual formula for the amplification of the images was proposed, based on heuristic arguments $$\begin{aligned}
&& A=\pi (u^2_o-u^2_i) \frac{\Delta \Theta}{2\pi} \\
&& \Delta \Theta= 2 \arctan \left( \frac{R_S}{D_{OS} \sin \beta}
\right),\end{aligned}$$ where $u_i$ and $u_o$ are the inner and outer impact parameters which bound the image, $\Delta \Theta$ is the amplitude of the arc described by the image, $R_S$ is the source radius.
According to this formula, a full Einstein ring would form only for a perfect alignment $\beta=0$. This is obviously wrong, since it is sufficient that just one point of the source is perfectly aligned to build a full Einstein ring. For a source far enough from perfect alignment, it can be shown that this approximate formula coincides with the standard formula we are going to derive in this subsection. However, it cannot be trusted in the high alignment regime and should be replaced by a more rigorous one.
@Oha already derived a formula valid for the Schwarzschild black hole. Here we rederive it for an arbitrary black hole. To describe the extension of the source in terms of our angles, we can attach polar coordinates to the lens frame, where $\gamma$ plays the role of the polar angle from the optical axis and we introduce an azimuthal angle $\phi$ around this axis. Then the solid angle element is given by $d\omega_S=d\gamma \sin \gamma
d\phi$. In the same way, we can attach polar coordinates to the observer obtaining an image element $d\omega_I=d\theta \sin \theta
d\phi$. The quantity $$\frac{d\omega_I}{d\omega_S}=\frac{\sin\theta}{\sin{\gamma} \frac{d
\gamma}{d\theta}}$$ represents the ratio between the angular extension of the image as it appears to the observer and the extension of the source as it appears to the black hole. The latter is related to the extension of the source as it appears to the observer by the simple quadratic ratio of the distances $D_{OS}^2/D_{LS}^2$. Finally we have $$\mu=\frac{D_{OS}^2}{D_{LS}^2}\frac{\sin\theta}{\sin{\gamma}
\frac{d \gamma}{d\theta}}. \label{mu}$$
The source distance is of course related to the other distances by simple trigonometry: $D_{OS}^2=D_{OL}^2+D_{LS}^2+2D_{OL}D_{LS}
\cos \gamma$. Moreover, since we are always considering black holes far from the observer, we can safely approximate $\sin\theta
\simeq\theta$. On the other hand, only for a high alignment situation, with $\gamma\simeq k\pi$, we can replace $\sin \gamma$ by $|\gamma-k\pi|$. In such cases, which correspond to standard and retro-lensing, we recover the usual formula for the magnification of spherically symmetric lenses, valid for small angles. Eq. (\[mu\]), on the contrary, is valid also for a general $\gamma$.
Now, using the lens equation, we can write $$\mu=-\frac{D_{OS}^2}{D_{LS}^2} \frac{\theta_m^2
e^{\frac{\overline{b}- \gamma}{\overline{a}}}(1+
e^{\frac{\overline{b}- \gamma}{\overline{a}}})}{\overline{a} \sin
\gamma},$$ where the sign coming out of the derivative correctly accounts for the parity of the image. For $\gamma \simeq 2n\pi$ we recover the magnification formula given by @Boz1, while for $\gamma
\simeq (2n-1)\pi$ we recover the formula given by @EirTor. However, this formula smoothly joins the two extreme cases, covering the whole range of $\gamma$. It can be obtained from the Kerr magnification formula given by @Boz2 in the limit of vanishing black hole spin. However, the derivation proposed here does not need to pass through the Kerr metric and can be applied to a generic spherically symmetric black hole.
The caustic points (points of formally infinite magnification) are exactly at $\gamma=k\pi$ where $\sin \gamma$ vanishes. Sources close to these points get the maximal amplification. That is why standard and retro-lensing were first studied as the most interesting physical cases.
To treat extended sources we need to integrate this formula over the angular extension of the source, eventually weighted by a surface brightness factor. If the source is far from a caustic point, then the magnification varies very little along the source surface and it makes almost no difference to approximate the source as a point. On the contrary, the magnification for point-like sources diverges on caustic points, while for realistic extended sources we get a finite result integrating over the source angular area. Then the source radius acts as an effective cut-off of the magnification.
In order to compare this formula with that of @HolWhe, we can use it in the limit of retro-lensing, with $\gamma \simeq
\pi$. The magnification becomes $$\mu_{retro}=-\frac{D_{OS}^2}{D_{LS}^2} \frac{\theta_m^2
e^{\frac{\overline{b}- \pi}{\overline{a}}}(1+
e^{\frac{\overline{b}- \pi}{\overline{a}}})}{\overline{a}
|\gamma-\pi|},\label{mur}$$ where we have neglected the dependence on $\gamma$ in the exponentials being much weaker than the dependence in the denominator.
Finally, the integral over the source extension, for a constant brightness source of radius $R_S$ is a standard calculation in terms of elliptic functions, first performed by @WitMao, and also given explicitly by @EirTor. It amounts to replacing the $|\gamma-\pi|^{-1}$ in Eq. (\[mur\]) by its integral over the source disk $D_S$ $$\begin{aligned}
&\frac{1}{\pi \gamma_S^2}&\int\limits_{D_S} \frac{1}{\gamma}\sin
\gamma
d\gamma d\phi= \frac{2Sign[\gamma_S-\gamma]}{\pi \gamma_S^2} \nonumber \\
&& \left[ (\gamma_S-\gamma) E\left(\frac{\pi}{2},-\frac{4\gamma_S
\gamma}{(\gamma_S-\gamma)^2} \right)+\right. \nonumber \\
&& \left.
(\gamma_S+\gamma)F\left(\frac{\pi}{2},-\frac{4\gamma_S
\gamma}{(\gamma_S-\gamma)^2} \right) \right],\label{Intmu}\end{aligned}$$ where for simplicity we have used $\gamma$ instead of $|\gamma-\pi|$ and we have defined $\gamma_S=R_S/D_{LS}$. $F$ and $E$ are the elliptic integrals of first and second kind.
With this formula, we can draw a retro-MACHO light curve for the situation described by @HolWhe, i.e. a black hole of 10 M$_\odot$ passing at 0.01 pc from the Sun, which deviates photons coming from the Sun backwards to the Earth. We can compare the curves in Fig. \[Fig Micro\] with those by @HolWhe to see how different the shape looks when the source starts to cover the caustic point (top curve). It is interesting to note that the absolute normalization of the curves by @HolWhe is slightly different from that by @DepRM and both are slightly different from ours. This might depend on minor approximations in the units of measure.
In the case of perfect alignment, Eq. (\[Intmu\]) gives $2/\gamma_S$. The maximum amplification is thus $$\mu_{max}=\frac{D_{OS}^2}{D_{LS}^2} \frac{4\theta_m^2
e^{\frac{\overline{b}- k\pi}{\overline{a}}}(1+
e^{\frac{\overline{b}- k\pi}{\overline{a}}})}{\overline{a}
\gamma_S},\label{mumax}$$ where we have also multiplied by two in order to sum the (equal) magnifications of the two images. In Sun retro-lensing, it amounts to $$\mu_{max}=1.29\times 10^{-14}
\left(\frac{M_L}{10M_\odot}\right)^2\left(\frac{R_S}{R_\odot}\right)^{-1}\left(\frac{D_{OL}}{0.01
pc }\right)^{-1}.$$
Now, let us consider an arbitrary geometry, where observer, source and lens are completely misaligned, so that $\gamma$ is far from $k\pi$. Then we have to use the general formula (\[mu\]) for the magnification with a value for $\sin \gamma$ which is a generic number between 0 and 1. To compare this situation with the perfect alignment, we can take the ratio between Eq. (\[mu\]) with $\sin
\gamma\simeq 1$ and one half of Eq. (\[mumax\]) (for a single image). Apart from the exponentials, which can weakly modify the order of magnitude, this ratio is generally of the order of $$\frac{\mu_{min}}{\mu_{max}}\sim\frac{\gamma_S}{2}\ll 1.$$
Thus intermediate lensing is generally disfavored with respect to standard and retro- gravitational lensing. However, this does not mean that intermediate lensing cannot be interesting from an observational point of view, as we shall see in Sect. 3.
Time delay in strong field gravitational lensing
------------------------------------------------
Time delay in strong field gravitational lensing provides a very important independent observable, which directly yields the distance of the lens with a good accuracy [@BozMan]. In fact the time delay between images with successive winding numbers is $$\Delta T=2\pi \frac{D_{OL} \theta_m}{c_0}, \label{TDR}$$ where $c_0$ is the speed of light. Thus, measuring $\theta_m$ and $\Delta T$ we can immediately obtain $D_{OL}$. To get an idea of how big the time delay is, we can write it as $$\Delta T=0.13 h \left(\frac{M}{2.87 \times 10^6 M_\odot} \right).$$
It is thus completely insignificant for solar mass black holes, even if they are close to our solar system. It is also quite small for the black hole at the center of our Galaxy, but it amounts to several days for supermassive black holes in other galaxies, being thus measurable, once we find a suitable variable source.
The calculations by @BozMan were made with the standard gravitational lensing configuration in mind. However, it is easy to see that in retro-lensing or any intermediate lensing configurations the time difference between relativistic images remains exactly the same, while only the (unobservable) absolute travel time of the photon changes.
Now let us consider the time delay between the first relativistic image and the direct image of the source. When the source is well aligned behind the lens, then the direct image becomes weakly lensed and the first relativistic image turns to the secondary weak field image. In this case, the Shapiro time delay formula applies (see @WGS for an application involving the Galactic center).
If the source is not behind the lens, then the time delay between the first relativistic image and the direct image is dominated by the geometric path difference $$\begin{aligned}
&\Delta T=& \frac{D_{OL}+D_{LS}-D_{OS}}{c}=\frac{D_{OL}}{c_0}
\left[ 1+\frac{\sin \beta -\sin \gamma}{\sin(\gamma-\beta)}
\right] \simeq
\nonumber \\
&& \simeq \frac{D_{OL}}{c_0}
\frac{1-\cos{\gamma}}{\sin{\gamma}}\beta, \label{TDSF-WF}\end{aligned}$$ where $\beta$ is the angular separation between the source and the black hole as seen by the observer. The last approximate equality is valid if $\beta \ll 1$. In principle, measuring $\beta$ and $\Delta T$, if we have a good knowledge of $\gamma$, we can use the time delay to estimate the distance to the lens. With respect to the time delay between consecutive relativistic images, this measure is surely easier, since we just need one relativistic image. However, we need a good knowledge of the angle $\gamma$, which was not required in the previous case.
Gravitational lensing of the star S2 by Sgr A\*
===============================================
S2 is the star with the minimum average distance from the galactic center discovered so far [@Sch1]. Its orbital motion has been very accurately reconstructed through proper motion and spectral measurements. Its orbital parameters, taken by @Sch are reported in Table 1. This star looks like a O8-B0 main sequence star of 15 M$_\odot$ with an apparent magnitude in the K-band (centered on $\lambda=2.2 \mu$m) of $m_K=13.9$. The extinction in the K band in the region of the galactic center amounts to 3.3 magnitudes [@Rie]. The orbital period and the major semiaxis fix the enclosed mass to $M_{enc}=3.3\times 10^6 M_\odot$, slightly larger than the central black hole mass, which is currently estimated to $M_{BH}=2.87\times 10^6 M_\odot$.
Orbital parameter Value
------------------- ----------------------
$a$ (pc) $4.54\times 10^{-3}$
$P$ (yr) 15.73
$e$ 0.87
$T_0$ (yr) 2002.31
$i$ (deg) 45.7
$\Omega$ (deg) 45.9
$\omega$ (deg) 244.7
: Orbital parameters for S2. $a$ is the major semiaxis, $P$ is the orbital period, $e$ is the eccentricity, $T_0$ is the epoch of periapse, $i$ is the inclination of the normal of the orbit with respect to the line of sight, $\Omega$ is the position angle of the ascending node, $\omega$ is the periapse anomaly with respect to the ascending node. Data taken from @Sch.
The value of the inclination of the orbit suggests that a high alignment with the observer-lens line does not occur during the motion of the star around Sgr A\*. This seems to rule out any possibility for standard or retro-gravitational lensing. Contrarily to what expected, @DepS2 claimed that the relativistic images of S2 in the central black hole are not far beyond instrumental sensitivities, even if the alignment is not favorable. In this section, we complete their analysis in the light of our formalism, including all the significant orbital parameters and drawing light curves for the relativistic images using our magnification formula. We can thus confirm their claim also predicting the best observability time for these images.
Rather than a retro-lensing configuration, S2 represents a case with an intermediate $\gamma$. The magnification of the images is thus well described by Eq. (\[mu\]). Since the radius of S2 is a few solar radii, $\gamma_S$ stays much smaller than $|\gamma-k\pi|$ (i.e. we are far enough from caustic points), allowing us to trust the point-like magnification without need to integrate it over the source surface. After some algebra, we can write down all the interesting quantities in terms of the orbital parameters of the system $$\begin{aligned}
&& D_{LS}=\frac{a(1-e^2)}{1+e \cos \phi} \\
&& D_{OL}\simeq D_{OS}=8kpc\\
&& \gamma=\arccos[\sin(\phi+\omega)\sin i], \label{gammaS2}\end{aligned}$$ where $\phi$ is the anomaly angle of the star starting from the periapse epoch, $i$ is the inclination of the orbit and $\omega$ is the periapse anomaly with respect to the ascending node. By the angular momentum conservation, we have $$L= M_{S2}\sqrt{G M_{enc}a (1-e^2)}=M_{S2}D_{LS}^2 \dot \phi.$$ By this equation, we can write a differential equation for $\dot
\phi$ $$\frac{\left[a(1-e^2) \right]^{3/2}}{\sqrt{G M_{enc}}(1+e \cos
\phi)^2}\dot \phi=1.$$
Integrating and inverting, we can get $\phi$ as a function of time, exploiting the initial condition $\phi(T_0)=0$, with $T_0$ given in Table 1. If the eccentricity of the orbit of S2 were negligible, $\phi(t)$ would just be a linear function of time, of the form $\phi(t)=\omega_0 t$, with $\omega_0=2\pi/T$ being a constant. This approximation was done for simplicity by @DepS2. However, in Fig. \[Fig phi\], we see that the high value of the eccentricity drastically modifies this function. To get accurate predictions, it is thus mandatory to take into account the angular motion of S2 correctly.
Finally, we can plot the magnification of the first two relativistic images, taking $M_{BH}=2.87\times 10^6 M_\odot$ as the mass of the black hole [@Sch] (@DepS2 used $M_{enc}$). The first relativistic image has $0<\gamma<\pi$. In practice it is the secondary image of weak field gravitational lensing which turns into a strong field image because of the high misalignment. It has a negative parity and is formed close to the black hole on the other side with respect to the direction of S2. The second relativstic image has $\pi<\gamma<2\pi$ and comes from light rays which take the “wrong” direction around the black hole, and are bound to turn once more around it in order to reach the observer. This image has positive parity and appears on the same side of S2. The next images are fainter and probably uninteresting for the moment.
In Fig. \[Fig S2mu\] we plot the magnifications of the first two images as functions of time. The periapse epoch is the most favorable for the observations since $D_{LS}$, which appears with a power of -2 in Eq. (\[mu\]), drops to its minimum value. This minimum value is still 1300 times larger than the Schwarzschild radius of the central black hole, so that we can safely treat the source as far from the lens. Since the angular velocity of S2 is maximal in the periapse epoch, the luminosity peak is relatively short. As S2 passed through the periapse in year 2002, we have to wait for the next periapse to get this luminosity peak. In Fig. \[Fig S2ing\], we have drawn the expected apparent magnitudes in the K band of the two first relativistic images at the epoch of the next periapse, supposing that the extinction value 3.3 in this band [@Rie] also applies from light coming to us from regions very close to the central black hole. The first relativistic image should stay brighter than 32 magnitudes from mid-August 2017 to mid-April 2018. The second relativistic image is typically 5.4 times fainter during the peak. A hard challenge for next generation instruments!
The observation of a relativistic image requires a strikingly high angular resolution together with a high flux sensitivity. Very impressive improvements in Very Long Baseline Interferometry have been performed in the last years in the radio bands. With the first detection of transatlantic fringes at 147 GHz [@Kri], the present world record has been established at 18 microarceseconds, which is just few times the angular size of the Schwarzschild radius of the black hole at the Galactic center. Further improvements can be obtained at sub-mm wavelengths.
A very important step for absolute sensitivity in the infrared band is the Next Generation Space Telescope (NGST), which will operate in the wavelength interval $0.6 - 27 \mu m$. At $\lambda=2.2 \mu m$, with a 3 hours exposure, it will be able to detect a flux corresponding to 32 magnitudes, just enough to catch the first S2 relativistic image. If coupled with other instruments still to come, then the detection of relativistic images would become a next future observational frontier. Furthermore, we cannot forget that the Galactic center is surrounded by a crowded clusters of stars, which can probably yield even better candidates than S2 for strong field gravitational lensing.
We close this section with some comments on possible time delay measurements. Since S2 is close to the center of Galaxy, but never aligned behind it, we can estimate the time delay between the first relativistic image and the direct image using Eq. (\[TDSF-WF\]). At the luminosity peak epoch, this delay amounts to 5.95 hours, thus being comparable to the exposure time. In the particular case of S2, no intrinsic variability has been reported up to now in the observed spectral bands. However, the principle of time delay measures could be applied to eventual new candidates or to S2 itself if any variability is detected in its flux. A measure of such a short time delay, requires a very good sampling over the whole period of variability. The precision of such estimate is limited by the effort undertaken to sample the two light curves to be compared. With a good knowledge of the orbit (in order to correctly estimate $\gamma$) and a sufficient sampling, which would depend on the future facilities and the characteristics of the source, it could be possible to establish the distance to the center of the Galaxy on a solid direct measurement.
Kerr gravitational lensing
==========================
What we have said up to now is true for spherically symmetric black holes (including the estimates on S2 gravitational lensing). In the case of spinning black holes, everything becomes much more complicated, but some very important facts already emerge from the analysis of quasi-equatorial motion, which has already been done analytically [@Boz2]. In fact, it is evident that the caustics are no longer aligned along the optical axis, but drift following the sense of rotation of the black hole. In addition, they are no longer point-like, but acquire finite extension. To get an idea of the importance of these changes, we can see that for $a=0.1$ the first retro-lensing caustic drifts from the optical axis by an angle of $10^\circ$ and acquires an extension of $2.4^\circ$ on the equatorial plane. The existence of extended caustics is accompanied by the formation of pairs of additional images when the source enters one of the caustics. These images are missed in the quasi-equatorial approach since they rather live far from the equatorial plane. A very important consequence is the fact that high magnifications can be attained much more easily. In fact, while in spherically symmetric metrics the caustics are point-like and coincide with the optical axis, in the Kerr metric they are distributed in all directions around the lens and have finite extension, so that it is much easier for a source to lie within a high amplification region for some relativistic images. Another complication comes because the usual hierarchy among relativistic images, which just follows the winding number, can be completely upset by the fact that the source lies in a caustic affecting images with higher winding number.
Summing up, the Kerr black hole lensing is much richer than spherically symmetric black hole lensing. It is also much more promising from a phenomenological point of view, since it is easier to have brighter images. And of course it is reasonable to expect that astrophysical black holes are born with a non-negligible spin. All these statements point to the importance of the investigation of the Kerr black hole lensing in the general case.
Kerr black hole lensing with S2
-------------------------------
As a practical example of the modifications that an eventual spin of the black hole in Sgr A\* would have on the light curve of the images of S2, let us calculate the magnification at the moments when S2 crosses the plane of the Galaxy. It is reasonable to assume that if the central black hole is spinning, its equatorial plane is very close to the rotation plane of the Galaxy. Using the orbital parameters of S2, we see that it has crossed the galactic plane in $t_1=2003.12$, where it had $\gamma=77.2^\circ$ and $D_{LS}=2.74\times 10^{-3}pc$, being east of the black hole, slightly behind it. It will cross again the galactic plane in $t_2=2018.00$ with $\gamma=102.5^\circ$ and $D_{LS}=6.93\times
10^{-4}pc$, being west of the black hole, slightly before it. For these crossing times, we can precisely calculate the magnifications of the equatorial images using the formula by @Boz2
$$\begin{aligned}
&\mu=&\frac{D_{OS}^2}{D_{OL}D_{LS}} \cdot\nonumber \\
&&\frac{\sqrt{u^2-a^2} u}{\overline{a} \left| \sqrt{u^2-a^2}
(D_{OL}+D_{LS}) C -D_{OL}D_{LS} S \right|}, \label{muKerr}\end{aligned}$$
where $u=\theta R_{Sch}/D_{OL}$, $C=\cos \overline{\phi}$, $S=\sin
\overline{\phi}$ and $\overline{\phi}$ is the phase of the photon oscillations on the equatorial plane, expressed by $$\overline{\phi}=-\frac{\overline{b}-\gamma}{\overline{a}}+\hat{b}.$$
The coefficients $\overline{a}$, $\overline{b}$ and $\hat b$ are all functions of the spin of the black hole $a$, (see @Boz2 for the whole derivation).
In Fig. \[Fig Kerr\], we plot the magnifications of the images as a function of the black hole spin, from $a=0$ (Schwarzschild) to $a=0.5$ (extremal Kerr black hole in our normalization). In Figs. \[Fig Kerr\]a and \[Fig Kerr\]b, we plot the image magnifications for the past time $t_1$, while in Fig. \[Fig Kerr\]c and \[Fig Kerr\]d we plot the image magnifications for the future time $t_2$. In the left figures we plot the magnifications of the images with negative parity. The brightest of them gives the most significant contribution (it is this image that is going to be most likely observable). The magnifications of the images with increasing winding number are plotted altogether. The higher the winding number, the lower the magnification. On the right figures we represent the magnification of the positive parity images. The most relevant of these was also represented in Figs. \[Fig S2mu\] and \[Fig S2ing\], where it was however fainter than the brightest negative parity image. Increasing the winding number, the magnification falls down. The images in Figs. \[Fig Kerr\]a and \[Fig Kerr\]d are formed by retrograde photons, i.e. winding oppositely to the sense of rotation of the black hole. The images in Figs. \[Fig Kerr\]b and \[Fig Kerr\]c are formed by co-rotating photons.
Now let us comment on the outcome of these figures. Increasing the spin of the black hole, we see that co-rotating images become brighter and retrograde images become fainter. The most striking features are the peaks in the magnification that occur at definite values of the black hole spin. Varying the black hole spin, the caustic points drift all around the trigonometric interval and for some values they meet the angular position of S2. Around these values, the magnification can become significantly higher than the normal value. We also see that the first caustics, corresponding to low winding numbers, move slowly and thus cross the position of S2 less times. The caustics corresponding to higher winding numbers move quickly. For this reason the fainter images meet caustics more often in the interval of variability of $a$.
If the true value of the spin of the central black hole is such that one of the images is close to a caustic point, that image will be highly amplified. By these plots, we see that in year 2018.00, for the particular value $a\sim 0.26$ the second negative parity image will overcome the brighter one.
Besides the luminosity of the relativistic images, the presence of a spin for the black hole also affects their position. In fact, image formed by retrograde photons appear farther from the central black hole up to 1.5 times, while co-rotating photons appear closer to it up to 2.6 times.
Finally, it must be kept in mind that since the caustics have extended areas, these caustic points are actually just the cusps of these caustics. This means that the images we are plotting in Fig. \[Fig Kerr\] change their parity at every caustic crossing. Global parity conservation is assured by creation or destruction of non equatorial images of the same parity of the image before the crossing. For these images we do not have an analytical treatment at the moment and all we can say is that they will appear at each caustic crossing and disappear when the source steps out of a caustic.
When a complete analytical treatment of the Kerr black hole lensing is available, not only we will have a clearer idea of the dynamics of all images, but we will be able to use the whole information about position and luminosity of all visible relativistic images to measure the spin of the black hole. We can imagine that such an estimate would be much cleaner and simpler than those relying on poorly understood models of the accretion disk or gas surrounding the central black hole, which are subject to very complicated physical processes and dynamics. However, for the details of such a fascinating measure, we still have to wait for further analytical progresses on the general Kerr lensing.
Sun retro-lensing by a Kerr black hole
--------------------------------------
Another context where Kerr black hole lensing was discussed was retro-lensing of the Sun by a nearby Kerr black hole [@DepRM]. As a first step, it was considered a geometry such that the whole event occurred within the equatorial plane of the black hole. Using the formulae by @Boz2, the positions of the images were correctly calculated. However, the magnification was estimated using the formula by @HolWhe, which was derived for a Schwarzschild black hole (being also inaccurate for central events as shown in Sect. 2). It is well known in different contexts [@GouLoe; @Bozp] that the modification to the magnification map come firstly from the change to the Jacobian function, due to the change in the lens model, and only secondly from the displacements of the image positions due to the modifications in the lens equation, which can be neglected in a first extent. As a consequence of this simplification @DepRM found no significant deviation from Schwarzschild retro-lensing. However, by the study of the equatorial case, we learn that the caustics drift and acquire finite extension, so that we expect that things are not so simple.
For the configuration considered in @HolWhe and @DepRM, we have strong magnification only if the source is close to a caustic. Very small misalignments can already kill the images to very small luminosities (it is sufficient to take a look at Fig. \[Fig Micro\] to get convinced; for quantitative estimates see @HolWhe). So, it is sufficient that the relevant caustic for retro-lensing moves out of the Earth orbit in order that the possibility of Sun retro-lensing disappears at all. With a black hole at distance 0.01 pc, this happens if the black hole spin is $a>0.00027$: quite a small value. If we have a black hole with a spin less than this, then the retro-lensing caustic area would be much smaller then the Sun angular radius as seen from the black hole, being effectively treatable as point-like. Then everything would work more or less like in the Schwarzschild case, with corrections to the magnification curves of the order of $\delta\mu/\mu \simeq a$, i.e. below the precision of SFL approximation.
Conclusions
===========
Gravitational lensing in strong fields is a very interesting subject from the theoretical point of view. Potentially it is a very appealing phenomenon which is completely embedded within the full general relativity. It would thus be an exceptional probe for physics in the regions close to the event horizons of black holes and would give very important feedback on the correct theory of gravitation. This justifies the very strong efforts that have been done by several groups to find possible candidate sources and lenses which could make this fascinating phenomenon manifest. The astrophysical cases investigated up to now are such that the relativistic images should be “almost” observable or should become observable within not many years. The most important problem is whether these theoretical configurations are likely or not. The case of S2, on the contrary, is a concrete case where definite predictions for the image luminosities can be done.
In this work we have extended the analytical framework of the Strong Field Limit analysis to cover all the possible geometric configurations, giving up any small angle approximations which are not automatically encoded in the tracks of the light rays. In this way, we have been capable to revisit the so called retro-lensing, in particular the Sun retro-lensing proposed by @HolWhe, with more valid mathematical instruments. Moreover, we can adequately cover any intermediate case, such that of the star S2 [@DepS2]. For the first relativistic image of this star, we predict an epoch of maximal luminosity (better than 32 magnitudes) between the end of year 2017 and the first part of 2018.
Using previous work on equatorial Kerr lensing [@Boz2], we have guessed how the presence of extended caustics and their drift from the optical axis can affect the brightness of the relativistic images of S2 at the epoch of its crossing through the galactic plane. We have also put upper limits on the black hole spin in Kerr retro-lensing of the Sun in order to have still significant images.
As a final remark, we can say that strong field gravitational lensing opens really fascinating perspectives, like testing the General Relativity, measuring the distance, the spin or other physical parameters of the black hole in the Galactic center and in the centers of other galaxies. The nice luminosity estimates for S2 encourage to look for more stars orbiting very close to the central black hole as new potential candidate sources for such an amazing phenomenon.
We thank the CERN theory department for hospitality. We also thank Gaetano Scarpetta for comments on the manuscript, Francesco De Paolis and Achille Nucita for useful discussions, and our referee for some nice suggestions.
——————————————————————
Atkinson, R.D. 1965, , 70, 517
Bhadra, A. 2003, 67, 103009
Bozza, V. 1999, , 348, 311
Bozza, V. 2002, , 66, 103001
Bozza, V. 2003, , 67, 103006
Bozza, V., Capozziello, S., Iovane, G. & Scarpetta, G. 2001, Gen. Rel. and Grav., 33, 1535
Bozza, V., & Mancini, L. 2004, Gen. Rel. and Grav., 36, 435
Chandrasekhar, S. 1983, [*Mathematical Theory of Black Holes*]{}, Clarendon Press, Oxford
Cunningham, C.T., & Bardeen, J.M. 1973, , 183, 237
Dabrowski, M.P., & Schunck, F.E. 2000, , 535, 316
Darwin, C. 1959, Proc. of the Royal Soc. of London, 249, 180
De Paolis, F., Geralico, A., Ingrosso, G. & Nucita, A.A. 2003, , 409, 809
De Paolis, F., Geralico, A., Ingrosso, G., Nucita, A.A. & Qadir, A. 2004, , 415, 1
Eiroa, E.F., Romero, G.E., & Torres, D.F. 2002, , 66, 024010
Eiroa, E.F. & Torres, D.F. 2003, gr-qc/0311013
Frittelli, S., Kling, T.P., & Newman, E.T. 2000, , 61, 064021
Frittelli, S., & Newman, E.T. 1999, , 59, 124001 Gould, A., & Loeb, A. 1992, , 396, 104
Holz, D.E. & Wheeler, J.A. 2002, , 578, 330
Krichbaum, T.P., Graham, D.A., Alef, W., et al. 2002, Proc. of the 6th European VLBI Network Symposium
Luminet, J.P. 1979, , 75, 228
Nemiroff, R.J. 1993, Amer. Jour. Phys., 61, 619
Ohanian 1987, Amer. Jour. Phys., 55, 428
Perlick, V. 2003, gr-qc/0307072
Petters, A.O. 2003, , 338, 457
Rieke, G.H., Rieke, M.J., & Paul, A.E. 1989, , 336, 752
Schödel, R., Ott, T., Genzel, R., et al. 2002, , 419, 694
Schödel, R., Ott, T., Genzel, R., et al. 2003, , 596, 1015
Vazquez, S.E., & Esteban, E.P. 2003, gr-qc/0308023.
Viergutz, S.U. 1993, , 272, 355
Virbhadra, K.S., & Ellis, G.F.R. 2000, , 62, 084003
Wex, N., Gil, J. & Sendyk, M. 1996, , 311, 746
Witt, H.J., & Mao, S. 1994, , 430, 505
[^1]: ARISE web page: arise.jpl.nasa.gov
[^2]: MAXIM web page: maxim.gsfc.nasa.gov
|
`arXiv:0901.0411`\
`DAMTP-2009-1`
**Two-loop Integrability of Planar\
$\mathcal{N}=6$ Superconformal Chern-Simons Theory**
<span style="font-variant:small-caps;">Benjamin I. Zwiebel</span>
*DAMTP, Centre for Mathematical Sciences\
University of Cambridge\
Wilberforce Road, Cambridge CB3 0WA, UK*
`b.zwiebel@damtp.cam.ac.uk`
**Abstract**
Bethe ansatz equations have been proposed for the asymptotic spectral problem of $AdS_4/CFT_3$. This proposal assumes integrability, but the previous verification of weak-coupling integrability covered only the ${\mathfrak{su}}(4)$ sector of the ABJM gauge theory. Here we derive the complete planar two-loop dilatation generator of $\mathcal{N}=6$ superconformal Chern-Simons theory from ${\mathfrak{osp}}(6|4)$ superconformal symmetry. For the ${\mathfrak{osp}}(4|2)$ sector, we prove integrability through a Yangian construction. We argue that integrability extends to the full planar two-loop dilatation generator, confirming the applicability of the Bethe equations at weak coupling. Further confirmation follows from an analytic computation of the two-loop twist-one spectrum.
Introduction
============
The $\mathcal{N}=6$ superconformal $U(N) \times U(N)$ Chern-Simons theory of Aharony, Bergman, Jafferis, and Maldacena has a ’t Hooft limit in which the dual description reduces to type IIA string theory on $AdS_4 \times CP^3$ [@Aharony:2008ug]. This limit is $N \to \infty$ and Chern-Simons level $k \to \infty$ with coupling $\lambda=N/k$ held fixed. Excitingly, the asymptotic spectral problem of this $AdS_4/CFT_3$ duality may be solvable by an ${\mathfrak{osp}}(6|4)$ Bethe ansatz, as first argued by Minahan and Zarembo based on their two-loop ($\mathcal{O}(\lambda^2)$) gauge theory calculation[@Minahan:2008hf] (also see [@Bak:2008cp]). Paralleling the case of $AdS_5/CFT_4$ [@Minahan:2002ve; @Beisert:2003tq; @Bena:2003wd; @Arutyunov:2004vx; @Staudacher:2004tk; @Beisert:2005tm; @Janik:2006dc; @Arutyunov:2006ak; @Beisert:2006ez], this apparent integrability is likely to lead to a better understanding of $AdS_4/CFT_3$ and the relationship between gauge/string duality and integrability.
There has been much related activity. Integrability of classical string theory on $AdS_4 \times CP^3$ has been proved for a subsector described by an $OSP(6|4)/(U(3) \times SO(1,3))$ coset sigma model [@Arutyunov:2008if; @Stefanski:2008ik]. The conjectured weak-coupling Bethe equations [@Minahan:2008hf] have been extended to an all-loop proposal [@Gromov:2008qe][^1], with the corresponding two-magnon S-matrix obtained in [@Ahn:2008aa]. This all-loop Bethe ansatz has passed strong-coupling tests in a $SU(2) \times SU(2)$ sector[@Astolfi:2008ji; @Sundin:2008vt], and seeming disagreement with semi-classical string computations [@McLoughlin:2008ms; @Alday:2008ut; @Krishnan:2008zs] is eliminated by a one-loop correction [@McLoughlin:2008he] to the magnon dispersion relation. Nonetheless, discrepancies between one-loop strong-coupling string calculations and the Bethe ansatz remain [@McLoughlin:2008he], and the classical integrability of the *complete* $AdS_4 \times CP^3$ Green-Schwarz action [@Gomis:2008jt] is unverified.
This work, however, will focus on the weak-coupling planar gauge theory. As is the case for $\mathcal{N}=4$ SYM, the spectral problem of computing anomalous dimensions is usefully formulated in terms of a spin chain. Local gauge-invariant operators are mapped to spin-chain states, and the anomalous part of the dilatation generator is mapped to a spin-chain Hamiltonian whose eigenvalues are the spectrum of anomalous dimensions. Then, if this Hamiltonian is integrable, one can replace the eigenvalue problem with a dramatically simpler problem of solving a system of Bethe equations, see [@Beisert:2004ry] for a review.
For the ABJM gauge theory, [@Minahan:2008hf] calculated the leading-order two-loop planar dilatation generator or Hamiltonian in the ${\mathfrak{su}}(4)$ sector, confirming two-loop integrability within that sector. The ${\mathfrak{su}}(4)$ sector corresponds to local operators composed of alternating scalars $\phi_i$ and $\bar{\phi}^i$, $i=1,\ldots4$, or a spin chain with alternating sites occupied by a spin transforming in the $\mathbf{4}$ or $\mathbf{\bar{4}}$ of ${\mathfrak{su}}(4)$. The spin chain is alternating because the $\phi_i$ transform as $(\mathbf{N}, \mathbf{\bar{N}})$ with respect to the $U(N) \times U(N)$ gauge group, while the $\bar{\phi}^i$ transform as $(\mathbf{\bar{N}},\mathbf{N})$.
To describe the full set of local operators, we must extend these conjugate finite-dimensional representations of the ${\mathfrak{su}}(4) = {\mathfrak{so}}(6)$ $R$ symmetry to two highest-weight irreducible representations of the $\mathcal{N}=6$ superconformal Lie algebra, ${\mathfrak{osp}}(6|4)$. In addition to the ${\mathfrak{su}}(4)$ scalars and their supersymmetric partners, $\bar{\psi}^i$ or $\psi_i$, these representations or modules includes arbitrarily many (symmetrized traceless) covariant derivatives acting on each scalar or fermion. The validity of the weak-coupling ${\mathfrak{osp}}(6|4)$ Bethe equation conjecture requires the significant assumption that two-loop integrability extends to the alternating spin chain with sites hosting these infinite-dimensional ${\mathfrak{osp}}(6|4)$ modules.
Before working on extending the two-loop ${\mathfrak{su}}(4)$ sector dilatation generator to the full ${\mathfrak{osp}}(6|4)$ spin chain, it is useful to review key steps used to compute the leading-order one-loop planar dilatation generator of $\mathcal{N}=4$ SYM. For that theory, the one-loop dilatation generator acts on two (identical) ${\mathfrak{psu}}(2,2|4)$ modules. An important observation is that the irreducible multiplets in the tensor product of two modules match up one-to-one with the corresponding multiplets in the ${\mathfrak{sl}}(2)$ sector[^2]. Combined with superconformal invariance, this enabled the lift of the ${\mathfrak{sl}}(2)$ sector dilatation generator[^3] to the complete one-loop dilatation generator [@Beisert:2003jj]. In fact, superconformal symmetry completely fixes the leading-order Hamiltonian (up to normalization); the ${\mathfrak{sl}}(2)$ sector one-loop dilatation generator was later computed [@Beisert:2004ry] just by requiring closure of the residual superconformal symmetry algebra of the sector, which includes an extra ${\mathfrak{su}}(1|1)$ symmetry generated by “hidden” supercharges.
We follow a similar path for the planar ABJM theory, using superconformal symmetry directly[^4] to compute the leading-order dilatation generator up to normalization[^5]. Again the first step involves restricting to a sector; we find the leading-order ${\mathfrak{osp}}(4|2)$ sector dilatation generator using residual superconformal symmetry. This sector corresponds to the set of local operators that are $1/12$ BPS at $\lambda=0$ with respect to a fixed pair of supercharges, which we label $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$. Similar to the case of the ${\mathfrak{sl}}(2)$ sector of $\mathcal{N}=4$ SYM, these supercharges generate an important additional ${\mathfrak{su}}(1|1)$, which commutes with ${\mathfrak{osp}}(4|2)$. By definition, all states in this sector are annihilated by these supercharges at $\lambda=0$, so we then have $$\delta {\mathfrak{D}}_2 = {\{\hat{{\mathfrak{Q}}}_1,\hat{{\mathfrak{S}}}_1\}}, \label{eq:introd2}$$ where subscripts refer to the order in $\lambda$. In other words, the two-loop dilatation generator within this sector is fixed once we know the *one-loop* corrections to these supercharges. Furthermore, requiring vanishing commutators with ${\mathfrak{osp}}(4|2)$ generators at leading order fixes $\hat{{\mathfrak{Q}}}_1$ and $\hat{{\mathfrak{S}}}_1$ up to overall normalization, giving the two-loop dilatation generator (the normalization can be identified from [@Minahan:2008hf]).
Superconformal invariance enables us to lift the leading-order dilatation generator within the ${\mathfrak{osp}}(4|2)$ sector to the full theory. However, in comparison to the analogous $\mathcal{N}=4$ SYM calculation, additional intermediate steps are needed. This is largely because here the leading-order Hamiltonian acts on three adjacent sites. The expression that immediately follows from the right side of (\[eq:introd2\]) is not in a convenient form for the lift. Instead we first prove integrability within this sector by constructing an ${\mathfrak{osp}}(4|2)$ Yangian that commutes with the Hamiltonian (The construction of the Yangian symmetry of the one-loop dilatation generator of $\mathcal{N}=4$ SYM was completed in [@Dolan:2003uh; @Dolan:2004ps]). This identifies the Hamiltonian as one that follows from an ${\mathfrak{osp}}(4|2)$ R-matrix construction. Now the Hamiltonian is written in terms of projectors onto irreducible modules appearing in the tensor products of two one-site modules, and now the lift becomes straightforward. Superconformal invariance, combined with some analysis of tensor products of one-site modules, allows us to lift this second expression for the ${\mathfrak{osp}}(4|2)$ Hamiltonian to the full two-loop ${\mathfrak{osp}}(6|4)$ spin-chain Hamiltonian.
In this work we do not explicitly verify that the complete Hamiltonian has an ${\mathfrak{osp}}(6|4)$ Yangian symmetry. However, it takes the exact form of an integrable alternating ${\mathfrak{osp}}(6|4)$ spin-chain Hamiltonian, assuming the existence of an ${\mathfrak{osp}}(6|4)$ R-matrix (This is parallel to a result of [@Beisert:2003yb] for $\mathcal{N}=4$ SYM). Combined with the explicit proof within the ${\mathfrak{osp}}(4|2)$ sector, this is convincing evidence that the complete two-loop planar model is integrable and that the two-loop spectral problem is solved by the Bethe ansatz.
In Section \[sec:alg\] we introduce the ABJM ${\mathfrak{osp}}(6|4)$ spin-chain model and the restriction to the ${\mathfrak{osp}}(4|2)$ sector, and Section \[sec:oneloop\] derives the two-loop dilatation generator for this sector. Here we also compute analytically the twist-one spectrum from the dilatation generator, which matches the Bethe ansatz prediction. The Yangian proof of integrability appears in Section \[sec:yangian\], and the following section computes the corresponding R-matrix expression for the Hamiltonian. Section \[sec:lift\] gives the unique lift to the complete two-loop planar dilatation generator, and we conclude with the following section.
The ${\mathfrak{osp}}(6|4)$ spin chain and its ${\mathfrak{osp}}(4|2)$ subsector \[sec:alg\]
============================================================================================
The ABJM gauge theory’s $\mathcal{N}=6$ superconformal symmetry, which has been verified in [@Benna:2008zy; @Bandres:2008ry], corresponds to the ${\mathfrak{osp}}(6|4)$ Lie algebra. After reviewing this algebra we introduce the corresponding module that is used for building the spin-chain description of gauge-invariant local operators. We then explain the restriction to the ${\mathfrak{osp}}(4|2)$ sector, and this sector’s algebra and spin module. Finally, we introduce a light-cone superspace basis for the ${\mathfrak{osp}}(4|2)$ module. For a recent general analysis of representations of the three dimensional superconformal groups, see [@Dolan:2008vc].
The complete algebra and spin module
------------------------------------
For the ${\mathfrak{osp}}(6|4)$ algebra generators we use one ${\mathfrak{su}}(2)$ spinor index, $\alpha, \beta = 1, 2$ and an ${\mathfrak{su}}(4)$ index $i, j=1, 2, 3, 4$. The ${\mathfrak{sp}}(4)$ elements are ${\mathfrak{su}}(2)$ Lorentz generators ${\mathfrak{L}}^\alpha{}_\beta$, translation and special conformal generators ${\mathfrak{P}}_{\alpha\beta}={\mathfrak{P}}_{\beta\alpha}$ and ${\mathfrak{K}}^{\alpha\beta}={\mathfrak{K}}^{\beta\alpha}$, and the dilatation generator ${\mathfrak{D}}$. The remaining bosonic elements are ${\mathfrak{su}}(4)$ ${\mathfrak{R}}$-symmetry generators ${\mathfrak{R}}^i{}_j$. Additionally, there are 24 supercharges with $${\mathfrak{Q}}_{ij,\alpha} = - {\mathfrak{Q}}_{ji,\alpha}, \quad {\mathfrak{S}}^{kl,\beta} = - {\mathfrak{S}}^{lk,\beta}.$$ The nonvanishing commutators with the ${\mathfrak{sp}}(4)$ generators are $$\begin{aligned}
{[{\mathfrak{L}}^\alpha{}_\beta,{\mathfrak{L}}^\gamma{}_\delta]} &= \delta^\alpha_\delta {\mathfrak{L}}^\gamma{}_\beta - \delta^\gamma_\beta {\mathfrak{L}}^\alpha{}_\delta, & {[{\mathfrak{L}}^\alpha{}_\beta,{\mathfrak{P}}_{\gamma\delta}]}& = 2 \delta^\alpha_{\{\gamma} {\mathfrak{P}}_{\delta\}\beta} - \delta^\alpha_\beta {\mathfrak{P}}_{\gamma\delta},
\notag \\
{[{\mathfrak{L}}^\alpha{}_\beta,{\mathfrak{K}}^{\gamma\delta}]}& = -2 \delta_\beta^{\{\gamma} {\mathfrak{K}}^{\delta\}\alpha} + \delta^\alpha_\beta {\mathfrak{K}}^{\gamma\delta}, &
{[{\mathfrak{L}}^\alpha{}_\beta,{\mathfrak{Q}}_\gamma]} &= \delta^\alpha_\gamma {\mathfrak{Q}}_\beta -{{{\textstyle\frac{1}{2}}}}\delta^\alpha_\beta {\mathfrak{Q}}_\gamma,
\notag \\
{[{\mathfrak{L}}^\alpha{}_\beta,{\mathfrak{S}}^\gamma]} & = -\delta^\gamma_\beta {\mathfrak{S}}^\alpha +{{{\textstyle\frac{1}{2}}}}\delta^\alpha_\beta {\mathfrak{S}}^\gamma, & {[{\mathfrak{K}}^{\alpha \beta},{\mathfrak{P}}_{\gamma\delta}]} & = 4\delta^{\{\alpha}_{\{\gamma} {\mathfrak{L}}^{\beta\}}{}_{\delta\}} + 4\delta^{\{\alpha}_{\{\gamma} \delta^{\beta\}}_{\delta\}} {\mathfrak{D}},
\notag \\
{[{\mathfrak{K}}^{\alpha \beta},{\mathfrak{Q}}_{kl, \gamma}]} & = {{{\textstyle\frac{1}{2}}}}\varepsilon_{klij}\big(\delta^\alpha_\gamma {\mathfrak{S}}^{ij, \beta} + \delta^\beta_\gamma {\mathfrak{S}}^{ij, \alpha} \big), & &
\notag \\
{[{\mathfrak{P}}_{\alpha \beta},{\mathfrak{S}}^{kl,\gamma}]} & = -{{{\textstyle\frac{1}{2}}}}\varepsilon^{klij}\big(\delta_\alpha^\gamma {\mathfrak{Q}}_{ij,\beta} + \delta_\beta^\gamma {\mathfrak{Q}}_{ij,\alpha} \big), &&\end{aligned}$$ and the dimensions of $\{{\mathfrak{P}}, {\mathfrak{K}}, {\mathfrak{Q}}, {\mathfrak{S}}\}$ are $\{1,-1,{{{\textstyle\frac{1}{2}}}},-{{{\textstyle\frac{1}{2}}}}\}$.
The commutators with the ${\mathfrak{so}}(6) = {\mathfrak{su}}(4)$ ${\mathfrak{R}}$ generators are $$\begin{aligned}
{[{\mathfrak{R}}^i{}_j,{\mathfrak{R}}^k{}_l]} & = \delta^i_l{\mathfrak{R}}^k{}_j - \delta^k_j{\mathfrak{R}}^i{}_l,& {[{\mathfrak{R}}^i{}_j,{\mathfrak{Q}}_{kl}]} &=2 \delta^i_{[l} {\mathfrak{Q}}_{k]j}-{{{\textstyle\frac{1}{2}}}}\delta^i_j {\mathfrak{Q}}_{kl}, \notag \\
{[{\mathfrak{R}}^i{}_j,{\mathfrak{S}}^{kl}]} & = -2 \delta_j^{[l} {\mathfrak{S}}^{k]i}+{{{\textstyle\frac{1}{2}}}}\delta^i_j {\mathfrak{S}}^{kl}. & &\end{aligned}$$ Finally, the nonvanishing anticommutators are $$\begin{aligned}
{\{{\mathfrak{Q}}_{ij,\alpha},{\mathfrak{Q}}_{kl,\beta}\}} & = -\varepsilon_{ijkl} {\mathfrak{P}}_{\alpha\beta}, &
{\{{\mathfrak{S}}^{ij,\alpha},{\mathfrak{S}}^{kl,\beta}\}} & = -\varepsilon^{ijkl} {\mathfrak{K}}^{\alpha\beta},
\notag \\
{\{{\mathfrak{Q}}_{ij,\beta},{\mathfrak{S}}^{kl,\gamma}\}} & = 4 \delta^\gamma_\beta \delta^{[k}_{[i}{\mathfrak{R}}^{l]}{}_{j]} + 2 \delta^k_{[j}\delta^l_{i]}{\mathfrak{L}} ^\gamma{}_\beta + 2 \delta^k_{[j}\delta^l_{i]}\delta^\gamma_\beta {\mathfrak{D}}. && \label{eq:anticommutators}\end{aligned}$$ There are multiple choices of positive roots of the Lie algebra. For instance, we can choose the raising generators as $${\mathfrak{L}}^2{}_1, \quad {\mathfrak{K}}, \quad {\mathfrak{R}}^i{}_j, \, i > j, \quad {\mathfrak{S}}.$$ and the Hermitian conjugate lowering generators $${\mathfrak{L}}^1{}_2, \quad {\mathfrak{P}}, \quad {\mathfrak{R}}^i{}_j, \, i < j, \quad {\mathfrak{Q}}.$$ The Cartan generators are then the diagonal generators of ${\mathfrak{L}}$ and ${\mathfrak{R}}$ (the traceless conditions mean that only $1+3$ of these are independent) and the dilatation generator.
The spin chain has alternating highest-weight modules. As stated in the introduction, this is because the matter fields are in bifundamental representations of the $U(N) \times U(N)$ gauge group. We also note that ABJM gauge theory has an additional ${\mathfrak{u}}(1)$ symmetry, under which the modules have alternating charge $\pm 1$. The first representation $\mathcal{V}_\phi$ has highest-weight element ${\mathopen{\big|}\phi_1^{(0,0)}\mathclose{\bigr \rangle}}$ and consists of $${\mathopen{\big|}\phi_i^{(n_1, n_2)}\mathclose{\bigr \rangle}}, \quad {\mathopen{\big|}(\bar{\psi}^i)^{(n_1, n_2)}\mathclose{\bigr \rangle}}, \quad n_i =0, 1, 2, \ldots$$ where the superscripts correspond to the number of ${\mathfrak{su}}(2)$ indices carried by covariant derivatives acting on the fields of the ABJM model, $\phi$ and $\bar{\psi}$. All Lorentz indices are symmetrized on individual module elements, as needed for an irreducible representation. Alternate sites of the chain, instead host a representation $\mathcal{V}_{\bar \phi}$ with highest-weight element ${\mathopen{\big|}(\bar{\phi}^4)^{(0,0)}\mathclose{\bigr \rangle}}$ spanned by $${\mathopen{\big|}(\bar{\phi}^i)^{(n_1, n_2)}\mathclose{\bigr \rangle}}, \quad {\mathopen{\big|}\psi_i^{(n_1, n_2)}\mathclose{\bigr \rangle}}, \quad n_i =0, 1, 2, \ldots$$ The ${\mathfrak{su}}(4)$ sector corresponds to the $\phi$ and $\bar{\phi}$ states with all $n_i=0$. Also, of course, $n_1+n_2$ is even (odd) for bosons (fermions).
The ${\mathfrak{osp}}(6|4)$ variations were given in [@Gaiotto:2008cg; @Hosomichi:2008jb]. We will not need the explicit action of the entire ${\mathfrak{osp}}(6|4)$ generators at leading order, which can be identified (up to a physically irrelevant choice of basis) straightforwardly by requiring closure of the algebra. However, a few details are useful. The diagonal Lorentz generators and the classical dilatation generator act (independently of ${\mathfrak{R}}$ symmetry indices) as \^1\_1[X\^[(n\_1, n\_2)]{}]{} [[&=&]{}]{}-\^2\_2[X\^[(n\_1, n\_2)]{}]{} = (n\_1-n\_2) [X\^[(n\_1, n\_2)]{}]{}, [\
]{}\_0 [X\^[(n\_1, n\_2)]{}]{} [[&=&]{}]{}(n\_1+n\_2+1)[X\^[(n\_1, n\_2)]{}]{}. \[eq:lorentanddilatationaction\] All unbarred elements transform in the $\mathbf{4}$ of ${\mathfrak{su}}(4)$ and barred elements transform (naturally) in the $\mathbf{\bar{4}}$, $$\label{eq:rsymaction}
{\mathfrak{R}}^i{}_j {\mathopen{\big|}X_k\mathclose{\bigr \rangle}} = \delta^i_k {\mathopen{\big|}X_j\mathclose{\bigr \rangle}} - {{{\textstyle\frac{1}{4}}}}\delta^i_j {\mathopen{\big|}X_k\mathclose{\bigr \rangle}}, \quad {\mathfrak{R}}^i{}_j {\mathopen{\big|}\bar{X}^k\mathclose{\bigr \rangle}} = -\delta^k_j {\mathopen{\big|}\bar{X}^i\mathclose{\bigr \rangle}} + {{{\textstyle\frac{1}{4}}}}\delta^i_j {\mathopen{\big|}\bar{X}^k\mathclose{\bigr \rangle}}.$$ Also, schematically we have the ${\mathfrak{R}}$-symmetry index dependence of the leading-order supercharges, $$\begin{aligned}
{\mathfrak{Q}}_{ij}{\mathopen{\big|}X_k\mathclose{\bigr \rangle}} & \sim \varepsilon_{ijkm}{\mathopen{\big|}\bar{Y}^m\mathclose{\bigr \rangle}}, & {\mathfrak{Q}}_{ij}{\mathopen{\big|}\bar{X}^k\mathclose{\bigr \rangle}} & \sim \delta^k_{[i}{\mathopen{\big|}Y_{j]}\mathclose{\bigr \rangle}},
\notag \\
{\mathfrak{S}}^{ij}{\mathopen{\big|}\bar{X}^k\mathclose{\bigr \rangle}} & \sim \varepsilon^{ijkm}{\mathopen{\big|}Y_m\mathclose{\bigr \rangle}}, & {\mathfrak{S}}^{ij}{\mathopen{\big|}X_k\mathclose{\bigr \rangle}} & \sim \delta_k^{[i}{\mathopen{\big|}\bar{Y}^{j]}\mathclose{\bigr \rangle}}. \label{eq:qaction}\end{aligned}$$ Of course, there is also dependence on the Lorentz indices, but we will not need these precise factors for the full algebra. Simply we note that all interactions allowed by quantum numbers appear with nonzero coefficient. So ${\mathfrak{S}}^\alpha$ has nonvanishing action only on module elements with $n_{\alpha}>0$.
For $\lambda=0$ the spin-chain states transform in the tensor product of the single-site ${\mathfrak{osp}}(6|4)$ representations, but beyond leading order they transform in a deformed representation; the ${\mathfrak{osp}}(6|4)$ generators act on the spin chain via interactions that couple multiple sites. Beside the manifest ${\mathfrak{R}}$ and ${\mathfrak{L}}$ symmetry generators, all generators receive corrections for $\lambda \neq 0$. Perturbatively, these interactions act on an increasing number of sites with each order in $\lambda$. Counting powers of the coupling constant implies that at $\mathcal{O}(\lambda)$, interactions can act on up to a total of four sites (e.g. two initial and two final sites), and up to six sites at $\mathcal{O}(\lambda^2)$. In the planar limit, which is our focus, these interactions act on adjacent modules.
Finally, ${\mathfrak{osp}}(6|4)$ has a quadratic Casimir $J^2$, \[eq:osp64jsquared\] J\^2 [[&=&]{}]{}([\[\_[ij,]{},\^[ij, ]{}\]]{} - 2 \^i\_j \^j\_i + 2 \^\_\^\_+ 4 \^2 - [{\_,\^}]{}). On highest-weight states, $J^2$ simplifies to J\^2 [[&=&]{}]{}(D(D+3) + s(s+2) + 3 R\^1\_1+2 R\^2\_2 + R\^3\_3 - \_[i=1]{}\^4 (R\^i\_i)\^2 ) [\
]{} [[&=&]{}]{}(D(D+3) + s(s+2) [\
&&]{} - q\_1(q\_1+2) - q\_2(q\_2+2) - (2 p + q\_1 + q\_2)\^2 - (2 p + q\_1 + q\_2)). Here $D$ is the dimension and $s$ is the Lorentz spin. The first expression uses eigenvalues of all diagonal entries of the traceless matrix of ${\mathfrak{R}}$-symmetry generators, while the second uses the standard ${\mathfrak{su}}(4)$ Dynkin labels, $$q_1 = R^2_2-R^1_1, \quad p=R^3_3 - R^2_2, \quad q_2 = R^4_4-R^3_3.$$ The ${\mathfrak{osp}}(6|4)$ spin $j$ satisfies $j(j+1) = J^2$, the eigenvalue of ${\mathfrak{J}}^2$. The tensor product of a conjugate pair of modules $\mathcal{V}_\phi$ and $\mathcal{V}_{\bar \phi}$ has one highest-weight state for each nonnegative integer spin $j$ (and no other highest-weight states), $$\label{eq:conjugateproduct}
\mathcal{V}_\phi \otimes \mathcal{V}_{\bar \phi} = \sum_{j=0}^\infty \mathcal{V}_j.$$ Similarly, a like pair of modules has has one highest-weight state with spin $(j-1/2)$ for each nonnegative integer $j$, $$\label{eq:likeproduct}
\mathcal{V}_\phi \otimes \mathcal{V}_\phi = \sum_{j=0}^\infty \mathcal{V}_{j-1/2}, \quad \mathcal{V}_{\bar \phi} \otimes \mathcal{V}_{\bar \phi} = \sum_{j=0}^\infty \mathcal{V}_{j-1/2}.$$
Restriction to the ${\mathfrak{osp}}(4|2)$ sector \[sec:restrictosp42\]
-----------------------------------------------------------------------
Consider the set of states that at leading order are annihilated by ${\mathfrak{Q}}_{12,2}$ and ${\mathfrak{S}}^{12,2}$, which is a $1/12$ BPS condition[^6]. According to the last algebra relation of (\[eq:anticommutators\]), they satisfy $$R^1{}_1 + R^2{}_2 - L^2{}_2 - D_0 = 0, \label{eq:1/12bps}$$ where ordinary font denotes the eigenvalue of the corresponding generator in Gothic font, and $D_0$ is the classical dimension of a state. Since states only mix with other states with the same Lorentz and ${\mathfrak{R}}$ symmetry quantum numbers, and we choose a renormalization scheme where only states with the same classical dimension mix, it follows that this set of states is closed to all orders in perturbation theory.
Under this restriction, ${\mathfrak{osp}}(6|4)$ reduces to the set of generators that commute with the left side of (\[eq:1/12bps\]), and from the above algebra relations we find a residual ${\mathfrak{u}}(1) \ltimes {\mathfrak{su}}(1|1) \times {\mathfrak{osp}}(4|2)$ algebra. Since at leading order the ${\mathfrak{su}}(1|1)$ algebra (generated by ${\mathfrak{Q}}_{12,2}$ and ${\mathfrak{S}}^{12,2}$) acts trivially, we call this sector the ${\mathfrak{osp}}(4|2)$ sector. $ {\mathfrak{osp}}(4|2)$ includes a ${\mathfrak{sl}}(2)$ subalgebra of ${\mathfrak{sp}}(4)$, two ${\mathfrak{su}}(2)$ algebras from the original ${\mathfrak{su}}(4)$ ${\mathfrak{R}}$ symmetry, and eight supercharges. More precisely, and introducing a convenient notation, the ${\mathfrak{osp}}(4|2)$ generators are related to those for the full ${\mathfrak{osp}}(6|4)$ theory as (all index variables run from 1 to 2) $$\begin{aligned}
{\mathfrak{J}}^{11} & = {{{\textstyle\frac{1}{2}}}}{\mathfrak{P}}_{11},
&
{\mathfrak{J}}^{22} & = {{{\textstyle\frac{1}{2}}}}{\mathfrak{K}}^{11},
&
{\mathfrak{J}}^{12} & = -{{\textstyle\frac{1}{2}}} {\mathfrak{L}} + {\mathfrak{D}} + {{{\textstyle\frac{1}{2}}}}\delta{\mathfrak{D}},
\notag \\
{\mathfrak{R}}^{ab}& = {{{\textstyle\frac{1}{2}}}}\varepsilon^{ac} {\mathfrak{R}}^b_c +{{{\textstyle\frac{1}{2}}}}\varepsilon^{bc} {\mathfrak{R}}^a_c,
&
\tilde{{\mathfrak{R}}}^{\mathfrak{ab}} & = {{{\textstyle\frac{1}{2}}}}\varepsilon^{\mathfrak{ac}} {\mathfrak{R}}^{\mathfrak{b}+2}_{\mathfrak{c}+2} +{{{\textstyle\frac{1}{2}}}}\varepsilon^{\mathfrak{bc}} {\mathfrak{R}}^{\mathfrak{a}+2}_{\mathfrak{c}+2}, & &
\notag \\
{\mathfrak{Q}}^{a1\mathfrak{b}} & = \varepsilon^{ac}\varepsilon^{\mathfrak{bd}} {\mathfrak{Q}}_{c(\mathfrak{d}+2),1},
&
{\mathfrak{Q}}^{a2\mathfrak{b}} & = - {\mathfrak{S}}^{a(\mathfrak{b}+2),1}. & & \label{eq:osp42generators}\end{aligned}$$ Here we have also introduced ${\mathfrak{L}}$, which generates an additional ${\mathfrak{u}}(1)$ and is related to ${\mathfrak{osp}}(6|4)$ generators as $${\mathfrak{L}} = {\mathfrak{R}}^1_1 + {\mathfrak{R}}^2_2 = -{\mathfrak{L}}^1_1 + {\mathfrak{D}}_0, \label{eq:definel}$$ The second equality is satisfied within this sector due to (\[eq:1/12bps\]). ${\mathfrak{L}}$ commutes with the ${\mathfrak{osp}}(4|2)$ generators. As we will see below, due to the restricted field content in this sector ${\mathfrak{L}}$ simply gives half of the number of spin-chain sites (the number of pairs of conjugate representations).
The rank-one subalgebras take the standard form $${[J^{AB},J^{CD}]} = \varepsilon^{CB}J^{AD} - \varepsilon^{AD}J^{CB}, \quad {[J^{AB},X^{C}]} = {{{\textstyle\frac{1}{2}}}}\varepsilon^{CB}X^{A} + {{{\textstyle\frac{1}{2}}}}\varepsilon^{CA}X^{B}, \label{eq:rankonealgebras}$$ and the anticommutators are $${\{{\mathfrak{Q}}^{a\beta\mathfrak{c}},{\mathfrak{Q}}^{d\epsilon\mathfrak{f}}\}} = -\varepsilon^{\beta\epsilon}\varepsilon^{\mathfrak{cf}} {\mathfrak{R}}^{ad} -\varepsilon^{ad}\varepsilon^{\beta\epsilon} \tilde{{\mathfrak{R}}}^{\mathfrak{cf}} + 2 \varepsilon^{ad}\varepsilon^{\mathfrak{cf}} {\mathfrak{J}}^{\beta\epsilon}. \label{eq:osp42anticom}$$ We use hatted notation for the ${\mathfrak{su}}(1|1)$ algebra supercharges (note the reversed order of ${\mathfrak{R}}$ indices for the second one), $$\lambda \hat{{\mathfrak{Q}}} = {\mathfrak{Q}}_{12,2}, \quad \lambda \hat{{\mathfrak{S}}} = {\mathfrak{S}}^{21,2}.$$ In contrast to the introduction, we have now included a factor of $\lambda$ in the definitions since these generators act nontrivially first at $\mathcal{O}(\lambda)$ in this sector. $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ are nilpotent and the only nonvanishing (anti)commutator for ${\mathfrak{su}}(1|1)$ is $$\label{eq:qhatshatacomm}
{\{\hat{{\mathfrak{Q}}},\hat{{\mathfrak{S}}}\}} = \frac{1}{\lambda^2} \delta {\mathfrak{D}} = \mathcal{H}.$$ Here we have introduced $\mathcal{H}$, the anomalous part of the dilatation generator divided by $\lambda^2$. We will also normalize the (one-loop) supercharges so that $$\hat{{\mathfrak{S}}} = (\hat{{\mathfrak{Q}}})^\dagger.$$ We can do this since only the product of their overall normalizations appears in the Hamiltonian.
Again, ${\mathfrak{su}}(1|1)$ and ${\mathfrak{osp}}(4|2)$ commute, and therefore $\delta {\mathfrak{D}}$ is a shared central charge. The ${\mathfrak{u}}(1)$ length generator ${\mathfrak{L}}$ satisfies $$\label{eq:lcomm}
{[{\mathfrak{L}},\hat{{\mathfrak{Q}}}]} = \hat{{\mathfrak{Q}}}, \quad {[{\mathfrak{L}},\hat{{\mathfrak{S}}}]} = -\hat{{\mathfrak{S}}}.$$ In this work we will only consider the leading (nonvanishing) contributions to the generators. So after this section, ${\mathfrak{Q}}$, ${\mathfrak{J}}$, $\hat{{\mathfrak{Q}}}$, $\hat{{\mathfrak{S}}}$ and $\mathcal{H}$ will refer only to $\mathcal{O}(\lambda^0)$ terms. We will still explicitly refer to the order of the leading anomalous piece of the dilatation generator, $\delta {\mathfrak{D}}_2$.
The ${\mathfrak{osp}}(4|2)$ module and leading-order representation
-------------------------------------------------------------------
The two modules of the ${\mathfrak{osp}}(4|2)$ sector $\mathcal{V}_\phi^{(4|2)}$ and $\mathcal{V}_{\bar \phi}^{(4|2)}$ can be obtained by acting with the generators (\[eq:osp42generators\]) on the highest weight states ${\mathopen{\big|}\phi_1^{(0,0)}\mathclose{\bigr \rangle}}$ and ${\mathopen{\big|}(\bar{\phi}^4)^{(0,0)}\mathclose{\bigr \rangle}}$. From (\[eq:rsymaction\] - \[eq:qaction\]), we conclude states in this sector have lower ${\mathfrak{R}}$ indices $1$ or $2$ or upper indices $3$ or $4$. Then, using (\[eq:lorentanddilatationaction\]), we see that (\[eq:1/12bps\]) implies that all states in this sector have only nonzero values for the first Lorentz excitation number, so we can use a single argument for the Lorentz indices of states as $${\mathopen{\big|}\phi_a^{(n)}\mathclose{\bigr \rangle}} = {\mathopen{\big|}\phi_a^{(2n, 0)}\mathclose{\bigr \rangle}} \sim \mathcal{D}_{11}^{n} \phi_a, \quad {\mathopen{\big|}\psi_a^{(n)}\mathclose{\bigr \rangle}} = {\mathopen{\big|}\psi_a^{(2n+1, 0)}\mathclose{\bigr \rangle}} \sim \mathcal{D}_{11}^{n} \psi_{1a}, \quad a=1,2.$$ Again, the conjugate fields have upper ${\mathfrak{R}}$ indices $3$, $4$ in the notation for the full theory; it is convenient to replace them with lower indices $\mathfrak{a,b}=1,2$, resulting in [|\_\^[(n)]{}]{} [[&=&]{}]{}\_ [(|\^[+2]{})\^[(2n, 0)]{}]{} \~\_ \_[11]{}\^[n]{}|\^[+2]{}, [\
]{} [|\_\^[(n)]{}]{} [[&=&]{}]{}\_ [(|\^[+2]{})\^[(2n+1, 0)]{}]{} \~\_ \_[11]{}\^[n]{}|\^[+2]{}\_1. (\[eq:definel\]) and (\[eq:lorentanddilatationaction\]) now imply in this sector that any state ${\mathopen{\big|}X\mathclose{\bigr \rangle}}$ satisfies $${\mathfrak{L}} {\mathopen{\big|}X\mathclose{\bigr \rangle}} = {{{\textstyle\frac{1}{2}}}}{\mathopen{\big|}X\mathclose{\bigr \rangle}}.$$ In other words, ${\mathfrak{L}}$ gives $L$ on an ${\mathfrak{osp}}(4|2)$ spin chain state of $2 L$ sites, as mentioned previously.
In the just-introduced notation, (\[eq:rsymaction\]) implies that the ${\mathfrak{su}}(2)$ subalgebras’ generators ${\mathfrak{R}}$ act canonically on the the unbarred states, and the $\tilde{{\mathfrak{R}}}$ act the same way on barred states, $${\mathfrak{R}}^{ab}{\mathopen{\big|}X_c\mathclose{\bigr \rangle}} = {{{\textstyle\frac{1}{2}}}}\delta^a_c \varepsilon^{bd} {\mathopen{\big|}X_d\mathclose{\bigr \rangle}} + {{{\textstyle\frac{1}{2}}}}\delta^b_c \varepsilon^{ad} {\mathopen{\big|}X_d\mathclose{\bigr \rangle}}, \quad \tilde{{\mathfrak{R}}}^{\mathfrak{ab}}{\mathopen{\big|}\bar{X}_{\mathfrak{c}}\mathclose{\bigr \rangle}} = {{{\textstyle\frac{1}{2}}}}\delta^{\mathfrak{a}}_{\mathfrak{c}} \varepsilon^{\mathfrak{bd}} {\mathopen{\big|}\bar{X}_{\mathfrak{d}}\mathclose{\bigr \rangle}} + {{{\textstyle\frac{1}{2}}}}\delta^{\mathfrak{b}}_{\mathfrak{c}} \varepsilon^{\mathfrak{ad}} {\mathopen{\big|}\bar{X}_{\mathfrak{d}}\mathclose{\bigr \rangle}}.$$ Closure of the algebra (\[eq:rankonealgebras\] - \[eq:osp42anticom\]) fixes the supercharges’ action (up to physically irrelevant possible changes of basis) as $$\begin{aligned}
{\mathfrak{Q}}^{a1\mathfrak{b}} {\mathopen{\big|}\phi_c^{(n)}\mathclose{\bigr \rangle}} & = \delta^a_c \varepsilon^{\mathfrak{bd}} \sqrt{2n + 1} {\mathopen{\big|}\bar{\psi}^{(n)}_{\mathfrak{d}}\mathclose{\bigr \rangle}}, & {\mathfrak{Q}}^{a1\mathfrak{b}} {\mathopen{\big|}\psi_c^{(n)}\mathclose{\bigr \rangle}} & = \delta^a_c \varepsilon^{\mathfrak{bd}} \sqrt{2n + 2} {\mathopen{\big|}\bar{\phi}^{(n+1)}_{\mathfrak{d}}\mathclose{\bigr \rangle}},
\notag \\
{\mathfrak{Q}}^{a2\mathfrak{b}} {\mathopen{\big|}\phi_c^{(n)}\mathclose{\bigr \rangle}} & = \delta^a_c \varepsilon^{\mathfrak{bd}} \sqrt{2n } {\mathopen{\big|}\bar{\psi}^{(n-1)}_{\mathfrak{d}}\mathclose{\bigr \rangle}}, & {\mathfrak{Q}}^{a2\mathfrak{b}} {\mathopen{\big|}\psi_c^{(n)}\mathclose{\bigr \rangle}} & = \delta^a_c \varepsilon^{\mathfrak{bd}} \sqrt{2n + 1} {\mathopen{\big|}\bar{\phi}^{(n)}_{\mathfrak{d}}\mathclose{\bigr \rangle}}
\notag \\
{\mathfrak{Q}}^{a1\mathfrak{b}} {\mathopen{\big|}\bar{\phi}_{\mathfrak{c}}^{(n)}\mathclose{\bigr \rangle}} & = - \delta^\mathfrak{b}_\mathfrak{c} \varepsilon^{ad} \sqrt{2n + 1} {\mathopen{\big|}\psi^{(n)}_{d}\mathclose{\bigr \rangle}}, & {\mathfrak{Q}}^{a1\mathfrak{b}} {\mathopen{\big|}\bar{\psi}_{\mathfrak{c}}^{(n)}\mathclose{\bigr \rangle}} & = - \delta^\mathfrak{b}_\mathfrak{c} \varepsilon^{ad} \sqrt{2n + 2} {\mathopen{\big|}\phi^{(n+1)}_{d}\mathclose{\bigr \rangle}},
\notag \\
{\mathfrak{Q}}^{a2\mathfrak{b}} {\mathopen{\big|}\bar{\phi}_{\mathfrak{c}}^{(n)}\mathclose{\bigr \rangle}} & = - \delta^\mathfrak{b}_\mathfrak{c} \varepsilon^{ad} \sqrt{2n } {\mathopen{\big|}\psi^{(n-1)}_{d}\mathclose{\bigr \rangle}}, & {\mathfrak{Q}}^{a2\mathfrak{b}} {\mathopen{\big|}\bar{\psi}_{\mathfrak{c}}^{(n)}\mathclose{\bigr \rangle}} & = - \delta^\mathfrak{b}_\mathfrak{c} \varepsilon^{ad} \sqrt{2n + 1} {\mathopen{\big|}\phi^{(n)}_{d}\mathclose{\bigr \rangle}}.\end{aligned}$$ Note the symmetry between the supercharge actions on barred and unbarred states, only differing by a minus sign and appropriate interchanges of Gothic and Latin indices. Finally, the action of the ${\mathfrak{J}}$ (independent of ${\mathfrak{R}}$ and $\tilde{{\mathfrak{R}}}$ indices) is the same for barred and unbarred states. Representing both $\phi_a$ and $\bar{\phi}_{\mathfrak{a}}$ with $\phi$, and $\psi_a$ and $\bar{\psi}_{\mathfrak{a}}$ with $\psi$, we have $$\begin{aligned}
{\mathfrak{J}}^{11} {\mathopen{\big|}\phi^{(n)}\mathclose{\bigr \rangle}} & = \sqrt{(n+{{{\textstyle\frac{1}{2}}}})(n+1)} {\mathopen{\big|}\phi^{(n+1)}\mathclose{\bigr \rangle}}, & {\mathfrak{J}}^{11} {\mathopen{\big|}\psi^{(n)}\mathclose{\bigr \rangle}} & = \sqrt{(n+1)(n+{{\textstyle\frac{3}{2}}})} {\mathopen{\big|}\psi^{(n+1)}\mathclose{\bigr \rangle}},
\notag \\
{\mathfrak{J}}^{22} {\mathopen{\big|}\phi^{(n)}\mathclose{\bigr \rangle}} & = \sqrt{(n-{{{\textstyle\frac{1}{2}}}})n} {\mathopen{\big|}\phi^{(n-1)}\mathclose{\bigr \rangle}}, & {\mathfrak{J}}^{22} {\mathopen{\big|}\psi^{(n)}\mathclose{\bigr \rangle}} & = \sqrt{n(n+{{{\textstyle\frac{1}{2}}}})} {\mathopen{\big|}\psi^{(n-1)}\mathclose{\bigr \rangle}},
\notag \\
{\mathfrak{J}}^{12} {\mathopen{\big|}\phi^{(n)}\mathclose{\bigr \rangle}} & = (n +{{\textstyle\frac{1}{4}}}) {\mathopen{\big|}\phi^{(n)}\mathclose{\bigr \rangle}}, & {\mathfrak{J}}^{12} {\mathopen{\big|}\psi^{(n)}\mathclose{\bigr \rangle}} & = (n +{{\textstyle\frac{3}{4}}}) {\mathopen{\big|}\psi^{(n)}\mathclose{\bigr \rangle}}.\end{aligned}$$ In the basis we have chosen, Hermiticity is manifest, $$({\mathfrak{Q}}^{a1 \mathfrak{b}})^\dagger = \varepsilon_{ac}\varepsilon_{\mathfrak{bd}}{\mathfrak{Q}}^{c2\mathfrak{d}}, \quad ({\mathfrak{J}}^{11})^\dagger = {\mathfrak{J}}^{22}, \quad ({\mathfrak{J}}^{12})^\dagger = {\mathfrak{J}}^{12}, \quad ({\mathfrak{X}}^{AB})^\dagger = \varepsilon_{AD}\varepsilon_{CB} {\mathfrak{X}}^{CD},$$ where ${\mathfrak{X}}$ denotes ${\mathfrak{R}}$ or $\tilde{{\mathfrak{R}}}$.
The quadratic Casimir for this sector is $${\mathfrak{J}}^2=-{{{\textstyle\frac{1}{4}}}}\varepsilon_{ad}\varepsilon_{cb} {\mathfrak{R}}^{ab}{\mathfrak{R}}^{cd} -{{{\textstyle\frac{1}{4}}}}\varepsilon_{\mathfrak{ad}}\varepsilon_{\mathfrak{cb}} \tilde{{\mathfrak{R}}}^{\mathfrak{ab}}{{\mathfrak{R}}}^{\mathfrak{cd}} + {{{\textstyle\frac{1}{2}}}}\varepsilon_{\alpha \delta}\varepsilon_{\gamma \beta} {\mathfrak{J}}^{\alpha \beta} {\mathfrak{J}}^{\gamma \delta} -{{{\textstyle\frac{1}{4}}}}\varepsilon_{ad}\varepsilon_{\beta \epsilon } \varepsilon_{\mathfrak{cd}} {\mathfrak{Q}}^{a\beta\mathfrak{c}}{\mathfrak{Q}}^{d\epsilon\mathfrak{d}}. \label{eq:defineosp42casimir}$$ It follows from the algebra commutation relations that highest-weight states, which are annihilated by ${\mathfrak{R}}^{22}$, $\tilde{{\mathfrak{R}}}^{22}$, ${\mathfrak{J}}^{22}$ and ${\mathfrak{Q}}^{a2\mathfrak{b}}$, have quadratic Casimir eigenvalue $$\label{eq:twositeCartan}
J^2 = - {{{\textstyle\frac{1}{2}}}}R^{12}(R^{12}-1)- {{{\textstyle\frac{1}{2}}}}\tilde{R}^{12}(\tilde{R}^{12} - 1) + J^{12}(J^{12} + 1).$$ As for ${\mathfrak{osp}}(6|4)$, a eigenstate of the quadratic Casimir has ${\mathfrak{osp}}(4|2)$ spin $j$ given by $J^2 = j(j+1)$.
It is a straightforward exercise to find the highest-weight states for two-site states. For a conjugate pair of modules, there is one highest-weight state (and irreducible highest-weight ${\mathfrak{osp}}(4|2)$ module) for each nonnegative integer spin $j$, matching precisely the result previously given for the full ${\mathfrak{osp}}(6|4)$ modules (\[eq:conjugateproduct\]), $$\label{eq:42conjugateproduct}
\mathcal{V}_\phi^{(4|2)} \otimes \mathcal{V}_{\bar \phi}^{(4|2)} = \sum_{j=0}^\infty \mathcal{V}^{(4|2)}_j.$$ The corresponding Cartan charges are $$\label{eq:twositedifferenthigestweight}
[ R^{12}, J^{12}, \tilde{R}^{12} ] = [-{{{\textstyle\frac{1}{2}}}}, {{{\textstyle\frac{1}{2}}}}, -{{{\textstyle\frac{1}{2}}}}] \quad \text{and} \quad [ R^{12}, J^{12}, \tilde{R}^{12} ] = [0, j, 0] \quad j =1, 2, \ldots$$ We will also need the case of two identical modules (which would appear as two next-nearest neighbor sites on the alternating chain). Again there is precise agreement with the ${\mathfrak{osp}}(6|4)$ result (\[eq:likeproduct\]), $$\label{eq:42likeproduct}
\mathcal{V}_\phi^{(4|2)} \otimes \mathcal{V}^{(4|2)}_\phi = \sum_{j=0}^\infty \mathcal{V}^{(4|2)}_{j-1/2}, \quad \mathcal{V}^{(4|2)}_{\bar \phi} \otimes \mathcal{V}^{(4|2)}_{\bar \phi} = \sum_{j=0}^\infty \mathcal{V}^{(4|2)}_{j-1/2}.$$ and the ${\mathfrak{osp}}(4|2)$ Cartan charges are $$\label{eq:twositeidenticalhigestweight}
[-1, {{{\textstyle\frac{1}{2}}}}, 0] \quad \text{and} \quad [0, j + {{{\textstyle\frac{1}{2}}}}, 0], \quad j = 0, 1, \ldots$$ Finally, the ${\mathfrak{osp}}(4|2)$ subsector has ${\mathfrak{sl}}(2)$ subsector(s) in which only the ${\mathfrak{J}}$ act nontrivially[^7]. The two modules are spanned by $${\mathopen{\big|}\phi_1^{(n)}\mathclose{\bigr \rangle}} \quad \text{and} \quad {\mathopen{\big|}\psi_1^{(n)}\mathclose{\bigr \rangle}}.$$ The ${\mathfrak{sl}}(2)$ sector two-site highest-weight states (which are descendants in the ${\mathfrak{osp}}(4|2)$ sector) have ${\mathfrak{sl}}(2)$ spins that take the same value as for the larger ${\mathfrak{osp}}(4|2)$ sector, and therefore as in ${\mathfrak{osp}}(6|4)$ as well.
Light-cone superspace basis
---------------------------
We will find it very useful to use a light-cone superspace [@Kogut:1969xa; @Brink:1982pd; @Mandelstam:1982cb; @Belitsky:2004yg] basis for the modules[^8]. This basis parameterizes $\mathcal{V}_{\phi}^{(4|2)}$ ($\mathcal{V}_{\bar \phi}^{(4|2)}$) with continuous variables: $x$, a ${\mathfrak{su}}(2)$ doublet $\theta^a$ ($\bar{\theta}^{\mathfrak{a}}$), and an anticommuting ${\mathfrak{su}}(2)$ doublet $\bar{\eta}^{\mathfrak{a}}$ ($\eta^a$). The spin-chain states are labeled ${\mathopen{\big|}x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}$, or ${\mathopen{\big|}x, \bar{\theta}, \eta\mathclose{\bigr \rangle}}$, which are defined via sums over the entire modules,
\[eq:definelightcone\] [x, , |]{} [[&=&]{}]{}\_[n=0]{}\^ [\_a\^[(n)]{}]{} + [|\_\^[(n)]{}]{}, [\
]{}[x, |, ]{} [[&=&]{}]{}\_[n=0]{}\^ [|\_\^[(n)]{}]{} + [\_a\^[(n)]{}]{}. In this basis the leading-order generators are represented by differential operators. As an example, we derive the representation for ${\mathfrak{Q}}^{a1\mathfrak{c}}$ acting on $\mathcal{V}_{\phi}^{(4|2)}$, \^[a1]{} [x, , |]{} [[&=&]{}]{}\_[n=0]{}\^ ( \^b \^[a1]{} [\_b\^[(n)]{}]{} - |\^ \^[a1]{} [\_\^[(n)]{}]{} ) [\
]{}[[&=&]{}]{} \_[n=0]{}\^ ( \^a \^ [|\_\^[(n)]{}]{} + \^[ab]{} |\^ [\_b\^[(n+1)]{}]{} ) [\
]{}[[&=&]{}]{}(\^a |\^ + \_x \^a |\^ ) [x, , |]{} . Our conventions for raising and lowering indices lead to, for $\mathcal{V}_{\phi}^{(4|2)}$, $$\begin{aligned}
\partial^{a} & = \varepsilon^{ab} \partial_{\theta^{b}} = \varepsilon^{ab} \partial_b, & \bar{\partial}^{\mathfrak{c}}& = \varepsilon^{\mathfrak{cd}} \partial_{\bar{\eta}^{\mathfrak{d}}} = \varepsilon^{\mathfrak{cd}} \bar{\partial}_{\mathfrak{d}}, \notag \\
\partial_{a} & = \partial^b \varepsilon_{ba}, & \partial_{\mathfrak{c}}& = \bar{\partial}^{\mathfrak{d}} \varepsilon_{\mathfrak{dc}} .\end{aligned}$$ For $\mathcal{V}_{\bar \phi}^{(4|2)}$, the same equations apply for $\theta$ replaced by $\eta$ and $\bar{\eta}$ replaced by $\bar{\theta}$. So, for example, $\partial^a$ is bosonic acting on $\mathcal{V}_{\phi}^{(4|2)}$, and fermionic on $\mathcal{V}_{\bar \phi}^{(4|2)}$. We repeat the above result for ${\mathfrak{Q}}^{a1\mathfrak{c}}$ and add the parallel expressions for the other supercharges and for $\mathcal{V}_{\bar \phi}^{(4|2)}$ (abbreviating the states by suppressing an $\eta$ or $\bar{\eta}$), $$\begin{aligned}
{\mathfrak{Q}}^{a1\mathfrak{c}} {\mathopen{\big|}x, \theta\mathclose{\bigr \rangle}} &=\Big(\theta^a \bar{\partial}^{\mathfrak{c}} + \partial_x \partial^a \bar{\eta}^{\mathfrak{c}} \Big) {\mathopen{\big|}x, \theta\mathclose{\bigr \rangle}} , & {\mathfrak{Q}}^{a2\mathfrak{c}} {\mathopen{\big|}x, \theta\mathclose{\bigr \rangle}} &=\Big(2 x \theta^a \bar{\partial}^{\mathfrak{c}} + (2 x \partial_x +1)\partial^a \bar{\eta}^{\mathfrak{c}} \Big) {\mathopen{\big|}x, \theta\mathclose{\bigr \rangle}} ,
\notag \\
{\mathfrak{Q}}^{a1\mathfrak{c}} {\mathopen{\big|}x, \bar{\theta}\mathclose{\bigr \rangle}} &=-\Big(\bar{\theta}^{\mathfrak{c}} \partial^{a} + \partial_x \bar{\partial}^{\mathfrak{c}} \eta^a \Big) {\mathopen{\big|}x, \bar{\theta}\mathclose{\bigr \rangle}} , & {\mathfrak{Q}}^{a2\mathfrak{c}} {\mathopen{\big|}x, \bar{\theta}\mathclose{\bigr \rangle}} &=-\Big(2 x \bar{\theta}^{\mathfrak{c}} \partial^{a} + (2 x \partial_x +1) \bar{\partial}^{\mathfrak{c}} \eta^a \Big) {\mathopen{\big|}x, \bar{\theta}\mathclose{\bigr \rangle}}.\end{aligned}$$ The remaining generators’ actions can be written the same way[^9] for both types of modules, $$\begin{aligned}
\label{eq:osp42lightcone}
{\mathfrak{R}}^{ab} &= \theta^{\{a} \partial^{b\}} + \eta^{\{a} \partial^{b\}}, & \bar{{\mathfrak{R}}}^{\mathfrak{cd}} &= \bar{\theta}^{\{\mathfrak{c}} \partial^{\mathfrak{d}\}} + \bar{\eta}^{\{\mathfrak{c}} \partial^{\mathfrak{d}\}},
\notag \\
{\mathfrak{J}}^{11} & = \partial_x, & {\mathfrak{J}}^{22} & = 2 x^2 \partial_x + x - 2 x \Big(\varepsilon_{ab} \eta^a \partial^b + \varepsilon_{\mathfrak{cd}} \bar{\eta}^c \bar{\partial}^{\mathfrak{d}}\Big),
\notag \\
{\mathfrak{J}}^{12} & = x \partial_x + \frac{1}{4} -\frac{1}{2} \Big(\varepsilon_{ab} \eta^a \partial^b + \varepsilon_{\mathfrak{cd}} \bar{\eta}^c \bar{\partial}^{\mathfrak{d}}\Big). & &\end{aligned}$$
Leading corrections in the ${\mathfrak{osp}}(4|2)$ sector \[sec:oneloop\]
=========================================================================
We will now compute the leading actions of $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$, which immediately give the two-loop dilatation generator.
Structure of $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ interactions
---------------------------------------------------------------------------
(\[eq:lcomm\]) implies that $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ are dynamic; they change the length of the spin chain as do spin-chain generators in $\mathcal{N}=4$ SYM [@Beisert:2003ys]. $\hat{{\mathfrak{Q}}}$ inserts two sites, and $\hat{{\mathfrak{S}}}$ removes two sites. Note that this means these supercharges have well-defined actions only on cyclic states, which is all that is required since these spin-chain states represent single-trace local operators. At one-loop, consistency with the coupling constant dependence of the interactions of the Lagrangian requires $\hat{{\mathfrak{Q}}}$ to replace one site with three, and $\hat{{\mathfrak{S}}}$ to replace three sites with one. Let $\mathcal{U}$ be the generator that shifts all sites by two to the right (with the last site $2 L$ going to site $2$ for example). By definition a cyclic alternating state ${\mathopen{\big|}Y\mathclose{\bigr \rangle}}$ satisfies $\mathcal{U}{\mathopen{\big|}Y\mathclose{\bigr \rangle}}={\mathopen{\big|}Y\mathclose{\bigr \rangle}}$. $\hat{{\mathfrak{Q}}}$ then acts as $$\hat{{\mathfrak{Q}}} {\mathopen{\big|}Y\mathclose{\bigr \rangle}} = \frac{L}{L +1} \sum_{i=0}^{L} \mathcal{U}^{-i} \Big(\hat{{\mathfrak{Q}}}(1) + \hat{{\mathfrak{Q}}}(2)\Big){\mathopen{\big|}Y\mathclose{\bigr \rangle}} .$$ and $\hat{{\mathfrak{Q}}}(i)$ gives the action of $\hat{{\mathfrak{Q}}}$ the $i$th site, which we determine below. Similarly $$\hat{{\mathfrak{S}}} {\mathopen{\big|}Y\mathclose{\bigr \rangle}} = \frac{L}{L -1} \sum_{i=0}^{L-2} \mathcal{U}^{-i} \Big(\hat{{\mathfrak{S}}}(1,2,3) + \hat{{\mathfrak{S}}}(2, 3, 4)\Big){\mathopen{\big|}Y\mathclose{\bigr \rangle}} .$$ In both these expressions a minus signs must be included for each crossing of two fermions (or for a supercharge crossing a fermion).
Constraints from manifest ${\mathfrak{R}}$ and $\tilde{{\mathfrak{R}}}$ symmetries and consistency with the classical scaling dimension assignments further severely restrict the supercharge actions. For instance, acting on a scalar initial state, we (almost) immediately can restrict to an ansatz for the action of $\hat{{\mathfrak{Q}}}$ on one site, \[eq:qhatansatz\] [\_a\^[(m)]{}]{} [[&=&]{}]{}\_[n+p < m]{} ( c\_1(m, n, p) \^[bc]{} [\_a\^[(n)]{} \_b\^[(p)]{} \_c\^[(m-n-p-1)]{}]{} [\
&&]{}- c\_1’(m, n, p) \^[bc]{} [\_c\^[(m-n-p-1)]{} \_b\^[(p)]{} \_a\^[(n)]{}]{} ) [\
&&]{}+ \^ (c\_2(m, n, p) [\_a\^[(n)]{} |\_\^[(p)]{} |\_\^[(m-n-p-1)]{}]{} - c\_2’(m, n, p) [|\_\^[(m-n-p-1)]{} |\_\^[(p)]{} \_a\^[(n)]{}]{}). [\
&&]{} Here and below summations will be over all nonnegative integers satisfying an inequality. In this case we have $n=0, 1, \ldots m-1$ and $p = 0, 1, \ldots m-n-1$. This ansatz (\[eq:qhatansatz\]) is the most general one consistent with the constraints mentioned above and two further observations. First, note the ${\mathfrak{su}}(2)$ identity, $$\varepsilon^{bc} {\mathopen{\big|}X_bY_cZ_a\mathclose{\bigr \rangle}} + \varepsilon^{bc} {\mathopen{\big|}X_aY_bZ_c\mathclose{\bigr \rangle}}= \varepsilon^{bc} {\mathopen{\big|}X_bY_aZ_c\mathclose{\bigr \rangle}},$$ which eliminates the need for another term. Second, a priori a scalar to three fermions interaction would be allowed, but this turns out to be inconsistent with the requirement that $\hat{{\mathfrak{Q}}}$ commutes with all of the ${\mathfrak{osp}}(4|2)$ generators (one can check this using the same methods we will now use to determine the $c_i$).
Constraints from commutator with ${\mathfrak{J}}^{11}$
------------------------------------------------------
Rather than considering the commutators with all ${\mathfrak{osp}}(4|2)$ generators, let us first just consider $${[\hat{{\mathfrak{Q}}},{\mathfrak{J}}^{11}]}=0 \text{ on cyclic states.}$$ As initially noted for $\mathcal{N}=4$ SYM [@Beisert:2003ys], the fact that the algebra only must be satisfied on cyclic spin chain states allows for “gauge transformations”. Here, consistency with the basic constraints used above and with alternating modules restricts such a gauge transformation to \[eq:gauge\] [\[,\^[11]{}\]]{}[X]{} [[&=&]{}]{}(-1)\^X g\_1 \^ [X|\_\^[(0)]{}|\_\^[(0)]{}]{} - g\_2 \^ [|\_\^[(0)]{}|\_\^[(0)]{}X]{} [\
&&]{}+ (-1)\^X g\_3 \^[ab]{}[X\_a\^[(0)]{}\_b\^[(0)]{}]{} - g\_4 \^[ab]{}[\_a\^[(0)]{}\_b\^[(0)]{}X]{}, [\
]{}[\[,\^[11]{}\]]{}[|[X]{}]{} [[&=&]{}]{}(-1)\^[|[X]{}]{} g\_2 \^ [|[X]{}|\_\^[(0)]{}|\_\^[(0)]{}]{} - g\_1 \^ [|\_\^[(0)]{}|\_\^[(0)]{}|[X]{}]{} [\
&&]{}+ (-1)\^[|[X]{}]{} g\_4 \^[ab]{}[|[X]{}\_a\^[(0)]{}\_b\^[(0)]{}]{} - g\_3 \^[ab]{}[\_a\^[(0)]{}\_b\^[(0)]{}|[X]{}]{}. $X$ represents any element of $\mathcal{V}_{\phi}^{(4|2)}$, $\bar{X}$ represents any element in $\mathcal{V}_{\bar \phi}^{(4|2)}$, and $(-1)^X$ gives $(-1)$ for fermionic $X$ and $1$ otherwise. To see that this interaction gives zero on cyclic states, consider the two terms with coefficient $g_1$. The first $g_1$ term inserts an $\tilde{{\mathfrak{R}}}$ singlet to the right of $\mathcal{V}_{\phi}^{(4|2)}$ sites. But on alternating cyclic (or infinite) chains, this is canceled by the second $g_1$ term, which inserts the same $\tilde{{\mathfrak{R}}}$ singlet to the left of $\mathcal{V}_{\bar \phi}^{(4|2)}$ sites with a relative minus sign. The cancellations for the other $g_i$ terms work in the same way.
Since ${\mathfrak{J}}^{11}$ does not affect ${\mathfrak{R}}$ charges, we obtain independent equations for the four coefficient functions appearing in (\[eq:qhatansatz\]). For instance the unprimed $c_1$ terms contribute \[eq:j11constraint\] [\[,\^[11]{}\]]{}\_[c\_1]{} [\_a\^[(m)]{}]{} [[&=&]{}]{}\_[n+p m]{} ( c\_1(m+1, n, p) [\
&&]{}- c\_1(m, n-1, p) - c\_1(m, n, p-1) [\
&&]{} -c\_1(m, n, p))\^[bc]{} [\_a\^[(n)]{}\_b\^[(p)]{}\_c\^[(m-n-p)]{}]{} [\
]{} [[&=&]{}]{}g\_3 \^[bc]{} [\_a\^[(0)]{} \_b\^[(0)]{} \_c\^[(m)]{}]{}. From acting on a scalar with $m=0$, we find the only coefficient with first argument $1$, $c_1(1, 0,0) = \sqrt{2} g_3$. But now it is straightforward to see that all coefficients are determined inductively. Assume all coefficients with first argument less than or equal to $m_0$ are known. Then (\[eq:j11constraint\]) determines all coefficients with first argument $m_0+1$ in terms of these known coefficients and $g_3$. The solution for all arguments is[^10] $$c_1(m,n, p) = \frac{\sqrt{2} g_3r_-(m)}{(m-n)r_-(n)r_+(p)r_-(m-n-p-1)}, \quad r_{\pm}(x) = \frac{\sqrt{x!}}{\sqrt{(2 x \pm 1)!!}} .$$ Repeating for the other three coefficient functions entering (\[eq:qhatansatz\]), again yields solutions determined inductively from a single coefficient. These solutions depend in total on the four $g_i$ coefficients appearing in (\[eq:gauge\]). Furthermore, (\[eq:gauge\]) implies that these are the only free parameters for $\hat{{\mathfrak{Q}}}$ for acting on scalars in $\mathcal{V}_{\bar \phi}^{(4|2)}$. For $\hat{{\mathfrak{Q}}}$ acting on fermions, in addition to structures paralleling the four terms of (\[eq:qhatansatz\]), there is an additional fermion-to-three-scalar interaction that is possible, which commutes exactly with ${\mathfrak{J}}^{11}$. Therefore, the ${\mathfrak{J}}^{11}$ constraint allows for two more free parameter (fermions and conjugate fermions), in addition to the four $g_i$. However, there are more constraints.
Anticommutator with ${\mathfrak{Q}}^{a1\mathfrak{b}}$ and solution for $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$
------------------------------------------------------------------------------------------------------------------------
Of course the ${\mathfrak{osp}}(4|2)$ supercharges relate coefficients unrelated by the ${\mathfrak{J}}^{11}$. Then requiring the anticommutator with ${\mathfrak{Q}}^{a1\mathfrak{b}}$ to vanish on cyclic states fixes all of the above independent coefficients to be proportional to a single free parameter; $\hat{{\mathfrak{Q}}}$ is determined by symmetry up to overall normalization. This free parameter is fixed by any single nonvanishing two-loop anomalous dimension, since $\hat{{\mathfrak{S}}} = (\hat{{\mathfrak{Q}}})^\dagger$, and (\[eq:qhatshatacomm\]) ${\{\hat{{\mathfrak{Q}}},\hat{{\mathfrak{S}}}\}} = \delta {\mathfrak{D}}_2$. The correct choice turns out to be $g_3=1/2$, as we will see below. With this normalization, the gauge transformation for the anticommutator between supercharges is \[eq:qqhatgauge\] [{,\^[a1]{}}]{}[X]{} [[&=&]{}]{}\^[ac]{} \^([\_c\^[(0)]{}|\^[(0)]{}\_X]{}- [X|\^[(0)]{}\_\_c\^[(0)]{}]{} ) , [\
]{}[{,\^[a1]{}}]{}[|[X]{}]{} [[&=&]{}]{}\^[ac]{} \^( [|\^[(0)]{}\_\_c\^[(0)]{}|[X]{}]{} -[|[X]{}\_c\^[(0)]{}|\^[(0)]{}\_]{} ). The corresponding complete solution for $\hat{{\mathfrak{Q}}}$ acting on $\mathcal{V}_{\phi}^{(4|2)}$ depends on three coefficient functions $c_i$ (note that $c_1$ is as previously, but $c_2$ is different from the $c_2$ appearing in (\[eq:qhatansatz\])), \[eq:qhatsolution\] [\_a\^[(m)]{}]{} [[&=&]{}]{}\_[n+p < m]{} c\_1(m, n, p) \^[bc]{} ( [\_a\^[(n)]{} \_b\^[(p)]{} \_c\^[(m-n-p-1)]{}]{} - [\_c\^[(m-n-p-1)]{} \_b\^[(p)]{} \_a\^[(n)]{}]{} ) [\
&&]{}+ c\_1(m, n, m-n-p-1) \^ ( [\_a\^[(n)]{} |\_\^[(p)]{} |\_\^[(m-n-p-1)]{}]{} - [|\_\^[(m-n-p-1)]{} |\_\^[(p)]{} \_a\^[(n)]{}]{}), [\
]{}[|\_\^[(m)]{}]{} [[&=&]{}]{}\_[n+p < m]{} c\_2(m, n, p) \^[bc]{} ( [|\_\^[(n)]{} \_b\^[(p)]{} \_c\^[(m-n-p-1)]{}]{} + [\_c\^[(m-n-p-1)]{} \_b\^[(p)]{} |\_\^[(n)]{}]{} ) [\
&&]{}+ c\_2(m, n, m-n-p-1) \^ ( [|\_\^[(n)]{} |\_\^[(p)]{} |\_\^[(m-n-p-1)]{}]{} + [|\_\^[(m-n-p-1)]{} |\_\^[(p)]{} |\_\^[(n)]{}]{}) [\
&&]{}+ \_[n+p m]{} c\_3(m, n, p) \^[bc]{} [\_b\^[(n)]{}|\_\^[(p)]{}\_c\^[(m-n-p)]{}]{}. The action on $\mathcal{V}_{\bar \phi}^{(4|2)}$ simply involves switching barred and unbarred module elements and an overall minus sign, \[eq:qhatsolutionconjugate\] [|\_\^[(m)]{}]{} [[&=&]{}]{}-\_[n+p < m]{} c\_1(m, n, p)\^( [|\_\^[(n)]{} |\_\^[(p)]{} |\_\^[(m-n-p-1)]{}]{} - [|\_\^[(m-n-p-1)]{} |\_\^[(p)]{} |\_\^[(n)]{}]{} ) [\
&&]{}+ c\_1(m, n, m-n-p-1) \^[bc]{} ( [|\_\^[(n)]{} \_b\^[(p)]{} \_c\^[(m-n-p-1)]{}]{} - [\_c\^[(m-n-p-1)]{} \_b\^[(p)]{} |\_\^[(n)]{}]{}), [\
]{}[\_a\^[(m)]{}]{} [[&=&]{}]{}-\_[n+p < m]{} (c\_2(m, n, p)\^( [\_a\^[(n)]{} |\_\^[(p)]{} |\_\^[(m-n-p-1)]{}]{} + [|\_\^[(m-n-p-1)]{} |\_\^[(p)]{} \_a\^[(n)]{}]{} ) [\
&&]{}+ c\_2(m, n, m-n-p-1) \^[bc]{} ( [\_a\^[(n)]{} \_b\^[(p)]{} \_c\^[(m-n-p-1)]{}]{} + [\_c\^[(m-n-p-1)]{} \_b\^[(p)]{} \_a\^[(n)]{}]{}) ) [\
&&]{}- \_[n+p m]{} c\_3(m, n, p) \^ [|\_\^[(n)]{}\_a\^[(p)]{}|\_\^[(m-n-p)]{}]{}. With the normalization $g_3=1/2$, $c_1$ becomes $$c_1(m,n, p) = \frac{ r_-(m)}{\sqrt{2}(m-n)r_-(n)r_+(p)r_-(m-n-p-1)}.$$ The next coefficient function takes a very similar form, just switching some $r_+$ and $r_-$ and an overall minus sign, $$c_2(m,n, p) = -\frac{ r_+(m)}{\sqrt{2}(m-n)r_+(n)r_+(p)r_-(m-n-p-1)}.$$ Finally, the last coefficient function takes a similar form, without the $\sqrt{2}(m-n)$ factor in the denominator, $$c_3(m,n, p) = -\frac{ r_+(m)}{r_-(n)r_-(p)r_-(m-n-p)}.$$ Initial (final) fermions lead to $r_+$ factors in the numerator (denominator) and initial (final) scalars to $r_-$ factors in the numerator (denominator).
Recall that the ${\mathfrak{su}}(1|1)$ supercharges are nilpotent. One could check that $\hat{{\mathfrak{Q}}}^2=0$ by working out all of its one-to-five site interactions in terms of the $c_i$. However, this is redundant; a few basic facts about the spin modules plus the vanishing commutators (up to gauge transformations) of $\hat{{\mathfrak{Q}}}$ with the ${\mathfrak{osp}}(4|2)$ generators already ensure that $\hat{{\mathfrak{Q}}}^2=0$. We leave it as an exercise for the reader to work out this argument.
Manifest Hermiticity of the leading ${\mathfrak{osp}}(4|2)$ generators implies that above we could have considered $\hat{{\mathfrak{S}}}$ instead of $\hat{{\mathfrak{Q}}}$, and we would have then found a unique solution to $\hat{{\mathfrak{S}}}$ (up to normalization). This was anticipated by the previously stated equality $\hat{{\mathfrak{S}}} = (\hat{{\mathfrak{Q}}})^\dagger$. Using this equality it is straightforward to work out the interactions of $\hat{{\mathfrak{S}}}$ by switching initial and final states of (\[eq:qhatsolution\]-\[eq:qhatsolutionconjugate\]). For instance, the first line of (\[eq:qhatsolution\]) implies [\_a\^[(m)]{}\_b\^[(n)]{}\_c\^[(p)]{}]{} [[&=&]{}]{}c\_1(m+n+p+1, m, n)\_[bc]{}[\_a\^[(m+n+p+1)]{}]{} [\
&&]{}+c\_1(m+n+p+1, p, n)\_[ab]{}[\_c\^[(m+n+p+1)]{}]{}. The complete solution for $\hat{{\mathfrak{S}}}$ is given in the light-cone basis in Appendix \[sec:lightcone\].
From the expression for $\hat{{\mathfrak{Q}}}$, we see that the (two-loop) ${\mathfrak{osp}}(4|2)$ sector has an additional discrete symmetry under *spin-chain* parity $\mathbf{p}$, which reverses the order of the spin chain sites with an extra signs for each crossing of fermions. After application of parity each site will have the opposite type of representation. Note that $\mathbf{p}$ is distinct from space-time parity, and, unlike the case of $\mathcal{N}=4$ SYM, this operation is distinct also from charge conjugation symmetry. While the leading order ${\mathfrak{osp}}(4|2)$ generators are (trivially) parity even, $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ are parity odd[^11].
Solution for $\hat{{\mathfrak{Q}}}$ in light-cone superspace basis
------------------------------------------------------------------
In the light-cone basis the $r_{\pm}$ factors are absorbed into the normalization of the states in the expansion of ${\mathopen{\big|}x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}$, while the $(m-n)$ factors in the denominators can be accounted for with an integral as \[eq:lightconeqhat\] [x, , |]{} [[&=&]{}]{}\_0\^x y ( \_[ab]{} \^a\_2 \^b\_3 + \_ \^\_2 \^\_3 ) [x, , |; y, |\_2, \_2;y, \_3, |\_3]{} [\
&&]{} + \_0\^x y ( \_[ab]{} \^a\_1 \^b\_2 + \_ \_1\^ \_2\^ ) [ y, \_1, |\_1; y, |\_2, \_2; x, , |]{} [\
&&]{} -\_ |\^ \_2\^ \_[cd]{} \_1\^c\_3\^d [ x, \_1, |\_1;x, |\_2, \_2; x, \_3, |\_3 ]{} . Here the subscripts label on which site the partial derivatives act. For example, in the first term $\partial_2^a=\varepsilon^{ac} \partial/\partial \eta_2^c$. The expression for acting on $\mathcal{V}_{\bar \phi}^{(4|2)}$ just follows from switching ${\mathfrak{su}}(2)$ indices in epsilon tensors and in the derivatives ($\partial^a \leftrightarrow \partial^{\mathfrak{a}}$), and switching all states with their conjugates (${\mathopen{\big|}x, \theta, \bar{\eta}\mathclose{\bigr \rangle}} \leftrightarrow {\mathopen{\big|}x, \bar{\theta}, \eta\mathclose{\bigr \rangle}}$). Also, there is an extra overall minus sign for acting on $\mathcal{V}_{\bar \phi}^{(4|2)}$. As an example, we check one term. Expanding the $\theta$ component of the first term of the first line of (\[eq:lightconeqhat\]) we find \[eq:checklightconeqhat\] [x, , |]{}\_ [[&=&]{}]{}\_0\^x y \_[ab]{} \^a\_2 \^b\_3 [x, , |; y, |\_2, \_2;y, \_3, |\_3]{} \_ [\
]{}[[&=&]{}]{}\_0\^x y \_[m\_1, m\_2,m\_3]{} x\^[m\_1]{}y\^[m\_2+m\_3]{} \^a \^[bc]{} [\_a\^[(m\_1)]{}\_b\^[(m\_2)]{}\_c\^[(m\_3)]{}]{} [\
]{}[[&=&]{}]{}\_[n+p < m]{} \^[bc]{} [\_a\^[(n)]{}\_b\^[(p)]{}\_c\^[(m-n-p-1)]{}]{} [\
]{}[[&=&]{}]{}\_m \^a \_[n+p < m]{} c\_1(m, n, p) \^[bc]{} [\_a\^[(n)]{}\_b\^[(p)]{}\_c\^[(m-n-p-1)]{}]{}. To reach the third line we did the integral, substituted for $m_3$ using $m=m_1+m_2+m_3+1$, and then replaced $m_1,m_2$ with $n, p$. The combinatoric factor simplifies to $\sqrt{(2m)!}c_1(m, n, p)/m!$, yielding the last line after reordering factors. On the other hand, acting with the first term of the first line of (\[eq:qhatsolution\]) on the left side of (\[eq:checklightconeqhat\]) clearly leads to the same result, as needed. One can the check remaining terms, involving also those with initial fermions, in a similar fashion.
Because the light-cone basis is not manifestly Hermitian, $\hat{{\mathfrak{S}}}$ takes a more involved form (requiring integration over two auxiliary variables, rather than just one), again see Appendix \[sec:lightcone\].
Hamiltonian, wrapping interactions, and twist-one spectrum
----------------------------------------------------------
The two loop dilatation generator for the ${\mathfrak{osp}}(4|2)$ sector now follows from the anticommutator (\[eq:qhatshatacomm\]) $${\{\hat{{\mathfrak{Q}}},\hat{{\mathfrak{S}}}\}} = \delta {\mathfrak{D}}_2. \label{eq:ham}$$ Since $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ are spin-chain parity odd, the Hamiltonian is parity even. We will not explicitly compute the expansion of the Hamiltonian in terms of interactions, since there is a simpler way to prove integrability. Still, we have used (\[eq:ham\]), (\[eq:qhatsolution\]-\[eq:qhatsolutionconjugate\]), and `Mathematica` to check the spectrum for many (cyclic) spin chain states of relatively low dimension and length, finding complete agreement with the Bethe ansatz predictions of [@Minahan:2008hf]. This is empirical confirmation of the integrability that we will prove in the next section.
It is important that there are physically equivalent expressions for the Hamiltonian which have different expansions in terms of local interactions. For periodic spin chains, including the cyclic spin chains of this work, there is always freedom to add chain derivatives to spin-chain generators including the Hamiltonian. Chain derivatives are nonzero interactions that vanish on periodic states. An example three-site chain derivative acts on a periodic spin chain of length $2L$ as $$\sum_{i=1}^{L} \Big({\mathfrak{L}}(2i-1){\mathfrak{X}}(2i,2i +1) -{\mathfrak{X}}(2i,2i +1){\mathfrak{L}}(2i+2)\Big),$$ where again ${\mathfrak{L}}$ is the length generator which just gives $1/2$ when acting on an individual site. ${\mathfrak{X}}$ here can be an arbitrary (length-preserving) two-site generator . If ${\mathfrak{X}}$ is fermionic, though, there would be extra signs. Of course, there is another chain derivative where ${\mathfrak{X}}$ acts on sites $(2i-1, 2i)$ instead. Similarly, there are physically equivalent expressions that act on a (superficially) different number of sites, due to interactions including spectator sites. For example a two-site generator ${\mathfrak{Z}}$ can be written equivalently as a three-site generator, $$\sum_{I=1}^{2L} {\mathfrak{Z}}(i, i+1) = 2 \sum_{i=1}^{2L} \Big({\mathfrak{Z}}(i, i+1) {\mathfrak{L}}(i+2)\Big).$$ Importantly (\[eq:qhatshatacomm\]) implies that there is no wrapping contribution for $\mathcal{N}=6$ Chern-Simons until four loops, as is also the case for $\mathcal{N}=4$ SYM [@Beisert:2005fw; @Kotikov:2007cy]. Each ${\mathfrak{osp}}(6|4)$ highest weight two-site state is in one-to-one correspondence with a descendant which is an ${\mathfrak{osp}}(4|2)$ sector highest-weight. Furthermore, in this sector ${\hat{{\mathfrak{S}}}}$ annihilates two-site states and $\hat{{\mathfrak{Q}}}$ combines non-BPS states with four-site states in long multiplets for $\lambda \neq 0$. Wrapping interactions for $\hat{{\mathfrak{Q}}}$ first appears when there are 3-to-5 site interactions, i.e. at $\mathcal{O}(\lambda^3)$, which leads to a wrapping contribution to the four-loop dilatation generator.
In fact, it is straightforward to compute the anomalous dimensions of two-site states at two loops. These states have twist one, and according to (\[eq:twositeCartan\]) there is one highest-weight twist-one state for each nonnegative integer ${\mathfrak{osp}}(4|2)$ spin $s$. A convenient representative of the $s$th multiplet is in the ${\mathfrak{sl}}(2)$ sector and has Lorentz spin $(s+1/2)$, $${\mathopen{\big|}{\mathnormal{\oldPsi}}_s\mathclose{\bigr \rangle}}= \sum_{m=0}^s (-1)^m \sqrt{\binom{2 s +1}{2m}} {\mathopen{\big|}\phi_1^{(m)} \psi_1^{(s-m)}\mathclose{\bigr \rangle}}.$$ ${\mathopen{\big|}{\mathnormal{\oldPsi}}_s\mathclose{\bigr \rangle}}$ is annihilated by ${\mathfrak{J}}^{22}$, and is therefore an eigenstate of the dilatation generator since there are no other such states with the same quantum numbers. Then the two-loop contribution to the anomalous dimension, $\Delta_{s,2}$, simply equals the coefficient of ${\mathopen{\big|}\phi_1^{(s)}\psi_1^{(0)}\mathclose{\bigr \rangle}}$ for $\mathcal{H}{\mathopen{\big|}{\mathnormal{\oldPsi}}_s\mathclose{\bigr \rangle}}$ divided by $(-1)^s \sqrt{2s+1}$. Since ${\hat{{\mathfrak{S}}}}$ annihilates two-site states, the Hamiltonian reduces to $ \hat{{\mathfrak{S}}}\hat{{\mathfrak{Q}}}$.
We can organize the contributions of $\hat{{\mathfrak{S}}}\hat{{\mathfrak{Q}}}$ as follows. There are (diagonal) terms from the product acting only on the first site, or only on the second site. As explained in Appendix \[sec:lightcone\], the one-site part of the Hamiltonian is $2 S_1(2m)$ for ${\mathopen{\big|}\phi^{(m)}\mathclose{\bigr \rangle}}$ and $2 S_1(2m +1)$ for ${\mathopen{\big|}\psi^{(m+1)}\mathclose{\bigr \rangle}}$. Also, there is a contribution from the fermion-to-three-boson interaction of $\hat{{\mathfrak{Q}}}$ combined with the conjugate interaction of $\hat{{\mathfrak{S}}}$, $${\mathopen{\big|}\phi_1^{(0)} \psi_1^{(s)}\mathclose{\bigr \rangle}} \rightarrow \varepsilon^{\mathfrak{bc}} {\mathopen{\big|}\phi_1^{(0)} \bar{\phi}_{\mathfrak{b}}^{(0)}\phi_1^{(s)}\bar{\phi}_{\mathfrak{c}}^{(0)}\mathclose{\bigr \rangle}} \rightarrow {\mathopen{\big|}\phi_1^{(s)} \psi_1^{(0)}\mathclose{\bigr \rangle}}.$$ Note that the second arrow refer to $\hat{{\mathfrak{S}}}$ acting on the last and first two sites, $\hat{{\mathfrak{S}}}(4,1,2)$. Finally, for the generic terms, $\hat{{\mathfrak{Q}}}$ inserts a ${\mathopen{\big|}\psi_1^{(0)}\mathclose{\bigr \rangle}}$ (and two additional module elements) and $\hat{{\mathfrak{S}}}$ replaces the other three sites with ${\mathopen{\big|}\phi_1^{(s)}\mathclose{\bigr \rangle}}$, yielding a multiple of ${\mathopen{\big|}\phi_1^{(s)}\psi_1^{(0)}\mathclose{\bigr \rangle}}$. Combining all of these contributions yields the $s$th two-loop anomalous dimension, \[eq:delta2sum\] \_[s,2]{} [[&=&]{}]{}\_[m=0]{}\^[s]{} A(m), [\
]{}A(m) [[&=&]{}]{}2 \_[ms]{} (S\_1(2s) + S\_1(1)) - + \_[n=0]{}\^[m-1]{} 2 c\_1(m,n, 0)c\_1(s, n, s-m) [\
&&]{}+ \_[n=0]{}\^[s-m-1]{} (- 2 c\_2(s-m, 0, n) c\_1(s, s-m-n-1, n) [\
&&]{}+ 8 c\_2(s-m, 0, n)c\_1(s, m, n) + 2 c\_2(s-m, n, 0)c\_1(s, m, n) ). It is simple to find the pattern numerically by evaluating the sum for low values of $s$, but in fact we have also done the sum analytically for arbitrary (nonnegative integer) $s$. For this it is significantly easier to use the light-cone superspace expressions for the supercharges, and we give more details about this in Appendix \[sec:lightcone\]. We find the spectrum in terms of the harmonic numbers and a generalized harmonic sum, $$\label{eq:twistonespectrum}
\Delta_s = 4 \lambda^2 \big(S_1(s) - S_{-1}(s)\big) + \mathcal{O}(\lambda^3),$$ which is similar to the twist-two spectrum of $\mathcal{N}=4$ SYM, $8 \lambda_{\mathcal{N}=4} S_1(s)$.
As we will see, the R-matrix construction requires at least four sites. Still the Bethe ansatz correctly gives this twist-one spectrum because it naturally accounts for the action of $\hat{{\mathfrak{Q}}}$ on two-site states described earlier. For zero-momentum solutions of the Bethe ansatz equations of [@Gromov:2008qe][^12], $\hat{{\mathfrak{Q}}}$ acts via $L \rightarrow (L+1)$ simultaneous with the removal of a single $u_3$ root at $0$, with all other roots unchanged. It is straightforward to check that this transformation does not change the energy or momentum (zero), and that it carries the same Cartan charges as $\hat{{\mathfrak{Q}}}$ does. It follows that the spectrum for $L=1$ states is the same as the spectrum of $L=2$ states without a $u_3$ root at zero (and only $u_4$, $\bar{u}_4$ and $u_3$ roots excited). A similar phenomenon occurs in the ${\mathfrak{psu}}(1,1|2)$ sector of $\mathcal{N}=4$ SYM [@Beisert:2005fw].
In Appendix \[sec:bethe\] we check analytically that the Bethe ansatz prediction gives precisely (\[eq:twistonespectrum\]). This is already very strong evidence in favor of the leading-order Bethe equations of [@Minahan:2008hf]. As found by [@Gromov:2008qe], the large $s$ behavior of $\Delta_s$ gives a cusp anomalous dimension of $f(\lambda) = 4 \lambda^2$, though this disagrees by a factor of four with the value give in [@Aharony:2008ug], based on [@Gaiotto:2007qi]. The author does not know the origin of this discrepancy, which was already noted in [@Gromov:2008qe].
Proof of ${\mathfrak{osp}}(4|2)$ sector integrability \[sec:yangian\]
=====================================================================
In this section we prove that the two-loop dilatation generator for the ${\mathfrak{osp}}(4|2)$ sector is integrable by constructing an ${\mathfrak{osp}}(4|2)$ Yangian that commutes with the leading-order ${\mathfrak{su}}(1|1)$ generators, and therefore with the two-loop dilatation generator.
Leading order Yangian
---------------------
A Yangian was used in a similar context for $\mathcal{N}=4$ SYM in [@Dolan:2003uh]. This is an infinite-dimensional symmetry algebra generated by the ordinary Lie algebra generators ${\mathfrak{J}}^A$ and bilocal products of Lie algebra generators $$\label{eq:definey}
{\mathfrak{Y}}^A = f^{A}{}_{CB} \sum_{i<j} {\mathfrak{J}}^B(i) {\mathfrak{J}}^C(j),$$ where $f^{A}{}_{CB}$ are the structure constants, with indices lowered (raised) using the (inverse) Cartan-Killing form. Note that the ${\mathfrak{Y}}^A $ are incompatible with periodic boundary conditions. As a result, generically the Yangian symmetry is only unbroken for infinite-length chains, as will be the case here. The $ {\mathfrak{Y}}^A$ manifestly transform in the adjoint of the Lie algebra. To consistently generate the entire algebra, the only additional requirement is that these ${\mathfrak{Y}}^A$ satisfy Serre relations $$\label{eq:serre}
3 {[{\mathfrak{Y}}^{[A },{[{\mathfrak{J}}^B,{\mathfrak{Y}}^{C\}\}\}} = - (-1)^{(EM)} f^{AK}{}_{D} f^B{}_{E}{}^{L} f^C{}_{F}{}^{M} f_{KLM}\{ {\mathfrak{J}}^D, {\mathfrak{J}}^E, {\mathfrak{J}}^F \},$$ where the curly brackets on the right side refer to the totally symmetric triple product, including a factor of $1/6$. Note that the indices on the left side are anti-symmetrized, with a factor of $1/6$. The signs and ordering of indices in the structure constants properly account for fermionic statistics for super Yangians [@Zwiebel:2006cb]. For ${\mathfrak{osp}}(4|2)$, a possible basis is $\{{\mathfrak{J}}^A \} = \{{\mathfrak{Q}}^{a\beta\mathfrak{c}},\, {\mathfrak{R}}^{ab}, \tilde{{\mathfrak{R}}}^{\mathfrak{ab}}, {\mathfrak{J}}^{\alpha\beta} \}$.
As reviewed in [@Dolan:2004ps], it is sufficient to check that the Serre relations are satisfied for a one-site chain. This is because the Yangian, a Hopf algebra, has a coproduct which gives the Yangian’s action on tensor products ((\[eq:definey\]) actually follows from the coproduct). If the Yangian’s Serre relations are satisfied on a single module, because of the coproduct they will be satisfied on chains of arbitrary length. For a one-site chain, the left-side of the Serre relation (\[eq:serre\]) vanishes, so we simply need to confirm that the right side vanishes. Programming the generators in $\texttt{Mathematica}$, we have confirmed that the Serre relations are satisfied when acting on any element of a single ${\mathfrak{osp}}(4|2)$ module (of either type)[^13], as required.
Vanishing commutator with $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$
---------------------------------------------------------------------------
We will now explicitly show that the Yangian generators ${\mathfrak{Y}}^{aa}$, which have the same ${\mathfrak{osp}}(4|2)$ indices as ${\mathfrak{R}}^{11}$ or ${\mathfrak{R}}^{22}$, commute with $\hat{{\mathfrak{Q}}}$ on infinite-length chains. At the end of this section we will infer from this that the ${\mathfrak{osp}}(4|2)$ Hamiltonian is integrable.
First, consider the commutators between $\hat{{\mathfrak{Q}}}$ and the ${\mathfrak{osp}}(4|2)$ Lie algebra generators. Because ${\mathfrak{R}}$ symmetry is manifest, the commutator of ${\mathfrak{R}}$ with $\hat{{\mathfrak{Q}}}$ vanishes locally, not just up to a gauge transformation. Also, ${\mathfrak{Q}}^{a2\mathfrak{b}}$ commutes with $\hat{{\mathfrak{Q}}}$ locally as well (this commutator has classical dimension $0$, and a gauge transformation that inserts two sites has a minimum dimension $1$, twice the dimension of scalars without derivatives). On the other hand, the commutator with ${\mathfrak{Q}}^{a1\mathfrak{b}}$ gives the gauge transformation (\[eq:qqhatgauge\]) that acts as [{\^[a1]{},}]{} [[&=&]{}]{}\_i \^[a]{}\_i -\^[a]{}\_i , [\
]{}\^[a]{}[x, , |]{} [[&=&]{}]{}\^a\_1|\^\_2 [0, \_1, |\_1;0, |\_2, \_2;x, , |]{}, [\
]{}\^[a]{}[x, , |]{} [[&=&]{}]{}|\^\_2\^a\_3 [x, , |; 0, |\_2, \_2; 0, \_3, |\_3;]{}, [\
]{}\^[a]{}[x, |, ]{} [[&=&]{}]{}|\^\_1\^a\_2 [0, |\_1, \_1;0, \_2, |\_2;x, |, ]{}, [\
]{}\^[a]{}[x, |, ]{} [[&=&]{}]{}\^a\_2 |\^\_3[x, |,; 0, \_2, |\_2; 0, |\_3, \_3;]{}. In terms of components, the $\acute{Z}$ and $\grave{Z}$ insert two scalars without derivatives on adjacent sites. Our convention for site indices is that $\hat{{\mathfrak{Q}}}_i$, $\acute{Z}_i$ or $\grave{Z}_i$ or act on site $i$ and inserts sites $i+1$ and $i+2$ . Therefore, the initial (and final) sites $1$ through $i-1$ are unaffected by $\hat{{\mathfrak{Q}}}_i$, while for $j >i$, an initial site $j$ becomes site $j+2$, with these sites otherwise unchanged.
Before focusing on the ${\mathfrak{Y}}^{aa}$, we consider general features of the commutator between $\hat{{\mathfrak{Q}}}$ and bilocal generators. Consider the commutator involving one-site (bosonic) generators ${\mathfrak{J}}^A$ and ${\mathfrak{J}}^B$, $$\label{eq:commbilocalqhat}
{[\sum_{i < j} {\mathfrak{J}}_i^A {\mathfrak{J}}_j^B,\hat{{\mathfrak{Q}}}]} = \sum_{i < j} {\mathfrak{J}}_i^A {[{\mathfrak{J}}^B,\hat{{\mathfrak{Q}}}]}_j + \sum_{i < j} {[{\mathfrak{J}}^A,\hat{{\mathfrak{Q}}}]}_i {\mathfrak{J}}_j^B + \text{local}.$$ Since $\hat{{\mathfrak{Q}}}$ has one-to-three site interactions, the commutator with an individual ${\mathfrak{J}}$ also gives a (spin-chain-local) one-to-three site generator, for which the subscript refers to the single site on which this new local generator acts. The terms of (\[eq:commbilocalqhat\]) emerge as follows. The commutator vanishes when the ${\mathfrak{J}}$ act on sites that $\hat{{\mathfrak{Q}}}$ does not act on or insert. The terms where ${\mathfrak{J}}^B$ acts on a site inserted or acted on by $\hat{{\mathfrak{Q}}}$ but ${\mathfrak{J}}^A$ acts completely to the left of $\hat{{\mathfrak{Q}}}$ simplifies to the first term of the right side of (\[eq:commbilocalqhat\]). The reflected terms, with ${\mathfrak{J}}^A$ and ${\mathfrak{J}}^B$ and right and left switched, yield the second term. Finally, there are terms on which both ${\mathfrak{J}}$ act on sites inserted by $\hat{{\mathfrak{Q}}}$, which we call local because these combine into a homogeneous one-to-three site interaction $$\label{eq:local}
\sum_{i} \Big( {\mathfrak{J}}^A_i ({\mathfrak{J}}^B_{i+1} +{\mathfrak{J}}^B_{i+2} ) +{\mathfrak{J}}^A_{i+1}{\mathfrak{J}}^B_{i+2} \Big) {\mathfrak{\hat{Q}}}_i.$$ Finally we turn to the ${\mathfrak{Y}}^{aa}$. From (\[eq:definey\]) we find 4 \^[aa]{} [[&=&]{}]{} \_[i < j]{} 2 \_[bc]{} \^[bc]{}\_i \^[ca]{}\_j - \_ \^[a1 ]{}\_i \^[a2 ]{}\_j + \_ \^[a2 ]{}\_i \^[a1 ]{}\_j [\
]{}[[&=&]{}]{}\^[aa]{}\_ + \^[aa]{}\_[\^1]{} + \^[aa]{}\_[\^2]{}. \[eq:Ycc\] where the factor of $4$ on the left side is for convenience. Using (\[eq:commbilocalqhat\]) and (\[eq:Ycc\]), the commutators described above, and taking into account statistics, we find 4[\[\^[aa]{},\]]{} [[&=&]{}]{}\_[i < j]{} \_ [{\^[a1]{},}]{}\_i \_j\^[a2]{} - \_[i < j]{} \_ \_i\^[a2]{} [{\^[a1]{},}]{}\_j + [\
]{}[[&=&]{}]{}\_[i < j]{} \_ (\_i\^[a]{} -\_i\^[a]{}) \_j\^[a2]{} - \_[i < j]{} \_ \_i\^[a2]{}(\_j\^[a]{} -\_j\^[a]{}) + [\
]{}[[&=&]{}]{}- \_[i]{} \_ \_i\^[a]{} \_[i+1]{}\^[a2]{} - \_i \_ \_i\^[a2]{}\_[i+1]{}\^[a]{} + . \[eq:bilocalYqhatcomm\] To reach the last line, we used the cancellations between $\acute{Z}$ and $\grave{Z}$ acting upon adjacent sites, similar to the cancellation explained after (\[eq:gauge\]). In Appendix \[sec:localycomm\] we evaluate the remaining local piece, which involves computing (\[eq:local\]) with the ${\mathfrak{J}}$ there replaced with the generators that appear in (\[eq:Ycc\]). The result leads to the precise cancellation on infinite-length chains, $${[{\mathfrak{Y}}^{aa},\hat{{\mathfrak{Q}}}]}=0.$$ Since $\hat{{\mathfrak{Q}}}$ commutes with all ${\mathfrak{osp}}(4|2)$ Lie algebra generators and because the Yangian generators transform in the adjoint of ${\mathfrak{osp}}(4|2)$, this is sufficient to imply that all generators commute with $\hat{{\mathfrak{Q}}}$. For instance, any of the fermionic Yangian generator can be written as $${\mathfrak{Y}}^{a\beta\mathfrak{c}} = \pm {[{\mathfrak{Q}}^{d\beta\mathfrak{c}},{\mathfrak{Y}}^{aa}]}, \quad d \neq a,$$ which then necessarily commute with $\hat{{\mathfrak{Q}}}$ since the ${\mathfrak{osp}}(4|2)$ supercharges do (on infinite-length chains, as needed). Similarly, one can extend this to the remaining Yangian generators. Hermiticity then implies that $\hat{{\mathfrak{S}}}$ also commutes with the Yangian. Therefore, the two-loop ${\mathfrak{osp}}(4|2)$ sector dilatation generator $$\delta{\mathfrak{D}}_2 = {\{\hat{{\mathfrak{Q}}},\hat{{\mathfrak{S}}}\}}$$ has an ${\mathfrak{osp}}(4|2)$ Yangian symmetry and is integrable.
R-matrix construction of the ${\mathfrak{osp}}(4|2)$ sector Hamiltonian \[sec:rmatrix\]
=======================================================================================
The ${\mathfrak{osp}}(4|2)$ sector spin chain and its Yangian symmetry can be restricted consistently to the ${\mathfrak{sl}}(2)$ sector. For such a ${\mathfrak{sl}}(2)$ alternating spin chain we can use the known universal ${\mathfrak{sl}}(2)$ R-matrix [@Kulish:1981gi] to construct a transfer matrix as a function of two spectral parameters, $u$ and $\alpha$. We will see below that in our case $\alpha=0$. As is well-known, the expansion about $u=\infty$ gives the Yangian symmetry, while the expansion about $u=0$ gives the complete set of local conserved charges, $\mathcal{Q}_I$, that commute with the Yangian on infinite-length chains. Then the Hamiltonian for the ${\mathfrak{sl}}(2)$ sector of the ABJM spin chain must be a linear combination of the $\mathcal{Q}_I$, up to physically irrelevant chain derivatives. The restriction to nearest- and next-nearest-neighbor interactions identifies this Hamiltonian uniquely, up to coefficients that can be fixed by acting on a few states.
Since the ${\mathfrak{sl}}(2)$ sector Hamiltonian originates from an R-matrix construction, we can conclude that the ${\mathfrak{osp}}(4|2)$ sector Hamiltonian also follows from an R-matrix construction. The argument is as follows. In Appendix \[sec:maptosl2\] we show that there is a unique lift from the ${\mathfrak{sl}}(2)$ sector Hamiltonian to the ${\mathfrak{osp}}(4|2)$ sector. The (Lie algebra invariant) R-matrix only acts on two sites at a time. Also, recall that there is a one-to-one map between irreducible modules in the tensor product of two-sites in the ${\mathfrak{sl}}(2)$ subsector and those in the ${\mathfrak{osp}}(4|2)$ sector. Then the ${\mathfrak{osp}}(4|2)$ sector Hamiltonian must take the form that would follow from an ${\mathfrak{osp}}(4|2)$ R-matrix construction (assuming the existence of such R-matrices). The Yangian construction of the previous section confirms the existence of R-matrices for the ${\mathfrak{osp}}(4|2)$ spin-chain modules.
In this section we review the general R-matrix construction of the transfer matrix and conserved charges for an alternating spin chain. Based on the above argument, we then apply this construction to the ${\mathfrak{osp}}(4|2)$ sector. We deduce the action of the ${\mathfrak{osp}}(4|2)$ R-matrix on the spin-chain modules from the universal ${\mathfrak{sl}}(2)$ R-matrix, obtaining another expression for the Hamiltonian. This new expression for the Hamiltonian will enable us to obtain the full ${\mathfrak{osp}}(6|4)$ two-loop dilatation generator in the next section.
The transfer matrix and local conserved charges
-----------------------------------------------
The following discussion parallels the recent construction for the alternating ${\mathfrak{su}}(4)$ spin chain [@Minahan:2008hf], and the original general construction of [@deVega:1991rc].
We start with a R-matrix, which satisfies the Yang-Baxter equation $$R_{12}(u-v)R_{13}(u)R_{23}(v)= R_{23}(v)R_{13}(u)R_{12}(u-v), \label{eq:ybe}$$ where $R_{ij}$ is the R-matrix acting on sites $i$ and $j$. For alternating chains, it is sufficient for the R-matrix to satisfy the Yang-Baxter for each of the $2^3=8$ possible ways to assign one of the two representations to sites $1,2,3$.
Now we consider an alternating chain with the two representations distinguished by the presence or absence of a bar, $1, \bar{2} \ldots (2L-1), \overline{2L}$. We build two monodromy matrices from the R-matrix, $$\mathcal{T}_a(u, \alpha) = \prod_{i=1}^L R_{a,2i-1}(u)R_{a, \bar{2i}}(u+\alpha), \quad \mathcal{T}_{\bar{b}}(u, \beta) = \prod_{i=1}^L R_{\bar{b},2i-1}(u+\beta)R_{\bar{b}, \bar{2i}}(u).$$ Since the R-matrices satisfy the Yang-Baxter equation, these monodromy matrices also satisfy the Yang-Baxter equations R\_[ab]{}(u-v) \_a(u) \_b(v)[[&=&]{}]{}\_b(v) \_a(u)R\_[ab]{}(u-v), [\
]{}R\_[|[a]{}|[b]{}]{}(u-v) \_[|[a]{}]{}(u) \_[|[b]{}]{}(v) [[&=&]{}]{}\_[|[b]{}]{}(v) \_[|[a]{}]{}(u)R\_[|[a]{}|[b]{}]{}(u-v). Moreover, if $\beta=-\alpha$, which we will choose from now on, the mixed Yang-Baxter equation is also satisfied $$R_{a\bar{b}}(u+\alpha-v) \mathcal{T}_{a}(u, \alpha)\mathcal{T}_{\bar{b}}(v, -\alpha)= \mathcal{T}_{\bar{b}}(v, -\alpha) \mathcal{T}_{a}(u,\alpha)R_{a\bar{b}}(u+\alpha-v).$$ Taking the trace, and using the invertibility of the R-matrix, we infer that the transfer matrices, $$T(u, \alpha) = \mathrm{Tr}_a \mathcal{T}_a(u, \alpha), \quad \bar{T}(u, -\alpha) = \mathrm{Tr}_{\bar{b}} \mathcal{T}_{\bar{b}}(u, -\alpha),$$ satisfy $${[T(u, \alpha),T(v, \alpha)]}=0, \quad {[\bar{T}(u, -\alpha),\bar{T}(v, -\alpha)]}=0, \quad {[T(u, \alpha),\bar{T}(v, -\alpha)]}=0.$$ In particular, this implies the existence of (up to) $2L$ commuting generators. The expansion of $T(u, \alpha)$ about $u=0$ gives $L$ commuting generators, which also commute with the $L$ commuting generators coming from the expansion of $\bar{T}(u, -\alpha)$ about $u=0$. However, we will only consider the first two terms in the expansions. The leading terms, the transfer matrices evaluated at zero spectral parameter, yield the generators \_1 [[&=&]{}]{}\_[i=1]{}\^[L-1]{}R\_[2i+3, 2i+1]{}(0) \_[i=1]{}\^L R\_[2i-1,2i]{}() , [\
]{} |\_1 [[&=&]{}]{}\_[i=1]{}\^[L]{}R\_[2i-1, 2i]{}(-) \_[i=1]{}\^[L-1]{} R\_[2i,2i+2]{}(0). Here we have stopped including bars to distinguish representations, which are of one type for odd-numbered sites, and the other for even-numbered sites. Also, these expressions require that, when acting on two identical representations, at $u=0$ the $R$-matrix is proportional to the permutation generator, which will be the case for the R-matrices we consider. Simplifying the product using $R^{-1}(\alpha) = R(-\alpha)$, another property of our R-matrices, we obtain the two-site shift generator $$\mathcal{Q}_1\bar{\mathcal{Q}}_1 = \prod_{i=1}^{L-1}R_{2i+3, 2i+1}(0)\prod_{i=1}^{L-1} R_{2i,2i+2}(0).$$ Expanding to $\mathcal{O}(u)$, the next charges are defined through T(u)[[&=&]{}]{}\_1 + u \_1 \_2 + …, [\
]{}|[T]{}(u) [[&=&]{}]{}|\_1 + u |\_1 |\_2 + … The charges have nearest-neighbor and next-nearest-neighbor contributions $$\mathcal{Q}_2 = (\mathcal{Q}_2)_{NN} + (\mathcal{Q}_2)_{NNN}, \quad \bar{\mathcal{Q}}_2 = (\bar{\mathcal{Q}}_2)_{NN} + (\bar{\mathcal{Q}}_2)_{NNN}.$$ The nearest-neighbor contribution can be chosen symmetrically as $$(\mathcal{Q}_2)_{NN} = \sum_{i=1}^{2L} R_{i,i+1}(-\alpha)R'_{i,i+1}(\alpha), \quad (\bar{\mathcal{Q}}_2)_{NN} = \sum_{i=1}^{2L} R_{i,i+1}(\alpha)R'_{i,i+1}(-\alpha)$$ provided the next nearest-neighbor contributions are (\_2)\_[NNN]{} [[&=&]{}]{}\_[i=1]{}\^L ( R\_[2i-1, 2i]{}()R\_[2i-1, 2i+1]{}’(0)R\_[2i-1, 2i+1]{}(0)R\_[2i-1, 2i]{}(-) [\
&&]{}+ R\_[2i, 2i+1]{}(-)R\_[2i-1, 2i+1]{}’(0)R\_[2i-1, 2i+1]{}(0)R\_[2i, 2i+1]{}()), [\
]{}(|\_2)\_[NNN]{} [[&=&]{}]{}\_[i=1]{}\^L (R\_[2i, 2i+1]{}(-)R\_[2i, 2i+2]{}’(0)R\_[2i, 2i+2]{}(0)R\_[2i, 2i+1]{}() [\
&&]{}+ R\_[2i+1, 2i+2]{}()R\_[2i, 2i+2]{}’(0)R\_[2i, 2i+2]{}(0)R\_[2i+1, 2i+2]{}(-)). To compute the next-nearest-neighbor terms we inserted a factor of $R(0)R^{-1}(0)=1$, and to obtain symmetric expressions we used the vanishing commutator between the $\mathcal{Q}_i$. There are additional possibilities that differ by chain derivatives.
The ${\mathfrak{osp}}(4|2)$ case
--------------------------------
As explained above, the relevant ${\mathfrak{osp}}(4|2)$ R-matrices are determined by the universal R-matrix of ${\mathfrak{sl}}(2)$ [@Kulish:1981gi]. This also occurred for the one-loop $\mathcal{N}=4$ SYM spin [@Beisert:2003yb]. The result is a sum over the irreducible representations of the tensor product of two sites, labeled by ${\mathfrak{osp}}(4|2)$ spin $j$, weighted by a certain ratio of Gamma functions, $$\label{eq:sl2universal}
R_{12}(u) = \sum_{j} (-1)^j f(cu) \frac{\Gamma(j+1+c u)\Gamma(1- c u)}{\Gamma(j+1-c u)\Gamma(1+ c u)}\mathcal{P}^{(j)}_{12}.$$ Here $\mathcal{P}^{(j)}$ is the projector that acts as the identity on ${\mathfrak{osp}}(4|2)$ states with spin $j$, and gives zero on all other states. Using (\[eq:42conjugateproduct\]) and (\[eq:42likeproduct\]), for the case of sites $1,2$ in alternate representations, the sum is over all nonnegative $j$, while for identical representations, $j$ takes values $n-{{{\textstyle\frac{1}{2}}}}$ for all nonnegative $n$. For our purposes, the function of the spectral parameter $f$ and the constant $c$ can be replaced simply with freedom in the normalization of the local charges. We set $c$ to 1 and choose $f$ to cancel any overall factors of $\pm i$ from the $(-1)^j$ factors. The Yang-Baxter equation (\[eq:ybe\]) is still satisfied even if we choose different (constant) values of $f$ for different pairs of representations. As in the previous section, with these conventions $R^{-1}(u)= R(-u)$, and $R(0)$ acts as the permutation operator on identical representations.
We are almost ready to simply insert the expression (\[eq:sl2universal\]) for the R-matrices into the expressions for $\mathcal{Q}_2$ and $\bar{\mathcal{Q}}_2$ given in the last subsection, but there are four coefficients to fix. These are $\alpha$, the coefficients of $\mathcal{Q}_2$ and $\bar{\mathcal{Q}}_2$, and the coefficient of the identity operator, which we are also free to add without spoiling integrability. In principle one could compute four eigenvalues to fix these coefficients. However, since the Hamiltonian is even under spin-chain parity, $\alpha$ must be zero. Also, symmetry under charge conjugation implies that $\mathcal{Q}_2$ and $\bar{\mathcal{Q}}_2$ have equal coefficients. We have found the final two coefficients by comparison with eigenvalues of the (previous expression for the) Hamiltonian. $\mathcal{Q}_2$ and $\bar{\mathcal{Q}}_2$ have coefficient $\lambda^2/2$ and the identity has coefficient $2 \lambda^2 \log 2$, \_2 [[&=&]{}]{}(\_2 + |\_2)\_[|=0]{} + 4 L 2 [\
]{}[[&=&]{}]{}\_[i=1]{}\^[2L]{} ( R\_[i, i+1]{}\^[(0)]{}R’\_[i, i+1]{}(0) + 2 2 + [\
&&]{} (R\_[i, i+1]{}\^[(0)]{} R\_[i, i+2]{}\^[(0)]{}R\_[i, i+2]{}’(0)R\_[i, i+1]{}\^[(0)]{} + R\_[i+1, i+2]{}\^[(0)]{} R\_[i, i+2]{}\^[(0)]{}R\_[i, i+2]{}’(0)R\_[i+1, i+2]{}\^[(0)]{}) ). $\mathcal{Q}_2$ and $\bar{\mathcal{Q}}_2$ combine nicely here, and we used the more compact notation $R^{(0)}$ for the R-matrix evaluated at zero spectral parameter, $R(0)$. Next, evaluating (\[eq:sl2universal\]) and its derivative at $u=0$, the dependence on $j$ reduces to factors of $(-1)^j$ and the harmonic numbers, $S_1(j)$ [^14], \[eq:osp42projectorham\] \_2 [[&=&]{}]{}\_[i=1]{}\^[2L]{} (2 2 + \_[j=0]{}\^S\_1(j)\^[(j)]{}\_[i, i+1]{} [\
&&]{} + \_[j\_1,j\_2,j\_3=0]{}\^(-1)\^[j\_1+j\_3]{} S\_1(j\_2-) (\^[(j\_1)]{}\_[i, i+1]{} \^[(j\_2-1/2)]{}\_[i, i+2]{}\^[(j\_3)]{}\_[i, i+1]{} + \^[(j\_1)]{}\_[i+1, i+2]{} \^[(j\_2-1/2)]{}\_[i, i+2]{}\^[(j\_3)]{}\_[i+1, i+2]{} ) ). [\
&&]{} Using the explicit form for the projectors given in Appendix \[sec:projectors\] and `Mathematica`, we have checked that this spin-chain Hamiltonian for the two-loop (${\mathfrak{osp}}(4|2)$ sector) dilatation generator leads to the correct two-magnon S-matrix, and that its spectrum for low numbers of excitations and short states agrees with Bethe ansatz predictions and the alternative expression for the Hamiltonian as the anticommutator of $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$. Still, the anticommutator of $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ gives a slightly more general form that applies to two-site states also, while this R-matrix expression requires chains of length four.
The lift to the complete ${\mathfrak{osp}}(6|4)$ chain \[sec:lift\]
===================================================================
Here we derive the two-loop planar ${\mathfrak{osp}}(6|4)$ dilatation generator using superconformal invariance. We then observe that this spin-chain Hamiltonian is integrable, assuming the existence of an ${\mathfrak{osp}}(6|4)$ R-matrix for like or conjugate spin-chain modules. We argue that there is no reason to doubt this assumption.
Unique lift from ${\mathfrak{osp}}(4|2)$ to ${\mathfrak{osp}}(6|4)$ \[sec:uniquelift\]
--------------------------------------------------------------------------------------
By adding chain derivatives to replace one-site or two-site interactions with three-site interactions, one can write the ${\mathfrak{osp}}(6|4)$ two-loop Hamiltonian completely in terms of a Hamiltonian density $\mathcal{H}_{i,i+1,i+2}$ of three-site to three-site interactions[^15], $$\mathcal{H}{\mathopen{\big|}X_1\ldots X_{2L}\mathclose{\bigr \rangle}} = \sum_{i=1}^{2L} \mathcal{H}_{i,i+1,i+2} {\mathopen{\big|}X_1\ldots X_{2L}\mathclose{\bigr \rangle}}.$$ ${\mathfrak{osp}}(6|4)$ invariance allows us to use the freedom to add chain derivatives so the Hamiltonian *density* commutes with the leading-order ${\mathfrak{osp}}(6|4)$ generators. Therefore the Hamiltonian is completely specified by the Hamiltonian density’s action on all three-site (without cyclicity condition) highest-weight states. Furthermore, the Hamiltonian density mixes highest-weight states only with other highest-weight states with the same ${\mathfrak{su}}(4)$ Cartan charges, classical dimension, and Lorentz spin. For fixed values of these five Cartan charges there are finitely many linearly-independent three-site highest-weight states, ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$. In the sector with basis ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$, the Hamiltonian density acts as $$\mathcal{H}_{123} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} = C^J_I {\mathopen{\big|}\Omega_J\mathclose{\bigr \rangle}},$$ for some coefficients $C^J_I$. The full set of such $C^J_I$ gives the Hamiltonian density and the Hamiltonian. If the Cartan charges of the$ {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ satisfy a BPS condition, then these states are in an ${\mathfrak{osp}}(4|2)$ sector[^16], and we have already determined the action of the Hamiltonian (density). If not, as we show below, one can act with a combination of supercharges $\prod {\mathfrak{Q}}$ (determined by the Cartan charges only) so that $$\big(\prod {\mathfrak{Q}}\big) {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} = M^{I'}_I {\mathopen{\big|}\Omega'_{I'}\mathclose{\bigr \rangle}}, \label{eq:maptoosp42}$$ where the ${\mathopen{\big|}\Omega_{I'}\mathclose{\bigr \rangle}}$ are contained within an ${\mathfrak{osp}}(4|2)$ sector and the matrix $M$ is invertible. We have $$\mathcal{H}_{123} {\mathopen{\big|}\Omega'_{I'}\mathclose{\bigr \rangle}} = C'^{J'}_{I'} {\mathopen{\big|}\Omega'_{J'}\mathclose{\bigr \rangle}},$$ for coefficients $C'^{J'}_{I'}$ determined by the known ${\mathfrak{osp}}(4|2)$ sector Hamiltonian. Then, the needed coefficients of the full Hamiltonian are given by $$C = M C' M^{-1}, \quad \text{or} \quad C^J_I = M^{I'}_{I} C'^{J'}_{I'} ((M)^{-1})^{J}_{J'}.$$ Therefore, as claimed, there is a unique lift of the ${\mathfrak{osp}}(4|2)$ sector Hamiltonian to ${\mathfrak{osp}}(6|4)$. We will give this Hamiltonian below, after first proving the existence of the invertible map (\[eq:maptoosp42\]). In Appendix \[sec:maptosl2\] we go one step further and use the same type of argument to show that there is a unique lift from the ${\mathfrak{sl}}(2)$ sector to the ${\mathfrak{osp}}(4|2)$ sector (and therefore to ${\mathfrak{osp}}(6|4)$).
Invertible map between ${\mathfrak{osp}}(6|4)$ and ${\mathfrak{osp}}(4|2)$ states \[sec:invertiblemap\]
-------------------------------------------------------------------------------------------------------
We will first show that for any basis ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ of three-site highest-weight states with identical Cartan charges there exists a product of supercharges that maps the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ to a ${\mathfrak{osp}}(4|2)$ sector. It will be straightforward afterward to show that this map is invertible.
Again we choose the roots of ${\mathfrak{osp}}(6|4)$ so that highest-weight states are annihilated by the following raising generators, $${\mathfrak{L}}^2_1, \quad {\mathfrak{K}}, \quad {\mathfrak{R}}^i_j \, \, i >j, \quad {\mathfrak{S}}. \label{eq:rraise}$$ Act on the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ with ${\mathfrak{Q}}_{12,1}$. If the Cartan charges of the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ satisfy the BPS condition for ${\mathfrak{Q}}_{12,1}$ and ${\mathfrak{S}}^{12,1}$, this will vanish, the ${\mathopen{\big|}\Omega\mathclose{\bigr \rangle}}_I$ are in an ${\mathfrak{osp}}(4|2)$ sector, and the needed map is trivial. So we can assume $ {\mathfrak{Q}}_{12,1} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}\neq 0$ .
Since the highest-weight states ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ are annihilated by the ${\mathfrak{R}}$ of (\[eq:rraise\]) it is a simple exercise to check that now the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ must transform as $\mathbf{4}$ or $\mathbf{\bar{4}}$ under ${\mathfrak{R}}$. This is because ${\mathfrak{Q}}_{12,1}$ has nonvanishing action only on lower ${\mathfrak{R}}$ indices $3$ or $4$ or upper indices $1$ or $2$, and because the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ have spin-chain length three. For simplicity, assume the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ transform as $\mathbf{4}$. With appropriate interchange of indices this argument can be repeated for the $\mathbf{\bar{4}}$ case.
Next consider $${\mathfrak{Q}}_{13,1} {\mathfrak{Q}}_{12,1} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}.$$ If these vanish then the ${\mathfrak{Q}}_{12,1}{\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ satisfy the BPS condition for ${\mathfrak{Q}}_{13,1}$ and ${\mathfrak{S}}^{13,1}$, and therefore ${\mathfrak{Q}}_{12,1}$ gives the required map to a ${\mathfrak{osp}}(4|2)$ sector[^17]. To see this use the last commutation relation of (\[eq:anticommutators\]) and the annihilation of the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ by the ${\mathfrak{R}}$ raising generator ${\mathfrak{R}}^3_2$. Similarly, if the $${\mathfrak{Q}}_{14,1}{\mathfrak{Q}}_{13,1} {\mathfrak{Q}}_{12,1} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$$ vanish, ${\mathfrak{Q}}_{13,1} {\mathfrak{Q}}_{12,1}$ gives the map to an ${\mathfrak{osp}}(4|2)$ subsector.
Finally, assume that ${\mathfrak{Q}}_{14,1}{\mathfrak{Q}}_{13,1} {\mathfrak{Q}}_{12,1} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} \neq 0$. With respect to ${\mathfrak{su}}(4)$ the scalars of the modules could transform as (fundamental, anti-fundamental, fundamental) or the conjugate. Since the argument would be essentially the same in either case, we assume the first possibility. For Lorentz spin $s$ and classical dimension $N + 3/2$ , the most general possibility is $${\mathfrak{Q}}_{14,1}{\mathfrak{Q}}_{13,1} {\mathfrak{Q}}_{12,1} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} = \sum_{n_{i,j}} a_I(n_{i, j}) {\mathopen{\big|}\phi_1^{(n_{1,1}, n_{2,1})} \psi_1^{(n_{1,2}, n_{2,2})}\phi_1^{(N + s -n_{1,1} - n_{1,2},N-s -n_{2,1} - n_{2,2})}\mathclose{\bigr \rangle}} \label{eq:genformdescendant}$$ where the $n_{i,j}$ are nonnegative integers such that all superscript arguments are also nonnegative integers consistent with spin statistics, and $a_I(n_{i,j})$ are some coefficients. Now ${\mathfrak{S}}^{12,2}$ must still annihilate (\[eq:genformdescendant\]), since it anticommutes with ${\mathfrak{Q}}_{13, 1}$ and ${\mathfrak{Q}}_{14, 1}$ and gives ${\mathfrak{R}}^2_1$ when anticommuted with ${\mathfrak{Q}}_{12, 1}$. ${\mathfrak{S}}^{12,2}$ acts on single-sites with lower $1$ indices as \^[12,2]{} [\_1\^[(n\_1, n\_2)]{}]{} [[&=&]{}]{}b(n\_2) [(|\^2)\^[(n\_1, n\_2-1)]{}]{}, b(n\_2) = 0 n\_2=0. [\
]{}\^[12,2]{} [\_1\^[(n\_1, n\_2)]{}]{} [[&=&]{}]{}c(n\_2) [(|\^2)\^[(n\_1, n\_2-1)]{}]{}, c(n\_2) = 0 n\_2=0, where all that matters here are the quantum number of the states and whether the coefficients $b$ and $c$ are nonzero. From this it follows that ${\mathfrak{S}}^{12,2}$ annihilates (\[eq:genformdescendant\]) only if all of the second Lorentz index excitation numbers $n_{2,1}$, $n_{2,2}$ and $N-s-n_{2,1}-n_{2,2}$ are zero. But then all of the states of (\[eq:genformdescendant\]) are clearly in an ${\mathfrak{osp}}(4|2)$ (${\mathfrak{sl}}(2)$) sector, and in this case ${\mathfrak{Q}}_{14,1}{\mathfrak{Q}}_{13,1} {\mathfrak{Q}}_{12,1}$ gives the required map.
We have shown that applying a product of supercharges (zero, one, two, or three depending on the Cartan charges) gives a map to an ${\mathfrak{osp}}(4|2)$ sector. This is as abbreviated in (\[eq:maptoosp42\]). To show that this map is invertible, we simply need to show the linear independence of the $$\big(\prod {\mathfrak{Q}}\big){\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}},$$ where as usual we are focusing on a linearly-independent basis given by three-site highest-weight states ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ with identical Cartan charges. If there were some linear combination of the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ that were annihilated by the relevant $ \prod {\mathfrak{Q}}$, the above construction implies this linear combination would satisfy another BPS condition. This is a contradiction because the BPS conditions depend only on the Cartan charges (and because the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ are assumed linearly independent). Therefore, the maps of (\[eq:maptoosp42\]) to the ${\mathfrak{osp}}(4|2)$ sector are invertible, completing the proof.
The two-loop ${\mathfrak{osp}}(6|4)$ spin-chain Hamiltonian and integrability
-----------------------------------------------------------------------------
Recall that there is a one-to-one mapping between highest-weight two-site states in the ${\mathfrak{osp}}(4|2)$ subsector and the full theory, and that these highest-weight states have the same value of ${\mathfrak{osp}}(4|2)$ or ${\mathfrak{osp}}(6|4)$ spin. Therefore, the unique lift is given by replacing projections onto ${\mathfrak{osp}}(4|2)$ spin in (\[eq:osp42projectorham\]) with projections onto the corresponding ${\mathfrak{osp}}(6|4)$ spin. Now the complete planar Hamiltonian is given by the same formal expression as for the ${\mathfrak{osp}}(4|2)$ sector, \[eq:completeham\] \_2 [[&=&]{}]{}\_[i=1]{}\^[2L]{} ( 2 2 + \_[j=0]{}\^S\_1(j)\^[(j)]{}\_[i, i+1]{} [\
&&]{} + \_[j\_1,j\_2,j\_3=0]{}\^(-1)\^[j\_1+j\_3]{} S\_1(j\_2-) (\^[(j\_1)]{}\_[i, i+1]{} \^[(j\_2-1/2)]{}\_[i, i+2]{}\^[(j\_3)]{}\_[i, i+1]{} + \^[(j\_1)]{}\_[i+1, i+2]{} \^[(j\_2-1/2)]{}\_[i, i+2]{}\^[(j\_3)]{}\_[i+1, i+2]{} ) ). [\
&&]{} Of course, the differences from the ${\mathfrak{osp}}(4|2)$ sector Hamiltonian are that the projectors act on the full ${\mathfrak{osp}}(6|4)$ modules, and the $j_i$ correspond to ${\mathfrak{osp}}(6|4)$ spin. Again, there is a sum over the spin-chain sites labeled by $i$, with $\mathcal{P}^{(j)}_{i,k}$ acting on sites $i$ and $k$. It would be nice to have the expressions for these projectors in components, extending those given for the ${\mathfrak{osp}}(4|2)$ sector in Appendix \[sec:projectors\].
Assuming the existence of an ${\mathfrak{osp}}(6|4)$ R-matrix acting on like or conjugate pairs of the two types of ${\mathfrak{osp}}(6|4)$ modules, a parallel derivation to the one given for ${\mathfrak{osp}}(4|2)$ in Section \[sec:rmatrix\] would apply, and would lead to the complete Hamiltonian given in (\[eq:completeham\]). Therefore, up to this assumption, we have shown that planar $\mathcal{N}=6$ superconformal Chern-Simons theory is integrable at two-loops. A similar assumption was used in the $\mathcal{N}=4$ SYM case [@Beisert:2003yb].
As noted earlier, the existence of Yangian symmetry for the ${\mathfrak{osp}}(4|2)$ sector implies the existence of the corresponding ${\mathfrak{osp}}(4|2)$ R-matrices for the modules appearing in the spin chain. This is indication that there is no problem constructing R-matrices for ${\mathfrak{osp}}$ algebras. It seems that it would be sufficient to confirm that the Serre relations (\[eq:serre\]) are satisfied for the complete ${\mathfrak{osp}}(6|4)$ modules, since that would imply the existence of a ${\mathfrak{osp}}(6|4)$ Yangian and the corresponding R-matrices. However, such a check is beyond the scope of this work.
In fact, it reasonable to assume even a *universal* ${\mathfrak{osp}}(6|4)$ R-matrix[^18], which would give the R-matrix for arbitrary ${\mathfrak{osp}}(6|4)$ representations. There is a general construction of universal R-matrices for Yangians of bosonic simple Lie algebras [@Khoroshkin:1994uk] later extended to ${\mathfrak{sl}}(m|n), m\neq n$ [@Stukopin:2005st]. The latter construction was further modified for a recent derivation of the leading-order spin-chain S-matrices of $\mathcal{N}=4$ SYM and ABJM [@Spill:2008yr]. It should be possible to apply a similar construction to ${\mathfrak{osp}}$ algebras as well[^19].
Finally, we emphasize that the invariance of the Hamiltonian density with respect to ${\mathfrak{osp}}(6|4)$ only applies at leading order. At the next order in $\lambda$, the Hamiltonian will only commute exactly with at most a proper subset of the ${\mathfrak{osp}}(6|4)$ supercharges, while the commutators with the other supercharges will vanish only when applied to cyclic alternating chains. This can be see already in the ${\mathfrak{osp}}(4|2)$ sector. While the expression (\[eq:osp42projectorham\]) for the Hamiltonian commutes with $\mathcal{O}(\lambda^0)$ ${\mathfrak{osp}}(4|2)$ generators manifestly, it only commutes with the $\mathcal{O}(\lambda^1)$ supercharges $\hat{{\mathfrak{Q}}}$ and $\hat{{\mathfrak{S}}}$ up to gauge transformations. A similar phenomenon occurs in the ${\mathfrak{psu}}(1,1|2)$ sector of $\mathcal{N}=4$ SYM [@Beisert:2007sk]. It can be traced back to the algebra of supersymmetry variations only closing up to gauge transformations, which appear for the spin-chain as (\[eq:qqhatgauge\]). Importantly, these gauge transformations appear at subleading order and do not obstruct the leading-order ${\mathfrak{osp}}(6|4)$ invariance used to lift the Hamiltonian to the full theory.
Conclusions \[sec:conc\]
========================
We have shown that the two-loop planar dilatation generator of $\mathcal{N}=6$ superconformal Chern-Simons theory is fixed by superconformal invariance up to overall normalization, and can be written compactly as in (\[eq:completeham\]). Through a Yangian construction we have proved integrability for the ${\mathfrak{osp}}(4|2)$ sector, and for the full model assuming the existence of an ${\mathfrak{osp}}(6|4)$ R-matrix. This confirms the conjectured two-loop ${\mathfrak{osp}}(6|4)$ Bethe equations of Minahan and Zarembo. We also analytically computed the twist-one spectrum of the model, both from the Bethe equations and from the dilatation generator, finding $\Delta_s= 4 \lambda^2(S_1(s)-S_{-1}(s)) + \mathcal{O}(\lambda^3)$.
It seems unreasonable to doubt complete two-loop integrability. Still, an explicit proof would be better. Constructing the ${\mathfrak{osp}}(6|4)$ Yangian that commutes with the Hamiltonian (\[eq:completeham\]) may be the simplest approach.
Further algebraic constructions of spin-chain generators for the ABJM gauge theory are possible. It would be good to calculate the complete $\mathcal{O}(\lambda)$ generators, extending the calculation here of two $\mathcal{O}(\lambda)$ supercharges acting within the ${\mathfrak{osp}}(4|2)$ sector. It would also be wonderful to obtain higher-loop corrections. For $\mathcal{N}=4$ SYM there is evidence of recursive structure to these corrections, at least in sectors of the theory [@Zwiebel:2008gr; @Bargheer:2008jt]. [@Bargheer:2008jt] also argues that such recursive structure appears in a compact sector for the ABJM spin chain (assuming higher-loop integrability). Perhaps this recursive structure extends to the ${\mathfrak{osp}}(4|2)$ sector, which could make an algebraic computation of the four-loop dilatation generator tractable. This would still require a direct field theory calculation of $h(\lambda)$, which appears in the one-magnon dispersion relation.
We found it useful to work with a representation of the spin module in terms of continuous variables, a light-cone superspace basis. We expect this and similar representations to be helpful for gaining new insights about the $\mathcal{N}=4$ SYM and the ABJM spin chains.
It is straightforward to lift the ${\mathfrak{osp}}(4|2)$ dilatation generator to the complete ${\mathfrak{osp}}(4|2)$ two-loop dilatation generator, including nonplanar corrections. $\hat{{\mathfrak{Q}}}$ ($\hat{{\mathfrak{S}}}$) has a unique nonplanar lift since it acts on one initial (final) module. The nonplanar two-loop dilatation generator than follows from the anticommutator (\[eq:qhatshatacomm\]), and this should match the ${\mathfrak{su}}(2) \times {\mathfrak{su}}(2)$ subsector result of [@Kristjansen:2008ib]. There is a very similar observation for the ${\mathfrak{psu}}(1,1|2)$ sector of $\mathcal{N}=4$ SYM [@Zwiebel:2005er]. Also for $\mathcal{N}=4$ SYM, an analysis of the gauge group structure of the Feynman diagrams that contribute to one-loop anomalous dimensions led to a unique lift from the planar limit to the full nonplanar theory [@Beisert:2003jj]. The product gauge group and bifundamental fields of ABJM probably make the corresponding two-loop lift here more difficult. Still, a lift from the nonplanar ${\mathfrak{osp}}(4|2)$ sector to the full theory should be possible, especially given the one-to-one maps of Section \[sec:lift\] between highest-weight three-site states of the ${\mathfrak{osp}}(4|2)$ and ${\mathfrak{osp}}(6|4)$ sectors.
The ${\mathfrak{osp}}(4|2)$ sector dilatation generator, including nonplanar corrections, should be useful for seeking the gauge duals of 1/12 BPS black holes in $AdS_4$. For recent related work and comments see [@Bhattacharya:2008bja]. It would also be interesting to calculate (nonplanar) anomalous dimensions of near ($1/12$) BPS states, as done previously [@Berkooz:2008gc] for near ($1/16$) BPS operators in $\mathcal{N}=4$ SYM.
Finally, this work’s confirmation of weak-coupling integrability further motivates study of the many topics related to the integrability of the ABJM gauge theory and its string theory dual. As for $AdS_5/CFT_4$, these topics range far beyond the weak-coupling spin chain.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I am very grateful to Niklas Beisert, Andrei Belitsky, Diego Hofman, and Eric Ragoucy for discussions. I also thank Georgios Papathanasiou and Marcus Spradlin for pointing out important typos in Appendix D of the first version of this work.
Additional light-cone superspace computations \[sec:lightcone\]
================================================================
Working in light-cone superspace given by (\[eq:definelightcone\]) often simplifies calculations. We gave the actions of $\hat{{\mathfrak{Q}}}$ in light-cone superspace in (\[eq:lightconeqhat\]), and in this appendix we give the corresponding expression for $\hat{{\mathfrak{S}}}$. We then use this expression to compute the one-site interactions of $\mathcal{H}$ and and the two-loop twist-one spectrum.
$\hat{{\mathfrak{S}}}$ in light-cone superspace
-----------------------------------------------
We abbreviate $${\mathopen{\big|}x_1, \theta_1, \bar{\eta}_1;x_2, \bar{\theta}_2, \eta_2 ;
x_3, \theta_3, \bar{\eta}_3\mathclose{\bigr \rangle}} = {\mathopen{\big|}1;\bar{2},3\mathclose{\bigr \rangle}}.$$ Then, $\hat{{\mathfrak{S}}}$ acts as \[eq:lightconeshat\] [1;|[2]{},3]{} [[&=&]{}]{}- \_0\^1 \_0\^1 [\
&&]{}(( \_[bc]{} \_2\^b \_3\^c \_[x\_2]{} + \_|\_2\^ |\_3\^ \_[x\_3]{} ) [(1-t\_2) x\_1 + t\_1 t\_2 x\_2 + (1-t\_1) t\_2x\_3, \_1, (1-t\_2) |\_1]{}\
&+& ( \_[bc]{} \_1\^b \_2\^c \_[x\_2]{} + \_|\_1\^ |\_2\^ \_[x\_1]{} ) [ (1-t\_1)t\_2 x\_1 + t\_1 t\_2 x\_2 + (1-t\_2) x\_3, \_3, (1-t\_2) |\_3]{}\
&+& |\^\_2 |\_ \_[bc]{} \_1\^b \_3\^c [ t\_1 t\_2 x\_1 + (1-t\_2) x\_2 + (1-t\_1) t\_2 x\_3,, t\_2 |]{} ). [\
&&]{} To check this, first expand both sides according to (\[eq:definelightcone\]). Using the Beta integral one can then show that this is equivalent to the action of $\hat{{\mathfrak{S}}}$ given previously (the Hermitian conjugate of the $\hat{{\mathfrak{Q}}}$ action (\[eq:qhatsolution\])). Since the light-cone superspace basis is not manifestly Hermitian, here $\hat{{\mathfrak{S}}}$ takes a more involved form than $\hat{{\mathfrak{Q}}}$ does. To obtain the action on the conjugate state, ${\mathopen{\big|}\bar{1};2\;\bar{3}\mathclose{\bigr \rangle}}$, remove the overall minus sign and replace all $\theta,\eta$ with $\bar{\theta},\bar{\eta}$ and vice-versa (interchanging all Latin and Gothic indices).
One-site interactions of $\mathcal{H}$
--------------------------------------
Because physical spin-chain states are cyclic and have at least two sites, in general one can not uniquely classify interactions as one-site interactions rather than two-site (or three-site) interactions. However, we can isolate the contribution to $\mathcal{H}$ from $\hat{{\mathfrak{Q}}}$ replacing one site with three new sites followed by $\hat{{\mathfrak{S}}}$ replacing the same three sites with a single site. This is what we mean here by $\mathcal{H}_{\text{one site}}$. Using the light-cone superspace expressions (\[eq:lightconeqhat\]) and (\[eq:lightconeshat\]), we find the following expression for $ \mathcal{H}_{\text{one site}}{\mathopen{\big|}x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}$, $$\begin{gathered}
\frac{1}{ 2 \pi} \int_0^1 \frac{\mathrm{d} t_1}{\sqrt{t_1}\sqrt{1-t_1}} \int_0^1 \frac{\mathrm{d}t_2}{t_2\sqrt{1-t_2}} \int_0^x \mathrm{d}\!y \, \times \notag \\
\bigg( 2 (\partial_{y_2} + \partial_{y_3}) {\mathopen{\big|}(1-t_2)x + t_1t_2 y_2 + (1-t_1)t_2 y_3, \theta, (1-t_2) \bar{\eta}\mathclose{\bigr \rangle}}_{|^{y_2,y_3=y}}
\notag \\
+ 2 \Big(\bar{\eta}^{\mathfrak{c}}\bar{\partial}_{\mathfrak{c}} \partial_x -\theta^c \partial_{c} \partial_{y_2} \Big){\mathopen{\big|}(1-t_1)t_2 x + t_1t_2y_2 + (1-t_2)y,\theta, (1-t_2) \bar{\eta}\mathclose{\bigr \rangle}}_{|^{y_2=y}}
\notag \\
+ 2 (\partial_{y_1} + \partial_{y_2}) {\mathopen{\big|}(1-t_1)t_2 y_1 + t_1t_2 y_2 + (1-t_2) x, \theta, (1-t_2) \bar{\eta}\mathclose{\bigr \rangle}}_{|^{y_1,y_2=y}} \bigg)
\notag \\
+ \frac{1}{2 \pi} \int_0^1 \frac{\mathrm{d}t_1}{\sqrt{t_1}\sqrt{1-t_1}} \int_0^1 \frac{\mathrm{d}t_2}{t_2\sqrt{1-t_2}} 2 \bar{\eta}^{\mathfrak{a}} \bar{\partial}_{\mathfrak{a}} {\mathopen{\big|}x, \theta, t_2\bar{\eta}\mathclose{\bigr \rangle}}.
\end{gathered}$$ Simplifying and doing integrals of derivatives and Beta integrals yields \_[1]{} [[&=&]{}]{}\_0\^1 ( 2 [x, , (1-t\_2) |]{} -2 [(1-t\_2)x , , (1-t\_2) |]{}) [\
&&]{} + \_0\^1 \_0\^1 [\
&&]{} ( |\^ |\_ -t\_1 ) ([x, , (1-t\_2) |]{} -[(1-t\_1)t\_2 x, , (1-t\_2)|]{} ) [\
&&]{} + 2 |\^ |\_ [x, , |]{}. We now expand the light-cone basis in components, with the $x^m \theta$ terms giving the action on ${\mathopen{\big|}\phi^{(m)}\mathclose{\bigr \rangle}}$ and the $x^m \bar{\eta}$ terms giving the action on ${\mathopen{\big|}\bar{\psi}^{(m)}\mathclose{\bigr \rangle}}$. The integrand of the middle two lines then simplifies to a finite sum of products of powers of the $t_i$ and $(1-t_i)$, so that the integrals again reduce to Beta integrals. Doing these integrals and the sum, and combining with contributions from the first and last line finally yields the result $$\mathcal{H}_{\text{one site}}{\mathopen{\big|}\phi_a^{(m)}\mathclose{\bigr \rangle}} = 2 S_1(2m) {\mathopen{\big|}\phi_a^{(m)}\mathclose{\bigr \rangle}}, \quad \mathcal{H}_{\text{one site}}{\mathopen{\big|}\bar{\psi}_{\mathfrak{a}}^{(m)}\mathclose{\bigr \rangle}} = 2 S_1(2m+1) {\mathopen{\big|}\bar{\psi}_{\mathfrak{a}}^{(m)}\mathclose{\bigr \rangle}}.$$ Of course, the same coefficients appear for the conjugate scalars and fermions. We also used the identity: $$S_1(m-{{{\textstyle\frac{1}{2}}}}) + S_1(m) + 2 \log 2 = 2 S_1(2m),$$ where the definition of the harmonic numbers that applies for nonintegers is the difference involving the digamma function $$S(x) = \psi(x+1)-\psi(1).$$
Two-loop twist-one spectrum
----------------------------
Working in light-cone superspace using (\[eq:lightconeqhat\]) and (\[eq:lightconeshat\]), we find the parallel expression to (\[eq:delta2sum\]), \[eq:delta2integral\] \_[s,2]{} [[&=&]{}]{}\_[m=0]{}\^[s]{} , [\
]{}(m) [[&=&]{}]{}\_[ms]{} (2 S(2s) + 2 S(1)) - [\
&&]{}+ y ((1-t\_1)t\_2y +(1-t\_2))\^[m-1]{} [\
&&]{}+ y (1-t\_1t\_2)\^[s-m-1]{}y\^[s-m-1]{} [\
&&]{}- y (t\_1+(1-t\_1)y)\^[s-m-1]{} [\
&&]{}- 4 y y\^[s-m-1]{}. The ranges of integration variables are all zero to one. The first integral should only be included in the sum for $m>0$, and the last three only for $m < s$. To obtain (\[eq:delta2integral\]) we expanded the light-cone basis in components and absorbed (most of) the wavefunction and light-cone basis normalizations into the integrands. Next, the $y$ integrals are elementary, and, using binomial expansions, all of the $t_i$ integrals can be done using the Beta integral. The sums can also be done[^20], provided we use the identity $$\frac{\sqrt{\pi}\Gamma(s+{{{\textstyle\frac{1}{2}}}})}{2}\sum_{m=1}^s\frac{(-1)^{m}}{m\Gamma(m+3/2)\Gamma(s-m+{{{\textstyle\frac{1}{2}}}})} = 2 S_{-1}(s) - S_1(s) + 2 -\frac{(-1)^s}{s+{{{\textstyle\frac{1}{2}}}}}.$$ An equivalent version for integer $s$ that is also valid for more general $s$ is $$\frac{{}_3F_2(1,{{\textstyle\frac{3}{2}}},s+1; s+2, s+{{\textstyle\frac{5}{2}}};1)}{(s+1)(2s+1)(2s+3)}=S_1(\frac{s-1}{2}) - S_1(\frac{s}{2}) + \frac{1}{s+{{{\textstyle\frac{1}{2}}}}}.$$ We have proved this identity through a somewhat involved calculation. Key steps include considering the difference of the identity at $s=s'$ and at $s=s'+1$, using the series expansion of hypergeometric functions, reintroducing Beta integrals (as well as another elementary integral), and switching orders of summation and integration. It would be nice to find an elegant proof of this identity, or better, a more elegant way to evaluate $\Delta_{s,2}$ directly from the light-cone superspace expressions for the supercharges. In any case, the final result is as given in (\[eq:twistonespectrum\]), $$\Delta_{s,2} = 4 \Big(S_1(s)-S_{-1}(s)\Big).$$
Bethe ansatz solution for two-loop twist-one spectrum \[sec:bethe\]
===================================================================
Here we work with the $\eta=-1$ (leading-order) Bethe equations of [@Gromov:2008qe]. The ${\mathfrak{sl}}(2)$ sector highest-weight state with Lorentz spin $(s+1/2)$ has $s$ pairs of roots $u_{4, k}=\bar{u}_{4, k}$. Labeling both types of roots $u_k$, the Bethe equations reduce to $$\frac{u_k+i/2}{u_k-i/2}=\prod_{j}\frac{u_k-u_j-i}{u_k-u_j+i}.$$ As done for the parallel calculation in $\mathcal{N}=4$ SYM [@Eden:2006rx] based on [@Korchemsky:1995be], we introduce the Baxter polynomial $$Q_s(u) = C_s\prod_k(u-u_k)$$ that satisfies $$T_s(u)Q_s(u) = (u+i/2) Q_s(u+i)-(u-i/2)Q_s(u-i)$$ for some auxiliary polynomial $T_s$. $C_s$ is a $u$-independent normalization factor. Matching powers of $u$ on each side requires $T_s$ to be independent of $u$, and for this equation to have a solution for all $s\geq0$ $T_s=(2 s+1)i$. The solution to a difference equation of this form is a Meixner polynomial (closely related to Jacobi and Krawtchouk polynomials) [@Koekoek:1996xx], $$Q_s(u)={}_2F_1(-s, iu + {{{\textstyle\frac{1}{2}}}}; 1; 2).$$ The two-loop contribution to the anomalous dimension is given by $$\sum_k \frac{2 \lambda^2}{u_k^2 + {{{\textstyle\frac{1}{4}}}}},$$ (the factor of $2$ is for the $u_4$ and $\bar{u}_4$ roots), which is proportional to the ratio of the coefficient of $u$ to the constant term in $Q_s(u+i/2)$. In terms of the expansion, $$\frac{Q_s(u+i/2)}{C_s} = \prod_k(u +i/2-u_k) = c^{(s)}_s u^s + c^{(s)}_{s-1} u^{s-1} + \ldots c^{(s)}_1 u + c^{(s)}_0,$$ this ratio is $4 i \, \lambda^2 c^{(s)}_1 / c^{(s)}_0$. The series expansion of the hypergeometric function gives $$Q_s(u +i/2)={}_2F_1(-s, iu ; 1; 2) = \sum_{k=0}^s \frac{(-s)_k (iu)_k}{(1)_k} \frac{2^k}{k!}, \quad (a)_k=\Gamma(a+k)/\Gamma(a),$$ which implies $$c^{(s)}_0 = 1, \quad c^{(s)}_1 = i \sum_{k=1}^s \frac{(-s)_k (k-1)! }{(1)_k} \frac{2^k}{ k!} = i \sum_{k=1}^s \frac{s! (-2)^k}{k(s-k)! k!}.$$ To find $c_1^{(s)}$ we evaluate c\^[(s)]{}\_1 - c\^[(s-1)]{}\_1 [[&=&]{}]{}i \_[k=1]{}\^s - i \_[k=1]{}\^[s-1]{} [\
]{}[[&=&]{}]{} (-2)\^s+ \_[k=1]{}\^[s-1]{} [\
]{}[[&=&]{}]{} ((-1)\^s -1), where we completed the binomial expansion of $(1- 2)^s$ to reach the last line. Since $c_1^{(0)} =0 $, $c_1^{(s)} = i\Big(S_{-1}(s) - S_1(s) \Big)$ and $$\Delta_{s,2} = 4 i \frac{c^{(s)}_1}{c^{(s)}_0} = 4 \Big(S_1(s) - S_{-1}(s)\Big).$$ As $s \rightarrow \infty$, $S_1(s) \rightarrow \log s$ while $S_{-1}(s) \rightarrow - \log 2$, so the cusp anomalous dimension has weak coupling expansion $$f(\lambda) = 4 \lambda^2 + \mathcal{O}(\lambda^3) .$$ We expect the methods of [@Kotikov:2008pv] could be used to obtain the four-loop correction to the twist-one spectrum, up to a currently unknown coefficient from $h(\lambda)$.
Local contribution to ${[{\mathfrak{Y}}^{aa},\hat{{\mathfrak{Q}}}]}$ \[sec:localycomm\]
=======================================================================================
In this appendix we complete the proof of ${\mathfrak{osp}}(4|2)$ Yangian symmetry of Section \[sec:yangian\] by computing what we called the local term of the commutator between $\hat{{\mathfrak{Q}}}$ and the ${\mathfrak{Y}}^{aa}$. Recall 4 \^[aa]{} [[&=&]{}]{} \_[i < j]{} 2 \_[bc]{} \^[bc]{}\_i \^[ca]{}\_j - \_ \^[a1 ]{}\_i \^[a2 ]{}\_j + \_ \^[a2 ]{}\_i \^[a1 ]{}\_j [\
]{}[[&=&]{}]{}\^[aa]{}\_ + \^[aa]{}\_[\^1]{} + \^[aa]{}\_[\^2]{}. So the local contributions of the form (\[eq:local\]) from ${\mathfrak{Y}}^{aa}_{{\mathfrak{R}}}$ are $$2 \varepsilon_{bc} \sum_{i} \Big( {\mathfrak{R}}^{ab}_i ({\mathfrak{R}}^{ac}_{i+1} +{\mathfrak{R}}^{ac}_{i+2} ) +{\mathfrak{R}}^{ab}_{i+1}{\mathfrak{R}}^{ac}_{i+2} \Big) {\mathfrak{\hat{Q}}}_i.$$ Using the expressions for the ${\mathfrak{R}}$ generators given in (\[eq:osp42lightcone\]), and the expression for $\hat{{\mathfrak{Q}}}$, \[eq:appqhat\] [x, , |]{} [[&=&]{}]{}\_0\^x y ( \_[ab]{} \^a\_2 \^b\_3 + \_ \^\_2 \^\_3 ) [x, , |; y, |\_2, \_2;y, \_3, |\_3]{} [\
&&]{} + \_0\^x y ( \_[ab]{} \^a\_1 \^b\_2 + \_ \_1\^ \_2\^ ) [ y, \_1, |\_1; y, |\_2, \_2; x, , |]{} [\
&&]{} -\_ |\^ \_2\^ \_[cd]{} \_1\^c\_3\^d [ x, \_1, |\_1;x, |\_2, \_2; x, \_3, |\_3 ]{} , we obtain 4 [\[\^[aa]{}\_,\]]{}\_ [x, , |]{} [[&=&]{}]{}\_0\^x y 2 \_2\^a \_3\^a [x, , |; y, |\_2, \_2;y, \_3, |\_3]{} [\
&&]{} + \_0\^x y 2 \_1\^a \_2\^a [ y, \_1, |\_1; y, |\_2, \_2; x, , |]{} [\
&&]{} - 2 \_|\^ |\_2\^ \_1\^a \_3\^a [ x, \_1, |\_1;x, |\_2, \_2; x, \_3, |\_3 ]{} . \[eq:rycom\] This above calculation is simplified using the vanishing of a single ${\mathfrak{R}}$ applied to ${\mathfrak{R}}$-singlet combinations of derivatives.
Next the local contributions from ${\mathfrak{Y}}^{aa}_{{\mathfrak{Q}}^1}$ are, after regrouping, $$\label{eq:twoparts}
- \varepsilon_{\mathfrak{bc}} \sum_{i} \Big(({\mathfrak{Q}}^{a1 \mathfrak{b}}_i + {\mathfrak{Q}}^{a1 \mathfrak{b}}_{i+1}){\mathfrak{Q}}^{a2 \mathfrak{c}}_{i+2} + {\mathfrak{Q}}^{a1 \mathfrak{b}}_i {\mathfrak{Q}}^{a2 \mathfrak{c}}_{i+1} \Big) {\mathfrak{\hat{Q}}}_i.$$ First we compute the action of the first term (including the two supercharge terms in parenthesis) on ${\mathopen{\big|}x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}$. The three lines of (\[eq:appqhat\]) yield respectively, $$\begin{aligned}
& & \int_0^x \mathrm{d}y \, \bigg(-\varepsilon_{\mathfrak{bc}} {\mathfrak{Q}}_1^{a1\mathfrak{b}} \Big(2 y \partial_2^a\bar{\partial}_3^{\mathfrak{c}} +(2 y \partial_{y_3}+1)\bar{\partial}_2^{\mathfrak{c}} \partial_3^a \Big) \label{eq:11}
\notag \\
& & -4 \partial_2^a\partial_3^a y \partial_{y_3} -2 \partial_2^a\partial_3^a \bigg) {\mathopen{\big|}x, \theta, \bar{\eta}; y, \bar{\theta}_2, \eta_2;y, \theta_3, \bar{\eta}_3\mathclose{\bigr \rangle}}
\\
&&-\varepsilon_{\mathfrak{bc}} \partial_1^a \bar{\partial}_2^{\mathfrak{b}}{\mathfrak{Q}}_3^{a2\mathfrak{c}} \Big({\mathopen{\big|}x,\theta_1, \bar{\eta}_1;x, \bar{\theta}_2, \eta_2;x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}-{\mathopen{\big|}0,\theta_1, \bar{\eta}_1;0, \bar{\theta}_2, \eta_2;x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}\Big) \label{eq:12}
\\
&& +2x \partial_1^a\partial_2^a \varepsilon_{\mathfrak{bc}}\bar{\eta}^{\mathfrak{b}} \bar{\partial}_3^{\mathfrak{c}} {\mathopen{\big|} x, \theta_1, \bar{\eta}_1;x, \bar{\theta}_2, \eta_2; x, \theta_3, \bar{\eta}_3 \mathclose{\bigr \rangle}}. \label{eq:13}
\end{aligned}$$ In the first line we introduced the notation $\partial_{y_i}$ ($i=3$ in this case), which should be understood as $\partial_y$ acting only on the $i$th site. Similarly, we compute the action of the last term of (\[eq:twoparts\]) on ${\mathopen{\big|}x, \theta, \bar{\eta}\mathclose{\bigr \rangle}}$, again splitting according to the three lines of $\hat{{\mathfrak{Q}}}$ in (\[eq:appqhat\]), yielding $$\begin{aligned}
& & \int_0^x \mathrm{d}y \, \varepsilon_{\mathfrak{bc}} {\mathfrak{Q}}_1^{a1\mathfrak{b}}\Big(-(2 y\partial_{y_2} +1)\bar{\partial}_2^{\mathfrak{c}}\partial_3^a + 2 y \partial_2^a \bar{\partial}_3^{\mathfrak{c}}\Big){\mathopen{\big|}x, \theta, \bar{\eta}; y, \bar{\theta}_2, \eta_2;y, \theta_3, \bar{\eta}_3\mathclose{\bigr \rangle}} \label{eq:21}
\\
& & + \int_0^x \mathrm{d}y \, 4 y \partial_{y_1} \partial_1^a \partial_2^a {\mathopen{\big|} y, \theta_1, \bar{\eta}_1; y, \bar{\theta}_2, \eta_2; x, \theta, \bar{\eta}\mathclose{\bigr \rangle}} \label{eq:22}
\\
& & + 2 x \varepsilon_{\mathfrak{bc}} \bar{\partial}_1^{\mathfrak{b}} \partial_2^a \partial_3^a \bar{\eta}^{\mathfrak{c}} {\mathopen{\big|} x, \theta_1, \bar{\eta}_1;x, \bar{\theta}_2, \eta_2; x, \theta_3, \bar{\eta}_3 \mathclose{\bigr \rangle}}. \label{eq:23}\end{aligned}$$ It remains to compute the local terms from the last part of ${\mathfrak{Y}}^{aa}$ (\[eq:Ycc\]), labeled ${\mathfrak{Y}}^{aa}_{{\mathfrak{Q}}^2}$. However, as we noted previously, $\hat{{\mathfrak{Q}}}$ is odd under spin-chain parity $\mathbf{p}$, and it is straightforward to check that the Yangian generators are as well. Therefore, we find $${[{\mathfrak{Y}}^{aa}_{{\mathfrak{Q}}^2},\hat{{\mathfrak{Q}}}]}_{\text{local}}= \mathbf{p} {[{\mathfrak{Y}}^{aa}_{{\mathfrak{Q}}^1},\hat{{\mathfrak{Q}}}]}_{\text{local}}\mathbf{p}^{-1},$$ where $4 {[{\mathfrak{Y}}^{aa}_{{\mathfrak{Q}}^1},\hat{{\mathfrak{Q}}}]}_{\text{local}}$ is given by the sum of (\[eq:11\]-\[eq:13\]) and (\[eq:21\]-\[eq:23\]). So now we add (\[eq:11\]-\[eq:13\]) and (\[eq:21\]-\[eq:23\]), their images under $\mathbf{p}$, and (\[eq:rycom\]) to give the full local contribution to the commutator. Many terms cancel[^21]. Also, the second term of (\[eq:11\]) combines with the first term of (\[eq:21\]) to give \[eq:last\] -2 \_0\^x y \_y(y \_ \_1\^[a1]{} |\_2\^ \_3\^a [x, , |;y, |\_2, \_2;y, \_3,|\_3]{} ) [[&=&]{}]{}\
-2 x \_ \_1\^[a1]{} |\_2\^ \_3\^a [x, , |;x, |\_2, \_2;x, \_3,|\_3]{}. & & All that remains is the last term of (\[eq:rycom\]), (\[eq:12\]) and its $\mathbf{p}$ image, and this last expression (\[eq:last\]) and its $\mathbf{p}$ image. We write out all these terms, expanding the remaining ${\mathfrak{Q}}$ factors, yielding $$\begin{aligned}
& & - 2 \varepsilon_{\mathfrak{bc}}\bar{\eta}^{\mathfrak{b}} \bar{\partial}_2^{\mathfrak{c}} \partial_1^a \partial_3^a {\mathopen{\big|} x, \theta_1, \bar{\eta}_1;x, \bar{\theta}_2, \eta_2; x, \theta_3, \bar{\eta}_3 \mathclose{\bigr \rangle}}
\notag \\
& & -\varepsilon_{\mathfrak{bc}} \partial_1^a \bar{\partial}_2^{\mathfrak{b}} (2 x \theta^a \bar{\partial}_3^{\mathfrak{c}} + 2 x \partial_{x_3} \partial_3^a \bar{\eta}^{\mathfrak{c}} + \partial_3^a \bar{\eta}^{\mathfrak{c}} ) {\mathopen{\big|}x,\theta_1, \bar{\eta}_1;x, \bar{\theta}_2, \eta_2;x, \theta, \bar{\eta}\mathclose{\bigr \rangle}} - \text{parity}
\notag \\
& & -2 x \varepsilon_{\mathfrak{bc}} (\theta^a \bar{\partial}_1^{\mathfrak{b}} + \partial_{x_1} \partial_1^a \bar{\eta}^\mathfrak{b}) \bar{\partial}_2^{\mathfrak{c}} \partial_3^a {\mathopen{\big|}x, \theta, \bar{\eta};x, \bar{\theta}_2, \eta_2;x, \theta_3,\bar{\eta}_3\mathclose{\bigr \rangle}} - \text{parity}
\notag \\
& & + \varepsilon_{\mathfrak{bc}} \partial_1^a \bar{\partial}_2^{\mathfrak{b}}{\mathfrak{Q}}_3^{a2\mathfrak{c}} {\mathopen{\big|}0,\theta_1, \bar{\eta}_1;0, \bar{\theta}_2, \eta_2;x, \theta, \bar{\eta}\mathclose{\bigr \rangle}} + \text{parity}
{\nonumber\\}{{\mathrel{}&=&\mathrel{}}}\varepsilon_{\mathfrak{bc}} \partial_1^a \bar{\partial}_2^{\mathfrak{b}}{\mathfrak{Q}}_3^{a2\mathfrak{c}} {\mathopen{\big|}0,\theta_1, \bar{\eta}_1;0, \bar{\theta}_2, \eta_2;x, \theta, \bar{\eta}\mathclose{\bigr \rangle}} - \varepsilon_{\mathfrak{bc}} {\mathfrak{Q}}_1^{a2\mathfrak{b}} \bar{\partial}_2^{\mathfrak{c}} \partial_3^a {\mathopen{\big|}x, \theta, \bar{\eta};0, \bar{\theta}_2, \eta_2;0,\theta_3, \bar{\eta}_3\mathclose{\bigr \rangle}}
\notag \\
& \mapsto & \sum_i \varepsilon_{\mathfrak{bc}} \acute{Z}^{a\mathfrak{b}}_i{\mathfrak{Q}}_{i+1}^{a2\mathfrak{c}} +\sum_i \varepsilon_{\mathfrak{bc}} {\mathfrak{Q}}_i^{a2\mathfrak{c}} \grave{Z}^{a\mathfrak{b}}_{i+1}.\end{aligned}$$ The initial expression simplified as follows. The first line canceled against the terms without $x$ coefficients in the second line, and the remaining terms proportional to $x$ canceled in four pairs, leaving only the fourth line. The second-to-last line is simply expanding the fourth line (the parity term). We used the symbol $\mapsto$ in the last line for two reasons. We have lifted the one-to-three local site interaction to its homogeneous action on an infinite chain. Also, this expression includes a second set of local actions, on initial sites ${\mathopen{\big|}x, \bar{\theta}, \eta\mathclose{\bigr \rangle}}.$ For these the calculation is completely analogous. Starting from the action of $\hat{{\mathfrak{Q}}}$ on such sites, [x, |, ]{} [[&=&]{}]{}-\_0\^x y ( \_ |\^\_2 |\^\_3 + \_[ab]{} \^a\_2 \^b\_3 ) [x, |, ; y, \_2, |\_2;y, |\_3, \_3]{} [\
&&]{} - \_0\^x y ( \_ |\^\_1 |\^\_2 + \_[ab]{} \^a\_1 \^b\_2 ) [ y, |\_1, \_1; y, \_2, |\_2; x, |, ]{} [\
&&]{} + \_[ab]{} \^[a]{} \_2\^b \_ |\^\_1 |\^\_3 [ x, |\_1, \_1;x, \_2, |\_2; x, |\_3, \_3 ]{} . We again can compute the contributions to the local terms of ${\mathfrak{Y}}_{{\mathfrak{R}}}$, ${\mathfrak{Y}}_{{\mathfrak{Q}}^1}$ and (using $\mathbf{p}$) ${\mathfrak{Y}}_{{\mathfrak{Q}}^2}$. After simplification, we find & & \_ |\_1\^ \_2\^a \^[a 2 ]{}\_3 [ 0, |\_1, \_1; 0, \_2, |\_2; x, |, ]{} + \_ \^[a 2 ]{}\_1 \_2\^a |\_3\^ [ x, |, ; 0, \_2, |\_2; 0, |\_3, \_3]{}\
& & \_i \_ \^[a]{}\_i\_[i+1]{}\^[a2]{} +\_i \_ \_i\^[a2]{} \^[a]{}\_[i+1]{}. As claimed, this gives a precise cancellation with the remaining contribution from bilocal terms of (\[eq:bilocalYqhatcomm\]).
Projectors in the spin-module representation \[sec:projectors\]
===============================================================
The results of this section parallel those given for the ${\mathfrak{psu}}(1,1|2)$ sector of $\mathcal{N}=4$ SYM spin chain in Appendix E of [@Zwiebel:2008gr]. The expression for the dilatation generator (\[eq:osp42projectorham\]) depends on certain ${\mathfrak{osp}}(4|2)$ invariant generators. These generators act on pairs of different types of modules, and on pairs of identical modules. First, for the case of different types of modules we introduce the weighted sum of projectors, $$\mathcal{P}(c) = \sum_{i=0}^{\infty} c_i \mathcal{P}^{(i)},$$ where again $\mathcal{P}^{(i)}$ is the projector for ${\mathfrak{osp}}(4|2)$ spin $i$. So for $c_{i'} = \delta_{ii'}$, $\mathcal{P}(c)$ reduces to $\mathcal{P}^{(i)}$. More generally, $\mathcal{P}(c)$ can be written in terms of six component functions that depend on the $c_i$ (we suppress the argument $c$ of $\mathcal{P}$). [\_a\^[(j)]{} \_b\^[(n-j)]{}]{} [[&=&]{}]{}\_[k=0]{}\^ n ( p\_1(n, j, k) [\_a\^[(k)]{} \_b\^[(n-k)]{}]{} + p\_2(n, j, k) [\_b\^[(k)]{} \_a\^[(n-k)]{}]{} ), [\
&&]{}+ \_[k=0]{}\^n p\_3(n, j, k) \_[ab]{}\^ [|\_\^[(k)]{} |\_\^[(n-k)]{}]{}, [\
]{}[\_a\^[(j)]{} |\_\^[(n-j)]{}]{} [[&=&]{}]{}\_[k=0]{}\^ n p\_4(n, j, k) [\_a\^[(k)]{} |\_\^[(n-k)]{}]{} + \_[k=0]{}\^ [n-1]{} p\_5(n, j, k) [|\_\^[(k)]{} \_a\^[(n-k-1)]{}]{}, [\
]{}[|\_\^[(j)]{} \_b\^[(n-j)]{}]{} [[&=&]{}]{}\_[k=0]{}\^n p\_6(n, j, k) [|\_\^[(k)]{} \_b\^[(n-k)]{}]{} + \_[k=0]{}\^ [n+1]{} p\_5(n+1, k, j) [\_b\^[(k)]{} |\_\^[(n-k+1)]{}]{}, [\
]{}[|\_\^[(j)]{} |\_\^[(n-j)]{}]{} [[&=&]{}]{}\_[k=0]{}\^n (p\_1(n, n-j, n-k) [|\_\^[(k)]{} |\_\^[(n-k)]{}]{} + p\_2(n, n-j, n-k) [|\_\^[(k)]{} |\_\^[(n-k)]{}]{}) [\
&&]{}+ \_[k=0]{}\^n p\_3(n, k, j) \_ \^[cd]{} [\_c\^[(k)]{} \_d\^[(n-k)]{}]{}. The $p_l$ for $l=1,2$ only differ by a few minus signs, $$\begin{gathered}
p_l(n, j, k) = \sqrt{\frac{\pi \Gamma(j+{{{\textstyle\frac{1}{2}}}})\Gamma(k+{{{\textstyle\frac{1}{2}}}})(2(n-j)+1)!(2(n-k)+1)!}{j!k!}} \sum_{i=0}^n \bigg( \frac{2^{j+k-2n}(n-i)!}{4(n+i+1)!}
\notag \\
{}_3F_2^{\text{reg}}(-{{{\textstyle\frac{1}{2}}}}-i, -i, -j;{{{\textstyle\frac{1}{2}}}}, 1-i-j+n;1)C_l(i) {}_3F_2^{\text{reg}}(-{{{\textstyle\frac{1}{2}}}}-i, -i, -k;{{{\textstyle\frac{1}{2}}}}, 1-i-k+n;1) \bigg), \notag\end{gathered}$$ C\_1(i) [[&=&]{}]{}(i-1) c\_[i-1]{} + (2 + \_[i,0]{})c\_i + c\_[i+1]{}, [\
]{}C\_2(i) [[&=&]{}]{}-(i-1) c\_[i-1]{} + (2 -\_[i,0]{})c\_i - c\_[i+1]{}. \[eq:definepl\] The regularized hypergeometric functions are defined as ordinary hypergeometric functions divided by gamma functions, as $${}_3F_2^{\text{reg}}(a_1, a_2, a_3; b_1, b_2; z) = \frac{{}_3F_2(a_1, a_2, a_3; b_1, b_2; z)}{\Gamma(b_1)\Gamma(b_2)}.$$ The remaining four component functions can then be written relatively compactly in terms of these two. p\_4(n, j, k) [[&=&]{}]{} p\_1(n, n-j, n-k) + p\_2(n, n-j, n-k) [\
&&]{}+ (n-j-1) p\_2(n, j+1, k), [\
]{}p\_3(n, j, k) [[&=&]{}]{}- p\_1(n, n-j, n-k) + p\_4(n, j, k), [\
]{}p\_5(n, j, k) [[&=&]{}]{}(n-k-1)( p\_2(n-1, j, k) - p\_3(n-1, k, j-1)), [\
]{}p\_6(n, j, k) [[&=&]{}]{} p\_1(n, j, k) - p\_3(n, j, k). Note that the first (second) term in the expression for $p_5$ should be set to zero when $n=j$ ($j=0$) (as written, they are $0 \times \infty$). For use in (\[eq:osp42projectorham\]), one only needs to set $c_i = (-1)^i$, or $c_i = S_1(i)$.
For identical representations we instead define $\mathcal{P}(c)$ as $$\mathcal{P}(c) = \sum_{i=0}^{\infty} c_{i} \mathcal{P}^{(i-1/2)}.$$ Recall that in this case the ${\mathfrak{osp}}(4|2)$ spin is half-integer valued. Now the expansion of $\mathcal{P}(c)$ is [\_a\^[(j)]{} \_b\^[(n-j)]{}]{} [[&=&]{}]{}\_[k=0]{}\^ n ( p\_7(n, j, k) [\_a\^[(k)]{} \_b\^[(n-k)]{}]{} + p\_8(n, j, k) [\_b\^[(k)]{} \_a\^[(n-k)]{}]{} ) [\
&&]{}+ \_[k=0]{}\^ [n-1]{} p\_9(n, j, k) \_[ab]{}\^ [|\_\^[(k)]{} |\_\^[(n-k-1)]{}]{} [\
]{}[\_a\^[(j)]{} |\_\^[(n-j)]{}]{} [[&=&]{}]{}\_[k=0]{}\^ n (p\_[10]{}(n, j, k) [\_a\^[(k)]{} |\_\^[(n-k)]{}]{} + p\_[11]{}(n, j, k) [|\_\^[(k)]{} \_a\^[(n-k)]{}]{}), [\
]{}[|\_\^[(j)]{}\_b\^[(n-j)]{} ]{} [[&=&]{}]{}\_[k=0]{}\^ n (p\_[10]{}(n, n-j, n-k) [|\_\^[(k)]{}\_b\^[(n-k)]{} ]{} + p\_[11]{}(n, n-j, n-k) [\_b\^[k]{} |\_\^[(n-k)]{}]{}), [\
]{}[|\_\^[(j)]{}|\_\^[(n-j)]{} ]{} [[&=&]{}]{}\_[k=0]{}\^ n (p\_[12]{}(n, j, k)[|\_\^[(k)]{}|\_\^[(n-k)]{} ]{} + p\_[13]{}(n, j, k)[|\_\^[(k)]{}|\_\^[(n-k)]{} ]{} ) [\
&&]{}+ \_[k=0]{}\^[n+1]{} p\_9(n+1, k, j) \_ \^[cd]{}[\_c\^[(k)]{} \_d\^[(n-k+1)]{}]{}. $p_{10}$ actually takes the same form as given in (\[eq:definepl\]), with $$C_{10}(i) = 2 c_i + 2 c_{i+1}.$$ For $p_m$, $m=7, 8$ there are some shifts of arguments, $$\begin{gathered}
p_m(n, j, k) = \sqrt{\frac{\pi \Gamma(j+{{{\textstyle\frac{1}{2}}}})\Gamma(k+{{{\textstyle\frac{1}{2}}}})(2(n-j))!(2(n-k))!}{j!k!}} \sum_{i=0}^n \bigg( \frac{2^{1+j+k-2n}(n-i)!}{4(n+i)!}
\notag \\
{}_3F_2^{\text{reg}}({{{\textstyle\frac{1}{2}}}}-i, -i, -j;{{{\textstyle\frac{1}{2}}}}, 1-i-j+n;1)D_m(i) {}_3F_2^{\text{reg}}({{{\textstyle\frac{1}{2}}}}-i, -i, -k;{{{\textstyle\frac{1}{2}}}}, 1-i-k+n;1) \bigg), \notag\end{gathered}$$ D\_7(i) [[&=&]{}]{}(i-1) c\_[i-1]{} + (2- \_[i, 0]{}) c\_i + c\_[i+1]{}, [\
]{}D\_8(i) [[&=&]{}]{}-(i-1) c\_[i-1]{} + (2- \_[i, 0]{})c\_i - c\_[i+1]{}. The remaining four component functions again can be written in terms of ones defined earlier. p\_9(n, j, k) [[&=&]{}]{}(n-k-1)( p\_7(n, k+1, j) [\
&&]{}- p\_[10]{}(n-1, n-k-1, n-j) ), [\
]{}p\_[11]{}(n, j, k) [[&=&]{}]{} p\_8(n+1, n-j+1, n-k) [\
&&]{}+ p\_9(n+1, n-k, n-j), [\
]{}p\_[12]{}(n, j, k) [[&=&]{}]{}p\_[10]{}(n, n-j, n-k) + p\_9(n+1, j+1, k) [\
]{}p\_[13]{}(n, j, k) [[&=&]{}]{}- p\_[11]{}(n,n-j, n-k) - p\_9(n+1, j+1, k). [\
&&]{} Note that the second term in the expression for $p_9$ should be set to zero for $j=0$. For the conjugate case of $\mathcal{V}_{\bar \phi}^{(4|2)}$, simply remove (add) a bar from (to) all (un)barred module elements, keeping the same component functions. For (\[eq:osp42projectorham\]), in these cases we need to evaluate $\mathcal{P}(c)$ only for $c_i = S_1(i-1/2)$.
Unique lift from ${\mathfrak{sl}}(2)$ to ${\mathfrak{osp}}(4|2)$ \[sec:maptosl2\]
=================================================================================
The general argument of Section \[sec:uniquelift\] can be straightforwardly applied to lift the Hamiltonian uniquely from the ${\mathfrak{sl}}(2)$ sector to the ${\mathfrak{osp}}(4|2)$ sector. However, we still need to construct invertible maps between the ${\mathfrak{osp}}(4|2)$ sector and the ${\mathfrak{sl}}(2)$ sector, as done between the ${\mathfrak{osp}}(6|4)$ and ${\mathfrak{osp}}(4|2)$ sectors in Section (\[sec:invertiblemap\]). We complete this construction here. Again, simply constructing the map from the larger to smaller sector will be sufficient. Invertibility will follow, as in the last paragraph of Section \[sec:invertiblemap\].
The ${\mathfrak{osp}}(4|2)$ raising generators are ${\mathfrak{R}}^{22}$, $\tilde{{\mathfrak{R}}}^{22}$, ${\mathfrak{J}}^{22}$, and ${\mathfrak{Q}}^{a2\mathfrak{b}}$. Let ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ be a linearly-independent basis of three-site highest-weight ${\mathfrak{osp}}(4|2)$ sector states with identical Cartan charges. Of course, we need not consider ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ already in the ${\mathfrak{sl}}(2)$ sector (actually there are no such three-site ${\mathfrak{osp}}(4|2)$ highest-weight states). So, first assume the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ transform with respect to ${\mathfrak{su}}(2)_{{\mathfrak{R}}} \otimes {\mathfrak{su}}(2)_{\tilde{{\mathfrak{R}}}}$ as $(\mathbf{3} ,\mathbf{2}),$ or equivalently that the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ have ${\mathfrak{su}}(2)$ Cartan eigenvalues $[R^{12}, \tilde{R}^{12}]$ = $[-1, -1/2]$ (The third Cartan charge is unimportant for this section). Then consider $${\mathfrak{Q}}^{211}{\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}.$$ These states have (${\mathfrak{su}}(2)$) Cartan charges $[-3/2, 0]$ and, equivalently, are in the ${\mathfrak{sl}}(2)$ subsector. However, a priori the ${\mathfrak{Q}}^{211}{\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ could be zero. But if they were zero, then the highest-weight states ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ would satisfy a second $1/12$ BPS condition with respect to ${\mathfrak{Q}}^{211}\sim {\mathfrak{Q}}_{14,1}$ and ${\mathfrak{Q}}^{122}\sim {\mathfrak{S}}^{14,1}$. This BPS condition(s) is inconsistent (for any possible value of $J^{12}$) with the ${\mathfrak{su}}(2)$ Cartan charges of the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$. Therefore, ${\mathfrak{Q}}^{211}{\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ is nonvanishing and in this case ${\mathfrak{Q}}^{211}$ gives the invertible map to the ${\mathfrak{sl}}(2)$ sector.
Up to interchanging ${\mathfrak{su}}(2)_{{\mathfrak{R}}}$ and ${\mathfrak{su}}(2)_{\tilde{{\mathfrak{R}}}}$ charges, the only other possibility for the Cartan charges of the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ is $[R^{12}, \tilde{R}^{12}]$ = $[-1/2, 0]$. Now descendants in the ${\mathfrak{sl}}(2)$ sector are given by $${\mathfrak{Q}}^{212}{\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} = -{\mathfrak{Q}}^{211} {\mathfrak{Q}}^{212}{\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}},$$ where the equality follows from the vanishing anticommutator between these supercharges. The same BPS condition as in the previous case would apply if the ${\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ were zero, and this is inconsistent with Cartan charges $[-1/2, 0]$. Therefore, we only need to show that it is impossible to simultaneously satisfy $${\mathfrak{Q}}^{212}{\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} =0 \quad \text{and} \quad {\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} \neq 0.$$ ${\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} $ have Cartan charges $[-1, 1/2]$. Also, they are annihilated by ${\mathfrak{Q}}^{121} =({\mathfrak{Q}}^{212})^\dagger $ because of the anticommutation relation (\[eq:osp42anticom\]) and because $\tilde{{\mathfrak{R}}}^{11}$ annihilates the ${\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ (they carries no $\tilde{{\mathfrak{R}}}$ charge). Therefore, if ${\mathfrak{Q}}^{212}$ annihilated the ${\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} $, ${\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}} $ would satisfy another BPS condition, which again turns out to be incompatible with their Cartan charges. We conclude that in this case ${\mathfrak{Q}}^{212}{\mathfrak{Q}}^{211} {\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ gives nonvanishing descendants in the ${\mathfrak{sl}}(2)$ sector, and therefore we can always construct the required invertible maps between the the ${\mathfrak{osp}}(4|2)$ sector and a ${\mathfrak{sl}}(2)$ sector.
[10]{}
O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, *“[$\mathcal{N}=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals]{}”*, , [`arXiv:0806.1218`](http://arXiv.org/abs/arXiv:0806.1218) \[hep-th\]. J. A. Minahan and K. Zarembo, *“[The Bethe ansatz for superconformal Chern-Simons]{}”*, , [`arXiv:0806.3951`](http://arXiv.org/abs/arXiv:0806.3951) \[hep-th\]. D. Bak and S. J. Rey, *“[Integrable Spin Chain in Superconformal Chern-Simons Theory]{}”*, , [`arXiv:0807.2063`](http://arXiv.org/abs/arXiv:0807.2063) \[hep-th\]. J. A. Minahan and K. Zarembo, *“The Bethe-ansatz for [$\mathcal{N}=\mathord{}$4]{} super Yang-Mills”*, , [`arXiv:hep-th/0212208`](http://arXiv.org/abs/arXiv:hep-th/0212208). N. Beisert, C. Kristjansen and M. Staudacher, *“The Dilatation Operator of [$\mathcal{N}=\mathord{}$4]{} Conformal Super Yang-Mills Theory”*, , [`arXiv:hep-th/0303060`](http://arXiv.org/abs/arXiv:hep-th/0303060). I. Bena, J. Polchinski and R. Roiban, *“Hidden symmetries of the [$AdS_5\times S^5$]{} superstring”*, , [`arXiv:hep-th/0305116`](http://arXiv.org/abs/arXiv:hep-th/0305116). G. Arutyunov, S. Frolov and M. Staudacher, *“Bethe ansatz for quantum strings”*, , [`arXiv:hep-th/0406256`](http://arXiv.org/abs/arXiv:hep-th/0406256). M. Staudacher, *“The factorized S-matrix of CFT/AdS”*, , [`arXiv:hep-th/0412188`](http://arXiv.org/abs/arXiv:hep-th/0412188). N. Beisert, *“The SU(2$|$2) Dynamic S-Matrix”*, [`arXiv:hep-th/0511082`](http://arXiv.org/abs/arXiv:hep-th/0511082). R. A. Janik, *“The $AdS_5\times S^5$ superstring worldsheet S-matrix and crossing symmetry”*, , [`arXiv:hep-th/0603038`](http://arXiv.org/abs/arXiv:hep-th/0603038). G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, *“The Off-shell Symmetry Algebra of the Light-cone $AdS_5\times S^5$ Superstring”*, , [`arXiv:hep-th/0609157`](http://arXiv.org/abs/arXiv:hep-th/0609157). N. Beisert, B. Eden and M. Staudacher, *“Transcendentality and Crossing”*, , [`arXiv:hep-th/0610251`](http://arXiv.org/abs/arXiv:hep-th/0610251). G. Arutyunov and S. Frolov, *“[Superstrings on $AdS_4 \times CP^3$ as a Coset Sigma-model]{}”*, , [`arXiv:0806.4940`](http://arXiv.org/abs/arXiv:0806.4940) \[hep-th\]. B. Stefa[ń]{}ski, Jr., *“[Green-Schwarz action for Type IIA strings on $AdS_4\times CP^3$]{}”*, , [`arXiv:0806.4948`](http://arXiv.org/abs/arXiv:0806.4948) \[hep-th\]. N. Gromov and P. Vieira, *“[The all loop $AdS_4/CFT_3$ Bethe ansatz]{}”*, , [`arXiv:0807.0777`](http://arXiv.org/abs/arXiv:0807.0777) \[hep-th\]. T. Nishioka and T. Takayanagi, *“[On Type IIA Penrose Limit and $\mathcal{N}=6$ Chern-Simons Theories]{}”*, , [`arXiv:0806.3391`](http://arXiv.org/abs/arXiv:0806.3391) \[hep-th\]. D. Gaiotto, S. Giombi and X. Yin, *“[Spin Chains in $\mathcal{N}=6$ Superconformal Chern-Simons-Matter Theory]{}”*, , [`arXiv:0806.4589`](http://arXiv.org/abs/arXiv:0806.4589) \[hep-th\]. G. Grignani, T. Harmark and M. Orselli, *“[The $SU(2) \times SU(2)$ sector in the string dual of $\mathcal{N}=6$ superconformal Chern-Simons theory]{}”*, , [`arXiv:0806.4959`](http://arXiv.org/abs/arXiv:0806.4959) \[hep-th\]. D. Berenstein and D. Trancanelli, *“[Three-dimensional $\mathcal{N}=6$ SCFT’s and their membrane dynamics]{}”*, , [`arXiv:0808.2503`](http://arXiv.org/abs/arXiv:0808.2503) \[hep-th\]. C. Ahn and R. I. Nepomechie, *“[$\mathcal{N}=6$ super Chern-Simons theory S-matrix and all-loop Bethe ansatz equations]{}”*, , [`arXiv:0807.1924`](http://arXiv.org/abs/arXiv:0807.1924) \[hep-th\]. D. Astolfi, V. G. M. Puletti, G. Grignani, T. Harmark and M. Orselli, *“[Finite-size corrections in the $SU(2) \times SU(2)$ sector of type IIA string theory on $AdS_4 \times CP^3$]{}”*, , [`arXiv:0807.1527`](http://arXiv.org/abs/arXiv:0807.1527) \[hep-th\]. P. Sundin, *“[The $AdS_4 \times CP^3$ string and its Bethe equations in the near plane wave limit]{}”*, , [`arXiv:0811.2775`](http://arXiv.org/abs/arXiv:0811.2775) \[hep-th\]. T. McLoughlin and R. Roiban, *“[Spinning strings at one-loop in $AdS_4 \times P^3$]{}”*, , [`arXiv:0807.3965`](http://arXiv.org/abs/arXiv:0807.3965) \[hep-th\]. L. F. Alday, G. Arutyunov and D. Bykov, *“[Semiclassical Quantization of Spinning Strings in $AdS_4 \times
CP^3$]{}”*, , [`arXiv:0807.4400`](http://arXiv.org/abs/arXiv:0807.4400) \[hep-th\]. C. Krishnan, *“[$AdS4/CFT_3$ at One Loop]{}”*, , [`arXiv0807.4561`](http://arXiv.org/abs/arXiv0807.4561) \[hep-th\]. T. McLoughlin, R. Roiban and A. A. Tseytlin, *“[Quantum spinning strings in $AdS_4 \times CP^3$: testing the Bethe Ansatz proposal]{}”*, , [`arXiv:0809.4038`](http://arXiv.org/abs/arXiv:0809.4038) \[hep-th\]. J. Gomis, D. Sorokin and L. Wulff, *“[The complete $AdS_4 \times CP^3$ superspace for the type IIA superstring and D-branes]{}”*, , [`arXiv:0811.1566`](http://arXiv.org/abs/arXiv:0811.1566) \[hep-th\]. N. Beisert, *“The Dilatation Operator of [$\mathcal{N}=\mathord{}$4]{} Super Yang-Mills Theory and Integrability”*, , [`arXiv:hep-th/0407277`](http://arXiv.org/abs/arXiv:hep-th/0407277). L. N. Lipatov, *“Evolution equations in QCD”*, in: *“Perspectives in hadronic physics”*, ed.: S. Boffi, C. Ciofi Degli Atti and M. Giannini, World Scientific (1998), Singapore. F. A. Dolan and H. Osborn, *“Superconformal symmetry, correlation functions and the operator product expansion”*, , [`arXiv:hep-th/0112251`](http://arXiv.org/abs/arXiv:hep-th/0112251). N. Beisert, *“The Complete One-Loop Dilatation Operator of [$\mathcal{N}=\mathord{}$4]{} Super Yang-Mills Theory”*, , [`arXiv:hep-th/0307015`](http://arXiv.org/abs/arXiv:hep-th/0307015). C. Sochichiu, *“[Dilatation operator in 3d]{}”*, , [`arXiv:0811.2669`](http://arXiv.org/abs/arXiv:0811.2669) \[hep-th\]. L. Dolan, C. R. Nappi and E. Witten, *“A Relation Between Approaches to Integrability in Superconformal Yang-Mills Theory”*, , [`arXiv:hep-th/0308089`](http://arXiv.org/abs/arXiv:hep-th/0308089). L. Dolan, C. R. Nappi and E. Witten, *“Yangian symmetry in $D=$4 superconformal Yang-Mills theory”*, [`arXiv:hep-th/0401243`](http://arXiv.org/abs/arXiv:hep-th/0401243), in: *“Quantum theory and symmetries”*, ed.: P. C. Argyres et al., World Scientific (2004), Singapore. N. Beisert and M. Staudacher, *“The [$\mathcal{N}=\mathord{}$4]{} SYM Integrable Super Spin Chain”*, , [`arXiv:hep-th/0307042`](http://arXiv.org/abs/arXiv:hep-th/0307042). M. Benna, I. Klebanov, T. Klose and M. Smedback, *“[Superconformal Chern-Simons Theories and $AdS_4/CFT_3$ Correspondence]{}”*, , [`arXiv:0806.1519`](http://arXiv.org/abs/arXiv:0806.1519) \[hep-th\]. M. A. Bandres, A. E. Lipstein and J. H. Schwarz, *“[Studies of the ABJM Theory in a Formulation with Manifest $SU(4)$ R-Symmetry]{}”*, , [`arXiv:0807.0880`](http://arXiv.org/abs/arXiv:0807.0880) \[hep-th\]. F. A. Dolan, *“[On Superconformal Characters and Partition Functions in Three Dimensions]{}”*, [`arXiv:0811.2740`](http://arXiv.org/abs/arXiv:0811.2740) \[hep-th\]. K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, *“[$\mathcal{N}=5,6$ Superconformal Chern-Simons Theories and M2-branes on Orbifolds]{}”*, , [`arXiv:0806.4977`](http://arXiv.org/abs/arXiv:0806.4977) \[hep-th\]. J. B. Kogut and D. E. Soper, *“[Quantum Electrodynamics in the Infinite Momentum Frame]{}”*, . L. Brink, O. Lindgren and B. E. W. Nilsson, *“[$\mathcal{N}=4$ Yang-Mills Theory on the Light Cone]{}”*, . S. Mandelstam, *“[Light Cone Superspace and the Ultraviolet Finiteness of the $\mathcal{N}=4$ Model]{}”*, . A. V. Belitsky, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, *“Quantum integrability in (super) Yang-Mills theory on the light-cone”*, , [`arXiv:hep-th/0403085`](http://arXiv.org/abs/arXiv:hep-th/0403085). N. Beisert, *“The SU(2$|$3) Dynamic Spin Chain”*, , [`arXiv:hep-th/0310252`](http://arXiv.org/abs/arXiv:hep-th/0310252). N. Beisert and M. Staudacher, *“Long-Range PSU(2,2$|$4) Bethe Ans[ä]{}tze for Gauge Theory and Strings”*, , [`arXiv:hep-th/0504190`](http://arXiv.org/abs/arXiv:hep-th/0504190). A. V. Kotikov, L. N. Lipatov, A. Rej, M. Staudacher and V. N. Velizhanin, *“[Dressing and Wrapping]{}”*, , [`arXiv:0704.3586`](http://arXiv.org/abs/arXiv:0704.3586) \[hep-th\]. D. Gaiotto and X. Yin, *“[Notes on superconformal Chern-Simons-matter theories]{}”*, , [`arXiv:0704.3740`](http://arXiv.org/abs/arXiv:0704.3740) \[hep-th\]. B. I. Zwiebel, *“Yangian symmetry at two-loops for the ${\mathfrak{su}}(2|1)$ sector of [$\mathcal{N}=\mathord{}$4]{} SYM”*, , [`arXiv:hep-th/0610283`](http://arXiv.org/abs/arXiv:hep-th/0610283). P. P. Kulish, N. Y. Reshetikhin and E. K. Sklyanin, *“Yang-Baxter equation and representation theory: I”*, . H. J. de Vega and F. Woynarovich, *“[New integrable quantum chains combining different kinds of spins]{}”*, . S. M. Khoroshkin and V. N. Tolstoi, *“[Yangian double and rational R matrix]{}”*, [`arXiv:hep-th/9406194`](http://arXiv.org/abs/arXiv:hep-th/9406194). V. Stukopin, *“Quantum Double of Yangian of Lie Superalgebra $A(m,n)$ and computation of Universal $R$-matrix”*, [`arXiv:math/0504302`](http://arXiv.org/abs/arXiv:math/0504302). F. Spill, *“[Weakly coupled $\mathcal{N}=4$ Super Yang-Mills and $\mathcal{N}=6$ Chern-Simons theories from ${\mathfrak{u}}(2|2)$ Yangian symmetry]{}”*, , [`arXiv:0810.3897`](http://arXiv.org/abs/arXiv:0810.3897) \[hep-th\]. N. Beisert and B. I. Zwiebel, *“[On Symmetry Enhancement in the ${\mathfrak{psu}}(1,1|2)$ Sector of $\mathcal{N}=4$ SYM]{}”*, , [`arXiv:0707.1031`](http://arXiv.org/abs/arXiv:0707.1031) \[hep-th\]. B. I. Zwiebel, *“[Iterative Structure of the $\mathcal{N}=4$ SYM Spin Chain]{}”*, , [`arXiv:0806.1786`](http://arXiv.org/abs/arXiv:0806.1786) \[hep-th\]. T. Bargheer, N. Beisert and F. Loebbert, *“[Boosting Nearest-Neighbour to Long-Range Integrable Spin Chains]{}”*, , [`arXiv:0807.5081`](http://arXiv.org/abs/arXiv:0807.5081) \[hep-th\]. C. Kristjansen, M. Orselli and K. Zoubos, *“[Non-planar ABJM Theory and Integrability]{}”*, , [`arXiv:0811.2150`](http://arXiv.org/abs/arXiv:0811.2150) \[hep-th\]. B. I. Zwiebel, *“[$\mathcal{N}=\mathord{}$4]{} SYM to two loops: Compact expressions for the non-compact symmetry algebra of the ${\mathfrak{su}}(1,1|2)$ sector”*, , [`arXiv:hep-th/0511109`](http://arXiv.org/abs/arXiv:hep-th/0511109). J. Bhattacharya and S. Minwalla, *“[Superconformal Indices for $\mathcal{N}=6$ Chern Simons Theories]{}”*, , [`arXiv:0806.3251`](http://arXiv.org/abs/arXiv:0806.3251) \[hep-th\]. M. Berkooz and D. Reichmann, *“[Weakly Renormalized Near $1/16$ SUSY Fermi Liquid Operators in $\mathcal{N} = 4$ SYM]{}”*, , [`arXiv:0807.0559`](http://arXiv.org/abs/arXiv:0807.0559) \[hep-th\]. B. Eden and M. Staudacher, *“Integrability and transcendentality”*, , [`arXiv:hep-th/0603157`](http://arXiv.org/abs/arXiv:hep-th/0603157). G. P. Korchemsky, *“[Quasiclassical QCD pomeron]{}”*, , [`arXiv:hep-th/9508025`](http://arXiv.org/abs/arXiv:hep-th/9508025). R. Koekoek and R. F. Swarttouw, *“[The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue]{}”*, [`arXiv:math/9602214`](http://arXiv.org/abs/arXiv:math/9602214). A. V. Kotikov, A. Rej and S. Zieme, *“[Analytic three-loop Solutions for $\mathcal{N}=4$ SYM Twist Operators]{}”*, , [`arXiv:0810.0691`](http://arXiv.org/abs/arXiv:0810.0691) \[hep-th\].
[^1]: The all-loop proposal depends on an unknown interpolating function that appears in the one-magnon dispersion relation [@Nishioka:2008gz; @Gaiotto:2008cg],[@Grignani:2008is; @Berenstein:2008dc].
[^2]: $\mathcal{N}=4$ SYM actually has two types of ${\mathfrak{sl}}(2)$ sectors, with bosonic and fermionic module elements respectively. For our purposes it is not important which sector was used for each of the following steps, so we will not again refer to this distinction.
[^3]: The one-loop ${\mathfrak{sl}}(2)$ sector dilatation generator is fixed by the twist-two spectrum, which was also computed earlier in [@Lipatov:1997vu; @Dolan:2001tt].
[^4]: As in the $\mathcal{N}=4$ SYM calculations, we also use basic structural properties of the gauge theory, such as the range of planar interactions.
[^5]: An alternative approach would be to use the construction of [@Sochichiu:2008tw] of the two-loop dilatation generator for arbitrary three-dimensional Lorentz-invariant renormalizable theories. That is likely to be a much more difficult approach since it would not take full advantage of superconformal symmetry.
[^6]: Beyond leading order, as we will see, generic such states acquire anomalous dimension so that they do not satisfy the exact $1/12$ BPS condition $R^1{}_1 + R^2{}_2 - L^2{}_2 - D = 0$. Here the full scaling dimension is $D=D_0 + \delta D$, which includes an anomalous contribution.
[^7]: Of course, ${\mathfrak{L}}$ also still acts within this sector, and is proportional to the action of ${\mathfrak{R}}^{12}$.
[^8]: I thank A. Belitsky for suggesting such a basis, actually in the context of $\mathcal{N}=4$ SYM.
[^9]: Provided we project the right sides to their bosonic components. For example, the first term for ${\mathfrak{R}}^{ab}$ only includes $\theta^{\{a}\varepsilon^{b\}c} \partial_{\theta^c}$ and *not* $\theta^{\{a}\varepsilon^{b\}c} \partial_{\eta^c}$.
[^10]: We could also write $r_{\pm}(x) = \sqrt{\frac{\sqrt{\pi/2}\Gamma(x+1)}{2^{x\pm {{{\textstyle\frac{1}{2}}}}}\Gamma(x+1\pm{{{\textstyle\frac{1}{2}}}})}} \rightarrow \sqrt{\frac{\sqrt{\pi}\Gamma(x+1)}{\Gamma(x+1\pm{{{\textstyle\frac{1}{2}}}})}}$. Up to a factor of $\sqrt{2}$ in one case, we can use the second expression since the powers of $2$ will cancel in the expressions for the $c_i$ given here and later.
[^11]: Here we have chosen not to define $\mathbf{p}$ with a factor of $(-1)^L$ for chains with $2L$ sites, which would make all generators parity even.
[^12]: With grading $\eta=1$. For the opposite grading, one adds rather than removes a $u_3$ root at zero.
[^13]: We have obtained extra confirmation by also performed a number of checks of the Serre relation on two-site alternating or homogeneous chains. Of course, as stated above, the Serre relations are guaranteed to be satisfied because of the one-site result.
[^14]: It is also possible to absorb the identity component into the next-nearest neighbor terms since a permutation squared equals the identity, as does the sum (with unit weight) over the projectors for next-nearest neighbors. The coefficient would then be $(-1)^{j_1+j_3}({{{\textstyle\frac{1}{2}}}}S_1(j_2-1/2) + \log 2)$, which is rational.
[^15]: The absence of one- or two-site interactions slightly simplifies our argument, but is not essential.
[^16]: There are twelve isomorphic ${\mathfrak{osp}}(4|2)$ sectors from different choices of the $1/12$ BPS condition.
[^17]: Note that either all or none of the ${\mathfrak{Q}}_{12,1}{\mathopen{\big|}\Omega_I\mathclose{\bigr \rangle}}$ satisfy the BPS condition since they all have the same Cartan charges.
[^18]: I thank E. Ragoucy for helpful related discussions.
[^19]: A degenerate Cartan matrix is an obstruction to constructing a R-matrix, but that is not a problem for ${\mathfrak{osp}}(6|4)$ since its Cartan matrix is invertible.
[^20]: We have used to evaluate these sums, symbolically as a function of $s$.
[^21]: Specifically, the cancellations are: last term of (\[eq:11\]) with first term of (\[eq:rycom\]) (and $\mathbf{p}$ image cancellation), first term of (\[eq:11\]) with second term of (\[eq:21\]), (\[eq:13\]) with $\mathbf{p}$ of (\[eq:23\]) (and $\mathbf{p}$ image), second-to-last term of (\[eq:11\]) with $\mathbf{p}$ of (\[eq:22\]) (and $\mathbf{p}$ image).
|
$ $
[[A random walk on Area Restricted Search]{}]{}
[[Simone Santini]{}]{}
[[Escuela Politécnica Superior\
Universidad Autónoma de Madrid]{}]{}
[[**XXVI**]{}]{}
$ $
[(C) Simone Santini, [2019]{}]{}
$ $
Author contact: **simone.santini@uam.es**
$ $
$ $
ARS, reward, and dopamine in the evolution of life
==================================================
The starting point of these notes is a talk to which I heard few years back in Milan, Italy. The speaker, Giuseppe Boccignone, showed us the pair of images in Figure \[wow\], which represent two paths in two dimensions. It is not hard to see that the macroscopic characteristics of these images are quite the same. Their most evident macroscopic feature is that they are heavily clustered: there are a bunch of points in a very restricted area and then suddenly the path jumps to a different area where a new cluster of points is created.
height 0.3
The images are so similar that one would have little trouble believing that they have been created by two instances of the same physical phenomenon. Yet, much to my amazement, Boccignone told us that this was not the case. The figure on the right shows the saccadic eye movements of a person looking at the picture that you can faintly see in the background; the picture on the left is the path followed by a spider monkey of the Yucatan peninsula while looking for food. It has nothing to do with the picture and was superimposed to it only to make the point more forcefully.
To find such a similarity in completely unrelated activities of two different species is as striking as it would be to find out that the ritualistic chant of a remote tribe in Papua has the same harmonic structure as the Goldberg Variations. Just as in the case of the tribe we would like to look for an explanation (maybe a previous contact with some Bach-loving explorer that has been incorporated into the rituals of the tribe), so in this case it is not too far-fetched to start looking for some common underlying mechanism.
We are encouraged in our endeavor by the fact that the two behaviors do have indeed something in common: they are both examples of *search*. Search for visual information in one case, search for food in the other. So, we are on a hunt for a common mechanism that guides *search* in a wide variety of species under the most diverse circumstances. The mechanism must be very general, since it should apply not only to different species but also to very different levels of abstraction (from search for food in physical space to search for information in conceptual space).
The behavior that we observe in these two examples is commonly known as *ARS (Area Restricted Search)*, a strategy that consists in [“[a concentration of searching effort around areas where rewards in the form of specific resources have been found in the past. When resources are encountered less frequently, behavior changes such that the search becomes less sensitive, but covers more space]{}”]{} .
As we shall see, the same basic mechanism permits ARS in a variety of cases and circumstances, from the foraging behavior of the nematode *C.elegans* to goal-directed cognition in people. You can have a personal experience of ARS by looking at Figure \[ARS-exp\] and following the instructions in the caption (read the caption before looking at the pictures).
width
ARS is incredibly widespread. Some form of it has been found in all major eumetazoan clades. To have an idea of what this entails, in Figure \[taxonomy\] I have drawn a very partial taxonomy of the animal kingdom.
[ $\displaystyle
\xymatrix@R=1em@C=4em{
& & \mbox{metazoa} \ar@{-}[dr] \ar@{-}[dl] \\
& \mbox{porifera} & & \mbox{eumetazoa} \ar@{-}[dr] \ar@{-}[dl] \\
& & \mbox{bilatera} \ar@{-}[dll] \ar@{-}[dl] \ar@{-}[d] \ar@{-}[drr] & & \mbox{radiata} \\
\mbox{lophotrochozoa} \ar@{-}[dd]& \mbox{platyzoa} & \mbox{ecydozoa} \ar@{-}[dr] \ar@{-}[dl] \ar@{-}[d] & & \mbox{deuterostomia} \ar@{-}[d] \\
& {} \ar@{..}[r] & \mbox{nemotoda}\ar@{..}[r] & {} & \mbox{chordata} \ar@{-}[dr] \ar@{-}[dl] \ar@{-}[d] \\
(mollusks) & & & {} \ar@{..}[r] & \mbox{craniata} \ar@{-}[dr] \ar@{-}[dl] \ar@{..}[r] & {} \\
& & & \mbox{vertebrata} & & \mbox{myxini}
}
$ ]{}
The clade on the left of the root, the *porifera* is composed of animals that do not have a real tissue: sponges and little else. The other clade, *eumetazoa* contains all other animals, from worms to mollusks to you and me. ARS can be observed, in some form or another, in the whole eumetazoa clade. This broad presence indicates that the mechanism behind ARS must have evolved quite early, since the major divisions of the eumetazoa clade are very ancient, and it is reasonable to assume that all forms of ARS derive from a mechanism that was put in place before this division. ARS is, in other words, one of the basic mechanisms of life.
ARS might even be more basic than the eumetazoa: there are molecular mechanisms in protozoa that could be precursors of ARS. The most primitive example is the [“[run and tumble]{}”]{} movement of *E.coli* and *Salmonella typhimurium*. The movement of these bacteria is controlled by a flagellar motor. *Runs* consist of forward motion (longish stretch of resource search), while *tumbles* are made of random turns that keep the bacteria more or less in the same place (exploiting local resources while they last). Receptor proteins in the membrane bind these behaviors to external stimuli . The mechanism on which ARS is based is, in its essential structure, fairly consistent across the whole spectrum of eumetazoans. Its fully formed presence in organisms with limited learning capabilities, such as *C.elegans* suggests that learning is not involved or, to the extent that it is (such as in mammals), it is based on a fully formed pre-learning machinery. This is what makes ARS so interesting: it is a basic mechanism that very different forms of life have adopted as a basic strategy to solve such diverse problems as looking for food in a Petri dish or trying to prove a mathematical theorem. Its omnipresence derives from the optimality of ARS as a search strategy in cases in which resources are [“[clumpy]{}”]{} and the information about the locations of the [“[clumps]{}”]{} is limited (we’ll see that in the next section). In the case of foraging animals, the resource is food, and the reward is finding something to eat. In the case of somebody looking at an image, the resource is information about its content, and the reward is understanding what the image is about[^1]. In all these cases, the animal or person moves locally around the same clump as long as it is rewarding to do so (viz. as long as locally one finds more food or new information), then starts moving rapidly to explore quickly new territory in search of a new clump.
(56,6)(0,0) (0,0)[ (2,1)[(2,1)[2]{}]{} (2,6)[(2,-1)[2]{}]{} (4,2)[(0,1)[3]{}]{} (4,5)[(2,1)[2]{}]{} (6,6)[(2,-1)[2]{}]{} (8,5)[(0,-1)[3]{}]{} (8,2)[(-2,-1)[2]{}]{} (6,1)[(-2,1)[2]{}]{} (4.4,2.4)[(0,1)[2.2]{}]{} (6,5.6)[(2,-1)[1.6]{}]{} (6,1.4)[(2,1)[1.6]{}]{} (8,5)[(2,1)[2]{}]{} (10,6)[(2,-1)[2]{}]{} (12,5)[(0,-1)[2.5]{}]{} (12,1.8)[(0,0)[NH$_2$]{}]{} (1.8,1)[(0,0)\[r\][HO]{}]{} (1.8,6)[(0,0)\[r\][HO]{}]{} (6,-2)[(0,0)[dopamine]{}]{} ]{} (20,0)[ (1,3)(4,0)[3]{}[(2,1)[2]{}]{} (3,4)(4,0)[3]{}[(2,-1)[2]{}]{} (0,0)(8,0)[2]{}[ (2.8,4.2)[(0,1)[1.4]{}]{} (3.2,4.2)[(0,1)[1.4]{}]{} (3,6)[(0,0)\[b\][O]{}]{} ]{} (9,3)[(0,-1)[2]{}]{} (8.75,1)[(0,0)\[lt\][NH$_3$ (+)]{}]{} (13.2,3)[(0,0)\[l\][O (-)]{}]{} (0.8,3)[(0,0)\[r\][(-) O]{}]{} (8.75,5.3)[(0,0)\[b\][H]{}]{} (9,3)[(0,1)[2]{}]{} (9,3)[(-1,4)[0.5]{}]{} (9,5)[(-1,0)[0.5]{}]{} (7,-2)[(0,0)[glutamate]{}]{} ]{} (40,0)[ (2,3)(4,0)[3]{}[(2,1)[2]{}]{} (4,4)(4,0)[2]{}[(2,-1)[2]{}]{} (3.8,4.2)[(0,1)[1.4]{}]{} (4.2,4.2)[(0,1)[1.4]{}]{} (4,6)[(0,0)\[b\][O]{}]{} (1.8,3)[(0,0)\[r\][HO]{}]{} (12.2,4)[(0,0)\[l\][NH$_2$]{}]{} (8,-2)[(0,0)[GABA]{}]{} ]{}
The [“[reward]{}”]{} that one is after can be something very concrete (food) or something very abstract (the [“[Eureka]{}”]{} moment of proving a theorem) but in all cases the nervous system makes the reward substantial by encoding it as the release of a very specific chemical: *dopamine* (figure \[dopamine\]). The organism in which the molecular basis of ARS is best understood is the nematode *C.elegans* . The neural circuitry consists of eight sensory neurons presynaptic to eight interneurons that coördinate forward and backward movements. The sensory neurons alter the turning frequency by releasing dopamine on the interneurons, modulating the reception of glutammate (Figure \[elegans\]).
height 15em [graphs/elegans\_da.jpg]{}
External administration of dopamine increases the turning frequency, while administration of a dopamine antagonist reduces it . A reasonable model of *C.elegans* behavior suggests that, while on food, the sensory neurons release dopamine, which, via the action of the glutammate, leads to increased switching behavior in the interneurons, resulting in more turns and, consequently, in a trajectory that stays local. When off food, the dopaminergic activity is reduced, and the interneurons reduce their switching frequency, leading to less turns and more ground covered
Although *C.elegans* is the only organism for which the neuromolecular mechanism of ARS is well understood, there is strong evidence of dopaminergic modultation of glutamatergic synapses throughout the major clades of the eumetazoans . In insects, for example, dopaminergic neurons in the abdominal ganglion are sparsely distributed, but show large branching patterns, indicative of neuromodulation .
The relation between dopamine and ARS has been documented throughout the invertebrates, especially in the fruit fly *Drosophila melanogaster* , in crustaceans , in *Aplysia* , etc. In all these cases, ARS is limited to food search which is, clearly, the search problem for which ARS first evolved.
In vertebrates, the modification of behavior by dopamine increases in complexity and begins to involve behavior not directly related to food. For example, in frogs and toads dopamine modulation is involved in the visuomotor focus on preys ; similar dopaminergic involvement in visuomotor coordination can be found in rats and humans . This finding is significant in that it indicates a strong relation between ARS and *inhibition of return*: the fact that viewers show significant latency in revisitig objects or regions of a scene that have already been investigated, united to the lingering of saccadic movements in regions of intertest .
The important change in vertebrates is the extension of ARS-like behavior to cover not only actions with an immediate reward, such as the search for food, but also situations in which the reward is projected or even in which the reward itself is a neural state. The detachment from the immediate food rewards is what makes it possible to adapt ARS to abstract functions such as *goal-directed cognition*. It seems, in other words, that when new problems arose that had the same abstract structure as search for food, animals, rather than developing a new mechanism, coöpted the dopaminergic modulation that guided food search to work on the new problem.
The most important neural structure associated with goal-directed cognition is the *basal ganglia* and, more specifically, the *striatum* (Figure \[striatum\]).
(0,0)(0,0) (1,-4)[(0,0)\[l\][1. lateral medial]{}]{} (1,-6)[(0,0)\[l\][2. globus pallidus]{}]{} (1,-8)[(0,0)\[l\][3. striatum]{}]{}
height 15em [graphs/basal-ganglia.jpg]{}
Information enters the basal ganglia through the striatum, and a great number of the inputs to the system are dopaminergic . The structure of the basal ganglia and much of their connectivity are maintained across vertebrates . The major change from anamniotes (fish and amphibians) to amniotes is the proliferation of dopaminergic neurons that input to the striatum , while the structure of the striatum stays pretty much the same.
The balance between glutammate and dopamine in the striatum is key to the proper functioning of ARS-like activities, and an imbalance between the two neurotransmitters is suspected in a number of pathologies affecting goal-directed cognition, including Parkinson’s, schizophrenia, and addiction. Many of these conditions can be regarded as radicalization of ARS in one direction or another (too local or too global) due to imperfect dopamine control (see Figure \[diseases\]).
(20,8)(0,0) (0,3)[ (0,0)(0,2)[2]{}[(1,0)[20]{}]{} (0,0)(10,0)[3]{}[(0,1)[2]{}]{} (5,1)[(0,0)[[Focused]{}]{}]{} (15,1)[(0,0)[[Diffused]{}]{}]{} (0.1,0.1)[(0,0)\[lb\][[too much DA]{}]{}]{} (19.9,0.1)[(0,0)\[rb\][[too little DA]{}]{}]{} (6,2.2)[(1,0)[8]{}]{} (8,2.7)(0.5,0)[8]{}[(1,0)[0.25]{}]{} (10,2.9)[(0,0)\[b\][schizophrenia]{}]{} ]{} (0,2)[ (0,0)[(0,1)[0.8]{}]{} (0,-0.1)[(0,0)\[t\][autism]{}]{} ]{} (7,2)[ (0,0)[(0,1)[0.8]{}]{} (0,-0.1)[(0,0)\[t\][addiction]{}]{} ]{} (18,2)[ (0,0)[(0,1)[0.8]{}]{} (0,-0.1)[(0,0)\[t\][parkinson’s]{}]{} ]{} (3,6)[ (0,0)[(0,-1)[0.8]{}]{} (0,0.1)[(0,0)\[b\][OCD, TS]{}]{} ]{} (16,6)[ (0,0)[(0,-1)[0.8]{}]{} (0,0.1)[(0,0)\[b\][ADHD]{}]{} ]{}
In the striatum, dopaminergic neurons modulate the glutammatergic input at the tips of spiny neurons. The action of the dopaminergic inputs appears to perform a neuromodulation of the strength of the glutammatergic inputs (Figure \[mammalian\]) The mechanism is similar to that described for *C.elegans* in Figure \[elegans\], but in the mammalian striatum the shape of the spiny neuron has specialized for this function and the inputs, several magnitudes higher in number, come primarily from connections to cortical neurons rather than directly from sensory neurons as in *C.Elegans* . However, at the level of the microcircuit, little has changed from nematodes to amniote vertebrates. The origin of the dopaminergic input does, however, mark a fundamental evolutionary shift in the activity range of ARS-like behavior. While in *C.elegans* or *Drosophila* the afferent dopaminergic signal is reliably related to the presence or absence of food, in higher vertebrates the signal may represent the *expectation of a reward*. The critical transition here is from a concrete (directly sensed) reward to its neural representation that is, from a physical reward to the abstract idea of a reward . As Hills puts it: [“[the evolutionary theory $[$of ARS$]$ is therefore completely consistent with the reward theory of dopamine, but adds the evolutionary hypothesis that the initial reward represented by the release of dopamine were food. Only later was this system co-opted to represent the expectation of a reward, which allows for goal-directed cognition]{}”]{} .
To conclude this brief *excursus* of ARS, we consider a region of relatively recent evolution that, outside of the basal ganglia, is heavily involved in goal-directed behavior: the *prefrontal cortex* (PFC). The PFC has clearly evolved much later than ARS; nevertheless, it is heavily involved in goal-directed behavior via massive connections to the striatum . Dopamine has been shown to be a factor in the sustained activation of the PFC . Most models of PFC see the rôle of dopamine as holding objects in attention long enough for appropriate behavior to be activated . Consistently with ARS, already known solutions mediated by the PFC are most typically tried when a problem has to be solved in a new situation .
The context in which goal-directed cognition takes place includes external and internal stimuli; ARS depends in part on the alignment of external stimuli with previous expectations. This is likely to be controlled by the connections between the PFC and the *Nucleus Accumbens* (NAcc) in the striatum, which modulates attention, eye movements, and the maintenance of working memory [@bertolucci:90; @floresco:99; @schultz:04]; dopamine has been identified as one of the main influences in the modulation of NAcc activity [@floresco:96]: novel stimuli lead to increase in dopamine in the NAcc and in the PFC [@berns:01].
height 18em [graphs/mammalian\_da.jpg]{}
So, in the evolution of vertebrates, we see a progressive extension of the rôle of ARS, from the dopaminergic control of visuomotor control in frogs and toads to the similarity mediated maintenance of ideas in working memory [@schultz:95]. ARS appears therefore to be one of the fundamental strategies in the animal kingdom, co-opted and adapted to a number of situations, from the [“[run and tumble]{}”]{} behavior of *E.coli* to the way we focus on and later abandon ideas when we think about a problem.
\* \* \*
This brief explanation of the evolutionary basis of ARS has been centered on its molecular mechanism, especially on the rôle of dopamine as neuromodulator. From now on, however, our focus will change: we shall try to understand the *exterior* characteristics of the behavior. ARS leads to a well identified patterns of motion either in the physical space (in the case of foraging), in the visual space (scanning an image), or in any number of abstract spaces. We shall study mathematically these patterns of motion and try to characterize them. Our methods will be based mostly on the study of random walks, of diffusion, and on the kinds of anomalous diffusion to which ARS leads.
Optimality of ARS {#genetic}
=================
The evolutionary success of ARS entails, according to the theory of natural selection, that ARS is an optimal strategy—if not globally, at least locally—for a large set of problems. In abstract terms, we have a space with certain resources placed in different parts of it; we need a strategy to navigate this space collecting the greatest amount of resources. This must be done without information on the placement of the resources. (If we can sense from afar where the resources are located, we simply walk there and get them: no search strategy is necessary.) The nature of the resources can be the most diverse: in the case of foraging (to which we shall mostly make reference), the resource is food; in the case of saccadic movements, it is the visual information that we get from the visual field, and so on. ARS is optimal if the resources are [“[patchy,]{}”]{} that is, if they are organized in resource-rich patches separated by areas of small or zero resource concentration. In the model that we shall develop in this section, we assume that the resources are consumed in the course of the activity, and that they are not replenished while the activity goes on. Resources are consumed simply by moving on top of them (assuming that, after walking on them, they would be consumed with a certain probability would not substantially change the model). In the example of food, this means that we have food distributed in patches (a grove, a pond, a herd, a school of fish...) and the forager moves inside the patch and between patches eating what it finds. We make a number of hypotheses. Firstly, we assume that the food doesn’t move around or if it does (as is the case of animal preys) its movement is not significant and food can be modeled as static. Secondly, we assume that the forager will eat all the food it can find as soon as it finds it (its eyesight is perfect and its appetite endless). Finally, food doesn’t grow back: once it has been eaten at a particular location, that location will remain barren for the rest of the forager’s activity.
In the case of saccades, we assume (as is often the case) that there are patchy areas in the visual field that are rich in information useful to interpret the scene (relevant or telling objects, faces, etc.). We also assume that once we have analyzed the information in a given area of the visual field, that information is remembered and it is not necessary to analyze it again. This is equivalent to the hypothesis that the food doesn’t grow back once it has been eaten.
\* \* \*
In this section, we want to check whether ARS emerges as an optimal solution to the foraging problem. We shall do this by implementing a genetic algorithm based on a competition among individuals whose characteristics are encoded in a string of bits called a *gene* [@mitchell:98].
Individuals move around a *foraging areas* under the guidance of their gene and collect food. Their *score*, which determines their fitness for survival, is the amount of food they have collected. The world in which these individuals move is a regular grid of patches of food of $p\times{p}$ ($p\in{\mathbb{N}}$) *pellets*, separated by barren areas without food (see Figure \[feedenvironment\]). Each pellet is an atomic unit of food, that it, it is either not consumed or consumed entirely. The consumption of each pellet increases the survival fitness by one unit.
(21,21)(-3,-3) (0,0)(0,5)[3]{}[ (0,0)(5,0)[3]{} ]{} (5,5)(0,5)[2]{}[ (0,0)(0.5,0)[10]{}[ (0,0)(0,0.05)[2]{}[(1,0)[0.25]{}]{}]{} ]{} (5,5)(5,0)[2]{}[ (0,0)(0,0.5)[10]{}[ (0,0)(0.05,0)[2]{}[(0,1)[0.25]{}]{}]{} ]{} (-1,1.5)(0,5)[3]{}[ (0,0)(-1,0)[3]{} ]{} (14,1.5)(0,5)[3]{}[ (0,0)(1,0)[3]{} ]{} (1.5,-1)(5,0)[3]{}[ (0,0)(0,-1)[3]{} ]{} (1.5,14)(5,0)[3]{}[ (0,0)(0,1)[3]{} ]{} (5,5)[ (-0.1,-0.1)[(0,0)\[tr\][$(0,0)$]{}]{} (0,-0.75)(3,0)[2]{}[(0,1)[0.5]{}]{} (0,-0.5)[(1,0)[3]{}]{} (1.5,-0.55)[(0,0)\[t\][$p$]{}]{} (0,-1.75)(5,0)[2]{}[(0,1)[0.5]{}]{} (0,-1.5)[(1,0)[5]{}]{} (2.5,-1.45)[(0,0)\[b\][$q$]{}]{} (-0.75,0)(0,3)[2]{}[(1,0)[0.5]{}]{} (-0.5,0)[(0,1)[3]{}]{} (-0.55,1.5)[(0,0)\[r\][$p$]{}]{} (-1.75,0)(0,5)[2]{}[(1,0)[0.5]{}]{} (-1.5,0)[(0,1)[5]{}]{} (-1.45,2.5)[(0,0)\[l\][$q$]{}]{} (1.5,1.5) (6.5,4)[(-2,-1)[4.8]{}]{} (6.5,4)[(0,0)\[lb\][$(x_0,y_0)$]{}]{} ]{}
Each patch is separated from the other by being placed in the lower-left corner of a larger square of $q\times{q}$ units ($q\in{\mathbb{N}}$, $q>p$) called a *plot*. The density of the food is $\rho=p^2/q^2$. The plots are created dynamically at run time as the walker steps on them for the first time, so that the foraging field is virtually infinite. In all the tests discussed below, $p$ is kept fixed ($p=16$), $\rho$ is a parameter that varies from $\rho=0.01$ to $\rho=0.95$, and $q$ is determined as $q=\lceil{p/\sqrt{\rho}}\rceil$. Each individual does a random walk (specified by certain parameters, as described below) starting at $(x_0,y_0)=(p/2,p/2)$ that is, in the center of the patch whose lower-left corner is the origin.
The walk parameters
-------------------
Each time the walker walks on a position containing a pellet, it [“[eats]{}”]{} it, incrementing its score (which determines its evolutionary fitness) by one. The pellet is removed, so that further visits to the location will not provide any food (Figure \[eating\]). The movement of each individual is a random walk whose statistical features depend on whether the individual is currently eating (status: on-food) or whether it has been without food for some time (status: off-food). The individual doesn’t go [“[off-food]{}”]{} immediately as soon as it steps on a location with no food: the individual has memory, so that it gradually changes its status from on-food to off-food during a certain number of time steps. The amount of time without food that it takes to go to the status off-food is controlled by a parameter in the gene of the individual.
(25,7)(0,-1) (2,0)(0,1)[7]{}[(1,0)[6]{}]{} (2,0)(1,0)[7]{}[(0,1)[6]{}]{} (-0.5,2.5)(0.25,0)[16]{}[(1,0)[0.125]{}]{} (3.5,2.5)(0.125,0.125)[8]{} (4.5,3.5)(0.125,-0.125)[16]{} (6.5,1.5)(0.125,0.125)[8]{} (7.5,2.5)(0,0.25)[8]{}[(0,1)[0.125]{}]{} (7.5,4.5)(0.25,0)[8]{}[(1,0)[0.125]{}]{} (9,4)[(0,0)\[tl\][path]{}]{} (14,0)[ (0,0)(0,1)[7]{}[(1,0)[6]{}]{} (0,0)(1,0)[7]{}[(0,1)[6]{}]{} (0,0)(1,0)[6]{} (0,1) (1,1) (2,1) (3,1) (5,1) (2,2) (4,2) (0,3) (1,3) (3,3) (4,3) (0,4)(1,0)[5]{} (0,5)(1,0)[6]{} (3,-0.5)[(0,0)\[t\][patch after the walk]{}]{} (3,-1)[(0,0)\[t\][(score: 8)]{}]{} ]{}
The general behavior of the random walk is the same regardless of whether the individual is on-food or off-food (the only thing that changes in the two cases is the numerical value of the parameters). Consider a generic situation in which the parameters are $(\alpha_0,l_0)$. The individual is coming from a direction $\theta$, being currently at location $(x,y)$. The individual chooses a deviation angle $\alpha$, selected with a Gaussian distribution centered at $\alpha_0$, and a length $l$ selected with an exponential distribution with average $l_0$, and performs a jump in a direction at an angle $\alpha$ from its current direction, and for a length $l$ (Figure \[jump\]). The parameters $(\alpha_0,l_0)$ characterize the statistics of the jump, they are encoded in the individual’s gene, and they take different values depending on whether the individual is on-food or off-food. When the individual is on-food, the jumps are done according to the parameters $(\alpha_f,l_f)$, while when the individual has been for some time off-food, the jumps are done according to $(\alpha_n,l_n)$. The switch from the on-food parameters to the off-food is done gradually when the individual is in an area without food. The gene defines a memory threshold $\tau$ to switch from the on-food to the off-food behavior. If $t$ is the number of step that the individual has been off-food ($t=0$ if the individual is on-food) then the parameters for the next jump will be $$\begin{array}{ll}
\begin{array}{rc}
(\alpha_f,l_f) &
\end{array} & \mbox{on food} \\
~ \\
\left.
\begin{array}{rc}
(\alpha_f + \frac{t}{\tau}(\alpha_n-\alpha_f), l_f + \frac{t}{\tau} (l_n-l_f)) & t < \tau \\
(\alpha_n,l_n) &t \ge \tau
\end{array}
\right\} & \mbox{off food}
\end{array}$$
(45,15)(0,-3.5) (0,0)[(1,0)[13]{}]{} (1,0)[(3,1)[6]{}]{} (7,2)(0.3,0.1)[18]{} (4.5,0.3)[(0,0)\[b\][$\theta$]{}]{} (7,1.8)[(0,0)\[tl\][$(x,y)$]{}]{} (7,2) (7,2)[(1,2)[2]{}]{} (7.1,2.1)[(1,2)[2]{}]{} (6.5,4.5)[(1,2)[1]{}]{} (6.5,4.5)[(-1,-2)[1]{}]{} (6.4,4.6)[(0,0)\[rb\][$l$]{}]{} (8.5,3.5)[(0,0)\[l\][$\alpha$]{}]{} (6,-3.5)[(0,0)[**(a)**]{}]{} (21,-3.5)[(0,0)[**(b)**]{}]{} (36,-3.5)[(0,0)[**(c)**]{}]{} (15,-3)
(5,5)(-2.5,-2.5) (-2.5,0)[(1,0)[5]{}]{} (0,-2.5)[(0,1)[5]{}]{} (1.000,0.000) (0.992,0.125) (0.969,0.249) (0.930,0.368) (0.876,0.482) (0.809,0.588) (0.729,0.685) (0.637,0.771) (0.536,0.844) (0.426,0.905) (0.309,0.951) (0.187,0.982) (0.063,0.998) (-0.063,0.998) (-0.187,0.982) (-0.309,0.951) (-0.426,0.905) (-0.536,0.844) (-0.637,0.771) (-0.729,0.685) (-0.809,0.588) (-0.876,0.482) (-0.930,0.368) (-0.969,0.249) (-0.992,0.125) (-1.000,0.000) (-0.992,-0.125) (-0.969,-0.249) (-0.930,-0.368) (-0.876,-0.482) (-0.809,-0.588) (-0.729,-0.685) (-0.637,-0.771) (-0.536,-0.844) (-0.426,-0.905) (-0.309,-0.951) (-0.187,-0.982) (-0.063,-0.998) (0.063,-0.998) (0.187,-0.982) (0.309,-0.951) (0.426,-0.905) (0.536,-0.844) (0.637,-0.771) (0.729,-0.685) (0.809,-0.588) (0.876,-0.482) (0.930,-0.368) (0.969,-0.249) (0.992,-0.125) (2.000,0.000) (1.996,0.126) (1.984,0.251) (1.965,0.375) (1.937,0.497) (1.902,0.618) (1.860,0.736) (1.810,0.852) (1.753,0.964) (1.689,1.072) (1.618,1.176) (1.541,1.275) (1.458,1.369) (1.369,1.458) (1.275,1.541) (1.176,1.618) (1.072,1.689) (0.964,1.753) (0.852,1.810) (0.736,1.860) (0.618,1.902) (0.497,1.937) (0.375,1.965) (0.251,1.984) (0.126,1.996) (0.000,2.000) (-0.126,1.996) (-0.251,1.984) (-0.375,1.965) (-0.497,1.937) (-0.618,1.902) (-0.736,1.860) (-0.852,1.810) (-0.964,1.753) (-1.072,1.689) (-1.176,1.618) (-1.275,1.541) (-1.369,1.458) (-1.458,1.369) (-1.541,1.275) (-1.618,1.176) (-1.689,1.072) (-1.753,0.964) (-1.810,0.852) (-1.860,0.736) (-1.902,0.618) (-1.937,0.497) (-1.965,0.375) (-1.984,0.251) (-1.996,0.126) (-2.000,0.000) (-1.996,-0.126) (-1.984,-0.251) (-1.965,-0.375) (-1.937,-0.497) (-1.902,-0.618) (-1.860,-0.736) (-1.810,-0.852) (-1.753,-0.964) (-1.689,-1.072) (-1.618,-1.176) (-1.541,-1.275) (-1.458,-1.369) (-1.369,-1.458) (-1.275,-1.541) (-1.176,-1.618) (-1.072,-1.689) (-0.964,-1.753) (-0.852,-1.810) (-0.736,-1.860) (-0.618,-1.902) (-0.497,-1.937) (-0.375,-1.965) (-0.251,-1.984) (-0.126,-1.996) (-0.000,-2.000) (0.126,-1.996) (0.251,-1.984) (0.375,-1.965) (0.497,-1.937) (0.618,-1.902) (0.736,-1.860) (0.852,-1.810) (0.964,-1.753) (1.072,-1.689) (1.176,-1.618) (1.275,-1.541) (1.369,-1.458) (1.458,-1.369) (1.541,-1.275) (1.618,-1.176) (1.689,-1.072) (1.753,-0.964) (1.810,-0.852) (1.860,-0.736) (1.902,-0.618) (1.937,-0.497) (1.965,-0.375) (1.984,-0.251) (1.996,-0.126) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.000, -0.000) ( -0.001, -0.000) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.001) ( -0.001, -0.002) ( -0.001, -0.002) ( -0.001, -0.002) ( -0.001, -0.002) ( -0.001, -0.002) ( -0.001, -0.002) ( -0.001, -0.002) ( -0.001, -0.003) ( -0.002, -0.003) ( -0.002, -0.003) ( -0.002, -0.003) ( -0.002, -0.004) ( -0.002, -0.004) ( -0.002, -0.004) ( -0.002, -0.004) ( -0.002, -0.005) ( -0.002, -0.005) ( -0.002, -0.005) ( -0.002, -0.006) ( -0.002, -0.006) ( -0.002, -0.007) ( -0.002, -0.007) ( -0.002, -0.008) ( -0.002, -0.008) ( -0.002, -0.009) ( -0.002, -0.009) ( -0.002, -0.010) ( -0.002, -0.010) ( -0.002, -0.011) ( -0.002, -0.012) ( -0.002, -0.012) ( -0.002, -0.013) ( -0.002, -0.014) ( -0.001, -0.015) ( -0.001, -0.016) ( -0.001, -0.017) ( -0.001, -0.018) ( -0.001, -0.019) ( -0.000, -0.020) ( 0.000, -0.021) ( 0.000, -0.022) ( 0.001, -0.023) ( 0.001, -0.024) ( 0.002, -0.026) ( 0.002, -0.027) ( 0.003, -0.029) ( 0.003, -0.030) ( 0.004, -0.032) ( 0.005, -0.033) ( 0.006, -0.035) ( 0.006, -0.037) ( 0.007, -0.038) ( 0.008, -0.040) ( 0.009, -0.042) ( 0.011, -0.044) ( 0.012, -0.046) ( 0.013, -0.048) ( 0.015, -0.051) ( 0.016, -0.053) ( 0.018, -0.055) ( 0.020, -0.058) ( 0.022, -0.060) ( 0.024, -0.063) ( 0.026, -0.065) ( 0.028, -0.068) ( 0.031, -0.071) ( 0.033, -0.074) ( 0.036, -0.077) ( 0.039, -0.080) ( 0.042, -0.083) ( 0.045, -0.086) ( 0.049, -0.089) ( 0.052, -0.092) ( 0.056, -0.095) ( 0.060, -0.098) ( 0.065, -0.102) ( 0.069, -0.105) ( 0.074, -0.109) ( 0.079, -0.112) ( 0.084, -0.116) ( 0.089, -0.119) ( 0.095, -0.123) ( 0.101, -0.126) ( 0.107, -0.130) ( 0.114, -0.133) ( 0.121, -0.137) ( 0.128, -0.140) ( 0.135, -0.144) ( 0.143, -0.147) ( 0.151, -0.151) ( 0.159, -0.154) ( 0.168, -0.157) ( 0.177, -0.161) ( 0.186, -0.164) ( 0.196, -0.167) ( 0.206, -0.170) ( 0.216, -0.173) ( 0.227, -0.176) ( 0.238, -0.178) ( 0.249, -0.181) ( 0.261, -0.183) ( 0.273, -0.185) ( 0.285, -0.187) ( 0.298, -0.189) ( 0.311, -0.191) ( 0.325, -0.192) ( 0.339, -0.193) ( 0.353, -0.194) ( 0.368, -0.195) ( 0.383, -0.195) ( 0.399, -0.195) ( 0.414, -0.195) ( 0.431, -0.194) ( 0.447, -0.193) ( 0.464, -0.192) ( 0.481, -0.190) ( 0.499, -0.188) ( 0.516, -0.186) ( 0.534, -0.183) ( 0.553, -0.180) ( 0.572, -0.176) ( 0.591, -0.172) ( 0.610, -0.167) ( 0.629, -0.162) ( 0.649, -0.156) ( 0.669, -0.150) ( 0.689, -0.143) ( 0.710, -0.135) ( 0.730, -0.127) ( 0.751, -0.119) ( 0.772, -0.110) ( 0.793, -0.100) ( 0.814, -0.090) ( 0.835, -0.079) ( 0.857, -0.067) ( 0.878, -0.055) ( 0.899, -0.042) ( 0.920, -0.029) ( 0.942, -0.015) ( 0.963, 0.000) ( 0.984, 0.015) ( 1.005, 0.032) ( 1.026, 0.048) ( 1.046, 0.066) ( 1.067, 0.084) ( 1.087, 0.103) ( 1.107, 0.122) ( 1.126, 0.142) ( 1.146, 0.163) ( 1.165, 0.184) ( 1.183, 0.207) ( 1.202, 0.229) ( 1.219, 0.253) ( 1.237, 0.276) ( 1.254, 0.301) ( 1.270, 0.326) ( 1.286, 0.352) ( 1.301, 0.378) ( 1.315, 0.405) ( 1.329, 0.432) ( 1.343, 0.460) ( 1.355, 0.488) ( 1.367, 0.517) ( 1.378, 0.546) ( 1.389, 0.575) ( 1.398, 0.605) ( 1.407, 0.635) ( 1.415, 0.666) ( 1.422, 0.697) ( 1.428, 0.728) ( 1.434, 0.759) ( 1.438, 0.791) ( 1.442, 0.822) ( 1.444, 0.854) ( 1.446, 0.886) ( 1.447, 0.918) ( 1.446, 0.950) ( 1.445, 0.982) ( 1.443, 1.014) ( 1.439, 1.046) ( 1.435, 1.078) ( 1.430, 1.109) ( 1.423, 1.140) ( 1.416, 1.172) ( 1.408, 1.202) ( 1.398, 1.233) ( 1.388, 1.263) ( 1.377, 1.293) ( 1.364, 1.322) ( 1.351, 1.351) ( 1.337, 1.379) ( 1.321, 1.407) ( 1.305, 1.435) ( 1.288, 1.461) ( 1.270, 1.487) ( 1.251, 1.512) ( 1.231, 1.537) ( 1.211, 1.561) ( 1.189, 1.584) ( 1.167, 1.606) ( 1.144, 1.628) ( 1.120, 1.648) ( 1.096, 1.668) ( 1.070, 1.687) ( 1.045, 1.705) ( 1.018, 1.721) ( 0.991, 1.737) ( 0.963, 1.752) ( 0.935, 1.766) ( 0.906, 1.779) ( 0.877, 1.791) ( 0.848, 1.801) ( 0.818, 1.811) ( 0.787, 1.819) ( 0.757, 1.827) ( 0.726, 1.833) ( 0.695, 1.838) ( 0.663, 1.842) ( 0.632, 1.845) ( 0.600, 1.847) ( 0.569, 1.848) ( 0.537, 1.848) ( 0.505, 1.846) ( 0.473, 1.844) ( 0.442, 1.840) ( 0.410, 1.835) ( 0.379, 1.830) ( 0.348, 1.823) ( 0.317, 1.815) ( 0.286, 1.806) ( 0.256, 1.796) ( 0.226, 1.785) ( 0.196, 1.774) ( 0.166, 1.761) ( 0.138, 1.747) ( 0.109, 1.733) ( 0.081, 1.717) ( 0.053, 1.701) ( 0.026, 1.684) ( 0.000, 1.666) ( -0.026, 1.647) ( -0.051, 1.628) ( -0.076, 1.608) ( -0.100, 1.587) ( -0.123, 1.566) ( -0.146, 1.544) ( -0.168, 1.521) ( -0.189, 1.498) ( -0.210, 1.475) ( -0.230, 1.450) ( -0.249, 1.426) ( -0.267, 1.401) ( -0.285, 1.376) ( -0.302, 1.350) ( -0.318, 1.324) ( -0.333, 1.298) ( -0.348, 1.272) ( -0.362, 1.245) ( -0.375, 1.218) ( -0.387, 1.191) ( -0.399, 1.164) ( -0.409, 1.137) ( -0.419, 1.110) ( -0.429, 1.083) ( -0.437, 1.056) ( -0.445, 1.029) ( -0.452, 1.002) ( -0.459, 0.975) ( -0.464, 0.948) ( -0.469, 0.921) ( -0.474, 0.895) ( -0.477, 0.868) ( -0.481, 0.842) ( -0.483, 0.817) ( -0.485, 0.791) ( -0.486, 0.766) ( -0.487, 0.741) ( -0.487, 0.716) ( -0.486, 0.692) ( -0.485, 0.668) ( -0.484, 0.644) ( -0.482, 0.621) ( -0.480, 0.599) ( -0.477, 0.576) ( -0.473, 0.554) ( -0.470, 0.533) ( -0.466, 0.512) ( -0.461, 0.491) ( -0.456, 0.471) ( -0.451, 0.451) ( -0.446, 0.432) ( -0.440, 0.413) ( -0.434, 0.395) ( -0.428, 0.377) ( -0.421, 0.360) ( -0.415, 0.343) ( -0.408, 0.327) ( -0.401, 0.311) ( -0.393, 0.295) ( -0.386, 0.281) ( -0.379, 0.266) ( -0.371, 0.252) ( -0.363, 0.239) ( -0.355, 0.226) ( -0.348, 0.213) ( -0.340, 0.201) ( -0.332, 0.189) ( -0.324, 0.178) ( -0.316, 0.167) ( -0.308, 0.157) ( -0.300, 0.147) ( -0.292, 0.137) ( -0.284, 0.128) ( -0.276, 0.119) ( -0.268, 0.111) ( -0.260, 0.103) ( -0.253, 0.095) ( -0.245, 0.088) ( -0.237, 0.081) ( -0.230, 0.075) ( -0.223, 0.068) ( -0.215, 0.063) ( -0.208, 0.057) ( -0.201, 0.052) ( -0.194, 0.047) ( -0.188, 0.042) ( -0.181, 0.037) ( -0.174, 0.033) ( -0.168, 0.029) ( -0.162, 0.026) ( -0.156, 0.022) ( -0.150, 0.019) ( -0.144, 0.016) ( -0.138, 0.013) ( -0.133, 0.010) ( -0.127, 0.008) ( -0.122, 0.006) ( -0.117, 0.004) ( -0.112, 0.002) (0,0)[(1,2)[1]{}]{} (1.1,2.1)[(0,0)\[lb\][$\alpha_0$]{}]{} (2,-0.1)[(0,0)\[t\]]{} (-2,-0.1)[(0,0)\[t\]]{} (1,-0.1)[(0,0)\[t\]]{} (-1,-0.1)[(0,0)\[t\]]{}
(30,-2)
(6,4)(0,0) (0,0)[(1,0)[6]{}]{} (0,0)[(0,1)[4]{}]{} ( 0.000, 3.000) ( 0.015, 2.985) ( 0.030, 2.970) ( 0.045, 2.955) ( 0.060, 2.941) ( 0.075, 2.926) ( 0.090, 2.911) ( 0.105, 2.897) ( 0.120, 2.882) ( 0.135, 2.868) ( 0.150, 2.854) ( 0.165, 2.839) ( 0.180, 2.825) ( 0.195, 2.811) ( 0.210, 2.797) ( 0.225, 2.783) ( 0.240, 2.769) ( 0.255, 2.756) ( 0.270, 2.742) ( 0.285, 2.728) ( 0.300, 2.715) ( 0.315, 2.701) ( 0.330, 2.688) ( 0.345, 2.674) ( 0.360, 2.661) ( 0.375, 2.647) ( 0.390, 2.634) ( 0.405, 2.621) ( 0.420, 2.608) ( 0.435, 2.595) ( 0.450, 2.582) ( 0.465, 2.569) ( 0.480, 2.556) ( 0.495, 2.544) ( 0.510, 2.531) ( 0.525, 2.518) ( 0.540, 2.506) ( 0.555, 2.493) ( 0.570, 2.481) ( 0.585, 2.469) ( 0.600, 2.456) ( 0.615, 2.444) ( 0.630, 2.432) ( 0.645, 2.420) ( 0.660, 2.408) ( 0.675, 2.396) ( 0.690, 2.384) ( 0.705, 2.372) ( 0.720, 2.360) ( 0.735, 2.348) ( 0.750, 2.336) ( 0.765, 2.325) ( 0.780, 2.313) ( 0.795, 2.302) ( 0.810, 2.290) ( 0.825, 2.279) ( 0.840, 2.267) ( 0.855, 2.256) ( 0.870, 2.245) ( 0.885, 2.234) ( 0.900, 2.222) ( 0.915, 2.211) ( 0.930, 2.200) ( 0.945, 2.189) ( 0.960, 2.178) ( 0.975, 2.168) ( 0.990, 2.157) ( 1.005, 2.146) ( 1.020, 2.135) ( 1.035, 2.125) ( 1.050, 2.114) ( 1.065, 2.104) ( 1.080, 2.093) ( 1.095, 2.083) ( 1.110, 2.072) ( 1.125, 2.062) ( 1.140, 2.052) ( 1.155, 2.041) ( 1.170, 2.031) ( 1.185, 2.021) ( 1.200, 2.011) ( 1.215, 2.001) ( 1.230, 1.991) ( 1.245, 1.981) ( 1.260, 1.971) ( 1.275, 1.961) ( 1.290, 1.952) ( 1.305, 1.942) ( 1.320, 1.932) ( 1.335, 1.922) ( 1.350, 1.913) ( 1.365, 1.903) ( 1.380, 1.894) ( 1.395, 1.884) ( 1.410, 1.875) ( 1.425, 1.866) ( 1.440, 1.856) ( 1.455, 1.847) ( 1.470, 1.838) ( 1.485, 1.829) ( 1.500, 1.820) ( 1.515, 1.811) ( 1.530, 1.801) ( 1.545, 1.793) ( 1.560, 1.784) ( 1.575, 1.775) ( 1.590, 1.766) ( 1.605, 1.757) ( 1.620, 1.748) ( 1.635, 1.740) ( 1.650, 1.731) ( 1.665, 1.722) ( 1.680, 1.714) ( 1.695, 1.705) ( 1.710, 1.697) ( 1.725, 1.688) ( 1.740, 1.680) ( 1.755, 1.671) ( 1.770, 1.663) ( 1.785, 1.655) ( 1.800, 1.646) ( 1.815, 1.638) ( 1.830, 1.630) ( 1.845, 1.622) ( 1.860, 1.614) ( 1.875, 1.606) ( 1.890, 1.598) ( 1.905, 1.590) ( 1.920, 1.582) ( 1.935, 1.574) ( 1.950, 1.566) ( 1.965, 1.558) ( 1.980, 1.551) ( 1.995, 1.543) ( 2.010, 1.535) ( 2.025, 1.527) ( 2.040, 1.520) ( 2.055, 1.512) ( 2.070, 1.505) ( 2.085, 1.497) ( 2.100, 1.490) ( 2.115, 1.482) ( 2.130, 1.475) ( 2.145, 1.468) ( 2.160, 1.460) ( 2.175, 1.453) ( 2.190, 1.446) ( 2.205, 1.439) ( 2.220, 1.431) ( 2.235, 1.424) ( 2.250, 1.417) ( 2.265, 1.410) ( 2.280, 1.403) ( 2.295, 1.396) ( 2.310, 1.389) ( 2.325, 1.382) ( 2.340, 1.375) ( 2.355, 1.368) ( 2.370, 1.362) ( 2.385, 1.355) ( 2.400, 1.348) ( 2.415, 1.341) ( 2.430, 1.335) ( 2.445, 1.328) ( 2.460, 1.321) ( 2.475, 1.315) ( 2.490, 1.308) ( 2.505, 1.302) ( 2.520, 1.295) ( 2.535, 1.289) ( 2.550, 1.282) ( 2.565, 1.276) ( 2.580, 1.269) ( 2.595, 1.263) ( 2.610, 1.257) ( 2.625, 1.251) ( 2.640, 1.244) ( 2.655, 1.238) ( 2.670, 1.232) ( 2.685, 1.226) ( 2.700, 1.220) ( 2.715, 1.214) ( 2.730, 1.208) ( 2.745, 1.202) ( 2.760, 1.196) ( 2.775, 1.190) ( 2.790, 1.184) ( 2.805, 1.178) ( 2.820, 1.172) ( 2.835, 1.166) ( 2.850, 1.160) ( 2.865, 1.154) ( 2.880, 1.149) ( 2.895, 1.143) ( 2.910, 1.137) ( 2.925, 1.132) ( 2.940, 1.126) ( 2.955, 1.120) ( 2.970, 1.115) ( 2.985, 1.109) ( 3.000, 1.104) ( 3.015, 1.098) ( 3.030, 1.093) ( 3.045, 1.087) ( 3.060, 1.082) ( 3.075, 1.076) ( 3.090, 1.071) ( 3.105, 1.066) ( 3.120, 1.060) ( 3.135, 1.055) ( 3.150, 1.050) ( 3.165, 1.045) ( 3.180, 1.039) ( 3.195, 1.034) ( 3.210, 1.029) ( 3.225, 1.024) ( 3.240, 1.019) ( 3.255, 1.014) ( 3.270, 1.009) ( 3.285, 1.004) ( 3.300, 0.999) ( 3.315, 0.994) ( 3.330, 0.989) ( 3.345, 0.984) ( 3.360, 0.979) ( 3.375, 0.974) ( 3.390, 0.969) ( 3.405, 0.964) ( 3.420, 0.959) ( 3.435, 0.955) ( 3.450, 0.950) ( 3.465, 0.945) ( 3.480, 0.940) ( 3.495, 0.936) ( 3.510, 0.931) ( 3.525, 0.926) ( 3.540, 0.922) ( 3.555, 0.917) ( 3.570, 0.913) ( 3.585, 0.908) ( 3.600, 0.904) ( 3.615, 0.899) ( 3.630, 0.895) ( 3.645, 0.890) ( 3.660, 0.886) ( 3.675, 0.881) ( 3.690, 0.877) ( 3.705, 0.873) ( 3.720, 0.868) ( 3.735, 0.864) ( 3.750, 0.860) ( 3.765, 0.855) ( 3.780, 0.851) ( 3.795, 0.847) ( 3.810, 0.842) ( 3.825, 0.838) ( 3.840, 0.834) ( 3.855, 0.830) ( 3.870, 0.826) ( 3.885, 0.822) ( 3.900, 0.818) ( 3.915, 0.814) ( 3.930, 0.809) ( 3.945, 0.805) ( 3.960, 0.801) ( 3.975, 0.797) ( 3.990, 0.793) ( 4.005, 0.789) ( 4.020, 0.786) ( 4.035, 0.782) ( 4.050, 0.778) ( 4.065, 0.774) ( 4.080, 0.770) ( 4.095, 0.766) ( 4.110, 0.762) ( 4.125, 0.759) ( 4.140, 0.755) ( 4.155, 0.751) ( 4.170, 0.747) ( 4.185, 0.743) ( 4.200, 0.740) ( 4.215, 0.736) ( 4.230, 0.732) ( 4.245, 0.729) ( 4.260, 0.725) ( 4.275, 0.722) ( 4.290, 0.718) ( 4.305, 0.714) ( 4.320, 0.711) ( 4.335, 0.707) ( 4.350, 0.704) ( 4.365, 0.700) ( 4.380, 0.697) ( 4.395, 0.693) ( 4.410, 0.690) ( 4.425, 0.686) ( 4.440, 0.683) ( 4.455, 0.680) ( 4.470, 0.676) ( 4.485, 0.673) ( 4.500, 0.669) ( 4.515, 0.666) ( 4.530, 0.663) ( 4.545, 0.659) ( 4.560, 0.656) ( 4.575, 0.653) ( 4.590, 0.650) ( 4.605, 0.646) ( 4.620, 0.643) ( 4.635, 0.640) ( 4.650, 0.637) ( 4.665, 0.634) ( 4.680, 0.630) ( 4.695, 0.627) ( 4.710, 0.624) ( 4.725, 0.621) ( 4.740, 0.618) ( 4.755, 0.615) ( 4.770, 0.612) ( 4.785, 0.609) ( 4.800, 0.606) ( 4.815, 0.603) ( 4.830, 0.600) ( 4.845, 0.597) ( 4.860, 0.594) ( 4.875, 0.591) ( 4.890, 0.588) ( 4.905, 0.585) ( 4.920, 0.582) ( 4.935, 0.579) ( 4.950, 0.576) ( 4.965, 0.573) ( 4.980, 0.570) ( 4.995, 0.568) ( 5.010, 0.565) ( 5.025, 0.562) ( 5.040, 0.559) ( 5.055, 0.556) ( 5.070, 0.554) ( 5.085, 0.551) ( 5.100, 0.548) ( 5.115, 0.545) ( 5.130, 0.543) ( 5.145, 0.540) ( 5.160, 0.537) ( 5.175, 0.535) ( 5.190, 0.532) ( 5.205, 0.529) ( 5.220, 0.527) ( 5.235, 0.524) ( 5.250, 0.521) ( 5.265, 0.519) ( 5.280, 0.516) ( 5.295, 0.514) ( 5.310, 0.511) ( 5.325, 0.508) ( 5.340, 0.506) ( 5.355, 0.503) ( 5.370, 0.501) ( 5.385, 0.498) ( 5.400, 0.496) ( 5.415, 0.493) ( 5.430, 0.491) ( 5.445, 0.489) ( 5.460, 0.486) ( 5.475, 0.484) ( 5.490, 0.481) ( 5.505, 0.479) ( 5.520, 0.476) ( 5.535, 0.474) ( 5.550, 0.472) ( 5.565, 0.469) ( 5.580, 0.467) ( 5.595, 0.465) ( 5.610, 0.462) ( 5.625, 0.460) ( 5.640, 0.458) ( 5.655, 0.455) ( 5.670, 0.453) ( 5.685, 0.451) ( 5.700, 0.449) ( 5.715, 0.446) ( 5.730, 0.444) ( 5.745, 0.442) ( 5.760, 0.440) ( 5.775, 0.438) ( 5.790, 0.435) ( 5.805, 0.433) ( 5.820, 0.431) ( 5.835, 0.429) ( 5.850, 0.427) ( 5.865, 0.425) ( 5.880, 0.423) ( 5.895, 0.420) ( 5.910, 0.418) ( 5.925, 0.416) ( 5.940, 0.414) ( 5.955, 0.412) ( 5.970, 0.410) ( 5.985, 0.408) ( -0.200, 3.000)[(0,0)\[r\][$l_0$]{}]{} ( 3.000, 0.000)(0,0.25)[8]{}[(0,1)[0.125]{}]{} ( 3.000, -0.200)[(0,0)\[t\][$1/l_0$]{}]{}
The parameters of the jump are reset to $(\alpha_f,l_f)$ as soon as the individual finds food. That is, there is a lingering memory that food was around there even if currently no food is found—a memory that fades away in a time $\tau$—but the absence of food is forgotten as soon as new food is found. Any moral or philosophical conclusion, be it positive or negative, that can be drawn from this hypothesis is beyond the scope of these notes.
Gene definition
---------------
Each individual is therefore characterized by five parameters: $(\alpha_f,l_f,\alpha_n,l_n,\tau)$. We represent each one as a 8-bit value (these values are scaled in order to compute the actual values of the parameters) and collect them in a 40-bit [“[gene.]{}”]{} We try to keep related parameters in nearby positions of the gene (this is believed to speed up the convergence of the algorithm). Calling $A_F$, $L_F$, $A_N$, $L_N$, and $T$ the 8-bit representations of the parameters we have the genetic representation of an individual in Figure \[gene\].
(25,3)(0,2) (0,4)(0,1)[2]{}[(1,0)[25]{}]{} (0,4)(5,0)[6]{}[(0,1)[1]{}]{} (0,3)(5,0)[6]{}[(0,1)[0.75]{}]{} (0,2.8)[(0,0)\[t\][0]{}]{} (5,2.8)[(0,0)\[t\][7]{}]{} (10,2.8)[(0,0)\[t\][15]{}]{} (15,2.8)[(0,0)\[t\][23]{}]{} (20,2.8)[(0,0)\[t\][31]{}]{} (25,2.8)[(0,0)\[t\][39]{}]{} (2.5,4.5)[(0,0)[$A_F$]{}]{} (7.5,4.5)[(0,0)[$L_F$]{}]{} (12.5,4.5)[(0,0)[$T$]{}]{} (17.5,4.5)[(0,0)[$A_N$]{}]{} (22.5,4.5)[(0,0)[$L_N$]{}]{} (0,2)[(1,0)[1]{}]{} (1.1,2)[(0,0)\[l\][bits]{}]{}
The jump parameters are derived from these 8-bit integers as $$\label{normal}
\begin{array}{ccc}
\displaystyle \alpha_f=2\pi\frac{A_F}{256} & \displaystyle l_f = \frac{L_F}{4} & \displaystyle \tau = \frac{T}{10} \\
& \\
\displaystyle \alpha_n=2\pi\frac{A_N}{256} & \displaystyle l_n = \frac{L_N}{4}
\end{array}$$ The scaling factors (except those for $\alpha_f$ and $\alpha_n$, which are derived from geometric considerations) have been determined by trial and error.
(20,12)(0,0) (4,1)[(0,1)[11]{}]{} (11,1)[(0,1)[11]{}]{} (4,0.8)[(0,0)\[t\][a]{}]{} (11,0.8)[(0,0)\[t\][b]{}]{} (0,2)[ (0,0)(0,1)[2]{}[(1,0)[14]{}]{} (0,0)(14,0)[2]{}[(0,1)[1]{}]{} (2,0.5)[(0,0)[$B_1$]{}]{} (7.5,0.5)[(0,0)[$A_2$]{}]{} (12.5,0.5)[(0,0)[$B_3$]{}]{} (16,0.5)[(0,0)\[l\][offspring 2]{}]{} ]{} (0,4)[ (0,0)(0,1)[2]{}[(1,0)[14]{}]{} (0,0)(14,0)[2]{}[(0,1)[1]{}]{} (2,0.5)[(0,0)[$A_1$]{}]{} (7.5,0.5)[(0,0)[$B_2$]{}]{} (12.5,0.5)[(0,0)[$A_3$]{}]{} (16,0.5)[(0,0)\[l\][offspring 1]{}]{} ]{} (0,7)[ (0,0)(0,1)[2]{}[(1,0)[14]{}]{} (0,0)(14,0)[2]{}[(0,1)[1]{}]{} (2,0.5)[(0,0)[$B_1$]{}]{} (7.5,0.5)[(0,0)[$B_2$]{}]{} (12.5,0.5)[(0,0)[$B_3$]{}]{} (16,0.5)[(0,0)\[l\][parent B]{}]{} ]{} (0,9)[ (0,0)(0,1)[2]{}[(1,0)[14]{}]{} (0,0)(14,0)[2]{}[(0,1)[1]{}]{} (2,0.5)[(0,0)[$A_1$]{}]{} (7.5,0.5)[(0,0)[$A_2$]{}]{} (12.5,0.5)[(0,0)[$A_3$]{}]{} (16,0.5)[(0,0)\[l\][parent A]{}]{} ]{}
The algorithm
-------------
The genetic algorithm is pretty standard. A *generation* is a set of individuals. Each individual is placed in the environment in the same initial position, and does a random walk of predetermined length, according to the parameters encoded in its gene, and collecting pellets of food as specified above. The environment is restored between individuals, so that each one has the same initial supply of food (this entails that there is no competition among the individuals). A point is scored for each pellet that is eaten. As a result, after all individuals have executed a random walk, individual number $k$, characterized by gene $\gamma_k$ has a score $s_k$, with $k=1,\ldots,G$, where $G$ is the number of individuals in a generation.
There are several methods to create the following generation of individuals. Since the performance of the algorithms seems to have little dependence in the specific method used, we use one of the simplest, based on the creation of an intermediate *gene pool*. The gene pool is a set ${\mathcal{P}}$ of $P$ individuals possibly replicated (generally $|P|=G$: the pool has the same size as the generations) such that the number of [“[copies]{}”]{} of an individual in the pool is proportional to its score. An easy algorithm for generating a pool is the *tournament*: we do $P$ comparisons of pairs of individuals taken at random from the generation: the individual with the highest score goes into the pool:
In order to build the next generation, pairs of genes are taken at random from the pool (with uniform distribution) and crossed to create two new individuals that will go into the next generation (this requires that $G$ be even). We use the method of the *double cut* to cross the genes. Two values $a,b\in[0,39]$ are chosen randomly. The two offspring are then generated as in Figure \[offsprings\], in which we assume $a<b$. We also define a small mutation probability: for each new gene, with a (small) probability $p$, we pick a random bit and flip it. Note that this method doesn’t guarantee that the best individual of a generation will pass unchanged to the next, so we actually use the crossing to create $G-2$ individuals to which we add the two best performers of the previous generation.
Results
-------
Figure \[paths\] shows typical paths from the best individual for various values of the density of food, while Table \[params\] shows the value of the parameters for the same individual.
[cc]{}
(800,800)(-160,-480) (-160,-480)(800,0)[2]{}[(0,1)[800]{}]{} (-160,-480)(0,800)[2]{}[(1,0)[800]{}]{} (0,160) (160,-480) (0,0) (-160,0) (320,-160) (480,0) (160,-320) (0,-320) (-160,-160) (-160,-480) (160,-160) (320,0) (480,-160) (320,-320) (-160,-320) (0,-160) (160,0) (0,-480) ( 7.78, 7.78) ( 8.49, 7.78) ( 8.49, 7.78) ( 9.19, 7.78) ( 9.90, 8.49) ( 9.90, 8.49) ( 9.90, 9.19) ( 9.90, 9.90) ( 9.90,10.61) ( 9.90,10.61) ( 9.90,10.61) (10.61, 9.90) (11.31, 9.90) (12.02, 9.19) (12.73, 9.19) (13.44, 8.49) (13.44, 8.49) (14.14, 8.49) (14.85, 7.78) (14.85, 7.78) (14.85, 7.78) (14.85, 7.78) (14.85, 8.49) (14.85, 9.19) (14.85, 9.90) (12.73, 6.36) (13.44, 7.07) (13.44, 7.78) (14.14, 8.49) (14.14, 9.19) (14.85, 9.90) (12.73, 6.36) (12.73, 6.36) ( 7.78, 3.54) ( 8.49, 4.24) ( 9.19, 4.24) ( 9.90, 4.95) (10.61, 4.95) (11.31, 5.66) (12.02, 5.66) (12.73, 6.36) ( 7.78, 2.83) ( 7.78, 3.54) ( 8.49, 0.00) ( 8.49, 0.71) ( 8.49, 1.41) ( 7.78, 2.12) ( 7.78, 2.83) ( 7.78, 0.00) ( 8.49, 0.00) ( 7.78, 0.00) ( 6.36, 0.00) ( 7.07, 0.00) ( 7.78, 0.00) ( 6.36, 0.00) ( 7.07, 0.00) ( 7.78, 0.00) ( 8.49, 0.00) ( 8.49, 0.00) ( 9.19, 0.00) ( 9.90, 0.00) ( 9.90, 0.00) ( 8.49, 0.00) ( 9.19, 0.00) ( 9.90, 0.00) ( 6.36, 0.71) ( 7.07, 0.71) ( 7.78, 0.00) ( 8.49, 0.00) ( 6.36, 0.71) ( 7.07, 0.71) ( 7.78, 1.41) ( 5.66, 3.54) ( 6.36, 2.83) ( 7.07, 2.12) ( 7.78, 1.41) ( 5.66, 3.54) ( 6.36, 3.54) ( 6.36, 3.54) ( 3.54, 3.54) ( 4.24, 3.54) ( 4.95, 3.54) ( 5.66, 3.54) ( 6.36, 3.54) ( 1.41, 2.83) ( 2.12, 2.83) ( 2.83, 3.54) ( 3.54, 3.54) ( 0.71, 0.71) ( 0.71, 1.41) ( 1.41, 2.12) ( 1.41, 2.83) ( 0.71, 0.71) ( 0.71, 0.71) ( 0.71, 1.41) ( 0.71, 2.12) ( 0.71, 2.83) ( 0.00, 1.41) ( 0.00, 2.12) ( 0.71, 2.83) (-0.71, 0.00) (-0.71, 0.71) ( 0.00, 1.41) (-0.71, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 0.71,-0.71) ( 0.00,-1.41) ( 0.71,-0.71) ( 0.00,-1.41) ( 0.00,-0.71) ( 0.00, 0.00) ( 0.00, 0.71) ( 0.00, 1.41) ( 0.00, 1.41) ( 0.71, 0.71) ( 1.41, 0.71) ( 2.12, 0.00) ( 2.83, 0.00) ( 3.54,-0.71) ( 3.54,-0.71) ( 4.24,-1.41) ( 4.95,-1.41) ( 5.66,-2.12) ( 6.36,-2.83) ( 6.36,-2.83) ( 7.07,-2.83) ( 7.78,-2.83) ( 8.49,-2.83) ( 9.19,-2.83) ( 9.90,-2.83) (10.61,-3.54) (11.31,-3.54) (12.02,-3.54) (12.73,-3.54) (13.44,-3.54) (13.44,-3.54) (14.14,-2.83) (14.85,-2.83) (15.56,-2.12) (16.26,-1.41) (16.97,-1.41) (17.68,-0.71) (18.38, 0.00) (19.09, 0.71) (19.80, 0.71) (20.51, 1.41) (11.31,-3.54) (12.02,-2.83) (12.73,-2.83) (13.44,-2.12) (14.14,-2.12) (14.85,-1.41) (15.56,-1.41) (16.26,-0.71) (16.97,-0.71) (17.68, 0.00) (18.38, 0.00) (19.09, 0.71) (19.80, 0.71) (20.51, 1.41) (11.31,-3.54) (12.02,-3.54) (12.73,-3.54) (13.44,-3.54) (14.14,-3.54) (14.85,-3.54) (15.56,-3.54) (16.26,-3.54) (16.97,-3.54) (17.68,-3.54) (18.38,-3.54) (19.09,-3.54) (19.80,-3.54) (20.51,-3.54) (21.21,-3.54) (21.92,-3.54) (22.63,-3.54) (23.33,-3.54) (23.33,-3.54) (22.63,-2.83) (22.63,-2.12) (21.92,-1.41) (21.21,-0.71) (20.51, 0.00) (20.51, 0.71) (19.80, 1.41) (19.09, 2.12) (18.38, 2.83) (18.38, 3.54) (17.68, 4.24) (16.97, 4.95) (16.97, 4.95) (16.26, 5.66) (15.56, 6.36) (14.85, 7.07) (14.85, 7.78) (14.14, 8.49) (13.44, 9.19) (12.73, 9.90) (12.02,10.61) (11.31,11.31) (10.61,12.02) (10.61,12.73) ( 9.90,13.44) ( 9.19,14.14) ( 8.49,14.85) ( 8.49,13.44) ( 8.49,14.14) ( 8.49,14.85) ( 8.49,12.73) ( 8.49,13.44) ( 8.49,12.73) ( 8.49,12.73) ( 9.19,12.73) ( 9.90,12.73) (10.61,13.44) (11.31,13.44) (11.31,13.44) ( 8.49,11.31) ( 9.19,12.02) ( 9.90,12.02) (10.61,12.73) (11.31,13.44) ( 7.78, 8.49) ( 7.78, 9.19) ( 7.78, 9.90) ( 8.49,10.61) ( 8.49,11.31) ( 7.78, 8.49) ( 7.78, 9.19) ( 7.78, 9.90) ( 5.66, 7.78) ( 6.36, 8.49) ( 7.07, 9.19) ( 7.78, 9.90) ( 6.36, 6.36) ( 6.36, 7.07) ( 5.66, 7.78) ( 6.36, 5.66) ( 6.36, 6.36) ( 5.66, 4.95) ( 6.36, 5.66) ( 5.66, 4.95) ( 6.36, 4.95) ( 7.07, 4.95) ( 7.78, 4.95) ( 7.78, 4.95) ( 8.49, 4.95) ( 8.49, 4.95) ( 9.19, 4.95) ( 9.90, 5.66) ( 9.90, 5.66) ( 9.19, 6.36) ( 9.19, 7.07) ( 8.49, 7.78) ( 7.78, 8.49) ( 7.78, 8.49) ( 8.49, 8.49) ( 9.19, 8.49) ( 9.90, 7.78) (10.61, 7.78) (10.61, 7.78) (11.31, 8.49) (11.31, 8.49) (11.31, 9.19) (10.61, 9.90) (10.61,10.61) ( 9.90,11.31) ( 9.90,10.61) ( 9.90,11.31) ( 9.90,10.61) ( 5.66,13.44) ( 6.36,12.73) ( 7.07,12.73) ( 7.78,12.02) ( 8.49,11.31) ( 9.19,11.31) ( 9.90,10.61) ( 5.66,13.44) ( 6.36,14.14) ( 7.07,14.14) ( 7.78,14.85) ( 6.36,15.56) ( 7.07,15.56) ( 7.78,14.85) ( 6.36,15.56) ( 7.07,15.56) ( 7.78,15.56) ( 8.49,15.56) ( 9.19,15.56) ( 9.90,15.56) ( 9.90,15.56) (10.61,16.26) (11.31,16.97) (12.02,16.97) (12.73,17.68) (13.44,18.38) ( 9.90,19.80) (10.61,19.80) (11.31,19.09) (12.02,19.09) (12.73,18.38) (13.44,18.38) ( 2.83,21.92) ( 3.54,21.92) ( 4.24,21.21) ( 4.95,21.21) ( 5.66,21.21) ( 6.36,21.21) ( 7.07,20.51) ( 7.78,20.51) ( 8.49,20.51) ( 9.19,19.80) ( 9.90,19.80) (-1.41,25.46) (-0.71,24.75) ( 0.00,24.04) ( 0.71,24.04) ( 1.41,23.33) ( 2.12,22.63) ( 2.83,21.92) (-1.41,25.46) (-1.41,26.16) (-1.41,26.87) (-2.12,27.58) (-2.12,28.28) (-2.12,28.99) (-2.12,29.70) (-2.12,30.41) (-2.83,31.11) (-2.83,31.82) (-2.83,32.53) (-2.83,32.53) (-2.12,31.82) (-1.41,31.82) (-0.71,31.11) ( 0.00,31.11) ( 0.71,30.41) ( 1.41,30.41) ( 2.12,29.70) ( 2.83,29.70) ( 3.54,28.99) ( 3.54,28.99) ( 4.24,29.70) ( 4.95,29.70) ( 5.66,30.41) ( 6.36,30.41) ( 7.07,31.11) ( 7.78,31.82) ( 8.49,31.82) ( 9.19,32.53) ( 9.90,33.23) (10.61,33.23) (11.31,33.94) (12.02,33.94) (12.73,34.65) ( 0.71,34.65) ( 1.41,34.65) ( 2.12,34.65) ( 2.83,34.65) ( 3.54,34.65) ( 4.24,34.65) ( 4.95,34.65) ( 5.66,34.65) ( 6.36,34.65) ( 7.07,34.65) ( 7.78,34.65) ( 8.49,34.65) ( 9.19,34.65) ( 9.90,34.65) (10.61,34.65) (11.31,34.65) (12.02,34.65) (12.73,34.65) ( 8.49,25.46) ( 7.78,26.16) ( 7.07,26.87) ( 6.36,27.58) ( 6.36,28.28) ( 5.66,28.99) ( 4.95,29.70) ( 4.24,30.41) ( 3.54,31.11) ( 2.83,31.82) ( 2.83,32.53) ( 2.12,33.23) ( 1.41,33.94) ( 0.71,34.65) ( 8.49,25.46) ( 9.19,25.46) ( 9.90,25.46) (10.61,25.46) (11.31,25.46) (12.02,25.46) (12.73,25.46) (13.44,25.46) (14.14,25.46) (14.85,24.75) (15.56,24.75) (16.26,24.75) (16.97,24.75) (17.68,24.75) (18.38,24.75) (19.09,24.75) (19.80,24.75) (20.51,24.75) (20.51,24.75) (19.80,25.46) (19.09,26.16) (18.38,26.87) (18.38,27.58) (17.68,28.28) (16.97,28.99) (16.26,29.70) (15.56,30.41) (14.85,31.11) (14.14,31.82) (13.44,32.53) (13.44,33.23) (12.73,33.94) (12.02,34.65) (11.31,35.36) (-1.41,29.70) (-0.71,29.70) ( 0.00,30.41) ( 0.71,30.41) ( 1.41,31.11) ( 2.12,31.11) ( 2.83,31.82) ( 3.54,31.82) ( 4.24,32.53) ( 4.95,32.53) ( 5.66,32.53) ( 6.36,33.23) ( 7.07,33.23) ( 7.78,33.94) ( 8.49,33.94) ( 9.19,34.65) ( 9.90,34.65) (10.61,35.36) (11.31,35.36) (-1.41,29.70) (-0.71,30.41) ( 0.00,31.11) ( 0.71,31.11) ( 1.41,31.82) ( 2.12,32.53) ( 2.83,33.23) ( 3.54,33.94) ( 4.24,34.65) ( 4.95,34.65) ( 5.66,35.36) ( 6.36,36.06) ( 7.07,36.77) ( 7.78,37.48) ( 8.49,38.18) ( 9.19,38.18) ( 9.90,38.89) (10.61,39.60) ( 3.54,24.75) ( 3.54,25.46) ( 4.24,26.16) ( 4.24,26.87) ( 4.95,27.58) ( 4.95,28.28) ( 5.66,28.99) ( 5.66,29.70) ( 6.36,30.41) ( 6.36,31.11) ( 7.07,31.82) ( 7.07,32.53) ( 7.78,33.23) ( 7.78,33.94) ( 8.49,34.65) ( 8.49,35.36) ( 9.19,36.06) ( 9.19,36.77) ( 9.90,37.48) ( 9.90,38.18) (10.61,38.89) (10.61,39.60) (-15.56,21.92) (-14.85,21.92) (-14.14,21.92) (-13.44,21.92) (-12.73,22.63) (-12.02,22.63) (-11.31,22.63) (-10.61,22.63) (-9.90,22.63) (-9.19,22.63) (-8.49,22.63) (-7.78,23.33) (-7.07,23.33) (-6.36,23.33) (-5.66,23.33) (-4.95,23.33) (-4.24,23.33) (-3.54,24.04) (-2.83,24.04) (-2.12,24.04) (-1.41,24.04) (-0.71,24.04) ( 0.00,24.04) ( 0.71,24.04) ( 1.41,24.75) ( 2.12,24.75) ( 2.83,24.75) ( 3.54,24.75) (-15.56,21.92) (-14.85,21.92) (-14.14,21.92) (-13.44,21.21) (-12.73,21.21) (-12.02,21.21) (-11.31,21.21) (-10.61,20.51) (-9.90,20.51) (-9.19,20.51) (-8.49,20.51) (-7.78,20.51) (-7.07,19.80) (-6.36,19.80) (-5.66,19.80) (-4.95,19.80) (-4.24,19.09) (-3.54,19.09) (-2.83,19.09) (-2.12,19.09) (-1.41,18.38) (-0.71,18.38) ( 0.00,18.38) (-18.38,16.97) (-17.68,16.97) (-16.97,16.97) (-16.26,16.97) (-15.56,16.97) (-14.85,16.97) (-14.14,16.97) (-13.44,17.68) (-12.73,17.68) (-12.02,17.68) (-11.31,17.68) (-10.61,17.68) (-9.90,17.68) (-9.19,17.68) (-8.49,17.68) (-7.78,17.68) (-7.07,17.68) (-6.36,17.68) (-5.66,17.68) (-4.95,17.68) (-4.24,18.38) (-3.54,18.38) (-2.83,18.38) (-2.12,18.38) (-1.41,18.38) (-0.71,18.38) ( 0.00,18.38) (-18.38,16.97) (-17.68,16.97) (-16.97,17.68) (-16.26,17.68) (-15.56,18.38) (-14.85,18.38) (-14.14,18.38) (-13.44,19.09) (-12.73,19.09) (-12.02,19.80) (-11.31,19.80) (-10.61,19.80) (-9.90,20.51) (-9.19,20.51) (-8.49,21.21) (-7.78,21.21) (-7.07,21.92) (-6.36,21.92) (-5.66,21.92) (-4.95,22.63) (-4.24,22.63) (-3.54,23.33) (-2.83,23.33) (-2.12,23.33) (-1.41,24.04) (-0.71,24.04) ( 0.00,24.75) ( 0.71,24.75) (-11.31, 7.78) (-10.61, 8.49) (-10.61, 9.19) (-9.90, 9.90) (-9.19,10.61) (-8.49,11.31) (-8.49,12.02) (-7.78,12.73) (-7.07,13.44) (-7.07,14.14) (-6.36,14.85) (-5.66,15.56) (-5.66,16.26) (-4.95,16.97) (-4.24,17.68) (-3.54,18.38) (-3.54,19.09) (-2.83,19.80) (-2.12,20.51) (-2.12,21.21) (-1.41,21.92) (-0.71,22.63) ( 0.00,23.33) ( 0.00,24.04) ( 0.71,24.75) (-33.94, 7.78) (-33.23, 7.78) (-32.53, 7.78) (-31.82, 7.78) (-31.11, 7.78) (-30.41, 7.78) (-29.70, 7.78) (-28.99, 7.78) (-28.28, 7.78) (-27.58, 7.78) (-26.87, 7.78) (-26.16, 7.78) (-25.46, 7.78) (-24.75, 7.78) (-24.04, 7.78) (-23.33, 7.78) (-22.63, 7.78) (-21.92, 7.78) (-21.21, 7.78) (-20.51, 7.78) (-19.80, 7.78) (-19.09, 7.78) (-18.38, 7.78) (-17.68, 7.78) (-16.97, 7.78) (-16.26, 7.78) (-15.56, 7.78) (-14.85, 7.78) (-14.14, 7.78) (-13.44, 7.78) (-12.73, 7.78) (-12.02, 7.78) (-11.31, 7.78) (-33.94, 7.78) (-33.23, 8.49) (-32.53, 9.19) (-31.82, 9.90) (-31.11,10.61) (-30.41,11.31) (-29.70,12.02) (-28.99,12.02) (-28.28,12.73) (-27.58,13.44) (-26.87,14.14) (-26.16,14.85) (-25.46,15.56) (-24.75,16.26) (-24.04,16.97) (-23.33,17.68) (-22.63,18.38) (-21.92,19.09) (-21.21,19.80) (-20.51,20.51) (-19.80,21.21) (-19.09,21.21) (-18.38,21.92) (-17.68,22.63) (-16.97,23.33) (-16.26,24.04) (-15.56,24.75) (-14.85,25.46) (-14.85,25.46) (-14.14,26.16) (-13.44,26.16) (-12.73,26.87) (-12.02,26.87) (-11.31,27.58) (-10.61,27.58) (-9.90,28.28) (-9.19,28.99) (-8.49,28.99) (-7.78,29.70) (-7.07,29.70) (-6.36,30.41) (-5.66,30.41) (-4.95,31.11) (-4.24,31.82) (-3.54,31.82) (-2.83,32.53) (-2.12,32.53) (-1.41,33.23) (-0.71,33.94) ( 0.00,33.94) ( 0.71,34.65) ( 1.41,34.65) ( 2.12,35.36) ( 2.83,35.36) ( 3.54,36.06) ( 4.24,36.77) ( 4.95,36.77) ( 5.66,37.48) ( 6.36,37.48) ( 7.07,38.18) ( 7.78,38.18) ( 8.49,38.89) (21.92,14.85) (21.21,15.56) (21.21,16.26) (20.51,16.97) (20.51,17.68) (19.80,18.38) (19.80,19.09) (19.09,19.80) (19.09,20.51) (18.38,21.21) (17.68,21.92) (17.68,22.63) (16.97,23.33) (16.97,24.04) (16.26,24.75) (16.26,25.46) (15.56,26.16) (15.56,26.87) (14.85,27.58) (14.14,28.28) (14.14,28.99) (13.44,29.70) (13.44,30.41) (12.73,31.11) (12.73,31.82) (12.02,32.53) (11.31,33.23) (11.31,33.94) (10.61,34.65) (10.61,35.36) ( 9.90,36.06) ( 9.90,36.77) ( 9.19,37.48) ( 9.19,38.18) ( 8.49,38.89) (21.92,14.85) (21.92,15.56) (21.21,16.26) (21.21,16.97) (21.21,17.68) (20.51,18.38) (20.51,19.09) (20.51,19.80) (19.80,20.51) (19.80,21.21) (19.80,21.92) (19.80,22.63) (19.09,23.33) (19.09,24.04) (19.09,24.75) (18.38,25.46) (18.38,26.16) (18.38,26.87) (17.68,27.58) (17.68,28.28) (17.68,28.99) (16.97,29.70) (16.97,30.41) (16.97,31.11) (16.26,31.82) (16.26,32.53) (16.26,33.23) (15.56,33.94) (15.56,34.65) (15.56,35.36) (15.56,36.06) (14.85,36.77) (14.85,37.48) (14.85,38.18) (14.14,38.89) (14.14,39.60) (14.14,40.31) (13.44,41.01) (13.44,41.72) (-5.66,23.33) (-4.95,24.04) (-4.24,24.75) (-3.54,25.46) (-2.83,26.16) (-2.12,26.87) (-1.41,27.58) (-0.71,28.28) ( 0.00,28.99) ( 0.71,29.70) ( 1.41,30.41) ( 2.12,31.11) ( 2.83,31.82) ( 3.54,32.53) ( 4.24,32.53) ( 4.95,33.23) ( 5.66,33.94) ( 6.36,34.65) ( 7.07,35.36) ( 7.78,36.06) ( 8.49,36.77) ( 9.19,37.48) ( 9.90,38.18) (10.61,38.89) (11.31,39.60) (12.02,40.31) (12.73,41.01) (13.44,41.72) (-26.87, 4.95) (-26.16, 5.66) (-25.46, 6.36) (-24.75, 7.07) (-24.04, 7.07) (-23.33, 7.78) (-22.63, 8.49) (-21.92, 9.19) (-21.21, 9.90) (-20.51,10.61) (-19.80,11.31) (-19.09,12.02) (-18.38,12.02) (-17.68,12.73) (-16.97,13.44) (-16.26,14.14) (-15.56,14.85) (-14.85,15.56) (-14.14,16.26) (-13.44,16.26) (-12.73,16.97) (-12.02,17.68) (-11.31,18.38) (-10.61,19.09) (-9.90,19.80) (-9.19,20.51) (-8.49,21.21) (-7.78,21.21) (-7.07,21.92) (-6.36,22.63) (-5.66,23.33) (-48.79,-14.85) (-48.08,-14.14) (-47.38,-13.44) (-46.67,-12.73) (-45.96,-12.02) (-45.25,-11.31) (-44.55,-11.31) (-43.84,-10.61) (-43.13,-9.90) (-42.43,-9.19) (-41.72,-8.49) (-41.01,-7.78) (-40.31,-7.07) (-39.60,-6.36) (-38.89,-5.66) (-38.18,-4.95) (-37.48,-4.95) (-36.77,-4.24) (-36.06,-3.54) (-35.36,-2.83) (-34.65,-2.12) (-33.94,-1.41) (-33.23,-0.71) (-32.53, 0.00) (-31.82, 0.71) (-31.11, 1.41) (-30.41, 1.41) (-29.70, 2.12) (-28.99, 2.83) (-28.28, 3.54) (-27.58, 4.24) (-26.87, 4.95) (-65.76,-41.72) (-65.05,-41.01) (-65.05,-40.31) (-64.35,-39.60) (-63.64,-38.89) (-63.64,-38.18) (-62.93,-37.48) (-62.93,-36.77) (-62.23,-36.06) (-61.52,-35.36) (-61.52,-34.65) (-60.81,-33.94) (-60.10,-33.23) (-60.10,-32.53) (-59.40,-31.82) (-59.40,-31.11) (-58.69,-30.41) (-57.98,-29.70) (-57.98,-28.99) (-57.28,-28.28) (-56.57,-27.58) (-56.57,-26.87) (-55.86,-26.16) (-55.15,-25.46) (-55.15,-24.75) (-54.45,-24.04) (-54.45,-23.33) (-53.74,-22.63) (-53.03,-21.92) (-53.03,-21.21) (-52.33,-20.51) (-51.62,-19.80) (-51.62,-19.09) (-50.91,-18.38) (-50.91,-17.68) (-50.20,-16.97) (-49.50,-16.26) (-49.50,-15.56) (-48.79,-14.85) (-53.74,-73.54) (-53.74,-72.83) (-54.45,-72.12) (-54.45,-71.42) (-55.15,-70.71) (-55.15,-70.00) (-55.15,-69.30) (-55.86,-68.59) (-55.86,-67.88) (-55.86,-67.18) (-56.57,-66.47) (-56.57,-65.76) (-57.28,-65.05) (-57.28,-64.35) (-57.28,-63.64) (-57.98,-62.93) (-57.98,-62.23) (-57.98,-61.52) (-58.69,-60.81) (-58.69,-60.10) (-59.40,-59.40) (-59.40,-58.69) (-59.40,-57.98) (-60.10,-57.28) (-60.10,-56.57) (-60.10,-55.86) (-60.81,-55.15) (-60.81,-54.45) (-61.52,-53.74) (-61.52,-53.03) (-61.52,-52.33) (-62.23,-51.62) (-62.23,-50.91) (-62.23,-50.20) (-62.93,-49.50) (-62.93,-48.79) (-63.64,-48.08) (-63.64,-47.38) (-63.64,-46.67) (-64.35,-45.96) (-64.35,-45.25) (-64.35,-44.55) (-65.05,-43.84) (-65.05,-43.13) (-65.76,-42.43) (-65.76,-41.72) (-53.74,-73.54) (-53.74,-72.83) (-53.74,-72.12) (-53.74,-71.42) (-53.74,-70.71) (-53.74,-70.00) (-53.74,-69.30) (-53.74,-68.59) (-53.74,-67.88) (-53.74,-67.18) (-53.74,-66.47) (-53.74,-65.76) (-53.74,-65.05) (-53.03,-64.35) (-53.03,-63.64) (-53.03,-62.93) (-53.03,-62.23) (-53.03,-61.52) (-53.03,-60.81) (-53.03,-60.10) (-53.03,-59.40) (-53.03,-58.69) (-53.03,-57.98) (-53.03,-57.28) (-53.03,-56.57) (-53.03,-55.86) (-53.03,-55.15) (-53.03,-54.45) (-53.03,-53.74) (-53.03,-53.03) (-53.03,-52.33) (-53.03,-51.62) (-53.03,-50.91) (-53.03,-50.20) (-53.03,-49.50) (-53.03,-48.79) (-53.03,-48.08) (-52.33,-47.38) (-52.33,-46.67) (-52.33,-45.96) (-52.33,-45.25) (-52.33,-44.55) (-52.33,-43.84) (-52.33,-43.13) (-52.33,-42.43) (-52.33,-41.72) (-52.33,-41.01) (-52.33,-40.31) (-52.33,-39.60) (-85.56,-48.79) (-84.85,-48.79) (-84.15,-48.08) (-83.44,-48.08) (-82.73,-48.08) (-82.02,-48.08) (-81.32,-47.38) (-80.61,-47.38) (-79.90,-47.38) (-79.20,-47.38) (-78.49,-46.67) (-77.78,-46.67) (-77.07,-46.67) (-76.37,-45.96) (-75.66,-45.96) (-74.95,-45.96) (-74.25,-45.96) (-73.54,-45.25) (-72.83,-45.25) (-72.12,-45.25) (-71.42,-44.55) (-70.71,-44.55) (-70.00,-44.55) (-69.30,-44.55) (-68.59,-43.84) (-67.88,-43.84) (-67.18,-43.84) (-66.47,-43.84) (-65.76,-43.13) (-65.05,-43.13) (-64.35,-43.13) (-63.64,-42.43) (-62.93,-42.43) (-62.23,-42.43) (-61.52,-42.43) (-60.81,-41.72) (-60.10,-41.72) (-59.40,-41.72) (-58.69,-41.01) (-57.98,-41.01) (-57.28,-41.01) (-56.57,-41.01) (-55.86,-40.31) (-55.15,-40.31) (-54.45,-40.31) (-53.74,-40.31) (-53.03,-39.60) (-52.33,-39.60) (-113.84,-70.71) (-113.14,-70.00) (-112.43,-69.30) (-111.72,-69.30) (-111.02,-68.59) (-110.31,-67.88) (-109.60,-67.18) (-108.89,-67.18) (-108.19,-66.47) (-107.48,-65.76) (-106.77,-65.05) (-106.07,-64.35) (-105.36,-64.35) (-104.65,-63.64) (-103.94,-62.93) (-103.24,-62.23) (-102.53,-62.23) (-101.82,-61.52) (-101.12,-60.81) (-100.41,-60.10) (-99.70,-60.10) (-98.99,-59.40) (-98.29,-58.69) (-97.58,-57.98) (-96.87,-57.28) (-96.17,-57.28) (-95.46,-56.57) (-94.75,-55.86) (-94.05,-55.15) (-93.34,-55.15) (-92.63,-54.45) (-91.92,-53.74) (-91.22,-53.03) (-90.51,-52.33) (-89.80,-52.33) (-89.10,-51.62) (-88.39,-50.91) (-87.68,-50.20) (-86.97,-50.20) (-86.27,-49.50) (-85.56,-48.79) (-117.38,-106.77) (-117.38,-106.07) (-117.38,-105.36) (-117.38,-104.65) (-117.38,-103.94) (-117.38,-103.24) (-116.67,-102.53) (-116.67,-101.82) (-116.67,-101.12) (-116.67,-100.41) (-116.67,-99.70) (-116.67,-98.99) (-116.67,-98.29) (-116.67,-97.58) (-116.67,-96.87) (-116.67,-96.17) (-115.97,-95.46) (-115.97,-94.75) (-115.97,-94.05) (-115.97,-93.34) (-115.97,-92.63) (-115.97,-91.92) (-115.97,-91.22) (-115.97,-90.51) (-115.97,-89.80) (-115.97,-89.10) (-115.26,-88.39) (-115.26,-87.68) (-115.26,-86.97) (-115.26,-86.27) (-115.26,-85.56) (-115.26,-84.85) (-115.26,-84.15) (-115.26,-83.44) (-115.26,-82.73) (-115.26,-82.02) (-114.55,-81.32) (-114.55,-80.61) (-114.55,-79.90) (-114.55,-79.20) (-114.55,-78.49) (-114.55,-77.78) (-114.55,-77.07) (-114.55,-76.37) (-114.55,-75.66) (-114.55,-74.95) (-113.84,-74.25) (-113.84,-73.54) (-113.84,-72.83) (-113.84,-72.12) (-113.84,-71.42) (-113.84,-70.71) (-117.38,-106.77) (-117.38,-106.07) (-117.38,-105.36) (-117.38,-104.65) (-117.38,-103.94) (-116.67,-103.24) (-116.67,-102.53) (-116.67,-101.82) (-116.67,-101.12) (-116.67,-100.41) (-116.67,-99.70) (-116.67,-98.99) (-116.67,-98.29) (-115.97,-97.58) (-115.97,-96.87) (-115.97,-96.17) (-115.97,-95.46) (-115.97,-94.75) (-115.97,-94.05) (-115.97,-93.34) (-115.97,-92.63) (-115.26,-91.92) (-115.26,-91.22) (-115.26,-90.51) (-115.26,-89.80) (-115.26,-89.10) (-115.26,-88.39) (-115.26,-87.68) (-115.26,-86.97) (-114.55,-86.27) (-114.55,-85.56) (-114.55,-84.85) (-114.55,-84.15) (-114.55,-83.44) (-114.55,-82.73) (-114.55,-82.02) (-114.55,-81.32) (-113.84,-80.61) (-113.84,-79.90) (-113.84,-79.20) (-113.84,-78.49) (-113.84,-77.78) (-113.84,-77.07) (-113.84,-76.37) (-113.84,-75.66) (-113.14,-74.95) (-113.14,-74.25) (-113.14,-73.54) (-113.14,-72.83) (-113.14,-72.12) (-113.14,-71.42) (-113.14,-70.71) (-113.14,-70.00) (-112.43,-69.30) (-112.43,-68.59) (-112.43,-67.88) (-112.43,-67.18) (-112.43,-66.47) (-132.94,-100.41) (-132.23,-99.70) (-132.23,-98.99) (-131.52,-98.29) (-131.52,-97.58) (-130.81,-96.87) (-130.11,-96.17) (-130.11,-95.46) (-129.40,-94.75) (-129.40,-94.05) (-128.69,-93.34) (-127.99,-92.63) (-127.99,-91.92) (-127.28,-91.22) (-127.28,-90.51) (-126.57,-89.80) (-125.87,-89.10) (-125.87,-88.39) (-125.16,-87.68) (-125.16,-86.97) (-124.45,-86.27) (-123.74,-85.56) (-123.74,-84.85) (-123.04,-84.15) (-123.04,-83.44) (-122.33,-82.73) (-121.62,-82.02) (-121.62,-81.32) (-120.92,-80.61) (-120.21,-79.90) (-120.21,-79.20) (-119.50,-78.49) (-119.50,-77.78) (-118.79,-77.07) (-118.09,-76.37) (-118.09,-75.66) (-117.38,-74.95) (-117.38,-74.25) (-116.67,-73.54) (-115.97,-72.83) (-115.97,-72.12) (-115.26,-71.42) (-115.26,-70.71) (-114.55,-70.00) (-113.84,-69.30) (-113.84,-68.59) (-113.14,-67.88) (-113.14,-67.18) (-112.43,-66.47) (-132.94,-100.41) (-132.23,-100.41) (-131.52,-100.41) (-130.81,-100.41) (-130.11,-100.41) (-129.40,-100.41) (-128.69,-100.41) (-127.99,-100.41) (-127.28,-100.41) (-126.57,-100.41) (-125.87,-100.41) (-125.16,-100.41) (-124.45,-100.41) (-123.74,-100.41) (-123.04,-100.41) (-122.33,-99.70) (-121.62,-99.70) (-120.92,-99.70) (-120.21,-99.70) (-119.50,-99.70) (-118.79,-99.70) (-118.09,-99.70) (-117.38,-99.70) (-116.67,-99.70) (-115.97,-99.70) (-115.26,-99.70) (-114.55,-99.70) (-113.84,-99.70) (-113.14,-99.70) (-112.43,-99.70) (-111.72,-99.70) (-111.02,-99.70) (-110.31,-99.70) (-109.60,-99.70) (-108.89,-99.70) (-108.19,-99.70) (-107.48,-99.70) (-106.77,-99.70) (-106.07,-99.70) (-105.36,-99.70) (-104.65,-99.70) (-103.94,-99.70) (-103.24,-99.70) (-102.53,-99.70) (-101.82,-98.99) (-101.12,-98.99) (-100.41,-98.99) (-99.70,-98.99) (-98.99,-98.99) (-98.29,-98.99) (-97.58,-98.99) (-96.87,-98.99) (-96.17,-98.99) (-95.46,-98.99) (-94.75,-98.99) (-94.05,-98.99) (-93.34,-98.99) (-92.63,-98.99) (-91.92,-98.99) (-91.92,-98.99) (-91.22,-99.70) (-90.51,-99.70) (-89.80,-100.41) (-89.10,-101.12) (-88.39,-101.12) (-87.68,-101.82) (-86.97,-102.53) (-86.27,-102.53) (-85.56,-103.24) (-84.85,-103.24) (-84.15,-103.94) (-83.44,-104.65) (-82.73,-104.65) (-82.02,-105.36) (-81.32,-106.07) (-80.61,-106.07) (-79.90,-106.77) (-79.20,-107.48) (-78.49,-107.48) (-77.78,-108.19) (-77.07,-108.89) (-76.37,-108.89) (-75.66,-109.60) (-74.95,-110.31) (-74.25,-110.31) (-73.54,-111.02) (-72.83,-111.02) (-72.12,-111.72) (-71.42,-112.43) (-70.71,-112.43) (-70.00,-113.14) (-69.30,-113.84) (-68.59,-113.84) (-67.88,-114.55) (-67.18,-115.26) (-66.47,-115.26) (-65.76,-115.97) (-65.05,-116.67) (-64.35,-116.67) (-63.64,-117.38) (-62.93,-118.09) (-62.23,-118.09) (-61.52,-118.79) (-60.81,-118.79) (-60.10,-119.50) (-59.40,-120.21) (-58.69,-120.21) (-57.98,-120.92) (-57.28,-121.62) (-56.57,-121.62) (-55.86,-122.33) (-31.82,-157.68) (-32.53,-156.98) (-32.53,-156.27) (-33.23,-155.56) (-33.94,-154.86) (-33.94,-154.15) (-34.65,-153.44) (-35.36,-152.74) (-35.36,-152.03) (-36.06,-151.32) (-36.77,-150.61) (-36.77,-149.91) (-37.48,-149.20) (-38.18,-148.49) (-38.89,-147.79) (-38.89,-147.08) (-39.60,-146.37) (-40.31,-145.66) (-40.31,-144.96) (-41.01,-144.25) (-41.72,-143.54) (-41.72,-142.84) (-42.43,-142.13) (-43.13,-141.42) (-43.13,-140.71) (-43.84,-140.01) (-44.55,-139.30) (-44.55,-138.59) (-45.25,-137.89) (-45.96,-137.18) (-45.96,-136.47) (-46.67,-135.76) (-47.38,-135.06) (-47.38,-134.35) (-48.08,-133.64) (-48.79,-132.94) (-48.79,-132.23) (-49.50,-131.52) (-50.20,-130.81) (-50.91,-130.11) (-50.91,-129.40) (-51.62,-128.69) (-52.33,-127.99) (-52.33,-127.28) (-53.03,-126.57) (-53.74,-125.87) (-53.74,-125.16) (-54.45,-124.45) (-55.15,-123.74) (-55.15,-123.04) (-55.86,-122.33) (-60.81,-193.75) (-60.10,-193.04) (-59.40,-192.33) (-59.40,-191.63) (-58.69,-190.92) (-57.98,-190.21) (-57.28,-189.50) (-56.57,-188.80) (-56.57,-188.09) (-55.86,-187.38) (-55.15,-186.68) (-54.45,-185.97) (-53.74,-185.26) (-53.74,-184.55) (-53.03,-183.85) (-52.33,-183.14) (-51.62,-182.43) (-50.91,-181.73) (-50.91,-181.02) (-50.20,-180.31) (-49.50,-179.61) (-48.79,-178.90) (-48.08,-178.19) (-48.08,-177.48) (-47.38,-176.78) (-46.67,-176.07) (-45.96,-175.36) (-45.25,-174.66) (-44.55,-173.95) (-44.55,-173.24) (-43.84,-172.53) (-43.13,-171.83) (-42.43,-171.12) (-41.72,-170.41) (-41.72,-169.71) (-41.01,-169.00) (-40.31,-168.29) (-39.60,-167.58) (-38.89,-166.88) (-38.89,-166.17) (-38.18,-165.46) (-37.48,-164.76) (-36.77,-164.05) (-36.06,-163.34) (-36.06,-162.63) (-35.36,-161.93) (-34.65,-161.22) (-33.94,-160.51) (-33.23,-159.81) (-33.23,-159.10) (-32.53,-158.39) (-31.82,-157.68) (-60.81,-193.75) (-60.10,-193.75) (-59.40,-194.45) (-58.69,-194.45) (-57.98,-195.16) (-57.28,-195.16) (-56.57,-195.87) (-55.86,-195.87) (-55.15,-196.58) (-54.45,-196.58) (-53.74,-196.58) (-53.03,-197.28) (-52.33,-197.28) (-51.62,-197.99) (-50.91,-197.99) (-50.20,-198.70) (-49.50,-198.70) (-48.79,-199.40) (-48.08,-199.40) (-47.38,-199.40) (-46.67,-200.11) (-45.96,-200.11) (-45.25,-200.82) (-44.55,-200.82) (-43.84,-201.53) (-43.13,-201.53) (-42.43,-202.23) (-41.72,-202.23) (-41.01,-202.23) (-40.31,-202.94) (-39.60,-202.94) (-38.89,-203.65) (-38.18,-203.65) (-37.48,-204.35) (-36.77,-204.35) (-36.06,-204.35) (-35.36,-205.06) (-34.65,-205.06) (-33.94,-205.77) (-33.23,-205.77) (-32.53,-206.48) (-31.82,-206.48) (-31.11,-207.18) (-30.41,-207.18) (-29.70,-207.18) (-28.99,-207.89) (-28.28,-207.89) (-27.58,-208.60) (-26.87,-208.60) (-26.16,-209.30) (-25.46,-209.30) (-24.75,-210.01) (-24.04,-210.01) (-23.33,-210.01) (-22.63,-210.72) (-21.92,-210.72) (-21.21,-211.42) (-20.51,-211.42) (-19.80,-212.13) (-19.09,-212.13) (-18.38,-212.84) (-17.68,-212.84) (-17.68,-212.84) (-16.97,-212.84) (-16.26,-212.84) (-15.56,-212.84) (-14.85,-212.13) (-14.14,-212.13) (-13.44,-212.13) (-12.73,-212.13) (-12.02,-212.13) (-11.31,-212.13) (-10.61,-212.13) (-9.90,-211.42) (-9.19,-211.42) (-8.49,-211.42) (-7.78,-211.42) (-7.07,-211.42) (-6.36,-211.42) (-5.66,-210.72) (-4.95,-210.72) (-4.24,-210.72) (-3.54,-210.72) (-2.83,-210.72) (-2.12,-210.72) (-1.41,-210.72) (-0.71,-210.01) ( 0.00,-210.01) ( 0.71,-210.01) ( 1.41,-210.01) ( 2.12,-210.01) ( 2.83,-210.01) ( 3.54,-210.01) ( 4.24,-209.30) ( 4.95,-209.30) ( 5.66,-209.30) ( 6.36,-209.30) ( 7.07,-209.30) ( 7.78,-209.30) ( 8.49,-208.60) ( 9.19,-208.60) ( 9.90,-208.60) (10.61,-208.60) (11.31,-208.60) (12.02,-208.60) (12.73,-208.60) (13.44,-207.89) (14.14,-207.89) (14.85,-207.89) (15.56,-207.89) (16.26,-207.89) (16.97,-207.89) (17.68,-207.89) (18.38,-207.18) (19.09,-207.18) (19.80,-207.18) (20.51,-207.18) (21.21,-207.18) (21.92,-207.18) (22.63,-206.48) (23.33,-206.48) (24.04,-206.48) (24.75,-206.48) (24.75,-206.48) (25.46,-207.18) (26.16,-207.18) (26.87,-207.89) (27.58,-207.89) (28.28,-208.60) (28.99,-208.60) (29.70,-209.30) (30.41,-210.01) (31.11,-210.01) (31.82,-210.72) (32.53,-210.72) (33.23,-211.42) (33.94,-211.42) (34.65,-212.13) (35.36,-212.13) (36.06,-212.84) (36.77,-213.55) (37.48,-213.55) (38.18,-214.25) (38.89,-214.25) (39.60,-214.96) (40.31,-214.96) (41.01,-215.67) (41.72,-216.37) (42.43,-216.37) (43.13,-217.08) (43.84,-217.08) (44.55,-217.79) (45.25,-217.79) (45.96,-218.50) (46.67,-218.50) (47.38,-219.20) (48.08,-219.91) (48.79,-219.91) (49.50,-220.62) (50.20,-220.62) (50.91,-221.32) (51.62,-221.32) (52.33,-222.03) (53.03,-222.74) (53.74,-222.74) (54.45,-223.45) (55.15,-223.45) (55.86,-224.15) (56.57,-224.15) (57.28,-224.86) (57.98,-224.86) (58.69,-225.57) (59.40,-226.27) (60.10,-226.27) (60.81,-226.98) (61.52,-226.98) (62.23,-227.69) (62.93,-227.69) (63.64,-228.40) (63.64,-228.40) (64.35,-228.40) (65.05,-228.40) (65.76,-228.40) (66.47,-228.40) (67.18,-228.40) (67.88,-228.40) (68.59,-229.10) (69.30,-229.10) (70.00,-229.10) (70.71,-229.10) (71.42,-229.10) (72.12,-229.10) (72.83,-229.10) (73.54,-229.10) (74.25,-229.10) (74.95,-229.10) (75.66,-229.10) (76.37,-229.10) (77.07,-229.81) (77.78,-229.81) (78.49,-229.81) (79.20,-229.81) (79.90,-229.81) (80.61,-229.81) (81.32,-229.81) (82.02,-229.81) (82.73,-229.81) (83.44,-229.81) (84.15,-229.81) (84.85,-229.81) (85.56,-229.81) (86.27,-230.52) (86.97,-230.52) (87.68,-230.52) (88.39,-230.52) (89.10,-230.52) (89.80,-230.52) (90.51,-230.52) (91.22,-230.52) (91.92,-230.52) (92.63,-230.52) (93.34,-230.52) (94.05,-230.52) (94.75,-231.22) (95.46,-231.22) (96.17,-231.22) (96.87,-231.22) (97.58,-231.22) (98.29,-231.22) (98.99,-231.22) (99.70,-231.22) (100.41,-231.22) (101.12,-231.22) (101.82,-231.22) (102.53,-231.22) (103.24,-231.93) (103.94,-231.93) (104.65,-231.93) (105.36,-231.93) (106.07,-231.93) (106.77,-231.93) (107.48,-231.93) (107.48,-231.93) (108.19,-231.22) (108.19,-230.52) (108.89,-229.81) (108.89,-229.10) (109.60,-228.40) (109.60,-227.69) (110.31,-226.98) (110.31,-226.27) (111.02,-225.57) (111.02,-224.86) (111.72,-224.15) (111.72,-223.45) (112.43,-222.74) (112.43,-222.03) (113.14,-221.32) (113.84,-220.62) (113.84,-219.91) (114.55,-219.20) (114.55,-218.50) (115.26,-217.79) (115.26,-217.08) (115.97,-216.37) (115.97,-215.67) (116.67,-214.96) (116.67,-214.25) (117.38,-213.55) (117.38,-212.84) (118.09,-212.13) (118.09,-211.42) (118.79,-210.72) (119.50,-210.01) (119.50,-209.30) (120.21,-208.60) (120.21,-207.89) (120.92,-207.18) (120.92,-206.48) (121.62,-205.77) (121.62,-205.06) (122.33,-204.35) (122.33,-203.65) (123.04,-202.94) (123.04,-202.23) (123.74,-201.53) (124.45,-200.82) (124.45,-200.11) (125.16,-199.40) (125.16,-198.70) (125.87,-197.99) (125.87,-197.28) (126.57,-196.58) (126.57,-195.87) (127.28,-195.16) (127.28,-194.45) (127.99,-193.75) (127.99,-193.04) (128.69,-192.33) (128.69,-191.63) (129.40,-190.92) (85.56,-187.38) (86.27,-187.38) (86.97,-187.38) (87.68,-187.38) (88.39,-187.38) (89.10,-187.38) (89.80,-187.38) (90.51,-188.09) (91.22,-188.09) (91.92,-188.09) (92.63,-188.09) (93.34,-188.09) (94.05,-188.09) (94.75,-188.09) (95.46,-188.09) (96.17,-188.09) (96.87,-188.09) (97.58,-188.09) (98.29,-188.09) (98.99,-188.80) (99.70,-188.80) (100.41,-188.80) (101.12,-188.80) (101.82,-188.80) (102.53,-188.80) (103.24,-188.80) (103.94,-188.80) (104.65,-188.80) (105.36,-188.80) (106.07,-188.80) (106.77,-188.80) (107.48,-188.80) (108.19,-189.50) (108.89,-189.50) (109.60,-189.50) (110.31,-189.50) (111.02,-189.50) (111.72,-189.50) (112.43,-189.50) (113.14,-189.50) (113.84,-189.50) (114.55,-189.50) (115.26,-189.50) (115.97,-189.50) (116.67,-190.21) (117.38,-190.21) (118.09,-190.21) (118.79,-190.21) (119.50,-190.21) (120.21,-190.21) (120.92,-190.21) (121.62,-190.21) (122.33,-190.21) (123.04,-190.21) (123.74,-190.21) (124.45,-190.21) (125.16,-190.92) (125.87,-190.92) (126.57,-190.92) (127.28,-190.92) (127.99,-190.92) (128.69,-190.92) (129.40,-190.92) (51.62,-214.96) (52.33,-214.25) (53.03,-213.55) (53.74,-213.55) (54.45,-212.84) (55.15,-212.13) (55.86,-211.42) (56.57,-210.72) (57.28,-210.72) (57.98,-210.01) (58.69,-209.30) (59.40,-208.60) (60.10,-207.89) (60.81,-207.18) (61.52,-207.18) (62.23,-206.48) (62.93,-205.77) (63.64,-205.06) (64.35,-204.35) (65.05,-204.35) (65.76,-203.65) (66.47,-202.94) (67.18,-202.23) (67.88,-201.53) (68.59,-201.53) (69.30,-200.82) (70.00,-200.11) (70.71,-199.40) (71.42,-198.70) (72.12,-197.99) (72.83,-197.99) (73.54,-197.28) (74.25,-196.58) (74.95,-195.87) (75.66,-195.16) (76.37,-195.16) (77.07,-194.45) (77.78,-193.75) (78.49,-193.04) (79.20,-192.33) (79.90,-192.33) (80.61,-191.63) (81.32,-190.92) (82.02,-190.21) (82.73,-189.50) (83.44,-188.80) (84.15,-188.80) (84.85,-188.09) (85.56,-187.38) (51.62,-214.96) (52.33,-215.67) (53.03,-216.37) (53.74,-216.37) (54.45,-217.08) (55.15,-217.79) (55.86,-218.50) (56.57,-219.20) (57.28,-219.20) (57.98,-219.91) (58.69,-220.62) (59.40,-221.32) (60.10,-222.03) (60.81,-222.03) (61.52,-222.74) (62.23,-223.45) (62.93,-224.15) (63.64,-224.86) (64.35,-224.86) (65.05,-225.57) (65.76,-226.27) (66.47,-226.98) (67.18,-227.69) (67.88,-227.69) (68.59,-228.40) (69.30,-229.10) (70.00,-229.81) (70.71,-230.52) (71.42,-231.22) (72.12,-231.22) (72.83,-231.93) (73.54,-232.64) (74.25,-233.35) (74.95,-234.05) (75.66,-234.05) (76.37,-234.76) (77.07,-235.47) (77.78,-236.17) (78.49,-236.88) (79.20,-236.88) (79.90,-237.59) (80.61,-238.29) (81.32,-239.00) (82.02,-239.71) (82.73,-239.71) (83.44,-240.42) (84.15,-241.12) (84.85,-241.83) (85.56,-242.54) (86.27,-242.54) (86.97,-243.24) (87.68,-243.95) (89.80,-289.91) (89.80,-289.21) (89.80,-288.50) (89.80,-287.79) (89.80,-287.09) (89.80,-286.38) (89.80,-285.67) (89.80,-284.96) (89.80,-284.26) (89.80,-283.55) (89.80,-282.84) (89.10,-282.14) (89.10,-281.43) (89.10,-280.72) (89.10,-280.01) (89.10,-279.31) (89.10,-278.60) (89.10,-277.89) (89.10,-277.19) (89.10,-276.48) (89.10,-275.77) (89.10,-275.06) (89.10,-274.36) (89.10,-273.65) (89.10,-272.94) (89.10,-272.24) (89.10,-271.53) (89.10,-270.82) (89.10,-270.11) (89.10,-269.41) (89.10,-268.70) (89.10,-267.99) (89.10,-267.29) (88.39,-266.58) (88.39,-265.87) (88.39,-265.17) (88.39,-264.46) (88.39,-263.75) (88.39,-263.04) (88.39,-262.34) (88.39,-261.63) (88.39,-260.92) (88.39,-260.22) (88.39,-259.51) (88.39,-258.80) (88.39,-258.09) (88.39,-257.39) (88.39,-256.68) (88.39,-255.97) (88.39,-255.27) (88.39,-254.56) (88.39,-253.85) (88.39,-253.14) (88.39,-252.44) (88.39,-251.73) (87.68,-251.02) (87.68,-250.32) (87.68,-249.61) (87.68,-248.90) (87.68,-248.19) (87.68,-247.49) (87.68,-246.78) (87.68,-246.07) (87.68,-245.37) (87.68,-244.66) (87.68,-243.95) (89.80,-289.91) (89.10,-289.21) (88.39,-288.50) (87.68,-287.79) (86.97,-287.09) (86.27,-286.38) (85.56,-285.67) (84.85,-284.96) (84.15,-284.26) (83.44,-283.55) (82.73,-282.84) (82.02,-282.14) (82.02,-281.43) (81.32,-280.72) (80.61,-280.01) (79.90,-279.31) (79.20,-278.60) (78.49,-277.89) (77.78,-277.19) (77.07,-276.48) (76.37,-275.77) (75.66,-275.06) (74.95,-274.36) (74.25,-273.65) (73.54,-272.94) (72.83,-272.24) (72.12,-271.53) (71.42,-270.82) (70.71,-270.11) (70.00,-269.41) (69.30,-268.70) (68.59,-267.99) (67.88,-267.29) (67.18,-266.58) (66.47,-265.87) (65.76,-265.17) (65.76,-264.46) (65.05,-263.75) (64.35,-263.04) (63.64,-262.34) (62.93,-261.63) (62.23,-260.92) (61.52,-260.22) (60.81,-259.51) (60.10,-258.80) (59.40,-258.09) (58.69,-257.39) (57.98,-256.68) (57.98,-256.68) (58.69,-255.97) (58.69,-255.27) (59.40,-254.56) (59.40,-253.85) (60.10,-253.14) (60.10,-252.44) (60.81,-251.73) (61.52,-251.02) (61.52,-250.32) (62.23,-249.61) (62.23,-248.90) (62.93,-248.19) (62.93,-247.49) (63.64,-246.78) (64.35,-246.07) (64.35,-245.37) (65.05,-244.66) (65.05,-243.95) (65.76,-243.24) (65.76,-242.54) (66.47,-241.83) (67.18,-241.12) (67.18,-240.42) (67.88,-239.71) (67.88,-239.00) (68.59,-238.29) (68.59,-237.59) (69.30,-236.88) (70.00,-236.17) (70.00,-235.47) (70.71,-234.76) (70.71,-234.05) (71.42,-233.35) (72.12,-232.64) (72.12,-231.93) (72.83,-231.22) (72.83,-230.52) (73.54,-229.81) (73.54,-229.10) (74.25,-228.40) (74.95,-227.69) (74.95,-226.98) (75.66,-226.27) (75.66,-225.57) (76.37,-224.86) (76.37,-224.15) (77.07,-223.45) (77.78,-222.74) (77.78,-222.03) (78.49,-221.32) (78.49,-220.62) (79.20,-219.91) (79.20,-219.20) (79.90,-218.50) (35.36,-227.69) (36.06,-227.69) (36.77,-227.69) (37.48,-226.98) (38.18,-226.98) (38.89,-226.98) (39.60,-226.98) (40.31,-226.98) (41.01,-226.27) (41.72,-226.27) (42.43,-226.27) (43.13,-226.27) (43.84,-226.27) (44.55,-225.57) (45.25,-225.57) (45.96,-225.57) (46.67,-225.57) (47.38,-224.86) (48.08,-224.86) (48.79,-224.86) (49.50,-224.86) (50.20,-224.86) (50.91,-224.15) (51.62,-224.15) (52.33,-224.15) (53.03,-224.15) (53.74,-224.15) (54.45,-223.45) (55.15,-223.45) (55.86,-223.45) (56.57,-223.45) (57.28,-223.45) (57.98,-222.74) (58.69,-222.74) (59.40,-222.74) (60.10,-222.74) (60.81,-222.74) (61.52,-222.03) (62.23,-222.03) (62.93,-222.03) (63.64,-222.03) (64.35,-222.03) (65.05,-221.32) (65.76,-221.32) (66.47,-221.32) (67.18,-221.32) (67.88,-221.32) (68.59,-220.62) (69.30,-220.62) (70.00,-220.62) (70.71,-220.62) (71.42,-219.91) (72.12,-219.91) (72.83,-219.91) (73.54,-219.91) (74.25,-219.91) (74.95,-219.20) (75.66,-219.20) (76.37,-219.20) (77.07,-219.20) (77.78,-219.20) (78.49,-218.50) (79.20,-218.50) (79.90,-218.50) (23.33,-272.94) (23.33,-272.24) (24.04,-271.53) (24.04,-270.82) (24.04,-270.11) (24.04,-269.41) (24.75,-268.70) (24.75,-267.99) (24.75,-267.29) (24.75,-266.58) (25.46,-265.87) (25.46,-265.17) (25.46,-264.46) (25.46,-263.75) (26.16,-263.04) (26.16,-262.34) (26.16,-261.63) (26.87,-260.92) (26.87,-260.22) (26.87,-259.51) (26.87,-258.80) (27.58,-258.09) (27.58,-257.39) (27.58,-256.68) (27.58,-255.97) (28.28,-255.27) (28.28,-254.56) (28.28,-253.85) (28.28,-253.14) (28.99,-252.44) (28.99,-251.73) (28.99,-251.02) (28.99,-250.32) (29.70,-249.61) (29.70,-248.90) (29.70,-248.19) (30.41,-247.49) (30.41,-246.78) (30.41,-246.07) (30.41,-245.37) (31.11,-244.66) (31.11,-243.95) (31.11,-243.24) (31.11,-242.54) (31.82,-241.83) (31.82,-241.12) (31.82,-240.42) (31.82,-239.71) (32.53,-239.00) (32.53,-238.29) (32.53,-237.59) (33.23,-236.88) (33.23,-236.17) (33.23,-235.47) (33.23,-234.76) (33.94,-234.05) (33.94,-233.35) (33.94,-232.64) (33.94,-231.93) (34.65,-231.22) (34.65,-230.52) (34.65,-229.81) (34.65,-229.10) (35.36,-228.40) (35.36,-227.69) (23.33,-272.94) (23.33,-272.24) (23.33,-271.53) (23.33,-270.82) (23.33,-270.11) (22.63,-269.41) (22.63,-268.70) (22.63,-267.99) (22.63,-267.29) (22.63,-266.58) (22.63,-265.87) (22.63,-265.17) (22.63,-264.46) (22.63,-263.75) (21.92,-263.04) (21.92,-262.34) (21.92,-261.63) (21.92,-260.92) (21.92,-260.22) (21.92,-259.51) (21.92,-258.80) (21.92,-258.09) (21.92,-257.39) (21.92,-256.68) (21.21,-255.97) (21.21,-255.27) (21.21,-254.56) (21.21,-253.85) (21.21,-253.14) (21.21,-252.44) (21.21,-251.73) (21.21,-251.02) (21.21,-250.32) (20.51,-249.61) (20.51,-248.90) (20.51,-248.19) (20.51,-247.49) (20.51,-246.78) (20.51,-246.07) (20.51,-245.37) (20.51,-244.66) (20.51,-243.95) (19.80,-243.24) (19.80,-242.54) (19.80,-241.83) (19.80,-241.12) (19.80,-240.42) (19.80,-239.71) (19.80,-239.00) (19.80,-238.29) (19.80,-237.59) (19.80,-236.88) (19.09,-236.17) (19.09,-235.47) (19.09,-234.76) (19.09,-234.05) (19.09,-233.35) (19.09,-232.64) (19.09,-231.93) (19.09,-231.22) (19.09,-230.52) (18.38,-229.81) (18.38,-229.10) (18.38,-228.40) (18.38,-227.69) (18.38,-226.98) (25.46,-268.70) (25.46,-267.99) (25.46,-267.29) (24.75,-266.58) (24.75,-265.87) (24.75,-265.17) (24.75,-264.46) (24.75,-263.75) (24.75,-263.04) (24.04,-262.34) (24.04,-261.63) (24.04,-260.92) (24.04,-260.22) (24.04,-259.51) (24.04,-258.80) (23.33,-258.09) (23.33,-257.39) (23.33,-256.68) (23.33,-255.97) (23.33,-255.27) (23.33,-254.56) (22.63,-253.85) (22.63,-253.14) (22.63,-252.44) (22.63,-251.73) (22.63,-251.02) (22.63,-250.32) (21.92,-249.61) (21.92,-248.90) (21.92,-248.19) (21.92,-247.49) (21.92,-246.78) (21.92,-246.07) (21.21,-245.37) (21.21,-244.66) (21.21,-243.95) (21.21,-243.24) (21.21,-242.54) (21.21,-241.83) (20.51,-241.12) (20.51,-240.42) (20.51,-239.71) (20.51,-239.00) (20.51,-238.29) (20.51,-237.59) (19.80,-236.88) (19.80,-236.17) (19.80,-235.47) (19.80,-234.76) (19.80,-234.05) (19.80,-233.35) (19.09,-232.64) (19.09,-231.93) (19.09,-231.22) (19.09,-230.52) (19.09,-229.81) (19.09,-229.10) (18.38,-228.40) (18.38,-227.69) (18.38,-226.98) (-17.68,-275.77) (-16.97,-275.77) (-16.26,-275.77) (-15.56,-275.77) (-14.85,-275.06) (-14.14,-275.06) (-13.44,-275.06) (-12.73,-275.06) (-12.02,-275.06) (-11.31,-275.06) (-10.61,-274.36) (-9.90,-274.36) (-9.19,-274.36) (-8.49,-274.36) (-7.78,-274.36) (-7.07,-274.36) (-6.36,-273.65) (-5.66,-273.65) (-4.95,-273.65) (-4.24,-273.65) (-3.54,-273.65) (-2.83,-273.65) (-2.12,-272.94) (-1.41,-272.94) (-0.71,-272.94) ( 0.00,-272.94) ( 0.71,-272.94) ( 1.41,-272.94) ( 2.12,-272.24) ( 2.83,-272.24) ( 3.54,-272.24) ( 4.24,-272.24) ( 4.95,-272.24) ( 5.66,-272.24) ( 6.36,-271.53) ( 7.07,-271.53) ( 7.78,-271.53) ( 8.49,-271.53) ( 9.19,-271.53) ( 9.90,-271.53) (10.61,-270.82) (11.31,-270.82) (12.02,-270.82) (12.73,-270.82) (13.44,-270.82) (14.14,-270.82) (14.85,-270.11) (15.56,-270.11) (16.26,-270.11) (16.97,-270.11) (17.68,-270.11) (18.38,-270.11) (19.09,-269.41) (19.80,-269.41) (20.51,-269.41) (21.21,-269.41) (21.92,-269.41) (22.63,-269.41) (23.33,-268.70) (24.04,-268.70) (24.75,-268.70) (25.46,-268.70) (-58.69,-295.57) (-57.98,-295.57) (-57.28,-294.86) (-56.57,-294.86) (-55.86,-294.16) (-55.15,-294.16) (-54.45,-293.45) (-53.74,-293.45) (-53.03,-292.74) (-52.33,-292.74) (-51.62,-292.04) (-50.91,-292.04) (-50.20,-291.33) (-49.50,-291.33) (-48.79,-290.62) (-48.08,-290.62) (-47.38,-289.91) (-46.67,-289.91) (-45.96,-289.21) (-45.25,-289.21) (-44.55,-288.50) (-43.84,-288.50) (-43.13,-287.79) (-42.43,-287.79) (-41.72,-287.09) (-41.01,-287.09) (-40.31,-286.38) (-39.60,-286.38) (-38.89,-285.67) (-38.18,-285.67) (-37.48,-285.67) (-36.77,-284.96) (-36.06,-284.96) (-35.36,-284.26) (-34.65,-284.26) (-33.94,-283.55) (-33.23,-283.55) (-32.53,-282.84) (-31.82,-282.84) (-31.11,-282.14) (-30.41,-282.14) (-29.70,-281.43) (-28.99,-281.43) (-28.28,-280.72) (-27.58,-280.72) (-26.87,-280.01) (-26.16,-280.01) (-25.46,-279.31) (-24.75,-279.31) (-24.04,-278.60) (-23.33,-278.60) (-22.63,-277.89) (-21.92,-277.89) (-21.21,-277.19) (-20.51,-277.19) (-19.80,-276.48) (-19.09,-276.48) (-18.38,-275.77) (-17.68,-275.77) (-58.69,-295.57) (-57.98,-296.28) (-57.28,-296.28) (-56.57,-296.98) (-55.86,-296.98) (-55.15,-297.69) (-54.45,-297.69) (-53.74,-298.40) (-53.03,-299.11) (-52.33,-299.11) (-51.62,-299.81) (-50.91,-299.81) (-50.20,-300.52) (-49.50,-301.23) (-48.79,-301.23) (-48.08,-301.93) (-47.38,-301.93) (-46.67,-302.64) (-45.96,-302.64) (-45.25,-303.35) (-44.55,-304.06) (-43.84,-304.06) (-43.13,-304.76) (-42.43,-304.76) (-41.72,-305.47) (-41.01,-305.47) (-40.31,-306.18) (-39.60,-306.88) (-38.89,-306.88) (-38.18,-307.59) (-37.48,-307.59) (-36.77,-308.30) (-36.06,-309.01) (-35.36,-309.01) (-34.65,-309.71) (-33.94,-309.71) (-33.23,-310.42) (-32.53,-310.42) (-31.82,-311.13) (-31.11,-311.83) (-30.41,-311.83) (-29.70,-312.54) (-28.99,-312.54) (-28.28,-313.25) (-27.58,-313.25) (-26.87,-313.96) (-26.16,-314.66) (-25.46,-314.66) (-24.75,-315.37) (-24.04,-315.37) (-23.33,-316.08) (-22.63,-316.78) (-21.92,-316.78) (-21.21,-317.49) (-20.51,-317.49) (-19.80,-318.20) (-19.09,-318.20) (-18.38,-318.91) ( 1.41,-356.38) ( 0.71,-355.67) ( 0.71,-354.97) ( 0.00,-354.26) ( 0.00,-353.55) (-0.71,-352.85) (-0.71,-352.14) (-1.41,-351.43) (-1.41,-350.72) (-2.12,-350.02) (-2.12,-349.31) (-2.83,-348.60) (-2.83,-347.90) (-3.54,-347.19) (-3.54,-346.48) (-4.24,-345.78) (-4.24,-345.07) (-4.95,-344.36) (-5.66,-343.65) (-5.66,-342.95) (-6.36,-342.24) (-6.36,-341.53) (-7.07,-340.83) (-7.07,-340.12) (-7.78,-339.41) (-7.78,-338.70) (-8.49,-338.00) (-8.49,-337.29) (-9.19,-336.58) (-9.19,-335.88) (-9.90,-335.17) (-9.90,-334.46) (-10.61,-333.75) (-10.61,-333.05) (-11.31,-332.34) (-11.31,-331.63) (-12.02,-330.93) (-12.73,-330.22) (-12.73,-329.51) (-13.44,-328.80) (-13.44,-328.10) (-14.14,-327.39) (-14.14,-326.68) (-14.85,-325.98) (-14.85,-325.27) (-15.56,-324.56) (-15.56,-323.85) (-16.26,-323.15) (-16.26,-322.44) (-16.97,-321.73) (-16.97,-321.03) (-17.68,-320.32) (-17.68,-319.61) (-18.38,-318.91) (-41.72,-369.82) (-41.01,-369.82) (-40.31,-369.11) (-39.60,-369.11) (-38.89,-369.11) (-38.18,-368.40) (-37.48,-368.40) (-36.77,-368.40) (-36.06,-368.40) (-35.36,-367.70) (-34.65,-367.70) (-33.94,-367.70) (-33.23,-366.99) (-32.53,-366.99) (-31.82,-366.99) (-31.11,-366.28) (-30.41,-366.28) (-29.70,-366.28) (-28.99,-365.57) (-28.28,-365.57) (-27.58,-365.57) (-26.87,-364.87) (-26.16,-364.87) (-25.46,-364.87) (-24.75,-364.87) (-24.04,-364.16) (-23.33,-364.16) (-22.63,-364.16) (-21.92,-363.45) (-21.21,-363.45) (-20.51,-363.45) (-19.80,-362.75) (-19.09,-362.75) (-18.38,-362.75) (-17.68,-362.04) (-16.97,-362.04) (-16.26,-362.04) (-15.56,-361.33) (-14.85,-361.33) (-14.14,-361.33) (-13.44,-361.33) (-12.73,-360.62) (-12.02,-360.62) (-11.31,-360.62) (-10.61,-359.92) (-9.90,-359.92) (-9.19,-359.92) (-8.49,-359.21) (-7.78,-359.21) (-7.07,-359.21) (-6.36,-358.50) (-5.66,-358.50) (-4.95,-358.50) (-4.24,-357.80) (-3.54,-357.80) (-2.83,-357.80) (-2.12,-357.80) (-1.41,-357.09) (-0.71,-357.09) ( 0.00,-357.09) ( 0.71,-356.38) ( 1.41,-356.38) (-31.82,-414.36) (-31.82,-413.66) (-31.82,-412.95) (-32.53,-412.24) (-32.53,-411.54) (-32.53,-410.83) (-32.53,-410.12) (-33.23,-409.41) (-33.23,-408.71) (-33.23,-408.00) (-33.23,-407.29) (-33.23,-406.59) (-33.94,-405.88) (-33.94,-405.17) (-33.94,-404.47) (-33.94,-403.76) (-34.65,-403.05) (-34.65,-402.34) (-34.65,-401.64) (-34.65,-400.93) (-34.65,-400.22) (-35.36,-399.52) (-35.36,-398.81) (-35.36,-398.10) (-35.36,-397.39) (-36.06,-396.69) (-36.06,-395.98) (-36.06,-395.27) (-36.06,-394.57) (-36.06,-393.86) (-36.77,-393.15) (-36.77,-392.44) (-36.77,-391.74) (-36.77,-391.03) (-37.48,-390.32) (-37.48,-389.62) (-37.48,-388.91) (-37.48,-388.20) (-37.48,-387.49) (-38.18,-386.79) (-38.18,-386.08) (-38.18,-385.37) (-38.18,-384.67) (-38.89,-383.96) (-38.89,-383.25) (-38.89,-382.54) (-38.89,-381.84) (-38.89,-381.13) (-39.60,-380.42) (-39.60,-379.72) (-39.60,-379.01) (-39.60,-378.30) (-40.31,-377.60) (-40.31,-376.89) (-40.31,-376.18) (-40.31,-375.47) (-40.31,-374.77) (-41.01,-374.06) (-41.01,-373.35) (-41.01,-372.65) (-41.01,-371.94) (-41.72,-371.23) (-41.72,-370.52) (-41.72,-369.82) (-75.66,-413.66) (-74.95,-413.66) (-74.25,-413.66) (-73.54,-413.66) (-72.83,-413.66) (-72.12,-413.66) (-71.42,-413.66) (-70.71,-413.66) (-70.00,-413.66) (-69.30,-413.66) (-68.59,-413.66) (-67.88,-413.66) (-67.18,-413.66) (-66.47,-413.66) (-65.76,-413.66) (-65.05,-413.66) (-64.35,-413.66) (-63.64,-413.66) (-62.93,-413.66) (-62.23,-413.66) (-61.52,-413.66) (-60.81,-413.66) (-60.10,-413.66) (-59.40,-413.66) (-58.69,-413.66) (-57.98,-413.66) (-57.28,-413.66) (-56.57,-413.66) (-55.86,-413.66) (-55.15,-413.66) (-54.45,-413.66) (-53.74,-413.66) (-53.03,-414.36) (-52.33,-414.36) (-51.62,-414.36) (-50.91,-414.36) (-50.20,-414.36) (-49.50,-414.36) (-48.79,-414.36) (-48.08,-414.36) (-47.38,-414.36) (-46.67,-414.36) (-45.96,-414.36) (-45.25,-414.36) (-44.55,-414.36) (-43.84,-414.36) (-43.13,-414.36) (-42.43,-414.36) (-41.72,-414.36) (-41.01,-414.36) (-40.31,-414.36) (-39.60,-414.36) (-38.89,-414.36) (-38.18,-414.36) (-37.48,-414.36) (-36.77,-414.36) (-36.06,-414.36) (-35.36,-414.36) (-34.65,-414.36) (-33.94,-414.36) (-33.23,-414.36) (-32.53,-414.36) (-31.82,-414.36) (-116.67,-433.46) (-115.97,-433.46) (-115.26,-432.75) (-114.55,-432.75) (-113.84,-432.04) (-113.14,-432.04) (-112.43,-431.34) (-111.72,-431.34) (-111.02,-430.63) (-110.31,-430.63) (-109.60,-429.92) (-108.89,-429.92) (-108.19,-429.21) (-107.48,-429.21) (-106.77,-428.51) (-106.07,-428.51) (-105.36,-427.80) (-104.65,-427.80) (-103.94,-427.09) (-103.24,-427.09) (-102.53,-426.39) (-101.82,-426.39) (-101.12,-425.68) (-100.41,-425.68) (-99.70,-424.97) (-98.99,-424.97) (-98.29,-424.26) (-97.58,-424.26) (-96.87,-423.56) (-96.17,-423.56) (-95.46,-423.56) (-94.75,-422.85) (-94.05,-422.85) (-93.34,-422.14) (-92.63,-422.14) (-91.92,-421.44) (-91.22,-421.44) (-90.51,-420.73) (-89.80,-420.73) (-89.10,-420.02) (-88.39,-420.02) (-87.68,-419.31) (-86.97,-419.31) (-86.27,-418.61) (-85.56,-418.61) (-84.85,-417.90) (-84.15,-417.90) (-83.44,-417.19) (-82.73,-417.19) (-82.02,-416.49) (-81.32,-416.49) (-80.61,-415.78) (-79.90,-415.78) (-79.20,-415.07) (-78.49,-415.07) (-77.78,-414.36) (-77.07,-414.36) (-76.37,-413.66) (-75.66,-413.66) (-116.67,-433.46) (-115.97,-433.46) (-115.26,-434.16) (-114.55,-434.16) (-113.84,-434.87) (-113.14,-434.87) (-112.43,-435.58) (-111.72,-435.58) (-111.02,-436.28) (-110.31,-436.28) (-109.60,-436.28) (-108.89,-436.99) (-108.19,-436.99) (-107.48,-437.70) (-106.77,-437.70) (-106.07,-438.41) (-105.36,-438.41) (-104.65,-439.11) (-103.94,-439.11) (-103.24,-439.11) (-102.53,-439.82) (-101.82,-439.82) (-101.12,-440.53) (-100.41,-440.53) (-99.70,-441.23) (-98.99,-441.23) (-98.29,-441.94) (-97.58,-441.94) (-96.87,-441.94) (-96.17,-442.65) (-95.46,-442.65) (-94.75,-443.36) (-94.05,-443.36) (-93.34,-444.06) (-92.63,-444.06) (-91.92,-444.06) (-91.22,-444.77) (-90.51,-444.77) (-89.80,-445.48) (-89.10,-445.48) (-88.39,-446.18) (-87.68,-446.18) (-86.97,-446.89) (-86.27,-446.89) (-85.56,-446.89) (-84.85,-447.60) (-84.15,-447.60) (-83.44,-448.31) (-82.73,-448.31) (-82.02,-449.01) (-81.32,-449.01) (-80.61,-449.72) (-79.90,-449.72) (-79.20,-449.72) (-78.49,-450.43) (-77.78,-450.43) (-77.07,-451.13) (-76.37,-451.13) (-75.66,-451.84) (-74.95,-451.84) (-74.25,-452.55) (-73.54,-452.55) (-73.54,-452.55) (-72.83,-452.55) (-72.12,-451.84) (-71.42,-451.84) (-70.71,-451.84) (-70.00,-451.84) (-69.30,-451.13) (-68.59,-451.13) (-67.88,-451.13) (-67.18,-450.43) (-66.47,-450.43) (-65.76,-450.43) (-65.05,-450.43) (-64.35,-449.72) (-63.64,-449.72) (-62.93,-449.72) (-62.23,-449.01) (-61.52,-449.01) (-60.81,-449.01) (-60.10,-448.31) (-59.40,-448.31) (-58.69,-448.31) (-57.98,-448.31) (-57.28,-447.60) (-56.57,-447.60) (-55.86,-447.60) (-55.15,-446.89) (-54.45,-446.89) (-53.74,-446.89) (-53.03,-446.89) (-52.33,-446.18) (-51.62,-446.18) (-50.91,-446.18) (-50.20,-445.48) (-49.50,-445.48) (-48.79,-445.48) (-48.08,-445.48) (-47.38,-444.77) (-46.67,-444.77) (-45.96,-444.77) (-45.25,-444.06) (-44.55,-444.06) (-43.84,-444.06) (-43.13,-444.06) (-42.43,-443.36) (-41.72,-443.36) (-41.01,-443.36) (-40.31,-442.65) (-39.60,-442.65) (-38.89,-442.65) (-38.18,-441.94) (-37.48,-441.94) (-36.77,-441.94) (-36.06,-441.94) (-35.36,-441.23) (-34.65,-441.23) (-33.94,-441.23) (-33.23,-440.53) (-32.53,-440.53) (-31.82,-440.53) (-31.11,-440.53) (-30.41,-439.82) (-29.70,-439.82) (-29.70,-439.82) (-28.99,-439.82) (-28.28,-439.11) (-27.58,-439.11) (-26.87,-439.11) (-26.16,-438.41) (-25.46,-438.41) (-24.75,-438.41) (-24.04,-437.70) (-23.33,-437.70) (-22.63,-437.70) (-21.92,-436.99) (-21.21,-436.99) (-20.51,-436.99) (-19.80,-436.28) (-19.09,-436.28) (-18.38,-436.28) (-17.68,-435.58) (-16.97,-435.58) (-16.26,-435.58) (-15.56,-434.87) (-14.85,-434.87) (-14.14,-434.87) (-13.44,-434.16) (-12.73,-434.16) (-12.02,-434.16) (-11.31,-433.46) (-10.61,-433.46) (-9.90,-433.46) (-9.19,-432.75) (-8.49,-432.75) (-7.78,-432.75) (-7.07,-432.04) (-6.36,-432.04) (-5.66,-432.04) (-4.95,-431.34) (-4.24,-431.34) (-3.54,-431.34) (-2.83,-430.63) (-2.12,-430.63) (-1.41,-430.63) (-0.71,-429.92) ( 0.00,-429.92) ( 0.71,-429.92) ( 1.41,-429.21) ( 2.12,-429.21) ( 2.83,-429.21) ( 3.54,-428.51) ( 4.24,-428.51) ( 4.95,-428.51) ( 5.66,-427.80) ( 6.36,-427.80) ( 7.07,-427.80) ( 7.78,-427.09) ( 8.49,-427.09) ( 9.19,-427.09) ( 9.90,-426.39) (10.61,-426.39) (10.61,-426.39) (10.61,-425.68) (10.61,-424.97) (11.31,-424.26) (11.31,-423.56) (11.31,-422.85) (11.31,-422.14) (12.02,-421.44) (12.02,-420.73) (12.02,-420.02) (12.02,-419.31) (12.02,-418.61) (12.73,-417.90) (12.73,-417.19) (12.73,-416.49) (12.73,-415.78) (13.44,-415.07) (13.44,-414.36) (13.44,-413.66) (13.44,-412.95) (14.14,-412.24) (14.14,-411.54) (14.14,-410.83) (14.14,-410.12) (14.14,-409.41) (14.85,-408.71) (14.85,-408.00) (14.85,-407.29) (14.85,-406.59) (15.56,-405.88) (15.56,-405.17) (15.56,-404.47) (15.56,-403.76) (15.56,-403.05) (16.26,-402.34) (16.26,-401.64) (16.26,-400.93) (16.26,-400.22) (16.97,-399.52) (16.97,-398.81) (16.97,-398.10) (16.97,-397.39) (16.97,-396.69) (17.68,-395.98) (17.68,-395.27) (17.68,-394.57) (17.68,-393.86) (18.38,-393.15) (18.38,-392.44) (18.38,-391.74) (18.38,-391.03) (19.09,-390.32) (19.09,-389.62) (19.09,-388.91) (19.09,-388.20) (19.09,-387.49) (19.80,-386.79) (19.80,-386.08) (19.80,-385.37) (19.80,-384.67) (20.51,-383.96) (20.51,-383.25) (20.51,-382.54) (18.38,-430.63) (18.38,-429.92) (18.38,-429.21) (18.38,-428.51) (18.38,-427.80) (18.38,-427.09) (18.38,-426.39) (18.38,-425.68) (18.38,-424.97) (18.38,-424.26) (18.38,-423.56) (18.38,-422.85) (19.09,-422.14) (19.09,-421.44) (19.09,-420.73) (19.09,-420.02) (19.09,-419.31) (19.09,-418.61) (19.09,-417.90) (19.09,-417.19) (19.09,-416.49) (19.09,-415.78) (19.09,-415.07) (19.09,-414.36) (19.09,-413.66) (19.09,-412.95) (19.09,-412.24) (19.09,-411.54) (19.09,-410.83) (19.09,-410.12) (19.09,-409.41) (19.09,-408.71) (19.09,-408.00) (19.09,-407.29) (19.09,-406.59) (19.80,-405.88) (19.80,-405.17) (19.80,-404.47) (19.80,-403.76) (19.80,-403.05) (19.80,-402.34) (19.80,-401.64) (19.80,-400.93) (19.80,-400.22) (19.80,-399.52) (19.80,-398.81) (19.80,-398.10) (19.80,-397.39) (19.80,-396.69) (19.80,-395.98) (19.80,-395.27) (19.80,-394.57) (19.80,-393.86) (19.80,-393.15) (19.80,-392.44) (19.80,-391.74) (19.80,-391.03) (20.51,-390.32) (20.51,-389.62) (20.51,-388.91) (20.51,-388.20) (20.51,-387.49) (20.51,-386.79) (20.51,-386.08) (20.51,-385.37) (20.51,-384.67) (20.51,-383.96) (20.51,-383.25) (20.51,-382.54) (-17.68,-401.64) (-16.97,-402.34) (-16.26,-403.05) (-15.56,-403.05) (-14.85,-403.76) (-14.14,-404.47) (-13.44,-405.17) (-12.73,-405.88) (-12.02,-405.88) (-11.31,-406.59) (-10.61,-407.29) (-9.90,-408.00) (-9.19,-408.71) (-8.49,-408.71) (-7.78,-409.41) (-7.07,-410.12) (-6.36,-410.83) (-5.66,-411.54) (-4.95,-411.54) (-4.24,-412.24) (-3.54,-412.95) (-2.83,-413.66) (-2.12,-414.36) (-1.41,-414.36) (-0.71,-415.07) ( 0.00,-415.78) ( 0.71,-416.49) ( 1.41,-417.19) ( 2.12,-417.90) ( 2.83,-417.90) ( 3.54,-418.61) ( 4.24,-419.31) ( 4.95,-420.02) ( 5.66,-420.73) ( 6.36,-420.73) ( 7.07,-421.44) ( 7.78,-422.14) ( 8.49,-422.85) ( 9.19,-423.56) ( 9.90,-423.56) (10.61,-424.26) (11.31,-424.97) (12.02,-425.68) (12.73,-426.39) (13.44,-426.39) (14.14,-427.09) (14.85,-427.80) (15.56,-428.51) (16.26,-429.21) (16.97,-429.21) (17.68,-429.92) (18.38,-430.63) (-17.68,-401.64) (-18.38,-400.93) (-18.38,-400.22) (-19.09,-399.52) (-19.09,-398.81) (-19.80,-398.10) (-19.80,-397.39) (-20.51,-396.69) (-20.51,-395.98) (-21.21,-395.27) (-21.21,-394.57) (-21.92,-393.86) (-21.92,-393.15) (-22.63,-392.44) (-22.63,-391.74) (-23.33,-391.03) (-23.33,-390.32) (-24.04,-389.62) (-24.04,-388.91) (-24.75,-388.20) (-24.75,-387.49) (-25.46,-386.79) (-25.46,-386.08) (-26.16,-385.37) (-26.16,-384.67) (-26.87,-383.96) (-26.87,-383.25) (-27.58,-382.54) (-27.58,-381.84) (-28.28,-381.13) (-28.99,-380.42) (-28.99,-379.72) (-29.70,-379.01) (-29.70,-378.30) (-30.41,-377.60) (-30.41,-376.89) (-31.11,-376.18) (-31.11,-375.47) (-31.82,-374.77) (-31.82,-374.06) (-32.53,-373.35) (-32.53,-372.65) (-33.23,-371.94) (-33.23,-371.23) (-33.94,-370.52) (-33.94,-369.82) (-34.65,-369.11) (-34.65,-368.40) (-35.36,-367.70) (-35.36,-366.99) (-36.06,-366.28) (-36.06,-365.57) (-36.77,-364.87) (-36.77,-364.16) (-37.48,-363.45) (-37.48,-362.75) (-38.18,-362.04) (-38.18,-361.33) (-38.89,-360.62) (-38.89,-360.62) (-38.18,-360.62) (-37.48,-360.62) (-36.77,-361.33) (-36.06,-361.33) (-35.36,-361.33) (-34.65,-361.33) (-33.94,-361.33) (-33.23,-362.04) (-32.53,-362.04) (-31.82,-362.04) (-31.11,-362.04) (-30.41,-362.04) (-29.70,-362.75) (-28.99,-362.75) (-28.28,-362.75) (-27.58,-362.75) (-26.87,-362.75) (-26.16,-363.45) (-25.46,-363.45) (-24.75,-363.45) (-24.04,-363.45) (-23.33,-363.45) (-22.63,-364.16) (-21.92,-364.16) (-21.21,-364.16) (-20.51,-364.16) (-19.80,-364.16) (-19.09,-364.87) (-18.38,-364.87) (-17.68,-364.87) (-16.97,-364.87) (-16.26,-364.87) (-15.56,-364.87) (-14.85,-365.57) (-14.14,-365.57) (-13.44,-365.57) (-12.73,-365.57) (-12.02,-365.57) (-11.31,-366.28) (-10.61,-366.28) (-9.90,-366.28) (-9.19,-366.28) (-8.49,-366.28) (-7.78,-366.99) (-7.07,-366.99) (-6.36,-366.99) (-5.66,-366.99) (-4.95,-366.99) (-4.24,-367.70) (-3.54,-367.70) (-2.83,-367.70) (-2.12,-367.70) (-1.41,-367.70) (-0.71,-368.40) ( 0.00,-368.40) ( 0.71,-368.40) ( 1.41,-368.40) ( 2.12,-368.40) ( 2.83,-369.11) ( 3.54,-369.11) ( 4.24,-369.11) ( 4.95,-369.11) ( 5.66,-369.11) ( 6.36,-369.82) ( 7.07,-369.82) ( 7.78,-369.82) ( 7.78,-369.82) ( 8.49,-369.11) ( 9.19,-368.40) ( 9.90,-368.40) (10.61,-367.70) (11.31,-366.99) (12.02,-366.28) (12.73,-366.28) (13.44,-365.57) (14.14,-364.87) (14.85,-364.16) (15.56,-363.45) (16.26,-363.45) (16.97,-362.75) (17.68,-362.04) (18.38,-361.33) (19.09,-360.62) (19.80,-360.62) (20.51,-359.92) (21.21,-359.21) (21.92,-358.50) (22.63,-358.50) (23.33,-357.80) (24.04,-357.09) (24.75,-356.38) (25.46,-355.67) (26.16,-355.67) (26.87,-354.97) (27.58,-354.26) (28.28,-353.55) (28.99,-352.85) (29.70,-352.85) (30.41,-352.14) (31.11,-351.43) (31.82,-350.72) (32.53,-350.72) (33.23,-350.02) (33.94,-349.31) (34.65,-348.60) (35.36,-347.90) (36.06,-347.90) (36.77,-347.19) (37.48,-346.48) (38.18,-345.78) (38.89,-345.07) (39.60,-345.07) (40.31,-344.36) (41.01,-343.65) (41.72,-342.95) (42.43,-342.95) (43.13,-342.24) (43.84,-341.53) (43.84,-341.53) (43.13,-340.83) (43.13,-340.12) (42.43,-339.41) (41.72,-338.70) (41.72,-338.00) (41.01,-337.29) (40.31,-336.58) (40.31,-335.88) (39.60,-335.17) (38.89,-334.46) (38.89,-333.75) (38.18,-333.05) (37.48,-332.34) (37.48,-331.63) (36.77,-330.93) (36.06,-330.22) (36.06,-329.51) (35.36,-328.80) (34.65,-328.10) (34.65,-327.39) (33.94,-326.68) (33.23,-325.98) (33.23,-325.27) (32.53,-324.56) (31.82,-323.85) (31.82,-323.15) (31.11,-322.44) (30.41,-321.73) (30.41,-321.03) (29.70,-320.32) (28.99,-319.61) (28.99,-318.91) (28.28,-318.20) (27.58,-317.49) (27.58,-316.78) (26.87,-316.08) (26.16,-315.37) (26.16,-314.66) (25.46,-313.96) (24.75,-313.25) (24.75,-312.54) (24.04,-311.83) (23.33,-311.13) (23.33,-310.42) (22.63,-309.71) (21.92,-309.01) (21.92,-308.30) (21.21,-307.59) (20.51,-306.88) (20.51,-306.18) (19.80,-305.47) (19.09,-304.76) (19.09,-304.06) (18.38,-303.35) (18.38,-303.35) (19.09,-303.35) (19.80,-302.64) (20.51,-302.64) (21.21,-302.64) (21.92,-302.64) (22.63,-301.93) (23.33,-301.93) (24.04,-301.93) (24.75,-301.23) (25.46,-301.23) (26.16,-301.23) (26.87,-300.52) (27.58,-300.52) (28.28,-300.52) (28.99,-300.52) (29.70,-299.81) (30.41,-299.81) (31.11,-299.81) (31.82,-299.11) (32.53,-299.11) (33.23,-299.11) (33.94,-299.11) (34.65,-298.40) (35.36,-298.40) (36.06,-298.40) (36.77,-297.69) (37.48,-297.69) (38.18,-297.69) (38.89,-296.98) (39.60,-296.98) (40.31,-296.98) (41.01,-296.98) (41.72,-296.28) (42.43,-296.28) (43.13,-296.28) (43.84,-295.57) (44.55,-295.57) (45.25,-295.57) (45.96,-294.86) (46.67,-294.86) (47.38,-294.86) (48.08,-294.86) (48.79,-294.16) (49.50,-294.16) (50.20,-294.16) (50.91,-293.45) (51.62,-293.45) (52.33,-293.45) (53.03,-293.45) (53.74,-292.74) (54.45,-292.74) (55.15,-292.74) (55.86,-292.04) (56.57,-292.04) (57.28,-292.04) (57.98,-291.33) (58.69,-291.33) (59.40,-291.33) (60.10,-291.33) (60.81,-290.62) (61.52,-290.62) (61.52,-290.62) (61.52,-289.91) (61.52,-289.21) (62.23,-288.50) (62.23,-287.79) (62.23,-287.09) (62.23,-286.38) (62.23,-285.67) (62.93,-284.96) (62.93,-284.26) (62.93,-283.55) (62.93,-282.84) (63.64,-282.14) (63.64,-281.43) (63.64,-280.72) (63.64,-280.01) (63.64,-279.31) (64.35,-278.60) (64.35,-277.89) (64.35,-277.19) (64.35,-276.48) (64.35,-275.77) (65.05,-275.06) (65.05,-274.36) (65.05,-273.65) (65.05,-272.94) (65.76,-272.24) (65.76,-271.53) (65.76,-270.82) (65.76,-270.11) (65.76,-269.41) (66.47,-268.70) (66.47,-267.99) (66.47,-267.29) (66.47,-266.58) (66.47,-265.87) (67.18,-265.17) (67.18,-264.46) (67.18,-263.75) (67.18,-263.04) (67.18,-262.34) (67.88,-261.63) (67.88,-260.92) (67.88,-260.22) (67.88,-259.51) (68.59,-258.80) (68.59,-258.09) (68.59,-257.39) (68.59,-256.68) (68.59,-255.97) (69.30,-255.27) (69.30,-254.56) (69.30,-253.85) (69.30,-253.14) (69.30,-252.44) (70.00,-251.73) (70.00,-251.02) (70.00,-250.32) (70.00,-249.61) (70.71,-248.90) (70.71,-248.19) (70.71,-247.49) (70.71,-246.78) (70.71,-246.07) (71.42,-245.37) (71.42,-244.66) (71.42,-243.95) (57.98,-283.55) (57.98,-282.84) (58.69,-282.14) (58.69,-281.43) (58.69,-280.72) (59.40,-280.01) (59.40,-279.31) (59.40,-278.60) (60.10,-277.89) (60.10,-277.19) (60.10,-276.48) (60.81,-275.77) (60.81,-275.06) (60.81,-274.36) (61.52,-273.65) (61.52,-272.94) (61.52,-272.24) (62.23,-271.53) (62.23,-270.82) (62.23,-270.11) (62.93,-269.41) (62.93,-268.70) (62.93,-267.99) (63.64,-267.29) (63.64,-266.58) (63.64,-265.87) (64.35,-265.17) (64.35,-264.46) (64.35,-263.75) (65.05,-263.04) (65.05,-262.34) (65.76,-261.63) (65.76,-260.92) (65.76,-260.22) (66.47,-259.51) (66.47,-258.80) (66.47,-258.09) (67.18,-257.39) (67.18,-256.68) (67.18,-255.97) (67.88,-255.27) (67.88,-254.56) (67.88,-253.85) (68.59,-253.14) (68.59,-252.44) (68.59,-251.73) (69.30,-251.02) (69.30,-250.32) (69.30,-249.61) (70.00,-248.90) (70.00,-248.19) (70.00,-247.49) (70.71,-246.78) (70.71,-246.07) (70.71,-245.37) (71.42,-244.66) (71.42,-243.95) (57.98,-283.55) (57.98,-282.84) (57.98,-282.14) (58.69,-281.43) (58.69,-280.72) (58.69,-280.01) (58.69,-279.31) (59.40,-278.60) (59.40,-277.89) (59.40,-277.19) (59.40,-276.48) (60.10,-275.77) (60.10,-275.06) (60.10,-274.36) (60.10,-273.65) (60.81,-272.94) (60.81,-272.24) (60.81,-271.53) (60.81,-270.82) (61.52,-270.11) (61.52,-269.41) (61.52,-268.70) (61.52,-267.99) (62.23,-267.29) (62.23,-266.58) (62.23,-265.87) (62.23,-265.17) (62.93,-264.46) (62.93,-263.75) (62.93,-263.04) (62.93,-262.34) (62.93,-261.63) (63.64,-260.92) (63.64,-260.22) (63.64,-259.51) (63.64,-258.80) (64.35,-258.09) (64.35,-257.39) (64.35,-256.68) (64.35,-255.97) (65.05,-255.27) (65.05,-254.56) (65.05,-253.85) (65.05,-253.14) (65.76,-252.44) (65.76,-251.73) (65.76,-251.02) (65.76,-250.32) (66.47,-249.61) (66.47,-248.90) (66.47,-248.19) (66.47,-247.49) (67.18,-246.78) (67.18,-246.07) (67.18,-245.37) (67.18,-244.66) (67.88,-243.95) (67.88,-243.24) (67.88,-242.54) (67.88,-241.83) (68.59,-241.12) (68.59,-240.42) (68.59,-239.71) (77.78,-285.67) (77.78,-284.96) (77.78,-284.26) (77.07,-283.55) (77.07,-282.84) (77.07,-282.14) (77.07,-281.43) (77.07,-280.72) (76.37,-280.01) (76.37,-279.31) (76.37,-278.60) (76.37,-277.89) (76.37,-277.19) (75.66,-276.48) (75.66,-275.77) (75.66,-275.06) (75.66,-274.36) (75.66,-273.65) (74.95,-272.94) (74.95,-272.24) (74.95,-271.53) (74.95,-270.82) (74.95,-270.11) (74.25,-269.41) (74.25,-268.70) (74.25,-267.99) (74.25,-267.29) (74.25,-266.58) (73.54,-265.87) (73.54,-265.17) (73.54,-264.46) (73.54,-263.75) (73.54,-263.04) (72.83,-262.34) (72.83,-261.63) (72.83,-260.92) (72.83,-260.22) (72.83,-259.51) (72.12,-258.80) (72.12,-258.09) (72.12,-257.39) (72.12,-256.68) (72.12,-255.97) (71.42,-255.27) (71.42,-254.56) (71.42,-253.85) (71.42,-253.14) (71.42,-252.44) (70.71,-251.73) (70.71,-251.02) (70.71,-250.32) (70.71,-249.61) (70.71,-248.90) (70.00,-248.19) (70.00,-247.49) (70.00,-246.78) (70.00,-246.07) (70.00,-245.37) (69.30,-244.66) (69.30,-243.95) (69.30,-243.24) (69.30,-242.54) (69.30,-241.83) (68.59,-241.12) (68.59,-240.42) (68.59,-239.71) (42.43,-256.68) (43.13,-257.39) (43.84,-258.09) (44.55,-258.09) (45.25,-258.80) (45.96,-259.51) (46.67,-260.22) (47.38,-260.92) (48.08,-261.63) (48.79,-261.63) (49.50,-262.34) (50.20,-263.04) (50.91,-263.75) (51.62,-264.46) (52.33,-264.46) (53.03,-265.17) (53.74,-265.87) (54.45,-266.58) (55.15,-267.29) (55.86,-267.99) (56.57,-267.99) (57.28,-268.70) (57.98,-269.41) (58.69,-270.11) (59.40,-270.82) (60.10,-270.82) (60.81,-271.53) (61.52,-272.24) (62.23,-272.94) (62.93,-273.65) (63.64,-274.36) (64.35,-274.36) (65.05,-275.06) (65.76,-275.77) (66.47,-276.48) (67.18,-277.19) (67.88,-277.89) (68.59,-277.89) (69.30,-278.60) (70.00,-279.31) (70.71,-280.01) (71.42,-280.72) (72.12,-280.72) (72.83,-281.43) (73.54,-282.14) (74.25,-282.84) (74.95,-283.55) (75.66,-284.26) (76.37,-284.26) (77.07,-284.96) (77.78,-285.67) (42.43,-256.68) (42.43,-255.97) (43.13,-255.27) (43.13,-254.56) (43.13,-253.85) (43.84,-253.14) (43.84,-252.44) (43.84,-251.73) (44.55,-251.02) (44.55,-250.32) (44.55,-249.61) (45.25,-248.90) (45.25,-248.19) (45.25,-247.49) (45.96,-246.78) (45.96,-246.07) (45.96,-245.37) (46.67,-244.66) (46.67,-243.95) (46.67,-243.24) (47.38,-242.54) (47.38,-241.83) (47.38,-241.12) (48.08,-240.42) (48.08,-239.71) (48.08,-239.00) (48.79,-238.29) (48.79,-237.59) (48.79,-236.88) (49.50,-236.17) (49.50,-235.47) (49.50,-234.76) (50.20,-234.05) (50.20,-233.35) (50.20,-232.64) (50.91,-231.93) (50.91,-231.22) (50.91,-230.52) (51.62,-229.81) (51.62,-229.10) (51.62,-228.40) (52.33,-227.69) (52.33,-226.98) (52.33,-226.27) (53.03,-225.57) (53.03,-224.86) (53.03,-224.15) (53.74,-223.45) (53.74,-222.74) (53.74,-222.03) (54.45,-221.32) (54.45,-220.62) (54.45,-219.91) (55.15,-219.20) (55.15,-218.50) (55.15,-217.79) (55.86,-217.08) (55.86,-216.37) (17.68,-235.47) (18.38,-235.47) (19.09,-234.76) (19.80,-234.76) (20.51,-234.05) (21.21,-234.05) (21.92,-233.35) (22.63,-233.35) (23.33,-232.64) (24.04,-232.64) (24.75,-231.93) (25.46,-231.93) (26.16,-231.22) (26.87,-231.22) (27.58,-230.52) (28.28,-230.52) (28.99,-229.81) (29.70,-229.81) (30.41,-229.10) (31.11,-229.10) (31.82,-228.40) (32.53,-228.40) (33.23,-227.69) (33.94,-227.69) (34.65,-226.98) (35.36,-226.98) (36.06,-226.27) (36.77,-226.27) (37.48,-225.57) (38.18,-225.57) (38.89,-224.86) (39.60,-224.86) (40.31,-224.15) (41.01,-224.15) (41.72,-223.45) (42.43,-223.45) (43.13,-222.74) (43.84,-222.74) (44.55,-222.03) (45.25,-222.03) (45.96,-221.32) (46.67,-221.32) (47.38,-220.62) (48.08,-220.62) (48.79,-219.91) (49.50,-219.91) (50.20,-219.20) (50.91,-219.20) (51.62,-218.50) (52.33,-218.50) (53.03,-217.79) (53.74,-217.79) (54.45,-217.08) (55.15,-217.08) (55.86,-216.37) (41.72,-272.94) (41.01,-272.24) (41.01,-271.53) (40.31,-270.82) (39.60,-270.11) (39.60,-269.41) (38.89,-268.70) (38.89,-267.99) (38.18,-267.29) (37.48,-266.58) (37.48,-265.87) (36.77,-265.17) (36.06,-264.46) (36.06,-263.75) (35.36,-263.04) (34.65,-262.34) (34.65,-261.63) (33.94,-260.92) (33.23,-260.22) (33.23,-259.51) (32.53,-258.80) (32.53,-258.09) (31.82,-257.39) (31.11,-256.68) (31.11,-255.97) (30.41,-255.27) (29.70,-254.56) (29.70,-253.85) (28.99,-253.14) (28.28,-252.44) (28.28,-251.73) (27.58,-251.02) (26.87,-250.32) (26.87,-249.61) (26.16,-248.90) (26.16,-248.19) (25.46,-247.49) (24.75,-246.78) (24.75,-246.07) (24.04,-245.37) (23.33,-244.66) (23.33,-243.95) (22.63,-243.24) (21.92,-242.54) (21.92,-241.83) (21.21,-241.12) (20.51,-240.42) (20.51,-239.71) (19.80,-239.00) (19.80,-238.29) (19.09,-237.59) (18.38,-236.88) (18.38,-236.17) (17.68,-235.47) ( 7.78,-301.93) ( 8.49,-301.23) ( 9.19,-300.52) ( 9.90,-299.81) (10.61,-299.81) (11.31,-299.11) (12.02,-298.40) (12.73,-297.69) (13.44,-296.98) (14.14,-296.28) (14.85,-295.57) (15.56,-295.57) (16.26,-294.86) (16.97,-294.16) (17.68,-293.45) (18.38,-292.74) (19.09,-292.04) (19.80,-291.33) (20.51,-291.33) (21.21,-290.62) (21.92,-289.91) (22.63,-289.21) (23.33,-288.50) (24.04,-287.79) (24.75,-287.79) (25.46,-287.09) (26.16,-286.38) (26.87,-285.67) (27.58,-284.96) (28.28,-284.26) (28.99,-283.55) (29.70,-283.55) (30.41,-282.84) (31.11,-282.14) (31.82,-281.43) (32.53,-280.72) (33.23,-280.01) (33.94,-279.31) (34.65,-279.31) (35.36,-278.60) (36.06,-277.89) (36.77,-277.19) (37.48,-276.48) (38.18,-275.77) (38.89,-275.06) (39.60,-275.06) (40.31,-274.36) (41.01,-273.65) (41.72,-272.94) ( 7.78,-301.93) ( 8.49,-301.93) ( 9.19,-301.93) ( 9.90,-301.93) (10.61,-301.93) (11.31,-301.93) (12.02,-301.93) (12.73,-301.23) (13.44,-301.23) (14.14,-301.23) (14.85,-301.23) (15.56,-301.23) (16.26,-301.23) (16.97,-301.23) (17.68,-301.23) (18.38,-301.23) (19.09,-301.23) (19.80,-301.23) (20.51,-301.23) (21.21,-300.52) (21.92,-300.52) (22.63,-300.52) (23.33,-300.52) (24.04,-300.52) (24.75,-300.52) (25.46,-300.52) (26.16,-300.52) (26.87,-300.52) (27.58,-300.52) (28.28,-300.52) (28.99,-300.52) (29.70,-299.81) (30.41,-299.81) (31.11,-299.81) (31.82,-299.81) (32.53,-299.81) (33.23,-299.81) (33.94,-299.81) (34.65,-299.81) (35.36,-299.81) (36.06,-299.81) (36.77,-299.81) (37.48,-299.81) (38.18,-299.11) (38.89,-299.11) (39.60,-299.11) (40.31,-299.11) (41.01,-299.11) (41.72,-299.11) (42.43,-299.11) (43.13,-299.11) (43.84,-299.11) (44.55,-299.11) (45.25,-299.11) (45.96,-299.11) (46.67,-298.40) (47.38,-298.40) (48.08,-298.40) (48.79,-298.40) (49.50,-298.40) (50.20,-298.40) (50.91,-298.40) (50.91,-298.40) (51.62,-298.40) (52.33,-298.40) (53.03,-298.40) (53.74,-298.40) (54.45,-298.40) (55.15,-297.69) (55.86,-297.69) (56.57,-297.69) (57.28,-297.69) (57.98,-297.69) (58.69,-297.69) (59.40,-297.69) (60.10,-297.69) (60.81,-297.69) (61.52,-297.69) (62.23,-297.69) (62.93,-297.69) (63.64,-296.98) (64.35,-296.98) (65.05,-296.98) (65.76,-296.98) (66.47,-296.98) (67.18,-296.98) (67.88,-296.98) (68.59,-296.98) (69.30,-296.98) (70.00,-296.98) (70.71,-296.98) (71.42,-296.98) (72.12,-296.28) (72.83,-296.28) (73.54,-296.28) (74.25,-296.28) (74.95,-296.28) (75.66,-296.28) (76.37,-296.28) (77.07,-296.28) (77.78,-296.28) (78.49,-296.28) (79.20,-296.28) (79.90,-296.28) (80.61,-295.57) (81.32,-295.57) (82.02,-295.57) (82.73,-295.57) (83.44,-295.57) (84.15,-295.57) (84.85,-295.57) (85.56,-295.57) (86.27,-295.57) (86.97,-295.57) (87.68,-295.57) (88.39,-295.57) (89.10,-294.86) (89.80,-294.86) (90.51,-294.86) (91.22,-294.86) (91.92,-294.86) (92.63,-294.86) (92.63,-294.86) (93.34,-294.86) (94.05,-294.16) (94.75,-294.16) (95.46,-294.16) (96.17,-293.45) (96.87,-293.45) (97.58,-292.74) (98.29,-292.74) (98.99,-292.74) (99.70,-292.04) (100.41,-292.04) (101.12,-292.04) (101.82,-291.33) (102.53,-291.33) (103.24,-291.33) (103.94,-290.62) (104.65,-290.62) (105.36,-290.62) (106.07,-289.91) (106.77,-289.91) (107.48,-289.21) (108.19,-289.21) (108.89,-289.21) (109.60,-288.50) (110.31,-288.50) (111.02,-288.50) (111.72,-287.79) (112.43,-287.79) (113.14,-287.79) (113.84,-287.09) (114.55,-287.09) (115.26,-286.38) (115.97,-286.38) (116.67,-286.38) (117.38,-285.67) (118.09,-285.67) (118.79,-285.67) (119.50,-284.96) (120.21,-284.96) (120.92,-284.96) (121.62,-284.26) (122.33,-284.26) (123.04,-283.55) (123.74,-283.55) (124.45,-283.55) (125.16,-282.84) (125.87,-282.84) (126.57,-282.84) (127.28,-282.14) (127.99,-282.14) (128.69,-282.14) (129.40,-281.43) (130.11,-281.43) (130.81,-281.43) (131.52,-280.72) (132.23,-280.72) (132.94,-280.01) (133.64,-280.01) (134.35,-280.01) (135.06,-279.31) (135.76,-279.31) (135.76,-279.31) (135.76,-278.60) (135.76,-277.89) (135.76,-277.19) (135.76,-276.48) (135.76,-275.77) (135.76,-275.06) (135.76,-274.36) (135.76,-273.65) (135.76,-272.94) (135.76,-272.24) (135.76,-271.53) (135.06,-270.82) (135.06,-270.11) (135.06,-269.41) (135.06,-268.70) (135.06,-267.99) (135.06,-267.29) (135.06,-266.58) (135.06,-265.87) (135.06,-265.17) (135.06,-264.46) (135.06,-263.75) (135.06,-263.04) (135.06,-262.34) (135.06,-261.63) (135.06,-260.92) (135.06,-260.22) (135.06,-259.51) (135.06,-258.80) (135.06,-258.09) (135.06,-257.39) (135.06,-256.68) (135.06,-255.97) (134.35,-255.27) (134.35,-254.56) (134.35,-253.85) (134.35,-253.14) (134.35,-252.44) (134.35,-251.73) (134.35,-251.02) (134.35,-250.32) (134.35,-249.61) (134.35,-248.90) (134.35,-248.19) (134.35,-247.49) (134.35,-246.78) (134.35,-246.07) (134.35,-245.37) (134.35,-244.66) (134.35,-243.95) (134.35,-243.24) (134.35,-242.54) (134.35,-241.83) (134.35,-241.12) (134.35,-240.42) (133.64,-239.71) (133.64,-239.00) (133.64,-238.29) (133.64,-237.59) (133.64,-236.88) (133.64,-236.17) (133.64,-235.47) (133.64,-234.76) (133.64,-234.05) (133.64,-233.35) (133.64,-232.64) (132.94,-276.48) (132.94,-275.77) (132.94,-275.06) (132.94,-274.36) (132.94,-273.65) (132.94,-272.94) (132.94,-272.24) (132.94,-271.53) (132.94,-270.82) (132.94,-270.11) (132.94,-269.41) (132.94,-268.70) (132.94,-267.99) (132.94,-267.29) (132.94,-266.58) (132.94,-265.87) (132.94,-265.17) (132.94,-264.46) (132.94,-263.75) (132.94,-263.04) (132.94,-262.34) (132.94,-261.63) (132.94,-260.92) (132.94,-260.22) (132.94,-259.51) (132.94,-258.80) (132.94,-258.09) (132.94,-257.39) (132.94,-256.68) (132.94,-255.97) (132.94,-255.27) (132.94,-254.56) (133.64,-253.85) (133.64,-253.14) (133.64,-252.44) (133.64,-251.73) (133.64,-251.02) (133.64,-250.32) (133.64,-249.61) (133.64,-248.90) (133.64,-248.19) (133.64,-247.49) (133.64,-246.78) (133.64,-246.07) (133.64,-245.37) (133.64,-244.66) (133.64,-243.95) (133.64,-243.24) (133.64,-242.54) (133.64,-241.83) (133.64,-241.12) (133.64,-240.42) (133.64,-239.71) (133.64,-239.00) (133.64,-238.29) (133.64,-237.59) (133.64,-236.88) (133.64,-236.17) (133.64,-235.47) (133.64,-234.76) (133.64,-234.05) (133.64,-233.35) (133.64,-232.64) (86.97,-274.36) (87.68,-274.36) (88.39,-274.36) (89.10,-274.36) (89.80,-274.36) (90.51,-274.36) (91.22,-274.36) (91.92,-274.36) (92.63,-274.36) (93.34,-274.36) (94.05,-274.36) (94.75,-275.06) (95.46,-275.06) (96.17,-275.06) (96.87,-275.06) (97.58,-275.06) (98.29,-275.06) (98.99,-275.06) (99.70,-275.06) (100.41,-275.06) (101.12,-275.06) (101.82,-275.06) (102.53,-275.06) (103.24,-275.06) (103.94,-275.06) (104.65,-275.06) (105.36,-275.06) (106.07,-275.06) (106.77,-275.06) (107.48,-275.06) (108.19,-275.06) (108.89,-275.06) (109.60,-275.06) (110.31,-275.77) (111.02,-275.77) (111.72,-275.77) (112.43,-275.77) (113.14,-275.77) (113.84,-275.77) (114.55,-275.77) (115.26,-275.77) (115.97,-275.77) (116.67,-275.77) (117.38,-275.77) (118.09,-275.77) (118.79,-275.77) (119.50,-275.77) (120.21,-275.77) (120.92,-275.77) (121.62,-275.77) (122.33,-275.77) (123.04,-275.77) (123.74,-275.77) (124.45,-275.77) (125.16,-275.77) (125.87,-276.48) (126.57,-276.48) (127.28,-276.48) (127.99,-276.48) (128.69,-276.48) (129.40,-276.48) (130.11,-276.48) (130.81,-276.48) (131.52,-276.48) (132.23,-276.48) (132.94,-276.48) (43.84,-263.75) (44.55,-263.75) (45.25,-263.75) (45.96,-264.46) (46.67,-264.46) (47.38,-264.46) (48.08,-264.46) (48.79,-265.17) (49.50,-265.17) (50.20,-265.17) (50.91,-265.17) (51.62,-265.87) (52.33,-265.87) (53.03,-265.87) (53.74,-265.87) (54.45,-266.58) (55.15,-266.58) (55.86,-266.58) (56.57,-266.58) (57.28,-267.29) (57.98,-267.29) (58.69,-267.29) (59.40,-267.29) (60.10,-267.99) (60.81,-267.99) (61.52,-267.99) (62.23,-267.99) (62.93,-268.70) (63.64,-268.70) (64.35,-268.70) (65.05,-268.70) (65.76,-269.41) (66.47,-269.41) (67.18,-269.41) (67.88,-269.41) (68.59,-270.11) (69.30,-270.11) (70.00,-270.11) (70.71,-270.11) (71.42,-270.82) (72.12,-270.82) (72.83,-270.82) (73.54,-270.82) (74.25,-271.53) (74.95,-271.53) (75.66,-271.53) (76.37,-271.53) (77.07,-272.24) (77.78,-272.24) (78.49,-272.24) (79.20,-272.24) (79.90,-272.94) (80.61,-272.94) (81.32,-272.94) (82.02,-272.94) (82.73,-273.65) (83.44,-273.65) (84.15,-273.65) (84.85,-273.65) (85.56,-274.36) (86.27,-274.36) (86.97,-274.36) ( 9.90,-229.81) (10.61,-230.52) (11.31,-231.22) (12.02,-231.93) (12.73,-232.64) (13.44,-233.35) (14.14,-234.05) (14.85,-234.76) (15.56,-235.47) (16.26,-236.17) (16.97,-236.88) (17.68,-237.59) (18.38,-238.29) (19.09,-239.00) (19.80,-239.71) (20.51,-240.42) (21.21,-241.12) (21.92,-241.83) (22.63,-242.54) (23.33,-243.24) (24.04,-243.95) (24.75,-244.66) (25.46,-245.37) (26.16,-246.07) (26.87,-246.78) (27.58,-247.49) (28.28,-248.19) (28.99,-248.90) (29.70,-249.61) (30.41,-250.32) (31.11,-251.02) (31.82,-251.73) (32.53,-252.44) (33.23,-253.14) (33.94,-253.85) (34.65,-254.56) (35.36,-255.27) (36.06,-255.97) (36.77,-256.68) (37.48,-257.39) (38.18,-258.09) (38.89,-258.80) (39.60,-259.51) (40.31,-260.22) (41.01,-260.92) (41.72,-261.63) (42.43,-262.34) (43.13,-263.04) (43.84,-263.75) ( 9.90,-229.81) (10.61,-229.10) (10.61,-228.40) (11.31,-227.69) (12.02,-226.98) (12.73,-226.27) (12.73,-225.57) (13.44,-224.86) (14.14,-224.15) (14.85,-223.45) (14.85,-222.74) (15.56,-222.03) (16.26,-221.32) (16.97,-220.62) (16.97,-219.91) (17.68,-219.20) (18.38,-218.50) (19.09,-217.79) (19.09,-217.08) (19.80,-216.37) (20.51,-215.67) (20.51,-214.96) (21.21,-214.25) (21.92,-213.55) (22.63,-212.84) (22.63,-212.13) (23.33,-211.42) (24.04,-210.72) (24.75,-210.01) (24.75,-209.30) (25.46,-208.60) (26.16,-207.89) (26.87,-207.18) (26.87,-206.48) (27.58,-205.77) (28.28,-205.06) (28.28,-204.35) (28.99,-203.65) (29.70,-202.94) (30.41,-202.23) (30.41,-201.53) (31.11,-200.82) (31.82,-200.11) (32.53,-199.40) (32.53,-198.70) (33.23,-197.99) (33.94,-197.28) (34.65,-196.58) (34.65,-195.87) (35.36,-195.16) (36.06,-194.45) (36.77,-193.75) (36.77,-193.04) (37.48,-192.33) (-2.83,-178.90) (-2.12,-178.90) (-1.41,-179.61) (-0.71,-179.61) ( 0.00,-179.61) ( 0.71,-180.31) ( 1.41,-180.31) ( 2.12,-180.31) ( 2.83,-181.02) ( 3.54,-181.02) ( 4.24,-181.02) ( 4.95,-181.73) ( 5.66,-181.73) ( 6.36,-181.73) ( 7.07,-182.43) ( 7.78,-182.43) ( 8.49,-182.43) ( 9.19,-183.14) ( 9.90,-183.14) (10.61,-183.14) (11.31,-183.85) (12.02,-183.85) (12.73,-183.85) (13.44,-184.55) (14.14,-184.55) (14.85,-184.55) (15.56,-185.26) (16.26,-185.26) (16.97,-185.26) (17.68,-185.97) (18.38,-185.97) (19.09,-185.97) (19.80,-186.68) (20.51,-186.68) (21.21,-186.68) (21.92,-187.38) (22.63,-187.38) (23.33,-187.38) (24.04,-188.09) (24.75,-188.09) (25.46,-188.09) (26.16,-188.80) (26.87,-188.80) (27.58,-188.80) (28.28,-189.50) (28.99,-189.50) (29.70,-189.50) (30.41,-190.21) (31.11,-190.21) (31.82,-190.21) (32.53,-190.92) (33.23,-190.92) (33.94,-190.92) (34.65,-191.63) (35.36,-191.63) (36.06,-191.63) (36.77,-192.33) (37.48,-192.33) (-48.79,-169.71) (-48.08,-169.71) (-47.38,-169.71) (-46.67,-170.41) (-45.96,-170.41) (-45.25,-170.41) (-44.55,-170.41) (-43.84,-170.41) (-43.13,-171.12) (-42.43,-171.12) (-41.72,-171.12) (-41.01,-171.12) (-40.31,-171.12) (-39.60,-171.83) (-38.89,-171.83) (-38.18,-171.83) (-37.48,-171.83) (-36.77,-171.83) (-36.06,-172.53) (-35.36,-172.53) (-34.65,-172.53) (-33.94,-172.53) (-33.23,-172.53) (-32.53,-173.24) (-31.82,-173.24) (-31.11,-173.24) (-30.41,-173.24) (-29.70,-173.24) (-28.99,-173.95) (-28.28,-173.95) (-27.58,-173.95) (-26.87,-173.95) (-26.16,-173.95) (-25.46,-174.66) (-24.75,-174.66) (-24.04,-174.66) (-23.33,-174.66) (-22.63,-174.66) (-21.92,-175.36) (-21.21,-175.36) (-20.51,-175.36) (-19.80,-175.36) (-19.09,-175.36) (-18.38,-176.07) (-17.68,-176.07) (-16.97,-176.07) (-16.26,-176.07) (-15.56,-176.07) (-14.85,-176.78) (-14.14,-176.78) (-13.44,-176.78) (-12.73,-176.78) (-12.02,-176.78) (-11.31,-177.48) (-10.61,-177.48) (-9.90,-177.48) (-9.19,-177.48) (-8.49,-177.48) (-7.78,-178.19) (-7.07,-178.19) (-6.36,-178.19) (-5.66,-178.19) (-4.95,-178.19) (-4.24,-178.90) (-3.54,-178.90) (-2.83,-178.90) (-91.92,-155.56) (-91.22,-155.56) (-90.51,-156.27) (-89.80,-156.27) (-89.10,-156.27) (-88.39,-156.98) (-87.68,-156.98) (-86.97,-156.98) (-86.27,-157.68) (-85.56,-157.68) (-84.85,-157.68) (-84.15,-158.39) (-83.44,-158.39) (-82.73,-158.39) (-82.02,-159.10) (-81.32,-159.10) (-80.61,-159.10) (-79.90,-159.81) (-79.20,-159.81) (-78.49,-159.81) (-77.78,-160.51) (-77.07,-160.51) (-76.37,-160.51) (-75.66,-161.22) (-74.95,-161.22) (-74.25,-161.22) (-73.54,-161.93) (-72.83,-161.93) (-72.12,-161.93) (-71.42,-162.63) (-70.71,-162.63) (-70.00,-162.63) (-69.30,-162.63) (-68.59,-163.34) (-67.88,-163.34) (-67.18,-163.34) (-66.47,-164.05) (-65.76,-164.05) (-65.05,-164.05) (-64.35,-164.76) (-63.64,-164.76) (-62.93,-164.76) (-62.23,-165.46) (-61.52,-165.46) (-60.81,-165.46) (-60.10,-166.17) (-59.40,-166.17) (-58.69,-166.17) (-57.98,-166.88) (-57.28,-166.88) (-56.57,-166.88) (-55.86,-167.58) (-55.15,-167.58) (-54.45,-167.58) (-53.74,-168.29) (-53.03,-168.29) (-52.33,-168.29) (-51.62,-169.00) (-50.91,-169.00) (-50.20,-169.00) (-49.50,-169.71) (-48.79,-169.71) (-91.92,-155.56) (-92.63,-154.86) (-92.63,-154.15) (-93.34,-153.44) (-94.05,-152.74) (-94.05,-152.03) (-94.75,-151.32) (-95.46,-150.61) (-96.17,-149.91) (-96.17,-149.20) (-96.87,-148.49) (-97.58,-147.79) (-97.58,-147.08) (-98.29,-146.37) (-98.99,-145.66) (-98.99,-144.96) (-99.70,-144.25) (-100.41,-143.54) (-100.41,-142.84) (-101.12,-142.13) (-101.82,-141.42) (-102.53,-140.71) (-102.53,-140.01) (-103.24,-139.30) (-103.94,-138.59) (-103.94,-137.89) (-104.65,-137.18) (-105.36,-136.47) (-105.36,-135.76) (-106.07,-135.06) (-106.77,-134.35) (-106.77,-133.64) (-107.48,-132.94) (-108.19,-132.23) (-108.89,-131.52) (-108.89,-130.81) (-109.60,-130.11) (-110.31,-129.40) (-110.31,-128.69) (-111.02,-127.99) (-111.72,-127.28) (-111.72,-126.57) (-112.43,-125.87) (-113.14,-125.16) (-113.14,-124.45) (-113.84,-123.74) (-114.55,-123.04) (-115.26,-122.33) (-115.26,-121.62) (-115.97,-120.92) (-116.67,-120.21) (-116.67,-119.50) (-117.38,-118.79) (-117.38,-118.79) (-116.67,-118.79) (-115.97,-118.09) (-115.26,-118.09) (-114.55,-117.38) (-113.84,-117.38) (-113.14,-117.38) (-112.43,-116.67) (-111.72,-116.67) (-111.02,-116.67) (-110.31,-115.97) (-109.60,-115.97) (-108.89,-115.26) (-108.19,-115.26) (-107.48,-115.26) (-106.77,-114.55) (-106.07,-114.55) (-105.36,-113.84) (-104.65,-113.84) (-103.94,-113.84) (-103.24,-113.14) (-102.53,-113.14) (-101.82,-113.14) (-101.12,-112.43) (-100.41,-112.43) (-99.70,-111.72) (-98.99,-111.72) (-98.29,-111.72) (-97.58,-111.02) (-96.87,-111.02) (-96.17,-111.02) (-95.46,-110.31) (-94.75,-110.31) (-94.05,-109.60) (-93.34,-109.60) (-92.63,-109.60) (-91.92,-108.89) (-91.22,-108.89) (-90.51,-108.19) (-89.80,-108.19) (-89.10,-108.19) (-88.39,-107.48) (-87.68,-107.48) (-86.97,-107.48) (-86.27,-106.77) (-85.56,-106.77) (-84.85,-106.07) (-84.15,-106.07) (-83.44,-106.07) (-82.73,-105.36) (-82.02,-105.36) (-81.32,-104.65) (-80.61,-104.65) (-79.90,-104.65) (-79.20,-103.94) (-78.49,-103.94) (-77.78,-103.94) (-77.07,-103.24) (-76.37,-103.24) (-75.66,-102.53) (-74.95,-102.53) (-74.95,-102.53) (-74.25,-101.82) (-74.25,-101.12) (-73.54,-100.41) (-73.54,-99.70) (-72.83,-98.99) (-72.83,-98.29) (-72.12,-97.58) (-71.42,-96.87) (-71.42,-96.17) (-70.71,-95.46) (-70.71,-94.75) (-70.00,-94.05) (-70.00,-93.34) (-69.30,-92.63) (-68.59,-91.92) (-68.59,-91.22) (-67.88,-90.51) (-67.88,-89.80) (-67.18,-89.10) (-67.18,-88.39) (-66.47,-87.68) (-65.76,-86.97) (-65.76,-86.27) (-65.05,-85.56) (-65.05,-84.85) (-64.35,-84.15) (-64.35,-83.44) (-63.64,-82.73) (-62.93,-82.02) (-62.93,-81.32) (-62.23,-80.61) (-62.23,-79.90) (-61.52,-79.20) (-61.52,-78.49) (-60.81,-77.78) (-60.10,-77.07) (-60.10,-76.37) (-59.40,-75.66) (-59.40,-74.95) (-58.69,-74.25) (-58.69,-73.54) (-57.98,-72.83) (-57.28,-72.12) (-57.28,-71.42) (-56.57,-70.71) (-56.57,-70.00) (-55.86,-69.30) (-55.86,-68.59) (-55.15,-67.88) (-54.45,-67.18) (-54.45,-66.47) (-53.74,-65.76) (-53.74,-65.05) (-53.03,-64.35) (-53.03,-63.64) (-52.33,-62.93) (-88.39,-84.85) (-87.68,-84.15) (-86.97,-84.15) (-86.27,-83.44) (-85.56,-83.44) (-84.85,-82.73) (-84.15,-82.02) (-83.44,-82.02) (-82.73,-81.32) (-82.02,-81.32) (-81.32,-80.61) (-80.61,-79.90) (-79.90,-79.90) (-79.20,-79.20) (-78.49,-78.49) (-77.78,-78.49) (-77.07,-77.78) (-76.37,-77.78) (-75.66,-77.07) (-74.95,-76.37) (-74.25,-76.37) (-73.54,-75.66) (-72.83,-75.66) (-72.12,-74.95) (-71.42,-74.25) (-70.71,-74.25) (-70.00,-73.54) (-69.30,-73.54) (-68.59,-72.83) (-67.88,-72.12) (-67.18,-72.12) (-66.47,-71.42) (-65.76,-71.42) (-65.05,-70.71) (-64.35,-70.00) (-63.64,-70.00) (-62.93,-69.30) (-62.23,-69.30) (-61.52,-68.59) (-60.81,-67.88) (-60.10,-67.88) (-59.40,-67.18) (-58.69,-66.47) (-57.98,-66.47) (-57.28,-65.76) (-56.57,-65.76) (-55.86,-65.05) (-55.15,-64.35) (-54.45,-64.35) (-53.74,-63.64) (-53.03,-63.64) (-52.33,-62.93) (-67.88,-121.62) (-68.59,-120.92) (-68.59,-120.21) (-69.30,-119.50) (-69.30,-118.79) (-70.00,-118.09) (-70.00,-117.38) (-70.71,-116.67) (-70.71,-115.97) (-71.42,-115.26) (-72.12,-114.55) (-72.12,-113.84) (-72.83,-113.14) (-72.83,-112.43) (-73.54,-111.72) (-73.54,-111.02) (-74.25,-110.31) (-74.25,-109.60) (-74.95,-108.89) (-75.66,-108.19) (-75.66,-107.48) (-76.37,-106.77) (-76.37,-106.07) (-77.07,-105.36) (-77.07,-104.65) (-77.78,-103.94) (-77.78,-103.24) (-78.49,-102.53) (-79.20,-101.82) (-79.20,-101.12) (-79.90,-100.41) (-79.90,-99.70) (-80.61,-98.99) (-80.61,-98.29) (-81.32,-97.58) (-82.02,-96.87) (-82.02,-96.17) (-82.73,-95.46) (-82.73,-94.75) (-83.44,-94.05) (-83.44,-93.34) (-84.15,-92.63) (-84.15,-91.92) (-84.85,-91.22) (-85.56,-90.51) (-85.56,-89.80) (-86.27,-89.10) (-86.27,-88.39) (-86.97,-87.68) (-86.97,-86.97) (-87.68,-86.27) (-87.68,-85.56) (-88.39,-84.85) (-91.92,-159.81) (-91.22,-159.10) (-91.22,-158.39) (-90.51,-157.68) (-89.80,-156.98) (-89.80,-156.27) (-89.10,-155.56) (-89.10,-154.86) (-88.39,-154.15) (-87.68,-153.44) (-87.68,-152.74) (-86.97,-152.03) (-86.27,-151.32) (-86.27,-150.61) (-85.56,-149.91) (-85.56,-149.20) (-84.85,-148.49) (-84.15,-147.79) (-84.15,-147.08) (-83.44,-146.37) (-82.73,-145.66) (-82.73,-144.96) (-82.02,-144.25) (-82.02,-143.54) (-81.32,-142.84) (-80.61,-142.13) (-80.61,-141.42) (-79.90,-140.71) (-79.20,-140.01) (-79.20,-139.30) (-78.49,-138.59) (-77.78,-137.89) (-77.78,-137.18) (-77.07,-136.47) (-77.07,-135.76) (-76.37,-135.06) (-75.66,-134.35) (-75.66,-133.64) (-74.95,-132.94) (-74.25,-132.23) (-74.25,-131.52) (-73.54,-130.81) (-73.54,-130.11) (-72.83,-129.40) (-72.12,-128.69) (-72.12,-127.99) (-71.42,-127.28) (-70.71,-126.57) (-70.71,-125.87) (-70.00,-125.16) (-70.00,-124.45) (-69.30,-123.74) (-68.59,-123.04) (-68.59,-122.33) (-67.88,-121.62) (-91.92,-159.81) (-91.22,-159.81) (-90.51,-159.81) (-89.80,-159.81) (-89.10,-159.10) (-88.39,-159.10) (-87.68,-159.10) (-86.97,-159.10) (-86.27,-159.10) (-85.56,-159.10) (-84.85,-159.10) (-84.15,-158.39) (-83.44,-158.39) (-82.73,-158.39) (-82.02,-158.39) (-81.32,-158.39) (-80.61,-158.39) (-79.90,-158.39) (-79.20,-157.68) (-78.49,-157.68) (-77.78,-157.68) (-77.07,-157.68) (-76.37,-157.68) (-75.66,-157.68) (-74.95,-157.68) (-74.25,-156.98) (-73.54,-156.98) (-72.83,-156.98) (-72.12,-156.98) (-71.42,-156.98) (-70.71,-156.98) (-70.00,-156.98) (-69.30,-156.27) (-68.59,-156.27) (-67.88,-156.27) (-67.18,-156.27) (-66.47,-156.27) (-65.76,-156.27) (-65.05,-156.27) (-64.35,-155.56) (-63.64,-155.56) (-62.93,-155.56) (-62.23,-155.56) (-61.52,-155.56) (-60.81,-155.56) (-60.10,-155.56) (-59.40,-154.86) (-58.69,-154.86) (-57.98,-154.86) (-57.28,-154.86) (-56.57,-154.86) (-55.86,-154.86) (-55.15,-154.86) (-54.45,-154.15) (-53.74,-154.15) (-53.03,-154.15) (-52.33,-154.15) (-51.62,-154.15) (-50.91,-154.15) (-50.20,-154.15) (-49.50,-153.44) (-48.79,-153.44) (-48.08,-153.44) (-47.38,-153.44) (-47.38,-153.44) (-46.67,-153.44) (-45.96,-152.74) (-45.25,-152.74) (-44.55,-152.03) (-43.84,-152.03) (-43.13,-151.32) (-42.43,-151.32) (-41.72,-150.61) (-41.01,-150.61) (-40.31,-149.91) (-39.60,-149.91) (-38.89,-149.20) (-38.18,-149.20) (-37.48,-148.49) (-36.77,-148.49) (-36.06,-148.49) (-35.36,-147.79) (-34.65,-147.79) (-33.94,-147.08) (-33.23,-147.08) (-32.53,-146.37) (-31.82,-146.37) (-31.11,-145.66) (-30.41,-145.66) (-29.70,-144.96) (-28.99,-144.96) (-28.28,-144.25) (-27.58,-144.25) (-26.87,-143.54) (-26.16,-143.54) (-25.46,-143.54) (-24.75,-142.84) (-24.04,-142.84) (-23.33,-142.13) (-22.63,-142.13) (-21.92,-141.42) (-21.21,-141.42) (-20.51,-140.71) (-19.80,-140.71) (-19.09,-140.01) (-18.38,-140.01) (-17.68,-139.30) (-16.97,-139.30) (-16.26,-138.59) (-15.56,-138.59) (-14.85,-138.59) (-14.14,-137.89) (-13.44,-137.89) (-12.73,-137.18) (-12.02,-137.18) (-11.31,-136.47) (-10.61,-136.47) (-9.90,-135.76) (-9.19,-135.76) (-8.49,-135.06) (-7.78,-135.06) (-7.07,-134.35) (-6.36,-134.35) (-5.66,-133.64) (-4.95,-133.64) (-4.95,-133.64) (-4.24,-133.64) (-3.54,-132.94) (-2.83,-132.94) (-2.12,-132.23) (-1.41,-132.23) (-0.71,-131.52) ( 0.00,-131.52) ( 0.71,-130.81) ( 1.41,-130.81) ( 2.12,-130.81) ( 2.83,-130.11) ( 3.54,-130.11) ( 4.24,-129.40) ( 4.95,-129.40) ( 5.66,-128.69) ( 6.36,-128.69) ( 7.07,-127.99) ( 7.78,-127.99) ( 8.49,-127.99) ( 9.19,-127.28) ( 9.90,-127.28) (10.61,-126.57) (11.31,-126.57) (12.02,-125.87) (12.73,-125.87) (13.44,-125.16) (14.14,-125.16) (14.85,-125.16) (15.56,-124.45) (16.26,-124.45) (16.97,-123.74) (17.68,-123.74) (18.38,-123.04) (19.09,-123.04) (19.80,-122.33) (20.51,-122.33) (21.21,-121.62) (21.92,-121.62) (22.63,-121.62) (23.33,-120.92) (24.04,-120.92) (24.75,-120.21) (25.46,-120.21) (26.16,-119.50) (26.87,-119.50) (27.58,-118.79) (28.28,-118.79) (28.99,-118.79) (29.70,-118.09) (30.41,-118.09) (31.11,-117.38) (31.82,-117.38) (32.53,-116.67) (33.23,-116.67) (33.94,-115.97) (34.65,-115.97) (34.65,-115.97) (34.65,-115.26) (35.36,-114.55) (35.36,-113.84) (36.06,-113.14) (36.06,-112.43) (36.06,-111.72) (36.77,-111.02) (36.77,-110.31) (37.48,-109.60) (37.48,-108.89) (37.48,-108.19) (38.18,-107.48) (38.18,-106.77) (38.89,-106.07) (38.89,-105.36) (38.89,-104.65) (39.60,-103.94) (39.60,-103.24) (39.60,-102.53) (40.31,-101.82) (40.31,-101.12) (41.01,-100.41) (41.01,-99.70) (41.01,-98.99) (41.72,-98.29) (41.72,-97.58) (42.43,-96.87) (42.43,-96.17) (42.43,-95.46) (43.13,-94.75) (43.13,-94.05) (43.84,-93.34) (43.84,-92.63) (43.84,-91.92) (44.55,-91.22) (44.55,-90.51) (45.25,-89.80) (45.25,-89.10) (45.25,-88.39) (45.96,-87.68) (45.96,-86.97) (46.67,-86.27) (46.67,-85.56) (46.67,-84.85) (47.38,-84.15) (47.38,-83.44) (47.38,-82.73) (48.08,-82.02) (48.08,-81.32) (48.79,-80.61) (48.79,-79.90) (48.79,-79.20) (49.50,-78.49) (49.50,-77.78) (50.20,-77.07) (50.20,-76.37) (50.20,-75.66) (50.91,-74.95) (50.91,-74.25) (51.62,-73.54) (51.62,-72.83) ( 6.36,-68.59) ( 7.07,-68.59) ( 7.78,-68.59) ( 8.49,-68.59) ( 9.19,-68.59) ( 9.90,-68.59) (10.61,-69.30) (11.31,-69.30) (12.02,-69.30) (12.73,-69.30) (13.44,-69.30) (14.14,-69.30) (14.85,-69.30) (15.56,-69.30) (16.26,-69.30) (16.97,-69.30) (17.68,-69.30) (18.38,-70.00) (19.09,-70.00) (19.80,-70.00) (20.51,-70.00) (21.21,-70.00) (21.92,-70.00) (22.63,-70.00) (23.33,-70.00) (24.04,-70.00) (24.75,-70.00) (25.46,-70.71) (26.16,-70.71) (26.87,-70.71) (27.58,-70.71) (28.28,-70.71) (28.99,-70.71) (29.70,-70.71) (30.41,-70.71) (31.11,-70.71) (31.82,-70.71) (32.53,-70.71) (33.23,-71.42) (33.94,-71.42) (34.65,-71.42) (35.36,-71.42) (36.06,-71.42) (36.77,-71.42) (37.48,-71.42) (38.18,-71.42) (38.89,-71.42) (39.60,-71.42) (40.31,-71.42) (41.01,-72.12) (41.72,-72.12) (42.43,-72.12) (43.13,-72.12) (43.84,-72.12) (44.55,-72.12) (45.25,-72.12) (45.96,-72.12) (46.67,-72.12) (47.38,-72.12) (48.08,-72.83) (48.79,-72.83) (49.50,-72.83) (50.20,-72.83) (50.91,-72.83) (51.62,-72.83) (-32.53,-89.80) (-31.82,-89.10) (-31.11,-89.10) (-30.41,-88.39) (-29.70,-88.39) (-28.99,-87.68) (-28.28,-87.68) (-27.58,-86.97) (-26.87,-86.97) (-26.16,-86.27) (-25.46,-86.27) (-24.75,-85.56) (-24.04,-84.85) (-23.33,-84.85) (-22.63,-84.15) (-21.92,-84.15) (-21.21,-83.44) (-20.51,-83.44) (-19.80,-82.73) (-19.09,-82.73) (-18.38,-82.02) (-17.68,-82.02) (-16.97,-81.32) (-16.26,-80.61) (-15.56,-80.61) (-14.85,-79.90) (-14.14,-79.90) (-13.44,-79.20) (-12.73,-79.20) (-12.02,-78.49) (-11.31,-78.49) (-10.61,-77.78) (-9.90,-77.78) (-9.19,-77.07) (-8.49,-76.37) (-7.78,-76.37) (-7.07,-75.66) (-6.36,-75.66) (-5.66,-74.95) (-4.95,-74.95) (-4.24,-74.25) (-3.54,-74.25) (-2.83,-73.54) (-2.12,-73.54) (-1.41,-72.83) (-0.71,-72.12) ( 0.00,-72.12) ( 0.71,-71.42) ( 1.41,-71.42) ( 2.12,-70.71) ( 2.83,-70.71) ( 3.54,-70.00) ( 4.24,-70.00) ( 4.95,-69.30) ( 5.66,-69.30) ( 6.36,-68.59) (-1.41,-122.33) (-2.12,-121.62) (-2.83,-120.92) (-3.54,-120.21) (-4.24,-119.50) (-4.95,-118.79) (-5.66,-118.09) (-6.36,-117.38) (-7.07,-116.67) (-7.78,-115.97) (-8.49,-115.26) (-9.19,-114.55) (-9.19,-113.84) (-9.90,-113.14) (-10.61,-112.43) (-11.31,-111.72) (-12.02,-111.02) (-12.73,-110.31) (-13.44,-109.60) (-14.14,-108.89) (-14.85,-108.19) (-15.56,-107.48) (-16.26,-106.77) (-16.97,-106.07) (-17.68,-105.36) (-18.38,-104.65) (-19.09,-103.94) (-19.80,-103.24) (-20.51,-102.53) (-21.21,-101.82) (-21.92,-101.12) (-22.63,-100.41) (-23.33,-99.70) (-24.04,-98.99) (-24.75,-98.29) (-24.75,-97.58) (-25.46,-96.87) (-26.16,-96.17) (-26.87,-95.46) (-27.58,-94.75) (-28.28,-94.05) (-28.99,-93.34) (-29.70,-92.63) (-30.41,-91.92) (-31.11,-91.22) (-31.82,-90.51) (-32.53,-89.80) (-36.77,-147.79) (-36.06,-147.08) (-35.36,-147.08) (-34.65,-146.37) (-33.94,-145.66) (-33.23,-144.96) (-32.53,-144.96) (-31.82,-144.25) (-31.11,-143.54) (-30.41,-143.54) (-29.70,-142.84) (-28.99,-142.13) (-28.28,-141.42) (-27.58,-141.42) (-26.87,-140.71) (-26.16,-140.01) (-25.46,-139.30) (-24.75,-139.30) (-24.04,-138.59) (-23.33,-137.89) (-22.63,-137.89) (-21.92,-137.18) (-21.21,-136.47) (-20.51,-135.76) (-19.80,-135.76) (-19.09,-135.06) (-18.38,-134.35) (-17.68,-134.35) (-16.97,-133.64) (-16.26,-132.94) (-15.56,-132.23) (-14.85,-132.23) (-14.14,-131.52) (-13.44,-130.81) (-12.73,-130.81) (-12.02,-130.11) (-11.31,-129.40) (-10.61,-128.69) (-9.90,-128.69) (-9.19,-127.99) (-8.49,-127.28) (-7.78,-126.57) (-7.07,-126.57) (-6.36,-125.87) (-5.66,-125.16) (-4.95,-125.16) (-4.24,-124.45) (-3.54,-123.74) (-2.83,-123.04) (-2.12,-123.04) (-1.41,-122.33) (-14.85,-189.50) (-15.56,-188.80) (-15.56,-188.09) (-16.26,-187.38) (-16.26,-186.68) (-16.97,-185.97) (-16.97,-185.26) (-17.68,-184.55) (-17.68,-183.85) (-18.38,-183.14) (-18.38,-182.43) (-19.09,-181.73) (-19.09,-181.02) (-19.80,-180.31) (-19.80,-179.61) (-20.51,-178.90) (-20.51,-178.19) (-21.21,-177.48) (-21.21,-176.78) (-21.92,-176.07) (-22.63,-175.36) (-22.63,-174.66) (-23.33,-173.95) (-23.33,-173.24) (-24.04,-172.53) (-24.04,-171.83) (-24.75,-171.12) (-24.75,-170.41) (-25.46,-169.71) (-25.46,-169.00) (-26.16,-168.29) (-26.16,-167.58) (-26.87,-166.88) (-26.87,-166.17) (-27.58,-165.46) (-27.58,-164.76) (-28.28,-164.05) (-28.28,-163.34) (-28.99,-162.63) (-28.99,-161.93) (-29.70,-161.22) (-30.41,-160.51) (-30.41,-159.81) (-31.11,-159.10) (-31.11,-158.39) (-31.82,-157.68) (-31.82,-156.98) (-32.53,-156.27) (-32.53,-155.56) (-33.23,-154.86) (-33.23,-154.15) (-33.94,-153.44) (-33.94,-152.74) (-34.65,-152.03) (-34.65,-151.32) (-35.36,-150.61) (-35.36,-149.91) (-36.06,-149.20) (-36.06,-148.49) (-36.77,-147.79) (-50.91,-214.96) (-50.20,-214.25) (-49.50,-214.25) (-48.79,-213.55) (-48.08,-212.84) (-47.38,-212.13) (-46.67,-212.13) (-45.96,-211.42) (-45.25,-210.72) (-44.55,-210.72) (-43.84,-210.01) (-43.13,-209.30) (-42.43,-209.30) (-41.72,-208.60) (-41.01,-207.89) (-40.31,-207.18) (-39.60,-207.18) (-38.89,-206.48) (-38.18,-205.77) (-37.48,-205.77) (-36.77,-205.06) (-36.06,-204.35) (-35.36,-203.65) (-34.65,-203.65) (-33.94,-202.94) (-33.23,-202.23) (-32.53,-202.23) (-31.82,-201.53) (-31.11,-200.82) (-30.41,-200.82) (-29.70,-200.11) (-28.99,-199.40) (-28.28,-198.70) (-27.58,-198.70) (-26.87,-197.99) (-26.16,-197.28) (-25.46,-197.28) (-24.75,-196.58) (-24.04,-195.87) (-23.33,-195.16) (-22.63,-195.16) (-21.92,-194.45) (-21.21,-193.75) (-20.51,-193.75) (-19.80,-193.04) (-19.09,-192.33) (-18.38,-192.33) (-17.68,-191.63) (-16.97,-190.92) (-16.26,-190.21) (-15.56,-190.21) (-14.85,-189.50) (-50.91,-214.96) (-50.20,-215.67) (-49.50,-215.67) (-48.79,-216.37) (-48.08,-216.37) (-47.38,-217.08) (-46.67,-217.08) (-45.96,-217.79) (-45.25,-217.79) (-44.55,-218.50) (-43.84,-218.50) (-43.13,-219.20) (-42.43,-219.20) (-41.72,-219.91) (-41.01,-219.91) (-40.31,-220.62) (-39.60,-220.62) (-38.89,-221.32) (-38.18,-221.32) (-37.48,-222.03) (-36.77,-222.03) (-36.06,-222.74) (-35.36,-222.74) (-34.65,-223.45) (-33.94,-223.45) (-33.23,-224.15) (-32.53,-224.15) (-31.82,-224.86) (-31.11,-224.86) (-30.41,-225.57) (-29.70,-225.57) (-28.99,-226.27) (-28.28,-226.27) (-27.58,-226.98) (-26.87,-226.98) (-26.16,-227.69) (-25.46,-227.69) (-24.75,-228.40) (-24.04,-228.40) (-23.33,-229.10) (-22.63,-229.10) (-21.92,-229.81) (-21.21,-229.81) (-20.51,-230.52) (-19.80,-230.52) (-19.09,-231.22) (-18.38,-231.22) (-17.68,-231.93) (-16.97,-231.93) (-16.26,-232.64) (-15.56,-232.64) (-14.85,-233.35) (-14.14,-233.35) (-13.44,-234.05) (-12.73,-234.05) (-12.02,-234.76) (-11.31,-234.76) (-10.61,-235.47) ( 3.54,-279.31) ( 3.54,-278.60) ( 2.83,-277.89) ( 2.83,-277.19) ( 2.83,-276.48) ( 2.12,-275.77) ( 2.12,-275.06) ( 2.12,-274.36) ( 1.41,-273.65) ( 1.41,-272.94) ( 1.41,-272.24) ( 0.71,-271.53) ( 0.71,-270.82) ( 0.71,-270.11) ( 0.00,-269.41) ( 0.00,-268.70) ( 0.00,-267.99) ( 0.00,-267.29) (-0.71,-266.58) (-0.71,-265.87) (-0.71,-265.17) (-1.41,-264.46) (-1.41,-263.75) (-1.41,-263.04) (-2.12,-262.34) (-2.12,-261.63) (-2.12,-260.92) (-2.83,-260.22) (-2.83,-259.51) (-2.83,-258.80) (-3.54,-258.09) (-3.54,-257.39) (-3.54,-256.68) (-4.24,-255.97) (-4.24,-255.27) (-4.24,-254.56) (-4.95,-253.85) (-4.95,-253.14) (-4.95,-252.44) (-5.66,-251.73) (-5.66,-251.02) (-5.66,-250.32) (-6.36,-249.61) (-6.36,-248.90) (-6.36,-248.19) (-7.07,-247.49) (-7.07,-246.78) (-7.07,-246.07) (-7.07,-245.37) (-7.78,-244.66) (-7.78,-243.95) (-7.78,-243.24) (-8.49,-242.54) (-8.49,-241.83) (-8.49,-241.12) (-9.19,-240.42) (-9.19,-239.71) (-9.19,-239.00) (-9.90,-238.29) (-9.90,-237.59) (-9.90,-236.88) (-10.61,-236.17) (-10.61,-235.47) (-25.46,-310.42) (-24.75,-309.71) (-24.04,-309.01) (-23.33,-308.30) (-22.63,-307.59) (-21.92,-306.88) (-21.21,-306.18) (-20.51,-305.47) (-20.51,-304.76) (-19.80,-304.06) (-19.09,-303.35) (-18.38,-302.64) (-17.68,-301.93) (-16.97,-301.23) (-16.26,-300.52) (-15.56,-299.81) (-14.85,-299.11) (-14.14,-298.40) (-13.44,-297.69) (-12.73,-296.98) (-12.02,-296.28) (-11.31,-295.57) (-11.31,-294.86) (-10.61,-294.16) (-9.90,-293.45) (-9.19,-292.74) (-8.49,-292.04) (-7.78,-291.33) (-7.07,-290.62) (-6.36,-289.91) (-5.66,-289.21) (-4.95,-288.50) (-4.24,-287.79) (-3.54,-287.09) (-2.83,-286.38) (-2.12,-285.67) (-1.41,-284.96) (-1.41,-284.26) (-0.71,-283.55) ( 0.00,-282.84) ( 0.71,-282.14) ( 1.41,-281.43) ( 2.12,-280.72) ( 2.83,-280.01) ( 3.54,-279.31) (-25.46,-310.42) (-24.75,-311.13) (-24.04,-311.13) (-23.33,-311.83) (-22.63,-311.83) (-21.92,-312.54) (-21.21,-313.25) (-20.51,-313.25) (-19.80,-313.96) (-19.09,-313.96) (-18.38,-314.66) (-17.68,-315.37) (-16.97,-315.37) (-16.26,-316.08) (-15.56,-316.08) (-14.85,-316.78) (-14.14,-317.49) (-13.44,-317.49) (-12.73,-318.20) (-12.02,-318.20) (-11.31,-318.91) (-10.61,-319.61) (-9.90,-319.61) (-9.19,-320.32) (-8.49,-320.32) (-7.78,-321.03) (-7.07,-321.73) (-6.36,-321.73) (-5.66,-322.44) (-4.95,-322.44) (-4.24,-323.15) (-3.54,-323.85) (-2.83,-323.85) (-2.12,-324.56) (-1.41,-324.56) (-0.71,-325.27) ( 0.00,-325.98) ( 0.71,-325.98) ( 1.41,-326.68) ( 2.12,-326.68) ( 2.83,-327.39) ( 3.54,-328.10) ( 4.24,-328.10) ( 4.95,-328.80) ( 5.66,-328.80) ( 6.36,-329.51) ( 7.07,-330.22) ( 7.78,-330.22) ( 8.49,-330.93) ( 9.19,-330.93) ( 9.90,-331.63) ( 9.90,-331.63) (10.61,-331.63) (11.31,-332.34) (12.02,-332.34) (12.73,-333.05) (13.44,-333.05) (14.14,-333.75) (14.85,-333.75) (15.56,-334.46) (16.26,-334.46) (16.97,-335.17) (17.68,-335.17) (18.38,-335.88) (19.09,-335.88) (19.80,-336.58) (20.51,-336.58) (21.21,-337.29) (21.92,-337.29) (22.63,-337.29) (23.33,-338.00) (24.04,-338.00) (24.75,-338.70) (25.46,-338.70) (26.16,-339.41) (26.87,-339.41) (27.58,-340.12) (28.28,-340.12) (28.99,-340.83) (29.70,-340.83) (30.41,-341.53) (31.11,-341.53) (31.82,-342.24) (32.53,-342.24) (33.23,-342.95) (33.94,-342.95) (34.65,-343.65) (35.36,-343.65) (36.06,-343.65) (36.77,-344.36) (37.48,-344.36) (38.18,-345.07) (38.89,-345.07) (39.60,-345.78) (40.31,-345.78) (41.01,-346.48) (41.72,-346.48) (42.43,-347.19) (43.13,-347.19) (43.84,-347.90) (44.55,-347.90) (45.25,-348.60) (45.96,-348.60) (46.67,-349.31) (47.38,-349.31) (47.38,-349.31) (48.08,-349.31) (48.79,-349.31) (49.50,-350.02) (50.20,-350.02) (50.91,-350.02) (51.62,-350.02) (52.33,-350.72) (53.03,-350.72) (53.74,-350.72) (54.45,-350.72) (55.15,-351.43) (55.86,-351.43) (56.57,-351.43) (57.28,-351.43) (57.98,-352.14) (58.69,-352.14) (59.40,-352.14) (60.10,-352.14) (60.81,-352.85) (61.52,-352.85) (62.23,-352.85) (62.93,-352.85) (63.64,-352.85) (64.35,-353.55) (65.05,-353.55) (65.76,-353.55) (66.47,-353.55) (67.18,-354.26) (67.88,-354.26) (68.59,-354.26) (69.30,-354.26) (70.00,-354.97) (70.71,-354.97) (71.42,-354.97) (72.12,-354.97) (72.83,-355.67) (73.54,-355.67) (74.25,-355.67) (74.95,-355.67) (75.66,-356.38) (76.37,-356.38) (77.07,-356.38) (77.78,-356.38) (78.49,-356.38) (79.20,-357.09) (79.90,-357.09) (80.61,-357.09) (81.32,-357.09) (82.02,-357.80) (82.73,-357.80) (83.44,-357.80) (84.15,-357.80) (84.85,-358.50) (85.56,-358.50) (86.27,-358.50) (86.97,-358.50) (87.68,-359.21) (88.39,-359.21) (89.10,-359.21) (89.80,-359.21) (90.51,-359.92) (91.22,-359.92) (91.92,-359.92) (91.92,-359.92) (92.63,-359.92) (93.34,-359.92) (94.05,-359.92) (94.75,-359.92) (95.46,-359.92) (96.17,-359.92) (96.87,-359.92) (97.58,-359.92) (98.29,-359.92) (98.99,-359.92) (99.70,-359.92) (100.41,-359.21) (101.12,-359.21) (101.82,-359.21) (102.53,-359.21) (103.24,-359.21) (103.94,-359.21) (104.65,-359.21) (105.36,-359.21) (106.07,-359.21) (106.77,-359.21) (107.48,-359.21) (108.19,-359.21) (108.89,-359.21) (109.60,-359.21) (110.31,-359.21) (111.02,-359.21) (111.72,-359.21) (112.43,-359.21) (113.14,-359.21) (113.84,-359.21) (114.55,-359.21) (115.26,-359.21) (115.97,-358.50) (116.67,-358.50) (117.38,-358.50) (118.09,-358.50) (118.79,-358.50) (119.50,-358.50) (120.21,-358.50) (120.92,-358.50) (121.62,-358.50) (122.33,-358.50) (123.04,-358.50) (123.74,-358.50) (124.45,-358.50) (125.16,-358.50) (125.87,-358.50) (126.57,-358.50) (127.28,-358.50) (127.99,-358.50) (128.69,-358.50) (129.40,-358.50) (130.11,-358.50) (130.81,-358.50) (131.52,-357.80) (132.23,-357.80) (132.94,-357.80) (133.64,-357.80) (134.35,-357.80) (135.06,-357.80) (135.76,-357.80) (136.47,-357.80) (137.18,-357.80) (137.89,-357.80) (138.59,-357.80) (138.59,-357.80) (139.30,-357.09) (140.01,-357.09) (140.71,-356.38) (141.42,-356.38) (142.13,-355.67) (142.84,-355.67) (143.54,-354.97) (144.25,-354.26) (144.96,-354.26) (145.66,-353.55) (146.37,-353.55) (147.08,-352.85) (147.79,-352.14) (148.49,-352.14) (149.20,-351.43) (149.91,-351.43) (150.61,-350.72) (151.32,-350.72) (152.03,-350.02) (152.74,-349.31) (153.44,-349.31) (154.15,-348.60) (154.86,-348.60) (155.56,-347.90) (156.27,-347.90) (156.98,-347.19) (157.68,-346.48) (158.39,-346.48) (159.10,-345.78) (159.81,-345.78) (160.51,-345.07) (161.22,-344.36) (161.93,-344.36) (162.63,-343.65) (163.34,-343.65) (164.05,-342.95) (164.76,-342.95) (165.46,-342.24) (166.17,-341.53) (166.88,-341.53) (167.58,-340.83) (168.29,-340.83) (169.00,-340.12) (169.71,-340.12) (170.41,-339.41) (171.12,-338.70) (171.83,-338.70) (172.53,-338.00) (173.24,-338.00) (173.95,-337.29) (174.66,-336.58) (175.36,-336.58) (176.07,-335.88) (176.78,-335.88) (177.48,-335.17) (178.19,-335.17) (178.90,-334.46) (178.90,-334.46) (178.90,-333.75) (178.90,-333.05) (178.19,-332.34) (178.19,-331.63) (178.19,-330.93) (178.19,-330.22) (177.48,-329.51) (177.48,-328.80) (177.48,-328.10) (177.48,-327.39) (177.48,-326.68) (176.78,-325.98) (176.78,-325.27) (176.78,-324.56) (176.78,-323.85) (176.78,-323.15) (176.07,-322.44) (176.07,-321.73) (176.07,-321.03) (176.07,-320.32) (175.36,-319.61) (175.36,-318.91) (175.36,-318.20) (175.36,-317.49) (175.36,-316.78) (174.66,-316.08) (174.66,-315.37) (174.66,-314.66) (174.66,-313.96) (174.66,-313.25) (173.95,-312.54) (173.95,-311.83) (173.95,-311.13) (173.95,-310.42) (173.24,-309.71) (173.24,-309.01) (173.24,-308.30) (173.24,-307.59) (173.24,-306.88) (172.53,-306.18) (172.53,-305.47) (172.53,-304.76) (172.53,-304.06) (172.53,-303.35) (171.83,-302.64) (171.83,-301.93) (171.83,-301.23) (171.83,-300.52) (171.12,-299.81) (171.12,-299.11) (171.12,-298.40) (171.12,-297.69) (171.12,-296.98) (170.41,-296.28) (170.41,-295.57) (170.41,-294.86) (170.41,-294.16) (170.41,-293.45) (169.71,-292.74) (169.71,-292.04) (169.71,-291.33) (169.71,-290.62) (169.00,-289.91) (169.00,-289.21) (169.00,-288.50) (190.92,-326.68) (190.21,-325.98) (190.21,-325.27) (189.50,-324.56) (189.50,-323.85) (188.80,-323.15) (188.80,-322.44) (188.09,-321.73) (187.38,-321.03) (187.38,-320.32) (186.68,-319.61) (186.68,-318.91) (185.97,-318.20) (185.97,-317.49) (185.26,-316.78) (184.55,-316.08) (184.55,-315.37) (183.85,-314.66) (183.85,-313.96) (183.14,-313.25) (183.14,-312.54) (182.43,-311.83) (181.73,-311.13) (181.73,-310.42) (181.02,-309.71) (181.02,-309.01) (180.31,-308.30) (180.31,-307.59) (179.61,-306.88) (178.90,-306.18) (178.90,-305.47) (178.19,-304.76) (178.19,-304.06) (177.48,-303.35) (176.78,-302.64) (176.78,-301.93) (176.07,-301.23) (176.07,-300.52) (175.36,-299.81) (175.36,-299.11) (174.66,-298.40) (173.95,-297.69) (173.95,-296.98) (173.24,-296.28) (173.24,-295.57) (172.53,-294.86) (172.53,-294.16) (171.83,-293.45) (171.12,-292.74) (171.12,-292.04) (170.41,-291.33) (170.41,-290.62) (169.71,-289.91) (169.71,-289.21) (169.00,-288.50) (155.56,-356.38) (156.27,-355.67) (156.98,-354.97) (157.68,-354.26) (158.39,-354.26) (159.10,-353.55) (159.81,-352.85) (160.51,-352.14) (161.22,-351.43) (161.93,-350.72) (162.63,-350.72) (163.34,-350.02) (164.05,-349.31) (164.76,-348.60) (165.46,-347.90) (166.17,-347.19) (166.88,-347.19) (167.58,-346.48) (168.29,-345.78) (169.00,-345.07) (169.71,-344.36) (170.41,-343.65) (171.12,-343.65) (171.83,-342.95) (172.53,-342.24) (173.24,-341.53) (173.95,-340.83) (174.66,-340.12) (175.36,-339.41) (176.07,-339.41) (176.78,-338.70) (177.48,-338.00) (178.19,-337.29) (178.90,-336.58) (179.61,-335.88) (180.31,-335.88) (181.02,-335.17) (181.73,-334.46) (182.43,-333.75) (183.14,-333.05) (183.85,-332.34) (184.55,-332.34) (185.26,-331.63) (185.97,-330.93) (186.68,-330.22) (187.38,-329.51) (188.09,-328.80) (188.80,-328.80) (189.50,-328.10) (190.21,-327.39) (190.92,-326.68) (155.56,-356.38) (156.27,-356.38) (156.98,-356.38) (157.68,-357.09) (158.39,-357.09) (159.10,-357.09) (159.81,-357.09) (160.51,-357.80) (161.22,-357.80) (161.93,-357.80) (162.63,-357.80) (163.34,-358.50) (164.05,-358.50) (164.76,-358.50) (165.46,-358.50) (166.17,-359.21) (166.88,-359.21) (167.58,-359.21) (168.29,-359.21) (169.00,-359.21) (169.71,-359.92) (170.41,-359.92) (171.12,-359.92) (171.83,-359.92) (172.53,-360.62) (173.24,-360.62) (173.95,-360.62) (174.66,-360.62) (175.36,-361.33) (176.07,-361.33) (176.78,-361.33) (177.48,-361.33) (178.19,-361.33) (178.90,-362.04) (179.61,-362.04) (180.31,-362.04) (181.02,-362.04) (181.73,-362.75) (182.43,-362.75) (183.14,-362.75) (183.85,-362.75) (184.55,-363.45) (185.26,-363.45) (185.97,-363.45) (186.68,-363.45) (187.38,-364.16) (188.09,-364.16) (188.80,-364.16) (189.50,-364.16) (190.21,-364.16) (190.92,-364.87) (191.63,-364.87) (192.33,-364.87) (193.04,-364.87) (193.75,-365.57) (194.45,-365.57) (195.16,-365.57) (195.87,-365.57) (196.58,-366.28) (197.28,-366.28) (197.99,-366.28) (198.70,-366.28) (199.40,-366.99) (200.11,-366.99) (200.82,-366.99) (200.82,-366.99) (201.53,-366.28) (202.23,-365.57) (202.94,-365.57) (203.65,-364.87) (204.35,-364.16) (205.06,-363.45) (205.77,-362.75) (206.48,-362.75) (207.18,-362.04) (207.89,-361.33) (208.60,-360.62) (209.30,-359.92) (210.01,-359.92) (210.72,-359.21) (211.42,-358.50) (212.13,-357.80) (212.84,-357.09) (213.55,-357.09) (214.25,-356.38) (214.96,-355.67) (215.67,-354.97) (216.37,-354.26) (217.08,-354.26) (217.79,-353.55) (218.50,-352.85) (219.20,-352.14) (219.91,-351.43) (220.62,-350.72) (221.32,-350.72) (222.03,-350.02) (222.74,-349.31) (223.45,-348.60) (224.15,-347.90) (224.86,-347.90) (225.57,-347.19) (226.27,-346.48) (226.98,-345.78) (227.69,-345.07) (228.40,-345.07) (229.10,-344.36) (229.81,-343.65) (230.52,-342.95) (231.22,-342.24) (231.93,-342.24) (232.64,-341.53) (233.35,-340.83) (234.05,-340.12) (234.76,-339.41) (235.47,-339.41) (236.17,-338.70) (236.88,-338.00) (236.88,-338.00) (236.17,-337.29) (236.17,-336.58) (235.47,-335.88) (235.47,-335.17) (234.76,-334.46) (234.76,-333.75) (234.05,-333.05) (234.05,-332.34) (233.35,-331.63) (233.35,-330.93) (232.64,-330.22) (232.64,-329.51) (231.93,-328.80) (231.93,-328.10) (231.22,-327.39) (231.22,-326.68) (230.52,-325.98) (230.52,-325.27) (229.81,-324.56) (229.81,-323.85) (229.10,-323.15) (229.10,-322.44) (228.40,-321.73) (228.40,-321.03) (227.69,-320.32) (227.69,-319.61) (226.98,-318.91) (226.98,-318.20) (226.27,-317.49) (225.57,-316.78) (225.57,-316.08) (224.86,-315.37) (224.86,-314.66) (224.15,-313.96) (224.15,-313.25) (223.45,-312.54) (223.45,-311.83) (222.74,-311.13) (222.74,-310.42) (222.03,-309.71) (222.03,-309.01) (221.32,-308.30) (221.32,-307.59) (220.62,-306.88) (220.62,-306.18) (219.91,-305.47) (219.91,-304.76) (219.20,-304.06) (219.20,-303.35) (218.50,-302.64) (218.50,-301.93) (217.79,-301.23) (217.79,-300.52) (217.08,-299.81) (217.08,-299.11) (216.37,-298.40) (216.37,-297.69) (215.67,-296.98) (215.67,-296.98) (216.37,-296.98) (217.08,-296.98) (217.79,-296.28) (218.50,-296.28) (219.20,-296.28) (219.91,-296.28) (220.62,-296.28) (221.32,-295.57) (222.03,-295.57) (222.74,-295.57) (223.45,-295.57) (224.15,-294.86) (224.86,-294.86) (225.57,-294.86) (226.27,-294.86) (226.98,-294.86) (227.69,-294.16) (228.40,-294.16) (229.10,-294.16) (229.81,-294.16) (230.52,-294.16) (231.22,-293.45) (231.93,-293.45) (232.64,-293.45) (233.35,-293.45) (234.05,-293.45) (234.76,-292.74) (235.47,-292.74) (236.17,-292.74) (236.88,-292.74) (237.59,-292.74) (238.29,-292.04) (239.00,-292.04) (239.71,-292.04) (240.42,-292.04) (241.12,-291.33) (241.83,-291.33) (242.54,-291.33) (243.24,-291.33) (243.95,-291.33) (244.66,-290.62) (245.37,-290.62) (246.07,-290.62) (246.78,-290.62) (247.49,-290.62) (248.19,-289.91) (248.90,-289.91) (249.61,-289.91) (250.32,-289.91) (251.02,-289.91) (251.73,-289.21) (252.44,-289.21) (253.14,-289.21) (253.85,-289.21) (254.56,-288.50) (255.27,-288.50) (255.97,-288.50) (256.68,-288.50) (257.39,-288.50) (258.09,-287.79) (258.80,-287.79) (259.51,-287.79) (259.51,-287.79) (260.22,-287.09) (260.92,-286.38) (261.63,-285.67) (262.34,-284.96) (263.04,-284.26) (263.75,-283.55) (264.46,-282.84) (265.17,-282.14) (265.87,-281.43) (266.58,-280.72) (267.29,-280.01) (267.99,-279.31) (268.70,-278.60) (269.41,-277.89) (270.11,-277.19) (270.82,-276.48) (271.53,-275.77) (272.24,-275.06) (272.94,-274.36) (273.65,-273.65) (274.36,-272.94) (275.06,-272.24) (275.77,-271.53) (276.48,-270.82) (277.19,-270.11) (277.89,-269.41) (278.60,-268.70) (279.31,-267.99) (280.01,-267.29) (280.72,-266.58) (281.43,-265.87) (282.14,-265.17) (282.84,-264.46) (283.55,-263.75) (284.26,-263.04) (284.96,-262.34) (285.67,-261.63) (286.38,-260.92) (287.09,-260.22) (287.79,-259.51) (288.50,-258.80) (289.21,-258.09) (289.91,-257.39) (290.62,-256.68) (291.33,-255.97) (291.33,-255.97) (290.62,-255.27) (289.91,-254.56) (289.21,-253.85) (289.21,-253.14) (288.50,-252.44) (287.79,-251.73) (287.09,-251.02) (286.38,-250.32) (285.67,-249.61) (284.96,-248.90) (284.96,-248.19) (284.26,-247.49) (283.55,-246.78) (282.84,-246.07) (282.14,-245.37) (281.43,-244.66) (280.72,-243.95) (280.72,-243.24) (280.01,-242.54) (279.31,-241.83) (278.60,-241.12) (277.89,-240.42) (277.19,-239.71) (276.48,-239.00) (276.48,-238.29) (275.77,-237.59) (275.06,-236.88) (274.36,-236.17) (273.65,-235.47) (272.94,-234.76) (272.24,-234.05) (272.24,-233.35) (271.53,-232.64) (270.82,-231.93) (270.11,-231.22) (269.41,-230.52) (268.70,-229.81) (267.99,-229.10) (267.99,-228.40) (267.29,-227.69) (266.58,-226.98) (265.87,-226.27) (265.17,-225.57) (264.46,-224.86) (263.75,-224.15) (263.75,-223.45) (263.04,-222.74) (262.34,-222.03) (261.63,-221.32) (261.63,-221.32) (262.34,-220.62) (263.04,-220.62) (263.75,-219.91) (264.46,-219.91) (265.17,-219.20) (265.87,-219.20) (266.58,-218.50) (267.29,-218.50) (267.99,-217.79) (268.70,-217.79) (269.41,-217.08) (270.11,-217.08) (270.82,-216.37) (271.53,-216.37) (272.24,-215.67) (272.94,-215.67) (273.65,-214.96) (274.36,-214.96) (275.06,-214.25) (275.77,-214.25) (276.48,-213.55) (277.19,-213.55) (277.89,-212.84) (278.60,-212.84) (279.31,-212.13) (280.01,-212.13) (280.72,-211.42) (281.43,-211.42) (282.14,-210.72) (282.84,-210.72) (283.55,-210.01) (284.26,-210.01) (284.96,-209.30) (285.67,-209.30) (286.38,-208.60) (287.09,-208.60) (287.79,-207.89) (288.50,-207.89) (289.21,-207.18) (289.91,-207.18) (290.62,-206.48) (291.33,-206.48) (292.04,-205.77) (292.74,-205.77) (293.45,-205.06) (294.16,-205.06) (294.86,-204.35) (295.57,-204.35) (296.28,-203.65) (296.98,-203.65) (297.69,-202.94) (298.40,-202.94) (299.11,-202.23) (299.81,-202.23) (300.52,-201.53) (301.23,-201.53) (301.93,-200.82) (301.93,-200.82) (302.64,-200.11) (302.64,-199.40) (303.35,-198.70) (303.35,-197.99) (304.06,-197.28) (304.06,-196.58) (304.76,-195.87) (304.76,-195.16) (305.47,-194.45) (305.47,-193.75) (306.18,-193.04) (306.18,-192.33) (306.88,-191.63) (306.88,-190.92) (307.59,-190.21) (307.59,-189.50) (308.30,-188.80) (308.30,-188.09) (309.01,-187.38) (309.01,-186.68) (309.71,-185.97) (309.71,-185.26) (310.42,-184.55) (310.42,-183.85) (311.13,-183.14) (311.13,-182.43) (311.83,-181.73) (311.83,-181.02) (312.54,-180.31) (312.54,-179.61) (313.25,-178.90) (313.25,-178.19) (313.96,-177.48) (313.96,-176.78) (314.66,-176.07) (314.66,-175.36) (315.37,-174.66) (315.37,-173.95) (316.08,-173.24) (316.08,-172.53) (316.78,-171.83) (316.78,-171.12) (317.49,-170.41) (317.49,-169.71) (318.20,-169.00) (318.20,-168.29) (318.91,-167.58) (318.91,-166.88) (319.61,-166.17) (319.61,-165.46) (320.32,-164.76) (320.32,-164.05) (321.03,-163.34) (321.03,-162.63) (321.73,-161.93) (321.73,-161.22) (322.44,-160.51) (278.60,-175.36) (279.31,-175.36) (280.01,-174.66) (280.72,-174.66) (281.43,-174.66) (282.14,-173.95) (282.84,-173.95) (283.55,-173.95) (284.26,-173.24) (284.96,-173.24) (285.67,-173.24) (286.38,-172.53) (287.09,-172.53) (287.79,-172.53) (288.50,-171.83) (289.21,-171.83) (289.91,-171.83) (290.62,-171.12) (291.33,-171.12) (292.04,-171.12) (292.74,-170.41) (293.45,-170.41) (294.16,-170.41) (294.86,-169.71) (295.57,-169.71) (296.28,-169.71) (296.98,-169.00) (297.69,-169.00) (298.40,-169.00) (299.11,-168.29) (299.81,-168.29) (300.52,-168.29) (301.23,-167.58) (301.93,-167.58) (302.64,-166.88) (303.35,-166.88) (304.06,-166.88) (304.76,-166.17) (305.47,-166.17) (306.18,-166.17) (306.88,-165.46) (307.59,-165.46) (308.30,-165.46) (309.01,-164.76) (309.71,-164.76) (310.42,-164.76) (311.13,-164.05) (311.83,-164.05) (312.54,-164.05) (313.25,-163.34) (313.96,-163.34) (314.66,-163.34) (315.37,-162.63) (316.08,-162.63) (316.78,-162.63) (317.49,-161.93) (318.20,-161.93) (318.91,-161.93) (319.61,-161.22) (320.32,-161.22) (321.03,-161.22) (321.73,-160.51) (322.44,-160.51) (269.41,-219.91) (269.41,-219.20) (269.41,-218.50) (270.11,-217.79) (270.11,-217.08) (270.11,-216.37) (270.11,-215.67) (270.11,-214.96) (270.82,-214.25) (270.82,-213.55) (270.82,-212.84) (270.82,-212.13) (270.82,-211.42) (271.53,-210.72) (271.53,-210.01) (271.53,-209.30) (271.53,-208.60) (272.24,-207.89) (272.24,-207.18) (272.24,-206.48) (272.24,-205.77) (272.24,-205.06) (272.94,-204.35) (272.94,-203.65) (272.94,-202.94) (272.94,-202.23) (272.94,-201.53) (273.65,-200.82) (273.65,-200.11) (273.65,-199.40) (273.65,-198.70) (273.65,-197.99) (274.36,-197.28) (274.36,-196.58) (274.36,-195.87) (274.36,-195.16) (274.36,-194.45) (275.06,-193.75) (275.06,-193.04) (275.06,-192.33) (275.06,-191.63) (275.06,-190.92) (275.77,-190.21) (275.77,-189.50) (275.77,-188.80) (275.77,-188.09) (275.77,-187.38) (276.48,-186.68) (276.48,-185.97) (276.48,-185.26) (276.48,-184.55) (277.19,-183.85) (277.19,-183.14) (277.19,-182.43) (277.19,-181.73) (277.19,-181.02) (277.89,-180.31) (277.89,-179.61) (277.89,-178.90) (277.89,-178.19) (277.89,-177.48) (278.60,-176.78) (278.60,-176.07) (278.60,-175.36) (269.41,-219.91) (269.41,-219.20) (269.41,-218.50) (268.70,-217.79) (268.70,-217.08) (268.70,-216.37) (268.70,-215.67) (268.70,-214.96) (268.70,-214.25) (267.99,-213.55) (267.99,-212.84) (267.99,-212.13) (267.99,-211.42) (267.99,-210.72) (267.29,-210.01) (267.29,-209.30) (267.29,-208.60) (267.29,-207.89) (267.29,-207.18) (266.58,-206.48) (266.58,-205.77) (266.58,-205.06) (266.58,-204.35) (266.58,-203.65) (266.58,-202.94) (265.87,-202.23) (265.87,-201.53) (265.87,-200.82) (265.87,-200.11) (265.87,-199.40) (265.17,-198.70) (265.17,-197.99) (265.17,-197.28) (265.17,-196.58) (265.17,-195.87) (264.46,-195.16) (264.46,-194.45) (264.46,-193.75) (264.46,-193.04) (264.46,-192.33) (264.46,-191.63) (263.75,-190.92) (263.75,-190.21) (263.75,-189.50) (263.75,-188.80) (263.75,-188.09) (263.04,-187.38) (263.04,-186.68) (263.04,-185.97) (263.04,-185.26) (263.04,-184.55) (262.34,-183.85) (262.34,-183.14) (262.34,-182.43) (262.34,-181.73) (262.34,-181.02) (262.34,-180.31) (261.63,-179.61) (261.63,-178.90) (261.63,-178.19) (261.63,-177.48) (261.63,-176.78) (260.92,-176.07) (260.92,-175.36) (260.92,-174.66) (283.55,-215.67) (282.84,-214.96) (282.84,-214.25) (282.14,-213.55) (282.14,-212.84) (281.43,-212.13) (281.43,-211.42) (280.72,-210.72) (280.72,-210.01) (280.01,-209.30) (279.31,-208.60) (279.31,-207.89) (278.60,-207.18) (278.60,-206.48) (277.89,-205.77) (277.89,-205.06) (277.19,-204.35) (277.19,-203.65) (276.48,-202.94) (276.48,-202.23) (275.77,-201.53) (275.06,-200.82) (275.06,-200.11) (274.36,-199.40) (274.36,-198.70) (273.65,-197.99) (273.65,-197.28) (272.94,-196.58) (272.94,-195.87) (272.24,-195.16) (271.53,-194.45) (271.53,-193.75) (270.82,-193.04) (270.82,-192.33) (270.11,-191.63) (270.11,-190.92) (269.41,-190.21) (269.41,-189.50) (268.70,-188.80) (267.99,-188.09) (267.99,-187.38) (267.29,-186.68) (267.29,-185.97) (266.58,-185.26) (266.58,-184.55) (265.87,-183.85) (265.87,-183.14) (265.17,-182.43) (265.17,-181.73) (264.46,-181.02) (263.75,-180.31) (263.75,-179.61) (263.04,-178.90) (263.04,-178.19) (262.34,-177.48) (262.34,-176.78) (261.63,-176.07) (261.63,-175.36) (260.92,-174.66) (267.99,-256.68) (267.99,-255.97) (268.70,-255.27) (268.70,-254.56) (269.41,-253.85) (269.41,-253.14) (269.41,-252.44) (270.11,-251.73) (270.11,-251.02) (270.11,-250.32) (270.82,-249.61) (270.82,-248.90) (271.53,-248.19) (271.53,-247.49) (271.53,-246.78) (272.24,-246.07) (272.24,-245.37) (272.24,-244.66) (272.94,-243.95) (272.94,-243.24) (273.65,-242.54) (273.65,-241.83) (273.65,-241.12) (274.36,-240.42) (274.36,-239.71) (274.36,-239.00) (275.06,-238.29) (275.06,-237.59) (275.77,-236.88) (275.77,-236.17) (275.77,-235.47) (276.48,-234.76) (276.48,-234.05) (277.19,-233.35) (277.19,-232.64) (277.19,-231.93) (277.89,-231.22) (277.89,-230.52) (277.89,-229.81) (278.60,-229.10) (278.60,-228.40) (279.31,-227.69) (279.31,-226.98) (279.31,-226.27) (280.01,-225.57) (280.01,-224.86) (280.01,-224.15) (280.72,-223.45) (280.72,-222.74) (281.43,-222.03) (281.43,-221.32) (281.43,-220.62) (282.14,-219.91) (282.14,-219.20) (282.14,-218.50) (282.84,-217.79) (282.84,-217.08) (283.55,-216.37) (283.55,-215.67) (267.99,-256.68) (268.70,-255.97) (269.41,-255.27) (270.11,-255.27) (270.82,-254.56) (271.53,-253.85) (272.24,-253.14) (272.94,-252.44) (273.65,-252.44) (274.36,-251.73) (275.06,-251.02) (275.77,-250.32) (276.48,-249.61) (277.19,-249.61) (277.89,-248.90) (278.60,-248.19) (279.31,-247.49) (280.01,-246.78) (280.72,-246.78) (281.43,-246.07) (282.14,-245.37) (282.84,-244.66) (283.55,-243.95) (284.26,-243.95) (284.96,-243.24) (285.67,-242.54) (286.38,-241.83) (287.09,-241.12) (287.79,-241.12) (288.50,-240.42) (289.21,-239.71) (289.91,-239.00) (290.62,-238.29) (291.33,-238.29) (292.04,-237.59) (292.74,-236.88) (293.45,-236.17) (294.16,-235.47) (294.86,-235.47) (295.57,-234.76) (296.28,-234.05) (296.98,-233.35) (297.69,-232.64) (298.40,-232.64) (299.11,-231.93) (299.81,-231.22) (300.52,-230.52) (301.23,-229.81) (301.93,-229.81) (302.64,-229.10) (303.35,-228.40) (303.35,-228.40) (303.35,-227.69) (303.35,-226.98) (302.64,-226.27) (302.64,-225.57) (302.64,-224.86) (302.64,-224.15) (301.93,-223.45) (301.93,-222.74) (301.93,-222.03) (301.93,-221.32) (301.23,-220.62) (301.23,-219.91) (301.23,-219.20) (301.23,-218.50) (300.52,-217.79) (300.52,-217.08) (300.52,-216.37) (300.52,-215.67) (299.81,-214.96) (299.81,-214.25) (299.81,-213.55) (299.81,-212.84) (299.11,-212.13) (299.11,-211.42) (299.11,-210.72) (299.11,-210.01) (298.40,-209.30) (298.40,-208.60) (298.40,-207.89) (298.40,-207.18) (298.40,-206.48) (297.69,-205.77) (297.69,-205.06) (297.69,-204.35) (297.69,-203.65) (296.98,-202.94) (296.98,-202.23) (296.98,-201.53) (296.98,-200.82) (296.28,-200.11) (296.28,-199.40) (296.28,-198.70) (296.28,-197.99) (295.57,-197.28) (295.57,-196.58) (295.57,-195.87) (295.57,-195.16) (294.86,-194.45) (294.86,-193.75) (294.86,-193.04) (294.86,-192.33) (294.16,-191.63) (294.16,-190.92) (294.16,-190.21) (294.16,-189.50) (293.45,-188.80) (293.45,-188.09) (293.45,-187.38) (293.45,-186.68) (292.74,-185.97) (292.74,-185.26) (292.74,-184.55) (292.74,-184.55) (293.45,-184.55) (294.16,-185.26) (294.86,-185.26) (295.57,-185.97) (296.28,-185.97) (296.98,-185.97) (297.69,-186.68) (298.40,-186.68) (299.11,-186.68) (299.81,-187.38) (300.52,-187.38) (301.23,-188.09) (301.93,-188.09) (302.64,-188.09) (303.35,-188.80) (304.06,-188.80) (304.76,-188.80) (305.47,-189.50) (306.18,-189.50) (306.88,-190.21) (307.59,-190.21) (308.30,-190.21) (309.01,-190.92) (309.71,-190.92) (310.42,-191.63) (311.13,-191.63) (311.83,-191.63) (312.54,-192.33) (313.25,-192.33) (313.96,-192.33) (314.66,-193.04) (315.37,-193.04) (316.08,-193.75) (316.78,-193.75) (317.49,-193.75) (318.20,-194.45) (318.91,-194.45) (319.61,-195.16) (320.32,-195.16) (321.03,-195.16) (321.73,-195.87) (322.44,-195.87) (323.15,-195.87) (323.85,-196.58) (324.56,-196.58) (325.27,-197.28) (325.98,-197.28) (326.68,-197.28) (327.39,-197.99) (328.10,-197.99) (328.80,-197.99) (329.51,-198.70) (330.22,-198.70) (330.93,-199.40) (331.63,-199.40) (331.63,-199.40) (332.34,-199.40) (333.05,-199.40) (333.75,-199.40) (334.46,-199.40) (335.17,-199.40) (335.88,-199.40) (336.58,-199.40) (337.29,-198.70) (338.00,-198.70) (338.70,-198.70) (339.41,-198.70) (340.12,-198.70) (340.83,-198.70) (341.53,-198.70) (342.24,-198.70) (342.95,-198.70) (343.65,-198.70) (344.36,-198.70) (345.07,-198.70) (345.78,-198.70) (346.48,-198.70) (347.19,-198.70) (347.90,-198.70) (348.60,-197.99) (349.31,-197.99) (350.02,-197.99) (350.72,-197.99) (351.43,-197.99) (352.14,-197.99) (352.85,-197.99) (353.55,-197.99) (354.26,-197.99) (354.97,-197.99) (355.67,-197.99) (356.38,-197.99) (357.09,-197.99) (357.80,-197.99) (358.50,-197.99) (359.21,-197.28) (359.92,-197.28) (360.62,-197.28) (361.33,-197.28) (362.04,-197.28) (362.75,-197.28) (363.45,-197.28) (364.16,-197.28) (364.87,-197.28) (365.57,-197.28) (366.28,-197.28) (366.99,-197.28) (367.70,-197.28) (368.40,-197.28) (369.11,-197.28) (369.82,-197.28) (370.52,-196.58) (371.23,-196.58) (371.94,-196.58) (372.65,-196.58) (373.35,-196.58) (374.06,-196.58) (374.77,-196.58) (375.47,-196.58) (375.47,-196.58) (376.18,-196.58) (376.89,-196.58) (377.60,-196.58) (378.30,-196.58) (379.01,-196.58) (379.72,-197.28) (380.42,-197.28) (381.13,-197.28) (381.84,-197.28) (382.54,-197.28) (383.25,-197.28) (383.96,-197.28) (384.67,-197.28) (385.37,-197.28) (386.08,-197.28) (386.79,-197.99) (387.49,-197.99) (388.20,-197.99) (388.91,-197.99) (389.62,-197.99) (390.32,-197.99) (391.03,-197.99) (391.74,-197.99) (392.44,-197.99) (393.15,-197.99) (393.86,-198.70) (394.57,-198.70) (395.27,-198.70) (395.98,-198.70) (396.69,-198.70) (397.39,-198.70) (398.10,-198.70) (398.81,-198.70) (399.52,-198.70) (400.22,-198.70) (400.93,-199.40) (401.64,-199.40) (402.34,-199.40) (403.05,-199.40) (403.76,-199.40) (404.47,-199.40) (405.17,-199.40) (405.88,-199.40) (406.59,-199.40) (407.29,-199.40) (408.00,-200.11) (408.71,-200.11) (409.41,-200.11) (410.12,-200.11) (410.83,-200.11) (411.54,-200.11) (412.24,-200.11) (412.95,-200.11) (413.66,-200.11) (414.36,-200.11) (415.07,-200.82) (415.78,-200.82) (416.49,-200.82) (417.19,-200.82) (417.90,-200.82) (418.61,-200.82) (418.61,-200.82) (418.61,-200.11) (419.31,-199.40) (419.31,-198.70) (419.31,-197.99) (420.02,-197.28) (420.02,-196.58) (420.02,-195.87) (420.73,-195.16) (420.73,-194.45) (420.73,-193.75) (421.44,-193.04) (421.44,-192.33) (421.44,-191.63) (422.14,-190.92) (422.14,-190.21) (422.14,-189.50) (422.14,-188.80) (422.85,-188.09) (422.85,-187.38) (422.85,-186.68) (423.56,-185.97) (423.56,-185.26) (423.56,-184.55) (424.26,-183.85) (424.26,-183.14) (424.26,-182.43) (424.97,-181.73) (424.97,-181.02) (424.97,-180.31) (425.68,-179.61) (425.68,-178.90) (425.68,-178.19) (426.39,-177.48) (426.39,-176.78) (426.39,-176.07) (427.09,-175.36) (427.09,-174.66) (427.09,-173.95) (427.80,-173.24) (427.80,-172.53) (427.80,-171.83) (428.51,-171.12) (428.51,-170.41) (428.51,-169.71) (429.21,-169.00) (429.21,-168.29) (429.21,-167.58) (429.21,-166.88) (429.92,-166.17) (429.92,-165.46) (429.92,-164.76) (430.63,-164.05) (430.63,-163.34) (430.63,-162.63) (431.34,-161.93) (431.34,-161.22) (431.34,-160.51) (432.04,-159.81) (432.04,-159.10) (432.04,-158.39) (432.75,-157.68) (432.75,-156.98) (387.49,-152.74) (388.20,-152.74) (388.91,-152.74) (389.62,-152.74) (390.32,-152.74) (391.03,-152.74) (391.74,-153.44) (392.44,-153.44) (393.15,-153.44) (393.86,-153.44) (394.57,-153.44) (395.27,-153.44) (395.98,-153.44) (396.69,-153.44) (397.39,-153.44) (398.10,-153.44) (398.81,-153.44) (399.52,-154.15) (400.22,-154.15) (400.93,-154.15) (401.64,-154.15) (402.34,-154.15) (403.05,-154.15) (403.76,-154.15) (404.47,-154.15) (405.17,-154.15) (405.88,-154.15) (406.59,-154.86) (407.29,-154.86) (408.00,-154.86) (408.71,-154.86) (409.41,-154.86) (410.12,-154.86) (410.83,-154.86) (411.54,-154.86) (412.24,-154.86) (412.95,-154.86) (413.66,-154.86) (414.36,-155.56) (415.07,-155.56) (415.78,-155.56) (416.49,-155.56) (417.19,-155.56) (417.90,-155.56) (418.61,-155.56) (419.31,-155.56) (420.02,-155.56) (420.73,-155.56) (421.44,-155.56) (422.14,-156.27) (422.85,-156.27) (423.56,-156.27) (424.26,-156.27) (424.97,-156.27) (425.68,-156.27) (426.39,-156.27) (427.09,-156.27) (427.80,-156.27) (428.51,-156.27) (429.21,-156.98) (429.92,-156.98) (430.63,-156.98) (431.34,-156.98) (432.04,-156.98) (432.75,-156.98) (344.36,-167.58) (345.07,-167.58) (345.78,-166.88) (346.48,-166.88) (347.19,-166.88) (347.90,-166.17) (348.60,-166.17) (349.31,-166.17) (350.02,-165.46) (350.72,-165.46) (351.43,-165.46) (352.14,-164.76) (352.85,-164.76) (353.55,-164.76) (354.26,-164.05) (354.97,-164.05) (355.67,-163.34) (356.38,-163.34) (357.09,-163.34) (357.80,-162.63) (358.50,-162.63) (359.21,-162.63) (359.92,-161.93) (360.62,-161.93) (361.33,-161.93) (362.04,-161.22) (362.75,-161.22) (363.45,-161.22) (364.16,-160.51) (364.87,-160.51) (365.57,-160.51) (366.28,-159.81) (366.99,-159.81) (367.70,-159.81) (368.40,-159.10) (369.11,-159.10) (369.82,-159.10) (370.52,-158.39) (371.23,-158.39) (371.94,-158.39) (372.65,-157.68) (373.35,-157.68) (374.06,-157.68) (374.77,-156.98) (375.47,-156.98) (376.18,-156.98) (376.89,-156.27) (377.60,-156.27) (378.30,-155.56) (379.01,-155.56) (379.72,-155.56) (380.42,-154.86) (381.13,-154.86) (381.84,-154.86) (382.54,-154.15) (383.25,-154.15) (383.96,-154.15) (384.67,-153.44) (385.37,-153.44) (386.08,-153.44) (386.79,-152.74) (387.49,-152.74) (369.82,-205.77) (369.11,-205.06) (369.11,-204.35) (368.40,-203.65) (367.70,-202.94) (367.70,-202.23) (366.99,-201.53) (366.28,-200.82) (366.28,-200.11) (365.57,-199.40) (364.87,-198.70) (364.87,-197.99) (364.16,-197.28) (363.45,-196.58) (363.45,-195.87) (362.75,-195.16) (362.04,-194.45) (362.04,-193.75) (361.33,-193.04) (360.62,-192.33) (360.62,-191.63) (359.92,-190.92) (359.21,-190.21) (359.21,-189.50) (358.50,-188.80) (357.80,-188.09) (357.80,-187.38) (357.09,-186.68) (356.38,-185.97) (356.38,-185.26) (355.67,-184.55) (354.97,-183.85) (354.97,-183.14) (354.26,-182.43) (353.55,-181.73) (353.55,-181.02) (352.85,-180.31) (352.14,-179.61) (352.14,-178.90) (351.43,-178.19) (350.72,-177.48) (350.72,-176.78) (350.02,-176.07) (349.31,-175.36) (349.31,-174.66) (348.60,-173.95) (347.90,-173.24) (347.90,-172.53) (347.19,-171.83) (346.48,-171.12) (346.48,-170.41) (345.78,-169.71) (345.07,-169.00) (345.07,-168.29) (344.36,-167.58) (352.85,-245.37) (352.85,-244.66) (353.55,-243.95) (353.55,-243.24) (354.26,-242.54) (354.26,-241.83) (354.97,-241.12) (354.97,-240.42) (354.97,-239.71) (355.67,-239.00) (355.67,-238.29) (356.38,-237.59) (356.38,-236.88) (357.09,-236.17) (357.09,-235.47) (357.09,-234.76) (357.80,-234.05) (357.80,-233.35) (358.50,-232.64) (358.50,-231.93) (359.21,-231.22) (359.21,-230.52) (359.21,-229.81) (359.92,-229.10) (359.92,-228.40) (360.62,-227.69) (360.62,-226.98) (361.33,-226.27) (361.33,-225.57) (361.33,-224.86) (362.04,-224.15) (362.04,-223.45) (362.75,-222.74) (362.75,-222.03) (363.45,-221.32) (363.45,-220.62) (363.45,-219.91) (364.16,-219.20) (364.16,-218.50) (364.87,-217.79) (364.87,-217.08) (365.57,-216.37) (365.57,-215.67) (365.57,-214.96) (366.28,-214.25) (366.28,-213.55) (366.99,-212.84) (366.99,-212.13) (367.70,-211.42) (367.70,-210.72) (367.70,-210.01) (368.40,-209.30) (368.40,-208.60) (369.11,-207.89) (369.11,-207.18) (369.82,-206.48) (369.82,-205.77) (352.85,-245.37) (353.55,-244.66) (354.26,-244.66) (354.97,-243.95) (355.67,-243.95) (356.38,-243.24) (357.09,-243.24) (357.80,-242.54) (358.50,-241.83) (359.21,-241.83) (359.92,-241.12) (360.62,-241.12) (361.33,-240.42) (362.04,-240.42) (362.75,-239.71) (363.45,-239.71) (364.16,-239.00) (364.87,-238.29) (365.57,-238.29) (366.28,-237.59) (366.99,-237.59) (367.70,-236.88) (368.40,-236.88) (369.11,-236.17) (369.82,-235.47) (370.52,-235.47) (371.23,-234.76) (371.94,-234.76) (372.65,-234.05) (373.35,-234.05) (374.06,-233.35) (374.77,-233.35) (375.47,-232.64) (376.18,-231.93) (376.89,-231.93) (377.60,-231.22) (378.30,-231.22) (379.01,-230.52) (379.72,-230.52) (380.42,-229.81) (381.13,-229.10) (381.84,-229.10) (382.54,-228.40) (383.25,-228.40) (383.96,-227.69) (384.67,-227.69) (385.37,-226.98) (386.08,-226.98) (386.79,-226.27) (387.49,-225.57) (388.20,-225.57) (388.91,-224.86) (389.62,-224.86) (390.32,-224.15) (391.03,-224.15) (391.74,-223.45) (391.74,-223.45) (392.44,-222.74) (393.15,-222.03) (393.86,-221.32) (394.57,-220.62) (395.27,-220.62) (395.98,-219.91) (396.69,-219.20) (397.39,-218.50) (398.10,-217.79) (398.81,-217.08) (399.52,-216.37) (400.22,-215.67) (400.93,-214.96) (401.64,-214.25) (402.34,-214.25) (403.05,-213.55) (403.76,-212.84) (404.47,-212.13) (405.17,-211.42) (405.88,-210.72) (406.59,-210.01) (407.29,-209.30) (408.00,-208.60) (408.71,-208.60) (409.41,-207.89) (410.12,-207.18) (410.83,-206.48) (411.54,-205.77) (412.24,-205.06) (412.95,-204.35) (413.66,-203.65) (414.36,-202.94) (415.07,-202.94) (415.78,-202.23) (416.49,-201.53) (417.19,-200.82) (417.90,-200.11) (418.61,-199.40) (419.31,-198.70) (420.02,-197.99) (420.73,-197.28) (421.44,-196.58) (422.14,-196.58) (422.85,-195.87) (423.56,-195.16) (424.26,-194.45) (424.97,-193.75) (384.67,-166.88) (385.37,-167.58) (386.08,-167.58) (386.79,-168.29) (387.49,-169.00) (388.20,-169.00) (388.91,-169.71) (389.62,-170.41) (390.32,-170.41) (391.03,-171.12) (391.74,-171.83) (392.44,-171.83) (393.15,-172.53) (393.86,-173.24) (394.57,-173.24) (395.27,-173.95) (395.98,-174.66) (396.69,-174.66) (397.39,-175.36) (398.10,-176.07) (398.81,-176.07) (399.52,-176.78) (400.22,-177.48) (400.93,-177.48) (401.64,-178.19) (402.34,-178.90) (403.05,-178.90) (403.76,-179.61) (404.47,-180.31) (405.17,-180.31) (405.88,-181.02) (406.59,-181.73) (407.29,-181.73) (408.00,-182.43) (408.71,-183.14) (409.41,-183.14) (410.12,-183.85) (410.83,-184.55) (411.54,-184.55) (412.24,-185.26) (412.95,-185.97) (413.66,-185.97) (414.36,-186.68) (415.07,-187.38) (415.78,-187.38) (416.49,-188.09) (417.19,-188.80) (417.90,-188.80) (418.61,-189.50) (419.31,-190.21) (420.02,-190.21) (420.73,-190.92) (421.44,-191.63) (422.14,-191.63) (422.85,-192.33) (423.56,-193.04) (424.26,-193.04) (424.97,-193.75) (384.67,-166.88) (384.67,-166.17) (383.96,-165.46) (383.96,-164.76) (383.96,-164.05) (383.25,-163.34) (383.25,-162.63) (382.54,-161.93) (382.54,-161.22) (382.54,-160.51) (381.84,-159.81) (381.84,-159.10) (381.84,-158.39) (381.13,-157.68) (381.13,-156.98) (381.13,-156.27) (380.42,-155.56) (380.42,-154.86) (379.72,-154.15) (379.72,-153.44) (379.72,-152.74) (379.01,-152.03) (379.01,-151.32) (379.01,-150.61) (378.30,-149.91) (378.30,-149.20) (378.30,-148.49) (377.60,-147.79) (377.60,-147.08) (376.89,-146.37) (376.89,-145.66) (376.89,-144.96) (376.18,-144.25) (376.18,-143.54) (376.18,-142.84) (375.47,-142.13) (375.47,-141.42) (374.77,-140.71) (374.77,-140.01) (374.77,-139.30) (374.06,-138.59) (374.06,-137.89) (374.06,-137.18) (373.35,-136.47) (373.35,-135.76) (373.35,-135.06) (372.65,-134.35) (372.65,-133.64) (371.94,-132.94) (371.94,-132.23) (371.94,-131.52) (371.23,-130.81) (371.23,-130.11) (371.23,-129.40) (370.52,-128.69) (370.52,-127.99) (370.52,-127.28) (369.82,-126.57) (369.82,-125.87) (369.11,-125.16) (369.11,-124.45) (369.11,-123.74) (368.40,-123.04) (368.40,-122.33) (368.40,-122.33) (369.11,-122.33) (369.82,-122.33) (370.52,-122.33) (371.23,-122.33) (371.94,-122.33) (372.65,-122.33) (373.35,-123.04) (374.06,-123.04) (374.77,-123.04) (375.47,-123.04) (376.18,-123.04) (376.89,-123.04) (377.60,-123.04) (378.30,-123.04) (379.01,-123.04) (379.72,-123.04) (380.42,-123.04) (381.13,-123.04) (381.84,-123.04) (382.54,-123.74) (383.25,-123.74) (383.96,-123.74) (384.67,-123.74) (385.37,-123.74) (386.08,-123.74) (386.79,-123.74) (387.49,-123.74) (388.20,-123.74) (388.91,-123.74) (389.62,-123.74) (390.32,-123.74) (391.03,-123.74) (391.74,-124.45) (392.44,-124.45) (393.15,-124.45) (393.86,-124.45) (394.57,-124.45) (395.27,-124.45) (395.98,-124.45) (396.69,-124.45) (397.39,-124.45) (398.10,-124.45) (398.81,-124.45) (399.52,-124.45) (400.22,-125.16) (400.93,-125.16) (401.64,-125.16) (402.34,-125.16) (403.05,-125.16) (403.76,-125.16) (404.47,-125.16) (405.17,-125.16) (405.88,-125.16) (406.59,-125.16) (407.29,-125.16) (408.00,-125.16) (408.71,-125.16) (409.41,-125.87) (410.12,-125.87) (410.83,-125.87) (411.54,-125.87) (412.24,-125.87) (412.95,-125.87) (413.66,-125.87) (413.66,-125.87) (414.36,-125.16) (414.36,-124.45) (415.07,-123.74) (415.78,-123.04) (416.49,-122.33) (416.49,-121.62) (417.19,-120.92) (417.90,-120.21) (418.61,-119.50) (418.61,-118.79) (419.31,-118.09) (420.02,-117.38) (420.73,-116.67) (420.73,-115.97) (421.44,-115.26) (422.14,-114.55) (422.85,-113.84) (422.85,-113.14) (423.56,-112.43) (424.26,-111.72) (424.97,-111.02) (424.97,-110.31) (425.68,-109.60) (426.39,-108.89) (426.39,-108.19) (427.09,-107.48) (427.80,-106.77) (428.51,-106.07) (428.51,-105.36) (429.21,-104.65) (429.92,-103.94) (430.63,-103.24) (430.63,-102.53) (431.34,-101.82) (432.04,-101.12) (432.75,-100.41) (432.75,-99.70) (433.46,-98.99) (434.16,-98.29) (434.87,-97.58) (434.87,-96.87) (435.58,-96.17) (436.28,-95.46) (436.99,-94.75) (436.99,-94.05) (437.70,-93.34) (438.41,-92.63) (439.11,-91.92) (439.11,-91.22) (439.82,-90.51) (404.47,-58.69) (405.17,-59.40) (405.88,-60.10) (406.59,-60.81) (407.29,-61.52) (408.00,-61.52) (408.71,-62.23) (409.41,-62.93) (410.12,-63.64) (410.83,-64.35) (411.54,-65.05) (412.24,-65.76) (412.95,-66.47) (413.66,-67.18) (414.36,-67.88) (415.07,-67.88) (415.78,-68.59) (416.49,-69.30) (417.19,-70.00) (417.90,-70.71) (418.61,-71.42) (419.31,-72.12) (420.02,-72.83) (420.73,-73.54) (421.44,-74.25) (422.14,-74.25) (422.85,-74.95) (423.56,-75.66) (424.26,-76.37) (424.97,-77.07) (425.68,-77.78) (426.39,-78.49) (427.09,-79.20) (427.80,-79.90) (428.51,-80.61) (429.21,-80.61) (429.92,-81.32) (430.63,-82.02) (431.34,-82.73) (432.04,-83.44) (432.75,-84.15) (433.46,-84.85) (434.16,-85.56) (434.87,-86.27) (435.58,-86.97) (436.28,-86.97) (436.99,-87.68) (437.70,-88.39) (438.41,-89.10) (439.11,-89.80) (439.82,-90.51) (404.47,-58.69) (404.47,-57.98) (405.17,-57.28) (405.17,-56.57) (405.88,-55.86) (405.88,-55.15) (406.59,-54.45) (406.59,-53.74) (407.29,-53.03) (407.29,-52.33) (408.00,-51.62) (408.00,-50.91) (408.71,-50.20) (408.71,-49.50) (409.41,-48.79) (409.41,-48.08) (410.12,-47.38) (410.12,-46.67) (410.83,-45.96) (410.83,-45.25) (411.54,-44.55) (411.54,-43.84) (412.24,-43.13) (412.24,-42.43) (412.95,-41.72) (412.95,-41.01) (413.66,-40.31) (413.66,-39.60) (414.36,-38.89) (414.36,-38.18) (415.07,-37.48) (415.07,-36.77) (415.78,-36.06) (415.78,-35.36) (416.49,-34.65) (416.49,-33.94) (417.19,-33.23) (417.19,-32.53) (417.90,-31.82) (417.90,-31.11) (418.61,-30.41) (418.61,-29.70) (419.31,-28.99) (419.31,-28.28) (420.02,-27.58) (420.02,-26.87) (420.73,-26.16) (420.73,-25.46) (421.44,-24.75) (421.44,-24.04) (422.14,-23.33) (422.14,-22.63) (422.85,-21.92) (422.85,-21.21) (423.56,-20.51) (423.56,-19.80) (424.26,-19.09) (424.26,-18.38) (424.97,-17.68) (424.97,-16.97) (392.44,-47.38) (393.15,-46.67) (393.86,-45.96) (394.57,-45.25) (395.27,-44.55) (395.98,-43.84) (396.69,-43.13) (397.39,-42.43) (398.10,-42.43) (398.81,-41.72) (399.52,-41.01) (400.22,-40.31) (400.93,-39.60) (401.64,-38.89) (402.34,-38.18) (403.05,-37.48) (403.76,-36.77) (404.47,-36.06) (405.17,-35.36) (405.88,-34.65) (406.59,-33.94) (407.29,-33.23) (408.00,-32.53) (408.71,-32.53) (409.41,-31.82) (410.12,-31.11) (410.83,-30.41) (411.54,-29.70) (412.24,-28.99) (412.95,-28.28) (413.66,-27.58) (414.36,-26.87) (415.07,-26.16) (415.78,-25.46) (416.49,-24.75) (417.19,-24.04) (417.90,-23.33) (418.61,-22.63) (419.31,-21.92) (420.02,-21.92) (420.73,-21.21) (421.44,-20.51) (422.14,-19.80) (422.85,-19.09) (423.56,-18.38) (424.26,-17.68) (424.97,-16.97) (392.44,-47.38) (393.15,-47.38) (393.86,-46.67) (394.57,-46.67) (395.27,-46.67) (395.98,-46.67) (396.69,-45.96) (397.39,-45.96) (398.10,-45.96) (398.81,-45.25) (399.52,-45.25) (400.22,-45.25) (400.93,-44.55) (401.64,-44.55) (402.34,-44.55) (403.05,-44.55) (403.76,-43.84) (404.47,-43.84) (405.17,-43.84) (405.88,-43.13) (406.59,-43.13) (407.29,-43.13) (408.00,-43.13) (408.71,-42.43) (409.41,-42.43) (410.12,-42.43) (410.83,-41.72) (411.54,-41.72) (412.24,-41.72) (412.95,-41.72) (413.66,-41.01) (414.36,-41.01) (415.07,-41.01) (415.78,-40.31) (416.49,-40.31) (417.19,-40.31) (417.90,-39.60) (418.61,-39.60) (419.31,-39.60) (420.02,-39.60) (420.73,-38.89) (421.44,-38.89) (422.14,-38.89) (422.85,-38.18) (423.56,-38.18) (424.26,-38.18) (424.97,-38.18) (425.68,-37.48) (426.39,-37.48) (427.09,-37.48) (427.80,-36.77) (428.51,-36.77) (429.21,-36.77) (429.92,-36.77) (430.63,-36.06) (431.34,-36.06) (432.04,-36.06) (432.75,-35.36) (433.46,-35.36) (434.16,-35.36) (434.87,-34.65) (435.58,-34.65) (436.28,-34.65) (436.99,-34.65) (437.70,-33.94) (438.41,-33.94) (438.41,-33.94) (439.11,-33.23) (439.11,-32.53) (439.82,-31.82) (440.53,-31.11) (440.53,-30.41) (441.23,-29.70) (441.94,-28.99) (442.65,-28.28) (442.65,-27.58) (443.36,-26.87) (444.06,-26.16) (444.06,-25.46) (444.77,-24.75) (445.48,-24.04) (445.48,-23.33) (446.18,-22.63) (446.89,-21.92) (447.60,-21.21) (447.60,-20.51) (448.31,-19.80) (449.01,-19.09) (449.01,-18.38) (449.72,-17.68) (450.43,-16.97) (450.43,-16.26) (451.13,-15.56) (451.84,-14.85) (452.55,-14.14) (452.55,-13.44) (453.26,-12.73) (453.96,-12.02) (453.96,-11.31) (454.67,-10.61) (455.38,-9.90) (455.38,-9.19) (456.08,-8.49) (456.79,-7.78) (457.50,-7.07) (457.50,-6.36) (458.21,-5.66) (458.91,-4.95) (458.91,-4.24) (459.62,-3.54) (460.33,-2.83) (460.33,-2.12) (461.03,-1.41) (461.74,-0.71) (462.45, 0.00) (462.45, 0.71) (463.15, 1.41) (463.86, 2.12) (463.86, 2.83) (464.57, 3.54) (422.85,-14.85) (423.56,-14.85) (424.26,-14.14) (424.97,-14.14) (425.68,-13.44) (426.39,-13.44) (427.09,-12.73) (427.80,-12.73) (428.51,-12.02) (429.21,-12.02) (429.92,-12.02) (430.63,-11.31) (431.34,-11.31) (432.04,-10.61) (432.75,-10.61) (433.46,-9.90) (434.16,-9.90) (434.87,-9.90) (435.58,-9.19) (436.28,-9.19) (436.99,-8.49) (437.70,-8.49) (438.41,-7.78) (439.11,-7.78) (439.82,-7.07) (440.53,-7.07) (441.23,-7.07) (441.94,-6.36) (442.65,-6.36) (443.36,-5.66) (444.06,-5.66) (444.77,-4.95) (445.48,-4.95) (446.18,-4.24) (446.89,-4.24) (447.60,-4.24) (448.31,-3.54) (449.01,-3.54) (449.72,-2.83) (450.43,-2.83) (451.13,-2.12) (451.84,-2.12) (452.55,-1.41) (453.26,-1.41) (453.96,-1.41) (454.67,-0.71) (455.38,-0.71) (456.08, 0.00) (456.79, 0.00) (457.50, 0.71) (458.21, 0.71) (458.91, 0.71) (459.62, 1.41) (460.33, 1.41) (461.03, 2.12) (461.74, 2.12) (462.45, 2.83) (463.15, 2.83) (463.86, 3.54) (464.57, 3.54) (441.94,-56.57) (441.94,-55.86) (441.23,-55.15) (441.23,-54.45) (440.53,-53.74) (440.53,-53.03) (439.82,-52.33) (439.82,-51.62) (439.11,-50.91) (439.11,-50.20) (438.41,-49.50) (438.41,-48.79) (438.41,-48.08) (437.70,-47.38) (437.70,-46.67) (436.99,-45.96) (436.99,-45.25) (436.28,-44.55) (436.28,-43.84) (435.58,-43.13) (435.58,-42.43) (434.87,-41.72) (434.87,-41.01) (434.16,-40.31) (434.16,-39.60) (434.16,-38.89) (433.46,-38.18) (433.46,-37.48) (432.75,-36.77) (432.75,-36.06) (432.04,-35.36) (432.04,-34.65) (431.34,-33.94) (431.34,-33.23) (430.63,-32.53) (430.63,-31.82) (430.63,-31.11) (429.92,-30.41) (429.92,-29.70) (429.21,-28.99) (429.21,-28.28) (428.51,-27.58) (428.51,-26.87) (427.80,-26.16) (427.80,-25.46) (427.09,-24.75) (427.09,-24.04) (426.39,-23.33) (426.39,-22.63) (426.39,-21.92) (425.68,-21.21) (425.68,-20.51) (424.97,-19.80) (424.97,-19.09) (424.26,-18.38) (424.26,-17.68) (423.56,-16.97) (423.56,-16.26) (422.85,-15.56) (422.85,-14.85) (399.52,-73.54) (400.22,-73.54) (400.93,-72.83) (401.64,-72.83) (402.34,-72.12) (403.05,-72.12) (403.76,-72.12) (404.47,-71.42) (405.17,-71.42) (405.88,-70.71) (406.59,-70.71) (407.29,-70.71) (408.00,-70.00) (408.71,-70.00) (409.41,-69.30) (410.12,-69.30) (410.83,-69.30) (411.54,-68.59) (412.24,-68.59) (412.95,-67.88) (413.66,-67.88) (414.36,-67.88) (415.07,-67.18) (415.78,-67.18) (416.49,-66.47) (417.19,-66.47) (417.90,-66.47) (418.61,-65.76) (419.31,-65.76) (420.02,-65.05) (420.73,-65.05) (421.44,-65.05) (422.14,-64.35) (422.85,-64.35) (423.56,-63.64) (424.26,-63.64) (424.97,-63.64) (425.68,-62.93) (426.39,-62.93) (427.09,-62.23) (427.80,-62.23) (428.51,-62.23) (429.21,-61.52) (429.92,-61.52) (430.63,-60.81) (431.34,-60.81) (432.04,-60.81) (432.75,-60.10) (433.46,-60.10) (434.16,-59.40) (434.87,-59.40) (435.58,-59.40) (436.28,-58.69) (436.99,-58.69) (437.70,-57.98) (438.41,-57.98) (439.11,-57.98) (439.82,-57.28) (440.53,-57.28) (441.23,-56.57) (441.94,-56.57) (405.88,-119.50) (405.88,-118.79) (405.88,-118.09) (405.88,-117.38) (405.17,-116.67) (405.17,-115.97) (405.17,-115.26) (405.17,-114.55) (405.17,-113.84) (405.17,-113.14) (405.17,-112.43) (404.47,-111.72) (404.47,-111.02) (404.47,-110.31) (404.47,-109.60) (404.47,-108.89) (404.47,-108.19) (404.47,-107.48) (404.47,-106.77) (403.76,-106.07) (403.76,-105.36) (403.76,-104.65) (403.76,-103.94) (403.76,-103.24) (403.76,-102.53) (403.76,-101.82) (403.05,-101.12) (403.05,-100.41) (403.05,-99.70) (403.05,-98.99) (403.05,-98.29) (403.05,-97.58) (403.05,-96.87) (402.34,-96.17) (402.34,-95.46) (402.34,-94.75) (402.34,-94.05) (402.34,-93.34) (402.34,-92.63) (402.34,-91.92) (401.64,-91.22) (401.64,-90.51) (401.64,-89.80) (401.64,-89.10) (401.64,-88.39) (401.64,-87.68) (401.64,-86.97) (400.93,-86.27) (400.93,-85.56) (400.93,-84.85) (400.93,-84.15) (400.93,-83.44) (400.93,-82.73) (400.93,-82.02) (400.93,-81.32) (400.22,-80.61) (400.22,-79.90) (400.22,-79.20) (400.22,-78.49) (400.22,-77.78) (400.22,-77.07) (400.22,-76.37) (399.52,-75.66) (399.52,-74.95) (399.52,-74.25) (399.52,-73.54) (359.92,-110.31) (360.62,-110.31) (361.33,-110.31) (362.04,-111.02) (362.75,-111.02) (363.45,-111.02) (364.16,-111.02) (364.87,-111.02) (365.57,-111.72) (366.28,-111.72) (366.99,-111.72) (367.70,-111.72) (368.40,-111.72) (369.11,-112.43) (369.82,-112.43) (370.52,-112.43) (371.23,-112.43) (371.94,-112.43) (372.65,-113.14) (373.35,-113.14) (374.06,-113.14) (374.77,-113.14) (375.47,-113.14) (376.18,-113.84) (376.89,-113.84) (377.60,-113.84) (378.30,-113.84) (379.01,-113.84) (379.72,-114.55) (380.42,-114.55) (381.13,-114.55) (381.84,-114.55) (382.54,-114.55) (383.25,-115.26) (383.96,-115.26) (384.67,-115.26) (385.37,-115.26) (386.08,-115.26) (386.79,-115.97) (387.49,-115.97) (388.20,-115.97) (388.91,-115.97) (389.62,-115.97) (390.32,-116.67) (391.03,-116.67) (391.74,-116.67) (392.44,-116.67) (393.15,-116.67) (393.86,-117.38) (394.57,-117.38) (395.27,-117.38) (395.98,-117.38) (396.69,-117.38) (397.39,-118.09) (398.10,-118.09) (398.81,-118.09) (399.52,-118.09) (400.22,-118.09) (400.93,-118.79) (401.64,-118.79) (402.34,-118.79) (403.05,-118.79) (403.76,-118.79) (404.47,-119.50) (405.17,-119.50) (405.88,-119.50) (325.98,-79.90) (326.68,-80.61) (327.39,-81.32) (328.10,-82.02) (328.80,-82.73) (329.51,-82.73) (330.22,-83.44) (330.93,-84.15) (331.63,-84.85) (332.34,-85.56) (333.05,-86.27) (333.75,-86.97) (334.46,-87.68) (335.17,-88.39) (335.88,-89.10) (336.58,-89.10) (337.29,-89.80) (338.00,-90.51) (338.70,-91.22) (339.41,-91.92) (340.12,-92.63) (340.83,-93.34) (341.53,-94.05) (342.24,-94.75) (342.95,-94.75) (343.65,-95.46) (344.36,-96.17) (345.07,-96.87) (345.78,-97.58) (346.48,-98.29) (347.19,-98.99) (347.90,-99.70) (348.60,-100.41) (349.31,-101.12) (350.02,-101.12) (350.72,-101.82) (351.43,-102.53) (352.14,-103.24) (352.85,-103.94) (353.55,-104.65) (354.26,-105.36) (354.97,-106.07) (355.67,-106.77) (356.38,-107.48) (357.09,-107.48) (357.80,-108.19) (358.50,-108.89) (359.21,-109.60) (359.92,-110.31) (325.98,-79.90) (325.98,-79.20) (325.98,-78.49) (325.98,-77.78) (325.98,-77.07) (326.68,-76.37) (326.68,-75.66) (326.68,-74.95) (326.68,-74.25) (326.68,-73.54) (326.68,-72.83) (326.68,-72.12) (326.68,-71.42) (326.68,-70.71) (327.39,-70.00) (327.39,-69.30) (327.39,-68.59) (327.39,-67.88) (327.39,-67.18) (327.39,-66.47) (327.39,-65.76) (327.39,-65.05) (327.39,-64.35) (328.10,-63.64) (328.10,-62.93) (328.10,-62.23) (328.10,-61.52) (328.10,-60.81) (328.10,-60.10) (328.10,-59.40) (328.10,-58.69) (328.10,-57.98) (328.10,-57.28) (328.80,-56.57) (328.80,-55.86) (328.80,-55.15) (328.80,-54.45) (328.80,-53.74) (328.80,-53.03) (328.80,-52.33) (328.80,-51.62) (328.80,-50.91) (329.51,-50.20) (329.51,-49.50) (329.51,-48.79) (329.51,-48.08) (329.51,-47.38) (329.51,-46.67) (329.51,-45.96) (329.51,-45.25) (329.51,-44.55) (330.22,-43.84) (330.22,-43.13) (330.22,-42.43) (330.22,-41.72) (330.22,-41.01) (330.22,-40.31) (330.22,-39.60) (330.22,-38.89) (330.22,-38.18) (330.93,-37.48) (330.93,-36.77) (330.93,-36.06) (330.93,-35.36) (330.93,-34.65) (302.64,-69.30) (303.35,-68.59) (304.06,-67.88) (304.06,-67.18) (304.76,-66.47) (305.47,-65.76) (306.18,-65.05) (306.88,-64.35) (307.59,-63.64) (307.59,-62.93) (308.30,-62.23) (309.01,-61.52) (309.71,-60.81) (310.42,-60.10) (310.42,-59.40) (311.13,-58.69) (311.83,-57.98) (312.54,-57.28) (313.25,-56.57) (313.96,-55.86) (313.96,-55.15) (314.66,-54.45) (315.37,-53.74) (316.08,-53.03) (316.78,-52.33) (316.78,-51.62) (317.49,-50.91) (318.20,-50.20) (318.91,-49.50) (319.61,-48.79) (319.61,-48.08) (320.32,-47.38) (321.03,-46.67) (321.73,-45.96) (322.44,-45.25) (323.15,-44.55) (323.15,-43.84) (323.85,-43.13) (324.56,-42.43) (325.27,-41.72) (325.98,-41.01) (325.98,-40.31) (326.68,-39.60) (327.39,-38.89) (328.10,-38.18) (328.80,-37.48) (329.51,-36.77) (329.51,-36.06) (330.22,-35.36) (330.93,-34.65) (302.64,-69.30) (303.35,-68.59) (304.06,-67.88) (304.06,-67.18) (304.76,-66.47) (305.47,-65.76) (306.18,-65.05) (306.88,-64.35) (307.59,-63.64) (307.59,-62.93) (308.30,-62.23) (309.01,-61.52) (309.71,-60.81) (310.42,-60.10) (310.42,-59.40) (311.13,-58.69) (311.83,-57.98) (312.54,-57.28) (313.25,-56.57) (313.96,-55.86) (313.96,-55.15) (314.66,-54.45) (315.37,-53.74) (316.08,-53.03) (316.78,-52.33) (316.78,-51.62) (317.49,-50.91) (318.20,-50.20) (318.91,-49.50) (319.61,-48.79) (319.61,-48.08) (320.32,-47.38) (321.03,-46.67) (321.73,-45.96) (322.44,-45.25) (323.15,-44.55) (323.15,-43.84) (323.85,-43.13) (324.56,-42.43) (325.27,-41.72) (325.98,-41.01) (325.98,-40.31) (326.68,-39.60) (327.39,-38.89) (328.10,-38.18) (328.80,-37.48) (329.51,-36.77) (329.51,-36.06) (330.22,-35.36) (330.93,-34.65) (289.91,-14.85) (290.62,-14.85) (291.33,-15.56) (292.04,-15.56) (292.74,-16.26) (293.45,-16.26) (294.16,-16.97) (294.86,-16.97) (295.57,-17.68) (296.28,-17.68) (296.98,-18.38) (297.69,-18.38) (298.40,-19.09) (299.11,-19.09) (299.81,-19.80) (300.52,-19.80) (301.23,-20.51) (301.93,-20.51) (302.64,-21.21) (303.35,-21.21) (304.06,-21.92) (304.76,-21.92) (305.47,-22.63) (306.18,-22.63) (306.88,-23.33) (307.59,-23.33) (308.30,-24.04) (309.01,-24.04) (309.71,-24.75) (310.42,-24.75) (311.13,-24.75) (311.83,-25.46) (312.54,-25.46) (313.25,-26.16) (313.96,-26.16) (314.66,-26.87) (315.37,-26.87) (316.08,-27.58) (316.78,-27.58) (317.49,-28.28) (318.20,-28.28) (318.91,-28.99) (319.61,-28.99) (320.32,-29.70) (321.03,-29.70) (321.73,-30.41) (322.44,-30.41) (323.15,-31.11) (323.85,-31.11) (324.56,-31.82) (325.27,-31.82) (325.98,-32.53) (326.68,-32.53) (327.39,-33.23) (328.10,-33.23) (328.80,-33.94) (329.51,-33.94) (330.22,-34.65) (330.93,-34.65) (289.91,-14.85) (289.21,-14.14) (288.50,-13.44) (287.79,-12.73) (287.09,-12.02) (287.09,-11.31) (286.38,-10.61) (285.67,-9.90) (284.96,-9.19) (284.26,-8.49) (283.55,-7.78) (282.84,-7.07) (282.14,-6.36) (281.43,-5.66) (280.72,-4.95) (280.72,-4.24) (280.01,-3.54) (279.31,-2.83) (278.60,-2.12) (277.89,-1.41) (277.19,-0.71) (276.48, 0.00) (275.77, 0.71) (275.06, 1.41) (274.36, 2.12) (274.36, 2.83) (273.65, 3.54) (272.94, 4.24) (272.24, 4.95) (271.53, 5.66) (270.82, 6.36) (270.11, 7.07) (269.41, 7.78) (268.70, 8.49) (267.99, 9.19) (267.99, 9.90) (267.29,10.61) (266.58,11.31) (265.87,12.02) (265.17,12.73) (264.46,13.44) (263.75,14.14) (263.04,14.85) (262.34,15.56) (261.63,16.26) (261.63,16.97) (260.92,17.68) (260.22,18.38) (259.51,19.09) (258.80,19.80) (258.80,19.80) (259.51,20.51) (260.22,21.21) (260.92,21.92) (260.92,22.63) (261.63,23.33) (262.34,24.04) (263.04,24.75) (263.75,25.46) (264.46,26.16) (265.17,26.87) (265.17,27.58) (265.87,28.28) (266.58,28.99) (267.29,29.70) (267.99,30.41) (268.70,31.11) (269.41,31.82) (269.41,32.53) (270.11,33.23) (270.82,33.94) (271.53,34.65) (272.24,35.36) (272.94,36.06) (272.94,36.77) (273.65,37.48) (274.36,38.18) (275.06,38.89) (275.77,39.60) (276.48,40.31) (277.19,41.01) (277.19,41.72) (277.89,42.43) (278.60,43.13) (279.31,43.84) (280.01,44.55) (280.72,45.25) (281.43,45.96) (281.43,46.67) (282.14,47.38) (282.84,48.08) (283.55,48.79) (284.26,49.50) (284.96,50.20) (285.67,50.91) (285.67,51.62) (286.38,52.33) (287.09,53.03) (287.79,53.74) (244.66,69.30) (245.37,69.30) (246.07,68.59) (246.78,68.59) (247.49,68.59) (248.19,67.88) (248.90,67.88) (249.61,67.18) (250.32,67.18) (251.02,67.18) (251.73,66.47) (252.44,66.47) (253.14,66.47) (253.85,65.76) (254.56,65.76) (255.27,65.76) (255.97,65.05) (256.68,65.05) (257.39,65.05) (258.09,64.35) (258.80,64.35) (259.51,63.64) (260.22,63.64) (260.92,63.64) (261.63,62.93) (262.34,62.93) (263.04,62.93) (263.75,62.23) (264.46,62.23) (265.17,62.23) (265.87,61.52) (266.58,61.52) (267.29,60.81) (267.99,60.81) (268.70,60.81) (269.41,60.10) (270.11,60.10) (270.82,60.10) (271.53,59.40) (272.24,59.40) (272.94,59.40) (273.65,58.69) (274.36,58.69) (275.06,57.98) (275.77,57.98) (276.48,57.98) (277.19,57.28) (277.89,57.28) (278.60,57.28) (279.31,56.57) (280.01,56.57) (280.72,56.57) (281.43,55.86) (282.14,55.86) (282.84,55.86) (283.55,55.15) (284.26,55.15) (284.96,54.45) (285.67,54.45) (286.38,54.45) (287.09,53.74) (287.79,53.74) (205.77,91.92) (206.48,91.22) (207.18,91.22) (207.89,90.51) (208.60,90.51) (209.30,89.80) (210.01,89.80) (210.72,89.10) (211.42,88.39) (212.13,88.39) (212.84,87.68) (213.55,87.68) (214.25,86.97) (214.96,86.27) (215.67,86.27) (216.37,85.56) (217.08,85.56) (217.79,84.85) (218.50,84.85) (219.20,84.15) (219.91,83.44) (220.62,83.44) (221.32,82.73) (222.03,82.73) (222.74,82.02) (223.45,81.32) (224.15,81.32) (224.86,80.61) (225.57,80.61) (226.27,79.90) (226.98,79.90) (227.69,79.20) (228.40,78.49) (229.10,78.49) (229.81,77.78) (230.52,77.78) (231.22,77.07) (231.93,76.37) (232.64,76.37) (233.35,75.66) (234.05,75.66) (234.76,74.95) (235.47,74.95) (236.17,74.25) (236.88,73.54) (237.59,73.54) (238.29,72.83) (239.00,72.83) (239.71,72.12) (240.42,71.42) (241.12,71.42) (241.83,70.71) (242.54,70.71) (243.24,70.00) (243.95,70.00) (244.66,69.30) (205.77,91.92) (205.77,92.63) (205.77,93.34) (205.77,94.05) (205.77,94.75) (205.77,95.46) (205.77,96.17) (205.77,96.87) (205.77,97.58) (205.77,98.29) (205.77,98.99) (205.06,99.70) (205.06,100.41) (205.06,101.12) (205.06,101.82) (205.06,102.53) (205.06,103.24) (205.06,103.94) (205.06,104.65) (205.06,105.36) (205.06,106.07) (205.06,106.77) (205.06,107.48) (205.06,108.19) (205.06,108.89) (205.06,109.60) (205.06,110.31) (205.06,111.02) (205.06,111.72) (205.06,112.43) (205.06,113.14) (205.06,113.84) (204.35,114.55) (204.35,115.26) (204.35,115.97) (204.35,116.67) (204.35,117.38) (204.35,118.09) (204.35,118.79) (204.35,119.50) (204.35,120.21) (204.35,120.92) (204.35,121.62) (204.35,122.33) (204.35,123.04) (204.35,123.74) (204.35,124.45) (204.35,125.16) (204.35,125.87) (204.35,126.57) (204.35,127.28) (204.35,127.99) (204.35,128.69) (203.65,129.40) (203.65,130.11) (203.65,130.81) (203.65,131.52) (203.65,132.23) (203.65,132.94) (203.65,133.64) (203.65,134.35) (203.65,135.06) (203.65,135.76) (203.65,136.47) (216.37,91.92) (216.37,92.63) (215.67,93.34) (215.67,94.05) (215.67,94.75) (215.67,95.46) (214.96,96.17) (214.96,96.87) (214.96,97.58) (214.25,98.29) (214.25,98.99) (214.25,99.70) (214.25,100.41) (213.55,101.12) (213.55,101.82) (213.55,102.53) (212.84,103.24) (212.84,103.94) (212.84,104.65) (212.84,105.36) (212.13,106.07) (212.13,106.77) (212.13,107.48) (211.42,108.19) (211.42,108.89) (211.42,109.60) (211.42,110.31) (210.72,111.02) (210.72,111.72) (210.72,112.43) (210.01,113.14) (210.01,113.84) (210.01,114.55) (210.01,115.26) (209.30,115.97) (209.30,116.67) (209.30,117.38) (208.60,118.09) (208.60,118.79) (208.60,119.50) (208.60,120.21) (207.89,120.92) (207.89,121.62) (207.89,122.33) (207.18,123.04) (207.18,123.74) (207.18,124.45) (207.18,125.16) (206.48,125.87) (206.48,126.57) (206.48,127.28) (205.77,127.99) (205.77,128.69) (205.77,129.40) (205.77,130.11) (205.06,130.81) (205.06,131.52) (205.06,132.23) (204.35,132.94) (204.35,133.64) (204.35,134.35) (204.35,135.06) (203.65,135.76) (203.65,136.47) (175.36,110.31) (176.07,110.31) (176.78,109.60) (177.48,109.60) (178.19,108.89) (178.90,108.89) (179.61,108.19) (180.31,108.19) (181.02,107.48) (181.73,107.48) (182.43,107.48) (183.14,106.77) (183.85,106.77) (184.55,106.07) (185.26,106.07) (185.97,105.36) (186.68,105.36) (187.38,104.65) (188.09,104.65) (188.80,103.94) (189.50,103.94) (190.21,103.94) (190.92,103.24) (191.63,103.24) (192.33,102.53) (193.04,102.53) (193.75,101.82) (194.45,101.82) (195.16,101.12) (195.87,101.12) (196.58,101.12) (197.28,100.41) (197.99,100.41) (198.70,99.70) (199.40,99.70) (200.11,98.99) (200.82,98.99) (201.53,98.29) (202.23,98.29) (202.94,98.29) (203.65,97.58) (204.35,97.58) (205.06,96.87) (205.77,96.87) (206.48,96.17) (207.18,96.17) (207.89,95.46) (208.60,95.46) (209.30,94.75) (210.01,94.75) (210.72,94.75) (211.42,94.05) (212.13,94.05) (212.84,93.34) (213.55,93.34) (214.25,92.63) (214.96,92.63) (215.67,91.92) (216.37,91.92) (137.89,137.89) (138.59,137.18) (139.30,137.18) (140.01,136.47) (140.71,135.76) (141.42,135.06) (142.13,135.06) (142.84,134.35) (143.54,133.64) (144.25,132.94) (144.96,132.94) (145.66,132.23) (146.37,131.52) (147.08,130.81) (147.79,130.81) (148.49,130.11) (149.20,129.40) (149.91,128.69) (150.61,128.69) (151.32,127.99) (152.03,127.28) (152.74,127.28) (153.44,126.57) (154.15,125.87) (154.86,125.16) (155.56,125.16) (156.27,124.45) (156.98,123.74) (157.68,123.04) (158.39,123.04) (159.10,122.33) (159.81,121.62) (160.51,120.92) (161.22,120.92) (161.93,120.21) (162.63,119.50) (163.34,119.50) (164.05,118.79) (164.76,118.09) (165.46,117.38) (166.17,117.38) (166.88,116.67) (167.58,115.97) (168.29,115.26) (169.00,115.26) (169.71,114.55) (170.41,113.84) (171.12,113.14) (171.83,113.14) (172.53,112.43) (173.24,111.72) (173.95,111.02) (174.66,111.02) (175.36,110.31) (137.89,137.89) (137.89,138.59) (138.59,139.30) (138.59,140.01) (138.59,140.71) (139.30,141.42) (139.30,142.13) (139.30,142.84) (140.01,143.54) (140.01,144.25) (140.01,144.96) (140.01,145.66) (140.71,146.37) (140.71,147.08) (140.71,147.79) (141.42,148.49) (141.42,149.20) (141.42,149.91) (142.13,150.61) (142.13,151.32) (142.13,152.03) (142.84,152.74) (142.84,153.44) (142.84,154.15) (143.54,154.86) (143.54,155.56) (143.54,156.27) (144.25,156.98) (144.25,157.68) (144.25,158.39) (144.25,159.10) (144.96,159.81) (144.96,160.51) (144.96,161.22) (145.66,161.93) (145.66,162.63) (145.66,163.34) (146.37,164.05) (146.37,164.76) (146.37,165.46) (147.08,166.17) (147.08,166.88) (147.08,167.58) (147.79,168.29) (147.79,169.00) (147.79,169.71) (148.49,170.41) (148.49,171.12) (148.49,171.83) (149.20,172.53) (149.20,173.24) (149.20,173.95) (149.20,174.66) (149.91,175.36) (149.91,176.07) (149.91,176.78) (150.61,177.48) (150.61,178.19) (150.61,178.90) (151.32,179.61) (151.32,180.31) (123.74,143.54) (124.45,144.25) (124.45,144.96) (125.16,145.66) (125.87,146.37) (126.57,147.08) (126.57,147.79) (127.28,148.49) (127.99,149.20) (128.69,149.91) (128.69,150.61) (129.40,151.32) (130.11,152.03) (130.81,152.74) (130.81,153.44) (131.52,154.15) (132.23,154.86) (132.94,155.56) (132.94,156.27) (133.64,156.98) (134.35,157.68) (135.06,158.39) (135.06,159.10) (135.76,159.81) (136.47,160.51) (137.18,161.22) (137.18,161.93) (137.89,162.63) (138.59,163.34) (139.30,164.05) (139.30,164.76) (140.01,165.46) (140.71,166.17) (141.42,166.88) (141.42,167.58) (142.13,168.29) (142.84,169.00) (143.54,169.71) (143.54,170.41) (144.25,171.12) (144.96,171.83) (145.66,172.53) (145.66,173.24) (146.37,173.95) (147.08,174.66) (147.79,175.36) (147.79,176.07) (148.49,176.78) (149.20,177.48) (149.91,178.19) (149.91,178.90) (150.61,179.61) (151.32,180.31) (123.74,143.54) (124.45,142.84) (125.16,142.84) (125.87,142.13) (126.57,141.42) (127.28,140.71) (127.99,140.71) (128.69,140.01) (129.40,139.30) (130.11,138.59) (130.81,138.59) (131.52,137.89) (132.23,137.18) (132.94,136.47) (133.64,136.47) (134.35,135.76) (135.06,135.06) (135.76,134.35) (136.47,134.35) (137.18,133.64) (137.89,132.94) (138.59,132.23) (139.30,132.23) (140.01,131.52) (140.71,130.81) (141.42,130.11) (142.13,130.11) (142.84,129.40) (143.54,128.69) (144.25,127.99) (144.96,127.99) (145.66,127.28) (146.37,126.57) (147.08,125.87) (147.79,125.87) (148.49,125.16) (149.20,124.45) (149.91,123.74) (150.61,123.74) (151.32,123.04) (152.03,122.33) (152.74,121.62) (153.44,121.62) (154.15,120.92) (154.86,120.21) (155.56,119.50) (156.27,119.50) (156.98,118.79) (157.68,118.09) (158.39,117.38) (159.10,117.38) (159.81,116.67) (159.81,116.67) (160.51,115.97) (161.22,115.97) (161.93,115.26) (162.63,114.55) (163.34,114.55) (164.05,113.84) (164.76,113.14) (165.46,113.14) (166.17,112.43) (166.88,111.72) (167.58,111.72) (168.29,111.02) (169.00,110.31) (169.71,110.31) (170.41,109.60) (171.12,108.89) (171.83,108.89) (172.53,108.19) (173.24,107.48) (173.95,107.48) (174.66,106.77) (175.36,106.07) (176.07,106.07) (176.78,105.36) (177.48,104.65) (178.19,104.65) (178.90,103.94) (179.61,103.24) (180.31,102.53) (181.02,102.53) (181.73,101.82) (182.43,101.12) (183.14,101.12) (183.85,100.41) (184.55,99.70) (185.26,99.70) (185.97,98.99) (186.68,98.29) (187.38,98.29) (188.09,97.58) (188.80,96.87) (189.50,96.87) (190.21,96.17) (190.92,95.46) (191.63,95.46) (192.33,94.75) (193.04,94.05) (193.75,94.05) (194.45,93.34) (195.16,92.63) (195.87,92.63) (196.58,91.92) (197.28,91.22) (197.99,91.22) (198.70,90.51) (220.62,51.62) (219.91,52.33) (219.91,53.03) (219.20,53.74) (219.20,54.45) (218.50,55.15) (218.50,55.86) (217.79,56.57) (217.08,57.28) (217.08,57.98) (216.37,58.69) (216.37,59.40) (215.67,60.10) (215.67,60.81) (214.96,61.52) (214.96,62.23) (214.25,62.93) (213.55,63.64) (213.55,64.35) (212.84,65.05) (212.84,65.76) (212.13,66.47) (212.13,67.18) (211.42,67.88) (210.72,68.59) (210.72,69.30) (210.01,70.00) (210.01,70.71) (209.30,71.42) (209.30,72.12) (208.60,72.83) (208.60,73.54) (207.89,74.25) (207.18,74.95) (207.18,75.66) (206.48,76.37) (206.48,77.07) (205.77,77.78) (205.77,78.49) (205.06,79.20) (204.35,79.90) (204.35,80.61) (203.65,81.32) (203.65,82.02) (202.94,82.73) (202.94,83.44) (202.23,84.15) (202.23,84.85) (201.53,85.56) (200.82,86.27) (200.82,86.97) (200.11,87.68) (200.11,88.39) (199.40,89.10) (199.40,89.80) (198.70,90.51) (198.70,11.31) (199.40,12.02) (199.40,12.73) (200.11,13.44) (200.11,14.14) (200.82,14.85) (200.82,15.56) (201.53,16.26) (201.53,16.97) (202.23,17.68) (202.23,18.38) (202.94,19.09) (203.65,19.80) (203.65,20.51) (204.35,21.21) (204.35,21.92) (205.06,22.63) (205.06,23.33) (205.77,24.04) (205.77,24.75) (206.48,25.46) (206.48,26.16) (207.18,26.87) (207.89,27.58) (207.89,28.28) (208.60,28.99) (208.60,29.70) (209.30,30.41) (209.30,31.11) (210.01,31.82) (210.01,32.53) (210.72,33.23) (210.72,33.94) (211.42,34.65) (211.42,35.36) (212.13,36.06) (212.84,36.77) (212.84,37.48) (213.55,38.18) (213.55,38.89) (214.25,39.60) (214.25,40.31) (214.96,41.01) (214.96,41.72) (215.67,42.43) (215.67,43.13) (216.37,43.84) (217.08,44.55) (217.08,45.25) (217.79,45.96) (217.79,46.67) (218.50,47.38) (218.50,48.08) (219.20,48.79) (219.20,49.50) (219.91,50.20) (219.91,50.91) (220.62,51.62) (198.70,11.31) (198.70,12.02) (199.40,12.73) (199.40,13.44) (199.40,14.14) (199.40,14.85) (200.11,15.56) (200.11,16.26) (200.11,16.97) (200.82,17.68) (200.82,18.38) (200.82,19.09) (200.82,19.80) (201.53,20.51) (201.53,21.21) (201.53,21.92) (202.23,22.63) (202.23,23.33) (202.23,24.04) (202.23,24.75) (202.94,25.46) (202.94,26.16) (202.94,26.87) (203.65,27.58) (203.65,28.28) (203.65,28.99) (203.65,29.70) (204.35,30.41) (204.35,31.11) (204.35,31.82) (205.06,32.53) (205.06,33.23) (205.06,33.94) (205.06,34.65) (205.77,35.36) (205.77,36.06) (205.77,36.77) (206.48,37.48) (206.48,38.18) (206.48,38.89) (206.48,39.60) (207.18,40.31) (207.18,41.01) (207.18,41.72) (207.89,42.43) (207.89,43.13) (207.89,43.84) (207.89,44.55) (208.60,45.25) (208.60,45.96) (208.60,46.67) (209.30,47.38) (209.30,48.08) (209.30,48.79) (209.30,49.50) (210.01,50.20) (210.01,50.91) (210.01,51.62) (210.72,52.33) (210.72,53.03) (210.72,53.74) (210.72,54.45) (211.42,55.15) (211.42,55.86) (169.71,46.67) (170.41,46.67) (171.12,46.67) (171.83,47.38) (172.53,47.38) (173.24,47.38) (173.95,47.38) (174.66,48.08) (175.36,48.08) (176.07,48.08) (176.78,48.08) (177.48,48.08) (178.19,48.79) (178.90,48.79) (179.61,48.79) (180.31,48.79) (181.02,49.50) (181.73,49.50) (182.43,49.50) (183.14,49.50) (183.85,49.50) (184.55,50.20) (185.26,50.20) (185.97,50.20) (186.68,50.20) (187.38,50.91) (188.09,50.91) (188.80,50.91) (189.50,50.91) (190.21,50.91) (190.92,51.62) (191.63,51.62) (192.33,51.62) (193.04,51.62) (193.75,51.62) (194.45,52.33) (195.16,52.33) (195.87,52.33) (196.58,52.33) (197.28,53.03) (197.99,53.03) (198.70,53.03) (199.40,53.03) (200.11,53.03) (200.82,53.74) (201.53,53.74) (202.23,53.74) (202.94,53.74) (203.65,54.45) (204.35,54.45) (205.06,54.45) (205.77,54.45) (206.48,54.45) (207.18,55.15) (207.89,55.15) (208.60,55.15) (209.30,55.15) (210.01,55.86) (210.72,55.86) (211.42,55.86) (138.59,16.97) (139.30,17.68) (140.01,18.38) (140.71,19.09) (141.42,19.80) (142.13,20.51) (142.84,21.21) (143.54,21.92) (144.25,22.63) (144.96,23.33) (145.66,24.04) (146.37,24.04) (147.08,24.75) (147.79,25.46) (148.49,26.16) (149.20,26.87) (149.91,27.58) (150.61,28.28) (151.32,28.99) (152.03,29.70) (152.74,30.41) (153.44,31.11) (154.15,31.82) (154.86,32.53) (155.56,33.23) (156.27,33.94) (156.98,34.65) (157.68,35.36) (158.39,36.06) (159.10,36.77) (159.81,37.48) (160.51,38.18) (161.22,38.89) (161.93,38.89) (162.63,39.60) (163.34,40.31) (164.05,41.01) (164.76,41.72) (165.46,42.43) (166.17,43.13) (166.88,43.84) (167.58,44.55) (168.29,45.25) (169.00,45.96) (169.71,46.67) (138.59,16.97) (139.30,16.97) (140.01,16.26) (140.71,16.26) (141.42,15.56) (142.13,15.56) (142.84,15.56) (143.54,14.85) (144.25,14.85) (144.96,14.14) (145.66,14.14) (146.37,13.44) (147.08,13.44) (147.79,13.44) (148.49,12.73) (149.20,12.73) (149.91,12.02) (150.61,12.02) (151.32,12.02) (152.03,11.31) (152.74,11.31) (153.44,10.61) (154.15,10.61) (154.86,10.61) (155.56, 9.90) (156.27, 9.90) (156.98, 9.19) (157.68, 9.19) (158.39, 9.19) (159.10, 8.49) (159.81, 8.49) (160.51, 7.78) (161.22, 7.78) (161.93, 7.07) (162.63, 7.07) (163.34, 7.07) (164.05, 6.36) (164.76, 6.36) (165.46, 5.66) (166.17, 5.66) (166.88, 5.66) (167.58, 4.95) (168.29, 4.95) (169.00, 4.24) (169.71, 4.24) (170.41, 4.24) (171.12, 3.54) (171.83, 3.54) (172.53, 2.83) (173.24, 2.83) (173.95, 2.83) (174.66, 2.12) (175.36, 2.12) (176.07, 1.41) (176.78, 1.41) (177.48, 0.71) (178.19, 0.71) (178.90, 0.71) (179.61, 0.00) (180.31, 0.00) (181.02,-0.71) (181.73,-0.71) (181.73,-0.71) (182.43,-0.71) (183.14,-1.41) (183.85,-1.41) (184.55,-2.12) (185.26,-2.12) (185.97,-2.83) (186.68,-2.83) (187.38,-3.54) (188.09,-3.54) (188.80,-3.54) (189.50,-4.24) (190.21,-4.24) (190.92,-4.95) (191.63,-4.95) (192.33,-5.66) (193.04,-5.66) (193.75,-6.36) (194.45,-6.36) (195.16,-6.36) (195.87,-7.07) (196.58,-7.07) (197.28,-7.78) (197.99,-7.78) (198.70,-8.49) (199.40,-8.49) (200.11,-9.19) (200.82,-9.19) (201.53,-9.19) (202.23,-9.90) (202.94,-9.90) (203.65,-10.61) (204.35,-10.61) (205.06,-11.31) (205.77,-11.31) (206.48,-12.02) (207.18,-12.02) (207.89,-12.73) (208.60,-12.73) (209.30,-12.73) (210.01,-13.44) (210.72,-13.44) (211.42,-14.14) (212.13,-14.14) (212.84,-14.85) (213.55,-14.85) (214.25,-15.56) (214.96,-15.56) (215.67,-15.56) (216.37,-16.26) (217.08,-16.26) (217.79,-16.97) (218.50,-16.97) (219.20,-17.68) (219.91,-17.68) (220.62,-18.38) (221.32,-18.38) (221.32,-18.38) (222.03,-19.09) (222.74,-19.09) (223.45,-19.80) (224.15,-19.80) (224.86,-20.51) (225.57,-20.51) (226.27,-21.21) (226.98,-21.21) (227.69,-21.92) (228.40,-21.92) (229.10,-22.63) (229.81,-22.63) (230.52,-23.33) (231.22,-23.33) (231.93,-24.04) (232.64,-24.04) (233.35,-24.75) (234.05,-24.75) (234.76,-25.46) (235.47,-26.16) (236.17,-26.16) (236.88,-26.87) (237.59,-26.87) (238.29,-27.58) (239.00,-27.58) (239.71,-28.28) (240.42,-28.28) (241.12,-28.99) (241.83,-28.99) (242.54,-29.70) (243.24,-29.70) (243.95,-30.41) (244.66,-30.41) (245.37,-31.11) (246.07,-31.11) (246.78,-31.82) (247.49,-31.82) (248.19,-32.53) (248.90,-33.23) (249.61,-33.23) (250.32,-33.94) (251.02,-33.94) (251.73,-34.65) (252.44,-34.65) (253.14,-35.36) (253.85,-35.36) (254.56,-36.06) (255.27,-36.06) (255.97,-36.77) (256.68,-36.77) (257.39,-37.48) (258.09,-37.48) (258.80,-38.18) (259.51,-38.18) (260.22,-38.89) (260.92,-38.89) (261.63,-39.60) (286.38,-74.95) (285.67,-74.25) (285.67,-73.54) (284.96,-72.83) (284.26,-72.12) (284.26,-71.42) (283.55,-70.71) (282.84,-70.00) (282.14,-69.30) (282.14,-68.59) (281.43,-67.88) (280.72,-67.18) (280.72,-66.47) (280.01,-65.76) (279.31,-65.05) (279.31,-64.35) (278.60,-63.64) (277.89,-62.93) (277.19,-62.23) (277.19,-61.52) (276.48,-60.81) (275.77,-60.10) (275.77,-59.40) (275.06,-58.69) (274.36,-57.98) (274.36,-57.28) (273.65,-56.57) (272.94,-55.86) (272.24,-55.15) (272.24,-54.45) (271.53,-53.74) (270.82,-53.03) (270.82,-52.33) (270.11,-51.62) (269.41,-50.91) (269.41,-50.20) (268.70,-49.50) (267.99,-48.79) (267.29,-48.08) (267.29,-47.38) (266.58,-46.67) (265.87,-45.96) (265.87,-45.25) (265.17,-44.55) (264.46,-43.84) (264.46,-43.13) (263.75,-42.43) (263.04,-41.72) (262.34,-41.01) (262.34,-40.31) (261.63,-39.60) (269.41,-118.79) (269.41,-118.09) (270.11,-117.38) (270.11,-116.67) (270.82,-115.97) (270.82,-115.26) (270.82,-114.55) (271.53,-113.84) (271.53,-113.14) (271.53,-112.43) (272.24,-111.72) (272.24,-111.02) (272.94,-110.31) (272.94,-109.60) (272.94,-108.89) (273.65,-108.19) (273.65,-107.48) (274.36,-106.77) (274.36,-106.07) (274.36,-105.36) (275.06,-104.65) (275.06,-103.94) (275.77,-103.24) (275.77,-102.53) (275.77,-101.82) (276.48,-101.12) (276.48,-100.41) (276.48,-99.70) (277.19,-98.99) (277.19,-98.29) (277.89,-97.58) (277.89,-96.87) (277.89,-96.17) (278.60,-95.46) (278.60,-94.75) (279.31,-94.05) (279.31,-93.34) (279.31,-92.63) (280.01,-91.92) (280.01,-91.22) (280.01,-90.51) (280.72,-89.80) (280.72,-89.10) (281.43,-88.39) (281.43,-87.68) (281.43,-86.97) (282.14,-86.27) (282.14,-85.56) (282.84,-84.85) (282.84,-84.15) (282.84,-83.44) (283.55,-82.73) (283.55,-82.02) (284.26,-81.32) (284.26,-80.61) (284.26,-79.90) (284.96,-79.20) (284.96,-78.49) (284.96,-77.78) (285.67,-77.07) (285.67,-76.37) (286.38,-75.66) (286.38,-74.95) (269.41,-118.79) (269.41,-118.09) (270.11,-117.38) (270.11,-116.67) (270.11,-115.97) (270.82,-115.26) (270.82,-114.55) (271.53,-113.84) (271.53,-113.14) (271.53,-112.43) (272.24,-111.72) (272.24,-111.02) (272.24,-110.31) (272.94,-109.60) (272.94,-108.89) (272.94,-108.19) (273.65,-107.48) (273.65,-106.77) (273.65,-106.07) (274.36,-105.36) (274.36,-104.65) (275.06,-103.94) (275.06,-103.24) (275.06,-102.53) (275.77,-101.82) (275.77,-101.12) (275.77,-100.41) (276.48,-99.70) (276.48,-98.99) (276.48,-98.29) (277.19,-97.58) (277.19,-96.87) (277.19,-96.17) (277.89,-95.46) (277.89,-94.75) (278.60,-94.05) (278.60,-93.34) (278.60,-92.63) (279.31,-91.92) (279.31,-91.22) (279.31,-90.51) (280.01,-89.80) (280.01,-89.10) (280.01,-88.39) (280.72,-87.68) (280.72,-86.97) (281.43,-86.27) (281.43,-85.56) (281.43,-84.85) (282.14,-84.15) (282.14,-83.44) (282.14,-82.73) (282.84,-82.02) (282.84,-81.32) (282.84,-80.61) (283.55,-79.90) (283.55,-79.20) (283.55,-78.49) (284.26,-77.78) (284.26,-77.07) (284.96,-76.37) (284.96,-75.66) (284.96,-74.95) (285.67,-74.25) (285.67,-73.54) (271.53,-115.97) (271.53,-115.26) (272.24,-114.55) (272.24,-113.84) (272.24,-113.14) (272.94,-112.43) (272.94,-111.72) (272.94,-111.02) (273.65,-110.31) (273.65,-109.60) (273.65,-108.89) (274.36,-108.19) (274.36,-107.48) (274.36,-106.77) (275.06,-106.07) (275.06,-105.36) (275.06,-104.65) (275.77,-103.94) (275.77,-103.24) (275.77,-102.53) (276.48,-101.82) (276.48,-101.12) (276.48,-100.41) (277.19,-99.70) (277.19,-98.99) (277.19,-98.29) (277.89,-97.58) (277.89,-96.87) (277.89,-96.17) (278.60,-95.46) (278.60,-94.75) (278.60,-94.05) (279.31,-93.34) (279.31,-92.63) (279.31,-91.92) (280.01,-91.22) (280.01,-90.51) (280.01,-89.80) (280.72,-89.10) (280.72,-88.39) (280.72,-87.68) (281.43,-86.97) (281.43,-86.27) (281.43,-85.56) (282.14,-84.85) (282.14,-84.15) (282.14,-83.44) (282.84,-82.73) (282.84,-82.02) (282.84,-81.32) (283.55,-80.61) (283.55,-79.90) (283.55,-79.20) (284.26,-78.49) (284.26,-77.78) (284.26,-77.07) (284.96,-76.37) (284.96,-75.66) (284.96,-74.95) (285.67,-74.25) (285.67,-73.54) (271.53,-115.97) (272.24,-115.26) (272.94,-114.55) (273.65,-113.84) (273.65,-113.14) (274.36,-112.43) (275.06,-111.72) (275.77,-111.02) (276.48,-110.31) (277.19,-109.60) (277.19,-108.89) (277.89,-108.19) (278.60,-107.48) (279.31,-106.77) (280.01,-106.07) (280.72,-105.36) (280.72,-104.65) (281.43,-103.94) (282.14,-103.24) (282.84,-102.53) (283.55,-101.82) (284.26,-101.12) (284.96,-100.41) (284.96,-99.70) (285.67,-98.99) (286.38,-98.29) (287.09,-97.58) (287.79,-96.87) (288.50,-96.17) (288.50,-95.46) (289.21,-94.75) (289.91,-94.05) (290.62,-93.34) (291.33,-92.63) (292.04,-91.92) (292.74,-91.22) (292.74,-90.51) (293.45,-89.80) (294.16,-89.10) (294.86,-88.39) (295.57,-87.68) (296.28,-86.97) (296.28,-86.27) (296.98,-85.56) (297.69,-84.85) (298.40,-84.15) (299.11,-83.44) (299.81,-82.73) (299.81,-82.02) (300.52,-81.32) (301.23,-80.61) (301.93,-79.90) (265.87,-52.33) (266.58,-53.03) (267.29,-53.74) (267.99,-53.74) (268.70,-54.45) (269.41,-55.15) (270.11,-55.86) (270.82,-55.86) (271.53,-56.57) (272.24,-57.28) (272.94,-57.98) (273.65,-57.98) (274.36,-58.69) (275.06,-59.40) (275.77,-60.10) (276.48,-60.10) (277.19,-60.81) (277.89,-61.52) (278.60,-62.23) (279.31,-62.93) (280.01,-62.93) (280.72,-63.64) (281.43,-64.35) (282.14,-65.05) (282.84,-65.05) (283.55,-65.76) (284.26,-66.47) (284.96,-67.18) (285.67,-67.18) (286.38,-67.88) (287.09,-68.59) (287.79,-69.30) (288.50,-69.30) (289.21,-70.00) (289.91,-70.71) (290.62,-71.42) (291.33,-72.12) (292.04,-72.12) (292.74,-72.83) (293.45,-73.54) (294.16,-74.25) (294.86,-74.25) (295.57,-74.95) (296.28,-75.66) (296.98,-76.37) (297.69,-76.37) (298.40,-77.07) (299.11,-77.78) (299.81,-78.49) (300.52,-78.49) (301.23,-79.20) (301.93,-79.90) (265.87,-52.33) (265.87,-51.62) (266.58,-50.91) (266.58,-50.20) (267.29,-49.50) (267.29,-48.79) (267.29,-48.08) (267.99,-47.38) (267.99,-46.67) (268.70,-45.96) (268.70,-45.25) (269.41,-44.55) (269.41,-43.84) (269.41,-43.13) (270.11,-42.43) (270.11,-41.72) (270.82,-41.01) (270.82,-40.31) (270.82,-39.60) (271.53,-38.89) (271.53,-38.18) (272.24,-37.48) (272.24,-36.77) (272.94,-36.06) (272.94,-35.36) (272.94,-34.65) (273.65,-33.94) (273.65,-33.23) (274.36,-32.53) (274.36,-31.82) (274.36,-31.11) (275.06,-30.41) (275.06,-29.70) (275.77,-28.99) (275.77,-28.28) (275.77,-27.58) (276.48,-26.87) (276.48,-26.16) (277.19,-25.46) (277.19,-24.75) (277.89,-24.04) (277.89,-23.33) (277.89,-22.63) (278.60,-21.92) (278.60,-21.21) (279.31,-20.51) (279.31,-19.80) (279.31,-19.09) (280.01,-18.38) (280.01,-17.68) (280.72,-16.97) (280.72,-16.26) (281.43,-15.56) (281.43,-14.85) (281.43,-14.14) (282.14,-13.44) (282.14,-12.73) (282.84,-12.02) (282.84,-11.31) (273.65,-55.86) (273.65,-55.15) (273.65,-54.45) (274.36,-53.74) (274.36,-53.03) (274.36,-52.33) (274.36,-51.62) (274.36,-50.91) (275.06,-50.20) (275.06,-49.50) (275.06,-48.79) (275.06,-48.08) (275.06,-47.38) (275.77,-46.67) (275.77,-45.96) (275.77,-45.25) (275.77,-44.55) (276.48,-43.84) (276.48,-43.13) (276.48,-42.43) (276.48,-41.72) (276.48,-41.01) (277.19,-40.31) (277.19,-39.60) (277.19,-38.89) (277.19,-38.18) (277.19,-37.48) (277.89,-36.77) (277.89,-36.06) (277.89,-35.36) (277.89,-34.65) (277.89,-33.94) (278.60,-33.23) (278.60,-32.53) (278.60,-31.82) (278.60,-31.11) (278.60,-30.41) (279.31,-29.70) (279.31,-28.99) (279.31,-28.28) (279.31,-27.58) (279.31,-26.87) (280.01,-26.16) (280.01,-25.46) (280.01,-24.75) (280.01,-24.04) (280.01,-23.33) (280.72,-22.63) (280.72,-21.92) (280.72,-21.21) (280.72,-20.51) (281.43,-19.80) (281.43,-19.09) (281.43,-18.38) (281.43,-17.68) (281.43,-16.97) (282.14,-16.26) (282.14,-15.56) (282.14,-14.85) (282.14,-14.14) (282.14,-13.44) (282.84,-12.73) (282.84,-12.02) (282.84,-11.31) (273.65,-55.86) (273.65,-55.15) (274.36,-54.45) (274.36,-53.74) (274.36,-53.03) (274.36,-52.33) (275.06,-51.62) (275.06,-50.91) (275.06,-50.20) (275.06,-49.50) (275.77,-48.79) (275.77,-48.08) (275.77,-47.38) (275.77,-46.67) (276.48,-45.96) (276.48,-45.25) (276.48,-44.55) (276.48,-43.84) (277.19,-43.13) (277.19,-42.43) (277.19,-41.72) (277.19,-41.01) (277.89,-40.31) (277.89,-39.60) (277.89,-38.89) (278.60,-38.18) (278.60,-37.48) (278.60,-36.77) (278.60,-36.06) (279.31,-35.36) (279.31,-34.65) (279.31,-33.94) (279.31,-33.23) (280.01,-32.53) (280.01,-31.82) (280.01,-31.11) (280.01,-30.41) (280.72,-29.70) (280.72,-28.99) (280.72,-28.28) (280.72,-27.58) (281.43,-26.87) (281.43,-26.16) (281.43,-25.46) (282.14,-24.75) (282.14,-24.04) (282.14,-23.33) (282.14,-22.63) (282.84,-21.92) (282.84,-21.21) (282.84,-20.51) (282.84,-19.80) (283.55,-19.09) (283.55,-18.38) (283.55,-17.68) (283.55,-16.97) (284.26,-16.26) (284.26,-15.56) (284.26,-14.85) (284.26,-14.14) (284.96,-13.44) (284.96,-12.73) (284.96,-12.02) (284.96,-11.31) (285.67,-10.61) (285.67,-9.90) (266.58,-49.50) (266.58,-48.79) (267.29,-48.08) (267.29,-47.38) (267.99,-46.67) (267.99,-45.96) (268.70,-45.25) (268.70,-44.55) (269.41,-43.84) (269.41,-43.13) (270.11,-42.43) (270.11,-41.72) (270.82,-41.01) (270.82,-40.31) (271.53,-39.60) (271.53,-38.89) (272.24,-38.18) (272.24,-37.48) (272.94,-36.77) (272.94,-36.06) (273.65,-35.36) (273.65,-34.65) (274.36,-33.94) (274.36,-33.23) (275.06,-32.53) (275.06,-31.82) (275.77,-31.11) (275.77,-30.41) (275.77,-29.70) (276.48,-28.99) (276.48,-28.28) (277.19,-27.58) (277.19,-26.87) (277.89,-26.16) (277.89,-25.46) (278.60,-24.75) (278.60,-24.04) (279.31,-23.33) (279.31,-22.63) (280.01,-21.92) (280.01,-21.21) (280.72,-20.51) (280.72,-19.80) (281.43,-19.09) (281.43,-18.38) (282.14,-17.68) (282.14,-16.97) (282.84,-16.26) (282.84,-15.56) (283.55,-14.85) (283.55,-14.14) (284.26,-13.44) (284.26,-12.73) (284.96,-12.02) (284.96,-11.31) (285.67,-10.61) (285.67,-9.90) (266.58,-49.50) (267.29,-48.79) (267.99,-48.79) (268.70,-48.08) (269.41,-48.08) (270.11,-47.38) (270.82,-47.38) (271.53,-46.67) (272.24,-46.67) (272.94,-45.96) (273.65,-45.25) (274.36,-45.25) (275.06,-44.55) (275.77,-44.55) (276.48,-43.84) (277.19,-43.84) (277.89,-43.13) (278.60,-42.43) (279.31,-42.43) (280.01,-41.72) (280.72,-41.72) (281.43,-41.01) (282.14,-41.01) (282.84,-40.31) (283.55,-40.31) (284.26,-39.60) (284.96,-38.89) (285.67,-38.89) (286.38,-38.18) (287.09,-38.18) (287.79,-37.48) (288.50,-37.48) (289.21,-36.77) (289.91,-36.06) (290.62,-36.06) (291.33,-35.36) (292.04,-35.36) (292.74,-34.65) (293.45,-34.65) (294.16,-33.94) (294.86,-33.94) (295.57,-33.23) (296.28,-32.53) (296.98,-32.53) (297.69,-31.82) (298.40,-31.82) (299.11,-31.11) (299.81,-31.11) (300.52,-30.41) (301.23,-29.70) (301.93,-29.70) (302.64,-28.99) (303.35,-28.99) (304.06,-28.28) (304.76,-28.28) (305.47,-27.58) (306.18,-27.58) (306.88,-26.87) (306.88,-26.87) (306.88,-26.16) (306.88,-25.46) (306.88,-24.75) (306.88,-24.04) (306.88,-23.33) (306.88,-22.63) (306.88,-21.92) (306.88,-21.21) (306.88,-20.51) (306.88,-19.80) (306.88,-19.09) (306.88,-18.38) (306.88,-17.68) (306.88,-16.97) (306.88,-16.26) (306.88,-15.56) (306.88,-14.85) (306.88,-14.14) (306.88,-13.44) (306.88,-12.73) (306.88,-12.02) (306.88,-11.31) (306.88,-10.61) (306.88,-9.90) (306.88,-9.19) (306.88,-8.49) (306.88,-7.78) (306.88,-7.07) (306.88,-6.36) (306.88,-5.66) (306.88,-4.95) (306.88,-4.24) (306.88,-3.54) (306.88,-2.83) (306.88,-2.12) (306.88,-1.41) (306.88,-0.71) (306.88, 0.00) (306.88, 0.71) (306.88, 1.41) (306.88, 2.12) (306.88, 2.83) (306.88, 3.54) (306.88, 4.24) (306.88, 4.95) (306.88, 5.66) (306.88, 6.36) (306.88, 7.07) (306.88, 7.78) (306.88, 8.49) (306.88, 9.19) (306.88, 9.90) (306.88,10.61) (306.88,11.31) (306.88,12.02) (306.88,12.73) (306.88,13.44) (306.88,14.14) (306.88,14.85) (306.88,15.56) (306.88,16.26) (306.88,16.97) (306.88,17.68) (306.88,17.68) (307.59,16.97) (308.30,16.97) (309.01,16.26) (309.71,16.26) (310.42,15.56) (311.13,14.85) (311.83,14.85) (312.54,14.14) (313.25,14.14) (313.96,13.44) (314.66,13.44) (315.37,12.73) (316.08,12.02) (316.78,12.02) (317.49,11.31) (318.20,11.31) (318.91,10.61) (319.61, 9.90) (320.32, 9.90) (321.03, 9.19) (321.73, 9.19) (322.44, 8.49) (323.15, 7.78) (323.85, 7.78) (324.56, 7.07) (325.27, 7.07) (325.98, 6.36) (326.68, 6.36) (327.39, 5.66) (328.10, 4.95) (328.80, 4.95) (329.51, 4.24) (330.22, 4.24) (330.93, 3.54) (331.63, 2.83) (332.34, 2.83) (333.05, 2.12) (333.75, 2.12) (334.46, 1.41) (335.17, 0.71) (335.88, 0.71) (336.58, 0.00) (337.29, 0.00) (338.00,-0.71) (338.70,-0.71) (339.41,-1.41) (340.12,-2.12) (340.83,-2.12) (341.53,-2.83) (342.24,-2.83) (342.95,-3.54) (342.95,-3.54) (343.65,-3.54) (344.36,-4.24) (345.07,-4.24) (345.78,-4.95) (346.48,-4.95) (347.19,-4.95) (347.90,-5.66) (348.60,-5.66) (349.31,-5.66) (350.02,-6.36) (350.72,-6.36) (351.43,-7.07) (352.14,-7.07) (352.85,-7.07) (353.55,-7.78) (354.26,-7.78) (354.97,-7.78) (355.67,-8.49) (356.38,-8.49) (357.09,-9.19) (357.80,-9.19) (358.50,-9.19) (359.21,-9.90) (359.92,-9.90) (360.62,-10.61) (361.33,-10.61) (362.04,-10.61) (362.75,-11.31) (363.45,-11.31) (364.16,-11.31) (364.87,-12.02) (365.57,-12.02) (366.28,-12.73) (366.99,-12.73) (367.70,-12.73) (368.40,-13.44) (369.11,-13.44) (369.82,-13.44) (370.52,-14.14) (371.23,-14.14) (371.94,-14.85) (372.65,-14.85) (373.35,-14.85) (374.06,-15.56) (374.77,-15.56) (375.47,-16.26) (376.18,-16.26) (376.89,-16.26) (377.60,-16.97) (378.30,-16.97) (379.01,-16.97) (379.72,-17.68) (380.42,-17.68) (381.13,-18.38) (381.84,-18.38) (382.54,-18.38) (383.25,-19.09) (383.96,-19.09) (384.67,-19.09) (385.37,-19.80) (386.08,-19.80) (386.79,-20.51) (387.49,-20.51) (387.49,-20.51) (388.20,-20.51) (388.91,-20.51) (389.62,-20.51) (390.32,-20.51) (391.03,-19.80) (391.74,-19.80) (392.44,-19.80) (393.15,-19.80) (393.86,-19.80) (394.57,-19.80) (395.27,-19.80) (395.98,-19.80) (396.69,-19.80) (397.39,-19.80) (398.10,-19.09) (398.81,-19.09) (399.52,-19.09) (400.22,-19.09) (400.93,-19.09) (401.64,-19.09) (402.34,-19.09) (403.05,-19.09) (403.76,-19.09) (404.47,-18.38) (405.17,-18.38) (405.88,-18.38) (406.59,-18.38) (407.29,-18.38) (408.00,-18.38) (408.71,-18.38) (409.41,-18.38) (410.12,-18.38) (410.83,-18.38) (411.54,-17.68) (412.24,-17.68) (412.95,-17.68) (413.66,-17.68) (414.36,-17.68) (415.07,-17.68) (415.78,-17.68) (416.49,-17.68) (417.19,-17.68) (417.90,-17.68) (418.61,-16.97) (419.31,-16.97) (420.02,-16.97) (420.73,-16.97) (421.44,-16.97) (422.14,-16.97) (422.85,-16.97) (423.56,-16.97) (424.26,-16.97) (424.97,-16.26) (425.68,-16.26) (426.39,-16.26) (427.09,-16.26) (427.80,-16.26) (428.51,-16.26) (429.21,-16.26) (429.92,-16.26) (430.63,-16.26) (431.34,-16.26) (432.04,-15.56) (432.75,-15.56) (433.46,-15.56) (434.16,-15.56) (434.87,-15.56) (434.87,-15.56) (435.58,-15.56) (436.28,-14.85) (436.99,-14.85) (437.70,-14.85) (438.41,-14.85) (439.11,-14.14) (439.82,-14.14) (440.53,-14.14) (441.23,-13.44) (441.94,-13.44) (442.65,-13.44) (443.36,-12.73) (444.06,-12.73) (444.77,-12.73) (445.48,-12.73) (446.18,-12.02) (446.89,-12.02) (447.60,-12.02) (448.31,-11.31) (449.01,-11.31) (449.72,-11.31) (450.43,-10.61) (451.13,-10.61) (451.84,-10.61) (452.55,-10.61) (453.26,-9.90) (453.96,-9.90) (454.67,-9.90) (455.38,-9.19) (456.08,-9.19) (456.79,-9.19) (457.50,-8.49) (458.21,-8.49) (458.91,-8.49) (459.62,-8.49) (460.33,-7.78) (461.03,-7.78) (461.74,-7.78) (462.45,-7.07) (463.15,-7.07) (463.86,-7.07) (464.57,-6.36) (465.28,-6.36) (465.98,-6.36) (466.69,-6.36) (467.40,-5.66) (468.10,-5.66) (468.81,-5.66) (469.52,-4.95) (470.23,-4.95) (470.93,-4.95) (471.64,-4.24) (472.35,-4.24) (473.05,-4.24) (473.76,-4.24) (474.47,-3.54) (475.18,-3.54) (475.88,-3.54) (476.59,-2.83) (477.30,-2.83) (477.30,-2.83) (477.30,-2.12) (478.00,-1.41) (478.00,-0.71) (478.00, 0.00) (478.71, 0.71) (478.71, 1.41) (478.71, 2.12) (479.42, 2.83) (479.42, 3.54) (479.42, 4.24) (480.13, 4.95) (480.13, 5.66) (480.13, 6.36) (480.83, 7.07) (480.83, 7.78) (480.83, 8.49) (481.54, 9.19) (481.54, 9.90) (481.54,10.61) (482.25,11.31) (482.25,12.02) (482.25,12.73) (482.95,13.44) (482.95,14.14) (482.95,14.85) (483.66,15.56) (483.66,16.26) (483.66,16.97) (484.37,17.68) (484.37,18.38) (484.37,19.09) (485.08,19.80) (485.08,20.51) (485.08,21.21) (485.78,21.92) (485.78,22.63) (485.78,23.33) (486.49,24.04) (486.49,24.75) (486.49,25.46) (487.20,26.16) (487.20,26.87) (487.20,27.58) (487.90,28.28) (487.90,28.99) (487.90,29.70) (488.61,30.41) (488.61,31.11) (488.61,31.82) (489.32,32.53) (489.32,33.23) (489.32,33.94) (490.02,34.65) (490.02,35.36) (490.02,36.06) (490.73,36.77) (490.73,37.48) (490.73,38.18) (491.44,38.89) (491.44,39.60) (471.64,-0.71) (471.64, 0.00) (472.35, 0.71) (472.35, 1.41) (473.05, 2.12) (473.05, 2.83) (473.76, 3.54) (473.76, 4.24) (474.47, 4.95) (474.47, 5.66) (475.18, 6.36) (475.18, 7.07) (475.88, 7.78) (475.88, 8.49) (476.59, 9.19) (476.59, 9.90) (477.30,10.61) (477.30,11.31) (478.00,12.02) (478.00,12.73) (478.71,13.44) (478.71,14.14) (479.42,14.85) (479.42,15.56) (480.13,16.26) (480.13,16.97) (480.83,17.68) (480.83,18.38) (481.54,19.09) (481.54,19.80) (482.25,20.51) (482.25,21.21) (482.95,21.92) (482.95,22.63) (483.66,23.33) (483.66,24.04) (484.37,24.75) (484.37,25.46) (485.08,26.16) (485.08,26.87) (485.78,27.58) (485.78,28.28) (486.49,28.99) (486.49,29.70) (487.20,30.41) (487.20,31.11) (487.90,31.82) (487.90,32.53) (488.61,33.23) (488.61,33.94) (489.32,34.65) (489.32,35.36) (490.02,36.06) (490.02,36.77) (490.73,37.48) (490.73,38.18) (491.44,38.89) (491.44,39.60) (471.64,-0.71) (472.35, 0.00) (473.05, 0.00) (473.76, 0.71) (474.47, 0.71) (475.18, 1.41) (475.88, 2.12) (476.59, 2.12) (477.30, 2.83) (478.00, 2.83) (478.71, 3.54) (479.42, 4.24) (480.13, 4.24) (480.83, 4.95) (481.54, 5.66) (482.25, 5.66) (482.95, 6.36) (483.66, 6.36) (484.37, 7.07) (485.08, 7.78) (485.78, 7.78) (486.49, 8.49) (487.20, 8.49) (487.90, 9.19) (488.61, 9.90) (489.32, 9.90) (490.02,10.61) (490.73,10.61) (491.44,11.31) (492.15,12.02) (492.85,12.02) (493.56,12.73) (494.27,13.44) (494.97,13.44) (495.68,14.14) (496.39,14.14) (497.10,14.85) (497.80,15.56) (498.51,15.56) (499.22,16.26) (499.92,16.26) (500.63,16.97) (501.34,17.68) (502.05,17.68) (502.75,18.38) (503.46,18.38) (504.17,19.09) (504.87,19.80) (505.58,19.80) (506.29,20.51) (507.00,21.21) (507.70,21.21) (508.41,21.92) (509.12,21.92) (509.82,22.63) (509.82,22.63) (509.82,23.33) (509.82,24.04) (509.12,24.75) (509.12,25.46) (509.12,26.16) (509.12,26.87) (509.12,27.58) (508.41,28.28) (508.41,28.99) (508.41,29.70) (508.41,30.41) (507.70,31.11) (507.70,31.82) (507.70,32.53) (507.70,33.23) (507.70,33.94) (507.00,34.65) (507.00,35.36) (507.00,36.06) (507.00,36.77) (507.00,37.48) (506.29,38.18) (506.29,38.89) (506.29,39.60) (506.29,40.31) (505.58,41.01) (505.58,41.72) (505.58,42.43) (505.58,43.13) (505.58,43.84) (504.87,44.55) (504.87,45.25) (504.87,45.96) (504.87,46.67) (504.87,47.38) (504.17,48.08) (504.17,48.79) (504.17,49.50) (504.17,50.20) (504.17,50.91) (503.46,51.62) (503.46,52.33) (503.46,53.03) (503.46,53.74) (502.75,54.45) (502.75,55.15) (502.75,55.86) (502.75,56.57) (502.75,57.28) (502.05,57.98) (502.05,58.69) (502.05,59.40) (502.05,60.10) (502.05,60.81) (501.34,61.52) (501.34,62.23) (501.34,62.93) (501.34,63.64) (500.63,64.35) (500.63,65.05) (500.63,65.76) (500.63,66.47) (500.63,67.18) (499.92,67.88) (499.92,68.59) (499.92,69.30) (499.92,69.30) (500.63,68.59) (501.34,67.88) (502.05,67.18) (502.75,66.47) (503.46,65.76) (504.17,65.05) (504.87,64.35) (505.58,63.64) (506.29,62.93) (507.00,62.23) (507.70,61.52) (508.41,60.81) (509.12,60.10) (509.82,59.40) (510.53,58.69) (511.24,57.98) (511.95,57.28) (512.65,56.57) (513.36,55.86) (514.07,55.15) (514.77,54.45) (515.48,53.74) (516.19,53.03) (516.90,52.33) (517.60,51.62) (518.31,50.91) (519.02,50.20) (519.72,49.50) (520.43,48.79) (521.14,48.08) (521.84,47.38) (522.55,46.67) (523.26,45.96) (523.97,45.25) (524.67,44.55) (525.38,43.84) (526.09,43.13) (526.79,42.43) (527.50,41.72) (528.21,41.01) (528.92,40.31) (529.62,39.60) (530.33,38.89) (530.33,-5.66) (530.33,-4.95) (530.33,-4.24) (530.33,-3.54) (530.33,-2.83) (530.33,-2.12) (530.33,-1.41) (530.33,-0.71) (530.33, 0.00) (530.33, 0.71) (530.33, 1.41) (530.33, 2.12) (530.33, 2.83) (530.33, 3.54) (530.33, 4.24) (530.33, 4.95) (530.33, 5.66) (530.33, 6.36) (530.33, 7.07) (530.33, 7.78) (530.33, 8.49) (530.33, 9.19) (530.33, 9.90) (530.33,10.61) (530.33,11.31) (530.33,12.02) (530.33,12.73) (530.33,13.44) (530.33,14.14) (530.33,14.85) (530.33,15.56) (530.33,16.26) (530.33,16.97) (530.33,17.68) (530.33,18.38) (530.33,19.09) (530.33,19.80) (530.33,20.51) (530.33,21.21) (530.33,21.92) (530.33,22.63) (530.33,23.33) (530.33,24.04) (530.33,24.75) (530.33,25.46) (530.33,26.16) (530.33,26.87) (530.33,27.58) (530.33,28.28) (530.33,28.99) (530.33,29.70) (530.33,30.41) (530.33,31.11) (530.33,31.82) (530.33,32.53) (530.33,33.23) (530.33,33.94) (530.33,34.65) (530.33,35.36) (530.33,36.06) (530.33,36.77) (530.33,37.48) (530.33,38.18) (530.33,38.89) (530.33,-5.66) (529.62,-4.95) (528.92,-4.24) (528.92,-3.54) (528.21,-2.83) (527.50,-2.12) (526.79,-1.41) (526.79,-0.71) (526.09, 0.00) (525.38, 0.71) (524.67, 1.41) (523.97, 2.12) (523.97, 2.83) (523.26, 3.54) (522.55, 4.24) (521.84, 4.95) (521.84, 5.66) (521.14, 6.36) (520.43, 7.07) (519.72, 7.78) (519.02, 8.49) (519.02, 9.19) (518.31, 9.90) (517.60,10.61) (516.90,11.31) (516.90,12.02) (516.19,12.73) (515.48,13.44) (514.77,14.14) (514.07,14.85) (514.07,15.56) (513.36,16.26) (512.65,16.97) (511.95,17.68) (511.24,18.38) (511.24,19.09) (510.53,19.80) (509.82,20.51) (509.12,21.21) (509.12,21.92) (508.41,22.63) (507.70,23.33) (507.00,24.04) (506.29,24.75) (506.29,25.46) (505.58,26.16) (504.87,26.87) (504.17,27.58) (504.17,28.28) (503.46,28.99) (502.75,29.70) (502.75,29.70) (502.75,30.41) (503.46,31.11) (503.46,31.82) (504.17,32.53) (504.17,33.23) (504.87,33.94) (504.87,34.65) (505.58,35.36) (505.58,36.06) (506.29,36.77) (506.29,37.48) (507.00,38.18) (507.00,38.89) (507.70,39.60) (507.70,40.31) (508.41,41.01) (508.41,41.72) (509.12,42.43) (509.12,43.13) (509.82,43.84) (509.82,44.55) (510.53,45.25) (510.53,45.96) (511.24,46.67) (511.24,47.38) (511.95,48.08) (511.95,48.79) (512.65,49.50) (512.65,50.20) (512.65,50.91) (513.36,51.62) (513.36,52.33) (514.07,53.03) (514.07,53.74) (514.77,54.45) (514.77,55.15) (515.48,55.86) (515.48,56.57) (516.19,57.28) (516.19,57.98) (516.90,58.69) (516.90,59.40) (517.60,60.10) (517.60,60.81) (518.31,61.52) (518.31,62.23) (519.02,62.93) (519.02,63.64) (519.72,64.35) (519.72,65.05) (520.43,65.76) (520.43,66.47) (521.14,67.18) (521.14,67.88) (521.84,68.59) (521.84,69.30) (522.55,70.00) (522.55,70.71) (501.34,31.82) (502.05,32.53) (502.05,33.23) (502.75,33.94) (502.75,34.65) (503.46,35.36) (503.46,36.06) (504.17,36.77) (504.17,37.48) (504.87,38.18) (504.87,38.89) (505.58,39.60) (506.29,40.31) (506.29,41.01) (507.00,41.72) (507.00,42.43) (507.70,43.13) (507.70,43.84) (508.41,44.55) (508.41,45.25) (509.12,45.96) (509.12,46.67) (509.82,47.38) (510.53,48.08) (510.53,48.79) (511.24,49.50) (511.24,50.20) (511.95,50.91) (511.95,51.62) (512.65,52.33) (512.65,53.03) (513.36,53.74) (513.36,54.45) (514.07,55.15) (514.77,55.86) (514.77,56.57) (515.48,57.28) (515.48,57.98) (516.19,58.69) (516.19,59.40) (516.90,60.10) (516.90,60.81) (517.60,61.52) (517.60,62.23) (518.31,62.93) (519.02,63.64) (519.02,64.35) (519.72,65.05) (519.72,65.76) (520.43,66.47) (520.43,67.18) (521.14,67.88) (521.14,68.59) (521.84,69.30) (521.84,70.00) (522.55,70.71) (501.34,31.82) (501.34,32.53) (501.34,33.23) (502.05,33.94) (502.05,34.65) (502.05,35.36) (502.05,36.06) (502.75,36.77) (502.75,37.48) (502.75,38.18) (502.75,38.89) (503.46,39.60) (503.46,40.31) (503.46,41.01) (503.46,41.72) (504.17,42.43) (504.17,43.13) (504.17,43.84) (504.17,44.55) (504.87,45.25) (504.87,45.96) (504.87,46.67) (504.87,47.38) (505.58,48.08) (505.58,48.79) (505.58,49.50) (505.58,50.20) (506.29,50.91) (506.29,51.62) (506.29,52.33) (506.29,53.03) (507.00,53.74) (507.00,54.45) (507.00,55.15) (507.00,55.86) (507.70,56.57) (507.70,57.28) (507.70,57.98) (507.70,58.69) (508.41,59.40) (508.41,60.10) (508.41,60.81) (508.41,61.52) (509.12,62.23) (509.12,62.93) (509.12,63.64) (509.12,64.35) (509.82,65.05) (509.82,65.76) (509.82,66.47) (509.82,67.18) (510.53,67.88) (510.53,68.59) (510.53,69.30) (510.53,70.00) (511.24,70.71) (511.24,71.42) (511.24,72.12) (511.24,72.83) (511.95,73.54) (511.95,74.25) (511.95,74.95) (479.42,42.43) (480.13,43.13) (480.83,43.84) (481.54,44.55) (482.25,45.25) (482.95,45.96) (483.66,46.67) (484.37,47.38) (485.08,48.08) (485.78,48.79) (486.49,49.50) (487.20,50.20) (487.90,50.91) (488.61,51.62) (489.32,52.33) (490.02,53.03) (490.73,53.74) (491.44,54.45) (492.15,55.15) (492.85,55.86) (493.56,56.57) (494.27,57.28) (494.97,57.98) (495.68,58.69) (496.39,59.40) (497.10,60.10) (497.80,60.81) (498.51,61.52) (499.22,62.23) (499.92,62.93) (500.63,63.64) (501.34,64.35) (502.05,65.05) (502.75,65.76) (503.46,66.47) (504.17,67.18) (504.87,67.88) (505.58,68.59) (506.29,69.30) (507.00,70.00) (507.70,70.71) (508.41,71.42) (509.12,72.12) (509.82,72.83) (510.53,73.54) (511.24,74.25) (511.95,74.95) (479.42,42.43) (480.13,42.43) (480.83,43.13) (481.54,43.13) (482.25,43.13) (482.95,43.84) (483.66,43.84) (484.37,44.55) (485.08,44.55) (485.78,44.55) (486.49,45.25) (487.20,45.25) (487.90,45.25) (488.61,45.96) (489.32,45.96) (490.02,45.96) (490.73,46.67) (491.44,46.67) (492.15,47.38) (492.85,47.38) (493.56,47.38) (494.27,48.08) (494.97,48.08) (495.68,48.08) (496.39,48.79) (497.10,48.79) (497.80,49.50) (498.51,49.50) (499.22,49.50) (499.92,50.20) (500.63,50.20) (501.34,50.20) (502.05,50.91) (502.75,50.91) (503.46,50.91) (504.17,51.62) (504.87,51.62) (505.58,52.33) (506.29,52.33) (507.00,52.33) (507.70,53.03) (508.41,53.03) (509.12,53.03) (509.82,53.74) (510.53,53.74) (511.24,53.74) (511.95,54.45) (512.65,54.45) (513.36,55.15) (514.07,55.15) (514.77,55.15) (515.48,55.86) (516.19,55.86) (516.90,55.86) (517.60,56.57) (518.31,56.57) (519.02,57.28) (519.72,57.28) (520.43,57.28) (521.14,57.98) (521.84,57.98) (521.84,57.98) (521.84,58.69) (521.14,59.40) (521.14,60.10) (520.43,60.81) (520.43,61.52) (520.43,62.23) (519.72,62.93) (519.72,63.64) (519.02,64.35) (519.02,65.05) (519.02,65.76) (518.31,66.47) (518.31,67.18) (517.60,67.88) (517.60,68.59) (517.60,69.30) (516.90,70.00) (516.90,70.71) (516.19,71.42) (516.19,72.12) (516.19,72.83) (515.48,73.54) (515.48,74.25) (514.77,74.95) (514.77,75.66) (514.77,76.37) (514.07,77.07) (514.07,77.78) (513.36,78.49) (513.36,79.20) (513.36,79.90) (512.65,80.61) (512.65,81.32) (511.95,82.02) (511.95,82.73) (511.95,83.44) (511.24,84.15) (511.24,84.85) (510.53,85.56) (510.53,86.27) (510.53,86.97) (509.82,87.68) (509.82,88.39) (509.12,89.10) (509.12,89.80) (509.12,90.51) (508.41,91.22) (508.41,91.92) (507.70,92.63) (507.70,93.34) (507.70,94.05) (507.00,94.75) (507.00,95.46) (506.29,96.17) (506.29,96.87) (506.29,97.58) (505.58,98.29) (505.58,98.99) (504.87,99.70) (504.87,100.41) (504.87,100.41) (505.58,99.70) (506.29,99.70) (507.00,98.99) (507.70,98.29) (508.41,98.29) (509.12,97.58) (509.82,96.87) (510.53,96.87) (511.24,96.17) (511.95,95.46) (512.65,95.46) (513.36,94.75) (514.07,94.05) (514.77,94.05) (515.48,93.34) (516.19,92.63) (516.90,92.63) (517.60,91.92) (518.31,91.22) (519.02,91.22) (519.72,90.51) (520.43,89.80) (521.14,89.80) (521.84,89.10) (522.55,88.39) (523.26,88.39) (523.97,87.68) (524.67,87.68) (525.38,86.97) (526.09,86.27) (526.79,86.27) (527.50,85.56) (528.21,84.85) (528.92,84.85) (529.62,84.15) (530.33,83.44) (531.04,83.44) (531.74,82.73) (532.45,82.02) (533.16,82.02) (533.87,81.32) (534.57,80.61) (535.28,80.61) (535.99,79.90) (536.69,79.20) (537.40,79.20) (538.11,78.49) (538.82,77.78) (539.52,77.78) (540.23,77.07) (540.94,76.37) (541.64,76.37) (542.35,75.66) (542.35,28.99) (542.35,29.70) (542.35,30.41) (542.35,31.11) (542.35,31.82) (542.35,32.53) (542.35,33.23) (542.35,33.94) (542.35,34.65) (542.35,35.36) (542.35,36.06) (542.35,36.77) (542.35,37.48) (542.35,38.18) (542.35,38.89) (542.35,39.60) (542.35,40.31) (542.35,41.01) (542.35,41.72) (542.35,42.43) (542.35,43.13) (542.35,43.84) (542.35,44.55) (542.35,45.25) (542.35,45.96) (542.35,46.67) (542.35,47.38) (542.35,48.08) (542.35,48.79) (542.35,49.50) (542.35,50.20) (542.35,50.91) (542.35,51.62) (542.35,52.33) (542.35,53.03) (542.35,53.74) (542.35,54.45) (542.35,55.15) (542.35,55.86) (542.35,56.57) (542.35,57.28) (542.35,57.98) (542.35,58.69) (542.35,59.40) (542.35,60.10) (542.35,60.81) (542.35,61.52) (542.35,62.23) (542.35,62.93) (542.35,63.64) (542.35,64.35) (542.35,65.05) (542.35,65.76) (542.35,66.47) (542.35,67.18) (542.35,67.88) (542.35,68.59) (542.35,69.30) (542.35,70.00) (542.35,70.71) (542.35,71.42) (542.35,72.12) (542.35,72.83) (542.35,73.54) (542.35,74.25) (542.35,74.95) (542.35,75.66) (508.41,57.98) (509.12,57.28) (509.82,56.57) (510.53,55.86) (511.24,55.86) (511.95,55.15) (512.65,54.45) (513.36,53.74) (514.07,53.03) (514.77,52.33) (515.48,51.62) (516.19,51.62) (516.90,50.91) (517.60,50.20) (518.31,49.50) (519.02,48.79) (519.72,48.08) (520.43,47.38) (521.14,47.38) (521.84,46.67) (522.55,45.96) (523.26,45.25) (523.97,44.55) (524.67,43.84) (525.38,43.84) (526.09,43.13) (526.79,42.43) (527.50,41.72) (528.21,41.01) (528.92,40.31) (529.62,39.60) (530.33,39.60) (531.04,38.89) (531.74,38.18) (532.45,37.48) (533.16,36.77) (533.87,36.06) (534.57,35.36) (535.28,35.36) (535.99,34.65) (536.69,33.94) (537.40,33.23) (538.11,32.53) (538.82,31.82) (539.52,31.11) (540.23,31.11) (540.94,30.41) (541.64,29.70) (542.35,28.99) (508.41,57.98) (509.12,58.69) (509.12,59.40) (509.82,60.10) (509.82,60.81) (510.53,61.52) (510.53,62.23) (511.24,62.93) (511.24,63.64) (511.95,64.35) (511.95,65.05) (512.65,65.76) (512.65,66.47) (513.36,67.18) (513.36,67.88) (514.07,68.59) (514.77,69.30) (514.77,70.00) (515.48,70.71) (515.48,71.42) (516.19,72.12) (516.19,72.83) (516.90,73.54) (516.90,74.25) (517.60,74.95) (517.60,75.66) (518.31,76.37) (518.31,77.07) (519.02,77.78) (519.72,78.49) (519.72,79.20) (520.43,79.90) (520.43,80.61) (521.14,81.32) (521.14,82.02) (521.84,82.73) (521.84,83.44) (522.55,84.15) (522.55,84.85) (523.26,85.56) (523.26,86.27) (523.97,86.97) (523.97,87.68) (524.67,88.39) (525.38,89.10) (525.38,89.80) (526.09,90.51) (526.09,91.22) (526.79,91.92) (526.79,92.63) (527.50,93.34) (527.50,94.05) (528.21,94.75) (528.21,95.46) (528.92,96.17) (528.92,96.87) (529.62,97.58) (486.49,79.90) (487.20,79.90) (487.90,80.61) (488.61,80.61) (489.32,81.32) (490.02,81.32) (490.73,81.32) (491.44,82.02) (492.15,82.02) (492.85,82.73) (493.56,82.73) (494.27,83.44) (494.97,83.44) (495.68,83.44) (496.39,84.15) (497.10,84.15) (497.80,84.85) (498.51,84.85) (499.22,84.85) (499.92,85.56) (500.63,85.56) (501.34,86.27) (502.05,86.27) (502.75,86.27) (503.46,86.97) (504.17,86.97) (504.87,87.68) (505.58,87.68) (506.29,87.68) (507.00,88.39) (507.70,88.39) (508.41,89.10) (509.12,89.10) (509.82,89.80) (510.53,89.80) (511.24,89.80) (511.95,90.51) (512.65,90.51) (513.36,91.22) (514.07,91.22) (514.77,91.22) (515.48,91.92) (516.19,91.92) (516.90,92.63) (517.60,92.63) (518.31,92.63) (519.02,93.34) (519.72,93.34) (520.43,94.05) (521.14,94.05) (521.84,94.05) (522.55,94.75) (523.26,94.75) (523.97,95.46) (524.67,95.46) (525.38,96.17) (526.09,96.17) (526.79,96.17) (527.50,96.87) (528.21,96.87) (528.92,97.58) (529.62,97.58) (499.92,36.77) (499.92,37.48) (499.22,38.18) (499.22,38.89) (499.22,39.60) (498.51,40.31) (498.51,41.01) (498.51,41.72) (498.51,42.43) (497.80,43.13) (497.80,43.84) (497.80,44.55) (497.10,45.25) (497.10,45.96) (497.10,46.67) (496.39,47.38) (496.39,48.08) (496.39,48.79) (495.68,49.50) (495.68,50.20) (495.68,50.91) (494.97,51.62) (494.97,52.33) (494.97,53.03) (494.97,53.74) (494.27,54.45) (494.27,55.15) (494.27,55.86) (493.56,56.57) (493.56,57.28) (493.56,57.98) (492.85,58.69) (492.85,59.40) (492.85,60.10) (492.15,60.81) (492.15,61.52) (492.15,62.23) (491.44,62.93) (491.44,63.64) (491.44,64.35) (491.44,65.05) (490.73,65.76) (490.73,66.47) (490.73,67.18) (490.02,67.88) (490.02,68.59) (490.02,69.30) (489.32,70.00) (489.32,70.71) (489.32,71.42) (488.61,72.12) (488.61,72.83) (488.61,73.54) (487.90,74.25) (487.90,74.95) (487.90,75.66) (487.90,76.37) (487.20,77.07) (487.20,77.78) (487.20,78.49) (486.49,79.20) (486.49,79.90) (472.35, 0.71) (473.05, 1.41) (473.76, 2.12) (473.76, 2.83) (474.47, 3.54) (475.18, 4.24) (475.88, 4.95) (475.88, 5.66) (476.59, 6.36) (477.30, 7.07) (478.00, 7.78) (478.00, 8.49) (478.71, 9.19) (479.42, 9.90) (480.13,10.61) (480.13,11.31) (480.83,12.02) (481.54,12.73) (482.25,13.44) (482.95,14.14) (482.95,14.85) (483.66,15.56) (484.37,16.26) (485.08,16.97) (485.08,17.68) (485.78,18.38) (486.49,19.09) (487.20,19.80) (487.20,20.51) (487.90,21.21) (488.61,21.92) (489.32,22.63) (489.32,23.33) (490.02,24.04) (490.73,24.75) (491.44,25.46) (492.15,26.16) (492.15,26.87) (492.85,27.58) (493.56,28.28) (494.27,28.99) (494.27,29.70) (494.97,30.41) (495.68,31.11) (496.39,31.82) (496.39,32.53) (497.10,33.23) (497.80,33.94) (498.51,34.65) (498.51,35.36) (499.22,36.06) (499.92,36.77) (472.35, 0.71) (473.05, 0.71) (473.76, 1.41) (474.47, 1.41) (475.18, 1.41) (475.88, 2.12) (476.59, 2.12) (477.30, 2.12) (478.00, 2.83) (478.71, 2.83) (479.42, 2.83) (480.13, 3.54) (480.83, 3.54) (481.54, 4.24) (482.25, 4.24) (482.95, 4.24) (483.66, 4.95) (484.37, 4.95) (485.08, 4.95) (485.78, 5.66) (486.49, 5.66) (487.20, 5.66) (487.90, 6.36) (488.61, 6.36) (489.32, 6.36) (490.02, 7.07) (490.73, 7.07) (491.44, 7.07) (492.15, 7.78) (492.85, 7.78) (493.56, 7.78) (494.27, 8.49) (494.97, 8.49) (495.68, 9.19) (496.39, 9.19) (497.10, 9.19) (497.80, 9.90) (498.51, 9.90) (499.22, 9.90) (499.92,10.61) (500.63,10.61) (501.34,10.61) (502.05,11.31) (502.75,11.31) (503.46,11.31) (504.17,12.02) (504.87,12.02) (505.58,12.02) (506.29,12.73) (507.00,12.73) (507.70,12.73) (508.41,13.44) (509.12,13.44) (509.82,14.14) (510.53,14.14) (511.24,14.14) (511.95,14.85) (512.65,14.85) (513.36,14.85) (514.07,15.56) (514.77,15.56) (514.77,15.56) (515.48,16.26) (516.19,16.97) (516.90,16.97) (517.60,17.68) (518.31,18.38) (519.02,19.09) (519.72,19.80) (520.43,20.51) (521.14,20.51) (521.84,21.21) (522.55,21.92) (523.26,22.63) (523.97,23.33) (524.67,23.33) (525.38,24.04) (526.09,24.75) (526.79,25.46) (527.50,26.16) (528.21,26.87) (528.92,26.87) (529.62,27.58) (530.33,28.28) (531.04,28.99) (531.74,29.70) (532.45,29.70) (533.16,30.41) (533.87,31.11) (534.57,31.82) (535.28,32.53) (535.99,32.53) (536.69,33.23) (537.40,33.94) (538.11,34.65) (538.82,35.36) (539.52,36.06) (540.23,36.06) (540.94,36.77) (541.64,37.48) (542.35,38.18) (543.06,38.89) (543.77,38.89) (544.47,39.60) (545.18,40.31) (545.89,41.01) (546.59,41.72) (547.30,42.43) (548.01,42.43) (548.71,43.13) (549.42,43.84) (514.77,74.95) (515.48,74.25) (516.19,73.54) (516.90,72.83) (517.60,72.12) (518.31,72.12) (519.02,71.42) (519.72,70.71) (520.43,70.00) (521.14,69.30) (521.84,68.59) (522.55,67.88) (523.26,67.18) (523.97,66.47) (524.67,65.76) (525.38,65.76) (526.09,65.05) (526.79,64.35) (527.50,63.64) (528.21,62.93) (528.92,62.23) (529.62,61.52) (530.33,60.81) (531.04,60.10) (531.74,59.40) (532.45,59.40) (533.16,58.69) (533.87,57.98) (534.57,57.28) (535.28,56.57) (535.99,55.86) (536.69,55.15) (537.40,54.45) (538.11,53.74) (538.82,53.03) (539.52,53.03) (540.23,52.33) (540.94,51.62) (541.64,50.91) (542.35,50.20) (543.06,49.50) (543.77,48.79) (544.47,48.08) (545.18,47.38) (545.89,46.67) (546.59,46.67) (547.30,45.96) (548.01,45.25) (548.71,44.55) (549.42,43.84) (514.77,74.95) (514.77,75.66) (514.77,76.37) (514.77,77.07) (515.48,77.78) (515.48,78.49) (515.48,79.20) (515.48,79.90) (515.48,80.61) (515.48,81.32) (515.48,82.02) (515.48,82.73) (516.19,83.44) (516.19,84.15) (516.19,84.85) (516.19,85.56) (516.19,86.27) (516.19,86.97) (516.19,87.68) (516.19,88.39) (516.90,89.10) (516.90,89.80) (516.90,90.51) (516.90,91.22) (516.90,91.92) (516.90,92.63) (516.90,93.34) (516.90,94.05) (517.60,94.75) (517.60,95.46) (517.60,96.17) (517.60,96.87) (517.60,97.58) (517.60,98.29) (517.60,98.99) (518.31,99.70) (518.31,100.41) (518.31,101.12) (518.31,101.82) (518.31,102.53) (518.31,103.24) (518.31,103.94) (518.31,104.65) (519.02,105.36) (519.02,106.07) (519.02,106.77) (519.02,107.48) (519.02,108.19) (519.02,108.89) (519.02,109.60) (519.02,110.31) (519.72,111.02) (519.72,111.72) (519.72,112.43) (519.72,113.14) (519.72,113.84) (519.72,114.55) (519.72,115.26) (519.72,115.97) (520.43,116.67) (520.43,117.38) (520.43,118.09) (520.43,118.79) (515.48,72.83) (515.48,73.54) (515.48,74.25) (515.48,74.95) (515.48,75.66) (516.19,76.37) (516.19,77.07) (516.19,77.78) (516.19,78.49) (516.19,79.20) (516.19,79.90) (516.19,80.61) (516.19,81.32) (516.19,82.02) (516.90,82.73) (516.90,83.44) (516.90,84.15) (516.90,84.85) (516.90,85.56) (516.90,86.27) (516.90,86.97) (516.90,87.68) (516.90,88.39) (516.90,89.10) (517.60,89.80) (517.60,90.51) (517.60,91.22) (517.60,91.92) (517.60,92.63) (517.60,93.34) (517.60,94.05) (517.60,94.75) (517.60,95.46) (518.31,96.17) (518.31,96.87) (518.31,97.58) (518.31,98.29) (518.31,98.99) (518.31,99.70) (518.31,100.41) (518.31,101.12) (518.31,101.82) (519.02,102.53) (519.02,103.24) (519.02,103.94) (519.02,104.65) (519.02,105.36) (519.02,106.07) (519.02,106.77) (519.02,107.48) (519.02,108.19) (519.02,108.89) (519.72,109.60) (519.72,110.31) (519.72,111.02) (519.72,111.72) (519.72,112.43) (519.72,113.14) (519.72,113.84) (519.72,114.55) (519.72,115.26) (520.43,115.97) (520.43,116.67) (520.43,117.38) (520.43,118.09) (520.43,118.79) (515.48,72.83) (514.77,73.54) (514.77,74.25) (514.07,74.95) (514.07,75.66) (513.36,76.37) (513.36,77.07) (512.65,77.78) (512.65,78.49) (511.95,79.20) (511.95,79.90) (511.24,80.61) (511.24,81.32) (510.53,82.02) (510.53,82.73) (509.82,83.44) (509.82,84.15) (509.12,84.85) (508.41,85.56) (508.41,86.27) (507.70,86.97) (507.70,87.68) (507.00,88.39) (507.00,89.10) (506.29,89.80) (506.29,90.51) (505.58,91.22) (505.58,91.92) (504.87,92.63) (504.87,93.34) (504.17,94.05) (504.17,94.75) (503.46,95.46) (503.46,96.17) (502.75,96.87) (502.75,97.58) (502.05,98.29) (501.34,98.99) (501.34,99.70) (500.63,100.41) (500.63,101.12) (499.92,101.82) (499.92,102.53) (499.22,103.24) (499.22,103.94) (498.51,104.65) (498.51,105.36) (497.80,106.07) (497.80,106.77) (497.10,107.48) (497.10,108.19) (496.39,108.89) (496.39,109.60) (495.68,110.31) (-160,-484)[(0,0)\[lt\][$\rho: 0.01$]{}]{}
&
(459,612)(-255,-306) (-255,-306)(459,0)[2]{}[(0,1)[612]{}]{} (-255,-306)(0,612)[2]{}[(1,0)[459]{}]{} (102,-51) (51,153) (-51,0) (-153,-102) (102,-153) (-102,204) (-51,-153) (-51,-306) (-102,102) (0,153) (-102,-204) (51,-102) (-102,0) (51,102) (-102,51) (-153,-153) (102,-102) (-51,-204) (51,-51) (-204,-204) (51,255) (-153,102) (-102,-51) (51,-153) (-255,-153) (51,51) (0,-255) (0,-204) (0,-306) (-102,-255) (-102,-306) (0,0) (-153,-204) (-204,-153) (-51,-102) (0,204) (-51,51) (-102,-102) (102,153) (51,-204) (-51,204) (0,-153) (-153,-255) (-51,-255) (-102,153) (-255,-204) (0,-51) (-51,-51) (-153,0) (102,0) (-204,-255) (0,-102) (153,204) (102,204) ( 7.78, 2.83) ( 7.78, 3.54) ( 7.78, 4.24) ( 7.78, 4.95) ( 7.78, 5.66) ( 7.78, 6.36) ( 7.78, 7.07) ( 7.78, 7.78) ( 7.78, 2.83) ( 7.78, 2.83) ( 8.49, 2.83) ( 9.19, 2.83) ( 9.90, 3.54) (10.61, 3.54) (10.61, 2.83) (10.61, 3.54) (10.61, 2.83) (11.31, 2.12) (12.02, 2.12) (12.73, 1.41) (12.73, 1.41) (12.73, 1.41) (12.73, 2.12) (12.73, 2.83) (12.73, 3.54) (12.73, 4.24) (12.73, 4.95) ( 8.49, 3.54) ( 9.19, 3.54) ( 9.90, 4.24) (10.61, 4.24) (11.31, 4.24) (12.02, 4.95) (12.73, 4.95) ( 6.36, 3.54) ( 7.07, 3.54) ( 7.78, 3.54) ( 8.49, 3.54) ( 6.36, 2.83) ( 6.36, 3.54) ( 5.66, 0.71) ( 5.66, 1.41) ( 6.36, 2.12) ( 6.36, 2.83) ( 5.66, 0.00) ( 5.66, 0.71) ( 5.66, 0.00) ( 5.66, 0.71) ( 5.66, 0.71) ( 6.36, 1.41) ( 7.07, 2.12) ( 7.78, 2.83) ( 7.78, 2.83) ( 8.49, 3.54) ( 8.49, 4.24) ( 9.19, 4.95) ( 9.19, 5.66) ( 9.90, 6.36) (10.61, 7.07) (10.61, 7.78) (11.31, 8.49) (10.61, 8.49) (11.31, 8.49) ( 9.90, 5.66) ( 9.90, 6.36) ( 9.90, 7.07) (10.61, 7.78) (10.61, 8.49) ( 9.90, 5.66) (10.61, 4.95) (11.31, 3.54) (11.31, 4.24) (10.61, 4.95) ( 9.90, 3.54) (10.61, 3.54) (11.31, 3.54) ( 9.90, 3.54) ( 9.90, 4.24) ( 9.90, 4.95) (10.61, 5.66) (10.61, 6.36) ( 8.49, 7.78) ( 9.19, 7.07) ( 9.90, 7.07) (10.61, 6.36) ( 7.78, 5.66) ( 7.78, 6.36) ( 8.49, 7.07) ( 8.49, 7.78) ( 7.78, 5.66) ( 8.49, 5.66) ( 9.19, 4.95) ( 9.90, 4.95) ( 9.90, 2.83) ( 9.90, 3.54) ( 9.90, 4.24) ( 9.90, 4.95) ( 9.90, 2.83) (10.61, 3.54) (10.61, 3.54) (11.31, 3.54) (11.31,-0.71) (11.31, 0.00) (11.31, 0.71) (11.31, 1.41) (11.31, 2.12) (11.31, 2.83) (11.31, 3.54) (11.31,-0.71) (12.02, 0.00) (12.73, 0.71) (13.44, 1.41) (14.14, 2.12) (14.14, 2.83) (14.85, 3.54) (15.56, 4.24) (16.26, 4.95) (16.97, 5.66) (17.68, 6.36) (17.68, 6.36) (18.38, 6.36) (19.09, 6.36) (19.80, 6.36) (20.51, 7.07) (21.21, 7.07) (21.92, 7.07) (22.63, 7.07) (23.33, 7.07) (24.04, 7.07) (24.75, 7.78) (25.46, 7.78) (26.16, 7.78) (26.87, 7.78) (27.58, 7.78) (28.28, 7.78) (28.99, 8.49) (29.70, 8.49) (30.41, 8.49) (24.75,-4.95) (24.75,-4.24) (25.46,-3.54) (25.46,-2.83) (26.16,-2.12) (26.16,-1.41) (26.87,-0.71) (26.87, 0.00) (26.87, 0.71) (27.58, 1.41) (27.58, 2.12) (28.28, 2.83) (28.28, 3.54) (28.28, 4.24) (28.99, 4.95) (28.99, 5.66) (29.70, 6.36) (29.70, 7.07) (30.41, 7.78) (30.41, 8.49) (30.41,-20.51) (30.41,-19.80) (29.70,-19.09) (29.70,-18.38) (29.70,-17.68) (28.99,-16.97) (28.99,-16.26) (28.28,-15.56) (28.28,-14.85) (28.28,-14.14) (27.58,-13.44) (27.58,-12.73) (27.58,-12.02) (26.87,-11.31) (26.87,-10.61) (26.87,-9.90) (26.16,-9.19) (26.16,-8.49) (25.46,-7.78) (25.46,-7.07) (25.46,-6.36) (24.75,-5.66) (24.75,-4.95) (30.41,-20.51) (31.11,-20.51) (31.82,-19.80) (32.53,-19.80) (33.23,-19.09) (33.94,-19.09) (34.65,-18.38) (35.36,-18.38) (36.06,-18.38) (36.77,-17.68) (37.48,-17.68) (38.18,-16.97) (38.89,-16.97) (39.60,-16.97) (40.31,-16.26) (41.01,-16.26) (41.72,-15.56) (42.43,-15.56) (43.13,-14.85) (43.84,-14.85) (44.55,-14.85) (45.25,-14.14) (45.96,-14.14) (46.67,-13.44) (47.38,-13.44) (48.08,-13.44) (48.79,-12.73) (49.50,-12.73) (50.20,-12.02) (50.91,-12.02) (51.62,-11.31) (52.33,-11.31) (48.79,-37.48) (48.79,-36.77) (48.79,-36.06) (48.79,-35.36) (49.50,-34.65) (49.50,-33.94) (49.50,-33.23) (49.50,-32.53) (49.50,-31.82) (49.50,-31.11) (49.50,-30.41) (49.50,-29.70) (50.20,-28.99) (50.20,-28.28) (50.20,-27.58) (50.20,-26.87) (50.20,-26.16) (50.20,-25.46) (50.20,-24.75) (50.91,-24.04) (50.91,-23.33) (50.91,-22.63) (50.91,-21.92) (50.91,-21.21) (50.91,-20.51) (50.91,-19.80) (51.62,-19.09) (51.62,-18.38) (51.62,-17.68) (51.62,-16.97) (51.62,-16.26) (51.62,-15.56) (51.62,-14.85) (51.62,-14.14) (52.33,-13.44) (52.33,-12.73) (52.33,-12.02) (52.33,-11.31) (52.33,-66.47) (52.33,-65.76) (52.33,-65.05) (52.33,-64.35) (52.33,-63.64) (51.62,-62.93) (51.62,-62.23) (51.62,-61.52) (51.62,-60.81) (51.62,-60.10) (51.62,-59.40) (51.62,-58.69) (51.62,-57.98) (50.91,-57.28) (50.91,-56.57) (50.91,-55.86) (50.91,-55.15) (50.91,-54.45) (50.91,-53.74) (50.91,-53.03) (50.91,-52.33) (50.20,-51.62) (50.20,-50.91) (50.20,-50.20) (50.20,-49.50) (50.20,-48.79) (50.20,-48.08) (50.20,-47.38) (50.20,-46.67) (49.50,-45.96) (49.50,-45.25) (49.50,-44.55) (49.50,-43.84) (49.50,-43.13) (49.50,-42.43) (49.50,-41.72) (49.50,-41.01) (48.79,-40.31) (48.79,-39.60) (48.79,-38.89) (48.79,-38.18) (48.79,-37.48) (52.33,-66.47) (53.03,-66.47) (53.74,-65.76) (54.45,-65.76) (55.15,-65.05) (55.86,-65.05) (56.57,-65.05) (57.28,-64.35) (57.98,-64.35) (58.69,-64.35) (59.40,-63.64) (60.10,-63.64) (60.81,-62.93) (61.52,-62.93) (62.23,-62.93) (62.93,-62.23) (63.64,-62.23) (64.35,-61.52) (65.05,-61.52) (65.76,-61.52) (66.47,-60.81) (67.18,-60.81) (67.88,-60.81) (68.59,-60.10) (69.30,-60.10) (70.00,-59.40) (70.71,-59.40) (71.42,-59.40) (72.12,-58.69) (72.83,-58.69) (73.54,-57.98) (74.25,-57.98) (74.95,-57.98) (75.66,-57.28) (76.37,-57.28) (77.07,-56.57) (77.78,-56.57) (78.49,-56.57) (79.20,-55.86) (79.90,-55.86) (80.61,-55.86) (81.32,-55.15) (82.02,-55.15) (82.73,-54.45) (83.44,-54.45) (52.33,-72.83) (53.03,-72.12) (53.74,-72.12) (54.45,-71.42) (55.15,-71.42) (55.86,-70.71) (56.57,-70.00) (57.28,-70.00) (57.98,-69.30) (58.69,-69.30) (59.40,-68.59) (60.10,-68.59) (60.81,-67.88) (61.52,-67.18) (62.23,-67.18) (62.93,-66.47) (63.64,-66.47) (64.35,-65.76) (65.05,-65.05) (65.76,-65.05) (66.47,-64.35) (67.18,-64.35) (67.88,-63.64) (68.59,-62.93) (69.30,-62.93) (70.00,-62.23) (70.71,-62.23) (71.42,-61.52) (72.12,-60.81) (72.83,-60.81) (73.54,-60.10) (74.25,-60.10) (74.95,-59.40) (75.66,-59.40) (76.37,-58.69) (77.07,-57.98) (77.78,-57.98) (78.49,-57.28) (79.20,-57.28) (79.90,-56.57) (80.61,-55.86) (81.32,-55.86) (82.02,-55.15) (82.73,-55.15) (83.44,-54.45) (13.44,-60.81) (14.14,-60.81) (14.85,-61.52) (15.56,-61.52) (16.26,-61.52) (16.97,-62.23) (17.68,-62.23) (18.38,-62.23) (19.09,-62.23) (19.80,-62.93) (20.51,-62.93) (21.21,-62.93) (21.92,-63.64) (22.63,-63.64) (23.33,-63.64) (24.04,-64.35) (24.75,-64.35) (25.46,-64.35) (26.16,-65.05) (26.87,-65.05) (27.58,-65.05) (28.28,-65.05) (28.99,-65.76) (29.70,-65.76) (30.41,-65.76) (31.11,-66.47) (31.82,-66.47) (32.53,-66.47) (33.23,-67.18) (33.94,-67.18) (34.65,-67.18) (35.36,-67.88) (36.06,-67.88) (36.77,-67.88) (37.48,-68.59) (38.18,-68.59) (38.89,-68.59) (39.60,-68.59) (40.31,-69.30) (41.01,-69.30) (41.72,-69.30) (42.43,-70.00) (43.13,-70.00) (43.84,-70.00) (44.55,-70.71) (45.25,-70.71) (45.96,-70.71) (46.67,-71.42) (47.38,-71.42) (48.08,-71.42) (48.79,-71.42) (49.50,-72.12) (50.20,-72.12) (50.91,-72.12) (51.62,-72.83) (52.33,-72.83) (13.44,-60.81) (14.14,-60.81) (14.85,-61.52) (15.56,-61.52) (16.26,-62.23) (16.97,-62.23) (17.68,-62.23) (18.38,-62.93) (19.09,-62.93) (19.80,-62.93) (20.51,-63.64) (21.21,-63.64) (21.92,-64.35) (22.63,-64.35) (23.33,-64.35) (24.04,-65.05) (24.75,-65.05) (25.46,-65.76) (26.16,-65.76) (26.87,-65.76) (27.58,-66.47) (28.28,-66.47) (28.99,-67.18) (29.70,-67.18) (30.41,-67.18) (31.11,-67.88) (31.82,-67.88) (32.53,-67.88) (33.23,-68.59) (33.94,-68.59) (34.65,-69.30) (35.36,-69.30) (36.06,-69.30) (36.77,-70.00) (37.48,-70.00) (38.18,-70.71) (38.89,-70.71) (39.60,-70.71) (40.31,-71.42) (41.01,-71.42) (41.72,-72.12) (42.43,-72.12) (43.13,-72.12) (43.84,-72.83) (44.55,-72.83) (45.25,-72.83) (45.96,-73.54) (46.67,-73.54) (47.38,-74.25) (48.08,-74.25) (48.79,-74.25) (49.50,-74.95) (50.20,-74.95) (50.91,-75.66) (51.62,-75.66) ( 8.49,-62.93) ( 9.19,-62.93) ( 9.90,-63.64) (10.61,-63.64) (11.31,-63.64) (12.02,-63.64) (12.73,-64.35) (13.44,-64.35) (14.14,-64.35) (14.85,-65.05) (15.56,-65.05) (16.26,-65.05) (16.97,-65.76) (17.68,-65.76) (18.38,-65.76) (19.09,-65.76) (19.80,-66.47) (20.51,-66.47) (21.21,-66.47) (21.92,-67.18) (22.63,-67.18) (23.33,-67.18) (24.04,-67.18) (24.75,-67.88) (25.46,-67.88) (26.16,-67.88) (26.87,-68.59) (27.58,-68.59) (28.28,-68.59) (28.99,-69.30) (29.70,-69.30) (30.41,-69.30) (31.11,-69.30) (31.82,-70.00) (32.53,-70.00) (33.23,-70.00) (33.94,-70.71) (34.65,-70.71) (35.36,-70.71) (36.06,-71.42) (36.77,-71.42) (37.48,-71.42) (38.18,-71.42) (38.89,-72.12) (39.60,-72.12) (40.31,-72.12) (41.01,-72.83) (41.72,-72.83) (42.43,-72.83) (43.13,-72.83) (43.84,-73.54) (44.55,-73.54) (45.25,-73.54) (45.96,-74.25) (46.67,-74.25) (47.38,-74.25) (48.08,-74.95) (48.79,-74.95) (49.50,-74.95) (50.20,-74.95) (50.91,-75.66) (51.62,-75.66) ( 8.49,-62.93) ( 9.19,-62.93) ( 9.90,-62.93) (10.61,-62.93) (11.31,-62.93) (12.02,-62.23) (12.73,-62.23) (13.44,-62.23) (14.14,-62.23) (14.85,-62.23) (15.56,-62.23) (16.26,-62.23) (16.97,-62.23) (17.68,-62.23) (18.38,-62.23) (19.09,-61.52) (19.80,-61.52) (20.51,-61.52) (21.21,-61.52) (21.92,-61.52) (22.63,-61.52) (23.33,-61.52) (24.04,-61.52) (24.75,-61.52) (25.46,-60.81) (26.16,-60.81) (26.87,-60.81) (27.58,-60.81) (28.28,-60.81) (28.99,-60.81) (29.70,-60.81) (30.41,-60.81) (31.11,-60.81) (31.82,-60.81) (32.53,-60.10) (33.23,-60.10) (33.94,-60.10) (34.65,-60.10) (35.36,-60.10) (36.06,-60.10) (36.77,-60.10) (37.48,-60.10) (38.18,-60.10) (38.89,-60.10) (39.60,-59.40) (40.31,-59.40) (41.01,-59.40) (41.72,-59.40) (42.43,-59.40) (43.13,-59.40) (43.84,-59.40) (44.55,-59.40) (45.25,-59.40) (45.96,-58.69) (46.67,-58.69) (47.38,-58.69) (48.08,-58.69) (48.79,-58.69) (49.50,-58.69) (50.20,-58.69) (50.91,-58.69) (51.62,-58.69) (52.33,-58.69) (53.03,-57.98) (53.74,-57.98) (54.45,-57.98) (55.15,-57.98) (55.86,-57.98) (10.61,-79.90) (11.31,-79.90) (12.02,-79.20) (12.73,-79.20) (13.44,-78.49) (14.14,-78.49) (14.85,-77.78) (15.56,-77.78) (16.26,-77.07) (16.97,-77.07) (17.68,-76.37) (18.38,-76.37) (19.09,-75.66) (19.80,-75.66) (20.51,-74.95) (21.21,-74.95) (21.92,-74.25) (22.63,-74.25) (23.33,-73.54) (24.04,-73.54) (24.75,-72.83) (25.46,-72.83) (26.16,-72.12) (26.87,-72.12) (27.58,-71.42) (28.28,-71.42) (28.99,-70.71) (29.70,-70.71) (30.41,-70.00) (31.11,-70.00) (31.82,-69.30) (32.53,-69.30) (33.23,-69.30) (33.94,-68.59) (34.65,-68.59) (35.36,-67.88) (36.06,-67.88) (36.77,-67.18) (37.48,-67.18) (38.18,-66.47) (38.89,-66.47) (39.60,-65.76) (40.31,-65.76) (41.01,-65.05) (41.72,-65.05) (42.43,-64.35) (43.13,-64.35) (43.84,-63.64) (44.55,-63.64) (45.25,-62.93) (45.96,-62.93) (46.67,-62.23) (47.38,-62.23) (48.08,-61.52) (48.79,-61.52) (49.50,-60.81) (50.20,-60.81) (50.91,-60.10) (51.62,-60.10) (52.33,-59.40) (53.03,-59.40) (53.74,-58.69) (54.45,-58.69) (55.15,-57.98) (55.86,-57.98) (10.61,-79.90) (10.61,-79.20) (10.61,-78.49) ( 9.90,-77.78) ( 9.90,-77.07) ( 9.90,-76.37) ( 9.90,-75.66) ( 9.19,-74.95) ( 9.19,-74.25) ( 9.19,-73.54) ( 9.19,-72.83) ( 9.19,-72.12) ( 8.49,-71.42) ( 8.49,-70.71) ( 8.49,-70.00) ( 8.49,-69.30) ( 7.78,-68.59) ( 7.78,-67.88) ( 7.78,-67.18) ( 7.78,-66.47) ( 7.07,-65.76) ( 7.07,-65.05) ( 7.07,-64.35) ( 7.07,-63.64) ( 7.07,-62.93) ( 6.36,-62.23) ( 6.36,-61.52) ( 6.36,-60.81) ( 6.36,-60.10) ( 5.66,-59.40) ( 5.66,-58.69) ( 5.66,-57.98) ( 5.66,-57.28) ( 5.66,-56.57) ( 4.95,-55.86) ( 4.95,-55.15) ( 4.95,-54.45) ( 4.95,-53.74) ( 4.24,-53.03) ( 4.24,-52.33) ( 4.24,-51.62) ( 4.24,-50.91) ( 3.54,-50.20) ( 3.54,-49.50) ( 3.54,-48.79) ( 3.54,-48.08) ( 3.54,-47.38) ( 2.83,-46.67) ( 2.83,-45.96) ( 2.83,-45.25) ( 2.83,-44.55) ( 2.12,-43.84) ( 2.12,-43.13) ( 2.12,-42.43) ( 2.12,-41.72) ( 2.12,-41.01) ( 1.41,-40.31) ( 1.41,-39.60) ( 1.41,-38.89) ( 1.41,-38.18) ( 0.71,-37.48) ( 0.71,-36.77) ( 0.71,-36.06) ( 0.71,-35.36) ( 0.00,-34.65) ( 0.00,-33.94) ( 0.00,-33.23) ( 0.00,-32.53) ( 0.00,-31.82) (-0.71,-31.11) (-0.71,-30.41) (-0.71,-29.70) (-0.71,-28.99) (-1.41,-28.28) (-1.41,-27.58) (-1.41,-26.87) (-47.38, 0.00) (-46.67,-0.71) (-45.96,-0.71) (-45.25,-1.41) (-44.55,-1.41) (-43.84,-2.12) (-43.13,-2.83) (-42.43,-2.83) (-41.72,-3.54) (-41.01,-3.54) (-40.31,-4.24) (-39.60,-4.24) (-38.89,-4.95) (-38.18,-5.66) (-37.48,-5.66) (-36.77,-6.36) (-36.06,-6.36) (-35.36,-7.07) (-34.65,-7.78) (-33.94,-7.78) (-33.23,-8.49) (-32.53,-8.49) (-31.82,-9.19) (-31.11,-9.19) (-30.41,-9.90) (-29.70,-10.61) (-28.99,-10.61) (-28.28,-11.31) (-27.58,-11.31) (-26.87,-12.02) (-26.16,-12.73) (-25.46,-12.73) (-24.75,-13.44) (-24.04,-13.44) (-23.33,-14.14) (-22.63,-14.14) (-21.92,-14.85) (-21.21,-15.56) (-20.51,-15.56) (-19.80,-16.26) (-19.09,-16.26) (-18.38,-16.97) (-17.68,-17.68) (-16.97,-17.68) (-16.26,-18.38) (-15.56,-18.38) (-14.85,-19.09) (-14.14,-19.09) (-13.44,-19.80) (-12.73,-20.51) (-12.02,-20.51) (-11.31,-21.21) (-10.61,-21.21) (-9.90,-21.92) (-9.19,-22.63) (-8.49,-22.63) (-7.78,-23.33) (-7.07,-23.33) (-6.36,-24.04) (-5.66,-24.04) (-4.95,-24.75) (-4.24,-25.46) (-3.54,-25.46) (-2.83,-26.16) (-2.12,-26.16) (-1.41,-26.87) (-47.38, 0.00) (-46.67, 0.00) (-45.96, 0.00) (-45.25, 0.00) (-44.55, 0.00) (-44.55, 0.00) (-44.55, 0.00) (-44.55, 0.71) (-44.55, 1.41) (-45.25, 2.12) (-45.25, 2.83) (-45.25, 3.54) (-45.25, 4.24) (-45.96, 4.95) (-45.96, 5.66) (-45.96, 4.95) (-45.96, 5.66) (-45.96, 3.54) (-45.96, 4.24) (-45.96, 4.95) (-45.96, 3.54) (-45.96, 3.54) (-45.96, 4.24) (-46.67, 4.95) (-46.67, 5.66) (-47.38, 6.36) (-47.38, 7.07) (-47.38, 7.78) (-48.08, 8.49) (-48.08, 9.19) (-48.79, 9.90) (-48.79,10.61) (-49.50, 7.78) (-49.50, 8.49) (-49.50, 9.19) (-48.79, 9.90) (-48.79,10.61) (-50.91, 4.95) (-50.91, 5.66) (-50.20, 6.36) (-50.20, 7.07) (-49.50, 7.78) (-50.91, 4.95) (-50.20, 4.24) (-49.50, 4.24) (-48.79, 3.54) (-49.50, 1.41) (-49.50, 2.12) (-48.79, 2.83) (-48.79, 3.54) (-49.50, 1.41) (-49.50,-1.41) (-49.50,-0.71) (-49.50, 0.00) (-49.50, 0.71) (-49.50, 1.41) (-49.50,-1.41) (-48.79,-2.12) (-48.08,-2.83) (-47.38,-3.54) (-46.67,-3.54) (-45.96,-4.24) (-45.25,-4.95) (-44.55,-5.66) (-52.33,-5.66) (-51.62,-5.66) (-50.91,-5.66) (-50.20,-5.66) (-49.50,-5.66) (-48.79,-5.66) (-48.08,-5.66) (-47.38,-5.66) (-46.67,-5.66) (-45.96,-5.66) (-45.25,-5.66) (-44.55,-5.66) (-52.33,-5.66) (-52.33,-4.95) (-52.33,-4.24) (-52.33,-3.54) (-52.33,-2.83) (-52.33,-2.12) (-52.33,-1.41) (-52.33,-0.71) (-52.33, 0.00) (-52.33, 0.71) (-52.33, 1.41) (-52.33, 1.41) (-52.33, 2.12) (-53.03, 2.83) (-53.03, 3.54) (-53.03, 4.24) (-53.03, 4.95) (-53.74, 5.66) (-53.74, 6.36) (-53.74, 7.07) (-53.74, 7.78) (-54.45, 8.49) (-54.45, 9.19) (-54.45, 9.90) (-54.45,10.61) (-55.15,11.31) (-55.15,12.02) (-55.15,12.73) (-55.15,13.44) (-55.86,14.14) (-55.86,14.85) (-70.71, 7.78) (-70.00, 7.78) (-69.30, 8.49) (-68.59, 8.49) (-67.88, 9.19) (-67.18, 9.19) (-66.47, 9.90) (-65.76, 9.90) (-65.05,10.61) (-64.35,10.61) (-63.64,11.31) (-62.93,11.31) (-62.23,12.02) (-61.52,12.02) (-60.81,12.73) (-60.10,12.73) (-59.40,13.44) (-58.69,13.44) (-57.98,14.14) (-57.28,14.14) (-56.57,14.85) (-55.86,14.85) (-85.56,19.80) (-84.85,19.09) (-84.15,18.38) (-83.44,18.38) (-82.73,17.68) (-82.02,16.97) (-81.32,16.26) (-80.61,15.56) (-79.90,15.56) (-79.20,14.85) (-78.49,14.14) (-77.78,13.44) (-77.07,12.73) (-76.37,12.02) (-75.66,12.02) (-74.95,11.31) (-74.25,10.61) (-73.54, 9.90) (-72.83, 9.19) (-72.12, 9.19) (-71.42, 8.49) (-70.71, 7.78) (-85.56,19.80) (-84.85,19.09) (-84.15,19.09) (-83.44,18.38) (-82.73,17.68) (-82.02,16.97) (-81.32,16.97) (-80.61,16.26) (-79.90,15.56) (-79.20,15.56) (-78.49,14.85) (-77.78,14.14) (-77.07,14.14) (-76.37,13.44) (-75.66,12.73) (-74.95,12.02) (-74.25,12.02) (-73.54,11.31) (-72.83,10.61) (-72.12,10.61) (-71.42, 9.90) (-70.71, 9.19) (-70.00, 8.49) (-69.30, 8.49) (-68.59, 7.78) (-89.80,19.80) (-89.10,19.09) (-88.39,19.09) (-87.68,18.38) (-86.97,18.38) (-86.27,17.68) (-85.56,17.68) (-84.85,16.97) (-84.15,16.26) (-83.44,16.26) (-82.73,15.56) (-82.02,15.56) (-81.32,14.85) (-80.61,14.85) (-79.90,14.14) (-79.20,14.14) (-78.49,13.44) (-77.78,12.73) (-77.07,12.73) (-76.37,12.02) (-75.66,12.02) (-74.95,11.31) (-74.25,11.31) (-73.54,10.61) (-72.83, 9.90) (-72.12, 9.90) (-71.42, 9.19) (-70.71, 9.19) (-70.00, 8.49) (-69.30, 8.49) (-68.59, 7.78) (-89.80,19.80) (-89.10,19.09) (-88.39,19.09) (-87.68,18.38) (-86.97,17.68) (-86.27,17.68) (-85.56,16.97) (-84.85,16.26) (-84.15,15.56) (-83.44,15.56) (-82.73,14.85) (-82.02,14.14) (-81.32,14.14) (-80.61,13.44) (-79.90,12.73) (-79.20,12.73) (-78.49,12.02) (-77.78,11.31) (-77.07,10.61) (-76.37,10.61) (-75.66, 9.90) (-74.95, 9.19) (-74.25, 9.19) (-73.54, 8.49) (-72.83, 7.78) (-72.12, 7.78) (-71.42, 7.07) (-70.71, 6.36) (-70.00, 5.66) (-69.30, 5.66) (-68.59, 4.95) (-67.88, 4.24) (-67.18, 4.24) (-66.47, 3.54) (-66.47, 3.54) (-66.47, 4.24) (-67.18, 4.95) (-67.18, 5.66) (-67.18, 6.36) (-67.18, 7.07) (-67.88, 7.78) (-67.88, 8.49) (-67.88, 9.19) (-68.59, 9.90) (-68.59,10.61) (-68.59,11.31) (-69.30,12.02) (-69.30,12.73) (-69.30,13.44) (-69.30,14.14) (-70.00,14.85) (-70.00,15.56) (-70.00,16.26) (-70.71,16.97) (-70.71,17.68) (-70.71,18.38) (-71.42,19.09) (-71.42,19.80) (-71.42,20.51) (-71.42,21.21) (-72.12,21.92) (-72.12,22.63) (-72.12,23.33) (-72.83,24.04) (-72.83,24.75) (-72.83,25.46) (-73.54,26.16) (-73.54,26.87) (-73.54,27.58) (-73.54,28.28) (-74.25,28.99) (-74.25,29.70) (-74.25,30.41) (-74.95,31.11) (-74.95,31.82) (-106.77,40.31) (-106.07,40.31) (-105.36,39.60) (-104.65,39.60) (-103.94,39.60) (-103.24,39.60) (-102.53,38.89) (-101.82,38.89) (-101.12,38.89) (-100.41,38.89) (-99.70,38.18) (-98.99,38.18) (-98.29,38.18) (-97.58,38.18) (-96.87,37.48) (-96.17,37.48) (-95.46,37.48) (-94.75,36.77) (-94.05,36.77) (-93.34,36.77) (-92.63,36.77) (-91.92,36.06) (-91.22,36.06) (-90.51,36.06) (-89.80,36.06) (-89.10,35.36) (-88.39,35.36) (-87.68,35.36) (-86.97,35.36) (-86.27,34.65) (-85.56,34.65) (-84.85,34.65) (-84.15,33.94) (-83.44,33.94) (-82.73,33.94) (-82.02,33.94) (-81.32,33.23) (-80.61,33.23) (-79.90,33.23) (-79.20,33.23) (-78.49,32.53) (-77.78,32.53) (-77.07,32.53) (-76.37,32.53) (-75.66,31.82) (-74.95,31.82) (-106.77,40.31) (-106.07,40.31) (-105.36,40.31) (-104.65,39.60) (-103.94,39.60) (-103.24,39.60) (-102.53,39.60) (-101.82,39.60) (-101.12,39.60) (-100.41,38.89) (-99.70,38.89) (-98.99,38.89) (-98.29,38.89) (-97.58,38.89) (-96.87,38.18) (-96.17,38.18) (-95.46,38.18) (-94.75,38.18) (-94.05,38.18) (-93.34,37.48) (-92.63,37.48) (-91.92,37.48) (-91.22,37.48) (-90.51,37.48) (-89.80,37.48) (-89.10,36.77) (-88.39,36.77) (-87.68,36.77) (-86.97,36.77) (-86.27,36.77) (-85.56,36.06) (-84.85,36.06) (-84.15,36.06) (-83.44,36.06) (-82.73,36.06) (-82.02,35.36) (-81.32,35.36) (-80.61,35.36) (-79.90,35.36) (-79.20,35.36) (-78.49,35.36) (-77.78,34.65) (-77.07,34.65) (-76.37,34.65) (-75.66,34.65) (-74.95,34.65) (-74.25,33.94) (-73.54,33.94) (-72.83,33.94) (-110.31,27.58) (-109.60,27.58) (-108.89,27.58) (-108.19,28.28) (-107.48,28.28) (-106.77,28.28) (-106.07,28.28) (-105.36,28.28) (-104.65,28.28) (-103.94,28.99) (-103.24,28.99) (-102.53,28.99) (-101.82,28.99) (-101.12,28.99) (-100.41,28.99) (-99.70,29.70) (-98.99,29.70) (-98.29,29.70) (-97.58,29.70) (-96.87,29.70) (-96.17,29.70) (-95.46,30.41) (-94.75,30.41) (-94.05,30.41) (-93.34,30.41) (-92.63,30.41) (-91.92,30.41) (-91.22,31.11) (-90.51,31.11) (-89.80,31.11) (-89.10,31.11) (-88.39,31.11) (-87.68,31.11) (-86.97,31.82) (-86.27,31.82) (-85.56,31.82) (-84.85,31.82) (-84.15,31.82) (-83.44,31.82) (-82.73,32.53) (-82.02,32.53) (-81.32,32.53) (-80.61,32.53) (-79.90,32.53) (-79.20,32.53) (-78.49,33.23) (-77.78,33.23) (-77.07,33.23) (-76.37,33.23) (-75.66,33.23) (-74.95,33.23) (-74.25,33.94) (-73.54,33.94) (-72.83,33.94) (-110.31,27.58) (-109.60,28.28) (-109.60,28.99) (-108.89,29.70) (-108.89,30.41) (-108.19,31.11) (-108.19,31.82) (-107.48,32.53) (-107.48,33.23) (-106.77,33.94) (-106.77,34.65) (-106.07,35.36) (-106.07,36.06) (-105.36,36.77) (-105.36,37.48) (-104.65,38.18) (-104.65,38.89) (-103.94,39.60) (-103.94,40.31) (-103.24,41.01) (-103.24,41.72) (-102.53,42.43) (-102.53,43.13) (-101.82,43.84) (-101.82,44.55) (-101.12,45.25) (-100.41,45.96) (-100.41,46.67) (-99.70,47.38) (-99.70,48.08) (-98.99,48.79) (-98.99,49.50) (-98.29,50.20) (-98.29,50.91) (-97.58,51.62) (-97.58,52.33) (-96.87,53.03) (-96.87,53.74) (-96.17,54.45) (-96.17,55.15) (-95.46,55.86) (-95.46,56.57) (-94.75,57.28) (-94.75,57.98) (-94.05,58.69) (-94.05,59.40) (-93.34,60.10) (-93.34,60.81) (-92.63,61.52) (-92.63,62.23) (-91.92,62.93) (-91.92,62.93) (-91.22,62.93) (-90.51,62.93) (-90.51,62.93) (-90.51,63.64) (-89.80,64.35) (-89.80,64.35) (-89.80,65.05) (-90.51,65.76) (-90.51,65.76) (-89.80,65.05) (-89.10,65.05) (-88.39,64.35) (-87.68,64.35) (-86.97,63.64) (-86.97,61.52) (-86.97,62.23) (-86.97,62.93) (-86.97,63.64) (-86.97,61.52) (-86.27,61.52) (-85.56,60.81) (-89.80,59.40) (-89.10,59.40) (-88.39,60.10) (-87.68,60.10) (-86.97,60.10) (-86.27,60.81) (-85.56,60.81) (-89.80,59.40) (-84.85,53.74) (-85.56,54.45) (-86.27,55.15) (-86.97,55.86) (-86.97,56.57) (-87.68,57.28) (-88.39,57.98) (-89.10,58.69) (-89.80,59.40) (-93.34,60.81) (-92.63,60.10) (-91.92,59.40) (-91.22,59.40) (-90.51,58.69) (-89.80,57.98) (-89.10,57.28) (-88.39,56.57) (-87.68,55.86) (-86.97,55.86) (-86.27,55.15) (-85.56,54.45) (-84.85,53.74) (-93.34,60.81) (-92.63,60.10) (-91.92,59.40) (-91.92,58.69) (-91.92,59.40) (-91.92,58.69) (-91.22,58.69) (-90.51,58.69) (-89.80,58.69) (-89.10,58.69) (-88.39,58.69) (-89.80,56.57) (-89.10,57.28) (-89.10,57.98) (-88.39,58.69) (-89.80,56.57) (-89.10,56.57) (-88.39,55.86) (-88.39,55.86) (-88.39,56.57) (-87.68,57.28) (-87.68,57.98) (-87.68,57.98) (-86.97,53.74) (-86.97,54.45) (-86.97,55.15) (-86.97,55.86) (-87.68,56.57) (-87.68,57.28) (-87.68,57.98) (-86.97,53.74) (-86.27,53.74) (-85.56,53.74) (-85.56,53.74) (-84.85,53.74) (-84.15,53.74) (-83.44,53.74) (-82.73,53.74) (-82.02,53.74) (-81.32,53.74) (-80.61,53.74) (-79.90,45.96) (-79.90,46.67) (-79.90,47.38) (-79.90,48.08) (-79.90,48.79) (-79.90,49.50) (-80.61,50.20) (-80.61,50.91) (-80.61,51.62) (-80.61,52.33) (-80.61,53.03) (-80.61,53.74) (-79.90,45.96) (-79.20,45.96) (-78.49,46.67) (-77.78,46.67) (-77.07,47.38) (-76.37,47.38) (-75.66,47.38) (-74.95,48.08) (-74.25,48.08) (-73.54,48.79) (-72.83,48.79) (-72.12,49.50) (-71.42,49.50) (-71.42,49.50) (-70.71,48.79) (-70.00,48.79) (-69.30,48.08) (-68.59,47.38) (-67.88,46.67) (-67.18,46.67) (-66.47,45.96) (-65.76,45.25) (-65.05,44.55) (-64.35,44.55) (-63.64,43.84) (-62.93,43.13) (-62.23,42.43) (-61.52,42.43) (-60.81,41.72) (-74.95,38.89) (-74.25,38.89) (-73.54,38.89) (-72.83,39.60) (-72.12,39.60) (-71.42,39.60) (-70.71,39.60) (-70.00,39.60) (-69.30,40.31) (-68.59,40.31) (-67.88,40.31) (-67.18,40.31) (-66.47,40.31) (-65.76,41.01) (-65.05,41.01) (-64.35,41.01) (-63.64,41.01) (-62.93,41.01) (-62.23,41.72) (-61.52,41.72) (-60.81,41.72) (-74.95,38.89) (-74.95,39.60) (-75.66,40.31) (-75.66,41.01) (-76.37,41.72) (-76.37,42.43) (-77.07,43.13) (-77.07,43.84) (-77.78,44.55) (-77.78,45.25) (-78.49,45.96) (-78.49,46.67) (-79.20,47.38) (-79.20,48.08) (-79.90,48.79) (-79.90,49.50) (-80.61,50.20) (-80.61,50.91) (-81.32,51.62) (-81.32,52.33) (-82.02,53.03) (-82.02,53.74) (-82.73,54.45) (-88.39,35.36) (-88.39,36.06) (-87.68,36.77) (-87.68,37.48) (-87.68,38.18) (-87.68,38.89) (-86.97,39.60) (-86.97,40.31) (-86.97,41.01) (-86.27,41.72) (-86.27,42.43) (-86.27,43.13) (-85.56,43.84) (-85.56,44.55) (-85.56,45.25) (-85.56,45.96) (-84.85,46.67) (-84.85,47.38) (-84.85,48.08) (-84.15,48.79) (-84.15,49.50) (-84.15,50.20) (-83.44,50.91) (-83.44,51.62) (-83.44,52.33) (-83.44,53.03) (-82.73,53.74) (-82.73,54.45) (-88.39,35.36) (-87.68,34.65) (-86.97,34.65) (-86.27,33.94) (-85.56,33.23) (-84.85,33.23) (-84.15,32.53) (-83.44,32.53) (-82.73,31.82) (-82.02,31.11) (-81.32,31.11) (-80.61,30.41) (-79.90,29.70) (-79.20,29.70) (-78.49,28.99) (-77.78,28.99) (-77.07,28.28) (-76.37,27.58) (-75.66,27.58) (-74.95,26.87) (-74.25,26.16) (-73.54,26.16) (-72.83,25.46) (-72.12,25.46) (-71.42,24.75) (-70.71,24.04) (-70.00,24.04) (-69.30,23.33) (-87.68,39.60) (-86.97,38.89) (-86.27,38.18) (-85.56,37.48) (-84.85,36.77) (-84.15,36.77) (-83.44,36.06) (-82.73,35.36) (-82.02,34.65) (-81.32,33.94) (-80.61,33.23) (-79.90,32.53) (-79.20,31.82) (-78.49,31.82) (-77.78,31.11) (-77.07,30.41) (-76.37,29.70) (-75.66,28.99) (-74.95,28.28) (-74.25,27.58) (-73.54,26.87) (-72.83,26.16) (-72.12,26.16) (-71.42,25.46) (-70.71,24.75) (-70.00,24.04) (-69.30,23.33) (-85.56,13.44) (-85.56,14.14) (-85.56,14.85) (-85.56,15.56) (-85.56,16.26) (-85.56,16.97) (-85.56,17.68) (-86.27,18.38) (-86.27,19.09) (-86.27,19.80) (-86.27,20.51) (-86.27,21.21) (-86.27,21.92) (-86.27,22.63) (-86.27,23.33) (-86.27,24.04) (-86.27,24.75) (-86.27,25.46) (-86.27,26.16) (-86.97,26.87) (-86.97,27.58) (-86.97,28.28) (-86.97,28.99) (-86.97,29.70) (-86.97,30.41) (-86.97,31.11) (-86.97,31.82) (-86.97,32.53) (-86.97,33.23) (-86.97,33.94) (-86.97,34.65) (-87.68,35.36) (-87.68,36.06) (-87.68,36.77) (-87.68,37.48) (-87.68,38.18) (-87.68,38.89) (-87.68,39.60) (-85.56,13.44) (-84.85,13.44) (-84.15,14.14) (-83.44,14.14) (-82.73,14.14) (-82.02,14.85) (-81.32,14.85) (-80.61,15.56) (-79.90,15.56) (-79.20,15.56) (-78.49,16.26) (-77.78,16.26) (-77.07,16.26) (-76.37,16.97) (-75.66,16.97) (-74.95,16.97) (-74.25,17.68) (-73.54,17.68) (-72.83,17.68) (-72.12,18.38) (-71.42,18.38) (-70.71,19.09) (-70.00,19.09) (-69.30,19.09) (-68.59,19.80) (-67.88,19.80) (-67.18,19.80) (-66.47,20.51) (-65.76,20.51) (-65.05,20.51) (-64.35,21.21) (-63.64,21.21) (-62.93,21.21) (-62.23,21.92) (-61.52,21.92) (-60.81,22.63) (-60.10,22.63) (-59.40,22.63) (-58.69,23.33) (-57.98,23.33) (-54.45,-5.66) (-54.45,-4.95) (-54.45,-4.24) (-54.45,-3.54) (-54.45,-2.83) (-55.15,-2.12) (-55.15,-1.41) (-55.15,-0.71) (-55.15, 0.00) (-55.15, 0.71) (-55.15, 1.41) (-55.15, 2.12) (-55.15, 2.83) (-55.86, 3.54) (-55.86, 4.24) (-55.86, 4.95) (-55.86, 5.66) (-55.86, 6.36) (-55.86, 7.07) (-55.86, 7.78) (-55.86, 8.49) (-56.57, 9.19) (-56.57, 9.90) (-56.57,10.61) (-56.57,11.31) (-56.57,12.02) (-56.57,12.73) (-56.57,13.44) (-56.57,14.14) (-57.28,14.85) (-57.28,15.56) (-57.28,16.26) (-57.28,16.97) (-57.28,17.68) (-57.28,18.38) (-57.28,19.09) (-57.28,19.80) (-57.98,20.51) (-57.98,21.21) (-57.98,21.92) (-57.98,22.63) (-57.98,23.33) (-54.45,-5.66) (-53.74,-5.66) (-53.03,-6.36) (-52.33,-6.36) (-51.62,-7.07) (-50.91,-7.07) (-50.20,-7.78) (-49.50,-7.78) (-48.79,-7.78) (-48.08,-8.49) (-47.38,-8.49) (-46.67,-9.19) (-45.96,-9.19) (-45.25,-9.90) (-44.55,-9.90) (-43.84,-9.90) (-43.13,-10.61) (-42.43,-10.61) (-41.72,-11.31) (-41.01,-11.31) (-40.31,-12.02) (-39.60,-12.02) (-38.89,-12.02) (-38.18,-12.73) (-37.48,-12.73) (-36.77,-13.44) (-36.06,-13.44) (-35.36,-14.14) (-34.65,-14.14) (-33.94,-14.14) (-33.23,-14.85) (-32.53,-14.85) (-31.82,-15.56) (-31.11,-15.56) (-30.41,-16.26) (-29.70,-16.26) (-28.99,-16.26) (-28.28,-16.97) (-27.58,-16.97) (-26.87,-17.68) (-26.16,-17.68) (-25.46,-18.38) (-24.75,-18.38) (-60.81,-22.63) (-60.10,-22.63) (-59.40,-22.63) (-58.69,-22.63) (-57.98,-22.63) (-57.28,-21.92) (-56.57,-21.92) (-55.86,-21.92) (-55.15,-21.92) (-54.45,-21.92) (-53.74,-21.92) (-53.03,-21.92) (-52.33,-21.92) (-51.62,-21.21) (-50.91,-21.21) (-50.20,-21.21) (-49.50,-21.21) (-48.79,-21.21) (-48.08,-21.21) (-47.38,-21.21) (-46.67,-21.21) (-45.96,-21.21) (-45.25,-20.51) (-44.55,-20.51) (-43.84,-20.51) (-43.13,-20.51) (-42.43,-20.51) (-41.72,-20.51) (-41.01,-20.51) (-40.31,-20.51) (-39.60,-19.80) (-38.89,-19.80) (-38.18,-19.80) (-37.48,-19.80) (-36.77,-19.80) (-36.06,-19.80) (-35.36,-19.80) (-34.65,-19.80) (-33.94,-19.80) (-33.23,-19.09) (-32.53,-19.09) (-31.82,-19.09) (-31.11,-19.09) (-30.41,-19.09) (-29.70,-19.09) (-28.99,-19.09) (-28.28,-19.09) (-27.58,-18.38) (-26.87,-18.38) (-26.16,-18.38) (-25.46,-18.38) (-24.75,-18.38) (-60.81,-22.63) (-60.81,-21.92) (-60.10,-21.21) (-60.10,-20.51) (-60.10,-19.80) (-59.40,-19.09) (-59.40,-18.38) (-59.40,-17.68) (-58.69,-16.97) (-58.69,-16.26) (-58.69,-15.56) (-58.69,-14.85) (-57.98,-14.14) (-57.98,-13.44) (-57.98,-12.73) (-57.28,-12.02) (-57.28,-11.31) (-57.28,-10.61) (-56.57,-9.90) (-56.57,-9.19) (-56.57,-8.49) (-55.86,-7.78) (-55.86,-7.07) (-55.86,-6.36) (-55.15,-5.66) (-55.15,-4.95) (-55.15,-4.24) (-55.15,-3.54) (-54.45,-2.83) (-54.45,-2.12) (-54.45,-1.41) (-53.74,-0.71) (-53.74, 0.00) (-53.74, 0.71) (-53.03, 1.41) (-53.03, 2.12) (-53.03, 2.83) (-52.33, 3.54) (-52.33, 4.24) (-52.33, 4.95) (-51.62, 5.66) (-51.62, 6.36) (-51.62, 7.07) (-51.62, 7.78) (-50.91, 8.49) (-50.91, 9.19) (-50.91, 9.90) (-50.20,10.61) (-50.20,11.31) (-50.20,12.02) (-49.50,12.73) (-49.50,13.44) (-49.50,13.44) (-49.50,14.14) (-49.50,14.85) (-48.79,15.56) (-48.79,16.26) (-48.79,16.97) (-53.74,18.38) (-53.03,18.38) (-52.33,17.68) (-51.62,17.68) (-50.91,17.68) (-50.20,17.68) (-49.50,16.97) (-48.79,16.97) (-53.74,18.38) (-53.03,19.09) (-53.03,19.80) (-52.33,20.51) (-51.62,21.21) (-51.62,21.92) (-50.91,22.63) (-50.20,23.33) (-49.50,24.04) (-49.50,24.75) (-48.79,25.46) (-48.79,25.46) (-48.79,26.16) (-48.08,26.87) (-48.08,27.58) (-47.38,28.28) (-47.38,28.99) (-46.67,29.70) (-46.67,30.41) (-45.96,31.11) (-45.96,31.82) (-45.25,32.53) (-45.25,33.23) (-44.55,33.94) (-44.55,34.65) (-44.55,34.65) (-44.55,35.36) (-45.25,36.06) (-45.25,36.77) (-45.25,37.48) (-45.96,38.18) (-45.96,38.89) (-45.96,39.60) (-46.67,40.31) (-46.67,41.01) (-46.67,41.72) (-47.38,42.43) (-47.38,43.13) (-48.08,43.84) (-48.08,44.55) (-48.08,45.25) (-48.79,45.96) (-48.79,46.67) (-48.79,47.38) (-49.50,48.08) (-49.50,48.79) (-65.76,41.72) (-65.05,41.72) (-64.35,42.43) (-63.64,42.43) (-62.93,43.13) (-62.23,43.13) (-61.52,43.84) (-60.81,43.84) (-60.10,43.84) (-59.40,44.55) (-58.69,44.55) (-57.98,45.25) (-57.28,45.25) (-56.57,45.96) (-55.86,45.96) (-55.15,46.67) (-54.45,46.67) (-53.74,46.67) (-53.03,47.38) (-52.33,47.38) (-51.62,48.08) (-50.91,48.08) (-50.20,48.79) (-49.50,48.79) (-65.76,41.72) (-65.76,42.43) (-65.76,43.13) (-65.76,43.84) (-65.76,44.55) (-65.76,45.25) (-65.76,45.96) (-65.76,46.67) (-65.76,47.38) (-65.76,48.08) (-65.76,48.79) (-65.76,49.50) (-65.76,50.20) (-65.76,50.91) (-65.76,51.62) (-65.76,52.33) (-66.47,53.03) (-66.47,53.74) (-66.47,54.45) (-66.47,55.15) (-66.47,55.86) (-66.47,56.57) (-66.47,57.28) (-66.47,57.98) (-66.47,58.69) (-66.47,59.40) (-66.47,60.10) (-66.47,60.81) (-66.47,61.52) (-66.47,62.23) (-66.47,62.93) (-66.47,62.93) (-67.18,63.64) (-67.88,64.35) (-67.88,65.05) (-68.59,65.76) (-69.30,66.47) (-70.00,67.18) (-70.00,67.88) (-70.71,68.59) (-71.42,69.30) (-72.12,70.00) (-72.12,70.71) (-72.83,71.42) (-73.54,72.12) (-74.25,72.83) (-74.95,73.54) (-74.95,74.25) (-75.66,74.95) (-76.37,75.66) (-77.07,76.37) (-77.07,77.07) (-77.78,77.78) (-78.49,78.49) (-79.20,79.20) (-79.20,79.90) (-79.90,80.61) (-80.61,81.32) (-69.30,53.74) (-69.30,54.45) (-70.00,55.15) (-70.00,55.86) (-70.71,56.57) (-70.71,57.28) (-70.71,57.98) (-71.42,58.69) (-71.42,59.40) (-72.12,60.10) (-72.12,60.81) (-72.83,61.52) (-72.83,62.23) (-72.83,62.93) (-73.54,63.64) (-73.54,64.35) (-74.25,65.05) (-74.25,65.76) (-74.25,66.47) (-74.95,67.18) (-74.95,67.88) (-75.66,68.59) (-75.66,69.30) (-75.66,70.00) (-76.37,70.71) (-76.37,71.42) (-77.07,72.12) (-77.07,72.83) (-77.07,73.54) (-77.78,74.25) (-77.78,74.95) (-78.49,75.66) (-78.49,76.37) (-79.20,77.07) (-79.20,77.78) (-79.20,78.49) (-79.90,79.20) (-79.90,79.90) (-80.61,80.61) (-80.61,81.32) (-69.30,53.74) (-68.59,54.45) (-67.88,55.15) (-67.18,55.86) (-66.47,56.57) (-65.76,57.28) (-65.76,57.98) (-65.05,58.69) (-64.35,59.40) (-63.64,60.10) (-62.93,60.81) (-62.23,61.52) (-61.52,62.23) (-60.81,62.93) (-60.10,63.64) (-59.40,64.35) (-59.40,65.05) (-58.69,65.76) (-57.98,66.47) (-57.28,67.18) (-56.57,67.88) (-55.86,68.59) (-55.15,69.30) (-54.45,70.00) (-53.74,70.71) (-53.03,71.42) (-52.33,72.12) (-52.33,72.83) (-51.62,73.54) (-50.91,74.25) (-50.20,74.95) (-49.50,75.66) (-48.79,76.37) (-48.79,76.37) (-48.08,76.37) (-47.38,76.37) (-46.67,76.37) (-45.96,76.37) (-45.25,76.37) (-44.55,75.66) (-43.84,75.66) (-43.13,75.66) (-42.43,75.66) (-41.72,75.66) (-41.01,75.66) (-40.31,75.66) (-39.60,75.66) (-38.89,75.66) (-38.18,75.66) (-37.48,74.95) (-36.77,74.95) (-36.06,74.95) (-35.36,74.95) (-34.65,74.95) (-33.94,74.95) (-33.23,74.95) (-32.53,74.95) (-31.82,74.95) (-31.11,74.95) (-30.41,74.95) (-29.70,74.25) (-28.99,74.25) (-28.28,74.25) (-27.58,74.25) (-26.87,74.25) (-26.16,74.25) (-25.46,74.25) (-24.75,74.25) (-24.04,74.25) (-23.33,74.25) (-22.63,74.25) (-21.92,73.54) (-21.21,73.54) (-20.51,73.54) (-19.80,73.54) (-19.09,73.54) (-18.38,73.54) (-17.68,73.54) (-16.97,73.54) (-16.26,73.54) (-15.56,73.54) (-14.85,72.83) (-14.14,72.83) (-13.44,72.83) (-12.73,72.83) (-12.02,72.83) (-11.31,72.83) (-29.70,38.89) (-28.99,39.60) (-28.99,40.31) (-28.28,41.01) (-28.28,41.72) (-27.58,42.43) (-27.58,43.13) (-26.87,43.84) (-26.87,44.55) (-26.16,45.25) (-26.16,45.96) (-25.46,46.67) (-25.46,47.38) (-24.75,48.08) (-24.04,48.79) (-24.04,49.50) (-23.33,50.20) (-23.33,50.91) (-22.63,51.62) (-22.63,52.33) (-21.92,53.03) (-21.92,53.74) (-21.21,54.45) (-21.21,55.15) (-20.51,55.86) (-19.80,56.57) (-19.80,57.28) (-19.09,57.98) (-19.09,58.69) (-18.38,59.40) (-18.38,60.10) (-17.68,60.81) (-17.68,61.52) (-16.97,62.23) (-16.97,62.93) (-16.26,63.64) (-16.26,64.35) (-15.56,65.05) (-14.85,65.76) (-14.85,66.47) (-14.14,67.18) (-14.14,67.88) (-13.44,68.59) (-13.44,69.30) (-12.73,70.00) (-12.73,70.71) (-12.02,71.42) (-12.02,72.12) (-11.31,72.83) (-70.71,31.82) (-70.00,31.82) (-69.30,31.82) (-68.59,32.53) (-67.88,32.53) (-67.18,32.53) (-66.47,32.53) (-65.76,32.53) (-65.05,32.53) (-64.35,33.23) (-63.64,33.23) (-62.93,33.23) (-62.23,33.23) (-61.52,33.23) (-60.81,33.23) (-60.10,33.94) (-59.40,33.94) (-58.69,33.94) (-57.98,33.94) (-57.28,33.94) (-56.57,33.94) (-55.86,34.65) (-55.15,34.65) (-54.45,34.65) (-53.74,34.65) (-53.03,34.65) (-52.33,34.65) (-51.62,35.36) (-50.91,35.36) (-50.20,35.36) (-49.50,35.36) (-48.79,35.36) (-48.08,36.06) (-47.38,36.06) (-46.67,36.06) (-45.96,36.06) (-45.25,36.06) (-44.55,36.06) (-43.84,36.77) (-43.13,36.77) (-42.43,36.77) (-41.72,36.77) (-41.01,36.77) (-40.31,36.77) (-39.60,37.48) (-38.89,37.48) (-38.18,37.48) (-37.48,37.48) (-36.77,37.48) (-36.06,37.48) (-35.36,38.18) (-34.65,38.18) (-33.94,38.18) (-33.23,38.18) (-32.53,38.18) (-31.82,38.18) (-31.11,38.89) (-30.41,38.89) (-29.70,38.89) (-70.71,31.82) (-70.71,32.53) (-70.71,33.23) (-70.71,33.94) (-70.71,34.65) (-70.71,35.36) (-70.71,36.06) (-70.71,36.77) (-70.71,37.48) (-70.71,38.18) (-70.71,38.89) (-70.71,39.60) (-70.71,40.31) (-70.71,41.01) (-70.71,41.72) (-70.71,42.43) (-70.71,43.13) (-70.71,43.84) (-70.71,44.55) (-70.71,45.25) (-70.71,45.96) (-70.71,46.67) (-70.71,47.38) (-70.71,48.08) (-70.71,48.79) (-70.71,49.50) (-70.71,50.20) (-70.71,50.91) (-70.71,51.62) (-70.71,52.33) (-70.71,53.03) (-70.71,53.74) (-70.71,54.45) (-70.71,55.15) (-70.71,55.86) (-70.71,56.57) (-70.71,57.28) (-70.71,57.98) (-70.71,58.69) (-70.71,59.40) (-70.71,60.10) (-70.71,60.81) (-70.71,61.52) (-70.71,62.23) (-70.71,62.93) (-70.71,63.64) (-70.71,64.35) (-70.71,65.05) (-70.71,65.76) (-70.71,66.47) (-70.71,67.18) (-70.71,67.88) (-70.71,68.59) (-70.71,69.30) (-70.71,70.00) (-70.71,70.71) (-70.71,71.42) (-70.71,72.12) (-70.71,72.83) (-70.71,73.54) (-70.71,74.25) (-70.71,74.95) (-70.71,75.66) (-70.71,75.66) (-70.71,76.37) (-71.42,77.07) (-71.42,77.78) (-71.42,78.49) (-72.12,79.20) (-72.12,79.90) (-72.12,80.61) (-72.83,81.32) (-72.83,82.02) (-72.83,82.73) (-73.54,83.44) (-73.54,84.15) (-73.54,84.85) (-74.25,85.56) (-74.25,86.27) (-74.25,86.97) (-74.95,87.68) (-74.95,88.39) (-74.95,89.10) (-75.66,89.80) (-75.66,90.51) (-75.66,91.22) (-76.37,91.92) (-76.37,92.63) (-76.37,93.34) (-77.07,94.05) (-77.07,94.75) (-77.07,95.46) (-77.78,96.17) (-77.78,96.87) (-77.78,97.58) (-78.49,98.29) (-78.49,98.99) (-79.20,99.70) (-79.20,100.41) (-79.20,101.12) (-79.90,101.82) (-79.90,102.53) (-79.90,103.24) (-80.61,103.94) (-80.61,104.65) (-80.61,105.36) (-81.32,106.07) (-81.32,106.77) (-81.32,107.48) (-82.02,108.19) (-82.02,108.89) (-82.02,109.60) (-82.73,110.31) (-82.73,111.02) (-82.73,111.72) (-83.44,112.43) (-83.44,113.14) (-83.44,113.84) (-84.15,114.55) (-84.15,115.26) (-84.15,115.97) (-84.85,116.67) (-84.85,117.38) (-84.85,118.09) (-85.56,118.79) (-85.56,119.50) (-132.94,138.59) (-132.23,138.59) (-131.52,137.89) (-130.81,137.89) (-130.11,137.18) (-129.40,137.18) (-128.69,137.18) (-127.99,136.47) (-127.28,136.47) (-126.57,135.76) (-125.87,135.76) (-125.16,135.76) (-124.45,135.06) (-123.74,135.06) (-123.04,134.35) (-122.33,134.35) (-121.62,134.35) (-120.92,133.64) (-120.21,133.64) (-119.50,132.94) (-118.79,132.94) (-118.09,132.94) (-117.38,132.23) (-116.67,132.23) (-115.97,131.52) (-115.26,131.52) (-114.55,131.52) (-113.84,130.81) (-113.14,130.81) (-112.43,130.11) (-111.72,130.11) (-111.02,130.11) (-110.31,129.40) (-109.60,129.40) (-108.89,128.69) (-108.19,128.69) (-107.48,127.99) (-106.77,127.99) (-106.07,127.99) (-105.36,127.28) (-104.65,127.28) (-103.94,126.57) (-103.24,126.57) (-102.53,126.57) (-101.82,125.87) (-101.12,125.87) (-100.41,125.16) (-99.70,125.16) (-98.99,125.16) (-98.29,124.45) (-97.58,124.45) (-96.87,123.74) (-96.17,123.74) (-95.46,123.74) (-94.75,123.04) (-94.05,123.04) (-93.34,122.33) (-92.63,122.33) (-91.92,122.33) (-91.22,121.62) (-90.51,121.62) (-89.80,120.92) (-89.10,120.92) (-88.39,120.92) (-87.68,120.21) (-86.97,120.21) (-86.27,119.50) (-85.56,119.50) (-132.94,138.59) (-132.23,137.89) (-131.52,137.18) (-130.81,136.47) (-130.11,135.76) (-129.40,135.06) (-128.69,134.35) (-127.99,133.64) (-127.28,132.94) (-126.57,132.23) (-125.87,131.52) (-125.16,130.81) (-124.45,130.11) (-123.74,129.40) (-123.04,128.69) (-122.33,127.99) (-121.62,127.28) (-120.92,126.57) (-120.21,125.87) (-119.50,125.16) (-118.79,124.45) (-118.09,123.74) (-117.38,123.04) (-116.67,122.33) (-115.97,121.62) (-115.26,120.92) (-114.55,120.21) (-113.84,119.50) (-113.14,119.50) (-112.43,118.79) (-111.72,118.09) (-111.02,117.38) (-110.31,116.67) (-109.60,115.97) (-108.89,115.26) (-108.19,114.55) (-107.48,113.84) (-106.77,113.14) (-106.07,112.43) (-105.36,111.72) (-104.65,111.02) (-103.94,110.31) (-103.24,109.60) (-102.53,108.89) (-101.82,108.19) (-101.12,107.48) (-100.41,106.77) (-99.70,106.07) (-98.99,105.36) (-98.29,104.65) (-97.58,103.94) (-96.87,103.24) (-96.17,102.53) (-95.46,101.82) (-94.75,101.12) (-94.05,100.41) (-93.34,99.70) (-93.34,99.70) (-93.34,100.41) (-93.34,101.12) (-92.63,101.82) (-92.63,102.53) (-92.63,103.24) (-92.63,103.94) (-92.63,104.65) (-92.63,105.36) (-91.92,106.07) (-91.92,106.77) (-91.92,107.48) (-91.92,108.19) (-91.92,108.89) (-91.92,109.60) (-91.22,110.31) (-91.22,111.02) (-91.22,111.72) (-91.22,112.43) (-91.22,113.14) (-91.22,113.84) (-90.51,114.55) (-90.51,115.26) (-90.51,115.97) (-90.51,116.67) (-90.51,117.38) (-90.51,118.09) (-89.80,118.79) (-89.80,119.50) (-89.80,120.21) (-89.80,120.92) (-89.80,121.62) (-89.10,122.33) (-89.10,123.04) (-89.10,123.74) (-89.10,124.45) (-89.10,125.16) (-89.10,125.87) (-88.39,126.57) (-88.39,127.28) (-88.39,127.99) (-88.39,128.69) (-88.39,129.40) (-88.39,130.11) (-87.68,130.81) (-87.68,131.52) (-87.68,132.23) (-87.68,132.94) (-87.68,133.64) (-87.68,134.35) (-86.97,135.06) (-86.97,135.76) (-86.97,136.47) (-86.97,137.18) (-86.97,137.89) (-86.27,138.59) (-86.27,139.30) (-86.27,140.01) (-86.27,140.71) (-86.27,141.42) (-86.27,142.13) (-85.56,142.84) (-85.56,143.54) (-85.56,144.25) (-85.56,144.96) (-85.56,145.66) (-85.56,146.37) (-84.85,147.08) (-84.85,147.79) (-84.85,148.49) (-84.85,149.20) (-84.85,149.91) (-84.85,150.61) (-84.15,151.32) (-84.15,152.03) (-84.15,152.74) (-84.15,153.44) (-84.15,154.15) (-84.15,154.86) (-83.44,155.56) (-83.44,156.27) (-83.44,156.98) (-83.44,156.98) (-82.73,157.68) (-82.73,158.39) (-82.02,159.10) (-82.02,159.81) (-81.32,160.51) (-80.61,161.22) (-80.61,161.93) (-79.90,162.63) (-79.90,163.34) (-79.20,164.05) (-78.49,164.76) (-78.49,165.46) (-77.78,166.17) (-77.78,166.88) (-77.07,167.58) (-76.37,168.29) (-76.37,169.00) (-75.66,169.71) (-75.66,170.41) (-74.95,171.12) (-74.95,171.83) (-74.25,172.53) (-73.54,173.24) (-73.54,173.95) (-72.83,174.66) (-72.83,175.36) (-72.12,176.07) (-71.42,176.78) (-71.42,177.48) (-70.71,178.19) (-70.71,178.90) (-70.00,179.61) (-69.30,180.31) (-69.30,181.02) (-68.59,181.73) (-68.59,182.43) (-67.88,183.14) (-67.18,183.85) (-67.18,184.55) (-66.47,185.26) (-66.47,185.97) (-65.76,186.68) (-65.05,187.38) (-65.05,188.09) (-64.35,188.80) (-64.35,189.50) (-63.64,190.21) (-62.93,190.92) (-62.93,191.63) (-62.23,192.33) (-62.23,193.04) (-61.52,193.75) (-61.52,194.45) (-60.81,195.16) (-60.10,195.87) (-60.10,196.58) (-59.40,197.28) (-59.40,197.99) (-58.69,198.70) (-57.98,199.40) (-57.98,200.11) (-57.28,200.82) (-57.28,201.53) (-56.57,202.23) (-55.86,202.94) (-55.86,203.65) (-55.15,204.35) (-55.15,205.06) (-54.45,205.77) (-54.45,205.77) (-53.74,206.48) (-53.03,206.48) (-52.33,207.18) (-51.62,207.89) (-50.91,208.60) (-50.20,208.60) (-49.50,209.30) (-48.79,210.01) (-48.08,210.72) (-47.38,210.72) (-46.67,211.42) (-45.96,212.13) (-45.25,212.13) (-44.55,212.84) (-43.84,213.55) (-43.13,214.25) (-42.43,214.25) (-41.72,214.96) (-41.01,215.67) (-40.31,216.37) (-39.60,216.37) (-38.89,217.08) (-38.18,217.79) (-37.48,217.79) (-36.77,218.50) (-36.06,219.20) (-35.36,219.91) (-34.65,219.91) (-33.94,220.62) (-33.23,221.32) (-32.53,222.03) (-31.82,222.03) (-31.11,222.74) (-30.41,223.45) (-29.70,223.45) (-28.99,224.15) (-28.28,224.86) (-27.58,225.57) (-26.87,225.57) (-26.16,226.27) (-25.46,226.98) (-24.75,227.69) (-24.04,227.69) (-23.33,228.40) (-22.63,229.10) (-21.92,229.10) (-21.21,229.81) (-20.51,230.52) (-19.80,231.22) (-19.09,231.22) (-18.38,231.93) (-17.68,232.64) (-16.97,233.35) (-16.26,233.35) (-15.56,234.05) (-14.85,234.76) (-14.14,234.76) (-13.44,235.47) (-12.73,236.17) (-12.02,236.88) (-11.31,236.88) (-10.61,237.59) (-9.90,238.29) (-9.19,239.00) (-8.49,239.00) (-7.78,239.71) (23.33,191.63) (22.63,192.33) (22.63,193.04) (21.92,193.75) (21.21,194.45) (21.21,195.16) (20.51,195.87) (19.80,196.58) (19.80,197.28) (19.09,197.99) (19.09,198.70) (18.38,199.40) (17.68,200.11) (17.68,200.82) (16.97,201.53) (16.26,202.23) (16.26,202.94) (15.56,203.65) (14.85,204.35) (14.85,205.06) (14.14,205.77) (13.44,206.48) (13.44,207.18) (12.73,207.89) (12.02,208.60) (12.02,209.30) (11.31,210.01) (11.31,210.72) (10.61,211.42) ( 9.90,212.13) ( 9.90,212.84) ( 9.19,213.55) ( 8.49,214.25) ( 8.49,214.96) ( 7.78,215.67) ( 7.07,216.37) ( 7.07,217.08) ( 6.36,217.79) ( 5.66,218.50) ( 5.66,219.20) ( 4.95,219.91) ( 4.24,220.62) ( 4.24,221.32) ( 3.54,222.03) ( 3.54,222.74) ( 2.83,223.45) ( 2.12,224.15) ( 2.12,224.86) ( 1.41,225.57) ( 0.71,226.27) ( 0.71,226.98) ( 0.00,227.69) (-0.71,228.40) (-0.71,229.10) (-1.41,229.81) (-2.12,230.52) (-2.12,231.22) (-2.83,231.93) (-3.54,232.64) (-3.54,233.35) (-4.24,234.05) (-4.24,234.76) (-4.95,235.47) (-5.66,236.17) (-5.66,236.88) (-6.36,237.59) (-7.07,238.29) (-7.07,239.00) (-7.78,239.71) (23.33,191.63) (23.33,192.33) (24.04,193.04) (24.04,193.75) (24.75,194.45) (24.75,195.16) (25.46,195.87) (25.46,196.58) (26.16,197.28) (26.16,197.99) (26.87,198.70) (26.87,199.40) (27.58,200.11) (27.58,200.82) (28.28,201.53) (28.28,202.23) (28.99,202.94) (28.99,203.65) (29.70,204.35) (29.70,205.06) (30.41,205.77) (30.41,206.48) (31.11,207.18) (31.11,207.89) (31.82,208.60) (31.82,209.30) (32.53,210.01) (32.53,210.72) (33.23,211.42) (33.23,212.13) (33.94,212.84) (33.94,213.55) (34.65,214.25) (34.65,214.96) (35.36,215.67) (35.36,216.37) (36.06,217.08) (36.06,217.79) (36.06,218.50) (36.77,219.20) (36.77,219.91) (37.48,220.62) (37.48,221.32) (38.18,222.03) (38.18,222.74) (38.89,223.45) (38.89,224.15) (39.60,224.86) (39.60,225.57) (40.31,226.27) (40.31,226.98) (41.01,227.69) (41.01,228.40) (41.72,229.10) (41.72,229.81) (42.43,230.52) (42.43,231.22) (43.13,231.93) (43.13,232.64) (43.84,233.35) (43.84,234.05) (44.55,234.76) (44.55,235.47) (45.25,236.17) (45.25,236.88) (45.96,237.59) (45.96,238.29) (46.67,239.00) (46.67,239.71) (47.38,240.42) (47.38,241.12) (48.08,241.83) (48.08,242.54) (48.79,243.24) (48.79,243.95) (48.79,243.95) (49.50,244.66) (50.20,244.66) (50.91,245.37) (51.62,246.07) (52.33,246.07) (53.03,246.78) (53.74,247.49) (54.45,247.49) (55.15,248.19) (55.86,248.90) (56.57,248.90) (57.28,249.61) (57.98,250.32) (58.69,250.32) (59.40,251.02) (60.10,251.73) (60.81,251.73) (61.52,252.44) (62.23,253.14) (62.93,253.14) (63.64,253.85) (64.35,254.56) (65.05,254.56) (65.76,255.27) (66.47,255.97) (67.18,255.97) (67.88,256.68) (68.59,257.39) (69.30,257.39) (70.00,258.09) (70.71,258.80) (71.42,258.80) (72.12,259.51) (72.83,260.22) (73.54,260.22) (74.25,260.92) (74.95,261.63) (75.66,261.63) (76.37,262.34) (77.07,263.04) (77.78,263.04) (78.49,263.75) (79.20,264.46) (79.90,264.46) (80.61,265.17) (81.32,265.87) (82.02,265.87) (82.73,266.58) (83.44,267.29) (84.15,267.29) (84.85,267.99) (85.56,268.70) (86.27,268.70) (86.97,269.41) (87.68,270.11) (88.39,270.11) (89.10,270.82) (89.80,271.53) (90.51,271.53) (91.22,272.24) (91.92,272.94) (92.63,272.94) (93.34,273.65) (94.05,274.36) (94.75,274.36) (95.46,275.06) (96.17,275.77) (96.87,275.77) (97.58,276.48) (115.97,221.32) (115.97,222.03) (115.26,222.74) (115.26,223.45) (115.26,224.15) (114.55,224.86) (114.55,225.57) (114.55,226.27) (113.84,226.98) (113.84,227.69) (113.84,228.40) (113.14,229.10) (113.14,229.81) (113.14,230.52) (112.43,231.22) (112.43,231.93) (112.43,232.64) (111.72,233.35) (111.72,234.05) (111.72,234.76) (111.02,235.47) (111.02,236.17) (111.02,236.88) (110.31,237.59) (110.31,238.29) (110.31,239.00) (109.60,239.71) (109.60,240.42) (109.60,241.12) (108.89,241.83) (108.89,242.54) (108.89,243.24) (108.19,243.95) (108.19,244.66) (108.19,245.37) (107.48,246.07) (107.48,246.78) (107.48,247.49) (106.77,248.19) (106.77,248.90) (106.77,249.61) (106.07,250.32) (106.07,251.02) (106.07,251.73) (105.36,252.44) (105.36,253.14) (105.36,253.85) (104.65,254.56) (104.65,255.27) (104.65,255.97) (103.94,256.68) (103.94,257.39) (103.94,258.09) (103.24,258.80) (103.24,259.51) (103.24,260.22) (102.53,260.92) (102.53,261.63) (102.53,262.34) (101.82,263.04) (101.82,263.75) (101.82,264.46) (101.12,265.17) (101.12,265.87) (101.12,266.58) (100.41,267.29) (100.41,267.99) (100.41,268.70) (99.70,269.41) (99.70,270.11) (99.70,270.82) (98.99,271.53) (98.99,272.24) (98.99,272.94) (98.29,273.65) (98.29,274.36) (98.29,275.06) (97.58,275.77) (97.58,276.48) (115.97,221.32) (116.67,222.03) (117.38,222.03) (118.09,222.74) (118.79,222.74) (119.50,223.45) (120.21,223.45) (120.92,224.15) (121.62,224.86) (122.33,224.86) (123.04,225.57) (123.74,225.57) (124.45,226.27) (125.16,226.27) (125.87,226.98) (126.57,226.98) (127.28,227.69) (127.99,228.40) (128.69,228.40) (129.40,229.10) (130.11,229.10) (130.81,229.81) (131.52,229.81) (132.23,230.52) (132.94,231.22) (133.64,231.22) (134.35,231.93) (135.06,231.93) (135.76,232.64) (136.47,232.64) (137.18,233.35) (137.89,234.05) (138.59,234.05) (139.30,234.76) (140.01,234.76) (140.71,235.47) (141.42,235.47) (142.13,236.17) (142.84,236.17) (143.54,236.88) (144.25,237.59) (144.96,237.59) (145.66,238.29) (146.37,238.29) (147.08,239.00) (147.79,239.00) (148.49,239.71) (149.20,240.42) (149.91,240.42) (150.61,241.12) (151.32,241.12) (152.03,241.83) (152.74,241.83) (153.44,242.54) (154.15,243.24) (154.86,243.24) (155.56,243.95) (156.27,243.95) (156.98,244.66) (157.68,244.66) (158.39,245.37) (159.10,245.37) (159.81,246.07) (160.51,246.78) (161.22,246.78) (161.93,247.49) (162.63,247.49) (163.34,248.19) (164.05,248.19) (164.76,248.90) (135.76,198.70) (136.47,199.40) (136.47,200.11) (137.18,200.82) (137.18,201.53) (137.89,202.23) (137.89,202.94) (138.59,203.65) (139.30,204.35) (139.30,205.06) (140.01,205.77) (140.01,206.48) (140.71,207.18) (141.42,207.89) (141.42,208.60) (142.13,209.30) (142.13,210.01) (142.84,210.72) (142.84,211.42) (143.54,212.13) (144.25,212.84) (144.25,213.55) (144.96,214.25) (144.96,214.96) (145.66,215.67) (145.66,216.37) (146.37,217.08) (147.08,217.79) (147.08,218.50) (147.79,219.20) (147.79,219.91) (148.49,220.62) (148.49,221.32) (149.20,222.03) (149.91,222.74) (149.91,223.45) (150.61,224.15) (150.61,224.86) (151.32,225.57) (152.03,226.27) (152.03,226.98) (152.74,227.69) (152.74,228.40) (153.44,229.10) (153.44,229.81) (154.15,230.52) (154.86,231.22) (154.86,231.93) (155.56,232.64) (155.56,233.35) (156.27,234.05) (156.27,234.76) (156.98,235.47) (157.68,236.17) (157.68,236.88) (158.39,237.59) (158.39,238.29) (159.10,239.00) (159.10,239.71) (159.81,240.42) (160.51,241.12) (160.51,241.83) (161.22,242.54) (161.22,243.24) (161.93,243.95) (162.63,244.66) (162.63,245.37) (163.34,246.07) (163.34,246.78) (164.05,247.49) (164.05,248.19) (164.76,248.90) (93.34,157.68) (94.05,158.39) (94.75,159.10) (95.46,159.81) (96.17,160.51) (96.87,161.22) (97.58,161.93) (98.29,162.63) (98.99,163.34) (99.70,164.05) (100.41,164.76) (101.12,165.46) (101.82,166.17) (102.53,166.88) (103.24,167.58) (103.94,167.58) (104.65,168.29) (105.36,169.00) (106.07,169.71) (106.77,170.41) (107.48,171.12) (108.19,171.83) (108.89,172.53) (109.60,173.24) (110.31,173.95) (111.02,174.66) (111.72,175.36) (112.43,176.07) (113.14,176.78) (113.84,177.48) (114.55,178.19) (115.26,178.90) (115.97,179.61) (116.67,180.31) (117.38,181.02) (118.09,181.73) (118.79,182.43) (119.50,183.14) (120.21,183.85) (120.92,184.55) (121.62,185.26) (122.33,185.97) (123.04,186.68) (123.74,187.38) (124.45,188.09) (125.16,188.09) (125.87,188.80) (126.57,189.50) (127.28,190.21) (127.99,190.92) (128.69,191.63) (129.40,192.33) (130.11,193.04) (130.81,193.75) (131.52,194.45) (132.23,195.16) (132.94,195.87) (133.64,196.58) (134.35,197.28) (135.06,197.99) (135.76,198.70) (37.48,159.81) (38.18,159.81) (38.89,159.81) (39.60,159.81) (40.31,159.81) (41.01,159.81) (41.72,159.81) (42.43,159.81) (43.13,159.81) (43.84,159.81) (44.55,159.81) (45.25,159.81) (45.96,159.81) (46.67,159.81) (47.38,159.10) (48.08,159.10) (48.79,159.10) (49.50,159.10) (50.20,159.10) (50.91,159.10) (51.62,159.10) (52.33,159.10) (53.03,159.10) (53.74,159.10) (54.45,159.10) (55.15,159.10) (55.86,159.10) (56.57,159.10) (57.28,159.10) (57.98,159.10) (58.69,159.10) (59.40,159.10) (60.10,159.10) (60.81,159.10) (61.52,159.10) (62.23,159.10) (62.93,159.10) (63.64,159.10) (64.35,159.10) (65.05,159.10) (65.76,158.39) (66.47,158.39) (67.18,158.39) (67.88,158.39) (68.59,158.39) (69.30,158.39) (70.00,158.39) (70.71,158.39) (71.42,158.39) (72.12,158.39) (72.83,158.39) (73.54,158.39) (74.25,158.39) (74.95,158.39) (75.66,158.39) (76.37,158.39) (77.07,158.39) (77.78,158.39) (78.49,158.39) (79.20,158.39) (79.90,158.39) (80.61,158.39) (81.32,158.39) (82.02,158.39) (82.73,158.39) (83.44,158.39) (84.15,157.68) (84.85,157.68) (85.56,157.68) (86.27,157.68) (86.97,157.68) (87.68,157.68) (88.39,157.68) (89.10,157.68) (89.80,157.68) (90.51,157.68) (91.22,157.68) (91.92,157.68) (92.63,157.68) (93.34,157.68) (37.48,159.81) (38.18,160.51) (38.89,160.51) (39.60,161.22) (40.31,161.22) (41.01,161.93) (41.72,161.93) (42.43,162.63) (43.13,162.63) (43.84,163.34) (44.55,163.34) (45.25,164.05) (45.96,164.05) (46.67,164.76) (47.38,164.76) (48.08,165.46) (48.79,165.46) (49.50,166.17) (50.20,166.17) (50.91,166.88) (51.62,166.88) (52.33,167.58) (53.03,167.58) (53.74,168.29) (54.45,168.29) (55.15,169.00) (55.86,169.00) (56.57,169.71) (57.28,169.71) (57.98,170.41) (58.69,170.41) (59.40,171.12) (60.10,171.12) (60.81,171.83) (61.52,171.83) (62.23,172.53) (62.93,172.53) (63.64,173.24) (64.35,173.95) (65.05,173.95) (65.76,174.66) (66.47,174.66) (67.18,175.36) (67.88,175.36) (68.59,176.07) (69.30,176.07) (70.00,176.78) (70.71,176.78) (71.42,177.48) (72.12,177.48) (72.83,178.19) (73.54,178.19) (74.25,178.90) (74.95,178.90) (75.66,179.61) (76.37,179.61) (77.07,180.31) (77.78,180.31) (78.49,181.02) (79.20,181.02) (79.90,181.73) (80.61,181.73) (81.32,182.43) (82.02,182.43) (82.73,183.14) (83.44,183.14) (84.15,183.85) (84.85,183.85) (85.56,184.55) (86.27,184.55) (86.97,185.26) (87.68,185.26) (88.39,185.97) (71.42,129.40) (71.42,130.11) (72.12,130.81) (72.12,131.52) (72.12,132.23) (72.12,132.94) (72.83,133.64) (72.83,134.35) (72.83,135.06) (73.54,135.76) (73.54,136.47) (73.54,137.18) (74.25,137.89) (74.25,138.59) (74.25,139.30) (74.25,140.01) (74.95,140.71) (74.95,141.42) (74.95,142.13) (75.66,142.84) (75.66,143.54) (75.66,144.25) (76.37,144.96) (76.37,145.66) (76.37,146.37) (76.37,147.08) (77.07,147.79) (77.07,148.49) (77.07,149.20) (77.78,149.91) (77.78,150.61) (77.78,151.32) (78.49,152.03) (78.49,152.74) (78.49,153.44) (78.49,154.15) (79.20,154.86) (79.20,155.56) (79.20,156.27) (79.90,156.98) (79.90,157.68) (79.90,158.39) (80.61,159.10) (80.61,159.81) (80.61,160.51) (80.61,161.22) (81.32,161.93) (81.32,162.63) (81.32,163.34) (82.02,164.05) (82.02,164.76) (82.02,165.46) (82.73,166.17) (82.73,166.88) (82.73,167.58) (82.73,168.29) (83.44,169.00) (83.44,169.71) (83.44,170.41) (84.15,171.12) (84.15,171.83) (84.15,172.53) (84.85,173.24) (84.85,173.95) (84.85,174.66) (84.85,175.36) (85.56,176.07) (85.56,176.78) (85.56,177.48) (86.27,178.19) (86.27,178.90) (86.27,179.61) (86.97,180.31) (86.97,181.02) (86.97,181.73) (86.97,182.43) (87.68,183.14) (87.68,183.85) (87.68,184.55) (88.39,185.26) (88.39,185.97) (73.54,69.30) (73.54,70.00) (73.54,70.71) (73.54,71.42) (73.54,72.12) (73.54,72.83) (73.54,73.54) (73.54,74.25) (73.54,74.95) (73.54,75.66) (73.54,76.37) (73.54,77.07) (73.54,77.78) (73.54,78.49) (73.54,79.20) (72.83,79.90) (72.83,80.61) (72.83,81.32) (72.83,82.02) (72.83,82.73) (72.83,83.44) (72.83,84.15) (72.83,84.85) (72.83,85.56) (72.83,86.27) (72.83,86.97) (72.83,87.68) (72.83,88.39) (72.83,89.10) (72.83,89.80) (72.83,90.51) (72.83,91.22) (72.83,91.92) (72.83,92.63) (72.83,93.34) (72.83,94.05) (72.83,94.75) (72.83,95.46) (72.83,96.17) (72.83,96.87) (72.83,97.58) (72.83,98.29) (72.83,98.99) (72.12,99.70) (72.12,100.41) (72.12,101.12) (72.12,101.82) (72.12,102.53) (72.12,103.24) (72.12,103.94) (72.12,104.65) (72.12,105.36) (72.12,106.07) (72.12,106.77) (72.12,107.48) (72.12,108.19) (72.12,108.89) (72.12,109.60) (72.12,110.31) (72.12,111.02) (72.12,111.72) (72.12,112.43) (72.12,113.14) (72.12,113.84) (72.12,114.55) (72.12,115.26) (72.12,115.97) (72.12,116.67) (72.12,117.38) (72.12,118.09) (72.12,118.79) (71.42,119.50) (71.42,120.21) (71.42,120.92) (71.42,121.62) (71.42,122.33) (71.42,123.04) (71.42,123.74) (71.42,124.45) (71.42,125.16) (71.42,125.87) (71.42,126.57) (71.42,127.28) (71.42,127.99) (71.42,128.69) (71.42,129.40) (101.82,17.68) (101.12,18.38) (101.12,19.09) (100.41,19.80) (100.41,20.51) (99.70,21.21) (99.70,21.92) (98.99,22.63) (98.99,23.33) (98.29,24.04) (98.29,24.75) (97.58,25.46) (96.87,26.16) (96.87,26.87) (96.17,27.58) (96.17,28.28) (95.46,28.99) (95.46,29.70) (94.75,30.41) (94.75,31.11) (94.05,31.82) (93.34,32.53) (93.34,33.23) (92.63,33.94) (92.63,34.65) (91.92,35.36) (91.92,36.06) (91.22,36.77) (91.22,37.48) (90.51,38.18) (90.51,38.89) (89.80,39.60) (89.10,40.31) (89.10,41.01) (88.39,41.72) (88.39,42.43) (87.68,43.13) (87.68,43.84) (86.97,44.55) (86.97,45.25) (86.27,45.96) (86.27,46.67) (85.56,47.38) (84.85,48.08) (84.85,48.79) (84.15,49.50) (84.15,50.20) (83.44,50.91) (83.44,51.62) (82.73,52.33) (82.73,53.03) (82.02,53.74) (82.02,54.45) (81.32,55.15) (80.61,55.86) (80.61,56.57) (79.90,57.28) (79.90,57.98) (79.20,58.69) (79.20,59.40) (78.49,60.10) (78.49,60.81) (77.78,61.52) (77.07,62.23) (77.07,62.93) (76.37,63.64) (76.37,64.35) (75.66,65.05) (75.66,65.76) (74.95,66.47) (74.95,67.18) (74.25,67.88) (74.25,68.59) (73.54,69.30) (101.82,17.68) (102.53,18.38) (103.24,18.38) (103.94,19.09) (104.65,19.09) (105.36,19.80) (106.07,20.51) (106.77,20.51) (107.48,21.21) (108.19,21.21) (108.89,21.92) (109.60,22.63) (110.31,22.63) (111.02,23.33) (111.72,23.33) (112.43,24.04) (113.14,24.75) (113.84,24.75) (114.55,25.46) (115.26,25.46) (115.97,26.16) (116.67,26.87) (117.38,26.87) (118.09,27.58) (118.79,27.58) (119.50,28.28) (120.21,28.99) (120.92,28.99) (121.62,29.70) (122.33,29.70) (123.04,30.41) (123.74,31.11) (124.45,31.11) (125.16,31.82) (125.87,31.82) (126.57,32.53) (127.28,33.23) (127.99,33.23) (128.69,33.94) (129.40,33.94) (130.11,34.65) (130.81,35.36) (131.52,35.36) (132.23,36.06) (132.94,36.06) (133.64,36.77) (134.35,37.48) (135.06,37.48) (135.76,38.18) (136.47,38.18) (137.18,38.89) (137.89,39.60) (138.59,39.60) (139.30,40.31) (140.01,40.31) (140.71,41.01) (141.42,41.72) (142.13,41.72) (142.84,42.43) (143.54,42.43) (144.25,43.13) (144.96,43.84) (145.66,43.84) (146.37,44.55) (147.08,44.55) (147.79,45.25) (148.49,45.96) (149.20,45.96) (149.91,46.67) (150.61,46.67) (151.32,47.38) (127.99,-2.83) (127.99,-2.12) (128.69,-1.41) (128.69,-0.71) (129.40, 0.00) (129.40, 0.71) (130.11, 1.41) (130.11, 2.12) (130.81, 2.83) (130.81, 3.54) (131.52, 4.24) (131.52, 4.95) (132.23, 5.66) (132.23, 6.36) (132.94, 7.07) (132.94, 7.78) (132.94, 8.49) (133.64, 9.19) (133.64, 9.90) (134.35,10.61) (134.35,11.31) (135.06,12.02) (135.06,12.73) (135.76,13.44) (135.76,14.14) (136.47,14.85) (136.47,15.56) (137.18,16.26) (137.18,16.97) (137.18,17.68) (137.89,18.38) (137.89,19.09) (138.59,19.80) (138.59,20.51) (139.30,21.21) (139.30,21.92) (140.01,22.63) (140.01,23.33) (140.71,24.04) (140.71,24.75) (141.42,25.46) (141.42,26.16) (142.13,26.87) (142.13,27.58) (142.13,28.28) (142.84,28.99) (142.84,29.70) (143.54,30.41) (143.54,31.11) (144.25,31.82) (144.25,32.53) (144.96,33.23) (144.96,33.94) (145.66,34.65) (145.66,35.36) (146.37,36.06) (146.37,36.77) (146.37,37.48) (147.08,38.18) (147.08,38.89) (147.79,39.60) (147.79,40.31) (148.49,41.01) (148.49,41.72) (149.20,42.43) (149.20,43.13) (149.91,43.84) (149.91,44.55) (150.61,45.25) (150.61,45.96) (151.32,46.67) (151.32,47.38) (85.56,-42.43) (86.27,-41.72) (86.97,-41.01) (87.68,-40.31) (88.39,-39.60) (89.10,-38.89) (89.80,-38.18) (90.51,-37.48) (91.22,-37.48) (91.92,-36.77) (92.63,-36.06) (93.34,-35.36) (94.05,-34.65) (94.75,-33.94) (95.46,-33.23) (96.17,-32.53) (96.87,-31.82) (97.58,-31.11) (98.29,-30.41) (98.99,-29.70) (99.70,-28.99) (100.41,-28.28) (101.12,-27.58) (101.82,-27.58) (102.53,-26.87) (103.24,-26.16) (103.94,-25.46) (104.65,-24.75) (105.36,-24.04) (106.07,-23.33) (106.77,-22.63) (107.48,-21.92) (108.19,-21.21) (108.89,-20.51) (109.60,-19.80) (110.31,-19.09) (111.02,-18.38) (111.72,-17.68) (112.43,-17.68) (113.14,-16.97) (113.84,-16.26) (114.55,-15.56) (115.26,-14.85) (115.97,-14.14) (116.67,-13.44) (117.38,-12.73) (118.09,-12.02) (118.79,-11.31) (119.50,-10.61) (120.21,-9.90) (120.92,-9.19) (121.62,-8.49) (122.33,-7.78) (123.04,-7.78) (123.74,-7.07) (124.45,-6.36) (125.16,-5.66) (125.87,-4.95) (126.57,-4.24) (127.28,-3.54) (127.99,-2.83) (29.70,-22.63) (30.41,-22.63) (31.11,-23.33) (31.82,-23.33) (32.53,-23.33) (33.23,-24.04) (33.94,-24.04) (34.65,-24.04) (35.36,-24.75) (36.06,-24.75) (36.77,-25.46) (37.48,-25.46) (38.18,-25.46) (38.89,-26.16) (39.60,-26.16) (40.31,-26.16) (41.01,-26.87) (41.72,-26.87) (42.43,-26.87) (43.13,-27.58) (43.84,-27.58) (44.55,-27.58) (45.25,-28.28) (45.96,-28.28) (46.67,-28.99) (47.38,-28.99) (48.08,-28.99) (48.79,-29.70) (49.50,-29.70) (50.20,-29.70) (50.91,-30.41) (51.62,-30.41) (52.33,-30.41) (53.03,-31.11) (53.74,-31.11) (54.45,-31.11) (55.15,-31.82) (55.86,-31.82) (56.57,-31.82) (57.28,-32.53) (57.98,-32.53) (58.69,-33.23) (59.40,-33.23) (60.10,-33.23) (60.81,-33.94) (61.52,-33.94) (62.23,-33.94) (62.93,-34.65) (63.64,-34.65) (64.35,-34.65) (65.05,-35.36) (65.76,-35.36) (66.47,-35.36) (67.18,-36.06) (67.88,-36.06) (68.59,-36.06) (69.30,-36.77) (70.00,-36.77) (70.71,-37.48) (71.42,-37.48) (72.12,-37.48) (72.83,-38.18) (73.54,-38.18) (74.25,-38.18) (74.95,-38.89) (75.66,-38.89) (76.37,-38.89) (77.07,-39.60) (77.78,-39.60) (78.49,-39.60) (79.20,-40.31) (79.90,-40.31) (80.61,-41.01) (81.32,-41.01) (82.02,-41.01) (82.73,-41.72) (83.44,-41.72) (84.15,-41.72) (84.85,-42.43) (85.56,-42.43) (29.70,-22.63) (30.41,-23.33) (31.11,-24.04) (31.82,-24.04) (32.53,-24.75) (33.23,-25.46) (33.94,-26.16) (34.65,-26.87) (35.36,-27.58) (36.06,-27.58) (36.77,-28.28) (37.48,-28.99) (38.18,-29.70) (38.89,-30.41) (39.60,-30.41) (40.31,-31.11) (41.01,-31.82) (41.72,-32.53) (42.43,-33.23) (43.13,-33.94) (43.84,-33.94) (44.55,-34.65) (45.25,-35.36) (45.96,-36.06) (46.67,-36.77) (47.38,-36.77) (48.08,-37.48) (48.79,-38.18) (49.50,-38.89) (50.20,-39.60) (50.91,-40.31) (51.62,-40.31) (52.33,-41.01) (53.03,-41.72) (53.74,-42.43) (54.45,-43.13) (55.15,-43.84) (55.86,-43.84) (56.57,-44.55) (57.28,-45.25) (57.98,-45.96) (58.69,-46.67) (59.40,-46.67) (60.10,-47.38) (60.81,-48.08) (61.52,-48.79) (62.23,-49.50) (62.93,-50.20) (63.64,-50.20) (64.35,-50.91) (65.05,-51.62) (65.76,-52.33) (66.47,-53.03) (67.18,-53.03) (67.88,-53.74) (68.59,-54.45) (69.30,-55.15) (70.00,-55.86) (70.71,-56.57) (71.42,-56.57) (72.12,-57.28) (72.83,-57.98) (24.75,-21.92) (25.46,-22.63) (26.16,-22.63) (26.87,-23.33) (27.58,-24.04) (28.28,-24.75) (28.99,-24.75) (29.70,-25.46) (30.41,-26.16) (31.11,-26.87) (31.82,-26.87) (32.53,-27.58) (33.23,-28.28) (33.94,-28.99) (34.65,-28.99) (35.36,-29.70) (36.06,-30.41) (36.77,-31.11) (37.48,-31.11) (38.18,-31.82) (38.89,-32.53) (39.60,-33.23) (40.31,-33.23) (41.01,-33.94) (41.72,-34.65) (42.43,-35.36) (43.13,-35.36) (43.84,-36.06) (44.55,-36.77) (45.25,-37.48) (45.96,-37.48) (46.67,-38.18) (47.38,-38.89) (48.08,-39.60) (48.79,-39.60) (49.50,-40.31) (50.20,-41.01) (50.91,-41.72) (51.62,-41.72) (52.33,-42.43) (53.03,-43.13) (53.74,-43.84) (54.45,-43.84) (55.15,-44.55) (55.86,-45.25) (56.57,-45.96) (57.28,-45.96) (57.98,-46.67) (58.69,-47.38) (59.40,-48.08) (60.10,-48.08) (60.81,-48.79) (61.52,-49.50) (62.23,-50.20) (62.93,-50.20) (63.64,-50.91) (64.35,-51.62) (65.05,-52.33) (65.76,-52.33) (66.47,-53.03) (67.18,-53.74) (67.88,-54.45) (68.59,-54.45) (69.30,-55.15) (70.00,-55.86) (70.71,-56.57) (71.42,-56.57) (72.12,-57.28) (72.83,-57.98) (20.51,-79.90) (20.51,-79.20) (20.51,-78.49) (20.51,-77.78) (20.51,-77.07) (20.51,-76.37) (20.51,-75.66) (21.21,-74.95) (21.21,-74.25) (21.21,-73.54) (21.21,-72.83) (21.21,-72.12) (21.21,-71.42) (21.21,-70.71) (21.21,-70.00) (21.21,-69.30) (21.21,-68.59) (21.21,-67.88) (21.21,-67.18) (21.21,-66.47) (21.21,-65.76) (21.92,-65.05) (21.92,-64.35) (21.92,-63.64) (21.92,-62.93) (21.92,-62.23) (21.92,-61.52) (21.92,-60.81) (21.92,-60.10) (21.92,-59.40) (21.92,-58.69) (21.92,-57.98) (21.92,-57.28) (21.92,-56.57) (21.92,-55.86) (22.63,-55.15) (22.63,-54.45) (22.63,-53.74) (22.63,-53.03) (22.63,-52.33) (22.63,-51.62) (22.63,-50.91) (22.63,-50.20) (22.63,-49.50) (22.63,-48.79) (22.63,-48.08) (22.63,-47.38) (22.63,-46.67) (23.33,-45.96) (23.33,-45.25) (23.33,-44.55) (23.33,-43.84) (23.33,-43.13) (23.33,-42.43) (23.33,-41.72) (23.33,-41.01) (23.33,-40.31) (23.33,-39.60) (23.33,-38.89) (23.33,-38.18) (23.33,-37.48) (23.33,-36.77) (24.04,-36.06) (24.04,-35.36) (24.04,-34.65) (24.04,-33.94) (24.04,-33.23) (24.04,-32.53) (24.04,-31.82) (24.04,-31.11) (24.04,-30.41) (24.04,-29.70) (24.04,-28.99) (24.04,-28.28) (24.04,-27.58) (24.04,-26.87) (24.75,-26.16) (24.75,-25.46) (24.75,-24.75) (24.75,-24.04) (24.75,-23.33) (24.75,-22.63) (24.75,-21.92) (20.51,-79.90) (21.21,-80.61) (21.92,-81.32) (22.63,-82.02) (23.33,-82.73) (24.04,-83.44) (24.75,-84.15) (25.46,-84.85) (26.16,-85.56) (26.87,-86.27) (27.58,-86.97) (28.28,-87.68) (28.99,-88.39) (29.70,-89.10) (30.41,-89.80) (31.11,-90.51) (31.82,-91.22) (32.53,-91.92) (33.23,-92.63) (33.94,-93.34) (34.65,-94.05) (35.36,-94.75) (36.06,-95.46) (36.77,-96.17) (37.48,-96.87) (38.18,-97.58) (38.89,-98.29) (39.60,-98.99) (40.31,-99.70) (41.01,-100.41) (41.72,-101.12) (42.43,-101.82) (43.13,-102.53) (43.84,-103.24) (44.55,-103.94) (45.25,-104.65) (45.96,-105.36) (46.67,-106.07) (47.38,-106.77) (48.08,-107.48) (48.79,-108.19) (49.50,-108.89) (50.20,-109.60) (50.91,-110.31) (51.62,-111.02) (52.33,-111.72) (53.03,-112.43) (53.74,-113.14) (54.45,-113.84) (55.15,-114.55) (55.86,-115.26) (56.57,-115.97) (57.28,-116.67) (57.98,-117.38) (58.69,-118.09) (59.40,-118.79) (59.40,-118.79) (60.10,-118.09) (60.10,-117.38) (60.81,-116.67) (60.81,-115.97) (61.52,-115.26) (61.52,-114.55) (62.23,-113.84) (62.23,-113.14) (62.93,-112.43) (63.64,-111.72) (63.64,-111.02) (64.35,-110.31) (64.35,-109.60) (65.05,-108.89) (65.05,-108.19) (65.76,-107.48) (65.76,-106.77) (66.47,-106.07) (67.18,-105.36) (67.18,-104.65) (67.88,-103.94) (67.88,-103.24) (68.59,-102.53) (68.59,-101.82) (69.30,-101.12) (69.30,-100.41) (70.00,-99.70) (70.71,-98.99) (70.71,-98.29) (71.42,-97.58) (71.42,-96.87) (72.12,-96.17) (72.12,-95.46) (72.83,-94.75) (72.83,-94.05) (73.54,-93.34) (73.54,-92.63) (74.25,-91.92) (74.95,-91.22) (74.95,-90.51) (75.66,-89.80) (75.66,-89.10) (76.37,-88.39) (76.37,-87.68) (77.07,-86.97) (77.07,-86.27) (77.78,-85.56) (78.49,-84.85) (78.49,-84.15) (79.20,-83.44) (79.20,-82.73) (79.90,-82.02) (79.90,-81.32) (80.61,-80.61) (80.61,-79.90) (81.32,-79.20) (82.02,-78.49) (82.02,-77.78) (82.73,-77.07) (82.73,-76.37) (83.44,-75.66) (83.44,-74.95) (84.15,-74.25) (84.15,-73.54) (84.85,-72.83) (85.56,-72.12) (85.56,-71.42) (86.27,-70.71) (86.27,-70.00) (86.97,-69.30) (86.97,-68.59) (87.68,-67.88) (87.68,-67.18) (88.39,-66.47) (88.39,-66.47) (89.10,-66.47) (89.80,-66.47) (90.51,-66.47) (91.22,-66.47) (91.92,-66.47) (92.63,-66.47) (93.34,-66.47) (94.05,-66.47) (94.75,-66.47) (95.46,-66.47) (96.17,-66.47) (96.87,-66.47) (97.58,-66.47) (98.29,-66.47) (98.99,-66.47) (99.70,-66.47) (100.41,-66.47) (101.12,-66.47) (101.82,-66.47) (102.53,-66.47) (103.24,-66.47) (103.94,-66.47) (104.65,-66.47) (105.36,-66.47) (106.07,-66.47) (106.77,-66.47) (107.48,-66.47) (108.19,-66.47) (108.89,-66.47) (109.60,-66.47) (110.31,-66.47) (111.02,-66.47) (111.72,-66.47) (112.43,-66.47) (113.14,-66.47) (113.84,-66.47) (114.55,-66.47) (115.26,-66.47) (115.97,-66.47) (116.67,-66.47) (117.38,-66.47) (118.09,-66.47) (118.79,-66.47) (119.50,-66.47) (120.21,-66.47) (120.92,-66.47) (121.62,-66.47) (122.33,-66.47) (123.04,-66.47) (123.74,-66.47) (124.45,-66.47) (125.16,-66.47) (125.87,-66.47) (126.57,-66.47) (127.28,-66.47) (127.99,-66.47) (128.69,-66.47) (129.40,-66.47) (130.11,-66.47) (130.81,-66.47) (131.52,-66.47) (132.23,-66.47) (132.94,-66.47) (133.64,-66.47) (134.35,-66.47) (135.06,-66.47) (135.76,-66.47) (136.47,-66.47) (137.18,-66.47) (137.89,-66.47) (138.59,-66.47) (139.30,-66.47) (140.01,-66.47) (140.71,-66.47) (141.42,-66.47) (142.13,-66.47) (142.84,-66.47) (143.54,-66.47) (144.25,-66.47) (144.96,-66.47) (145.66,-66.47) (146.37,-66.47) (108.89,-111.72) (109.60,-111.02) (110.31,-110.31) (110.31,-109.60) (111.02,-108.89) (111.72,-108.19) (112.43,-107.48) (113.14,-106.77) (113.84,-106.07) (113.84,-105.36) (114.55,-104.65) (115.26,-103.94) (115.97,-103.24) (116.67,-102.53) (117.38,-101.82) (117.38,-101.12) (118.09,-100.41) (118.79,-99.70) (119.50,-98.99) (120.21,-98.29) (120.92,-97.58) (120.92,-96.87) (121.62,-96.17) (122.33,-95.46) (123.04,-94.75) (123.74,-94.05) (124.45,-93.34) (124.45,-92.63) (125.16,-91.92) (125.87,-91.22) (126.57,-90.51) (127.28,-89.80) (127.28,-89.10) (127.99,-88.39) (128.69,-87.68) (129.40,-86.97) (130.11,-86.27) (130.81,-85.56) (130.81,-84.85) (131.52,-84.15) (132.23,-83.44) (132.94,-82.73) (133.64,-82.02) (134.35,-81.32) (134.35,-80.61) (135.06,-79.90) (135.76,-79.20) (136.47,-78.49) (137.18,-77.78) (137.89,-77.07) (137.89,-76.37) (138.59,-75.66) (139.30,-74.95) (140.01,-74.25) (140.71,-73.54) (141.42,-72.83) (141.42,-72.12) (142.13,-71.42) (142.84,-70.71) (143.54,-70.00) (144.25,-69.30) (144.96,-68.59) (144.96,-67.88) (145.66,-67.18) (146.37,-66.47) (61.52,-143.54) (62.23,-142.84) (62.93,-142.84) (63.64,-142.13) (64.35,-141.42) (65.05,-141.42) (65.76,-140.71) (66.47,-140.01) (67.18,-140.01) (67.88,-139.30) (68.59,-138.59) (69.30,-138.59) (70.00,-137.89) (70.71,-137.18) (71.42,-137.18) (72.12,-136.47) (72.83,-135.76) (73.54,-135.76) (74.25,-135.06) (74.95,-134.35) (75.66,-134.35) (76.37,-133.64) (77.07,-132.94) (77.78,-132.94) (78.49,-132.23) (79.20,-131.52) (79.90,-131.52) (80.61,-130.81) (81.32,-130.11) (82.02,-130.11) (82.73,-129.40) (83.44,-128.69) (84.15,-128.69) (84.85,-127.99) (85.56,-127.28) (86.27,-126.57) (86.97,-126.57) (87.68,-125.87) (88.39,-125.16) (89.10,-125.16) (89.80,-124.45) (90.51,-123.74) (91.22,-123.74) (91.92,-123.04) (92.63,-122.33) (93.34,-122.33) (94.05,-121.62) (94.75,-120.92) (95.46,-120.92) (96.17,-120.21) (96.87,-119.50) (97.58,-119.50) (98.29,-118.79) (98.99,-118.09) (99.70,-118.09) (100.41,-117.38) (101.12,-116.67) (101.82,-116.67) (102.53,-115.97) (103.24,-115.26) (103.94,-115.26) (104.65,-114.55) (105.36,-113.84) (106.07,-113.84) (106.77,-113.14) (107.48,-112.43) (108.19,-112.43) (108.89,-111.72) (60.81,-145.66) (60.81,-144.96) (61.52,-144.25) (61.52,-143.54) (60.81,-145.66) (61.52,-146.37) (59.40,-145.66) (60.10,-145.66) (60.81,-146.37) (61.52,-146.37) (59.40,-145.66) (60.10,-144.96) (60.10,-144.25) (60.81,-143.54) (60.81,-143.54) (57.98,-145.66) (58.69,-144.96) (59.40,-144.96) (60.10,-144.25) (60.81,-143.54) (56.57,-146.37) (57.28,-146.37) (57.98,-145.66) (56.57,-147.79) (56.57,-147.08) (56.57,-146.37) (56.57,-147.79) (57.28,-147.08) (57.98,-146.37) (58.69,-145.66) (58.69,-145.66) (59.40,-145.66) (60.10,-144.96) (60.81,-144.96) (59.40,-145.66) (60.10,-145.66) (60.81,-144.96) (59.40,-145.66) (60.10,-145.66) (60.81,-146.37) (61.52,-146.37) (62.23,-146.37) (62.93,-147.08) (63.64,-147.08) (64.35,-147.08) (65.05,-147.79) (65.76,-147.79) (66.47,-147.79) (67.18,-148.49) (67.88,-148.49) (55.86,-156.98) (56.57,-156.27) (57.28,-156.27) (57.98,-155.56) (58.69,-154.86) (59.40,-154.15) (60.10,-154.15) (60.81,-153.44) (61.52,-152.74) (62.23,-152.74) (62.93,-152.03) (63.64,-151.32) (64.35,-151.32) (65.05,-150.61) (65.76,-149.91) (66.47,-149.20) (67.18,-149.20) (67.88,-148.49) (34.65,-155.56) (35.36,-155.56) (36.06,-155.56) (36.77,-155.56) (37.48,-155.56) (38.18,-155.56) (38.89,-155.56) (39.60,-155.56) (40.31,-156.27) (41.01,-156.27) (41.72,-156.27) (42.43,-156.27) (43.13,-156.27) (43.84,-156.27) (44.55,-156.27) (45.25,-156.27) (45.96,-156.27) (46.67,-156.27) (47.38,-156.27) (48.08,-156.27) (48.79,-156.27) (49.50,-156.27) (50.20,-156.27) (50.91,-156.98) (51.62,-156.98) (52.33,-156.98) (53.03,-156.98) (53.74,-156.98) (54.45,-156.98) (55.15,-156.98) (55.86,-156.98) (34.65,-155.56) (34.65,-154.86) (35.36,-154.15) (35.36,-153.44) (35.36,-152.74) (35.36,-152.03) (36.06,-151.32) (36.06,-150.61) (36.06,-149.91) (36.06,-149.20) (36.77,-148.49) (36.77,-147.79) (36.77,-147.08) (37.48,-146.37) (37.48,-145.66) (37.48,-144.96) (37.48,-144.25) (38.18,-143.54) (38.18,-142.84) (38.18,-142.13) (38.18,-141.42) (38.89,-140.71) (38.89,-140.01) (38.89,-139.30) (38.89,-138.59) (39.60,-137.89) (39.60,-137.18) (39.60,-136.47) (40.31,-135.76) (40.31,-135.06) (40.31,-134.35) (40.31,-133.64) (41.01,-132.94) (41.01,-132.23) (41.01,-131.52) (41.01,-130.81) (41.72,-130.11) (41.72,-129.40) (41.72,-129.40) (42.43,-128.69) (42.43,-127.99) (43.13,-127.28) (43.13,-126.57) (43.84,-125.87) (44.55,-125.16) (44.55,-124.45) (45.25,-123.74) (45.96,-123.04) (45.96,-122.33) (46.67,-121.62) (46.67,-120.92) (47.38,-120.21) (48.08,-119.50) (48.08,-118.79) (48.79,-118.09) (48.79,-117.38) (49.50,-116.67) (50.20,-115.97) (50.20,-115.26) (50.91,-114.55) (51.62,-113.84) (51.62,-113.14) (52.33,-112.43) (52.33,-111.72) (53.03,-111.02) (53.74,-110.31) (53.74,-109.60) (54.45,-108.89) (54.45,-108.19) (55.15,-107.48) (55.86,-106.77) (55.86,-106.07) (56.57,-105.36) (57.28,-104.65) (57.28,-103.94) (57.98,-103.24) (57.98,-102.53) (58.69,-101.82) (58.69,-101.82) (58.69,-101.12) (58.69,-100.41) (58.69,-99.70) (58.69,-98.99) (58.69,-98.99) (58.69,-98.29) (57.98,-97.58) (57.98,-97.58) (57.98,-97.58) (57.98,-96.87) (58.69,-96.17) (58.69,-95.46) (58.69,-94.75) (59.40,-94.05) (59.40,-93.34) (59.40,-93.34) (59.40,-92.63) (60.10,-91.92) (60.10,-91.22) (60.81,-90.51) (60.81,-90.51) (60.81,-89.80) (60.81,-89.10) (60.81,-88.39) (60.81,-87.68) (60.81,-87.68) (60.10,-86.97) (60.10,-86.27) (59.40,-85.56) (53.74,-87.68) (54.45,-87.68) (55.15,-86.97) (55.86,-86.97) (56.57,-86.97) (57.28,-86.27) (57.98,-86.27) (58.69,-85.56) (59.40,-85.56) (52.33,-86.97) (53.03,-86.97) (53.74,-87.68) (52.33,-86.97) (52.33,-86.27) (52.33,-85.56) (51.62,-84.85) (51.62,-84.15) (51.62,-83.44) (46.67,-86.97) (47.38,-86.27) (48.08,-86.27) (48.79,-85.56) (49.50,-84.85) (50.20,-84.15) (50.91,-84.15) (51.62,-83.44) (38.89,-91.92) (39.60,-91.22) (40.31,-91.22) (41.01,-90.51) (41.72,-89.80) (42.43,-89.80) (43.13,-89.10) (43.84,-89.10) (44.55,-88.39) (45.25,-87.68) (45.96,-87.68) (46.67,-86.97) (25.46,-85.56) (26.16,-85.56) (26.87,-86.27) (27.58,-86.27) (28.28,-86.97) (28.99,-86.97) (29.70,-87.68) (30.41,-87.68) (31.11,-88.39) (31.82,-88.39) (32.53,-89.10) (33.23,-89.10) (33.94,-89.80) (34.65,-89.80) (35.36,-90.51) (36.06,-90.51) (36.77,-91.22) (37.48,-91.22) (38.18,-91.92) (38.89,-91.92) (25.46,-85.56) (26.16,-85.56) (26.87,-85.56) (27.58,-85.56) (28.28,-85.56) (28.99,-85.56) (29.70,-85.56) (30.41,-86.27) (31.11,-86.27) (31.82,-86.27) (32.53,-86.27) (33.23,-86.27) (33.94,-86.27) (34.65,-86.27) (35.36,-86.27) (36.06,-86.27) (36.77,-86.27) (37.48,-86.27) (38.18,-86.27) (38.89,-86.27) (39.60,-86.97) (40.31,-86.97) (41.01,-86.97) (41.72,-86.97) (42.43,-86.97) (43.13,-86.97) (43.84,-86.97) (28.99,-99.70) (29.70,-98.99) (30.41,-98.29) (31.11,-97.58) (31.82,-97.58) (32.53,-96.87) (33.23,-96.17) (33.94,-95.46) (34.65,-94.75) (35.36,-94.05) (36.06,-93.34) (36.77,-93.34) (37.48,-92.63) (38.18,-91.92) (38.89,-91.22) (39.60,-90.51) (40.31,-89.80) (41.01,-89.10) (41.72,-89.10) (42.43,-88.39) (43.13,-87.68) (43.84,-86.97) ( 3.54,-108.89) ( 4.24,-108.89) ( 4.95,-108.19) ( 5.66,-108.19) ( 6.36,-108.19) ( 7.07,-107.48) ( 7.78,-107.48) ( 8.49,-106.77) ( 9.19,-106.77) ( 9.90,-106.77) (10.61,-106.07) (11.31,-106.07) (12.02,-106.07) (12.73,-105.36) (13.44,-105.36) (14.14,-105.36) (14.85,-104.65) (15.56,-104.65) (16.26,-104.65) (16.97,-103.94) (17.68,-103.94) (18.38,-103.24) (19.09,-103.24) (19.80,-103.24) (20.51,-102.53) (21.21,-102.53) (21.92,-102.53) (22.63,-101.82) (23.33,-101.82) (24.04,-101.82) (24.75,-101.12) (25.46,-101.12) (26.16,-100.41) (26.87,-100.41) (27.58,-100.41) (28.28,-99.70) (28.99,-99.70) ( 3.54,-108.89) ( 3.54,-108.19) ( 2.83,-107.48) ( 2.83,-106.77) ( 2.12,-106.07) ( 2.12,-105.36) ( 1.41,-104.65) ( 1.41,-103.94) ( 0.71,-103.24) ( 0.71,-102.53) ( 0.00,-101.82) ( 0.00,-101.12) (-0.71,-100.41) (-0.71,-99.70) (-1.41,-98.99) (-1.41,-98.29) (-2.12,-97.58) (-2.12,-96.87) (-2.12,-96.17) (-2.83,-95.46) (-2.83,-94.75) (-3.54,-94.05) (-3.54,-93.34) (-4.24,-92.63) (-4.24,-91.92) (-4.95,-91.22) (-4.95,-90.51) (-5.66,-89.80) (-5.66,-89.10) (-6.36,-88.39) (-6.36,-87.68) (-7.07,-86.97) (-7.07,-86.27) (-7.78,-85.56) (-7.78,-84.85) (-8.49,-84.15) (-8.49,-83.44) (-22.63,-111.72) (-22.63,-111.02) (-21.92,-110.31) (-21.92,-109.60) (-21.21,-108.89) (-21.21,-108.19) (-20.51,-107.48) (-20.51,-106.77) (-19.80,-106.07) (-19.80,-105.36) (-19.09,-104.65) (-19.09,-103.94) (-18.38,-103.24) (-18.38,-102.53) (-17.68,-101.82) (-17.68,-101.12) (-16.97,-100.41) (-16.97,-99.70) (-16.26,-98.99) (-16.26,-98.29) (-15.56,-97.58) (-15.56,-96.87) (-14.85,-96.17) (-14.85,-95.46) (-14.14,-94.75) (-14.14,-94.05) (-13.44,-93.34) (-13.44,-92.63) (-12.73,-91.92) (-12.73,-91.22) (-12.02,-90.51) (-12.02,-89.80) (-11.31,-89.10) (-11.31,-88.39) (-10.61,-87.68) (-10.61,-86.97) (-9.90,-86.27) (-9.90,-85.56) (-9.19,-84.85) (-9.19,-84.15) (-8.49,-83.44) (-31.82,-144.96) (-31.82,-144.25) (-31.11,-143.54) (-31.11,-142.84) (-31.11,-142.13) (-31.11,-141.42) (-30.41,-140.71) (-30.41,-140.01) (-30.41,-139.30) (-30.41,-138.59) (-29.70,-137.89) (-29.70,-137.18) (-29.70,-136.47) (-28.99,-135.76) (-28.99,-135.06) (-28.99,-134.35) (-28.99,-133.64) (-28.28,-132.94) (-28.28,-132.23) (-28.28,-131.52) (-27.58,-130.81) (-27.58,-130.11) (-27.58,-129.40) (-27.58,-128.69) (-26.87,-127.99) (-26.87,-127.28) (-26.87,-126.57) (-26.87,-125.87) (-26.16,-125.16) (-26.16,-124.45) (-26.16,-123.74) (-25.46,-123.04) (-25.46,-122.33) (-25.46,-121.62) (-25.46,-120.92) (-24.75,-120.21) (-24.75,-119.50) (-24.75,-118.79) (-24.04,-118.09) (-24.04,-117.38) (-24.04,-116.67) (-24.04,-115.97) (-23.33,-115.26) (-23.33,-114.55) (-23.33,-113.84) (-23.33,-113.14) (-22.63,-112.43) (-22.63,-111.72) (-39.60,-182.43) (-39.60,-181.73) (-39.60,-181.02) (-38.89,-180.31) (-38.89,-179.61) (-38.89,-178.90) (-38.89,-178.19) (-38.89,-177.48) (-38.18,-176.78) (-38.18,-176.07) (-38.18,-175.36) (-38.18,-174.66) (-38.18,-173.95) (-37.48,-173.24) (-37.48,-172.53) (-37.48,-171.83) (-37.48,-171.12) (-36.77,-170.41) (-36.77,-169.71) (-36.77,-169.00) (-36.77,-168.29) (-36.77,-167.58) (-36.06,-166.88) (-36.06,-166.17) (-36.06,-165.46) (-36.06,-164.76) (-36.06,-164.05) (-35.36,-163.34) (-35.36,-162.63) (-35.36,-161.93) (-35.36,-161.22) (-35.36,-160.51) (-34.65,-159.81) (-34.65,-159.10) (-34.65,-158.39) (-34.65,-157.68) (-34.65,-156.98) (-33.94,-156.27) (-33.94,-155.56) (-33.94,-154.86) (-33.94,-154.15) (-33.23,-153.44) (-33.23,-152.74) (-33.23,-152.03) (-33.23,-151.32) (-33.23,-150.61) (-32.53,-149.91) (-32.53,-149.20) (-32.53,-148.49) (-32.53,-147.79) (-32.53,-147.08) (-31.82,-146.37) (-31.82,-145.66) (-31.82,-144.96) (-36.77,-224.86) (-36.77,-224.15) (-36.77,-223.45) (-36.77,-222.74) (-36.77,-222.03) (-36.77,-221.32) (-36.77,-220.62) (-36.77,-219.91) (-37.48,-219.20) (-37.48,-218.50) (-37.48,-217.79) (-37.48,-217.08) (-37.48,-216.37) (-37.48,-215.67) (-37.48,-214.96) (-37.48,-214.25) (-37.48,-213.55) (-37.48,-212.84) (-37.48,-212.13) (-37.48,-211.42) (-37.48,-210.72) (-37.48,-210.01) (-37.48,-209.30) (-38.18,-208.60) (-38.18,-207.89) (-38.18,-207.18) (-38.18,-206.48) (-38.18,-205.77) (-38.18,-205.06) (-38.18,-204.35) (-38.18,-203.65) (-38.18,-202.94) (-38.18,-202.23) (-38.18,-201.53) (-38.18,-200.82) (-38.18,-200.11) (-38.18,-199.40) (-38.18,-198.70) (-38.89,-197.99) (-38.89,-197.28) (-38.89,-196.58) (-38.89,-195.87) (-38.89,-195.16) (-38.89,-194.45) (-38.89,-193.75) (-38.89,-193.04) (-38.89,-192.33) (-38.89,-191.63) (-38.89,-190.92) (-38.89,-190.21) (-38.89,-189.50) (-38.89,-188.80) (-38.89,-188.09) (-39.60,-187.38) (-39.60,-186.68) (-39.60,-185.97) (-39.60,-185.26) (-39.60,-184.55) (-39.60,-183.85) (-39.60,-183.14) (-39.60,-182.43) (-8.49,-262.34) (-9.19,-261.63) (-9.90,-260.92) (-9.90,-260.22) (-10.61,-259.51) (-11.31,-258.80) (-12.02,-258.09) (-12.02,-257.39) (-12.73,-256.68) (-13.44,-255.97) (-14.14,-255.27) (-14.14,-254.56) (-14.85,-253.85) (-15.56,-253.14) (-16.26,-252.44) (-16.26,-251.73) (-16.97,-251.02) (-17.68,-250.32) (-18.38,-249.61) (-18.38,-248.90) (-19.09,-248.19) (-19.80,-247.49) (-20.51,-246.78) (-20.51,-246.07) (-21.21,-245.37) (-21.92,-244.66) (-22.63,-243.95) (-22.63,-243.24) (-23.33,-242.54) (-24.04,-241.83) (-24.75,-241.12) (-24.75,-240.42) (-25.46,-239.71) (-26.16,-239.00) (-26.87,-238.29) (-26.87,-237.59) (-27.58,-236.88) (-28.28,-236.17) (-28.99,-235.47) (-28.99,-234.76) (-29.70,-234.05) (-30.41,-233.35) (-31.11,-232.64) (-31.11,-231.93) (-31.82,-231.22) (-32.53,-230.52) (-33.23,-229.81) (-33.23,-229.10) (-33.94,-228.40) (-34.65,-227.69) (-35.36,-226.98) (-35.36,-226.27) (-36.06,-225.57) (-36.77,-224.86) (-8.49,-262.34) (-7.78,-262.34) (-7.07,-261.63) (-6.36,-261.63) (-5.66,-260.92) (-4.95,-260.92) (-4.24,-260.22) (-3.54,-260.22) (-2.83,-259.51) (-2.12,-259.51) (-1.41,-258.80) (-0.71,-258.80) ( 0.00,-258.09) ( 0.71,-258.09) ( 1.41,-257.39) ( 2.12,-257.39) ( 2.83,-256.68) ( 3.54,-256.68) ( 4.24,-255.97) ( 4.95,-255.97) ( 5.66,-255.27) ( 6.36,-255.27) ( 7.07,-254.56) ( 7.78,-254.56) ( 8.49,-253.85) ( 9.19,-253.85) ( 9.90,-253.14) (10.61,-253.14) (11.31,-252.44) (12.02,-252.44) (12.73,-251.73) (13.44,-251.73) (14.14,-251.02) (14.85,-251.02) (15.56,-250.32) (16.26,-250.32) (16.97,-249.61) (17.68,-249.61) (18.38,-248.90) (19.09,-248.90) (19.80,-248.19) (20.51,-248.19) (21.21,-247.49) (21.92,-247.49) (22.63,-246.78) (23.33,-246.78) (24.04,-246.07) (24.75,-246.07) (25.46,-245.37) (26.16,-245.37) (26.87,-244.66) (27.58,-244.66) (28.28,-243.95) (28.99,-243.95) (29.70,-243.24) (30.41,-243.24) (31.11,-242.54) (31.82,-242.54) (32.53,-241.83) (33.23,-241.83) (33.94,-241.12) (34.65,-241.12) (35.36,-240.42) (36.06,-240.42) (36.77,-239.71) ( 8.49,-286.38) ( 9.19,-285.67) ( 9.19,-284.96) ( 9.90,-284.26) ( 9.90,-283.55) (10.61,-282.84) (11.31,-282.14) (11.31,-281.43) (12.02,-280.72) (12.02,-280.01) (12.73,-279.31) (13.44,-278.60) (13.44,-277.89) (14.14,-277.19) (14.14,-276.48) (14.85,-275.77) (15.56,-275.06) (15.56,-274.36) (16.26,-273.65) (16.97,-272.94) (16.97,-272.24) (17.68,-271.53) (17.68,-270.82) (18.38,-270.11) (19.09,-269.41) (19.09,-268.70) (19.80,-267.99) (19.80,-267.29) (20.51,-266.58) (21.21,-265.87) (21.21,-265.17) (21.92,-264.46) (21.92,-263.75) (22.63,-263.04) (23.33,-262.34) (23.33,-261.63) (24.04,-260.92) (24.04,-260.22) (24.75,-259.51) (25.46,-258.80) (25.46,-258.09) (26.16,-257.39) (26.16,-256.68) (26.87,-255.97) (27.58,-255.27) (27.58,-254.56) (28.28,-253.85) (28.28,-253.14) (28.99,-252.44) (29.70,-251.73) (29.70,-251.02) (30.41,-250.32) (31.11,-249.61) (31.11,-248.90) (31.82,-248.19) (31.82,-247.49) (32.53,-246.78) (33.23,-246.07) (33.23,-245.37) (33.94,-244.66) (33.94,-243.95) (34.65,-243.24) (35.36,-242.54) (35.36,-241.83) (36.06,-241.12) (36.06,-240.42) (36.77,-239.71) (-47.38,-290.62) (-46.67,-290.62) (-45.96,-290.62) (-45.25,-290.62) (-44.55,-290.62) (-43.84,-290.62) (-43.13,-290.62) (-42.43,-289.91) (-41.72,-289.91) (-41.01,-289.91) (-40.31,-289.91) (-39.60,-289.91) (-38.89,-289.91) (-38.18,-289.91) (-37.48,-289.91) (-36.77,-289.91) (-36.06,-289.91) (-35.36,-289.91) (-34.65,-289.91) (-33.94,-289.91) (-33.23,-289.21) (-32.53,-289.21) (-31.82,-289.21) (-31.11,-289.21) (-30.41,-289.21) (-29.70,-289.21) (-28.99,-289.21) (-28.28,-289.21) (-27.58,-289.21) (-26.87,-289.21) (-26.16,-289.21) (-25.46,-289.21) (-24.75,-289.21) (-24.04,-288.50) (-23.33,-288.50) (-22.63,-288.50) (-21.92,-288.50) (-21.21,-288.50) (-20.51,-288.50) (-19.80,-288.50) (-19.09,-288.50) (-18.38,-288.50) (-17.68,-288.50) (-16.97,-288.50) (-16.26,-288.50) (-15.56,-288.50) (-14.85,-288.50) (-14.14,-287.79) (-13.44,-287.79) (-12.73,-287.79) (-12.02,-287.79) (-11.31,-287.79) (-10.61,-287.79) (-9.90,-287.79) (-9.19,-287.79) (-8.49,-287.79) (-7.78,-287.79) (-7.07,-287.79) (-6.36,-287.79) (-5.66,-287.79) (-4.95,-287.09) (-4.24,-287.09) (-3.54,-287.09) (-2.83,-287.09) (-2.12,-287.09) (-1.41,-287.09) (-0.71,-287.09) ( 0.00,-287.09) ( 0.71,-287.09) ( 1.41,-287.09) ( 2.12,-287.09) ( 2.83,-287.09) ( 3.54,-287.09) ( 4.24,-286.38) ( 4.95,-286.38) ( 5.66,-286.38) ( 6.36,-286.38) ( 7.07,-286.38) ( 7.78,-286.38) ( 8.49,-286.38) (-47.38,-290.62) (-47.38,-289.91) (-48.79,-294.86) (-48.79,-294.16) (-48.08,-293.45) (-48.08,-292.74) (-48.08,-292.04) (-48.08,-291.33) (-47.38,-290.62) (-47.38,-289.91) (-48.79,-294.86) (-48.08,-294.86) (-47.38,-294.86) (-47.38,-294.86) (-47.38,-294.16) (-47.38,-293.45) (-47.38,-293.45) (-46.67,-293.45) (-45.96,-294.86) (-45.96,-294.16) (-46.67,-293.45) (-45.96,-294.86) (-45.25,-294.16) (-45.25,-293.45) (-44.55,-292.74) (-44.55,-292.74) (-49.50,-289.91) (-48.79,-290.62) (-48.08,-290.62) (-47.38,-291.33) (-46.67,-291.33) (-45.96,-292.04) (-45.25,-292.04) (-44.55,-292.74) (-49.50,-289.91) (-48.79,-289.91) (-48.08,-289.91) (-47.38,-289.91) (-46.67,-289.91) (-45.96,-289.91) (-45.25,-289.91) (-44.55,-289.91) (-43.84,-289.91) (-43.13,-289.91) (-42.43,-289.91) (-41.72,-289.91) (-41.01,-289.91) (-40.31,-289.91) (-39.60,-289.91) (-39.60,-289.91) (-38.89,-290.62) (-38.18,-290.62) (-37.48,-291.33) (-36.77,-291.33) (-36.06,-292.04) (-35.36,-292.74) (-34.65,-292.74) (-33.94,-293.45) (-33.23,-294.16) (-32.53,-294.16) (-31.82,-294.86) (-31.11,-294.86) (-30.41,-295.57) (-43.84,-290.62) (-43.13,-290.62) (-42.43,-291.33) (-41.72,-291.33) (-41.01,-291.33) (-40.31,-292.04) (-39.60,-292.04) (-38.89,-292.74) (-38.18,-292.74) (-37.48,-292.74) (-36.77,-293.45) (-36.06,-293.45) (-35.36,-293.45) (-34.65,-294.16) (-33.94,-294.16) (-33.23,-294.86) (-32.53,-294.86) (-31.82,-294.86) (-31.11,-295.57) (-30.41,-295.57) (-43.84,-290.62) (-43.13,-290.62) (-42.43,-289.91) (-42.43,-289.91) (-41.72,-289.91) (-41.01,-289.91) (-40.31,-289.91) (-39.60,-289.91) (-38.89,-289.91) (-38.18,-290.62) (-37.48,-290.62) (-36.77,-290.62) (-36.06,-290.62) (-35.36,-290.62) (-34.65,-290.62) (-32.53,-297.69) (-32.53,-296.98) (-33.23,-296.28) (-33.23,-295.57) (-33.23,-294.86) (-33.23,-294.16) (-33.94,-293.45) (-33.94,-292.74) (-33.94,-292.04) (-34.65,-291.33) (-34.65,-290.62) (-32.53,-297.69) (-32.53,-296.98) (-31.82,-296.28) (-31.82,-295.57) (-31.11,-294.86) (-31.11,-294.16) (-30.41,-293.45) (-30.41,-292.74) (-30.41,-292.04) (-29.70,-291.33) (-29.70,-290.62) (-28.99,-289.91) (-28.99,-289.21) (-28.28,-288.50) (-28.28,-287.79) (-27.58,-287.09) (-27.58,-286.38) (-27.58,-286.38) (-27.58,-285.67) (-26.87,-284.96) (-26.87,-284.26) (-26.16,-283.55) (-26.16,-282.84) (-25.46,-282.14) (-25.46,-281.43) (-24.75,-280.72) (-24.75,-280.01) (-24.04,-279.31) (-24.04,-278.60) (-24.04,-277.89) (-23.33,-277.19) (-23.33,-276.48) (-22.63,-275.77) (-22.63,-275.06) (-21.92,-274.36) (-21.92,-273.65) (-21.21,-272.94) (-21.21,-272.24) (-20.51,-271.53) (-20.51,-270.82) (-19.80,-270.11) (-19.80,-269.41) (-19.80,-269.41) (-19.80,-268.70) (-19.80,-267.99) (-19.80,-267.29) (-19.80,-266.58) (-19.80,-265.87) (-19.80,-265.17) (-19.80,-264.46) (-19.09,-263.75) (-19.09,-263.04) (-19.09,-262.34) (-19.09,-261.63) (-19.09,-260.92) (-19.09,-260.22) (-19.09,-259.51) (-19.09,-258.80) (-19.09,-258.09) (-19.09,-257.39) (-19.09,-256.68) (-19.09,-255.97) (-19.09,-255.27) (-19.09,-254.56) (-19.09,-253.85) (-19.09,-253.14) (-18.38,-252.44) (-18.38,-251.73) (-18.38,-251.02) (-18.38,-250.32) (-18.38,-249.61) (-18.38,-248.90) (-18.38,-248.19) (-18.38,-247.49) (-37.48,-230.52) (-36.77,-231.22) (-36.06,-231.93) (-35.36,-232.64) (-34.65,-233.35) (-33.94,-233.35) (-33.23,-234.05) (-32.53,-234.76) (-31.82,-235.47) (-31.11,-236.17) (-30.41,-236.88) (-29.70,-237.59) (-28.99,-238.29) (-28.28,-239.00) (-27.58,-239.00) (-26.87,-239.71) (-26.16,-240.42) (-25.46,-241.12) (-24.75,-241.83) (-24.04,-242.54) (-23.33,-243.24) (-22.63,-243.95) (-21.92,-244.66) (-21.21,-244.66) (-20.51,-245.37) (-19.80,-246.07) (-19.09,-246.78) (-18.38,-247.49) (-37.48,-230.52) (-36.77,-231.22) (-36.06,-231.22) (-35.36,-231.93) (-34.65,-231.93) (-33.94,-232.64) (-33.23,-232.64) (-32.53,-233.35) (-31.82,-234.05) (-31.11,-234.05) (-30.41,-234.76) (-29.70,-234.76) (-28.99,-235.47) (-28.28,-235.47) (-27.58,-236.17) (-26.87,-236.88) (-26.16,-236.88) (-25.46,-237.59) (-24.75,-237.59) (-24.04,-238.29) (-23.33,-238.29) (-22.63,-239.00) (-21.92,-239.71) (-21.21,-239.71) (-20.51,-240.42) (-19.80,-240.42) (-19.09,-241.12) (-18.38,-241.12) (-17.68,-241.83) (-16.97,-242.54) (-16.26,-242.54) (-15.56,-243.24) (-14.85,-243.24) (-14.14,-243.95) (-13.44,-243.95) (-12.73,-244.66) (-44.55,-234.76) (-43.84,-234.76) (-43.13,-235.47) (-42.43,-235.47) (-41.72,-235.47) (-41.01,-236.17) (-40.31,-236.17) (-39.60,-236.17) (-38.89,-236.17) (-38.18,-236.88) (-37.48,-236.88) (-36.77,-236.88) (-36.06,-237.59) (-35.36,-237.59) (-34.65,-237.59) (-33.94,-238.29) (-33.23,-238.29) (-32.53,-238.29) (-31.82,-239.00) (-31.11,-239.00) (-30.41,-239.00) (-29.70,-239.71) (-28.99,-239.71) (-28.28,-239.71) (-27.58,-239.71) (-26.87,-240.42) (-26.16,-240.42) (-25.46,-240.42) (-24.75,-241.12) (-24.04,-241.12) (-23.33,-241.12) (-22.63,-241.83) (-21.92,-241.83) (-21.21,-241.83) (-20.51,-242.54) (-19.80,-242.54) (-19.09,-242.54) (-18.38,-243.24) (-17.68,-243.24) (-16.97,-243.24) (-16.26,-243.24) (-15.56,-243.95) (-14.85,-243.95) (-14.14,-243.95) (-13.44,-244.66) (-12.73,-244.66) (-44.55,-234.76) (-43.84,-235.47) (-43.13,-235.47) (-42.43,-236.17) (-41.72,-236.88) (-41.01,-236.88) (-40.31,-237.59) (-39.60,-238.29) (-38.89,-238.29) (-38.18,-239.00) (-37.48,-239.71) (-36.77,-239.71) (-36.06,-240.42) (-35.36,-241.12) (-34.65,-241.83) (-33.94,-241.83) (-33.23,-242.54) (-32.53,-243.24) (-31.82,-243.24) (-31.11,-243.95) (-30.41,-244.66) (-29.70,-244.66) (-28.99,-245.37) (-28.28,-246.07) (-27.58,-246.07) (-26.87,-246.78) (-26.16,-247.49) (-25.46,-247.49) (-24.75,-248.19) (-24.04,-248.90) (-23.33,-248.90) (-22.63,-249.61) (-21.92,-250.32) (-21.21,-250.32) (-20.51,-251.02) (-19.80,-251.73) (-19.09,-252.44) (-18.38,-252.44) (-17.68,-253.14) (-16.97,-253.85) (-16.26,-253.85) (-15.56,-254.56) (-14.85,-255.27) (-14.14,-255.27) (-13.44,-255.97) (-13.44,-255.97) (-14.14,-255.27) (-14.14,-254.56) (-14.85,-253.85) (-15.56,-253.14) (-16.26,-252.44) (-16.26,-251.73) (-16.97,-251.02) (-17.68,-250.32) (-17.68,-249.61) (-18.38,-248.90) (-19.09,-248.19) (-19.80,-247.49) (-19.80,-246.78) (-20.51,-246.07) (-21.21,-245.37) (-21.92,-244.66) (-21.92,-243.95) (-22.63,-243.24) (-23.33,-242.54) (-23.33,-241.83) (-24.04,-241.12) (-24.75,-240.42) (-25.46,-239.71) (-25.46,-239.00) (-26.16,-238.29) (-26.87,-237.59) (-26.87,-236.88) (-27.58,-236.17) (-28.28,-235.47) (-28.99,-234.76) (-28.99,-234.05) (-29.70,-233.35) (-30.41,-232.64) (-30.41,-231.93) (-31.11,-231.22) (-31.82,-230.52) (-32.53,-229.81) (-32.53,-229.10) (-33.23,-228.40) (-33.94,-227.69) (-34.65,-226.98) (-34.65,-226.27) (-35.36,-225.57) (-36.06,-224.86) (-36.06,-224.15) (-36.77,-223.45) (-37.48,-222.74) (-38.18,-222.03) (-38.18,-221.32) (-38.89,-220.62) (-69.30,-252.44) (-68.59,-251.73) (-67.88,-251.02) (-67.18,-250.32) (-66.47,-249.61) (-65.76,-248.90) (-65.05,-248.19) (-64.35,-247.49) (-63.64,-246.78) (-62.93,-246.07) (-62.23,-245.37) (-61.52,-244.66) (-61.52,-243.95) (-60.81,-243.24) (-60.10,-242.54) (-59.40,-241.83) (-58.69,-241.12) (-57.98,-240.42) (-57.28,-239.71) (-56.57,-239.00) (-55.86,-238.29) (-55.15,-237.59) (-54.45,-236.88) (-53.74,-236.17) (-53.03,-235.47) (-52.33,-234.76) (-51.62,-234.05) (-50.91,-233.35) (-50.20,-232.64) (-49.50,-231.93) (-48.79,-231.22) (-48.08,-230.52) (-47.38,-229.81) (-46.67,-229.10) (-46.67,-228.40) (-45.96,-227.69) (-45.25,-226.98) (-44.55,-226.27) (-43.84,-225.57) (-43.13,-224.86) (-42.43,-224.15) (-41.72,-223.45) (-41.01,-222.74) (-40.31,-222.03) (-39.60,-221.32) (-38.89,-220.62) (-116.67,-237.59) (-115.97,-237.59) (-115.26,-238.29) (-114.55,-238.29) (-113.84,-238.29) (-113.14,-239.00) (-112.43,-239.00) (-111.72,-239.00) (-111.02,-239.71) (-110.31,-239.71) (-109.60,-239.71) (-108.89,-239.71) (-108.19,-240.42) (-107.48,-240.42) (-106.77,-240.42) (-106.07,-241.12) (-105.36,-241.12) (-104.65,-241.12) (-103.94,-241.83) (-103.24,-241.83) (-102.53,-241.83) (-101.82,-242.54) (-101.12,-242.54) (-100.41,-242.54) (-99.70,-243.24) (-98.99,-243.24) (-98.29,-243.24) (-97.58,-243.24) (-96.87,-243.95) (-96.17,-243.95) (-95.46,-243.95) (-94.75,-244.66) (-94.05,-244.66) (-93.34,-244.66) (-92.63,-245.37) (-91.92,-245.37) (-91.22,-245.37) (-90.51,-246.07) (-89.80,-246.07) (-89.10,-246.07) (-88.39,-246.78) (-87.68,-246.78) (-86.97,-246.78) (-86.27,-246.78) (-85.56,-247.49) (-84.85,-247.49) (-84.15,-247.49) (-83.44,-248.19) (-82.73,-248.19) (-82.02,-248.19) (-81.32,-248.90) (-80.61,-248.90) (-79.90,-248.90) (-79.20,-249.61) (-78.49,-249.61) (-77.78,-249.61) (-77.07,-250.32) (-76.37,-250.32) (-75.66,-250.32) (-74.95,-250.32) (-74.25,-251.02) (-73.54,-251.02) (-72.83,-251.02) (-72.12,-251.73) (-71.42,-251.73) (-70.71,-251.73) (-70.00,-252.44) (-69.30,-252.44) (-116.67,-237.59) (-115.97,-238.29) (-115.26,-238.29) (-114.55,-239.00) (-113.84,-239.71) (-113.14,-239.71) (-112.43,-240.42) (-111.72,-241.12) (-111.02,-241.12) (-110.31,-241.83) (-109.60,-242.54) (-108.89,-243.24) (-108.19,-243.24) (-107.48,-243.95) (-106.77,-244.66) (-106.07,-244.66) (-105.36,-245.37) (-104.65,-246.07) (-103.94,-246.07) (-103.24,-246.78) (-102.53,-247.49) (-101.82,-247.49) (-101.12,-248.19) (-100.41,-248.90) (-99.70,-248.90) (-98.99,-249.61) (-98.29,-250.32) (-97.58,-251.02) (-96.87,-251.02) (-96.17,-251.73) (-95.46,-252.44) (-94.75,-252.44) (-94.05,-253.14) (-93.34,-253.85) (-92.63,-253.85) (-91.92,-254.56) (-91.22,-255.27) (-90.51,-255.27) (-89.80,-255.97) (-89.10,-256.68) (-88.39,-256.68) (-87.68,-257.39) (-86.97,-258.09) (-86.27,-258.80) (-85.56,-258.80) (-84.85,-259.51) (-84.15,-260.22) (-83.44,-260.22) (-82.73,-260.92) (-82.02,-261.63) (-81.32,-261.63) (-80.61,-262.34) (-79.90,-263.04) (-79.20,-263.04) (-78.49,-263.75) (-77.78,-264.46) (-77.07,-264.46) (-76.37,-265.17) (-75.66,-265.87) (-74.95,-266.58) (-74.25,-266.58) (-73.54,-267.29) (-72.83,-267.99) (-72.12,-267.99) (-71.42,-268.70) (-71.42,-268.70) (-71.42,-267.99) (-71.42,-267.29) (-71.42,-266.58) (-71.42,-265.87) (-71.42,-265.17) (-70.71,-264.46) (-70.71,-263.75) (-70.71,-263.04) (-70.71,-262.34) (-70.71,-261.63) (-70.71,-260.92) (-70.71,-260.22) (-70.71,-259.51) (-70.71,-258.80) (-70.71,-258.09) (-70.71,-257.39) (-70.71,-256.68) (-70.00,-255.97) (-70.00,-255.27) (-70.00,-254.56) (-70.00,-253.85) (-70.00,-253.14) (-70.00,-252.44) (-70.00,-251.73) (-70.00,-251.02) (-70.00,-250.32) (-70.00,-249.61) (-70.00,-248.90) (-69.30,-248.19) (-69.30,-247.49) (-69.30,-246.78) (-69.30,-246.07) (-69.30,-245.37) (-69.30,-244.66) (-69.30,-243.95) (-69.30,-243.24) (-69.30,-242.54) (-69.30,-241.83) (-69.30,-241.12) (-69.30,-240.42) (-68.59,-239.71) (-68.59,-239.00) (-68.59,-238.29) (-68.59,-237.59) (-68.59,-236.88) (-68.59,-236.17) (-68.59,-235.47) (-68.59,-234.76) (-68.59,-234.05) (-68.59,-233.35) (-68.59,-232.64) (-68.59,-231.93) (-67.88,-231.22) (-67.88,-230.52) (-67.88,-229.81) (-67.88,-229.10) (-67.88,-228.40) (-67.88,-227.69) (-67.88,-226.98) (-67.88,-226.27) (-67.88,-225.57) (-67.88,-224.86) (-67.88,-224.15) (-67.18,-223.45) (-67.18,-222.74) (-67.18,-222.03) (-67.18,-221.32) (-67.18,-220.62) (-67.18,-219.91) (-67.18,-219.20) (-67.18,-218.50) (-67.18,-217.79) (-67.18,-217.08) (-67.18,-216.37) (-67.18,-215.67) (-66.47,-214.96) (-66.47,-214.25) (-66.47,-213.55) (-66.47,-212.84) (-66.47,-212.13) (-66.47,-211.42) (-66.47,-211.42) (-66.47,-210.72) (-66.47,-210.01) (-66.47,-209.30) (-67.18,-208.60) (-67.18,-207.89) (-67.18,-207.18) (-67.18,-206.48) (-67.18,-205.77) (-67.18,-205.06) (-67.18,-204.35) (-67.18,-203.65) (-67.88,-202.94) (-67.88,-202.23) (-67.88,-201.53) (-67.88,-200.82) (-67.88,-200.11) (-67.88,-199.40) (-67.88,-198.70) (-67.88,-197.99) (-68.59,-197.28) (-68.59,-196.58) (-68.59,-195.87) (-68.59,-195.16) (-68.59,-194.45) (-68.59,-193.75) (-68.59,-193.04) (-68.59,-192.33) (-69.30,-191.63) (-69.30,-190.92) (-69.30,-190.21) (-69.30,-189.50) (-69.30,-188.80) (-69.30,-188.09) (-69.30,-187.38) (-69.30,-186.68) (-70.00,-185.97) (-70.00,-185.26) (-70.00,-184.55) (-70.00,-183.85) (-70.00,-183.14) (-70.00,-182.43) (-70.00,-181.73) (-70.00,-181.02) (-70.71,-180.31) (-70.71,-179.61) (-70.71,-178.90) (-70.71,-178.19) (-70.71,-177.48) (-70.71,-176.78) (-70.71,-176.07) (-70.71,-175.36) (-71.42,-174.66) (-71.42,-173.95) (-71.42,-173.24) (-71.42,-172.53) (-71.42,-171.83) (-71.42,-171.12) (-71.42,-170.41) (-71.42,-169.71) (-72.12,-169.00) (-72.12,-168.29) (-72.12,-167.58) (-72.12,-166.88) (-72.12,-166.17) (-72.12,-165.46) (-72.12,-164.76) (-72.12,-164.05) (-72.83,-163.34) (-72.83,-162.63) (-72.83,-161.93) (-72.83,-161.22) (-72.83,-160.51) (-72.83,-159.81) (-72.83,-159.10) (-72.83,-158.39) (-73.54,-157.68) (-73.54,-156.98) (-73.54,-156.27) (-73.54,-155.56) (-73.54,-155.56) (-74.25,-154.86) (-74.95,-154.15) (-75.66,-153.44) (-76.37,-152.74) (-77.07,-152.03) (-77.78,-151.32) (-78.49,-150.61) (-79.20,-149.91) (-79.90,-149.20) (-80.61,-148.49) (-81.32,-147.79) (-82.02,-147.08) (-82.73,-146.37) (-83.44,-145.66) (-84.15,-144.96) (-84.85,-144.25) (-85.56,-143.54) (-86.27,-142.84) (-86.97,-142.13) (-87.68,-141.42) (-88.39,-140.71) (-89.10,-140.01) (-89.80,-139.30) (-90.51,-138.59) (-91.22,-137.89) (-91.92,-137.18) (-92.63,-136.47) (-93.34,-135.76) (-93.34,-135.06) (-94.05,-134.35) (-94.75,-133.64) (-95.46,-132.94) (-96.17,-132.23) (-96.87,-131.52) (-97.58,-130.81) (-98.29,-130.11) (-98.99,-129.40) (-99.70,-128.69) (-100.41,-127.99) (-101.12,-127.28) (-101.82,-126.57) (-102.53,-125.87) (-103.24,-125.16) (-103.94,-124.45) (-104.65,-123.74) (-105.36,-123.04) (-106.07,-122.33) (-106.77,-121.62) (-107.48,-120.92) (-108.19,-120.21) (-108.89,-119.50) (-109.60,-118.79) (-110.31,-118.09) (-111.02,-117.38) (-111.72,-116.67) (-112.43,-115.97) (-113.14,-115.26) (-113.84,-114.55) (-167.58,-128.69) (-166.88,-128.69) (-166.17,-127.99) (-165.46,-127.99) (-164.76,-127.99) (-164.05,-127.99) (-163.34,-127.28) (-162.63,-127.28) (-161.93,-127.28) (-161.22,-127.28) (-160.51,-126.57) (-159.81,-126.57) (-159.10,-126.57) (-158.39,-126.57) (-157.68,-125.87) (-156.98,-125.87) (-156.27,-125.87) (-155.56,-125.87) (-154.86,-125.16) (-154.15,-125.16) (-153.44,-125.16) (-152.74,-124.45) (-152.03,-124.45) (-151.32,-124.45) (-150.61,-124.45) (-149.91,-123.74) (-149.20,-123.74) (-148.49,-123.74) (-147.79,-123.74) (-147.08,-123.04) (-146.37,-123.04) (-145.66,-123.04) (-144.96,-123.04) (-144.25,-122.33) (-143.54,-122.33) (-142.84,-122.33) (-142.13,-122.33) (-141.42,-121.62) (-140.71,-121.62) (-140.01,-121.62) (-139.30,-120.92) (-138.59,-120.92) (-137.89,-120.92) (-137.18,-120.92) (-136.47,-120.21) (-135.76,-120.21) (-135.06,-120.21) (-134.35,-120.21) (-133.64,-119.50) (-132.94,-119.50) (-132.23,-119.50) (-131.52,-119.50) (-130.81,-118.79) (-130.11,-118.79) (-129.40,-118.79) (-128.69,-118.79) (-127.99,-118.09) (-127.28,-118.09) (-126.57,-118.09) (-125.87,-117.38) (-125.16,-117.38) (-124.45,-117.38) (-123.74,-117.38) (-123.04,-116.67) (-122.33,-116.67) (-121.62,-116.67) (-120.92,-116.67) (-120.21,-115.97) (-119.50,-115.97) (-118.79,-115.97) (-118.09,-115.97) (-117.38,-115.26) (-116.67,-115.26) (-115.97,-115.26) (-115.26,-115.26) (-114.55,-114.55) (-113.84,-114.55) (-167.58,-128.69) (-167.58,-127.99) (-166.88,-127.28) (-166.88,-126.57) (-166.17,-125.87) (-166.17,-125.16) (-165.46,-124.45) (-165.46,-123.74) (-164.76,-123.04) (-164.76,-122.33) (-164.05,-121.62) (-164.05,-120.92) (-163.34,-120.21) (-163.34,-119.50) (-162.63,-118.79) (-162.63,-118.09) (-161.93,-117.38) (-161.93,-116.67) (-161.22,-115.97) (-161.22,-115.26) (-161.22,-114.55) (-160.51,-113.84) (-160.51,-113.14) (-159.81,-112.43) (-159.81,-111.72) (-159.10,-111.02) (-159.10,-110.31) (-158.39,-109.60) (-158.39,-108.89) (-157.68,-108.19) (-157.68,-107.48) (-156.98,-106.77) (-156.98,-106.07) (-156.27,-105.36) (-156.27,-104.65) (-155.56,-103.94) (-155.56,-103.24) (-155.56,-102.53) (-154.86,-101.82) (-154.86,-101.12) (-154.15,-100.41) (-154.15,-99.70) (-153.44,-98.99) (-153.44,-98.29) (-152.74,-97.58) (-152.74,-96.87) (-152.03,-96.17) (-152.03,-95.46) (-151.32,-94.75) (-151.32,-94.05) (-150.61,-93.34) (-150.61,-92.63) (-149.91,-91.92) (-149.91,-91.22) (-149.20,-90.51) (-149.20,-89.80) (-149.20,-89.10) (-148.49,-88.39) (-148.49,-87.68) (-147.79,-86.97) (-147.79,-86.27) (-147.08,-85.56) (-147.08,-84.85) (-146.37,-84.15) (-146.37,-83.44) (-145.66,-82.73) (-145.66,-82.02) (-144.96,-81.32) (-144.96,-80.61) (-144.25,-79.90) (-144.25,-79.20) (-143.54,-78.49) (-143.54,-77.78) (-142.84,-77.07) (-142.84,-76.37) (-142.84,-76.37) (-142.13,-76.37) (-141.42,-76.37) (-140.71,-76.37) (-140.01,-76.37) (-139.30,-76.37) (-138.59,-76.37) (-137.89,-76.37) (-137.18,-76.37) (-136.47,-76.37) (-135.76,-76.37) (-135.06,-76.37) (-134.35,-76.37) (-133.64,-76.37) (-132.94,-76.37) (-132.23,-76.37) (-131.52,-76.37) (-130.81,-76.37) (-130.11,-76.37) (-129.40,-76.37) (-128.69,-76.37) (-127.99,-76.37) (-127.28,-76.37) (-126.57,-76.37) (-125.87,-76.37) (-125.16,-76.37) (-124.45,-76.37) (-123.74,-76.37) (-123.04,-76.37) (-122.33,-76.37) (-121.62,-76.37) (-120.92,-76.37) (-120.21,-76.37) (-119.50,-76.37) (-118.79,-76.37) (-118.09,-76.37) (-117.38,-76.37) (-116.67,-76.37) (-115.97,-76.37) (-115.26,-76.37) (-114.55,-76.37) (-113.84,-76.37) (-113.14,-75.66) (-112.43,-75.66) (-111.72,-75.66) (-111.02,-75.66) (-110.31,-75.66) (-109.60,-75.66) (-108.89,-75.66) (-108.19,-75.66) (-107.48,-75.66) (-106.77,-75.66) (-106.07,-75.66) (-105.36,-75.66) (-104.65,-75.66) (-103.94,-75.66) (-103.24,-75.66) (-102.53,-75.66) (-101.82,-75.66) (-101.12,-75.66) (-100.41,-75.66) (-99.70,-75.66) (-98.99,-75.66) (-98.29,-75.66) (-97.58,-75.66) (-96.87,-75.66) (-96.17,-75.66) (-95.46,-75.66) (-94.75,-75.66) (-94.05,-75.66) (-93.34,-75.66) (-92.63,-75.66) (-91.92,-75.66) (-91.22,-75.66) (-90.51,-75.66) (-89.80,-75.66) (-89.10,-75.66) (-88.39,-75.66) (-87.68,-75.66) (-86.97,-75.66) (-86.27,-75.66) (-85.56,-75.66) (-84.85,-75.66) (-145.66,-75.66) (-144.96,-75.66) (-144.25,-75.66) (-143.54,-75.66) (-142.84,-75.66) (-142.13,-75.66) (-141.42,-75.66) (-140.71,-75.66) (-140.01,-75.66) (-139.30,-75.66) (-138.59,-75.66) (-137.89,-75.66) (-137.18,-75.66) (-136.47,-75.66) (-135.76,-75.66) (-135.06,-75.66) (-134.35,-75.66) (-133.64,-75.66) (-132.94,-75.66) (-132.23,-75.66) (-131.52,-75.66) (-130.81,-75.66) (-130.11,-75.66) (-129.40,-75.66) (-128.69,-75.66) (-127.99,-75.66) (-127.28,-75.66) (-126.57,-75.66) (-125.87,-75.66) (-125.16,-75.66) (-124.45,-75.66) (-123.74,-75.66) (-123.04,-75.66) (-122.33,-75.66) (-121.62,-75.66) (-120.92,-75.66) (-120.21,-75.66) (-119.50,-75.66) (-118.79,-75.66) (-118.09,-75.66) (-117.38,-75.66) (-116.67,-75.66) (-115.97,-75.66) (-115.26,-75.66) (-114.55,-75.66) (-113.84,-75.66) (-113.14,-75.66) (-112.43,-75.66) (-111.72,-75.66) (-111.02,-75.66) (-110.31,-75.66) (-109.60,-75.66) (-108.89,-75.66) (-108.19,-75.66) (-107.48,-75.66) (-106.77,-75.66) (-106.07,-75.66) (-105.36,-75.66) (-104.65,-75.66) (-103.94,-75.66) (-103.24,-75.66) (-102.53,-75.66) (-101.82,-75.66) (-101.12,-75.66) (-100.41,-75.66) (-99.70,-75.66) (-98.99,-75.66) (-98.29,-75.66) (-97.58,-75.66) (-96.87,-75.66) (-96.17,-75.66) (-95.46,-75.66) (-94.75,-75.66) (-94.05,-75.66) (-93.34,-75.66) (-92.63,-75.66) (-91.92,-75.66) (-91.22,-75.66) (-90.51,-75.66) (-89.80,-75.66) (-89.10,-75.66) (-88.39,-75.66) (-87.68,-75.66) (-86.97,-75.66) (-86.27,-75.66) (-85.56,-75.66) (-84.85,-75.66) (-145.66,-75.66) (-144.96,-75.66) (-144.25,-75.66) (-143.54,-75.66) (-142.84,-75.66) (-142.13,-75.66) (-141.42,-75.66) (-140.71,-75.66) (-140.01,-75.66) (-139.30,-75.66) (-138.59,-75.66) (-137.89,-75.66) (-137.18,-75.66) (-136.47,-75.66) (-135.76,-75.66) (-135.06,-74.95) (-134.35,-74.95) (-133.64,-74.95) (-132.94,-74.95) (-132.23,-74.95) (-131.52,-74.95) (-130.81,-74.95) (-130.11,-74.95) (-129.40,-74.95) (-128.69,-74.95) (-127.99,-74.95) (-127.28,-74.95) (-126.57,-74.95) (-125.87,-74.95) (-125.16,-74.95) (-124.45,-74.95) (-123.74,-74.95) (-123.04,-74.95) (-122.33,-74.95) (-121.62,-74.95) (-120.92,-74.95) (-120.21,-74.95) (-119.50,-74.95) (-118.79,-74.95) (-118.09,-74.95) (-117.38,-74.95) (-116.67,-74.95) (-115.97,-74.95) (-115.26,-74.25) (-114.55,-74.25) (-113.84,-74.25) (-113.14,-74.25) (-112.43,-74.25) (-111.72,-74.25) (-111.02,-74.25) (-110.31,-74.25) (-109.60,-74.25) (-108.89,-74.25) (-108.19,-74.25) (-107.48,-74.25) (-106.77,-74.25) (-106.07,-74.25) (-105.36,-74.25) (-104.65,-74.25) (-103.94,-74.25) (-103.24,-74.25) (-102.53,-74.25) (-101.82,-74.25) (-101.12,-74.25) (-100.41,-74.25) (-99.70,-74.25) (-98.99,-74.25) (-98.29,-74.25) (-97.58,-74.25) (-96.87,-74.25) (-96.17,-74.25) (-95.46,-73.54) (-94.75,-73.54) (-94.05,-73.54) (-93.34,-73.54) (-92.63,-73.54) (-91.92,-73.54) (-91.22,-73.54) (-90.51,-73.54) (-89.80,-73.54) (-89.10,-73.54) (-88.39,-73.54) (-87.68,-73.54) (-86.97,-73.54) (-86.27,-73.54) (-85.56,-73.54) (-121.62,-118.79) (-120.92,-118.09) (-120.21,-117.38) (-120.21,-116.67) (-119.50,-115.97) (-118.79,-115.26) (-118.09,-114.55) (-117.38,-113.84) (-117.38,-113.14) (-116.67,-112.43) (-115.97,-111.72) (-115.26,-111.02) (-114.55,-110.31) (-114.55,-109.60) (-113.84,-108.89) (-113.14,-108.19) (-112.43,-107.48) (-111.72,-106.77) (-111.72,-106.07) (-111.02,-105.36) (-110.31,-104.65) (-109.60,-103.94) (-108.89,-103.24) (-108.89,-102.53) (-108.19,-101.82) (-107.48,-101.12) (-106.77,-100.41) (-106.07,-99.70) (-106.07,-98.99) (-105.36,-98.29) (-104.65,-97.58) (-103.94,-96.87) (-103.94,-96.17) (-103.24,-95.46) (-102.53,-94.75) (-101.82,-94.05) (-101.12,-93.34) (-101.12,-92.63) (-100.41,-91.92) (-99.70,-91.22) (-98.99,-90.51) (-98.29,-89.80) (-98.29,-89.10) (-97.58,-88.39) (-96.87,-87.68) (-96.17,-86.97) (-95.46,-86.27) (-95.46,-85.56) (-94.75,-84.85) (-94.05,-84.15) (-93.34,-83.44) (-92.63,-82.73) (-92.63,-82.02) (-91.92,-81.32) (-91.22,-80.61) (-90.51,-79.90) (-89.80,-79.20) (-89.80,-78.49) (-89.10,-77.78) (-88.39,-77.07) (-87.68,-76.37) (-86.97,-75.66) (-86.97,-74.95) (-86.27,-74.25) (-85.56,-73.54) (-169.00,-147.79) (-168.29,-147.08) (-167.58,-147.08) (-166.88,-146.37) (-166.17,-146.37) (-165.46,-145.66) (-164.76,-144.96) (-164.05,-144.96) (-163.34,-144.25) (-162.63,-143.54) (-161.93,-143.54) (-161.22,-142.84) (-160.51,-142.84) (-159.81,-142.13) (-159.10,-141.42) (-158.39,-141.42) (-157.68,-140.71) (-156.98,-140.71) (-156.27,-140.01) (-155.56,-139.30) (-154.86,-139.30) (-154.15,-138.59) (-153.44,-138.59) (-152.74,-137.89) (-152.03,-137.18) (-151.32,-137.18) (-150.61,-136.47) (-149.91,-135.76) (-149.20,-135.76) (-148.49,-135.06) (-147.79,-135.06) (-147.08,-134.35) (-146.37,-133.64) (-145.66,-133.64) (-144.96,-132.94) (-144.25,-132.94) (-143.54,-132.23) (-142.84,-131.52) (-142.13,-131.52) (-141.42,-130.81) (-140.71,-130.81) (-140.01,-130.11) (-139.30,-129.40) (-138.59,-129.40) (-137.89,-128.69) (-137.18,-127.99) (-136.47,-127.99) (-135.76,-127.28) (-135.06,-127.28) (-134.35,-126.57) (-133.64,-125.87) (-132.94,-125.87) (-132.23,-125.16) (-131.52,-125.16) (-130.81,-124.45) (-130.11,-123.74) (-129.40,-123.74) (-128.69,-123.04) (-127.99,-123.04) (-127.28,-122.33) (-126.57,-121.62) (-125.87,-121.62) (-125.16,-120.92) (-124.45,-120.21) (-123.74,-120.21) (-123.04,-119.50) (-122.33,-119.50) (-121.62,-118.79) (-218.50,-119.50) (-217.79,-120.21) (-217.08,-120.21) (-216.37,-120.92) (-215.67,-120.92) (-214.96,-121.62) (-214.25,-121.62) (-213.55,-122.33) (-212.84,-123.04) (-212.13,-123.04) (-211.42,-123.74) (-210.72,-123.74) (-210.01,-124.45) (-209.30,-124.45) (-208.60,-125.16) (-207.89,-125.87) (-207.18,-125.87) (-206.48,-126.57) (-205.77,-126.57) (-205.06,-127.28) (-204.35,-127.28) (-203.65,-127.99) (-202.94,-128.69) (-202.23,-128.69) (-201.53,-129.40) (-200.82,-129.40) (-200.11,-130.11) (-199.40,-130.11) (-198.70,-130.81) (-197.99,-131.52) (-197.28,-131.52) (-196.58,-132.23) (-195.87,-132.23) (-195.16,-132.94) (-194.45,-132.94) (-193.75,-133.64) (-193.04,-134.35) (-192.33,-134.35) (-191.63,-135.06) (-190.92,-135.06) (-190.21,-135.76) (-189.50,-135.76) (-188.80,-136.47) (-188.09,-137.18) (-187.38,-137.18) (-186.68,-137.89) (-185.97,-137.89) (-185.26,-138.59) (-184.55,-138.59) (-183.85,-139.30) (-183.14,-140.01) (-182.43,-140.01) (-181.73,-140.71) (-181.02,-140.71) (-180.31,-141.42) (-179.61,-141.42) (-178.90,-142.13) (-178.19,-142.84) (-177.48,-142.84) (-176.78,-143.54) (-176.07,-143.54) (-175.36,-144.25) (-174.66,-144.25) (-173.95,-144.96) (-173.24,-145.66) (-172.53,-145.66) (-171.83,-146.37) (-171.12,-146.37) (-170.41,-147.08) (-169.71,-147.08) (-169.00,-147.79) (-207.89,-176.78) (-207.89,-176.07) (-207.89,-175.36) (-208.60,-174.66) (-208.60,-173.95) (-208.60,-173.24) (-208.60,-172.53) (-208.60,-171.83) (-208.60,-171.12) (-209.30,-170.41) (-209.30,-169.71) (-209.30,-169.00) (-209.30,-168.29) (-209.30,-167.58) (-210.01,-166.88) (-210.01,-166.17) (-210.01,-165.46) (-210.01,-164.76) (-210.01,-164.05) (-210.72,-163.34) (-210.72,-162.63) (-210.72,-161.93) (-210.72,-161.22) (-210.72,-160.51) (-210.72,-159.81) (-211.42,-159.10) (-211.42,-158.39) (-211.42,-157.68) (-211.42,-156.98) (-211.42,-156.27) (-212.13,-155.56) (-212.13,-154.86) (-212.13,-154.15) (-212.13,-153.44) (-212.13,-152.74) (-212.13,-152.03) (-212.84,-151.32) (-212.84,-150.61) (-212.84,-149.91) (-212.84,-149.20) (-212.84,-148.49) (-213.55,-147.79) (-213.55,-147.08) (-213.55,-146.37) (-213.55,-145.66) (-213.55,-144.96) (-214.25,-144.25) (-214.25,-143.54) (-214.25,-142.84) (-214.25,-142.13) (-214.25,-141.42) (-214.25,-140.71) (-214.96,-140.01) (-214.96,-139.30) (-214.96,-138.59) (-214.96,-137.89) (-214.96,-137.18) (-215.67,-136.47) (-215.67,-135.76) (-215.67,-135.06) (-215.67,-134.35) (-215.67,-133.64) (-215.67,-132.94) (-216.37,-132.23) (-216.37,-131.52) (-216.37,-130.81) (-216.37,-130.11) (-216.37,-129.40) (-217.08,-128.69) (-217.08,-127.99) (-217.08,-127.28) (-217.08,-126.57) (-217.08,-125.87) (-217.79,-125.16) (-217.79,-124.45) (-217.79,-123.74) (-217.79,-123.04) (-217.79,-122.33) (-217.79,-121.62) (-218.50,-120.92) (-218.50,-120.21) (-218.50,-119.50) (-207.89,-176.78) (-207.18,-176.78) (-206.48,-176.78) (-205.77,-177.48) (-205.06,-177.48) (-204.35,-177.48) (-203.65,-177.48) (-202.94,-177.48) (-202.23,-177.48) (-201.53,-178.19) (-200.82,-178.19) (-200.11,-178.19) (-199.40,-178.19) (-198.70,-178.19) (-197.99,-178.90) (-197.28,-178.90) (-196.58,-178.90) (-195.87,-178.90) (-195.16,-178.90) (-194.45,-179.61) (-193.75,-179.61) (-193.04,-179.61) (-192.33,-179.61) (-191.63,-179.61) (-190.92,-179.61) (-190.21,-180.31) (-189.50,-180.31) (-188.80,-180.31) (-188.09,-180.31) (-187.38,-180.31) (-186.68,-181.02) (-185.97,-181.02) (-185.26,-181.02) (-184.55,-181.02) (-183.85,-181.02) (-183.14,-181.02) (-182.43,-181.73) (-181.73,-181.73) (-181.02,-181.73) (-180.31,-181.73) (-179.61,-181.73) (-178.90,-182.43) (-178.19,-182.43) (-177.48,-182.43) (-176.78,-182.43) (-176.07,-182.43) (-175.36,-183.14) (-174.66,-183.14) (-173.95,-183.14) (-173.24,-183.14) (-172.53,-183.14) (-171.83,-183.14) (-171.12,-183.85) (-170.41,-183.85) (-169.71,-183.85) (-169.00,-183.85) (-168.29,-183.85) (-167.58,-184.55) (-166.88,-184.55) (-166.17,-184.55) (-165.46,-184.55) (-164.76,-184.55) (-164.05,-184.55) (-163.34,-185.26) (-162.63,-185.26) (-161.93,-185.26) (-161.22,-185.26) (-160.51,-185.26) (-159.81,-185.97) (-159.10,-185.97) (-158.39,-185.97) (-157.68,-185.97) (-156.98,-185.97) (-156.27,-186.68) (-155.56,-186.68) (-154.86,-186.68) (-154.15,-186.68) (-153.44,-186.68) (-152.74,-186.68) (-152.03,-187.38) (-151.32,-187.38) (-150.61,-187.38) (-206.48,-189.50) (-205.77,-189.50) (-205.06,-189.50) (-204.35,-189.50) (-203.65,-189.50) (-202.94,-189.50) (-202.23,-189.50) (-201.53,-189.50) (-200.82,-189.50) (-200.11,-189.50) (-199.40,-189.50) (-198.70,-189.50) (-197.99,-189.50) (-197.28,-189.50) (-196.58,-188.80) (-195.87,-188.80) (-195.16,-188.80) (-194.45,-188.80) (-193.75,-188.80) (-193.04,-188.80) (-192.33,-188.80) (-191.63,-188.80) (-190.92,-188.80) (-190.21,-188.80) (-189.50,-188.80) (-188.80,-188.80) (-188.09,-188.80) (-187.38,-188.80) (-186.68,-188.80) (-185.97,-188.80) (-185.26,-188.80) (-184.55,-188.80) (-183.85,-188.80) (-183.14,-188.80) (-182.43,-188.80) (-181.73,-188.80) (-181.02,-188.80) (-180.31,-188.80) (-179.61,-188.80) (-178.90,-188.80) (-178.19,-188.09) (-177.48,-188.09) (-176.78,-188.09) (-176.07,-188.09) (-175.36,-188.09) (-174.66,-188.09) (-173.95,-188.09) (-173.24,-188.09) (-172.53,-188.09) (-171.83,-188.09) (-171.12,-188.09) (-170.41,-188.09) (-169.71,-188.09) (-169.00,-188.09) (-168.29,-188.09) (-167.58,-188.09) (-166.88,-188.09) (-166.17,-188.09) (-165.46,-188.09) (-164.76,-188.09) (-164.05,-188.09) (-163.34,-188.09) (-162.63,-188.09) (-161.93,-188.09) (-161.22,-188.09) (-160.51,-188.09) (-159.81,-187.38) (-159.10,-187.38) (-158.39,-187.38) (-157.68,-187.38) (-156.98,-187.38) (-156.27,-187.38) (-155.56,-187.38) (-154.86,-187.38) (-154.15,-187.38) (-153.44,-187.38) (-152.74,-187.38) (-152.03,-187.38) (-151.32,-187.38) (-150.61,-187.38) (-206.48,-189.50) (-205.77,-189.50) (-205.06,-189.50) (-204.35,-189.50) (-203.65,-189.50) (-202.94,-189.50) (-202.23,-189.50) (-201.53,-189.50) (-200.82,-189.50) (-200.11,-189.50) (-199.40,-189.50) (-198.70,-189.50) (-197.99,-189.50) (-197.28,-189.50) (-196.58,-189.50) (-195.87,-189.50) (-195.16,-189.50) (-194.45,-189.50) (-193.75,-189.50) (-193.04,-189.50) (-192.33,-189.50) (-191.63,-189.50) (-190.92,-189.50) (-190.21,-189.50) (-189.50,-189.50) (-188.80,-189.50) (-188.09,-189.50) (-187.38,-189.50) (-186.68,-189.50) (-185.97,-189.50) (-185.26,-189.50) (-184.55,-189.50) (-183.85,-189.50) (-183.14,-189.50) (-182.43,-189.50) (-181.73,-189.50) (-181.02,-189.50) (-180.31,-189.50) (-179.61,-189.50) (-178.90,-189.50) (-178.19,-189.50) (-177.48,-189.50) (-176.78,-189.50) (-176.07,-189.50) (-175.36,-189.50) (-174.66,-189.50) (-173.95,-189.50) (-173.24,-189.50) (-172.53,-189.50) (-171.83,-189.50) (-171.12,-189.50) (-170.41,-189.50) (-169.71,-189.50) (-169.00,-189.50) (-168.29,-189.50) (-167.58,-189.50) (-166.88,-189.50) (-166.17,-189.50) (-165.46,-189.50) (-164.76,-189.50) (-164.05,-189.50) (-163.34,-189.50) (-162.63,-189.50) (-161.93,-189.50) (-161.22,-189.50) (-160.51,-189.50) (-159.81,-189.50) (-159.10,-189.50) (-158.39,-189.50) (-157.68,-189.50) (-156.98,-189.50) (-156.27,-189.50) (-155.56,-189.50) (-154.86,-189.50) (-154.15,-189.50) (-153.44,-189.50) (-152.74,-189.50) (-152.03,-189.50) (-151.32,-189.50) (-150.61,-189.50) (-149.91,-189.50) (-149.20,-189.50) (-148.49,-189.50) (-148.49,-189.50) (-147.79,-189.50) (-147.79,-189.50) (-147.79,-189.50) (-147.08,-189.50) (-146.37,-190.21) (-145.66,-190.21) (-144.96,-190.21) (-144.25,-190.92) (-143.54,-190.92) (-142.84,-190.92) (-142.13,-191.63) (-141.42,-191.63) (-143.54,-192.33) (-142.84,-192.33) (-142.13,-191.63) (-141.42,-191.63) (-143.54,-192.33) (-141.42,-196.58) (-141.42,-195.87) (-142.13,-195.16) (-142.13,-194.45) (-142.84,-193.75) (-142.84,-193.04) (-143.54,-192.33) (-141.42,-196.58) (-141.42,-195.87) (-140.71,-195.16) (-140.71,-194.45) (-140.01,-193.75) (-140.01,-193.04) (-139.30,-192.33) (-139.30,-192.33) (-138.59,-197.99) (-138.59,-197.28) (-138.59,-196.58) (-138.59,-195.87) (-138.59,-195.16) (-139.30,-194.45) (-139.30,-193.75) (-139.30,-193.04) (-139.30,-192.33) (-138.59,-197.99) (-137.89,-197.99) (-137.18,-197.99) (-136.47,-197.99) (-135.76,-197.99) (-135.76,-201.53) (-135.76,-200.82) (-135.76,-200.11) (-135.76,-199.40) (-135.76,-198.70) (-135.76,-197.99) (-135.76,-201.53) (-135.06,-201.53) (-134.35,-200.82) (-133.64,-200.82) (-132.94,-200.82) (-132.23,-200.82) (-131.52,-200.11) (-130.81,-200.11) (-130.11,-200.11) (-129.40,-199.40) (-128.69,-199.40) (-130.81,-213.55) (-130.81,-212.84) (-130.81,-212.13) (-130.81,-211.42) (-130.11,-210.72) (-130.11,-210.01) (-130.11,-209.30) (-130.11,-208.60) (-130.11,-207.89) (-130.11,-207.18) (-130.11,-206.48) (-129.40,-205.77) (-129.40,-205.06) (-129.40,-204.35) (-129.40,-203.65) (-129.40,-202.94) (-129.40,-202.23) (-128.69,-201.53) (-128.69,-200.82) (-128.69,-200.11) (-128.69,-199.40) (-130.81,-213.55) (-130.11,-214.25) (-129.40,-214.25) (-128.69,-214.96) (-127.99,-215.67) (-127.28,-215.67) (-126.57,-216.37) (-125.87,-217.08) (-125.16,-217.08) (-124.45,-217.79) (-123.74,-217.79) (-123.04,-218.50) (-122.33,-219.20) (-121.62,-219.20) (-120.92,-219.91) (-120.21,-220.62) (-119.50,-220.62) (-118.79,-221.32) (-118.09,-222.03) (-117.38,-222.03) (-116.67,-222.74) (-138.59,-227.69) (-137.89,-227.69) (-137.18,-227.69) (-136.47,-226.98) (-135.76,-226.98) (-135.06,-226.98) (-134.35,-226.98) (-133.64,-226.27) (-132.94,-226.27) (-132.23,-226.27) (-131.52,-226.27) (-130.81,-226.27) (-130.11,-225.57) (-129.40,-225.57) (-128.69,-225.57) (-127.99,-225.57) (-127.28,-224.86) (-126.57,-224.86) (-125.87,-224.86) (-125.16,-224.86) (-124.45,-224.15) (-123.74,-224.15) (-123.04,-224.15) (-122.33,-224.15) (-121.62,-224.15) (-120.92,-223.45) (-120.21,-223.45) (-119.50,-223.45) (-118.79,-223.45) (-118.09,-222.74) (-117.38,-222.74) (-116.67,-222.74) (-138.59,-227.69) (-138.59,-226.98) (-138.59,-226.27) (-138.59,-225.57) (-138.59,-224.86) (-137.89,-224.15) (-137.89,-223.45) (-137.89,-222.74) (-137.89,-222.03) (-137.89,-221.32) (-137.89,-220.62) (-137.89,-219.91) (-137.89,-219.20) (-137.89,-218.50) (-137.18,-217.79) (-137.18,-217.08) (-137.18,-216.37) (-137.18,-215.67) (-137.18,-214.96) (-137.18,-214.25) (-137.18,-213.55) (-137.18,-212.84) (-136.47,-212.13) (-136.47,-211.42) (-136.47,-210.72) (-136.47,-210.01) (-136.47,-209.30) (-136.47,-208.60) (-136.47,-207.89) (-136.47,-207.18) (-136.47,-206.48) (-135.76,-205.77) (-135.76,-205.06) (-135.76,-204.35) (-135.76,-203.65) (-135.76,-202.94) (-135.76,-202.94) (-135.76,-202.23) (-136.47,-201.53) (-136.47,-200.82) (-137.18,-200.11) (-137.18,-199.40) (-137.18,-198.70) (-137.89,-197.99) (-137.89,-197.28) (-138.59,-196.58) (-138.59,-195.87) (-139.30,-195.16) (-139.30,-194.45) (-139.30,-193.75) (-140.01,-193.04) (-140.01,-192.33) (-140.71,-191.63) (-140.71,-190.92) (-140.71,-190.21) (-141.42,-189.50) (-141.42,-188.80) (-142.13,-188.09) (-142.13,-187.38) (-142.84,-186.68) (-142.84,-185.97) (-142.84,-185.26) (-143.54,-184.55) (-143.54,-183.85) (-144.25,-183.14) (-144.25,-182.43) (-144.25,-181.73) (-144.96,-181.02) (-144.96,-180.31) (-145.66,-179.61) (-145.66,-178.90) (-146.37,-178.19) (-146.37,-177.48) (-146.37,-176.78) (-147.08,-176.07) (-147.08,-175.36) (-147.79,-174.66) (-147.79,-173.95) (-178.90,-189.50) (-178.19,-189.50) (-177.48,-188.80) (-176.78,-188.80) (-176.07,-188.09) (-175.36,-188.09) (-174.66,-187.38) (-173.95,-187.38) (-173.24,-186.68) (-172.53,-186.68) (-171.83,-185.97) (-171.12,-185.97) (-170.41,-185.26) (-169.71,-185.26) (-169.00,-184.55) (-168.29,-184.55) (-167.58,-183.85) (-166.88,-183.85) (-166.17,-183.14) (-165.46,-183.14) (-164.76,-182.43) (-164.05,-182.43) (-163.34,-181.73) (-162.63,-181.73) (-161.93,-181.02) (-161.22,-181.02) (-160.51,-180.31) (-159.81,-180.31) (-159.10,-179.61) (-158.39,-179.61) (-157.68,-178.90) (-156.98,-178.90) (-156.27,-178.19) (-155.56,-178.19) (-154.86,-177.48) (-154.15,-177.48) (-153.44,-176.78) (-152.74,-176.78) (-152.03,-176.07) (-151.32,-176.07) (-150.61,-175.36) (-149.91,-175.36) (-149.20,-174.66) (-148.49,-174.66) (-147.79,-173.95) (-210.72,-165.46) (-210.01,-166.17) (-209.30,-166.88) (-208.60,-166.88) (-207.89,-167.58) (-207.18,-168.29) (-206.48,-169.00) (-205.77,-169.00) (-205.06,-169.71) (-204.35,-170.41) (-203.65,-171.12) (-202.94,-171.12) (-202.23,-171.83) (-201.53,-172.53) (-200.82,-173.24) (-200.11,-173.24) (-199.40,-173.95) (-198.70,-174.66) (-197.99,-175.36) (-197.28,-175.36) (-196.58,-176.07) (-195.87,-176.78) (-195.16,-177.48) (-194.45,-177.48) (-193.75,-178.19) (-193.04,-178.90) (-192.33,-179.61) (-191.63,-179.61) (-190.92,-180.31) (-190.21,-181.02) (-189.50,-181.73) (-188.80,-181.73) (-188.09,-182.43) (-187.38,-183.14) (-186.68,-183.85) (-185.97,-183.85) (-185.26,-184.55) (-184.55,-185.26) (-183.85,-185.97) (-183.14,-185.97) (-182.43,-186.68) (-181.73,-187.38) (-181.02,-188.09) (-180.31,-188.09) (-179.61,-188.80) (-178.90,-189.50) (-190.92,-204.35) (-191.63,-203.65) (-191.63,-202.94) (-192.33,-202.23) (-192.33,-201.53) (-193.04,-200.82) (-193.04,-200.11) (-193.75,-199.40) (-193.75,-198.70) (-194.45,-197.99) (-194.45,-197.28) (-195.16,-196.58) (-195.16,-195.87) (-195.87,-195.16) (-195.87,-194.45) (-196.58,-193.75) (-196.58,-193.04) (-197.28,-192.33) (-197.28,-191.63) (-197.99,-190.92) (-197.99,-190.21) (-198.70,-189.50) (-198.70,-188.80) (-199.40,-188.09) (-199.40,-187.38) (-200.11,-186.68) (-200.11,-185.97) (-200.82,-185.26) (-200.82,-184.55) (-201.53,-183.85) (-201.53,-183.14) (-202.23,-182.43) (-202.23,-181.73) (-202.94,-181.02) (-202.94,-180.31) (-203.65,-179.61) (-203.65,-178.90) (-204.35,-178.19) (-204.35,-177.48) (-205.06,-176.78) (-205.06,-176.07) (-205.77,-175.36) (-205.77,-174.66) (-206.48,-173.95) (-206.48,-173.24) (-207.18,-172.53) (-207.18,-171.83) (-207.89,-171.12) (-207.89,-170.41) (-208.60,-169.71) (-208.60,-169.00) (-209.30,-168.29) (-209.30,-167.58) (-210.01,-166.88) (-210.01,-166.17) (-210.72,-165.46) (-190.92,-204.35) (-190.21,-203.65) (-189.50,-202.94) (-188.80,-202.94) (-188.09,-202.23) (-187.38,-201.53) (-186.68,-200.82) (-185.97,-200.11) (-185.26,-199.40) (-184.55,-199.40) (-183.85,-198.70) (-183.14,-197.99) (-182.43,-197.28) (-181.73,-196.58) (-181.02,-195.87) (-180.31,-195.87) (-179.61,-195.16) (-178.90,-194.45) (-178.19,-193.75) (-177.48,-193.04) (-176.78,-192.33) (-176.07,-192.33) (-175.36,-191.63) (-174.66,-190.92) (-173.95,-190.21) (-173.24,-189.50) (-172.53,-188.80) (-171.83,-188.80) (-171.12,-188.09) (-170.41,-187.38) (-169.71,-186.68) (-169.00,-185.97) (-168.29,-185.26) (-167.58,-185.26) (-166.88,-184.55) (-166.17,-183.85) (-165.46,-183.14) (-164.76,-182.43) (-164.05,-181.73) (-163.34,-181.73) (-162.63,-181.02) (-161.93,-180.31) (-161.22,-179.61) (-160.51,-178.90) (-159.81,-178.19) (-159.10,-178.19) (-158.39,-177.48) (-157.68,-176.78) (-156.98,-176.07) (-156.27,-175.36) (-155.56,-174.66) (-154.86,-174.66) (-154.15,-173.95) (-153.44,-173.24) (-152.74,-172.53) (-127.99,-222.74) (-127.99,-222.03) (-128.69,-221.32) (-128.69,-220.62) (-129.40,-219.91) (-129.40,-219.20) (-130.11,-218.50) (-130.11,-217.79) (-130.81,-217.08) (-130.81,-216.37) (-131.52,-215.67) (-131.52,-214.96) (-132.23,-214.25) (-132.23,-213.55) (-132.94,-212.84) (-132.94,-212.13) (-133.64,-211.42) (-133.64,-210.72) (-134.35,-210.01) (-134.35,-209.30) (-135.06,-208.60) (-135.06,-207.89) (-135.76,-207.18) (-135.76,-206.48) (-136.47,-205.77) (-136.47,-205.06) (-137.18,-204.35) (-137.18,-203.65) (-137.89,-202.94) (-137.89,-202.23) (-138.59,-201.53) (-138.59,-200.82) (-139.30,-200.11) (-139.30,-199.40) (-140.01,-198.70) (-140.01,-197.99) (-140.71,-197.28) (-140.71,-196.58) (-141.42,-195.87) (-141.42,-195.16) (-142.13,-194.45) (-142.13,-193.75) (-142.84,-193.04) (-142.84,-192.33) (-143.54,-191.63) (-143.54,-190.92) (-144.25,-190.21) (-144.25,-189.50) (-144.96,-188.80) (-144.96,-188.09) (-145.66,-187.38) (-145.66,-186.68) (-146.37,-185.97) (-146.37,-185.26) (-147.08,-184.55) (-147.08,-183.85) (-147.79,-183.14) (-147.79,-182.43) (-148.49,-181.73) (-148.49,-181.02) (-149.20,-180.31) (-149.20,-179.61) (-149.91,-178.90) (-149.91,-178.19) (-150.61,-177.48) (-150.61,-176.78) (-151.32,-176.07) (-151.32,-175.36) (-152.03,-174.66) (-152.03,-173.95) (-152.74,-173.24) (-152.74,-172.53) (-127.99,-222.74) (-127.28,-222.74) (-126.57,-222.03) (-125.87,-222.03) (-125.16,-221.32) (-124.45,-221.32) (-123.74,-220.62) (-123.04,-220.62) (-122.33,-219.91) (-121.62,-219.91) (-120.92,-219.20) (-120.21,-219.20) (-119.50,-218.50) (-118.79,-218.50) (-118.09,-217.79) (-117.38,-217.79) (-116.67,-217.79) (-115.97,-217.08) (-115.26,-217.08) (-114.55,-216.37) (-113.84,-216.37) (-113.14,-215.67) (-112.43,-215.67) (-111.72,-214.96) (-111.02,-214.96) (-110.31,-214.25) (-109.60,-214.25) (-108.89,-213.55) (-108.19,-213.55) (-107.48,-212.84) (-106.77,-212.84) (-106.07,-212.84) (-105.36,-212.13) (-104.65,-212.13) (-103.94,-211.42) (-103.24,-211.42) (-102.53,-210.72) (-101.82,-210.72) (-101.12,-210.01) (-100.41,-210.01) (-99.70,-209.30) (-98.99,-209.30) (-98.29,-208.60) (-97.58,-208.60) (-96.87,-207.89) (-96.17,-207.89) (-95.46,-207.89) (-94.75,-207.18) (-94.05,-207.18) (-93.34,-206.48) (-92.63,-206.48) (-91.92,-205.77) (-91.22,-205.77) (-90.51,-205.06) (-89.80,-205.06) (-89.10,-204.35) (-88.39,-204.35) (-87.68,-203.65) (-86.97,-203.65) (-86.27,-202.94) (-85.56,-202.94) (-84.85,-202.94) (-84.15,-202.23) (-83.44,-202.23) (-82.73,-201.53) (-82.02,-201.53) (-81.32,-200.82) (-80.61,-200.82) (-79.90,-200.11) (-79.20,-200.11) (-78.49,-199.40) (-77.78,-199.40) (-77.07,-198.70) (-76.37,-198.70) (-75.66,-197.99) (-74.95,-197.99) (-124.45,-225.57) (-123.74,-224.86) (-123.04,-224.86) (-122.33,-224.15) (-121.62,-224.15) (-120.92,-223.45) (-120.21,-223.45) (-119.50,-222.74) (-118.79,-222.74) (-118.09,-222.03) (-117.38,-221.32) (-116.67,-221.32) (-115.97,-220.62) (-115.26,-220.62) (-114.55,-219.91) (-113.84,-219.91) (-113.14,-219.20) (-112.43,-219.20) (-111.72,-218.50) (-111.02,-217.79) (-110.31,-217.79) (-109.60,-217.08) (-108.89,-217.08) (-108.19,-216.37) (-107.48,-216.37) (-106.77,-215.67) (-106.07,-215.67) (-105.36,-214.96) (-104.65,-214.25) (-103.94,-214.25) (-103.24,-213.55) (-102.53,-213.55) (-101.82,-212.84) (-101.12,-212.84) (-100.41,-212.13) (-99.70,-212.13) (-98.99,-211.42) (-98.29,-210.72) (-97.58,-210.72) (-96.87,-210.01) (-96.17,-210.01) (-95.46,-209.30) (-94.75,-209.30) (-94.05,-208.60) (-93.34,-207.89) (-92.63,-207.89) (-91.92,-207.18) (-91.22,-207.18) (-90.51,-206.48) (-89.80,-206.48) (-89.10,-205.77) (-88.39,-205.77) (-87.68,-205.06) (-86.97,-204.35) (-86.27,-204.35) (-85.56,-203.65) (-84.85,-203.65) (-84.15,-202.94) (-83.44,-202.94) (-82.73,-202.23) (-82.02,-202.23) (-81.32,-201.53) (-80.61,-200.82) (-79.90,-200.82) (-79.20,-200.11) (-78.49,-200.11) (-77.78,-199.40) (-77.07,-199.40) (-76.37,-198.70) (-75.66,-198.70) (-74.95,-197.99) (-124.45,-225.57) (-125.16,-224.86) (-125.16,-224.15) (-125.87,-223.45) (-125.87,-222.74) (-126.57,-222.03) (-126.57,-221.32) (-127.28,-220.62) (-127.28,-219.91) (-127.99,-219.20) (-127.99,-218.50) (-128.69,-217.79) (-129.40,-217.08) (-129.40,-216.37) (-130.11,-215.67) (-130.11,-214.96) (-130.81,-214.25) (-130.81,-213.55) (-131.52,-212.84) (-131.52,-212.13) (-132.23,-211.42) (-132.23,-210.72) (-132.94,-210.01) (-133.64,-209.30) (-133.64,-208.60) (-134.35,-207.89) (-134.35,-207.18) (-135.06,-206.48) (-135.06,-205.77) (-135.76,-205.06) (-135.76,-204.35) (-136.47,-203.65) (-136.47,-202.94) (-137.18,-202.23) (-137.18,-201.53) (-137.89,-200.82) (-138.59,-200.11) (-138.59,-199.40) (-139.30,-198.70) (-139.30,-197.99) (-140.01,-197.28) (-140.01,-196.58) (-140.71,-195.87) (-140.71,-195.16) (-141.42,-194.45) (-141.42,-193.75) (-142.13,-193.04) (-142.84,-192.33) (-142.84,-191.63) (-143.54,-190.92) (-143.54,-190.21) (-144.25,-189.50) (-144.25,-188.80) (-144.96,-188.09) (-144.96,-187.38) (-145.66,-186.68) (-145.66,-185.97) (-146.37,-185.26) (-147.08,-184.55) (-147.08,-183.85) (-147.79,-183.14) (-147.79,-182.43) (-148.49,-181.73) (-148.49,-181.02) (-149.20,-180.31) (-149.20,-179.61) (-149.91,-178.90) (-149.91,-178.19) (-150.61,-177.48) (-202.94,-153.44) (-202.23,-153.44) (-201.53,-154.15) (-200.82,-154.15) (-200.11,-154.86) (-199.40,-154.86) (-198.70,-155.56) (-197.99,-155.56) (-197.28,-156.27) (-196.58,-156.27) (-195.87,-156.98) (-195.16,-156.98) (-194.45,-157.68) (-193.75,-157.68) (-193.04,-157.68) (-192.33,-158.39) (-191.63,-158.39) (-190.92,-159.10) (-190.21,-159.10) (-189.50,-159.81) (-188.80,-159.81) (-188.09,-160.51) (-187.38,-160.51) (-186.68,-161.22) (-185.97,-161.22) (-185.26,-161.22) (-184.55,-161.93) (-183.85,-161.93) (-183.14,-162.63) (-182.43,-162.63) (-181.73,-163.34) (-181.02,-163.34) (-180.31,-164.05) (-179.61,-164.05) (-178.90,-164.76) (-178.19,-164.76) (-177.48,-165.46) (-176.78,-165.46) (-176.07,-165.46) (-175.36,-166.17) (-174.66,-166.17) (-173.95,-166.88) (-173.24,-166.88) (-172.53,-167.58) (-171.83,-167.58) (-171.12,-168.29) (-170.41,-168.29) (-169.71,-169.00) (-169.00,-169.00) (-168.29,-169.71) (-167.58,-169.71) (-166.88,-169.71) (-166.17,-170.41) (-165.46,-170.41) (-164.76,-171.12) (-164.05,-171.12) (-163.34,-171.83) (-162.63,-171.83) (-161.93,-172.53) (-161.22,-172.53) (-160.51,-173.24) (-159.81,-173.24) (-159.10,-173.24) (-158.39,-173.95) (-157.68,-173.95) (-156.98,-174.66) (-156.27,-174.66) (-155.56,-175.36) (-154.86,-175.36) (-154.15,-176.07) (-153.44,-176.07) (-152.74,-176.78) (-152.03,-176.78) (-151.32,-177.48) (-150.61,-177.48) (-202.94,-153.44) (-202.23,-153.44) (-201.53,-154.15) (-200.82,-154.15) (-200.11,-154.15) (-199.40,-154.86) (-198.70,-154.86) (-197.99,-155.56) (-197.28,-155.56) (-196.58,-155.56) (-195.87,-156.27) (-195.16,-156.27) (-194.45,-156.27) (-193.75,-156.98) (-193.04,-156.98) (-192.33,-157.68) (-191.63,-157.68) (-190.92,-157.68) (-190.21,-158.39) (-189.50,-158.39) (-188.80,-158.39) (-188.09,-159.10) (-187.38,-159.10) (-186.68,-159.81) (-185.97,-159.81) (-185.26,-159.81) (-184.55,-160.51) (-183.85,-160.51) (-183.14,-160.51) (-182.43,-161.22) (-181.73,-161.22) (-181.02,-161.93) (-180.31,-161.93) (-179.61,-161.93) (-178.90,-162.63) (-178.19,-162.63) (-177.48,-162.63) (-176.78,-163.34) (-176.07,-163.34) (-175.36,-163.34) (-174.66,-164.05) (-173.95,-164.05) (-173.24,-164.76) (-172.53,-164.76) (-171.83,-164.76) (-171.12,-165.46) (-170.41,-165.46) (-169.71,-165.46) (-169.00,-166.17) (-168.29,-166.17) (-167.58,-166.88) (-166.88,-166.88) (-166.17,-166.88) (-165.46,-167.58) (-164.76,-167.58) (-164.05,-167.58) (-163.34,-168.29) (-162.63,-168.29) (-161.93,-169.00) (-161.22,-169.00) (-160.51,-169.00) (-159.81,-169.71) (-159.10,-169.71) (-158.39,-169.71) (-157.68,-170.41) (-156.98,-170.41) (-156.27,-171.12) (-155.56,-171.12) (-154.86,-171.12) (-154.15,-171.83) (-153.44,-171.83) (-152.74,-171.83) (-152.03,-172.53) (-151.32,-172.53) (-150.61,-173.24) (-149.91,-173.24) (-149.20,-173.24) (-148.49,-173.95) (-147.79,-173.95) (-201.53,-147.79) (-200.82,-147.79) (-200.11,-148.49) (-199.40,-148.49) (-198.70,-149.20) (-197.99,-149.20) (-197.28,-149.91) (-196.58,-149.91) (-195.87,-150.61) (-195.16,-150.61) (-194.45,-151.32) (-193.75,-151.32) (-193.04,-152.03) (-192.33,-152.03) (-191.63,-152.74) (-190.92,-152.74) (-190.21,-153.44) (-189.50,-153.44) (-188.80,-154.15) (-188.09,-154.15) (-187.38,-154.86) (-186.68,-154.86) (-185.97,-155.56) (-185.26,-155.56) (-184.55,-156.27) (-183.85,-156.27) (-183.14,-156.98) (-182.43,-156.98) (-181.73,-157.68) (-181.02,-157.68) (-180.31,-158.39) (-179.61,-158.39) (-178.90,-159.10) (-178.19,-159.10) (-177.48,-159.81) (-176.78,-159.81) (-176.07,-160.51) (-175.36,-160.51) (-174.66,-160.51) (-173.95,-161.22) (-173.24,-161.22) (-172.53,-161.93) (-171.83,-161.93) (-171.12,-162.63) (-170.41,-162.63) (-169.71,-163.34) (-169.00,-163.34) (-168.29,-164.05) (-167.58,-164.05) (-166.88,-164.76) (-166.17,-164.76) (-165.46,-165.46) (-164.76,-165.46) (-164.05,-166.17) (-163.34,-166.17) (-162.63,-166.88) (-161.93,-166.88) (-161.22,-167.58) (-160.51,-167.58) (-159.81,-168.29) (-159.10,-168.29) (-158.39,-169.00) (-157.68,-169.00) (-156.98,-169.71) (-156.27,-169.71) (-155.56,-170.41) (-154.86,-170.41) (-154.15,-171.12) (-153.44,-171.12) (-152.74,-171.83) (-152.03,-171.83) (-151.32,-172.53) (-150.61,-172.53) (-149.91,-173.24) (-149.20,-173.24) (-148.49,-173.95) (-147.79,-173.95) (-201.53,-147.79) (-200.82,-147.79) (-200.11,-147.79) (-199.40,-147.79) (-199.40,-147.79) (-198.70,-148.49) (-198.70,-148.49) (-198.70,-147.79) (-198.70,-147.08) (-198.70,-146.37) (-199.40,-146.37) (-198.70,-146.37) (-199.40,-146.37) (-199.40,-145.66) (-198.70,-144.96) (-198.70,-144.96) (-198.70,-144.25) (-198.70,-143.54) (-198.70,-143.54) (-201.53,-142.84) (-200.82,-142.84) (-200.11,-142.84) (-199.40,-143.54) (-198.70,-143.54) (-201.53,-142.84) (-200.82,-142.13) (-200.82,-141.42) (-200.11,-140.71) (-200.11,-140.01) (-199.40,-139.30) (-199.40,-139.30) (-199.40,-138.59) (-199.40,-137.89) (-198.70,-137.18) (-198.70,-136.47) (-198.70,-136.47) (-198.70,-135.76) (-199.40,-135.06) (-199.40,-134.35) (-200.11,-133.64) (-200.11,-132.94) (-200.82,-132.23) (-200.82,-131.52) (-206.48,-136.47) (-205.77,-135.76) (-205.06,-135.06) (-204.35,-134.35) (-203.65,-134.35) (-202.94,-133.64) (-202.23,-132.94) (-201.53,-132.23) (-200.82,-131.52) (-214.96,-144.96) (-214.25,-144.25) (-213.55,-143.54) (-212.84,-142.84) (-212.13,-142.13) (-211.42,-141.42) (-210.72,-140.71) (-210.01,-140.01) (-209.30,-139.30) (-208.60,-138.59) (-207.89,-137.89) (-207.18,-137.18) (-206.48,-136.47) (-224.86,-158.39) (-224.15,-157.68) (-224.15,-156.98) (-223.45,-156.27) (-222.74,-155.56) (-222.03,-154.86) (-222.03,-154.15) (-221.32,-153.44) (-220.62,-152.74) (-219.91,-152.03) (-219.91,-151.32) (-219.20,-150.61) (-218.50,-149.91) (-217.79,-149.20) (-217.79,-148.49) (-217.08,-147.79) (-216.37,-147.08) (-215.67,-146.37) (-215.67,-145.66) (-214.96,-144.96) (-226.98,-176.78) (-226.98,-176.07) (-226.98,-175.36) (-226.98,-174.66) (-226.98,-173.95) (-226.27,-173.24) (-226.27,-172.53) (-226.27,-171.83) (-226.27,-171.12) (-226.27,-170.41) (-226.27,-169.71) (-226.27,-169.00) (-226.27,-168.29) (-226.27,-167.58) (-225.57,-166.88) (-225.57,-166.17) (-225.57,-165.46) (-225.57,-164.76) (-225.57,-164.05) (-225.57,-163.34) (-225.57,-162.63) (-225.57,-161.93) (-224.86,-161.22) (-224.86,-160.51) (-224.86,-159.81) (-224.86,-159.10) (-224.86,-158.39) (-226.98,-176.78) (-226.27,-176.78) (-225.57,-177.48) (-224.86,-177.48) (-224.15,-178.19) (-223.45,-178.19) (-222.74,-178.90) (-222.03,-178.90) (-221.32,-179.61) (-220.62,-179.61) (-219.91,-180.31) (-219.20,-180.31) (-218.50,-180.31) (-217.79,-181.02) (-217.08,-181.02) (-216.37,-181.73) (-215.67,-181.73) (-214.96,-182.43) (-214.25,-182.43) (-213.55,-183.14) (-212.84,-183.14) (-212.13,-183.85) (-211.42,-183.85) (-210.72,-183.85) (-210.01,-184.55) (-209.30,-184.55) (-208.60,-185.26) (-207.89,-185.26) (-207.18,-185.97) (-206.48,-185.97) (-205.77,-186.68) (-205.06,-186.68) (-204.35,-187.38) (-203.65,-187.38) (-229.81,-184.55) (-229.10,-184.55) (-228.40,-184.55) (-227.69,-184.55) (-226.98,-184.55) (-226.27,-185.26) (-225.57,-185.26) (-224.86,-185.26) (-224.15,-185.26) (-223.45,-185.26) (-222.74,-185.26) (-222.03,-185.26) (-221.32,-185.26) (-220.62,-185.26) (-219.91,-185.97) (-219.20,-185.97) (-218.50,-185.97) (-217.79,-185.97) (-217.08,-185.97) (-216.37,-185.97) (-215.67,-185.97) (-214.96,-185.97) (-214.25,-185.97) (-213.55,-185.97) (-212.84,-186.68) (-212.13,-186.68) (-211.42,-186.68) (-210.72,-186.68) (-210.01,-186.68) (-209.30,-186.68) (-208.60,-186.68) (-207.89,-186.68) (-207.18,-186.68) (-206.48,-187.38) (-205.77,-187.38) (-205.06,-187.38) (-204.35,-187.38) (-203.65,-187.38) (-229.81,-184.55) (-229.10,-183.85) (-228.40,-183.85) (-227.69,-183.14) (-226.98,-182.43) (-226.27,-181.73) (-225.57,-181.73) (-224.86,-181.02) (-224.15,-180.31) (-223.45,-179.61) (-222.74,-179.61) (-222.03,-178.90) (-221.32,-178.19) (-220.62,-177.48) (-219.91,-177.48) (-219.20,-176.78) (-218.50,-176.07) (-217.79,-175.36) (-217.08,-175.36) (-216.37,-174.66) (-215.67,-173.95) (-214.96,-173.24) (-214.25,-173.24) (-213.55,-172.53) (-212.84,-171.83) (-212.13,-171.12) (-211.42,-171.12) (-210.72,-170.41) (-210.01,-169.71) (-209.30,-169.00) (-208.60,-169.00) (-207.89,-168.29) (-207.18,-167.58) (-206.48,-166.88) (-205.77,-166.88) (-205.06,-166.17) (-204.35,-165.46) (-183.85,-192.33) (-184.55,-191.63) (-185.26,-190.92) (-185.26,-190.21) (-185.97,-189.50) (-186.68,-188.80) (-187.38,-188.09) (-187.38,-187.38) (-188.09,-186.68) (-188.80,-185.97) (-189.50,-185.26) (-189.50,-184.55) (-190.21,-183.85) (-190.92,-183.14) (-191.63,-182.43) (-191.63,-181.73) (-192.33,-181.02) (-193.04,-180.31) (-193.75,-179.61) (-193.75,-178.90) (-194.45,-178.19) (-195.16,-177.48) (-195.87,-176.78) (-196.58,-176.07) (-196.58,-175.36) (-197.28,-174.66) (-197.99,-173.95) (-198.70,-173.24) (-198.70,-172.53) (-199.40,-171.83) (-200.11,-171.12) (-200.82,-170.41) (-200.82,-169.71) (-201.53,-169.00) (-202.23,-168.29) (-202.94,-167.58) (-202.94,-166.88) (-203.65,-166.17) (-204.35,-165.46) (-183.85,-192.33) (-183.85,-191.63) (-183.85,-190.92) (-183.85,-190.21) (-184.55,-189.50) (-184.55,-188.80) (-184.55,-188.09) (-184.55,-187.38) (-184.55,-186.68) (-184.55,-185.97) (-184.55,-185.26) (-185.26,-184.55) (-185.26,-183.85) (-185.26,-183.14) (-185.26,-182.43) (-185.26,-181.73) (-185.26,-181.02) (-185.26,-180.31) (-185.97,-179.61) (-185.97,-178.90) (-185.97,-178.19) (-185.97,-177.48) (-185.97,-176.78) (-185.97,-176.07) (-185.97,-175.36) (-186.68,-174.66) (-186.68,-173.95) (-186.68,-173.24) (-186.68,-172.53) (-186.68,-171.83) (-186.68,-171.12) (-186.68,-170.41) (-187.38,-169.71) (-187.38,-169.00) (-187.38,-168.29) (-187.38,-167.58) (-187.38,-166.88) (-187.38,-166.17) (-187.38,-165.46) (-188.09,-164.76) (-188.09,-164.05) (-188.09,-163.34) (-188.09,-162.63) (-188.09,-161.93) (-188.09,-161.22) (-188.09,-160.51) (-188.80,-159.81) (-188.80,-159.10) (-188.80,-158.39) (-188.80,-157.68) (-188.80,-156.98) (-188.80,-156.27) (-188.80,-155.56) (-189.50,-154.86) (-189.50,-154.15) (-189.50,-153.44) (-189.50,-152.74) (-190.92,-152.74) (-190.21,-152.74) (-189.50,-152.74) (-190.92,-152.74) (-190.92,-152.03) (-190.92,-151.32) (-190.92,-150.61) (-190.92,-149.91) (-191.63,-150.61) (-190.92,-149.91) (-192.33,-151.32) (-191.63,-150.61) (-191.63,-152.74) (-191.63,-152.03) (-192.33,-151.32) (-191.63,-152.74) (-191.63,-152.03) (-191.63,-151.32) (-191.63,-150.61) (-190.92,-149.91) (-190.92,-149.20) (-190.92,-148.49) (-190.92,-147.79) (-190.92,-147.79) (-190.92,-147.08) (-190.92,-146.37) (-190.92,-147.79) (-190.92,-147.08) (-190.92,-146.37) (-190.92,-147.79) (-190.21,-147.79) (-189.50,-147.79) (-188.80,-147.08) (-188.09,-147.08) (-187.38,-147.08) (-186.68,-147.08) (-185.97,-146.37) (-185.26,-146.37) (-184.55,-146.37) (-183.85,-146.37) (-183.14,-145.66) (-182.43,-145.66) (-188.80,-159.81) (-188.80,-159.10) (-188.09,-158.39) (-188.09,-157.68) (-187.38,-156.98) (-187.38,-156.27) (-186.68,-155.56) (-186.68,-154.86) (-185.97,-154.15) (-185.97,-153.44) (-185.97,-152.74) (-185.26,-152.03) (-185.26,-151.32) (-184.55,-150.61) (-184.55,-149.91) (-183.85,-149.20) (-183.85,-148.49) (-183.14,-147.79) (-183.14,-147.08) (-182.43,-146.37) (-182.43,-145.66) (-194.45,-181.73) (-194.45,-181.02) (-193.75,-180.31) (-193.75,-179.61) (-193.75,-178.90) (-193.75,-178.19) (-193.04,-177.48) (-193.04,-176.78) (-193.04,-176.07) (-193.04,-175.36) (-192.33,-174.66) (-192.33,-173.95) (-192.33,-173.24) (-192.33,-172.53) (-191.63,-171.83) (-191.63,-171.12) (-191.63,-170.41) (-191.63,-169.71) (-190.92,-169.00) (-190.92,-168.29) (-190.92,-167.58) (-190.92,-166.88) (-190.21,-166.17) (-190.21,-165.46) (-190.21,-164.76) (-190.21,-164.05) (-189.50,-163.34) (-189.50,-162.63) (-189.50,-161.93) (-189.50,-161.22) (-188.80,-160.51) (-188.80,-159.81) (-194.45,-181.73) (-193.75,-182.43) (-193.04,-182.43) (-192.33,-183.14) (-191.63,-183.85) (-190.92,-184.55) (-190.21,-184.55) (-189.50,-185.26) (-188.80,-185.97) (-188.09,-185.97) (-187.38,-186.68) (-186.68,-187.38) (-185.97,-188.09) (-185.26,-188.09) (-184.55,-188.80) (-183.85,-189.50) (-183.14,-189.50) (-182.43,-190.21) (-181.73,-190.92) (-181.02,-190.92) (-180.31,-191.63) (-179.61,-192.33) (-178.90,-193.04) (-178.19,-193.04) (-177.48,-193.75) (-176.78,-194.45) (-176.07,-194.45) (-175.36,-195.16) (-174.66,-195.87) (-173.95,-196.58) (-173.24,-196.58) (-172.53,-197.28) (-171.83,-197.99) (-171.12,-197.99) (-170.41,-198.70) (-169.71,-199.40) (-169.00,-200.11) (-168.29,-200.11) (-167.58,-200.82) (-167.58,-200.82) (-168.29,-200.11) (-168.29,-199.40) (-169.00,-198.70) (-169.00,-197.99) (-169.71,-197.28) (-169.71,-196.58) (-170.41,-195.87) (-170.41,-195.16) (-171.12,-194.45) (-171.12,-193.75) (-171.83,-193.04) (-171.83,-192.33) (-172.53,-191.63) (-172.53,-190.92) (-173.24,-190.21) (-173.24,-189.50) (-173.95,-188.80) (-173.95,-188.09) (-174.66,-187.38) (-174.66,-186.68) (-175.36,-185.97) (-175.36,-185.26) (-176.07,-184.55) (-176.78,-183.85) (-176.78,-183.14) (-177.48,-182.43) (-177.48,-181.73) (-178.19,-181.02) (-178.19,-180.31) (-178.90,-179.61) (-178.90,-178.90) (-179.61,-178.19) (-179.61,-177.48) (-180.31,-176.78) (-180.31,-176.07) (-181.02,-175.36) (-181.02,-174.66) (-181.73,-173.95) (-181.73,-173.24) (-182.43,-172.53) (-182.43,-171.83) (-183.14,-171.12) (-183.14,-170.41) (-183.85,-169.71) (-228.40,-180.31) (-227.69,-180.31) (-226.98,-180.31) (-226.27,-179.61) (-225.57,-179.61) (-224.86,-179.61) (-224.15,-179.61) (-223.45,-178.90) (-222.74,-178.90) (-222.03,-178.90) (-221.32,-178.90) (-220.62,-178.19) (-219.91,-178.19) (-219.20,-178.19) (-218.50,-178.19) (-217.79,-177.48) (-217.08,-177.48) (-216.37,-177.48) (-215.67,-177.48) (-214.96,-176.78) (-214.25,-176.78) (-213.55,-176.78) (-212.84,-176.78) (-212.13,-176.78) (-211.42,-176.07) (-210.72,-176.07) (-210.01,-176.07) (-209.30,-176.07) (-208.60,-175.36) (-207.89,-175.36) (-207.18,-175.36) (-206.48,-175.36) (-205.77,-174.66) (-205.06,-174.66) (-204.35,-174.66) (-203.65,-174.66) (-202.94,-173.95) (-202.23,-173.95) (-201.53,-173.95) (-200.82,-173.95) (-200.11,-173.24) (-199.40,-173.24) (-198.70,-173.24) (-197.99,-173.24) (-197.28,-173.24) (-196.58,-172.53) (-195.87,-172.53) (-195.16,-172.53) (-194.45,-172.53) (-193.75,-171.83) (-193.04,-171.83) (-192.33,-171.83) (-191.63,-171.83) (-190.92,-171.12) (-190.21,-171.12) (-189.50,-171.12) (-188.80,-171.12) (-188.09,-170.41) (-187.38,-170.41) (-186.68,-170.41) (-185.97,-170.41) (-185.26,-169.71) (-184.55,-169.71) (-183.85,-169.71) (-228.40,-180.31) (-227.69,-179.61) (-226.98,-178.90) (-226.27,-178.19) (-225.57,-177.48) (-224.86,-176.78) (-224.15,-176.07) (-223.45,-175.36) (-222.74,-174.66) (-222.03,-173.95) (-221.32,-173.24) (-220.62,-172.53) (-219.91,-171.83) (-219.20,-171.12) (-218.50,-170.41) (-217.79,-169.71) (-217.08,-169.00) (-216.37,-168.29) (-215.67,-167.58) (-214.96,-166.88) (-214.25,-166.17) (-213.55,-165.46) (-212.84,-164.76) (-212.13,-164.05) (-211.42,-163.34) (-210.72,-162.63) (-210.01,-162.63) (-209.30,-161.93) (-208.60,-161.22) (-207.89,-160.51) (-207.18,-159.81) (-206.48,-159.10) (-205.77,-158.39) (-205.06,-157.68) (-204.35,-156.98) (-203.65,-156.27) (-202.94,-155.56) (-202.23,-154.86) (-201.53,-154.15) (-200.82,-153.44) (-200.11,-152.74) (-199.40,-152.03) (-198.70,-151.32) (-197.99,-150.61) (-197.28,-149.91) (-196.58,-149.20) (-195.87,-148.49) (-195.16,-147.79) (-194.45,-147.08) (-193.75,-146.37) (-193.04,-145.66) (-192.33,-144.96) (-192.33,-144.96) (-192.33,-144.25) (-191.63,-143.54) (-191.63,-143.54) (-190.92,-143.54) (-190.21,-143.54) (-189.50,-143.54) (-189.50,-143.54) (-187.38,-150.61) (-187.38,-149.91) (-188.09,-149.20) (-188.09,-148.49) (-188.09,-147.79) (-188.09,-147.08) (-188.80,-146.37) (-188.80,-145.66) (-188.80,-144.96) (-189.50,-144.25) (-189.50,-143.54) (-187.38,-150.61) (-186.68,-149.91) (-186.68,-149.20) (-185.97,-148.49) (-185.26,-147.79) (-184.55,-147.08) (-184.55,-146.37) (-183.85,-145.66) (-183.14,-144.96) (-183.14,-144.25) (-182.43,-143.54) (-181.73,-142.84) (-181.02,-142.13) (-181.02,-141.42) (-180.31,-140.71) (-255,-310)[(0,0)\[lt\][$\rho:0.1$]{}]{}
\
(483,391)(-368,-161) (-368,-161)(483,0)[2]{}[(0,1)[391]{}]{} (-368,-161)(0,391)[2]{}[(1,0)[483]{}]{} (46,-23) (-345,-46) (-207,-138) (-69,-23) (-253,23) (-138,69) (-207,-161) (-276,0) (-230,23) (-92,46) (-115,138) (-46,115) (-230,-46) (-299,-92) (-345,-92) (-138,138) (-207,138) (23,46) (-23,23) (-322,-46) (-253,0) (-207,69) (-276,23) (-299,46) (-138,207) (-230,92) (-253,92) (-69,46) (-138,92) (-115,-23) (46,46) (-230,0) (-92,161) (92,23) (23,23) (-46,138) (-184,46) (0,0) (-253,-138) (-115,-46) (-184,115) (-46,46) (-184,-115) (-368,-92) (-368,23) (-345,-23) (-322,-92) (-253,-115) (23,0) (-299,23) (-23,161) (-69,115) (-92,-23) (0,23) (-253,-23) (-92,92) (-92,-46) (-276,69) (-46,69) (-207,92) (-138,-69) (-299,-69) (-299,92) (-69,0) (-207,161) (46,0) (-276,-92) (-299,-23) (-230,69) (-299,0) (-322,-115) (-23,138) (0,46) (-299,-46) (-299,69) ( 6.36, 6.36) ( 7.07, 6.36) ( 7.78, 5.66) ( 8.49, 5.66) ( 7.78, 4.95) ( 8.49, 5.66) ( 7.78, 4.95) ( 7.78, 4.95) ( 7.78, 5.66) ( 7.78, 6.36) ( 7.78, 7.07) ( 7.78, 7.78) ( 7.78, 8.49) ( 7.78, 9.19) ( 7.78, 9.90) ( 7.78,10.61) ( 7.78,11.31) ( 7.78,12.02) ( 8.49,12.73) ( 8.49,13.44) ( 8.49,14.14) ( 8.49,14.85) ( 8.49,15.56) ( 8.49,16.26) ( 8.49,16.97) ( 8.49,17.68) ( 8.49,18.38) ( 8.49,19.09) ( 8.49,19.80) ( 8.49,20.51) ( 8.49,21.21) ( 8.49,21.92) ( 8.49,22.63) ( 8.49,23.33) ( 8.49,24.04) ( 8.49,24.75) ( 8.49,25.46) ( 8.49,26.16) ( 9.19,26.87) ( 9.19,27.58) ( 9.19,28.28) ( 9.19,28.99) ( 9.19,29.70) ( 9.19,30.41) ( 9.19,31.11) ( 9.19,31.82) ( 9.19,32.53) ( 9.19,33.23) ( 9.19,33.94) ( 9.19,34.65) ( 9.19,35.36) ( 9.19,36.06) ( 9.19,36.77) ( 9.19,37.48) ( 9.19,38.18) ( 9.19,38.89) ( 9.19,39.60) ( 9.19,40.31) ( 9.19,41.01) ( 9.90,41.72) ( 9.90,42.43) ( 9.90,43.13) ( 9.90,43.84) ( 9.90,44.55) ( 9.90,45.25) ( 9.90,45.96) ( 9.90,46.67) ( 9.90,47.38) ( 9.90,48.08) ( 9.90,48.79) ( 9.90,49.50) ( 9.90,50.20) ( 9.90,50.91) ( 9.90,51.62) ( 9.90,52.33) ( 9.90,53.03) ( 9.90,53.74) ( 9.90,54.45) ( 9.90,55.15) (10.61,55.86) (10.61,56.57) (10.61,57.28) (10.61,57.98) (10.61,58.69) (10.61,59.40) (10.61,60.10) (10.61,60.81) (10.61,61.52) (10.61,62.23) (10.61,62.93) (25.46, 4.95) (25.46, 5.66) (24.75, 6.36) (24.75, 7.07) (24.75, 7.78) (24.75, 8.49) (24.04, 9.19) (24.04, 9.90) (24.04,10.61) (24.04,11.31) (23.33,12.02) (23.33,12.73) (23.33,13.44) (23.33,14.14) (22.63,14.85) (22.63,15.56) (22.63,16.26) (22.63,16.97) (21.92,17.68) (21.92,18.38) (21.92,19.09) (21.92,19.80) (21.21,20.51) (21.21,21.21) (21.21,21.92) (21.21,22.63) (20.51,23.33) (20.51,24.04) (20.51,24.75) (20.51,25.46) (19.80,26.16) (19.80,26.87) (19.80,27.58) (19.80,28.28) (19.09,28.99) (19.09,29.70) (19.09,30.41) (19.09,31.11) (18.38,31.82) (18.38,32.53) (18.38,33.23) (18.38,33.94) (17.68,34.65) (17.68,35.36) (17.68,36.06) (16.97,36.77) (16.97,37.48) (16.97,38.18) (16.97,38.89) (16.26,39.60) (16.26,40.31) (16.26,41.01) (16.26,41.72) (15.56,42.43) (15.56,43.13) (15.56,43.84) (15.56,44.55) (14.85,45.25) (14.85,45.96) (14.85,46.67) (14.85,47.38) (14.14,48.08) (14.14,48.79) (14.14,49.50) (14.14,50.20) (13.44,50.91) (13.44,51.62) (13.44,52.33) (13.44,53.03) (12.73,53.74) (12.73,54.45) (12.73,55.15) (12.73,55.86) (12.02,56.57) (12.02,57.28) (12.02,57.98) (12.02,58.69) (11.31,59.40) (11.31,60.10) (11.31,60.81) (11.31,61.52) (10.61,62.23) (10.61,62.93) (25.46, 4.95) (25.46, 5.66) (25.46, 6.36) (22.63, 7.78) (23.33, 7.78) (24.04, 7.07) (24.75, 7.07) (25.46, 6.36) (22.63, 7.78) (22.63, 7.78) (23.33, 8.49) (23.33, 9.19) (24.04, 9.90) (24.04,10.61) (24.75,11.31) (24.75,12.02) (25.46,12.73) (26.16,13.44) (26.16,14.14) (26.87,14.85) (26.87,15.56) (27.58,16.26) (28.28,16.97) (28.28,17.68) (28.99,18.38) (28.99,19.09) (29.70,19.80) (29.70,20.51) (30.41,21.21) (31.11,21.92) (31.11,22.63) (31.82,23.33) (31.82,24.04) (32.53,24.75) (33.23,25.46) (33.23,26.16) (33.94,26.87) (33.94,27.58) (34.65,28.28) (34.65,28.99) (35.36,29.70) (36.06,30.41) (36.06,31.11) (36.77,31.82) (36.77,32.53) (37.48,33.23) (38.18,33.94) (38.18,34.65) (38.89,35.36) (38.89,36.06) (39.60,36.77) (39.60,37.48) (40.31,38.18) (41.01,38.89) (41.01,39.60) (41.72,40.31) (41.72,41.01) (42.43,41.72) (43.13,42.43) (43.13,43.13) (43.84,43.84) (43.84,44.55) (44.55,45.25) (44.55,45.96) (45.25,46.67) (45.96,47.38) (45.96,48.08) (46.67,48.79) (46.67,49.50) (47.38,50.20) (48.08,50.91) (48.08,51.62) (48.79,52.33) (48.79,53.03) (49.50,53.74) (49.50,54.45) (50.20,55.15) (50.91,55.86) (50.91,56.57) (51.62,57.28) (51.62,57.98) (52.33,58.69) (52.33,58.69) (53.03,58.69) (53.74,59.40) (53.74,59.40) (53.74,60.10) (54.45,60.81) (54.45,60.81) ( 0.71,37.48) ( 1.41,37.48) ( 2.12,38.18) ( 2.83,38.18) ( 3.54,38.89) ( 4.24,38.89) ( 4.95,39.60) ( 5.66,39.60) ( 6.36,39.60) ( 7.07,40.31) ( 7.78,40.31) ( 8.49,41.01) ( 9.19,41.01) ( 9.90,41.72) (10.61,41.72) (11.31,42.43) (12.02,42.43) (12.73,42.43) (13.44,43.13) (14.14,43.13) (14.85,43.84) (15.56,43.84) (16.26,44.55) (16.97,44.55) (17.68,44.55) (18.38,45.25) (19.09,45.25) (19.80,45.96) (20.51,45.96) (21.21,46.67) (21.92,46.67) (22.63,46.67) (23.33,47.38) (24.04,47.38) (24.75,48.08) (25.46,48.08) (26.16,48.79) (26.87,48.79) (27.58,48.79) (28.28,49.50) (28.99,49.50) (29.70,50.20) (30.41,50.20) (31.11,50.91) (31.82,50.91) (32.53,51.62) (33.23,51.62) (33.94,51.62) (34.65,52.33) (35.36,52.33) (36.06,53.03) (36.77,53.03) (37.48,53.74) (38.18,53.74) (38.89,53.74) (39.60,54.45) (40.31,54.45) (41.01,55.15) (41.72,55.15) (42.43,55.86) (43.13,55.86) (43.84,55.86) (44.55,56.57) (45.25,56.57) (45.96,57.28) (46.67,57.28) (47.38,57.98) (48.08,57.98) (48.79,58.69) (49.50,58.69) (50.20,58.69) (50.91,59.40) (51.62,59.40) (52.33,60.10) (53.03,60.10) (53.74,60.81) (54.45,60.81) (-1.41,36.77) (-0.71,36.77) ( 0.00,37.48) ( 0.71,37.48) (-1.41,36.77) (-0.71,37.48) ( 0.00,37.48) ( 0.71,38.18) ( 1.41,38.89) ( 2.12,39.60) ( 2.83,39.60) ( 3.54,40.31) ( 0.00,27.58) ( 0.00,28.28) ( 0.71,28.99) ( 0.71,29.70) ( 0.71,30.41) ( 0.71,31.11) ( 1.41,31.82) ( 1.41,32.53) ( 1.41,33.23) ( 1.41,33.94) ( 2.12,34.65) ( 2.12,35.36) ( 2.12,36.06) ( 2.83,36.77) ( 2.83,37.48) ( 2.83,38.18) ( 2.83,38.89) ( 3.54,39.60) ( 3.54,40.31) ( 0.00,27.58) ( 0.71,27.58) ( 1.41,28.28) ( 2.12,28.28) ( 2.83,28.99) ( 4.95,25.46) ( 4.24,26.16) ( 4.24,26.87) ( 3.54,27.58) ( 3.54,28.28) ( 2.83,28.99) ( 0.00,24.75) ( 0.71,24.75) ( 1.41,24.75) ( 2.12,24.75) ( 2.83,25.46) ( 3.54,25.46) ( 4.24,25.46) ( 4.95,25.46) ( 0.00,24.75) ( 0.00,25.46) (-0.71,24.75) ( 0.00,25.46) (-0.71,24.75) (-0.71,25.46) ( 0.00,26.16) ( 0.00,26.87) ( 0.71,27.58) ( 0.71,28.28) ( 1.41,28.99) ( 1.41,29.70) ( 2.12,30.41) ( 2.12,31.11) ( 2.83,31.82) ( 1.41,32.53) ( 2.12,32.53) ( 2.83,31.82) ( 1.41,29.70) ( 1.41,30.41) ( 1.41,31.11) ( 1.41,31.82) ( 1.41,32.53) ( 1.41,29.70) ( 1.41,29.70) ( 0.71,30.41) ( 0.00,31.11) (-0.71,31.82) (-0.71,32.53) (-1.41,33.23) (-2.12,33.94) (-2.83,34.65) (-3.54,35.36) (-4.24,36.06) (-4.24,36.77) (-4.95,37.48) (-5.66,38.18) (-6.36,38.89) (-7.07,39.60) (-7.78,40.31) (-7.78,41.01) (-8.49,41.72) (-9.19,42.43) (-9.90,43.13) (-10.61,43.84) (-11.31,44.55) (-11.31,45.25) (-12.02,45.96) (-12.73,46.67) (-13.44,47.38) (-14.14,48.08) (-14.85,48.79) (-14.85,49.50) (-15.56,50.20) (-16.26,50.91) (-16.97,51.62) (-17.68,52.33) (-18.38,53.03) (-18.38,53.74) (-19.09,54.45) (-19.80,55.15) (-20.51,55.86) (-21.21,56.57) (-21.92,57.28) (-21.92,57.98) (-22.63,58.69) (-23.33,59.40) (-24.04,60.10) (-24.75,60.81) (-25.46,61.52) (-25.46,62.23) (-26.16,62.93) (-26.87,63.64) (-27.58,64.35) (-28.28,65.05) (-28.99,65.76) (-28.99,66.47) (-29.70,67.18) (-30.41,67.88) (-31.11,68.59) (-31.82,69.30) (-32.53,70.00) (-32.53,70.71) (-33.23,71.42) (-33.94,72.12) (-34.65,72.83) (-33.94,69.30) (-33.94,70.00) (-33.94,70.71) (-34.65,71.42) (-34.65,72.12) (-34.65,72.83) (-33.94,69.30) (-33.23,70.00) (-32.53,70.71) (-33.94,70.71) (-33.23,70.71) (-32.53,70.71) (-33.94,70.71) (-33.23,71.42) (-32.53,72.12) (-31.82,72.83) (-31.82,72.83) (-31.82,73.54) (-31.82,74.25) (-31.82,74.95) (-32.53,72.83) (-32.53,73.54) (-31.82,74.25) (-31.82,74.95) (-33.94,69.30) (-33.94,70.00) (-33.23,70.71) (-33.23,71.42) (-32.53,72.12) (-32.53,72.83) (-37.48,59.40) (-37.48,60.10) (-36.77,60.81) (-36.77,61.52) (-36.77,62.23) (-36.06,62.93) (-36.06,63.64) (-36.06,64.35) (-35.36,65.05) (-35.36,65.76) (-34.65,66.47) (-34.65,67.18) (-34.65,67.88) (-33.94,68.59) (-33.94,69.30) (-37.48,59.40) (-36.77,59.40) (-36.06,58.69) (-35.36,58.69) (-34.65,57.98) (-33.94,57.98) (-34.65,52.33) (-34.65,53.03) (-34.65,53.74) (-34.65,54.45) (-34.65,55.15) (-33.94,55.86) (-33.94,56.57) (-33.94,57.28) (-33.94,57.98) (-34.65,52.33) (-33.94,52.33) (-33.23,52.33) (-32.53,52.33) (-32.53,52.33) (-31.82,51.62) (-31.11,50.91) (-30.41,50.20) (-29.70,49.50) (-28.99,48.79) (-28.99,48.79) (-28.28,48.08) (-27.58,47.38) (-26.87,46.67) (-26.16,45.96) (-25.46,45.25) (-24.75,44.55) (-24.04,43.84) (-23.33,43.84) (-22.63,43.13) (-21.92,42.43) (-21.21,41.72) (-20.51,41.01) (-19.80,40.31) (-19.09,39.60) (-18.38,38.89) (-17.68,38.18) (-16.97,37.48) (-16.26,36.77) (-15.56,36.06) (-14.85,35.36) (-14.14,34.65) (-13.44,33.94) (-12.73,33.23) (-12.02,33.23) (-11.31,32.53) (-10.61,31.82) (-9.90,31.11) (-9.19,30.41) (-8.49,29.70) (-7.78,28.99) (-7.07,28.28) (-6.36,27.58) (-5.66,26.87) (-4.95,26.16) (-4.24,25.46) (-3.54,24.75) (-2.83,24.04) (-2.12,23.33) (-1.41,23.33) (-0.71,22.63) ( 0.00,21.92) ( 0.71,21.21) ( 1.41,20.51) ( 2.12,19.80) ( 2.83,19.09) ( 3.54,18.38) ( 4.24,17.68) ( 4.95,16.97) ( 5.66,16.26) ( 6.36,15.56) ( 7.07,14.85) ( 7.78,14.14) ( 8.49,13.44) ( 9.19,12.73) ( 9.90,12.73) (10.61,12.02) (11.31,11.31) (12.02,10.61) (12.73, 9.90) (13.44, 9.19) (14.14, 8.49) (14.85, 7.78) (12.73, 7.78) (13.44, 7.78) (14.14, 7.78) (14.85, 7.78) (12.73, 7.78) (12.73, 8.49) (12.73, 9.19) (12.73, 9.90) (12.73,10.61) (12.73,11.31) ( 8.49,13.44) ( 9.19,13.44) ( 9.90,12.73) (10.61,12.73) (11.31,12.02) (12.02,12.02) (12.73,11.31) ( 8.49,13.44) ( 9.19,13.44) ( 9.90,12.73) (10.61,12.73) ( 8.49, 9.90) ( 9.19,10.61) ( 9.19,11.31) ( 9.90,12.02) (10.61,12.73) ( 8.49, 8.49) ( 8.49, 9.19) ( 8.49, 9.90) ( 7.78, 7.78) ( 8.49, 8.49) ( 7.78, 7.78) ( 8.49, 7.78) ( 9.19, 7.78) ( 9.90, 7.78) (10.61, 7.07) (11.31, 7.07) (12.02, 7.07) (12.73, 7.07) (13.44, 7.07) (14.14, 7.07) (14.85, 6.36) (15.56, 6.36) (16.26, 6.36) (16.97, 6.36) (17.68, 6.36) (18.38, 6.36) (19.09, 5.66) (19.80, 5.66) (20.51, 5.66) (21.21, 5.66) (21.92, 5.66) (22.63, 5.66) (23.33, 5.66) (24.04, 4.95) (24.75, 4.95) (25.46, 4.95) (26.16, 4.95) (26.87, 4.95) (27.58, 4.95) (28.28, 4.24) (28.99, 4.24) (29.70, 4.24) (30.41, 4.24) (31.11, 4.24) (31.82, 4.24) (32.53, 3.54) (33.23, 3.54) (33.94, 3.54) (34.65, 3.54) (35.36, 3.54) (36.06, 3.54) (36.77, 3.54) (37.48, 2.83) (38.18, 2.83) (38.89, 2.83) (39.60, 2.83) (40.31, 2.83) (41.01, 2.83) (41.72, 2.12) (42.43, 2.12) (43.13, 2.12) (43.84, 2.12) (44.55, 2.12) (45.25, 2.12) (45.96, 1.41) (46.67, 1.41) (47.38, 1.41) (48.08, 1.41) (48.79, 1.41) (49.50, 1.41) (50.20, 0.71) (50.91, 0.71) (51.62, 0.71) (52.33, 0.71) (53.03, 0.71) (53.74, 0.71) (54.45, 0.71) (55.15, 0.00) (55.86, 0.00) (56.57, 0.00) (57.28, 0.00) (57.98, 0.00) (58.69, 0.00) (59.40,-0.71) (60.10,-0.71) (60.81,-0.71) (61.52,-0.71) (62.23,-0.71) (62.93,-0.71) (63.64,-1.41) (64.35,-1.41) (65.05,-1.41) (65.76,-1.41) (65.76,-1.41) (66.47,-0.71) (66.47, 0.00) (67.18, 0.71) (67.88, 1.41) (68.59, 2.12) (68.59, 2.83) (69.30, 3.54) (70.00, 4.24) (70.71, 4.95) (70.71, 5.66) (71.42, 6.36) (72.12, 7.07) (72.83, 7.78) (72.83, 8.49) (73.54, 9.19) (74.25, 9.90) (74.95,10.61) (74.95,11.31) (75.66,12.02) (76.37,12.73) (77.07,13.44) (77.07,14.14) (77.78,14.85) (78.49,15.56) (78.49,16.26) (79.20,16.97) (79.90,17.68) (80.61,18.38) (80.61,19.09) (81.32,19.80) (82.02,20.51) (82.73,21.21) (82.73,21.92) (83.44,22.63) (84.15,23.33) (84.85,24.04) (84.85,24.75) (85.56,25.46) (86.27,26.16) (86.97,26.87) (86.97,27.58) (87.68,28.28) (88.39,28.99) (88.39,29.70) (89.10,30.41) (89.80,31.11) (90.51,31.82) (90.51,32.53) (91.22,33.23) (91.92,33.94) (92.63,34.65) (92.63,35.36) (93.34,36.06) (94.05,36.77) (94.75,37.48) (94.75,38.18) (95.46,38.89) (96.17,39.60) (96.87,40.31) (96.87,41.01) (97.58,41.72) (98.29,42.43) (98.99,43.13) (98.99,43.84) (99.70,44.55) (54.45, 3.54) (55.15, 4.24) (55.86, 4.95) (56.57, 5.66) (57.28, 6.36) (57.98, 7.07) (58.69, 7.07) (59.40, 7.78) (60.10, 8.49) (60.81, 9.19) (61.52, 9.90) (62.23,10.61) (62.93,11.31) (63.64,12.02) (64.35,12.73) (65.05,13.44) (65.76,13.44) (66.47,14.14) (67.18,14.85) (67.88,15.56) (68.59,16.26) (69.30,16.97) (70.00,17.68) (70.71,18.38) (71.42,19.09) (72.12,19.80) (72.83,20.51) (73.54,20.51) (74.25,21.21) (74.95,21.92) (75.66,22.63) (76.37,23.33) (77.07,24.04) (77.78,24.75) (78.49,25.46) (79.20,26.16) (79.90,26.87) (80.61,27.58) (81.32,27.58) (82.02,28.28) (82.73,28.99) (83.44,29.70) (84.15,30.41) (84.85,31.11) (85.56,31.82) (86.27,32.53) (86.97,33.23) (87.68,33.94) (88.39,33.94) (89.10,34.65) (89.80,35.36) (90.51,36.06) (91.22,36.77) (91.92,37.48) (92.63,38.18) (93.34,38.89) (94.05,39.60) (94.75,40.31) (95.46,41.01) (96.17,41.01) (96.87,41.72) (97.58,42.43) (98.29,43.13) (98.99,43.84) (99.70,44.55) (52.33, 0.71) (53.03, 1.41) (53.03, 2.12) (53.74, 2.83) (54.45, 3.54) (51.62, 0.00) (52.33, 0.71) (51.62, 0.00) (51.62, 0.00) (51.62, 0.71) (50.91, 1.41) (50.91, 2.12) (50.20, 2.83) (50.20, 3.54) (50.20, 4.24) (49.50, 4.95) (49.50, 5.66) (48.79, 6.36) (48.79, 7.07) (48.79, 7.78) (48.08, 8.49) (48.08, 9.19) (47.38, 9.90) (47.38,10.61) (47.38,11.31) (46.67,12.02) (46.67,12.73) (45.96,13.44) (45.96,14.14) (45.96,14.85) (45.25,15.56) (45.25,16.26) (44.55,16.97) (44.55,17.68) (44.55,18.38) (43.84,19.09) (43.84,19.80) (43.13,20.51) (43.13,21.21) (43.13,21.92) (42.43,22.63) (42.43,23.33) (41.72,24.04) (41.72,24.75) (41.72,25.46) (41.01,26.16) (41.01,26.87) (40.31,27.58) (40.31,28.28) (39.60,28.99) (39.60,29.70) (39.60,30.41) (38.89,31.11) (38.89,31.82) (38.18,32.53) (38.18,33.23) (38.18,33.94) (37.48,34.65) (37.48,35.36) (36.77,36.06) (36.77,36.77) (36.77,37.48) (36.06,38.18) (36.06,38.89) (35.36,39.60) (35.36,40.31) (35.36,41.01) (34.65,41.72) (34.65,42.43) (33.94,43.13) (33.94,43.84) (33.94,44.55) (33.23,45.25) (33.23,45.96) (32.53,46.67) (32.53,47.38) (32.53,48.08) (31.82,48.79) (31.82,49.50) (31.11,50.20) (31.11,50.91) (31.11,51.62) (30.41,52.33) (30.41,53.03) (29.70,53.74) (29.70,54.45) (27.58,50.91) (28.28,51.62) (28.28,52.33) (28.99,53.03) (28.99,53.74) (29.70,54.45) (27.58,50.91) (28.28,50.20) (28.99,50.20) (29.70,49.50) (28.99,46.67) (28.99,47.38) (28.99,48.08) (29.70,48.79) (29.70,49.50) (28.99,45.96) (28.99,46.67) (27.58,44.55) (28.28,45.25) (28.99,45.96) (14.85,49.50) (15.56,49.50) (16.26,48.79) (16.97,48.79) (17.68,48.08) (18.38,48.08) (19.09,48.08) (19.80,47.38) (20.51,47.38) (21.21,47.38) (21.92,46.67) (22.63,46.67) (23.33,45.96) (24.04,45.96) (24.75,45.96) (25.46,45.25) (26.16,45.25) (26.87,44.55) (27.58,44.55) (14.85,49.50) (15.56,49.50) (16.26,48.79) (16.97,48.79) ( 9.90,48.79) (10.61,48.79) (11.31,48.79) (12.02,48.79) (12.73,48.79) (13.44,48.79) (14.14,48.79) (14.85,48.79) (15.56,48.79) (16.26,48.79) (16.97,48.79) ( 7.78,49.50) ( 8.49,49.50) ( 9.19,48.79) ( 9.90,48.79) ( 7.78,49.50) ( 8.49,48.79) ( 9.19,48.79) ( 9.90,48.08) (10.61,47.38) ( 9.90,46.67) (10.61,47.38) ( 9.90,46.67) (10.61,46.67) (11.31,47.38) (11.31,47.38) (12.02,47.38) (12.73,47.38) (13.44,47.38) (13.44,45.96) (13.44,46.67) (13.44,47.38) (13.44,45.96) (13.44,46.67) (12.73,47.38) (10.61,46.67) (11.31,46.67) (12.02,47.38) (12.73,47.38) ( 9.90,45.96) (10.61,46.67) ( 7.78,46.67) ( 8.49,46.67) ( 9.19,45.96) ( 9.90,45.96) ( 7.78,45.96) ( 7.78,46.67) ( 6.36,43.84) ( 7.07,44.55) ( 7.07,45.25) ( 7.78,45.96) ( 6.36,43.84) ( 5.66,44.55) ( 4.95,45.25) ( 4.24,45.96) ( 3.54,46.67) ( 2.83,47.38) ( 2.12,48.08) ( 1.41,48.79) ( 1.41,49.50) ( 0.71,50.20) ( 0.00,50.91) (-0.71,51.62) (-1.41,52.33) (-2.12,53.03) (-2.83,53.74) (-3.54,54.45) (-4.24,55.15) (-4.95,55.86) (-5.66,56.57) (-6.36,57.28) (-7.07,57.98) (-7.78,58.69) (-8.49,59.40) (-8.49,60.10) (-9.19,60.81) (-9.90,61.52) (-10.61,62.23) (-11.31,62.93) (-12.02,63.64) (-12.73,64.35) (-13.44,65.05) (-14.14,65.76) (-14.85,66.47) (-15.56,67.18) (-16.26,67.88) (-16.97,68.59) (-17.68,69.30) (-18.38,70.00) (-19.09,70.71) (-19.09,71.42) (-19.80,72.12) (-20.51,72.83) (-21.21,73.54) (-21.92,74.25) (-22.63,74.95) (-23.33,75.66) (-24.04,76.37) (-24.75,77.07) (-25.46,77.78) (-26.16,78.49) (-26.87,79.20) (-27.58,79.90) (-28.28,80.61) (-28.99,81.32) (-28.99,82.02) (-29.70,82.73) (-30.41,83.44) (-31.11,84.15) (-31.82,84.85) (-32.53,85.56) (-33.23,86.27) (-33.94,86.97) (-33.94,86.97) (-33.94,87.68) (-33.94,88.39) (-33.94,89.10) (-33.94,89.80) (-33.94,90.51) (-33.94,91.22) (-33.94,91.92) (-33.94,92.63) (-33.94,93.34) (-33.94,94.05) (-33.94,94.75) (-33.94,95.46) (-33.94,96.17) (-33.94,96.87) (-33.94,97.58) (-33.94,98.29) (-33.94,98.99) (-33.94,99.70) (-33.94,100.41) (-33.94,101.12) (-33.94,101.82) (-33.94,102.53) (-33.94,103.24) (-33.94,103.94) (-33.94,104.65) (-33.94,105.36) (-33.94,106.07) (-33.94,106.77) (-33.94,107.48) (-33.94,108.19) (-33.94,108.89) (-33.94,109.60) (-33.94,110.31) (-33.94,111.02) (-33.94,111.72) (-33.94,112.43) (-33.94,113.14) (-33.94,113.84) (-33.94,114.55) (-33.94,115.26) (-33.94,115.97) (-34.65,116.67) (-34.65,117.38) (-34.65,118.09) (-34.65,118.79) (-34.65,119.50) (-34.65,120.21) (-34.65,120.92) (-34.65,121.62) (-34.65,122.33) (-34.65,123.04) (-34.65,123.74) (-34.65,124.45) (-34.65,125.16) (-34.65,125.87) (-34.65,126.57) (-34.65,127.28) (-34.65,127.99) (-34.65,128.69) (-34.65,129.40) (-34.65,130.11) (-34.65,130.81) (-34.65,131.52) (-34.65,132.23) (-34.65,132.94) (-34.65,133.64) (-34.65,134.35) (-34.65,135.06) (-34.65,135.76) (-34.65,136.47) (-34.65,137.18) (-34.65,137.89) (-34.65,138.59) (-34.65,139.30) (-34.65,140.01) (-34.65,140.71) (-34.65,141.42) (-34.65,142.13) (-34.65,142.84) (-34.65,143.54) (-34.65,144.25) (-34.65,144.96) (-34.65,145.66) (-34.65,144.96) (-34.65,145.66) (-35.36,144.96) (-34.65,144.96) (-35.36,144.96) (-34.65,144.96) (-72.83,97.58) (-72.12,98.29) (-71.42,98.99) (-71.42,99.70) (-70.71,100.41) (-70.00,101.12) (-69.30,101.82) (-68.59,102.53) (-68.59,103.24) (-67.88,103.94) (-67.18,104.65) (-66.47,105.36) (-65.76,106.07) (-65.76,106.77) (-65.05,107.48) (-64.35,108.19) (-63.64,108.89) (-62.93,109.60) (-62.23,110.31) (-62.23,111.02) (-61.52,111.72) (-60.81,112.43) (-60.10,113.14) (-59.40,113.84) (-59.40,114.55) (-58.69,115.26) (-57.98,115.97) (-57.28,116.67) (-56.57,117.38) (-56.57,118.09) (-55.86,118.79) (-55.15,119.50) (-54.45,120.21) (-53.74,120.92) (-53.74,121.62) (-53.03,122.33) (-52.33,123.04) (-51.62,123.74) (-50.91,124.45) (-50.91,125.16) (-50.20,125.87) (-49.50,126.57) (-48.79,127.28) (-48.08,127.99) (-48.08,128.69) (-47.38,129.40) (-46.67,130.11) (-45.96,130.81) (-45.25,131.52) (-45.25,132.23) (-44.55,132.94) (-43.84,133.64) (-43.13,134.35) (-42.43,135.06) (-41.72,135.76) (-41.72,136.47) (-41.01,137.18) (-40.31,137.89) (-39.60,138.59) (-38.89,139.30) (-38.89,140.01) (-38.18,140.71) (-37.48,141.42) (-36.77,142.13) (-36.06,142.84) (-36.06,143.54) (-35.36,144.25) (-34.65,144.96) (-72.83,97.58) (-72.12,98.29) (-71.42,98.99) (-70.71,98.99) (-70.00,99.70) (-69.30,100.41) (-68.59,101.12) (-67.88,101.12) (-67.18,101.82) (-66.47,102.53) (-65.76,103.24) (-65.05,103.94) (-64.35,103.94) (-63.64,104.65) (-62.93,105.36) (-62.23,106.07) (-61.52,106.07) (-60.81,106.77) (-60.10,107.48) (-59.40,108.19) (-58.69,108.89) (-57.98,108.89) (-57.28,109.60) (-56.57,110.31) (-55.86,111.02) (-55.15,111.02) (-54.45,111.72) (-53.74,112.43) (-53.03,113.14) (-52.33,113.84) (-51.62,113.84) (-50.91,114.55) (-50.20,115.26) (-49.50,115.97) (-48.79,115.97) (-48.08,116.67) (-47.38,117.38) (-46.67,118.09) (-45.96,118.09) (-45.25,118.79) (-44.55,119.50) (-43.84,120.21) (-43.13,120.92) (-42.43,120.92) (-41.72,121.62) (-41.01,122.33) (-40.31,123.04) (-39.60,123.04) (-38.89,123.74) (-38.18,124.45) (-37.48,125.16) (-36.77,125.87) (-36.06,125.87) (-35.36,126.57) (-34.65,127.28) (-33.94,127.99) (-33.23,127.99) (-32.53,128.69) (-31.82,129.40) (-31.11,130.11) (-30.41,130.81) (-29.70,130.81) (-28.99,131.52) (-28.28,132.23) (-27.58,132.94) (-26.87,132.94) (-26.16,133.64) (-25.46,134.35) (-82.73,110.31) (-82.02,110.31) (-81.32,111.02) (-80.61,111.02) (-79.90,111.72) (-79.20,111.72) (-78.49,112.43) (-77.78,112.43) (-77.07,112.43) (-76.37,113.14) (-75.66,113.14) (-74.95,113.84) (-74.25,113.84) (-73.54,113.84) (-72.83,114.55) (-72.12,114.55) (-71.42,115.26) (-70.71,115.26) (-70.00,115.97) (-69.30,115.97) (-68.59,115.97) (-67.88,116.67) (-67.18,116.67) (-66.47,117.38) (-65.76,117.38) (-65.05,117.38) (-64.35,118.09) (-63.64,118.09) (-62.93,118.79) (-62.23,118.79) (-61.52,119.50) (-60.81,119.50) (-60.10,119.50) (-59.40,120.21) (-58.69,120.21) (-57.98,120.92) (-57.28,120.92) (-56.57,121.62) (-55.86,121.62) (-55.15,121.62) (-54.45,122.33) (-53.74,122.33) (-53.03,123.04) (-52.33,123.04) (-51.62,123.04) (-50.91,123.74) (-50.20,123.74) (-49.50,124.45) (-48.79,124.45) (-48.08,125.16) (-47.38,125.16) (-46.67,125.16) (-45.96,125.87) (-45.25,125.87) (-44.55,126.57) (-43.84,126.57) (-43.13,127.28) (-42.43,127.28) (-41.72,127.28) (-41.01,127.99) (-40.31,127.99) (-39.60,128.69) (-38.89,128.69) (-38.18,128.69) (-37.48,129.40) (-36.77,129.40) (-36.06,130.11) (-35.36,130.11) (-34.65,130.81) (-33.94,130.81) (-33.23,130.81) (-32.53,131.52) (-31.82,131.52) (-31.11,132.23) (-30.41,132.23) (-29.70,132.23) (-28.99,132.94) (-28.28,132.94) (-27.58,133.64) (-26.87,133.64) (-26.16,134.35) (-25.46,134.35) (-61.52,50.91) (-61.52,51.62) (-62.23,52.33) (-62.23,53.03) (-62.23,53.74) (-62.93,54.45) (-62.93,55.15) (-62.93,55.86) (-63.64,56.57) (-63.64,57.28) (-64.35,57.98) (-64.35,58.69) (-64.35,59.40) (-65.05,60.10) (-65.05,60.81) (-65.05,61.52) (-65.76,62.23) (-65.76,62.93) (-65.76,63.64) (-66.47,64.35) (-66.47,65.05) (-66.47,65.76) (-67.18,66.47) (-67.18,67.18) (-67.88,67.88) (-67.88,68.59) (-67.88,69.30) (-68.59,70.00) (-68.59,70.71) (-68.59,71.42) (-69.30,72.12) (-69.30,72.83) (-69.30,73.54) (-70.00,74.25) (-70.00,74.95) (-70.00,75.66) (-70.71,76.37) (-70.71,77.07) (-71.42,77.78) (-71.42,78.49) (-71.42,79.20) (-72.12,79.90) (-72.12,80.61) (-72.12,81.32) (-72.83,82.02) (-72.83,82.73) (-72.83,83.44) (-73.54,84.15) (-73.54,84.85) (-73.54,85.56) (-74.25,86.27) (-74.25,86.97) (-74.95,87.68) (-74.95,88.39) (-74.95,89.10) (-75.66,89.80) (-75.66,90.51) (-75.66,91.22) (-76.37,91.92) (-76.37,92.63) (-76.37,93.34) (-77.07,94.05) (-77.07,94.75) (-77.07,95.46) (-77.78,96.17) (-77.78,96.87) (-78.49,97.58) (-78.49,98.29) (-78.49,98.99) (-79.20,99.70) (-79.20,100.41) (-79.20,101.12) (-79.90,101.82) (-79.90,102.53) (-79.90,103.24) (-80.61,103.94) (-80.61,104.65) (-80.61,105.36) (-81.32,106.07) (-81.32,106.77) (-82.02,107.48) (-82.02,108.19) (-82.02,108.89) (-82.73,109.60) (-82.73,110.31) (-61.52,50.91) (-60.81,50.91) (-60.81,50.91) (-51.62,-7.78) (-51.62,-7.07) (-51.62,-6.36) (-51.62,-5.66) (-52.33,-4.95) (-52.33,-4.24) (-52.33,-3.54) (-52.33,-2.83) (-52.33,-2.12) (-52.33,-1.41) (-53.03,-0.71) (-53.03, 0.00) (-53.03, 0.71) (-53.03, 1.41) (-53.03, 2.12) (-53.03, 2.83) (-53.74, 3.54) (-53.74, 4.24) (-53.74, 4.95) (-53.74, 5.66) (-53.74, 6.36) (-53.74, 7.07) (-53.74, 7.78) (-54.45, 8.49) (-54.45, 9.19) (-54.45, 9.90) (-54.45,10.61) (-54.45,11.31) (-54.45,12.02) (-55.15,12.73) (-55.15,13.44) (-55.15,14.14) (-55.15,14.85) (-55.15,15.56) (-55.15,16.26) (-55.15,16.97) (-55.86,17.68) (-55.86,18.38) (-55.86,19.09) (-55.86,19.80) (-55.86,20.51) (-55.86,21.21) (-56.57,21.92) (-56.57,22.63) (-56.57,23.33) (-56.57,24.04) (-56.57,24.75) (-56.57,25.46) (-57.28,26.16) (-57.28,26.87) (-57.28,27.58) (-57.28,28.28) (-57.28,28.99) (-57.28,29.70) (-57.28,30.41) (-57.98,31.11) (-57.98,31.82) (-57.98,32.53) (-57.98,33.23) (-57.98,33.94) (-57.98,34.65) (-58.69,35.36) (-58.69,36.06) (-58.69,36.77) (-58.69,37.48) (-58.69,38.18) (-58.69,38.89) (-58.69,39.60) (-59.40,40.31) (-59.40,41.01) (-59.40,41.72) (-59.40,42.43) (-59.40,43.13) (-59.40,43.84) (-60.10,44.55) (-60.10,45.25) (-60.10,45.96) (-60.10,46.67) (-60.10,47.38) (-60.10,48.08) (-60.81,48.79) (-60.81,49.50) (-60.81,50.20) (-60.81,50.91) (-110.31,-2.83) (-109.60,-2.83) (-108.89,-2.83) (-108.19,-2.83) (-107.48,-2.83) (-106.77,-2.83) (-106.07,-3.54) (-105.36,-3.54) (-104.65,-3.54) (-103.94,-3.54) (-103.24,-3.54) (-102.53,-3.54) (-101.82,-3.54) (-101.12,-3.54) (-100.41,-3.54) (-99.70,-3.54) (-98.99,-3.54) (-98.29,-3.54) (-97.58,-4.24) (-96.87,-4.24) (-96.17,-4.24) (-95.46,-4.24) (-94.75,-4.24) (-94.05,-4.24) (-93.34,-4.24) (-92.63,-4.24) (-91.92,-4.24) (-91.22,-4.24) (-90.51,-4.24) (-89.80,-4.24) (-89.10,-4.95) (-88.39,-4.95) (-87.68,-4.95) (-86.97,-4.95) (-86.27,-4.95) (-85.56,-4.95) (-84.85,-4.95) (-84.15,-4.95) (-83.44,-4.95) (-82.73,-4.95) (-82.02,-4.95) (-81.32,-4.95) (-80.61,-5.66) (-79.90,-5.66) (-79.20,-5.66) (-78.49,-5.66) (-77.78,-5.66) (-77.07,-5.66) (-76.37,-5.66) (-75.66,-5.66) (-74.95,-5.66) (-74.25,-5.66) (-73.54,-5.66) (-72.83,-5.66) (-72.12,-6.36) (-71.42,-6.36) (-70.71,-6.36) (-70.00,-6.36) (-69.30,-6.36) (-68.59,-6.36) (-67.88,-6.36) (-67.18,-6.36) (-66.47,-6.36) (-65.76,-6.36) (-65.05,-6.36) (-64.35,-6.36) (-63.64,-7.07) (-62.93,-7.07) (-62.23,-7.07) (-61.52,-7.07) (-60.81,-7.07) (-60.10,-7.07) (-59.40,-7.07) (-58.69,-7.07) (-57.98,-7.07) (-57.28,-7.07) (-56.57,-7.07) (-55.86,-7.07) (-55.15,-7.78) (-54.45,-7.78) (-53.74,-7.78) (-53.03,-7.78) (-52.33,-7.78) (-51.62,-7.78) (-130.81,-59.40) (-130.81,-58.69) (-130.11,-57.98) (-130.11,-57.28) (-130.11,-56.57) (-129.40,-55.86) (-129.40,-55.15) (-128.69,-54.45) (-128.69,-53.74) (-128.69,-53.03) (-127.99,-52.33) (-127.99,-51.62) (-127.99,-50.91) (-127.28,-50.20) (-127.28,-49.50) (-127.28,-48.79) (-126.57,-48.08) (-126.57,-47.38) (-125.87,-46.67) (-125.87,-45.96) (-125.87,-45.25) (-125.16,-44.55) (-125.16,-43.84) (-125.16,-43.13) (-124.45,-42.43) (-124.45,-41.72) (-124.45,-41.01) (-123.74,-40.31) (-123.74,-39.60) (-123.04,-38.89) (-123.04,-38.18) (-123.04,-37.48) (-122.33,-36.77) (-122.33,-36.06) (-122.33,-35.36) (-121.62,-34.65) (-121.62,-33.94) (-121.62,-33.23) (-120.92,-32.53) (-120.92,-31.82) (-120.92,-31.11) (-120.21,-30.41) (-120.21,-29.70) (-119.50,-28.99) (-119.50,-28.28) (-119.50,-27.58) (-118.79,-26.87) (-118.79,-26.16) (-118.79,-25.46) (-118.09,-24.75) (-118.09,-24.04) (-118.09,-23.33) (-117.38,-22.63) (-117.38,-21.92) (-116.67,-21.21) (-116.67,-20.51) (-116.67,-19.80) (-115.97,-19.09) (-115.97,-18.38) (-115.97,-17.68) (-115.26,-16.97) (-115.26,-16.26) (-115.26,-15.56) (-114.55,-14.85) (-114.55,-14.14) (-113.84,-13.44) (-113.84,-12.73) (-113.84,-12.02) (-113.14,-11.31) (-113.14,-10.61) (-113.14,-9.90) (-112.43,-9.19) (-112.43,-8.49) (-112.43,-7.78) (-111.72,-7.07) (-111.72,-6.36) (-111.02,-5.66) (-111.02,-4.95) (-111.02,-4.24) (-110.31,-3.54) (-110.31,-2.83) (-130.81,-59.40) (-130.11,-59.40) (-129.40,-60.10) (-128.69,-60.10) (-127.99,-60.81) (-127.99,-60.81) (-127.99,-60.81) (-127.28,-60.10) (-126.57,-59.40) (-125.87,-59.40) (-125.16,-58.69) (-124.45,-57.98) (-123.74,-57.28) (-123.04,-56.57) (-122.33,-56.57) (-121.62,-55.86) (-120.92,-55.15) (-120.21,-54.45) (-119.50,-53.74) (-118.79,-53.74) (-118.09,-53.03) (-117.38,-52.33) (-116.67,-51.62) (-115.97,-51.62) (-115.26,-50.91) (-114.55,-50.20) (-113.84,-49.50) (-113.14,-48.79) (-112.43,-48.79) (-111.72,-48.08) (-111.02,-47.38) (-110.31,-46.67) (-109.60,-45.96) (-108.89,-45.96) (-108.19,-45.25) (-107.48,-44.55) (-106.77,-43.84) (-106.07,-43.13) (-105.36,-43.13) (-104.65,-42.43) (-103.94,-41.72) (-103.24,-41.01) (-102.53,-40.31) (-101.82,-40.31) (-101.12,-39.60) (-100.41,-38.89) (-99.70,-38.18) (-98.99,-37.48) (-98.29,-37.48) (-97.58,-36.77) (-96.87,-36.06) (-96.17,-35.36) (-95.46,-34.65) (-94.75,-34.65) (-94.05,-33.94) (-93.34,-33.23) (-92.63,-32.53) (-91.92,-32.53) (-91.22,-31.82) (-90.51,-31.11) (-89.80,-30.41) (-89.10,-29.70) (-88.39,-29.70) (-87.68,-28.99) (-86.97,-28.28) (-86.27,-27.58) (-85.56,-26.87) (-84.85,-26.87) (-84.15,-26.16) (-83.44,-25.46) (-82.73,-24.75) (-82.02,-24.04) (-81.32,-24.04) (-80.61,-23.33) (-79.90,-22.63) (-79.90,-22.63) (-79.20,-22.63) (-78.49,-22.63) (-77.78,-22.63) (-76.37,-24.75) (-77.07,-24.04) (-77.07,-23.33) (-77.78,-22.63) (-100.41,-42.43) (-99.70,-41.72) (-98.99,-41.72) (-98.29,-41.01) (-97.58,-40.31) (-96.87,-39.60) (-96.17,-39.60) (-95.46,-38.89) (-94.75,-38.18) (-94.05,-37.48) (-93.34,-37.48) (-92.63,-36.77) (-91.92,-36.06) (-91.22,-35.36) (-90.51,-35.36) (-89.80,-34.65) (-89.10,-33.94) (-88.39,-33.94) (-87.68,-33.23) (-86.97,-32.53) (-86.27,-31.82) (-85.56,-31.82) (-84.85,-31.11) (-84.15,-30.41) (-83.44,-29.70) (-82.73,-29.70) (-82.02,-28.99) (-81.32,-28.28) (-80.61,-27.58) (-79.90,-27.58) (-79.20,-26.87) (-78.49,-26.16) (-77.78,-25.46) (-77.07,-25.46) (-76.37,-24.75) (-100.41,-42.43) (-100.41,-42.43) (-99.70,-41.72) (-98.99,-41.01) (-98.29,-40.31) (-97.58,-39.60) (-96.87,-38.89) (-96.17,-38.18) (-95.46,-37.48) (-94.75,-36.77) (-94.05,-36.06) (-93.34,-35.36) (-93.34,-34.65) (-92.63,-33.94) (-91.92,-33.23) (-91.22,-32.53) (-90.51,-31.82) (-89.80,-31.11) (-89.10,-30.41) (-88.39,-29.70) (-87.68,-28.99) (-86.97,-28.28) (-86.27,-27.58) (-85.56,-26.87) (-84.85,-26.16) (-84.15,-25.46) (-83.44,-24.75) (-82.73,-24.04) (-82.02,-23.33) (-81.32,-22.63) (-80.61,-21.92) (-79.90,-21.21) (-79.90,-20.51) (-79.20,-19.80) (-78.49,-19.09) (-77.78,-18.38) (-77.07,-17.68) (-76.37,-16.97) (-75.66,-16.26) (-74.95,-15.56) (-74.25,-14.85) (-73.54,-14.14) (-72.83,-13.44) (-72.12,-12.73) (-71.42,-12.02) (-70.71,-11.31) (-70.00,-10.61) (-69.30,-9.90) (-68.59,-9.19) (-67.88,-8.49) (-67.18,-7.78) (-66.47,-7.07) (-65.76,-6.36) (-65.76,-5.66) (-65.05,-4.95) (-64.35,-4.24) (-63.64,-3.54) (-62.93,-2.83) (-62.23,-2.12) (-61.52,-1.41) (-60.81,-0.71) (-60.10, 0.00) (-59.40, 0.71) (-58.69, 1.41) (-58.69, 1.41) (-58.69, 2.12) (-57.98, 2.83) (-57.98, 2.83) (-57.28, 2.83) (-56.57, 2.83) (-55.86, 2.83) (-55.86, 2.83) (-55.86, 2.83) (-56.57, 3.54) (-56.57, 4.24) (-57.28, 4.95) (-57.28, 5.66) (-57.98, 6.36) (-58.69, 7.07) (-58.69, 7.78) (-59.40, 8.49) (-59.40, 9.19) (-60.10, 9.90) (-60.81,10.61) (-60.81,11.31) (-61.52,12.02) (-61.52,12.73) (-62.23,13.44) (-62.93,14.14) (-62.93,14.85) (-63.64,15.56) (-63.64,16.26) (-64.35,16.97) (-65.05,17.68) (-65.05,18.38) (-65.76,19.09) (-65.76,19.80) (-66.47,20.51) (-67.18,21.21) (-67.18,21.92) (-67.88,22.63) (-67.88,23.33) (-68.59,24.04) (-69.30,24.75) (-69.30,25.46) (-70.00,26.16) (-70.00,26.87) (-70.71,27.58) (-71.42,28.28) (-71.42,28.99) (-72.12,29.70) (-72.12,30.41) (-72.83,31.11) (-73.54,31.82) (-73.54,32.53) (-74.25,33.23) (-74.25,33.94) (-74.95,34.65) (-75.66,35.36) (-75.66,36.06) (-76.37,36.77) (-76.37,37.48) (-77.07,38.18) (-77.78,38.89) (-77.78,39.60) (-78.49,40.31) (-78.49,41.01) (-79.20,41.72) (-79.90,42.43) (-79.90,43.13) (-80.61,43.84) (-80.61,44.55) (-81.32,45.25) (-82.02,45.96) (-82.02,46.67) (-82.73,47.38) (-82.73,48.08) (-83.44,48.79) (-84.15,49.50) (-84.15,50.20) (-84.85,50.91) (-84.85,51.62) (-85.56,52.33) (-84.85,50.91) (-84.85,51.62) (-85.56,52.33) (-84.85,50.91) (-84.85,51.62) (-84.85,48.79) (-84.85,49.50) (-84.85,50.20) (-84.85,50.91) (-84.85,51.62) (-84.85,48.79) (-84.15,48.79) (-83.44,48.79) (-82.73,48.79) (-82.73,48.79) (-82.02,48.08) (-81.32,47.38) (-81.32,47.38) (-81.32,47.38) (-82.02,48.08) (-82.02,48.79) (-82.73,49.50) (-83.44,50.20) (-84.15,50.91) (-84.15,51.62) (-84.85,52.33) (-85.56,53.03) (-86.27,53.74) (-86.27,54.45) (-86.97,55.15) (-87.68,55.86) (-88.39,56.57) (-88.39,57.28) (-89.10,57.98) (-89.80,58.69) (-90.51,59.40) (-90.51,60.10) (-91.22,60.81) (-91.92,61.52) (-92.63,62.23) (-92.63,62.93) (-93.34,63.64) (-94.05,64.35) (-94.75,65.05) (-94.75,65.76) (-95.46,66.47) (-96.17,67.18) (-96.87,67.88) (-96.87,68.59) (-97.58,69.30) (-98.29,70.00) (-98.99,70.71) (-98.99,71.42) (-99.70,72.12) (-100.41,72.83) (-101.12,73.54) (-101.12,74.25) (-101.82,74.95) (-102.53,75.66) (-103.24,76.37) (-103.24,77.07) (-103.94,77.78) (-104.65,78.49) (-105.36,79.20) (-105.36,79.90) (-106.07,80.61) (-106.77,81.32) (-107.48,82.02) (-107.48,82.73) (-108.19,83.44) (-108.89,84.15) (-109.60,84.85) (-109.60,85.56) (-110.31,86.27) (-111.02,86.97) (-111.72,87.68) (-111.72,88.39) (-112.43,89.10) (-113.14,89.80) (-113.84,90.51) (-113.84,91.22) (-114.55,91.92) (-115.26,92.63) (-115.97,93.34) (-115.97,94.05) (-116.67,94.75) (-117.38,95.46) (-117.38,95.46) (-117.38,96.17) (-117.38,96.87) (-117.38,97.58) (-117.38,98.29) (-116.67,98.99) (-116.67,99.70) (-116.67,100.41) (-116.67,101.12) (-116.67,101.82) (-116.67,102.53) (-116.67,103.24) (-116.67,103.94) (-115.97,104.65) (-115.97,105.36) (-115.97,106.07) (-115.97,106.77) (-115.97,107.48) (-115.97,108.19) (-115.97,108.89) (-115.97,109.60) (-115.26,110.31) (-115.26,111.02) (-115.26,111.72) (-115.26,112.43) (-115.26,113.14) (-115.26,113.84) (-115.26,114.55) (-115.26,115.26) (-114.55,115.97) (-114.55,116.67) (-114.55,117.38) (-114.55,118.09) (-114.55,118.79) (-114.55,119.50) (-114.55,120.21) (-114.55,120.92) (-113.84,121.62) (-113.84,122.33) (-113.84,123.04) (-113.84,123.74) (-113.84,124.45) (-113.84,125.16) (-113.84,125.87) (-113.84,126.57) (-113.84,127.28) (-113.14,127.99) (-113.14,128.69) (-113.14,129.40) (-113.14,130.11) (-113.14,130.81) (-113.14,131.52) (-113.14,132.23) (-113.14,132.94) (-112.43,133.64) (-112.43,134.35) (-112.43,135.06) (-112.43,135.76) (-112.43,136.47) (-112.43,137.18) (-112.43,137.89) (-112.43,138.59) (-111.72,139.30) (-111.72,140.01) (-111.72,140.71) (-111.72,141.42) (-111.72,142.13) (-111.72,142.84) (-111.72,143.54) (-111.72,144.25) (-111.02,144.96) (-111.02,145.66) (-111.02,146.37) (-111.02,147.08) (-111.02,147.79) (-111.02,148.49) (-111.02,149.20) (-111.02,149.91) (-110.31,150.61) (-110.31,151.32) (-110.31,152.03) (-110.31,152.74) (-110.31,153.44) (-110.31,153.44) (-109.60,152.74) (-108.89,152.74) (-108.19,152.03) (-107.48,152.03) (-106.77,151.32) (-106.07,150.61) (-105.36,150.61) (-104.65,149.91) (-103.94,149.91) (-103.24,149.20) (-102.53,148.49) (-101.82,148.49) (-101.12,147.79) (-100.41,147.79) (-99.70,147.08) (-98.99,147.08) (-98.29,146.37) (-97.58,145.66) (-96.87,145.66) (-96.17,144.96) (-95.46,144.96) (-94.75,144.25) (-94.05,143.54) (-93.34,143.54) (-92.63,142.84) (-91.92,142.84) (-91.22,142.13) (-90.51,141.42) (-89.80,141.42) (-89.10,140.71) (-88.39,140.71) (-87.68,140.01) (-86.97,139.30) (-86.27,139.30) (-85.56,138.59) (-84.85,138.59) (-84.15,137.89) (-83.44,137.89) (-82.73,137.18) (-82.02,136.47) (-81.32,136.47) (-80.61,135.76) (-79.90,135.76) (-79.20,135.06) (-78.49,134.35) (-77.78,134.35) (-77.07,133.64) (-76.37,133.64) (-75.66,132.94) (-74.95,132.23) (-74.25,132.23) (-73.54,131.52) (-72.83,131.52) (-72.12,130.81) (-71.42,130.11) (-70.71,130.11) (-70.00,129.40) (-69.30,129.40) (-68.59,128.69) (-67.88,127.99) (-67.18,127.99) (-66.47,127.28) (-65.76,127.28) (-65.05,126.57) (-64.35,126.57) (-63.64,125.87) (-62.93,125.16) (-62.23,125.16) (-61.52,124.45) (-60.81,124.45) (-60.10,123.74) (-59.40,123.04) (-58.69,123.04) (-57.98,122.33) (-57.28,122.33) (-56.57,121.62) (-57.98,118.79) (-57.98,119.50) (-57.28,120.21) (-57.28,120.92) (-56.57,121.62) (-57.98,118.79) (-57.28,118.79) (-56.57,118.79) (-55.86,118.79) (-54.45,116.67) (-55.15,117.38) (-55.15,118.09) (-55.86,118.79) (-54.45,116.67) (-53.74,116.67) (-53.74,116.67) (-53.74,116.67) (-53.03,117.38) (-53.03,118.09) (-52.33,118.79) (-52.33,119.50) (-51.62,120.21) (-50.91,120.92) (-50.91,121.62) (-50.20,122.33) (-50.20,123.04) (-49.50,123.74) (-48.79,124.45) (-48.79,125.16) (-48.08,125.87) (-48.08,126.57) (-47.38,127.28) (-46.67,127.99) (-46.67,128.69) (-45.96,129.40) (-45.96,130.11) (-45.25,130.81) (-44.55,131.52) (-44.55,132.23) (-43.84,132.94) (-43.84,133.64) (-43.13,134.35) (-42.43,135.06) (-42.43,135.76) (-41.72,136.47) (-41.72,137.18) (-41.01,137.89) (-40.31,138.59) (-40.31,139.30) (-39.60,140.01) (-39.60,140.71) (-38.89,141.42) (-38.18,142.13) (-38.18,142.84) (-37.48,143.54) (-37.48,144.25) (-36.77,144.96) (-36.06,145.66) (-36.06,146.37) (-35.36,147.08) (-35.36,147.79) (-34.65,148.49) (-33.94,149.20) (-33.94,149.91) (-33.23,150.61) (-33.23,151.32) (-32.53,152.03) (-31.82,152.74) (-31.82,153.44) (-31.11,154.15) (-31.11,154.86) (-30.41,155.56) (-29.70,156.27) (-29.70,156.98) (-28.99,157.68) (-28.99,158.39) (-28.28,159.10) (-27.58,159.81) (-27.58,160.51) (-26.87,161.22) (-26.87,161.93) (-26.16,162.63) (-25.46,163.34) (-25.46,164.05) (-24.75,164.76) (-24.75,165.46) (-24.04,166.17) (-23.33,166.88) (-23.33,167.58) (-22.63,168.29) (-22.63,169.00) (-21.92,169.71) (-21.92,169.71) (-21.92,170.41) (-22.63,164.76) (-22.63,165.46) (-22.63,166.17) (-22.63,166.88) (-22.63,167.58) (-21.92,168.29) (-21.92,169.00) (-21.92,169.71) (-21.92,170.41) (-22.63,164.76) (-21.92,164.05) (-21.21,164.05) (-20.51,163.34) (-21.92,161.93) (-21.21,162.63) (-20.51,163.34) (-21.92,159.81) (-21.92,160.51) (-21.92,161.22) (-21.92,161.93) (-75.66,177.48) (-74.95,177.48) (-74.25,176.78) (-73.54,176.78) (-72.83,176.78) (-72.12,176.07) (-71.42,176.07) (-70.71,176.07) (-70.00,175.36) (-69.30,175.36) (-68.59,175.36) (-67.88,174.66) (-67.18,174.66) (-66.47,174.66) (-65.76,173.95) (-65.05,173.95) (-64.35,173.95) (-63.64,173.24) (-62.93,173.24) (-62.23,173.24) (-61.52,172.53) (-60.81,172.53) (-60.10,172.53) (-59.40,171.83) (-58.69,171.83) (-57.98,171.83) (-57.28,171.12) (-56.57,171.12) (-55.86,171.12) (-55.15,170.41) (-54.45,170.41) (-53.74,170.41) (-53.03,169.71) (-52.33,169.71) (-51.62,169.71) (-50.91,169.00) (-50.20,169.00) (-49.50,169.00) (-48.79,169.00) (-48.08,168.29) (-47.38,168.29) (-46.67,168.29) (-45.96,167.58) (-45.25,167.58) (-44.55,167.58) (-43.84,166.88) (-43.13,166.88) (-42.43,166.88) (-41.72,166.17) (-41.01,166.17) (-40.31,166.17) (-39.60,165.46) (-38.89,165.46) (-38.18,165.46) (-37.48,164.76) (-36.77,164.76) (-36.06,164.76) (-35.36,164.05) (-34.65,164.05) (-33.94,164.05) (-33.23,163.34) (-32.53,163.34) (-31.82,163.34) (-31.11,162.63) (-30.41,162.63) (-29.70,162.63) (-28.99,161.93) (-28.28,161.93) (-27.58,161.93) (-26.87,161.22) (-26.16,161.22) (-25.46,161.22) (-24.75,160.51) (-24.04,160.51) (-23.33,160.51) (-22.63,159.81) (-21.92,159.81) (-127.99,207.89) (-127.28,207.18) (-126.57,207.18) (-125.87,206.48) (-125.16,206.48) (-124.45,205.77) (-123.74,205.77) (-123.04,205.06) (-122.33,204.35) (-121.62,204.35) (-120.92,203.65) (-120.21,203.65) (-119.50,202.94) (-118.79,202.23) (-118.09,202.23) (-117.38,201.53) (-116.67,201.53) (-115.97,200.82) (-115.26,200.82) (-114.55,200.11) (-113.84,199.40) (-113.14,199.40) (-112.43,198.70) (-111.72,198.70) (-111.02,197.99) (-110.31,197.28) (-109.60,197.28) (-108.89,196.58) (-108.19,196.58) (-107.48,195.87) (-106.77,195.87) (-106.07,195.16) (-105.36,194.45) (-104.65,194.45) (-103.94,193.75) (-103.24,193.75) (-102.53,193.04) (-101.82,193.04) (-101.12,192.33) (-100.41,191.63) (-99.70,191.63) (-98.99,190.92) (-98.29,190.92) (-97.58,190.21) (-96.87,189.50) (-96.17,189.50) (-95.46,188.80) (-94.75,188.80) (-94.05,188.09) (-93.34,188.09) (-92.63,187.38) (-91.92,186.68) (-91.22,186.68) (-90.51,185.97) (-89.80,185.97) (-89.10,185.26) (-88.39,184.55) (-87.68,184.55) (-86.97,183.85) (-86.27,183.85) (-85.56,183.14) (-84.85,183.14) (-84.15,182.43) (-83.44,181.73) (-82.73,181.73) (-82.02,181.02) (-81.32,181.02) (-80.61,180.31) (-79.90,179.61) (-79.20,179.61) (-78.49,178.90) (-77.78,178.90) (-77.07,178.19) (-76.37,178.19) (-75.66,177.48) (-127.99,207.89) (-127.28,207.18) (-126.57,206.48) (-128.69,206.48) (-127.99,206.48) (-127.28,206.48) (-126.57,206.48) (-128.69,206.48) (-127.99,207.18) (-127.28,207.18) (-126.57,207.89) (-126.57,207.89) (-125.87,207.89) (-125.16,208.60) (-124.45,208.60) (-124.45,208.60) (-123.74,209.30) (-123.74,209.30) (-123.74,210.01) (-123.74,210.72) (-124.45,209.30) (-124.45,210.01) (-123.74,210.72) (-125.87,210.72) (-125.16,210.01) (-124.45,209.30) (-125.87,210.72) (-125.16,210.01) (-124.45,209.30) (-84.85,161.93) (-85.56,162.63) (-86.27,163.34) (-86.97,164.05) (-86.97,164.76) (-87.68,165.46) (-88.39,166.17) (-89.10,166.88) (-89.80,167.58) (-90.51,168.29) (-90.51,169.00) (-91.22,169.71) (-91.92,170.41) (-92.63,171.12) (-93.34,171.83) (-94.05,172.53) (-94.05,173.24) (-94.75,173.95) (-95.46,174.66) (-96.17,175.36) (-96.87,176.07) (-97.58,176.78) (-97.58,177.48) (-98.29,178.19) (-98.99,178.90) (-99.70,179.61) (-100.41,180.31) (-101.12,181.02) (-101.12,181.73) (-101.82,182.43) (-102.53,183.14) (-103.24,183.85) (-103.94,184.55) (-104.65,185.26) (-104.65,185.97) (-105.36,186.68) (-106.07,187.38) (-106.77,188.09) (-107.48,188.80) (-108.19,189.50) (-108.19,190.21) (-108.89,190.92) (-109.60,191.63) (-110.31,192.33) (-111.02,193.04) (-111.72,193.75) (-111.72,194.45) (-112.43,195.16) (-113.14,195.87) (-113.84,196.58) (-114.55,197.28) (-115.26,197.99) (-115.26,198.70) (-115.97,199.40) (-116.67,200.11) (-117.38,200.82) (-118.09,201.53) (-118.79,202.23) (-118.79,202.94) (-119.50,203.65) (-120.21,204.35) (-120.92,205.06) (-121.62,205.77) (-122.33,206.48) (-122.33,207.18) (-123.04,207.89) (-123.74,208.60) (-124.45,209.30) (-85.56,160.51) (-85.56,161.22) (-84.85,161.93) (-85.56,160.51) (-85.56,161.22) (-85.56,161.93) (-84.85,160.51) (-84.85,161.22) (-85.56,161.93) (-84.85,160.51) (-84.15,161.22) (-83.44,161.93) (-82.73,162.63) (-82.73,162.63) (-82.02,163.34) (-82.02,164.05) (-81.32,164.76) (-82.73,164.76) (-82.02,164.76) (-81.32,164.76) (-82.73,164.76) (-82.02,164.76) (-81.32,165.46) (-81.32,165.46) (-81.32,166.17) (-81.32,166.88) (-80.61,164.76) (-80.61,165.46) (-81.32,166.17) (-81.32,166.88) (-80.61,164.76) (-79.90,164.76) (-79.20,165.46) (-78.49,165.46) (-78.49,165.46) (-77.78,165.46) (-77.07,165.46) (-76.37,165.46) (-76.37,165.46) (-75.66,166.17) (-75.66,166.88) (-74.95,167.58) (-130.81,149.91) (-130.11,149.91) (-129.40,150.61) (-128.69,150.61) (-127.99,150.61) (-127.28,151.32) (-126.57,151.32) (-125.87,151.32) (-125.16,152.03) (-124.45,152.03) (-123.74,152.03) (-123.04,152.03) (-122.33,152.74) (-121.62,152.74) (-120.92,152.74) (-120.21,153.44) (-119.50,153.44) (-118.79,153.44) (-118.09,154.15) (-117.38,154.15) (-116.67,154.15) (-115.97,154.86) (-115.26,154.86) (-114.55,154.86) (-113.84,155.56) (-113.14,155.56) (-112.43,155.56) (-111.72,156.27) (-111.02,156.27) (-110.31,156.27) (-109.60,156.27) (-108.89,156.98) (-108.19,156.98) (-107.48,156.98) (-106.77,157.68) (-106.07,157.68) (-105.36,157.68) (-104.65,158.39) (-103.94,158.39) (-103.24,158.39) (-102.53,159.10) (-101.82,159.10) (-101.12,159.10) (-100.41,159.81) (-99.70,159.81) (-98.99,159.81) (-98.29,160.51) (-97.58,160.51) (-96.87,160.51) (-96.17,161.22) (-95.46,161.22) (-94.75,161.22) (-94.05,161.22) (-93.34,161.93) (-92.63,161.93) (-91.92,161.93) (-91.22,162.63) (-90.51,162.63) (-89.80,162.63) (-89.10,163.34) (-88.39,163.34) (-87.68,163.34) (-86.97,164.05) (-86.27,164.05) (-85.56,164.05) (-84.85,164.76) (-84.15,164.76) (-83.44,164.76) (-82.73,165.46) (-82.02,165.46) (-81.32,165.46) (-80.61,165.46) (-79.90,166.17) (-79.20,166.17) (-78.49,166.17) (-77.78,166.88) (-77.07,166.88) (-76.37,166.88) (-75.66,167.58) (-74.95,167.58) (-133.64,148.49) (-132.94,148.49) (-132.23,149.20) (-131.52,149.20) (-130.81,149.91) (-133.64,148.49) (-133.64,149.20) (-133.64,149.91) (-133.64,150.61) (-133.64,151.32) (-134.35,150.61) (-133.64,151.32) (-136.47,149.91) (-135.76,149.91) (-135.06,150.61) (-134.35,150.61) (-136.47,149.91) (-136.47,150.61) (-136.47,150.61) (-135.76,149.91) (-135.06,149.20) (-134.35,148.49) (-135.76,147.79) (-135.06,147.79) (-134.35,148.49) (-136.47,147.79) (-135.76,147.79) (-136.47,147.79) (-135.76,147.79) (-123.74,87.68) (-123.74,88.39) (-123.74,89.10) (-124.45,89.80) (-124.45,90.51) (-124.45,91.22) (-124.45,91.92) (-124.45,92.63) (-125.16,93.34) (-125.16,94.05) (-125.16,94.75) (-125.16,95.46) (-125.16,96.17) (-125.87,96.87) (-125.87,97.58) (-125.87,98.29) (-125.87,98.99) (-125.87,99.70) (-126.57,100.41) (-126.57,101.12) (-126.57,101.82) (-126.57,102.53) (-126.57,103.24) (-127.28,103.94) (-127.28,104.65) (-127.28,105.36) (-127.28,106.07) (-127.28,106.77) (-127.99,107.48) (-127.99,108.19) (-127.99,108.89) (-127.99,109.60) (-127.99,110.31) (-128.69,111.02) (-128.69,111.72) (-128.69,112.43) (-128.69,113.14) (-128.69,113.84) (-129.40,114.55) (-129.40,115.26) (-129.40,115.97) (-129.40,116.67) (-129.40,117.38) (-130.11,118.09) (-130.11,118.79) (-130.11,119.50) (-130.11,120.21) (-130.11,120.92) (-130.81,121.62) (-130.81,122.33) (-130.81,123.04) (-130.81,123.74) (-130.81,124.45) (-131.52,125.16) (-131.52,125.87) (-131.52,126.57) (-131.52,127.28) (-131.52,127.99) (-132.23,128.69) (-132.23,129.40) (-132.23,130.11) (-132.23,130.81) (-132.23,131.52) (-132.94,132.23) (-132.94,132.94) (-132.94,133.64) (-132.94,134.35) (-132.94,135.06) (-133.64,135.76) (-133.64,136.47) (-133.64,137.18) (-133.64,137.89) (-133.64,138.59) (-134.35,139.30) (-134.35,140.01) (-134.35,140.71) (-134.35,141.42) (-134.35,142.13) (-135.06,142.84) (-135.06,143.54) (-135.06,144.25) (-135.06,144.96) (-135.06,145.66) (-135.76,146.37) (-135.76,147.08) (-135.76,147.79) (-173.95,51.62) (-173.24,52.33) (-172.53,52.33) (-171.83,53.03) (-171.12,53.74) (-170.41,54.45) (-169.71,54.45) (-169.00,55.15) (-168.29,55.86) (-167.58,55.86) (-166.88,56.57) (-166.17,57.28) (-165.46,57.98) (-164.76,57.98) (-164.05,58.69) (-163.34,59.40) (-162.63,59.40) (-161.93,60.10) (-161.22,60.81) (-160.51,61.52) (-159.81,61.52) (-159.10,62.23) (-158.39,62.93) (-157.68,63.64) (-156.98,63.64) (-156.27,64.35) (-155.56,65.05) (-154.86,65.05) (-154.15,65.76) (-153.44,66.47) (-152.74,67.18) (-152.03,67.18) (-151.32,67.88) (-150.61,68.59) (-149.91,68.59) (-149.20,69.30) (-148.49,70.00) (-147.79,70.71) (-147.08,70.71) (-146.37,71.42) (-145.66,72.12) (-144.96,72.12) (-144.25,72.83) (-143.54,73.54) (-142.84,74.25) (-142.13,74.25) (-141.42,74.95) (-140.71,75.66) (-140.01,75.66) (-139.30,76.37) (-138.59,77.07) (-137.89,77.78) (-137.18,77.78) (-136.47,78.49) (-135.76,79.20) (-135.06,79.90) (-134.35,79.90) (-133.64,80.61) (-132.94,81.32) (-132.23,81.32) (-131.52,82.02) (-130.81,82.73) (-130.11,83.44) (-129.40,83.44) (-128.69,84.15) (-127.99,84.85) (-127.28,84.85) (-126.57,85.56) (-125.87,86.27) (-125.16,86.97) (-124.45,86.97) (-123.74,87.68) (-174.66,50.91) (-173.95,51.62) (-176.78,50.91) (-176.07,50.91) (-175.36,50.91) (-174.66,50.91) (-176.78,50.91) (-176.07,50.91) (-175.36,50.91) (-175.36,50.91) (-175.36,50.91) (-174.66,50.91) (-173.95,51.62) (-173.24,51.62) (-172.53,52.33) (-171.83,52.33) (-171.12,53.03) (-170.41,53.03) (-169.71,53.74) (-169.00,53.74) (-168.29,54.45) (-167.58,54.45) (-166.88,54.45) (-166.17,55.15) (-165.46,55.15) (-164.76,55.86) (-164.05,55.86) (-163.34,56.57) (-162.63,56.57) (-161.93,57.28) (-161.22,57.28) (-160.51,57.98) (-159.81,57.98) (-159.10,57.98) (-158.39,58.69) (-157.68,58.69) (-156.98,59.40) (-156.27,59.40) (-155.56,60.10) (-154.86,60.10) (-154.15,60.81) (-153.44,60.81) (-152.74,61.52) (-152.03,61.52) (-151.32,61.52) (-150.61,62.23) (-149.91,62.23) (-149.20,62.93) (-148.49,62.93) (-147.79,63.64) (-147.08,63.64) (-146.37,64.35) (-145.66,64.35) (-144.96,65.05) (-144.25,65.05) (-143.54,65.76) (-142.84,65.76) (-142.13,65.76) (-141.42,66.47) (-140.71,66.47) (-140.01,67.18) (-139.30,67.18) (-138.59,67.88) (-137.89,67.88) (-137.18,68.59) (-136.47,68.59) (-135.76,69.30) (-135.06,69.30) (-134.35,69.30) (-133.64,70.00) (-132.94,70.00) (-132.23,70.71) (-131.52,70.71) (-130.81,71.42) (-130.11,71.42) (-129.40,72.12) (-128.69,72.12) (-127.99,72.83) (-127.28,72.83) (-126.57,72.83) (-125.87,73.54) (-125.16,73.54) (-124.45,74.25) (-123.74,74.25) (-123.04,74.95) (-122.33,74.95) (-121.62,75.66) (-120.92,75.66) (-120.21,76.37) (-119.50,76.37) (-163.34,115.97) (-162.63,115.26) (-161.93,114.55) (-161.22,113.84) (-160.51,113.14) (-159.81,112.43) (-159.10,112.43) (-158.39,111.72) (-157.68,111.02) (-156.98,110.31) (-156.27,109.60) (-155.56,108.89) (-154.86,108.19) (-154.15,107.48) (-153.44,106.77) (-152.74,106.07) (-152.03,106.07) (-151.32,105.36) (-150.61,104.65) (-149.91,103.94) (-149.20,103.24) (-148.49,102.53) (-147.79,101.82) (-147.08,101.12) (-146.37,100.41) (-145.66,99.70) (-144.96,99.70) (-144.25,98.99) (-143.54,98.29) (-142.84,97.58) (-142.13,96.87) (-141.42,96.17) (-140.71,95.46) (-140.01,94.75) (-139.30,94.05) (-138.59,93.34) (-137.89,92.63) (-137.18,92.63) (-136.47,91.92) (-135.76,91.22) (-135.06,90.51) (-134.35,89.80) (-133.64,89.10) (-132.94,88.39) (-132.23,87.68) (-131.52,86.97) (-130.81,86.27) (-130.11,86.27) (-129.40,85.56) (-128.69,84.85) (-127.99,84.15) (-127.28,83.44) (-126.57,82.73) (-125.87,82.02) (-125.16,81.32) (-124.45,80.61) (-123.74,79.90) (-123.04,79.90) (-122.33,79.20) (-121.62,78.49) (-120.92,77.78) (-120.21,77.07) (-119.50,76.37) (-163.34,115.97) (-164.05,116.67) (-164.76,117.38) (-164.76,118.09) (-165.46,118.79) (-166.17,119.50) (-166.88,120.21) (-167.58,120.92) (-168.29,121.62) (-168.29,122.33) (-169.00,123.04) (-169.71,123.74) (-170.41,124.45) (-171.12,125.16) (-171.12,125.87) (-171.83,126.57) (-172.53,127.28) (-173.24,127.99) (-173.95,128.69) (-173.95,129.40) (-174.66,130.11) (-175.36,130.81) (-176.07,131.52) (-176.78,132.23) (-177.48,132.94) (-177.48,133.64) (-178.19,134.35) (-178.90,135.06) (-179.61,135.76) (-180.31,136.47) (-180.31,137.18) (-181.02,137.89) (-181.73,138.59) (-182.43,139.30) (-183.14,140.01) (-183.85,140.71) (-183.85,141.42) (-184.55,142.13) (-185.26,142.84) (-185.97,143.54) (-186.68,144.25) (-186.68,144.96) (-187.38,145.66) (-188.09,146.37) (-188.80,147.08) (-189.50,147.79) (-190.21,148.49) (-190.21,149.20) (-190.92,149.91) (-191.63,150.61) (-192.33,151.32) (-193.04,152.03) (-193.04,152.74) (-193.75,153.44) (-194.45,154.15) (-195.16,154.86) (-195.87,155.56) (-195.87,156.27) (-196.58,156.98) (-197.28,157.68) (-197.99,158.39) (-198.70,159.10) (-199.40,159.81) (-199.40,160.51) (-200.11,161.22) (-200.82,161.93) (-197.99,158.39) (-198.70,159.10) (-199.40,159.81) (-199.40,160.51) (-200.11,161.22) (-200.82,161.93) (-200.82,95.46) (-200.82,96.17) (-200.82,96.87) (-200.82,97.58) (-200.82,98.29) (-200.82,98.99) (-200.82,99.70) (-200.82,100.41) (-200.82,101.12) (-200.82,101.82) (-200.82,102.53) (-200.82,103.24) (-200.11,103.94) (-200.11,104.65) (-200.11,105.36) (-200.11,106.07) (-200.11,106.77) (-200.11,107.48) (-200.11,108.19) (-200.11,108.89) (-200.11,109.60) (-200.11,110.31) (-200.11,111.02) (-200.11,111.72) (-200.11,112.43) (-200.11,113.14) (-200.11,113.84) (-200.11,114.55) (-200.11,115.26) (-200.11,115.97) (-200.11,116.67) (-200.11,117.38) (-200.11,118.09) (-200.11,118.79) (-199.40,119.50) (-199.40,120.21) (-199.40,120.92) (-199.40,121.62) (-199.40,122.33) (-199.40,123.04) (-199.40,123.74) (-199.40,124.45) (-199.40,125.16) (-199.40,125.87) (-199.40,126.57) (-199.40,127.28) (-199.40,127.99) (-199.40,128.69) (-199.40,129.40) (-199.40,130.11) (-199.40,130.81) (-199.40,131.52) (-199.40,132.23) (-199.40,132.94) (-199.40,133.64) (-199.40,134.35) (-198.70,135.06) (-198.70,135.76) (-198.70,136.47) (-198.70,137.18) (-198.70,137.89) (-198.70,138.59) (-198.70,139.30) (-198.70,140.01) (-198.70,140.71) (-198.70,141.42) (-198.70,142.13) (-198.70,142.84) (-198.70,143.54) (-198.70,144.25) (-198.70,144.96) (-198.70,145.66) (-198.70,146.37) (-198.70,147.08) (-198.70,147.79) (-198.70,148.49) (-198.70,149.20) (-198.70,149.91) (-197.99,150.61) (-197.99,151.32) (-197.99,152.03) (-197.99,152.74) (-197.99,153.44) (-197.99,154.15) (-197.99,154.86) (-197.99,155.56) (-197.99,156.27) (-197.99,156.98) (-197.99,157.68) (-197.99,158.39) (-200.82,95.46) (-200.11,95.46) (-199.40,95.46) (-198.70,95.46) (-197.99,92.63) (-197.99,93.34) (-197.99,94.05) (-198.70,94.75) (-198.70,95.46) (-197.99,92.63) (-197.28,92.63) (-196.58,92.63) (-196.58,92.63) (-195.87,92.63) (-195.16,92.63) (-194.45,92.63) (-193.75,92.63) (-192.33,90.51) (-193.04,91.22) (-193.04,91.92) (-193.75,92.63) (-253.85,84.85) (-253.14,84.85) (-252.44,84.85) (-251.73,84.85) (-251.02,84.85) (-250.32,84.85) (-249.61,85.56) (-248.90,85.56) (-248.19,85.56) (-247.49,85.56) (-246.78,85.56) (-246.07,85.56) (-245.37,85.56) (-244.66,85.56) (-243.95,85.56) (-243.24,85.56) (-242.54,85.56) (-241.83,86.27) (-241.12,86.27) (-240.42,86.27) (-239.71,86.27) (-239.00,86.27) (-238.29,86.27) (-237.59,86.27) (-236.88,86.27) (-236.17,86.27) (-235.47,86.27) (-234.76,86.27) (-234.05,86.97) (-233.35,86.97) (-232.64,86.97) (-231.93,86.97) (-231.22,86.97) (-230.52,86.97) (-229.81,86.97) (-229.10,86.97) (-228.40,86.97) (-227.69,86.97) (-226.98,86.97) (-226.27,87.68) (-225.57,87.68) (-224.86,87.68) (-224.15,87.68) (-223.45,87.68) (-222.74,87.68) (-222.03,87.68) (-221.32,87.68) (-220.62,87.68) (-219.91,87.68) (-219.20,88.39) (-218.50,88.39) (-217.79,88.39) (-217.08,88.39) (-216.37,88.39) (-215.67,88.39) (-214.96,88.39) (-214.25,88.39) (-213.55,88.39) (-212.84,88.39) (-212.13,88.39) (-211.42,89.10) (-210.72,89.10) (-210.01,89.10) (-209.30,89.10) (-208.60,89.10) (-207.89,89.10) (-207.18,89.10) (-206.48,89.10) (-205.77,89.10) (-205.06,89.10) (-204.35,89.10) (-203.65,89.80) (-202.94,89.80) (-202.23,89.80) (-201.53,89.80) (-200.82,89.80) (-200.11,89.80) (-199.40,89.80) (-198.70,89.80) (-197.99,89.80) (-197.28,89.80) (-196.58,89.80) (-195.87,90.51) (-195.16,90.51) (-194.45,90.51) (-193.75,90.51) (-193.04,90.51) (-192.33,90.51) (-280.72,31.82) (-280.01,32.53) (-280.01,33.23) (-279.31,33.94) (-279.31,34.65) (-278.60,35.36) (-278.60,36.06) (-277.89,36.77) (-277.89,37.48) (-277.19,38.18) (-277.19,38.89) (-276.48,39.60) (-276.48,40.31) (-275.77,41.01) (-275.77,41.72) (-275.06,42.43) (-275.06,43.13) (-274.36,43.84) (-274.36,44.55) (-273.65,45.25) (-273.65,45.96) (-272.94,46.67) (-272.94,47.38) (-272.24,48.08) (-272.24,48.79) (-271.53,49.50) (-271.53,50.20) (-270.82,50.91) (-270.82,51.62) (-270.11,52.33) (-270.11,53.03) (-269.41,53.74) (-269.41,54.45) (-268.70,55.15) (-268.70,55.86) (-267.99,56.57) (-267.99,57.28) (-267.29,57.98) (-267.29,58.69) (-266.58,59.40) (-266.58,60.10) (-265.87,60.81) (-265.87,61.52) (-265.17,62.23) (-265.17,62.93) (-264.46,63.64) (-264.46,64.35) (-263.75,65.05) (-263.75,65.76) (-263.04,66.47) (-263.04,67.18) (-262.34,67.88) (-262.34,68.59) (-261.63,69.30) (-261.63,70.00) (-260.92,70.71) (-260.92,71.42) (-260.22,72.12) (-260.22,72.83) (-259.51,73.54) (-259.51,74.25) (-258.80,74.95) (-258.80,75.66) (-258.09,76.37) (-258.09,77.07) (-257.39,77.78) (-257.39,78.49) (-256.68,79.20) (-256.68,79.90) (-255.97,80.61) (-255.97,81.32) (-255.27,82.02) (-255.27,82.73) (-254.56,83.44) (-254.56,84.15) (-253.85,84.85) (-280.72,31.82) (-280.72,32.53) (-280.72,33.23) (-280.72,33.94) (-280.72,34.65) (-280.72,35.36) (-280.72,36.06) (-280.72,36.77) (-281.43,37.48) (-281.43,38.18) (-281.43,38.89) (-281.43,39.60) (-281.43,40.31) (-281.43,41.01) (-281.43,41.72) (-281.43,42.43) (-281.43,43.13) (-281.43,43.84) (-281.43,44.55) (-281.43,45.25) (-281.43,45.96) (-281.43,46.67) (-282.14,47.38) (-282.14,48.08) (-282.14,48.79) (-282.14,49.50) (-282.14,50.20) (-282.14,50.91) (-282.14,51.62) (-282.14,52.33) (-282.14,53.03) (-282.14,53.74) (-282.14,54.45) (-282.14,55.15) (-282.14,55.86) (-282.14,56.57) (-282.84,57.28) (-282.84,57.98) (-282.84,58.69) (-282.84,59.40) (-282.84,60.10) (-282.84,60.81) (-282.84,61.52) (-282.84,62.23) (-282.84,62.93) (-282.84,63.64) (-282.84,64.35) (-282.84,65.05) (-282.84,65.76) (-282.84,66.47) (-282.84,67.18) (-283.55,67.88) (-283.55,68.59) (-283.55,69.30) (-283.55,70.00) (-283.55,70.71) (-283.55,71.42) (-283.55,72.12) (-283.55,72.83) (-283.55,73.54) (-283.55,74.25) (-283.55,74.95) (-283.55,75.66) (-283.55,76.37) (-283.55,77.07) (-284.26,77.78) (-284.26,78.49) (-284.26,79.20) (-284.26,79.90) (-284.26,80.61) (-284.26,81.32) (-284.26,82.02) (-284.26,82.73) (-284.26,83.44) (-284.26,84.15) (-284.26,84.85) (-284.26,85.56) (-284.26,86.27) (-284.26,86.97) (-284.96,87.68) (-284.96,88.39) (-284.96,89.10) (-284.96,89.80) (-284.96,90.51) (-284.96,91.22) (-284.96,91.92) (-284.96,92.63) (-286.38,91.92) (-285.67,91.92) (-284.96,92.63) (-288.50,91.92) (-287.79,91.92) (-287.09,91.92) (-286.38,91.92) (-288.50,91.92) (-287.79,92.63) (-288.50,93.34) (-287.79,92.63) (-288.50,93.34) (-287.79,93.34) (-287.09,93.34) (-286.38,93.34) (-285.67,93.34) (-283.55,89.80) (-284.26,90.51) (-284.26,91.22) (-284.96,91.92) (-284.96,92.63) (-285.67,93.34) (-291.33,29.70) (-291.33,30.41) (-291.33,31.11) (-291.33,31.82) (-290.62,32.53) (-290.62,33.23) (-290.62,33.94) (-290.62,34.65) (-290.62,35.36) (-290.62,36.06) (-290.62,36.77) (-290.62,37.48) (-289.91,38.18) (-289.91,38.89) (-289.91,39.60) (-289.91,40.31) (-289.91,41.01) (-289.91,41.72) (-289.91,42.43) (-289.91,43.13) (-289.21,43.84) (-289.21,44.55) (-289.21,45.25) (-289.21,45.96) (-289.21,46.67) (-289.21,47.38) (-289.21,48.08) (-289.21,48.79) (-288.50,49.50) (-288.50,50.20) (-288.50,50.91) (-288.50,51.62) (-288.50,52.33) (-288.50,53.03) (-288.50,53.74) (-287.79,54.45) (-287.79,55.15) (-287.79,55.86) (-287.79,56.57) (-287.79,57.28) (-287.79,57.98) (-287.79,58.69) (-287.79,59.40) (-287.09,60.10) (-287.09,60.81) (-287.09,61.52) (-287.09,62.23) (-287.09,62.93) (-287.09,63.64) (-287.09,64.35) (-287.09,65.05) (-286.38,65.76) (-286.38,66.47) (-286.38,67.18) (-286.38,67.88) (-286.38,68.59) (-286.38,69.30) (-286.38,70.00) (-285.67,70.71) (-285.67,71.42) (-285.67,72.12) (-285.67,72.83) (-285.67,73.54) (-285.67,74.25) (-285.67,74.95) (-285.67,75.66) (-284.96,76.37) (-284.96,77.07) (-284.96,77.78) (-284.96,78.49) (-284.96,79.20) (-284.96,79.90) (-284.96,80.61) (-284.96,81.32) (-284.26,82.02) (-284.26,82.73) (-284.26,83.44) (-284.26,84.15) (-284.26,84.85) (-284.26,85.56) (-284.26,86.27) (-284.26,86.97) (-283.55,87.68) (-283.55,88.39) (-283.55,89.10) (-283.55,89.80) (-291.33,29.70) (-290.62,28.99) (-289.91,26.87) (-289.91,27.58) (-290.62,28.28) (-290.62,28.99) (-289.91,26.87) (-289.91,27.58) (-290.62,23.33) (-290.62,24.04) (-290.62,24.75) (-290.62,25.46) (-289.91,26.16) (-289.91,26.87) (-289.91,27.58) (-290.62,23.33) (-289.91,22.63) (-289.21,22.63) (-288.50,21.92) (-288.50,21.92) (-287.79,21.92) (-287.09,21.92) (-286.38,21.92) (-285.67,21.92) (-284.96,21.92) (-284.26,21.92) (-283.55,21.92) (-282.84,21.92) (-282.14,21.92) (-281.43,21.92) (-280.72,21.92) (-280.01,21.92) (-279.31,21.92) (-278.60,21.92) (-277.89,21.21) (-277.19,21.21) (-276.48,21.21) (-275.77,21.21) (-275.06,21.21) (-274.36,21.21) (-273.65,21.21) (-272.94,21.21) (-272.24,21.21) (-271.53,21.21) (-270.82,21.21) (-270.11,21.21) (-269.41,21.21) (-268.70,21.21) (-267.99,21.21) (-267.29,21.21) (-266.58,21.21) (-265.87,21.21) (-265.17,21.21) (-264.46,21.21) (-263.75,21.21) (-263.04,21.21) (-262.34,21.21) (-261.63,21.21) (-260.92,21.21) (-260.22,21.21) (-259.51,21.21) (-258.80,21.21) (-258.09,21.21) (-257.39,20.51) (-256.68,20.51) (-255.97,20.51) (-255.27,20.51) (-254.56,20.51) (-253.85,20.51) (-253.14,20.51) (-252.44,20.51) (-251.73,20.51) (-251.02,20.51) (-250.32,20.51) (-249.61,20.51) (-248.90,20.51) (-248.19,20.51) (-247.49,20.51) (-246.78,20.51) (-246.07,20.51) (-245.37,20.51) (-244.66,20.51) (-243.95,20.51) (-243.24,20.51) (-242.54,20.51) (-241.83,20.51) (-241.12,20.51) (-240.42,20.51) (-239.71,20.51) (-239.00,20.51) (-238.29,20.51) (-237.59,20.51) (-236.88,19.80) (-236.17,19.80) (-235.47,19.80) (-234.76,19.80) (-234.05,19.80) (-233.35,19.80) (-232.64,19.80) (-231.93,19.80) (-231.22,19.80) (-230.52,19.80) (-229.81,19.80) (-229.10,19.80) (-228.40,19.80) (-227.69,19.80) (-226.98,19.80) (-226.98,19.80) (-226.98,20.51) (-226.27,21.21) (-226.27,21.92) (-225.57,22.63) (-225.57,23.33) (-225.57,24.04) (-224.86,24.75) (-224.86,25.46) (-224.15,26.16) (-224.15,26.87) (-224.15,27.58) (-223.45,28.28) (-223.45,28.99) (-222.74,29.70) (-222.74,30.41) (-222.74,31.11) (-222.03,31.82) (-222.03,32.53) (-222.03,33.23) (-221.32,33.94) (-221.32,34.65) (-220.62,35.36) (-220.62,36.06) (-220.62,36.77) (-219.91,37.48) (-219.91,38.18) (-219.20,38.89) (-219.20,39.60) (-219.20,40.31) (-218.50,41.01) (-218.50,41.72) (-217.79,42.43) (-217.79,43.13) (-217.79,43.84) (-217.08,44.55) (-217.08,45.25) (-216.37,45.96) (-216.37,46.67) (-216.37,47.38) (-215.67,48.08) (-215.67,48.79) (-214.96,49.50) (-214.96,50.20) (-214.96,50.91) (-214.25,51.62) (-214.25,52.33) (-213.55,53.03) (-213.55,53.74) (-213.55,54.45) (-212.84,55.15) (-212.84,55.86) (-212.13,56.57) (-212.13,57.28) (-212.13,57.98) (-211.42,58.69) (-211.42,59.40) (-211.42,60.10) (-210.72,60.81) (-210.72,61.52) (-210.01,62.23) (-210.01,62.93) (-210.01,63.64) (-209.30,64.35) (-209.30,65.05) (-208.60,65.76) (-208.60,66.47) (-208.60,67.18) (-207.89,67.88) (-207.89,68.59) (-207.18,69.30) (-207.18,70.00) (-207.18,70.71) (-206.48,71.42) (-206.48,72.12) (-205.77,72.83) (-205.77,73.54) (-205.77,73.54) (-205.06,73.54) (-204.35,73.54) (-204.35,73.54) (-204.35,73.54) (-205.06,74.25) (-205.06,74.95) (-205.77,75.66) (-205.77,76.37) (-206.48,77.07) (-206.48,77.78) (-207.18,78.49) (-207.18,79.20) (-207.89,79.90) (-207.89,80.61) (-208.60,81.32) (-209.30,82.02) (-209.30,82.73) (-210.01,83.44) (-210.01,84.15) (-210.72,84.85) (-210.72,85.56) (-211.42,86.27) (-211.42,86.97) (-212.13,87.68) (-212.84,88.39) (-212.84,89.10) (-213.55,89.80) (-213.55,90.51) (-214.25,91.22) (-214.25,91.92) (-214.96,92.63) (-214.96,93.34) (-215.67,94.05) (-215.67,94.75) (-216.37,95.46) (-217.08,96.17) (-217.08,96.87) (-217.79,97.58) (-217.79,98.29) (-218.50,98.99) (-218.50,99.70) (-219.20,100.41) (-219.20,101.12) (-219.91,101.82) (-218.50,99.70) (-219.20,100.41) (-219.20,101.12) (-219.91,101.82) (-218.50,99.70) (-217.79,100.41) (-217.08,101.12) (-216.37,101.82) (-216.37,101.82) (-216.37,102.53) (-219.91,100.41) (-219.20,101.12) (-218.50,101.12) (-217.79,101.82) (-217.08,101.82) (-216.37,102.53) (-219.91,100.41) (-219.91,101.12) (-220.62,101.82) (-220.62,101.82) (-219.91,101.82) (-223.45,94.75) (-223.45,95.46) (-222.74,96.17) (-222.74,96.87) (-222.03,97.58) (-222.03,98.29) (-221.32,98.99) (-221.32,99.70) (-220.62,100.41) (-220.62,101.12) (-219.91,101.82) (-221.32,90.51) (-221.32,91.22) (-222.03,91.92) (-222.03,92.63) (-222.74,93.34) (-222.74,94.05) (-223.45,94.75) (-256.68,38.89) (-255.97,39.60) (-255.97,40.31) (-255.27,41.01) (-254.56,41.72) (-254.56,42.43) (-253.85,43.13) (-253.14,43.84) (-253.14,44.55) (-252.44,45.25) (-251.73,45.96) (-251.02,46.67) (-251.02,47.38) (-250.32,48.08) (-249.61,48.79) (-249.61,49.50) (-248.90,50.20) (-248.19,50.91) (-248.19,51.62) (-247.49,52.33) (-246.78,53.03) (-246.78,53.74) (-246.07,54.45) (-245.37,55.15) (-245.37,55.86) (-244.66,56.57) (-243.95,57.28) (-243.95,57.98) (-243.24,58.69) (-242.54,59.40) (-241.83,60.10) (-241.83,60.81) (-241.12,61.52) (-240.42,62.23) (-240.42,62.93) (-239.71,63.64) (-239.00,64.35) (-239.00,65.05) (-238.29,65.76) (-237.59,66.47) (-237.59,67.18) (-236.88,67.88) (-236.17,68.59) (-236.17,69.30) (-235.47,70.00) (-234.76,70.71) (-234.05,71.42) (-234.05,72.12) (-233.35,72.83) (-232.64,73.54) (-232.64,74.25) (-231.93,74.95) (-231.22,75.66) (-231.22,76.37) (-230.52,77.07) (-229.81,77.78) (-229.81,78.49) (-229.10,79.20) (-228.40,79.90) (-228.40,80.61) (-227.69,81.32) (-226.98,82.02) (-226.98,82.73) (-226.27,83.44) (-225.57,84.15) (-224.86,84.85) (-224.86,85.56) (-224.15,86.27) (-223.45,86.97) (-223.45,87.68) (-222.74,88.39) (-222.03,89.10) (-222.03,89.80) (-221.32,90.51) (-256.68,38.89) (-256.68,39.60) (-255.97,40.31) (-255.97,41.01) (-255.27,41.72) (-255.27,42.43) (-255.27,43.13) (-254.56,43.84) (-254.56,44.55) (-253.85,45.25) (-253.85,45.96) (-253.85,46.67) (-253.14,47.38) (-253.14,48.08) (-253.14,48.79) (-252.44,49.50) (-252.44,50.20) (-251.73,50.91) (-251.73,51.62) (-251.73,52.33) (-251.02,53.03) (-251.02,53.74) (-250.32,54.45) (-250.32,55.15) (-250.32,55.86) (-249.61,56.57) (-249.61,57.28) (-248.90,57.98) (-248.90,58.69) (-248.90,59.40) (-248.19,60.10) (-248.19,60.81) (-247.49,61.52) (-247.49,62.23) (-247.49,62.93) (-246.78,63.64) (-246.78,64.35) (-246.07,65.05) (-246.07,65.76) (-246.07,66.47) (-245.37,67.18) (-245.37,67.88) (-245.37,68.59) (-244.66,69.30) (-244.66,70.00) (-243.95,70.71) (-243.95,71.42) (-243.95,72.12) (-243.24,72.83) (-243.24,73.54) (-242.54,74.25) (-242.54,74.95) (-242.54,75.66) (-241.83,76.37) (-241.83,77.07) (-241.12,77.78) (-241.12,78.49) (-241.12,79.20) (-240.42,79.90) (-240.42,80.61) (-239.71,81.32) (-239.71,82.02) (-239.71,82.73) (-239.00,83.44) (-239.00,84.15) (-238.29,84.85) (-238.29,85.56) (-238.29,86.27) (-237.59,86.97) (-237.59,87.68) (-237.59,88.39) (-236.88,89.10) (-236.88,89.80) (-236.17,90.51) (-236.17,91.22) (-236.17,91.92) (-235.47,92.63) (-235.47,93.34) (-234.76,94.05) (-234.76,94.75) (-234.76,32.53) (-234.76,33.23) (-234.76,33.94) (-234.76,34.65) (-234.76,35.36) (-234.76,36.06) (-234.76,36.77) (-234.76,37.48) (-234.76,38.18) (-234.76,38.89) (-234.76,39.60) (-234.76,40.31) (-234.76,41.01) (-234.76,41.72) (-234.76,42.43) (-234.76,43.13) (-234.76,43.84) (-234.76,44.55) (-234.76,45.25) (-234.76,45.96) (-234.76,46.67) (-234.76,47.38) (-234.76,48.08) (-234.76,48.79) (-234.76,49.50) (-234.76,50.20) (-234.76,50.91) (-234.76,51.62) (-234.76,52.33) (-234.76,53.03) (-234.76,53.74) (-234.76,54.45) (-234.76,55.15) (-234.76,55.86) (-234.76,56.57) (-234.76,57.28) (-234.76,57.98) (-234.76,58.69) (-234.76,59.40) (-234.76,60.10) (-234.76,60.81) (-234.76,61.52) (-234.76,62.23) (-234.76,62.93) (-234.76,63.64) (-234.76,64.35) (-234.76,65.05) (-234.76,65.76) (-234.76,66.47) (-234.76,67.18) (-234.76,67.88) (-234.76,68.59) (-234.76,69.30) (-234.76,70.00) (-234.76,70.71) (-234.76,71.42) (-234.76,72.12) (-234.76,72.83) (-234.76,73.54) (-234.76,74.25) (-234.76,74.95) (-234.76,75.66) (-234.76,76.37) (-234.76,77.07) (-234.76,77.78) (-234.76,78.49) (-234.76,79.20) (-234.76,79.90) (-234.76,80.61) (-234.76,81.32) (-234.76,82.02) (-234.76,82.73) (-234.76,83.44) (-234.76,84.15) (-234.76,84.85) (-234.76,85.56) (-234.76,86.27) (-234.76,86.97) (-234.76,87.68) (-234.76,88.39) (-234.76,89.10) (-234.76,89.80) (-234.76,90.51) (-234.76,91.22) (-234.76,91.92) (-234.76,92.63) (-234.76,93.34) (-234.76,94.05) (-234.76,94.75) (-293.45,45.96) (-292.74,45.96) (-292.04,45.96) (-291.33,45.25) (-290.62,45.25) (-289.91,45.25) (-289.21,45.25) (-288.50,44.55) (-287.79,44.55) (-287.09,44.55) (-286.38,44.55) (-285.67,43.84) (-284.96,43.84) (-284.26,43.84) (-283.55,43.84) (-282.84,43.84) (-282.14,43.13) (-281.43,43.13) (-280.72,43.13) (-280.01,43.13) (-279.31,42.43) (-278.60,42.43) (-277.89,42.43) (-277.19,42.43) (-276.48,42.43) (-275.77,41.72) (-275.06,41.72) (-274.36,41.72) (-273.65,41.72) (-272.94,41.01) (-272.24,41.01) (-271.53,41.01) (-270.82,41.01) (-270.11,40.31) (-269.41,40.31) (-268.70,40.31) (-267.99,40.31) (-267.29,40.31) (-266.58,39.60) (-265.87,39.60) (-265.17,39.60) (-264.46,39.60) (-263.75,38.89) (-263.04,38.89) (-262.34,38.89) (-261.63,38.89) (-260.92,38.18) (-260.22,38.18) (-259.51,38.18) (-258.80,38.18) (-258.09,38.18) (-257.39,37.48) (-256.68,37.48) (-255.97,37.48) (-255.27,37.48) (-254.56,36.77) (-253.85,36.77) (-253.14,36.77) (-252.44,36.77) (-251.73,36.06) (-251.02,36.06) (-250.32,36.06) (-249.61,36.06) (-248.90,36.06) (-248.19,35.36) (-247.49,35.36) (-246.78,35.36) (-246.07,35.36) (-245.37,34.65) (-244.66,34.65) (-243.95,34.65) (-243.24,34.65) (-242.54,34.65) (-241.83,33.94) (-241.12,33.94) (-240.42,33.94) (-239.71,33.94) (-239.00,33.23) (-238.29,33.23) (-237.59,33.23) (-236.88,33.23) (-236.17,32.53) (-235.47,32.53) (-234.76,32.53) (-293.45,45.96) (-292.74,46.67) (-292.04,46.67) (-291.33,47.38) (-291.33,47.38) (-290.62,47.38) (-291.33,46.67) (-290.62,47.38) (-291.33,46.67) (-349.31,29.70) (-348.60,29.70) (-347.90,30.41) (-347.19,30.41) (-346.48,30.41) (-345.78,30.41) (-345.07,31.11) (-344.36,31.11) (-343.65,31.11) (-342.95,31.82) (-342.24,31.82) (-341.53,31.82) (-340.83,32.53) (-340.12,32.53) (-339.41,32.53) (-338.70,32.53) (-338.00,33.23) (-337.29,33.23) (-336.58,33.23) (-335.88,33.94) (-335.17,33.94) (-334.46,33.94) (-333.75,33.94) (-333.05,34.65) (-332.34,34.65) (-331.63,34.65) (-330.93,35.36) (-330.22,35.36) (-329.51,35.36) (-328.80,35.36) (-328.10,36.06) (-327.39,36.06) (-326.68,36.06) (-325.98,36.77) (-325.27,36.77) (-324.56,36.77) (-323.85,37.48) (-323.15,37.48) (-322.44,37.48) (-321.73,37.48) (-321.03,38.18) (-320.32,38.18) (-319.61,38.18) (-318.91,38.89) (-318.20,38.89) (-317.49,38.89) (-316.78,38.89) (-316.08,39.60) (-315.37,39.60) (-314.66,39.60) (-313.96,40.31) (-313.25,40.31) (-312.54,40.31) (-311.83,41.01) (-311.13,41.01) (-310.42,41.01) (-309.71,41.01) (-309.01,41.72) (-308.30,41.72) (-307.59,41.72) (-306.88,42.43) (-306.18,42.43) (-305.47,42.43) (-304.76,42.43) (-304.06,43.13) (-303.35,43.13) (-302.64,43.13) (-301.93,43.84) (-301.23,43.84) (-300.52,43.84) (-299.81,43.84) (-299.11,44.55) (-298.40,44.55) (-297.69,44.55) (-296.98,45.25) (-296.28,45.25) (-295.57,45.25) (-294.86,45.96) (-294.16,45.96) (-293.45,45.96) (-292.74,45.96) (-292.04,46.67) (-291.33,46.67) (-331.63,-27.58) (-331.63,-26.87) (-332.34,-26.16) (-332.34,-25.46) (-332.34,-24.75) (-333.05,-24.04) (-333.05,-23.33) (-333.05,-22.63) (-333.05,-21.92) (-333.75,-21.21) (-333.75,-20.51) (-333.75,-19.80) (-334.46,-19.09) (-334.46,-18.38) (-334.46,-17.68) (-335.17,-16.97) (-335.17,-16.26) (-335.17,-15.56) (-335.88,-14.85) (-335.88,-14.14) (-335.88,-13.44) (-335.88,-12.73) (-336.58,-12.02) (-336.58,-11.31) (-336.58,-10.61) (-337.29,-9.90) (-337.29,-9.19) (-337.29,-8.49) (-338.00,-7.78) (-338.00,-7.07) (-338.00,-6.36) (-338.70,-5.66) (-338.70,-4.95) (-338.70,-4.24) (-338.70,-3.54) (-339.41,-2.83) (-339.41,-2.12) (-339.41,-1.41) (-340.12,-0.71) (-340.12, 0.00) (-340.12, 0.71) (-340.83, 1.41) (-340.83, 2.12) (-340.83, 2.83) (-341.53, 3.54) (-341.53, 4.24) (-341.53, 4.95) (-342.24, 5.66) (-342.24, 6.36) (-342.24, 7.07) (-342.24, 7.78) (-342.95, 8.49) (-342.95, 9.19) (-342.95, 9.90) (-343.65,10.61) (-343.65,11.31) (-343.65,12.02) (-344.36,12.73) (-344.36,13.44) (-344.36,14.14) (-345.07,14.85) (-345.07,15.56) (-345.07,16.26) (-345.07,16.97) (-345.78,17.68) (-345.78,18.38) (-345.78,19.09) (-346.48,19.80) (-346.48,20.51) (-346.48,21.21) (-347.19,21.92) (-347.19,22.63) (-347.19,23.33) (-347.90,24.04) (-347.90,24.75) (-347.90,25.46) (-347.90,26.16) (-348.60,26.87) (-348.60,27.58) (-348.60,28.28) (-349.31,28.99) (-349.31,29.70) (-340.83,-85.56) (-340.83,-84.85) (-340.83,-84.15) (-340.83,-83.44) (-340.12,-82.73) (-340.12,-82.02) (-340.12,-81.32) (-340.12,-80.61) (-340.12,-79.90) (-340.12,-79.20) (-339.41,-78.49) (-339.41,-77.78) (-339.41,-77.07) (-339.41,-76.37) (-339.41,-75.66) (-339.41,-74.95) (-338.70,-74.25) (-338.70,-73.54) (-338.70,-72.83) (-338.70,-72.12) (-338.70,-71.42) (-338.70,-70.71) (-338.70,-70.00) (-338.00,-69.30) (-338.00,-68.59) (-338.00,-67.88) (-338.00,-67.18) (-338.00,-66.47) (-338.00,-65.76) (-337.29,-65.05) (-337.29,-64.35) (-337.29,-63.64) (-337.29,-62.93) (-337.29,-62.23) (-337.29,-61.52) (-336.58,-60.81) (-336.58,-60.10) (-336.58,-59.40) (-336.58,-58.69) (-336.58,-57.98) (-336.58,-57.28) (-336.58,-56.57) (-335.88,-55.86) (-335.88,-55.15) (-335.88,-54.45) (-335.88,-53.74) (-335.88,-53.03) (-335.88,-52.33) (-335.17,-51.62) (-335.17,-50.91) (-335.17,-50.20) (-335.17,-49.50) (-335.17,-48.79) (-335.17,-48.08) (-334.46,-47.38) (-334.46,-46.67) (-334.46,-45.96) (-334.46,-45.25) (-334.46,-44.55) (-334.46,-43.84) (-333.75,-43.13) (-333.75,-42.43) (-333.75,-41.72) (-333.75,-41.01) (-333.75,-40.31) (-333.75,-39.60) (-333.75,-38.89) (-333.05,-38.18) (-333.05,-37.48) (-333.05,-36.77) (-333.05,-36.06) (-333.05,-35.36) (-333.05,-34.65) (-332.34,-33.94) (-332.34,-33.23) (-332.34,-32.53) (-332.34,-31.82) (-332.34,-31.11) (-332.34,-30.41) (-331.63,-29.70) (-331.63,-28.99) (-331.63,-28.28) (-331.63,-27.58) (-340.83,-85.56) (-340.12,-85.56) (-339.41,-85.56) (-338.70,-85.56) (-338.70,-86.97) (-338.70,-86.27) (-338.70,-85.56) (-338.70,-86.97) (-338.70,-86.97) (-338.70,-86.27) (-338.70,-85.56) (-338.70,-84.85) (-338.70,-84.15) (-338.70,-83.44) (-338.70,-82.73) (-338.70,-82.02) (-338.70,-81.32) (-338.70,-80.61) (-338.70,-79.90) (-338.70,-79.20) (-338.00,-78.49) (-338.00,-77.78) (-338.00,-77.07) (-338.00,-76.37) (-338.00,-75.66) (-338.00,-74.95) (-338.00,-74.25) (-338.00,-73.54) (-338.00,-72.83) (-338.00,-72.12) (-338.00,-71.42) (-338.00,-70.71) (-338.00,-70.71) (-337.29,-70.71) (-336.58,-70.71) (-335.88,-70.71) (-335.17,-70.71) (-334.46,-70.71) (-333.75,-70.71) (-333.05,-70.71) (-332.34,-70.71) (-331.63,-70.00) (-330.93,-70.00) (-330.22,-70.00) (-329.51,-70.00) (-328.80,-70.00) (-328.10,-70.00) (-327.39,-70.00) (-326.68,-70.00) (-325.98,-70.00) (-325.27,-70.00) (-324.56,-70.00) (-323.85,-70.00) (-323.15,-70.00) (-322.44,-70.00) (-321.73,-70.00) (-321.03,-70.00) (-320.32,-69.30) (-319.61,-69.30) (-318.91,-69.30) (-318.20,-69.30) (-317.49,-69.30) (-316.78,-69.30) (-316.08,-69.30) (-315.37,-69.30) (-314.66,-69.30) (-301.93,-98.99) (-301.93,-98.29) (-302.64,-97.58) (-302.64,-96.87) (-303.35,-96.17) (-303.35,-95.46) (-304.06,-94.75) (-304.06,-94.05) (-304.06,-93.34) (-304.76,-92.63) (-304.76,-91.92) (-305.47,-91.22) (-305.47,-90.51) (-306.18,-89.80) (-306.18,-89.10) (-306.18,-88.39) (-306.88,-87.68) (-306.88,-86.97) (-307.59,-86.27) (-307.59,-85.56) (-308.30,-84.85) (-308.30,-84.15) (-308.30,-83.44) (-309.01,-82.73) (-309.01,-82.02) (-309.71,-81.32) (-309.71,-80.61) (-310.42,-79.90) (-310.42,-79.20) (-310.42,-78.49) (-311.13,-77.78) (-311.13,-77.07) (-311.83,-76.37) (-311.83,-75.66) (-312.54,-74.95) (-312.54,-74.25) (-312.54,-73.54) (-313.25,-72.83) (-313.25,-72.12) (-313.96,-71.42) (-313.96,-70.71) (-314.66,-70.00) (-314.66,-69.30) (-301.93,-98.99) (-301.23,-98.29) (-301.23,-97.58) (-300.52,-96.87) (-299.81,-96.17) (-299.11,-95.46) (-299.11,-94.75) (-298.40,-94.05) (-297.69,-93.34) (-297.69,-92.63) (-296.98,-91.92) (-296.28,-91.22) (-296.28,-90.51) (-295.57,-89.80) (-294.86,-89.10) (-294.16,-88.39) (-294.16,-87.68) (-293.45,-86.97) (-292.74,-86.27) (-292.74,-85.56) (-292.04,-84.85) (-291.33,-84.15) (-291.33,-83.44) (-290.62,-82.73) (-289.91,-82.02) (-289.21,-81.32) (-289.21,-80.61) (-288.50,-79.90) (-287.79,-79.20) (-287.79,-78.49) (-287.09,-77.78) (-286.38,-77.07) (-286.38,-76.37) (-285.67,-75.66) (-284.96,-74.95) (-284.26,-74.25) (-284.26,-73.54) (-283.55,-72.83) (-282.84,-72.12) (-282.84,-71.42) (-282.14,-70.71) (-281.43,-70.00) (-281.43,-69.30) (-280.72,-68.59) (-280.01,-67.88) (-279.31,-67.18) (-279.31,-66.47) (-278.60,-65.76) (-278.60,-65.76) (-277.89,-66.47) (-277.19,-66.47) (-276.48,-67.18) (-275.77,-67.88) (-275.06,-68.59) (-274.36,-68.59) (-273.65,-69.30) (-272.94,-70.00) (-272.24,-70.71) (-271.53,-70.71) (-270.82,-71.42) (-270.11,-72.12) (-269.41,-72.12) (-268.70,-72.83) (-267.99,-73.54) (-267.29,-74.25) (-266.58,-74.25) (-265.87,-74.95) (-265.17,-75.66) (-264.46,-75.66) (-263.75,-76.37) (-263.04,-77.07) (-262.34,-77.78) (-261.63,-77.78) (-260.92,-78.49) (-260.22,-79.20) (-259.51,-79.90) (-258.80,-79.90) (-258.09,-80.61) (-257.39,-81.32) (-256.68,-81.32) (-255.97,-82.02) (-255.27,-82.73) (-254.56,-83.44) (-253.85,-83.44) (-253.14,-84.15) (-252.44,-84.85) (-251.73,-85.56) (-251.02,-85.56) (-250.32,-86.27) (-249.61,-86.97) (-248.90,-86.97) (-248.19,-87.68) (-247.49,-88.39) (-246.78,-89.10) (-246.07,-89.10) (-245.37,-89.80) (-244.66,-90.51) (-243.95,-90.51) (-243.24,-91.22) (-242.54,-91.92) (-241.83,-92.63) (-241.12,-92.63) (-240.42,-93.34) (-239.71,-94.05) (-239.00,-94.75) (-238.29,-94.75) (-237.59,-95.46) (-237.59,-95.46) (-236.88,-95.46) (-236.17,-96.17) (-235.47,-96.17) (-234.76,-96.17) (-234.05,-96.17) (-233.35,-96.87) (-232.64,-96.87) (-231.93,-96.87) (-231.22,-97.58) (-230.52,-97.58) (-229.81,-97.58) (-229.10,-97.58) (-228.40,-98.29) (-227.69,-98.29) (-226.98,-98.29) (-226.27,-98.99) (-225.57,-98.99) (-224.86,-98.99) (-224.15,-98.99) (-223.45,-99.70) (-222.74,-99.70) (-222.03,-99.70) (-221.32,-100.41) (-220.62,-100.41) (-219.91,-100.41) (-219.20,-101.12) (-218.50,-101.12) (-217.79,-101.12) (-217.08,-101.12) (-216.37,-101.82) (-215.67,-101.82) (-214.96,-101.82) (-214.25,-102.53) (-213.55,-102.53) (-212.84,-102.53) (-212.13,-102.53) (-211.42,-103.24) (-210.72,-103.24) (-210.01,-103.24) (-209.30,-103.94) (-208.60,-103.94) (-207.89,-103.94) (-207.18,-103.94) (-206.48,-104.65) (-205.77,-104.65) (-205.06,-104.65) (-204.35,-105.36) (-203.65,-105.36) (-202.94,-105.36) (-202.23,-105.36) (-201.53,-106.07) (-200.82,-106.07) (-200.11,-106.07) (-199.40,-106.77) (-198.70,-106.77) (-197.99,-106.77) (-197.28,-106.77) (-196.58,-107.48) (-195.87,-107.48) (-195.16,-107.48) (-194.45,-108.19) (-193.75,-108.19) (-193.04,-108.19) (-192.33,-108.89) (-191.63,-108.89) (-190.92,-108.89) (-190.21,-108.89) (-189.50,-109.60) (-188.80,-109.60) (-188.09,-109.60) (-187.38,-110.31) (-186.68,-110.31) (-185.97,-110.31) (-185.26,-110.31) (-184.55,-111.02) (-183.85,-111.02) (-183.14,-111.02) (-182.43,-111.72) (-181.73,-111.72) (-181.02,-111.72) (-180.31,-111.72) (-179.61,-112.43) (-178.90,-112.43) (-178.90,-112.43) (-190.92,-118.79) (-190.21,-118.09) (-189.50,-118.09) (-188.80,-117.38) (-188.09,-117.38) (-187.38,-116.67) (-186.68,-116.67) (-185.97,-115.97) (-185.26,-115.97) (-184.55,-115.26) (-183.85,-115.26) (-183.14,-114.55) (-182.43,-114.55) (-181.73,-113.84) (-181.02,-113.84) (-180.31,-113.14) (-179.61,-113.14) (-178.90,-112.43) (-195.87,-149.91) (-195.87,-149.20) (-195.87,-148.49) (-195.87,-147.79) (-195.16,-147.08) (-195.16,-146.37) (-195.16,-145.66) (-195.16,-144.96) (-195.16,-144.25) (-195.16,-143.54) (-194.45,-142.84) (-194.45,-142.13) (-194.45,-141.42) (-194.45,-140.71) (-194.45,-140.01) (-194.45,-139.30) (-193.75,-138.59) (-193.75,-137.89) (-193.75,-137.18) (-193.75,-136.47) (-193.75,-135.76) (-193.75,-135.06) (-193.75,-134.35) (-193.04,-133.64) (-193.04,-132.94) (-193.04,-132.23) (-193.04,-131.52) (-193.04,-130.81) (-193.04,-130.11) (-192.33,-129.40) (-192.33,-128.69) (-192.33,-127.99) (-192.33,-127.28) (-192.33,-126.57) (-192.33,-125.87) (-191.63,-125.16) (-191.63,-124.45) (-191.63,-123.74) (-191.63,-123.04) (-191.63,-122.33) (-191.63,-121.62) (-190.92,-120.92) (-190.92,-120.21) (-190.92,-119.50) (-190.92,-118.79) (-195.87,-149.91) (-195.16,-149.91) (-194.45,-149.20) (-193.75,-149.20) (-193.04,-148.49) (-192.33,-148.49) (-192.33,-148.49) (-191.63,-148.49) (-193.75,-151.32) (-193.04,-150.61) (-193.04,-149.91) (-192.33,-149.20) (-191.63,-148.49) (-192.33,-155.56) (-192.33,-154.86) (-193.04,-154.15) (-193.04,-153.44) (-193.04,-152.74) (-193.75,-152.03) (-193.75,-151.32) (-192.33,-155.56) (-191.63,-154.86) (-190.92,-154.86) (-190.21,-154.15) (-189.50,-153.44) (-243.95,-124.45) (-243.24,-125.16) (-242.54,-125.16) (-241.83,-125.87) (-241.12,-125.87) (-240.42,-126.57) (-239.71,-126.57) (-239.00,-127.28) (-238.29,-127.28) (-237.59,-127.99) (-236.88,-127.99) (-236.17,-128.69) (-235.47,-128.69) (-234.76,-129.40) (-234.05,-129.40) (-233.35,-130.11) (-232.64,-130.81) (-231.93,-130.81) (-231.22,-131.52) (-230.52,-131.52) (-229.81,-132.23) (-229.10,-132.23) (-228.40,-132.94) (-227.69,-132.94) (-226.98,-133.64) (-226.27,-133.64) (-225.57,-134.35) (-224.86,-134.35) (-224.15,-135.06) (-223.45,-135.06) (-222.74,-135.76) (-222.03,-136.47) (-221.32,-136.47) (-220.62,-137.18) (-219.91,-137.18) (-219.20,-137.89) (-218.50,-137.89) (-217.79,-138.59) (-217.08,-138.59) (-216.37,-139.30) (-215.67,-139.30) (-214.96,-140.01) (-214.25,-140.01) (-213.55,-140.71) (-212.84,-140.71) (-212.13,-141.42) (-211.42,-141.42) (-210.72,-142.13) (-210.01,-142.84) (-209.30,-142.84) (-208.60,-143.54) (-207.89,-143.54) (-207.18,-144.25) (-206.48,-144.25) (-205.77,-144.96) (-205.06,-144.96) (-204.35,-145.66) (-203.65,-145.66) (-202.94,-146.37) (-202.23,-146.37) (-201.53,-147.08) (-200.82,-147.08) (-200.11,-147.79) (-199.40,-148.49) (-198.70,-148.49) (-197.99,-149.20) (-197.28,-149.20) (-196.58,-149.91) (-195.87,-149.91) (-195.16,-150.61) (-194.45,-150.61) (-193.75,-151.32) (-193.04,-151.32) (-192.33,-152.03) (-191.63,-152.03) (-190.92,-152.74) (-190.21,-152.74) (-189.50,-153.44) (-243.95,-124.45) (-243.24,-124.45) (-242.54,-124.45) (-241.83,-123.74) (-241.12,-123.74) (-240.42,-123.74) (-240.42,-123.74) (-239.71,-124.45) (-239.00,-124.45) (-238.29,-125.16) (-237.59,-125.87) (-237.59,-125.87) (-237.59,-125.16) (-237.59,-124.45) (-237.59,-124.45) (-236.88,-124.45) (-238.29,-128.69) (-238.29,-127.99) (-237.59,-127.28) (-237.59,-126.57) (-237.59,-125.87) (-236.88,-125.16) (-236.88,-124.45) (-238.29,-128.69) (-237.59,-128.69) (-236.88,-128.69) (-236.17,-128.69) (-235.47,-128.69) (-248.90,-118.79) (-248.19,-119.50) (-247.49,-119.50) (-246.78,-120.21) (-246.07,-120.92) (-245.37,-121.62) (-244.66,-121.62) (-243.95,-122.33) (-243.24,-123.04) (-242.54,-123.74) (-241.83,-123.74) (-241.12,-124.45) (-240.42,-125.16) (-239.71,-125.87) (-239.00,-125.87) (-238.29,-126.57) (-237.59,-127.28) (-236.88,-127.99) (-236.17,-127.99) (-235.47,-128.69) (-248.90,-118.79) (-248.90,-118.09) (-248.90,-117.38) (-248.90,-116.67) (-249.61,-115.97) (-249.61,-115.26) (-249.61,-114.55) (-249.61,-113.84) (-249.61,-113.14) (-249.61,-112.43) (-249.61,-111.72) (-250.32,-111.02) (-250.32,-110.31) (-250.32,-109.60) (-250.32,-108.89) (-250.32,-108.19) (-250.32,-107.48) (-251.02,-106.77) (-251.02,-106.07) (-251.02,-105.36) (-251.02,-104.65) (-251.02,-103.94) (-251.02,-103.24) (-251.02,-102.53) (-251.73,-101.82) (-251.73,-101.12) (-251.73,-100.41) (-251.73,-99.70) (-251.73,-98.99) (-251.73,-98.29) (-251.73,-97.58) (-252.44,-96.87) (-252.44,-96.17) (-252.44,-95.46) (-252.44,-94.75) (-252.44,-94.05) (-252.44,-93.34) (-253.14,-92.63) (-253.14,-91.92) (-253.14,-91.22) (-253.14,-90.51) (-253.14,-89.80) (-253.14,-89.10) (-253.14,-88.39) (-253.85,-87.68) (-253.85,-86.97) (-253.85,-86.27) (-253.85,-85.56) (-296.98,-70.71) (-296.28,-70.71) (-295.57,-71.42) (-294.86,-71.42) (-294.16,-71.42) (-293.45,-72.12) (-292.74,-72.12) (-292.04,-72.12) (-291.33,-72.83) (-290.62,-72.83) (-289.91,-72.83) (-289.21,-73.54) (-288.50,-73.54) (-287.79,-73.54) (-287.09,-74.25) (-286.38,-74.25) (-285.67,-74.95) (-284.96,-74.95) (-284.26,-74.95) (-283.55,-75.66) (-282.84,-75.66) (-282.14,-75.66) (-281.43,-76.37) (-280.72,-76.37) (-280.01,-76.37) (-279.31,-77.07) (-278.60,-77.07) (-277.89,-77.07) (-277.19,-77.78) (-276.48,-77.78) (-275.77,-77.78) (-275.06,-78.49) (-274.36,-78.49) (-273.65,-78.49) (-272.94,-79.20) (-272.24,-79.20) (-271.53,-79.20) (-270.82,-79.90) (-270.11,-79.90) (-269.41,-79.90) (-268.70,-80.61) (-267.99,-80.61) (-267.29,-80.61) (-266.58,-81.32) (-265.87,-81.32) (-265.17,-81.32) (-264.46,-82.02) (-263.75,-82.02) (-263.04,-82.73) (-262.34,-82.73) (-261.63,-82.73) (-260.92,-83.44) (-260.22,-83.44) (-259.51,-83.44) (-258.80,-84.15) (-258.09,-84.15) (-257.39,-84.15) (-256.68,-84.85) (-255.97,-84.85) (-255.27,-84.85) (-254.56,-85.56) (-253.85,-85.56) (-354.97,-80.61) (-354.26,-80.61) (-353.55,-80.61) (-352.85,-79.90) (-352.14,-79.90) (-351.43,-79.90) (-350.72,-79.90) (-350.02,-79.90) (-349.31,-79.90) (-348.60,-79.20) (-347.90,-79.20) (-347.19,-79.20) (-346.48,-79.20) (-345.78,-79.20) (-345.07,-79.20) (-344.36,-78.49) (-343.65,-78.49) (-342.95,-78.49) (-342.24,-78.49) (-341.53,-78.49) (-340.83,-78.49) (-340.12,-77.78) (-339.41,-77.78) (-338.70,-77.78) (-338.00,-77.78) (-337.29,-77.78) (-336.58,-77.78) (-335.88,-77.07) (-335.17,-77.07) (-334.46,-77.07) (-333.75,-77.07) (-333.05,-77.07) (-332.34,-77.07) (-331.63,-76.37) (-330.93,-76.37) (-330.22,-76.37) (-329.51,-76.37) (-328.80,-76.37) (-328.10,-76.37) (-327.39,-75.66) (-326.68,-75.66) (-325.98,-75.66) (-325.27,-75.66) (-324.56,-75.66) (-323.85,-74.95) (-323.15,-74.95) (-322.44,-74.95) (-321.73,-74.95) (-321.03,-74.95) (-320.32,-74.95) (-319.61,-74.25) (-318.91,-74.25) (-318.20,-74.25) (-317.49,-74.25) (-316.78,-74.25) (-316.08,-74.25) (-315.37,-73.54) (-314.66,-73.54) (-313.96,-73.54) (-313.25,-73.54) (-312.54,-73.54) (-311.83,-73.54) (-311.13,-72.83) (-310.42,-72.83) (-309.71,-72.83) (-309.01,-72.83) (-308.30,-72.83) (-307.59,-72.83) (-306.88,-72.12) (-306.18,-72.12) (-305.47,-72.12) (-304.76,-72.12) (-304.06,-72.12) (-303.35,-72.12) (-302.64,-71.42) (-301.93,-71.42) (-301.23,-71.42) (-300.52,-71.42) (-299.81,-71.42) (-299.11,-71.42) (-298.40,-70.71) (-297.69,-70.71) (-296.98,-70.71) (-354.97,-80.61) (-354.97,-79.90) (-355.67,-79.20) (-355.67,-78.49) (-355.67,-80.61) (-355.67,-79.90) (-355.67,-79.20) (-355.67,-78.49) (-355.67,-80.61) (-354.97,-80.61) (-354.26,-80.61) (-353.55,-80.61) (-352.85,-82.73) (-352.85,-82.02) (-353.55,-81.32) (-353.55,-80.61) (-352.85,-82.73) (-352.85,-82.02) (-352.85,-81.32) (-352.85,-80.61) (-353.55,-81.32) (-352.85,-80.61) (-353.55,-83.44) (-353.55,-82.73) (-353.55,-82.02) (-353.55,-81.32) (-353.55,-83.44) (-352.85,-83.44) (-352.14,-83.44) (-351.43,-83.44) (-356.38,-87.68) (-355.67,-86.97) (-354.97,-86.27) (-354.26,-85.56) (-353.55,-85.56) (-352.85,-84.85) (-352.14,-84.15) (-351.43,-83.44) (-356.38,-87.68) (-356.38,-87.68) (-355.67,-86.97) (-354.97,-86.27) (-354.26,-85.56) (-353.55,-84.85) (-352.85,-84.15) (-352.85,-83.44) (-352.14,-82.73) (-351.43,-82.02) (-350.72,-81.32) (-350.02,-80.61) (-349.31,-79.90) (-348.60,-79.20) (-347.90,-78.49) (-347.19,-77.78) (-346.48,-77.07) (-346.48,-76.37) (-345.78,-75.66) (-345.07,-74.95) (-344.36,-74.25) (-343.65,-73.54) (-342.95,-72.83) (-342.24,-72.12) (-341.53,-71.42) (-340.83,-70.71) (-340.12,-70.00) (-339.41,-69.30) (-339.41,-68.59) (-338.70,-67.88) (-338.00,-67.18) (-337.29,-66.47) (-336.58,-65.76) (-335.88,-65.05) (-335.17,-64.35) (-334.46,-63.64) (-333.75,-62.93) (-333.05,-62.23) (-332.34,-61.52) (-332.34,-60.81) (-331.63,-60.10) (-330.93,-59.40) (-330.22,-58.69) (-329.51,-57.98) (-328.80,-57.28) (-328.10,-56.57) (-327.39,-55.86) (-326.68,-55.15) (-325.98,-54.45) (-325.98,-53.74) (-325.27,-53.03) (-324.56,-52.33) (-323.85,-51.62) (-323.15,-50.91) (-322.44,-50.20) (-321.73,-49.50) (-321.03,-48.79) (-320.32,-48.08) (-319.61,-47.38) (-318.91,-46.67) (-318.91,-45.96) (-318.20,-45.25) (-317.49,-44.55) (-316.78,-43.84) (-316.08,-43.13) (-315.37,-42.43) (-315.37,-42.43) (-314.66,-41.72) (-314.66,-41.72) (-314.66,-41.72) (-313.96,-41.72) (-313.25,-41.01) (-312.54,-41.01) (-311.83,-41.01) (-311.13,-40.31) (-310.42,-40.31) (-309.71,-40.31) (-309.01,-39.60) (-308.30,-39.60) (-307.59,-39.60) (-306.88,-38.89) (-306.18,-38.89) (-305.47,-38.89) (-304.76,-38.89) (-304.06,-38.18) (-303.35,-38.18) (-302.64,-38.18) (-301.93,-37.48) (-301.23,-37.48) (-300.52,-37.48) (-299.81,-36.77) (-299.11,-36.77) (-298.40,-36.77) (-297.69,-36.06) (-296.98,-36.06) (-296.28,-36.06) (-295.57,-35.36) (-294.86,-35.36) (-294.86,-35.36) (-302.64,-28.99) (-301.93,-29.70) (-301.23,-30.41) (-300.52,-30.41) (-299.81,-31.11) (-299.11,-31.82) (-298.40,-32.53) (-297.69,-33.23) (-296.98,-33.94) (-296.28,-33.94) (-295.57,-34.65) (-294.86,-35.36) (-322.44,-10.61) (-321.73,-11.31) (-321.03,-12.02) (-320.32,-12.73) (-319.61,-13.44) (-318.91,-14.14) (-318.20,-14.85) (-317.49,-14.85) (-316.78,-15.56) (-316.08,-16.26) (-315.37,-16.97) (-314.66,-17.68) (-313.96,-18.38) (-313.25,-19.09) (-312.54,-19.80) (-311.83,-20.51) (-311.13,-21.21) (-310.42,-21.92) (-309.71,-22.63) (-309.01,-23.33) (-308.30,-24.04) (-307.59,-24.04) (-306.88,-24.75) (-306.18,-25.46) (-305.47,-26.16) (-304.76,-26.87) (-304.06,-27.58) (-303.35,-28.28) (-302.64,-28.99) (-322.44,-10.61) (-321.73,-10.61) (-321.03,-11.31) (-320.32,-11.31) (-319.61,-11.31) (-318.91,-12.02) (-318.20,-12.02) (-317.49,-12.02) (-316.78,-12.73) (-316.08,-12.73) (-315.37,-12.73) (-314.66,-13.44) (-313.96,-13.44) (-313.25,-14.14) (-312.54,-14.14) (-311.83,-14.14) (-311.13,-14.85) (-310.42,-14.85) (-309.71,-14.85) (-309.01,-15.56) (-308.30,-15.56) (-307.59,-15.56) (-306.88,-16.26) (-306.18,-16.26) (-305.47,-16.26) (-304.76,-16.97) (-304.06,-16.97) (-303.35,-16.97) (-302.64,-17.68) (-301.93,-17.68) (-301.23,-17.68) (-300.52,-18.38) (-299.81,-18.38) (-299.11,-18.38) (-298.40,-19.09) (-297.69,-19.09) (-296.98,-19.09) (-296.28,-19.80) (-295.57,-19.80) (-294.86,-20.51) (-294.16,-20.51) (-293.45,-20.51) (-292.74,-21.21) (-292.04,-21.21) (-291.33,-21.21) (-290.62,-21.92) (-289.91,-21.92) (-289.21,-21.92) (-288.50,-22.63) (-287.79,-22.63) (-288.50,-23.33) (-287.79,-22.63) (-288.50,-23.33) (-287.79,-22.63) (-287.09,-21.92) (-286.38,-21.92) (-285.67,-21.21) (-284.96,-20.51) (-284.26,-19.80) (-283.55,-19.09) (-282.84,-18.38) (-282.14,-18.38) (-281.43,-17.68) (-280.72,-16.97) (-280.01,-16.26) (-279.31,-15.56) (-278.60,-15.56) (-277.89,-14.85) (-277.19,-14.14) (-276.48,-13.44) (-275.77,-12.73) (-275.06,-12.02) (-274.36,-12.02) (-273.65,-11.31) (-272.94,-10.61) (-272.24,-9.90) (-271.53,-9.19) (-270.82,-9.19) (-270.11,-8.49) (-269.41,-7.78) (-268.70,-7.07) (-267.99,-6.36) (-267.29,-5.66) (-266.58,-5.66) (-265.87,-4.95) (-265.17,-4.24) (-264.46,-3.54) (-263.75,-2.83) (-263.04,-2.83) (-262.34,-2.12) (-261.63,-1.41) (-260.92,-0.71) (-260.22, 0.00) (-259.51, 0.71) (-258.80, 0.71) (-258.09, 1.41) (-257.39, 2.12) (-256.68, 2.83) (-255.97, 3.54) (-255.27, 3.54) (-254.56, 4.24) (-253.85, 4.95) (-253.14, 5.66) (-252.44, 6.36) (-251.73, 7.07) (-251.02, 7.07) (-250.32, 7.78) (-249.61, 8.49) (-248.90, 9.19) (-248.19, 9.90) (-247.49, 9.90) (-246.78,10.61) (-246.07,11.31) (-245.37,12.02) (-244.66,12.73) (-243.95,13.44) (-243.24,13.44) (-242.54,14.14) (-241.83,14.85) (-241.83,14.85) (-241.12,15.56) (-241.12,16.26) (-240.42,16.97) (-217.79,-36.77) (-217.79,-36.06) (-218.50,-35.36) (-218.50,-34.65) (-219.20,-33.94) (-219.20,-33.23) (-219.91,-32.53) (-219.91,-31.82) (-219.91,-31.11) (-220.62,-30.41) (-220.62,-29.70) (-221.32,-28.99) (-221.32,-28.28) (-221.32,-27.58) (-222.03,-26.87) (-222.03,-26.16) (-222.74,-25.46) (-222.74,-24.75) (-223.45,-24.04) (-223.45,-23.33) (-223.45,-22.63) (-224.15,-21.92) (-224.15,-21.21) (-224.86,-20.51) (-224.86,-19.80) (-225.57,-19.09) (-225.57,-18.38) (-225.57,-17.68) (-226.27,-16.97) (-226.27,-16.26) (-226.98,-15.56) (-226.98,-14.85) (-226.98,-14.14) (-227.69,-13.44) (-227.69,-12.73) (-228.40,-12.02) (-228.40,-11.31) (-229.10,-10.61) (-229.10,-9.90) (-229.10,-9.19) (-229.81,-8.49) (-229.81,-7.78) (-230.52,-7.07) (-230.52,-6.36) (-231.22,-5.66) (-231.22,-4.95) (-231.22,-4.24) (-231.93,-3.54) (-231.93,-2.83) (-232.64,-2.12) (-232.64,-1.41) (-232.64,-0.71) (-233.35, 0.00) (-233.35, 0.71) (-234.05, 1.41) (-234.05, 2.12) (-234.76, 2.83) (-234.76, 3.54) (-234.76, 4.24) (-235.47, 4.95) (-235.47, 5.66) (-236.17, 6.36) (-236.17, 7.07) (-236.88, 7.78) (-236.88, 8.49) (-236.88, 9.19) (-237.59, 9.90) (-237.59,10.61) (-238.29,11.31) (-238.29,12.02) (-238.29,12.73) (-239.00,13.44) (-239.00,14.14) (-239.71,14.85) (-239.71,15.56) (-240.42,16.26) (-240.42,16.97) (-217.79,-36.77) (-217.79,-36.06) (-217.08,-35.36) (-217.08,-34.65) (-217.08,-33.94) (-216.37,-33.23) (-216.37,-32.53) (-216.37,-32.53) (-217.08,-31.82) (-217.08,-31.11) (-217.79,-30.41) (-218.50,-29.70) (-260.92,11.31) (-260.22,10.61) (-259.51, 9.90) (-258.80, 9.19) (-258.09, 8.49) (-257.39, 7.78) (-256.68, 7.07) (-255.97, 6.36) (-255.27, 5.66) (-254.56, 4.95) (-253.85, 4.24) (-253.14, 3.54) (-252.44, 2.83) (-251.73, 2.12) (-251.02, 1.41) (-250.32, 1.41) (-249.61, 0.71) (-248.90, 0.00) (-248.19,-0.71) (-247.49,-1.41) (-246.78,-2.12) (-246.07,-2.83) (-245.37,-3.54) (-244.66,-4.24) (-243.95,-4.95) (-243.24,-5.66) (-242.54,-6.36) (-241.83,-7.07) (-241.12,-7.78) (-240.42,-8.49) (-239.71,-9.19) (-239.00,-9.90) (-238.29,-10.61) (-237.59,-11.31) (-236.88,-12.02) (-236.17,-12.73) (-235.47,-13.44) (-234.76,-14.14) (-234.05,-14.85) (-233.35,-15.56) (-232.64,-16.26) (-231.93,-16.97) (-231.22,-17.68) (-230.52,-18.38) (-229.81,-19.09) (-229.10,-19.09) (-228.40,-19.80) (-227.69,-20.51) (-226.98,-21.21) (-226.27,-21.92) (-225.57,-22.63) (-224.86,-23.33) (-224.15,-24.04) (-223.45,-24.75) (-222.74,-25.46) (-222.03,-26.16) (-221.32,-26.87) (-220.62,-27.58) (-219.91,-28.28) (-219.20,-28.99) (-218.50,-29.70) (-260.92,11.31) (-260.22,11.31) (-259.51,11.31) (-258.80,11.31) (-258.80,11.31) (-258.80,12.02) (-259.51,12.73) (-259.51,13.44) (-260.22,14.14) (-260.22,14.85) (-260.92,15.56) (-260.92,16.26) (-261.63,16.97) (-261.63,17.68) (-262.34,18.38) (-262.34,19.09) (-263.04,19.80) (-263.04,20.51) (-263.75,21.21) (-263.75,21.92) (-264.46,22.63) (-264.46,23.33) (-265.17,24.04) (-265.17,24.75) (-265.87,25.46) (-265.87,26.16) (-266.58,26.87) (-266.58,27.58) (-267.29,28.28) (-267.29,28.99) (-267.99,29.70) (-267.99,30.41) (-268.70,31.11) (-268.70,31.82) (-266.58,26.87) (-266.58,27.58) (-267.29,28.28) (-267.29,28.99) (-267.99,29.70) (-267.99,30.41) (-268.70,31.11) (-268.70,31.82) (-266.58,26.87) (-267.29,27.58) (-267.29,28.28) (-267.99,28.99) (-267.99,28.99) (-267.29,28.28) (-266.58,27.58) (-265.87,26.87) (-267.99,27.58) (-267.29,27.58) (-266.58,26.87) (-265.87,26.87) (-267.99,27.58) (-267.29,27.58) (-266.58,27.58) (-265.87,27.58) (-265.17,27.58) (-264.46,27.58) (-264.46,25.46) (-264.46,26.16) (-264.46,26.87) (-264.46,27.58) (-264.46,25.46) (-263.75,24.75) (-263.04,24.04) (-262.34,24.04) (-261.63,23.33) (-260.92,22.63) (-260.92,21.92) (-260.92,22.63) (-260.92,21.92) (-260.22,21.92) (-259.51,21.92) (-258.80,21.92) (-258.09,21.92) (-257.39,21.21) (-256.68,21.21) (-255.97,21.21) (-255.27,21.21) (-254.56,21.21) (-253.85,21.21) (-253.14,21.21) (-252.44,21.21) (-251.73,21.21) (-251.02,20.51) (-250.32,20.51) (-249.61,20.51) (-248.90,20.51) (-248.19,20.51) (-247.49,20.51) (-246.78,20.51) (-246.07,20.51) (-245.37,19.80) (-244.66,19.80) (-243.95,19.80) (-243.24,19.80) (-242.54,19.80) (-242.54,19.80) (-241.83,19.80) (-241.12,19.80) (-240.42,19.80) (-239.71,19.80) (-239.00,20.51) (-238.29,20.51) (-237.59,20.51) (-236.88,20.51) (-236.17,20.51) (-235.47,20.51) (-234.76,20.51) (-234.05,20.51) (-233.35,20.51) (-232.64,21.21) (-231.93,21.21) (-231.22,21.21) (-230.52,21.21) (-229.81,21.21) (-229.10,21.21) (-228.40,21.21) (-227.69,21.21) (-226.98,21.21) (-226.27,21.92) (-225.57,21.92) (-224.86,21.92) (-224.15,21.92) (-223.45,21.92) (-222.74,21.92) (-222.03,21.92) (-221.32,21.92) (-220.62,21.92) (-219.91,22.63) (-219.20,22.63) (-218.50,22.63) (-217.79,22.63) (-217.08,22.63) (-216.37,22.63) (-215.67,22.63) (-214.96,22.63) (-214.25,22.63) (-213.55,23.33) (-212.84,23.33) (-212.13,23.33) (-211.42,23.33) (-210.72,23.33) (-238.29,-7.78) (-237.59,-7.07) (-236.88,-6.36) (-236.17,-5.66) (-235.47,-4.95) (-235.47,-4.24) (-234.76,-3.54) (-234.05,-2.83) (-233.35,-2.12) (-232.64,-1.41) (-231.93,-0.71) (-231.22, 0.00) (-230.52, 0.71) (-229.81, 1.41) (-229.81, 2.12) (-229.10, 2.83) (-228.40, 3.54) (-227.69, 4.24) (-226.98, 4.95) (-226.27, 5.66) (-225.57, 6.36) (-224.86, 7.07) (-224.86, 7.78) (-224.15, 8.49) (-223.45, 9.19) (-222.74, 9.90) (-222.03,10.61) (-221.32,11.31) (-220.62,12.02) (-219.91,12.73) (-219.20,13.44) (-219.20,14.14) (-218.50,14.85) (-217.79,15.56) (-217.08,16.26) (-216.37,16.97) (-215.67,17.68) (-214.96,18.38) (-214.25,19.09) (-213.55,19.80) (-213.55,20.51) (-212.84,21.21) (-212.13,21.92) (-211.42,22.63) (-210.72,23.33) (-238.29,-7.78) (-237.59,-7.78) (-237.59,-7.78) (-237.59,-7.07) (-237.59,-6.36) (-247.49, 0.00) (-246.78,-0.71) (-246.07,-0.71) (-245.37,-1.41) (-244.66,-2.12) (-243.95,-2.12) (-243.24,-2.83) (-242.54,-2.83) (-241.83,-3.54) (-241.12,-4.24) (-240.42,-4.24) (-239.71,-4.95) (-239.00,-5.66) (-238.29,-5.66) (-237.59,-6.36) (-247.49, 0.00) (-246.78, 0.00) (-248.90, 0.00) (-248.19, 0.00) (-247.49, 0.00) (-246.78, 0.00) (-248.90, 0.00) (-248.19, 0.00) (-247.49, 0.71) (-246.78, 0.71) (-246.07, 0.71) (-245.37, 1.41) (-244.66, 1.41) (-244.66, 1.41) (-243.95, 1.41) (-243.24, 1.41) (-242.54, 1.41) (-241.83, 0.00) (-241.83, 0.71) (-242.54, 1.41) (-241.83, 0.00) (-241.12, 0.00) (-240.42, 0.71) (-240.42, 0.71) (-239.71, 1.41) (-239.00, 2.12) (-238.29, 2.83) (-238.29, 2.83) (-247.49, 7.78) (-246.78, 7.07) (-246.07, 7.07) (-245.37, 6.36) (-244.66, 6.36) (-243.95, 5.66) (-243.24, 5.66) (-242.54, 4.95) (-241.83, 4.95) (-241.12, 4.24) (-240.42, 4.24) (-239.71, 3.54) (-239.00, 3.54) (-238.29, 2.83) (-247.49, 7.78) (-246.78, 7.78) (-246.07, 7.78) (-245.37, 7.78) (-244.66, 7.78) (-243.95, 7.78) (-243.95, 7.78) (-243.24, 7.07) (-242.54, 6.36) (-242.54, 6.36) (-242.54, 7.07) (-241.83, 7.78) (-241.83, 8.49) (-243.95, 8.49) (-243.24, 8.49) (-242.54, 8.49) (-241.83, 8.49) (-243.95, 8.49) (-243.95, 8.49) (-243.95, 9.19) (-244.66, 9.90) (-244.66,10.61) (-245.37,11.31) (-245.37,12.02) (-245.37,12.73) (-246.07,13.44) (-246.07,14.14) (-246.07,14.85) (-246.78,15.56) (-246.78,16.26) (-247.49,16.97) (-247.49,17.68) (-247.49,17.68) (-247.49,18.38) (-247.49,19.09) (-247.49,19.80) (-247.49,20.51) (-247.49,21.21) (-247.49,21.92) (-248.19,22.63) (-248.19,23.33) (-248.19,24.04) (-248.19,24.75) (-248.19,25.46) (-248.19,26.16) (-248.19,26.87) (-248.19,27.58) (-248.19,28.28) (-248.19,28.99) (-248.19,29.70) (-248.19,30.41) (-248.19,31.11) (-248.19,31.82) (-248.90,32.53) (-248.90,33.23) (-248.90,33.94) (-248.90,34.65) (-248.90,35.36) (-248.90,36.06) (-248.90,36.77) (-248.90,35.36) (-248.90,36.06) (-248.90,36.77) (-249.61,35.36) (-248.90,35.36) (-249.61,35.36) (-249.61,36.06) (-249.61,36.77) (-250.32,34.65) (-250.32,35.36) (-249.61,36.06) (-249.61,36.77) (-251.73,30.41) (-251.73,31.11) (-251.02,31.82) (-251.02,32.53) (-251.02,33.23) (-250.32,33.94) (-250.32,34.65) (-251.73,30.41) (-251.02,30.41) (-250.32,30.41) (-249.61,29.70) (-248.90,29.70) (-368,-165)[(0,0)\[lt\][$\rho: 0.5$]{}]{}
&
(612,950)(-544,-289) (-544,-289)(612,0)[2]{}[(0,1)[850]{}]{} (-544,-289)(0,850)[2]{}[(1,0)[612]{}]{} (-408,408) (-204,-119) (-34,0) (-136,-68) (-221,-102) (-459,306) (-272,-153) (-374,306) (-493,153) (-442,425) (-272,374) (-221,391) (-255,391) (-136,17) (-170,476) (-510,170) (-238,340) (-425,289) (-17,0) (-153,-34) (-153,493) (-170,-85) (-408,357) (-204,-34) (-374,408) (0,-272) (-306,255) (-459,357) (-340,187) (-272,85) (-476,425) (-425,374) (0,-221) (-68,-238) (51,34) (-153,-272) (-102,17) (-85,-187) (-204,442) (-34,-17) (-510,187) (-391,153) (-187,-17) (-493,306) (-374,187) (-272,221) (-306,136) (-238,0) (-289,221) (-510,306) (-68,-153) (51,17) (-408,153) (-459,102) (-102,-255) (-476,306) (-187,-255) (-221,-34) (-306,374) (-442,323) (-425,255) (-187,-187) (-323,272) (-272,425) (-357,289) (-442,357) (-459,170) (-102,-238) (17,-187) (-187,-204) (-136,-51) (-119,-255) (-306,323) (17,-17) (-442,340) (-255,17) (-272,391) (-306,102) (-357,442) (-255,374) (-408,255) (34,34) (-238,357) (-170,0) (-476,187) (-85,0) (-187,-102) (-425,102) (-374,170) (-102,-153) (-17,34) (-170,-289) (-153,-17) (0,-255) (-306,204) (-459,340) (-442,306) (-34,68) (-442,459) (-187,442) (17,0) (-374,357) (-68,-221) (-170,510) (-221,-136) (-272,408) (-408,221) (-476,136) (-136,527) (-510,136) (-102,-51) (34,-204) (-408,323) (-204,-68) (-459,442) (-221,-17) (-221,-153) (-442,289) (0,34) (-425,340) (-323,204) (-340,340) (-204,476) (-68,-272) (-51,-17) (-17,-204) (-289,153) (-102,51) (-85,-221) (-204,-17) (-170,-221) (-68,34) (-102,-34) (-408,493) (-493,272) (-119,-187) (-391,408) (17,-34) (-357,221) (-306,170) (-34,-289) (-340,238) (-425,459) (-51,-204) (-204,17) (-391,68) (-17,-255) (-221,-119) (-85,-17) (-170,-68) (-102,-221) (17,-204) (-136,-34) (-119,-272) (0,-187) (-306,272) (-204,408) (-153,544) (-374,391) (-238,374) (-442,68) (-170,17) (-34,-238) (-68,68) (-119,-17) (-408,391) (-34,51) (-391,102) (-391,374) (-238,408) (-306,221) (-459,323) (-255,-136) (-272,357) (-442,408) (-323,136) (-357,408) (-136,0) (-170,527) (-391,136) (-425,170) (-187,-68) (-442,374) (-459,255) (-391,476) (-187,-51) (-204,-51) (-459,425) (-221,425) (-255,-170) (-289,306) (-340,170) (-221,-68) (0,51) (-289,255) (-238,425) (-187,0) (-493,221) (-102,0) (-408,187) (-408,204) (-425,391) (34,-170) (-51,34) (-153,-289) (-119,-204) (-408,289) (-425,85) (-476,340) (-442,187) (-340,374) (-221,-187) (-306,187) (-204,391) (0,0) (-289,34) (-204,-102) (-34,17) (-34,-170) (-153,-204) (-17,-221) (-85,-255) (-68,0) (-391,442) (0,-170) (-204,425) (-136,-238) (-306,68) (-340,204) (34,0) (-425,442) (-544,170) (-527,221) (-34,-221) (-408,85) (40.31, 9.90) (40.31,10.61) (41.01,11.31) (41.01,12.02) (41.72,12.73) (41.72,13.44) (42.43,14.14) (42.43,14.85) (43.13,15.56) (43.13,16.26) (43.84,16.97) (43.84,17.68) (44.55,18.38) (44.55,19.09) (44.55,19.80) (45.25,20.51) (45.25,21.21) (45.96,21.92) (45.96,22.63) (46.67,23.33) (46.67,24.04) (47.38,24.75) (47.38,25.46) (48.08,26.16) (48.08,26.87) (48.79,27.58) (48.79,28.28) (49.50,28.99) (49.50,29.70) (49.50,30.41) (50.20,31.11) (50.20,31.82) (50.91,32.53) (50.91,33.23) (51.62,33.94) (51.62,34.65) (52.33,35.36) (52.33,36.06) (53.03,36.77) (53.03,37.48) (53.74,38.18) (53.74,38.89) (33.94,10.61) (34.65,11.31) (34.65,12.02) (35.36,12.73) (36.06,13.44) (36.06,14.14) (36.77,14.85) (37.48,15.56) (38.18,16.26) (38.18,16.97) (38.89,17.68) (39.60,18.38) (39.60,19.09) (40.31,19.80) (41.01,20.51) (41.01,21.21) (41.72,21.92) (42.43,22.63) (43.13,23.33) (43.13,24.04) (43.84,24.75) (44.55,25.46) (44.55,26.16) (45.25,26.87) (45.96,27.58) (45.96,28.28) (46.67,28.99) (47.38,29.70) (48.08,30.41) (48.08,31.11) (48.79,31.82) (49.50,32.53) (49.50,33.23) (50.20,33.94) (50.91,34.65) (50.91,35.36) (51.62,36.06) (52.33,36.77) (53.03,37.48) (53.03,38.18) (53.74,38.89) (33.94,10.61) (33.94,11.31) (34.65,12.02) (34.65,12.73) (35.36,13.44) (35.36,14.14) (35.36,14.85) (36.06,15.56) (36.06,16.26) (36.06,16.97) (36.77,17.68) (36.77,18.38) (37.48,19.09) (37.48,19.80) (37.48,20.51) (38.18,21.21) (38.18,21.92) (38.89,22.63) (38.89,23.33) (38.89,24.04) (39.60,24.75) (39.60,25.46) (39.60,26.16) (40.31,26.87) (40.31,27.58) (41.01,28.28) (41.01,28.99) (41.01,29.70) (41.72,30.41) (41.72,31.11) (42.43,31.82) (42.43,32.53) (42.43,33.23) (43.13,33.94) (43.13,34.65) (43.84,35.36) (43.84,36.06) (43.84,36.77) (44.55,37.48) (44.55,38.18) (44.55,38.89) (45.25,39.60) (45.25,40.31) (45.96,41.01) (45.96,41.72) (20.51,15.56) (21.21,16.26) (21.92,16.97) (22.63,17.68) (23.33,18.38) (24.04,19.09) (24.75,19.80) (25.46,20.51) (26.16,21.21) (26.87,21.92) (27.58,22.63) (28.28,23.33) (28.99,24.04) (29.70,24.75) (30.41,25.46) (31.11,26.16) (31.82,26.87) (32.53,27.58) (33.23,28.28) (33.23,28.99) (33.94,29.70) (34.65,30.41) (35.36,31.11) (36.06,31.82) (36.77,32.53) (37.48,33.23) (38.18,33.94) (38.89,34.65) (39.60,35.36) (40.31,36.06) (41.01,36.77) (41.72,37.48) (42.43,38.18) (43.13,38.89) (43.84,39.60) (44.55,40.31) (45.25,41.01) (45.96,41.72) (20.51,15.56) (21.21,16.26) (21.92,16.97) (22.63,17.68) (23.33,17.68) (24.04,18.38) (24.75,19.09) (25.46,19.80) (26.16,20.51) (26.87,21.21) (27.58,21.21) (28.28,21.92) (28.99,22.63) (29.70,23.33) (30.41,24.04) (31.11,24.75) (31.82,24.75) (32.53,25.46) (33.23,26.16) (33.94,26.87) (34.65,27.58) (35.36,28.28) (36.06,28.28) (36.77,28.99) (37.48,29.70) (38.18,30.41) (38.89,31.11) (39.60,31.82) (40.31,31.82) (41.01,32.53) (41.72,33.23) (42.43,33.94) (43.13,34.65) (43.84,35.36) (44.55,35.36) (45.25,36.06) (45.96,36.77) (46.67,37.48) (12.73,37.48) (13.44,37.48) (14.14,37.48) (14.85,37.48) (15.56,37.48) (16.26,37.48) (16.97,37.48) (17.68,37.48) (18.38,37.48) (19.09,37.48) (19.80,37.48) (20.51,37.48) (21.21,37.48) (21.92,37.48) (22.63,37.48) (23.33,37.48) (24.04,37.48) (24.75,37.48) (25.46,37.48) (26.16,37.48) (26.87,37.48) (27.58,37.48) (28.28,37.48) (28.99,37.48) (29.70,37.48) (30.41,37.48) (31.11,37.48) (31.82,37.48) (32.53,37.48) (33.23,37.48) (33.94,37.48) (34.65,37.48) (35.36,37.48) (36.06,37.48) (36.77,37.48) (37.48,37.48) (38.18,37.48) (38.89,37.48) (39.60,37.48) (40.31,37.48) (41.01,37.48) (41.72,37.48) (42.43,37.48) (43.13,37.48) (43.84,37.48) (44.55,37.48) (45.25,37.48) (45.96,37.48) (46.67,37.48) ( 1.41, 5.66) ( 1.41, 6.36) ( 2.12, 7.07) ( 2.12, 7.78) ( 2.12, 8.49) ( 2.83, 9.19) ( 2.83, 9.90) ( 2.83,10.61) ( 3.54,11.31) ( 3.54,12.02) ( 4.24,12.73) ( 4.24,13.44) ( 4.24,14.14) ( 4.95,14.85) ( 4.95,15.56) ( 4.95,16.26) ( 5.66,16.97) ( 5.66,17.68) ( 5.66,18.38) ( 6.36,19.09) ( 6.36,19.80) ( 6.36,20.51) ( 7.07,21.21) ( 7.07,21.92) ( 7.78,22.63) ( 7.78,23.33) ( 7.78,24.04) ( 8.49,24.75) ( 8.49,25.46) ( 8.49,26.16) ( 9.19,26.87) ( 9.19,27.58) ( 9.19,28.28) ( 9.90,28.99) ( 9.90,29.70) ( 9.90,30.41) (10.61,31.11) (10.61,31.82) (11.31,32.53) (11.31,33.23) (11.31,33.94) (12.02,34.65) (12.02,35.36) (12.02,36.06) (12.73,36.77) (12.73,37.48) ( 1.41, 5.66) ( 1.41, 6.36) ( 0.71, 7.07) ( 0.71, 7.78) ( 0.00, 8.49) ( 0.00, 9.19) (-0.71, 9.90) (-0.71,10.61) (-1.41,11.31) (-1.41,12.02) (-2.12,12.73) (-2.12,13.44) (-2.83,14.14) (-2.83,14.85) (-3.54,15.56) (-3.54,16.26) (-4.24,16.97) (-4.24,17.68) (-4.95,18.38) (-4.95,19.09) (-5.66,19.80) (-5.66,20.51) (-6.36,21.21) (-6.36,21.92) (-7.07,22.63) (-7.07,23.33) (-7.78,24.04) (-7.78,24.75) (-8.49,25.46) (-8.49,26.16) (-9.19,26.87) (-9.19,27.58) (-9.90,28.28) (-9.90,28.99) (-10.61,29.70) (-10.61,30.41) (-11.31,31.11) (-11.31,31.82) (-12.02,32.53) (-12.02,33.23) (-12.73,33.94) (-12.73,34.65) (-12.73,34.65) (-12.02,35.36) (-11.31,36.06) (-10.61,36.77) (-9.90,37.48) (-9.19,38.18) (-8.49,38.89) (-7.78,39.60) (-7.07,40.31) (-6.36,41.01) (-5.66,41.72) (-4.95,42.43) (-4.24,43.13) (-3.54,43.84) (-2.83,44.55) (-2.12,45.25) (-1.41,45.96) (-0.71,46.67) ( 0.00,46.67) ( 0.71,47.38) ( 1.41,48.08) ( 2.12,48.79) ( 2.83,49.50) ( 3.54,50.20) ( 4.24,50.91) ( 4.95,51.62) ( 5.66,52.33) ( 6.36,53.03) ( 7.07,53.74) ( 7.78,54.45) ( 8.49,55.15) ( 9.19,55.86) ( 9.90,56.57) (10.61,57.28) (11.31,57.98) (12.02,58.69) (12.73,59.40) (-19.80,56.57) (-19.09,56.57) (-18.38,56.57) (-17.68,56.57) (-16.97,56.57) (-16.26,56.57) (-15.56,57.28) (-14.85,57.28) (-14.14,57.28) (-13.44,57.28) (-12.73,57.28) (-12.02,57.28) (-11.31,57.28) (-10.61,57.28) (-9.90,57.28) (-9.19,57.28) (-8.49,57.28) (-7.78,57.28) (-7.07,57.98) (-6.36,57.98) (-5.66,57.98) (-4.95,57.98) (-4.24,57.98) (-3.54,57.98) (-2.83,57.98) (-2.12,57.98) (-1.41,57.98) (-0.71,57.98) ( 0.00,57.98) ( 0.71,58.69) ( 1.41,58.69) ( 2.12,58.69) ( 2.83,58.69) ( 3.54,58.69) ( 4.24,58.69) ( 4.95,58.69) ( 5.66,58.69) ( 6.36,58.69) ( 7.07,58.69) ( 7.78,58.69) ( 8.49,58.69) ( 9.19,59.40) ( 9.90,59.40) (10.61,59.40) (11.31,59.40) (12.02,59.40) (12.73,59.40) (-21.92,20.51) (-21.92,21.21) (-21.92,21.92) (-21.92,22.63) (-21.92,23.33) (-21.92,24.04) (-21.92,24.75) (-21.92,25.46) (-21.92,26.16) (-21.21,26.87) (-21.21,27.58) (-21.21,28.28) (-21.21,28.99) (-21.21,29.70) (-21.21,30.41) (-21.21,31.11) (-21.21,31.82) (-21.21,32.53) (-21.21,33.23) (-21.21,33.94) (-21.21,34.65) (-21.21,35.36) (-21.21,36.06) (-21.21,36.77) (-21.21,37.48) (-21.21,38.18) (-20.51,38.89) (-20.51,39.60) (-20.51,40.31) (-20.51,41.01) (-20.51,41.72) (-20.51,42.43) (-20.51,43.13) (-20.51,43.84) (-20.51,44.55) (-20.51,45.25) (-20.51,45.96) (-20.51,46.67) (-20.51,47.38) (-20.51,48.08) (-20.51,48.79) (-20.51,49.50) (-20.51,50.20) (-19.80,50.91) (-19.80,51.62) (-19.80,52.33) (-19.80,53.03) (-19.80,53.74) (-19.80,54.45) (-19.80,55.15) (-19.80,55.86) (-19.80,56.57) (-21.92,20.51) (-22.63,21.21) (-23.33,21.92) (-23.33,22.63) (-24.04,23.33) (-24.75,24.04) (-25.46,24.75) (-25.46,25.46) (-26.16,26.16) (-26.87,26.87) (-27.58,27.58) (-27.58,28.28) (-28.28,28.99) (-28.99,29.70) (-29.70,30.41) (-29.70,31.11) (-30.41,31.82) (-31.11,32.53) (-31.82,33.23) (-31.82,33.94) (-32.53,34.65) (-33.23,35.36) (-33.94,36.06) (-34.65,36.77) (-34.65,37.48) (-35.36,38.18) (-36.06,38.89) (-36.77,39.60) (-36.77,40.31) (-37.48,41.01) (-38.18,41.72) (-38.89,42.43) (-38.89,43.13) (-39.60,43.84) (-40.31,44.55) (-41.01,45.25) (-41.01,45.96) (-41.72,46.67) (-42.43,47.38) (-42.43,47.38) (-41.72,48.08) (-41.72,48.79) (-41.01,49.50) (-41.01,50.20) (-40.31,50.91) (-39.60,51.62) (-39.60,52.33) (-38.89,53.03) (-38.89,53.74) (-38.18,54.45) (-37.48,55.15) (-37.48,55.86) (-36.77,56.57) (-36.77,57.28) (-36.06,57.98) (-35.36,58.69) (-35.36,59.40) (-34.65,60.10) (-34.65,60.81) (-33.94,61.52) (-33.23,62.23) (-33.23,62.93) (-32.53,63.64) (-32.53,64.35) (-31.82,65.05) (-31.11,65.76) (-31.11,66.47) (-30.41,67.18) (-30.41,67.88) (-29.70,68.59) (-28.99,69.30) (-28.99,70.00) (-28.28,70.71) (-28.28,71.42) (-27.58,72.12) (-26.87,72.83) (-26.87,73.54) (-26.16,74.25) (-26.16,74.95) (-25.46,75.66) (-57.98,68.59) (-57.28,68.59) (-56.57,68.59) (-55.86,69.30) (-55.15,69.30) (-54.45,69.30) (-53.74,69.30) (-53.03,70.00) (-52.33,70.00) (-51.62,70.00) (-50.91,70.00) (-50.20,70.00) (-49.50,70.71) (-48.79,70.71) (-48.08,70.71) (-47.38,70.71) (-46.67,70.71) (-45.96,71.42) (-45.25,71.42) (-44.55,71.42) (-43.84,71.42) (-43.13,72.12) (-42.43,72.12) (-41.72,72.12) (-41.01,72.12) (-40.31,72.12) (-39.60,72.83) (-38.89,72.83) (-38.18,72.83) (-37.48,72.83) (-36.77,73.54) (-36.06,73.54) (-35.36,73.54) (-34.65,73.54) (-33.94,73.54) (-33.23,74.25) (-32.53,74.25) (-31.82,74.25) (-31.11,74.25) (-30.41,74.25) (-29.70,74.95) (-28.99,74.95) (-28.28,74.95) (-27.58,74.95) (-26.87,75.66) (-26.16,75.66) (-25.46,75.66) (-45.96,34.65) (-45.96,35.36) (-46.67,36.06) (-46.67,36.77) (-46.67,37.48) (-47.38,38.18) (-47.38,38.89) (-47.38,39.60) (-48.08,40.31) (-48.08,41.01) (-48.79,41.72) (-48.79,42.43) (-48.79,43.13) (-49.50,43.84) (-49.50,44.55) (-49.50,45.25) (-50.20,45.96) (-50.20,46.67) (-50.20,47.38) (-50.91,48.08) (-50.91,48.79) (-50.91,49.50) (-51.62,50.20) (-51.62,50.91) (-51.62,51.62) (-52.33,52.33) (-52.33,53.03) (-53.03,53.74) (-53.03,54.45) (-53.03,55.15) (-53.74,55.86) (-53.74,56.57) (-53.74,57.28) (-54.45,57.98) (-54.45,58.69) (-54.45,59.40) (-55.15,60.10) (-55.15,60.81) (-55.15,61.52) (-55.86,62.23) (-55.86,62.93) (-56.57,63.64) (-56.57,64.35) (-56.57,65.05) (-57.28,65.76) (-57.28,66.47) (-57.28,67.18) (-57.98,67.88) (-57.98,68.59) (-62.93, 4.95) (-62.23, 5.66) (-62.23, 6.36) (-61.52, 7.07) (-61.52, 7.78) (-60.81, 8.49) (-60.81, 9.19) (-60.10, 9.90) (-59.40,10.61) (-59.40,11.31) (-58.69,12.02) (-58.69,12.73) (-57.98,13.44) (-57.98,14.14) (-57.28,14.85) (-56.57,15.56) (-56.57,16.26) (-55.86,16.97) (-55.86,17.68) (-55.15,18.38) (-55.15,19.09) (-54.45,19.80) (-53.74,20.51) (-53.74,21.21) (-53.03,21.92) (-53.03,22.63) (-52.33,23.33) (-52.33,24.04) (-51.62,24.75) (-50.91,25.46) (-50.91,26.16) (-50.20,26.87) (-50.20,27.58) (-49.50,28.28) (-49.50,28.99) (-48.79,29.70) (-48.08,30.41) (-48.08,31.11) (-47.38,31.82) (-47.38,32.53) (-46.67,33.23) (-46.67,33.94) (-45.96,34.65) (-62.93, 4.95) (-62.93, 5.66) (-62.93, 6.36) (-62.93, 7.07) (-62.23, 7.78) (-62.23, 8.49) (-62.23, 9.19) (-62.23, 9.90) (-62.23,10.61) (-62.23,11.31) (-61.52,12.02) (-61.52,12.73) (-61.52,13.44) (-61.52,14.14) (-61.52,14.85) (-61.52,15.56) (-61.52,16.26) (-60.81,16.97) (-60.81,17.68) (-60.81,18.38) (-60.81,19.09) (-60.81,19.80) (-60.81,20.51) (-60.10,21.21) (-60.10,21.92) (-60.10,22.63) (-60.10,23.33) (-60.10,24.04) (-60.10,24.75) (-59.40,25.46) (-59.40,26.16) (-59.40,26.87) (-59.40,27.58) (-59.40,28.28) (-59.40,28.99) (-59.40,29.70) (-58.69,30.41) (-58.69,31.11) (-58.69,31.82) (-58.69,32.53) (-58.69,33.23) (-58.69,33.94) (-57.98,34.65) (-57.98,35.36) (-57.98,36.06) (-57.98,36.77) (-64.35, 1.41) (-64.35, 2.12) (-64.35, 2.83) (-63.64, 3.54) (-63.64, 4.24) (-63.64, 4.95) (-63.64, 5.66) (-63.64, 6.36) (-63.64, 7.07) (-62.93, 7.78) (-62.93, 8.49) (-62.93, 9.19) (-62.93, 9.90) (-62.93,10.61) (-62.23,11.31) (-62.23,12.02) (-62.23,12.73) (-62.23,13.44) (-62.23,14.14) (-62.23,14.85) (-61.52,15.56) (-61.52,16.26) (-61.52,16.97) (-61.52,17.68) (-61.52,18.38) (-61.52,19.09) (-60.81,19.80) (-60.81,20.51) (-60.81,21.21) (-60.81,21.92) (-60.81,22.63) (-60.10,23.33) (-60.10,24.04) (-60.10,24.75) (-60.10,25.46) (-60.10,26.16) (-60.10,26.87) (-59.40,27.58) (-59.40,28.28) (-59.40,28.99) (-59.40,29.70) (-59.40,30.41) (-58.69,31.11) (-58.69,31.82) (-58.69,32.53) (-58.69,33.23) (-58.69,33.94) (-58.69,34.65) (-57.98,35.36) (-57.98,36.06) (-57.98,36.77) (-90.51,21.92) (-89.80,21.21) (-89.10,20.51) (-88.39,20.51) (-87.68,19.80) (-86.97,19.09) (-86.27,18.38) (-85.56,18.38) (-84.85,17.68) (-84.15,16.97) (-83.44,16.26) (-82.73,15.56) (-82.02,15.56) (-81.32,14.85) (-80.61,14.14) (-79.90,13.44) (-79.20,12.73) (-78.49,12.73) (-77.78,12.02) (-77.07,11.31) (-76.37,10.61) (-75.66,10.61) (-74.95, 9.90) (-74.25, 9.19) (-73.54, 8.49) (-72.83, 7.78) (-72.12, 7.78) (-71.42, 7.07) (-70.71, 6.36) (-70.00, 5.66) (-69.30, 4.95) (-68.59, 4.95) (-67.88, 4.24) (-67.18, 3.54) (-66.47, 2.83) (-65.76, 2.83) (-65.05, 2.12) (-64.35, 1.41) (-90.51,21.92) (-90.51,22.63) (-90.51,23.33) (-90.51,24.04) (-90.51,24.75) (-90.51,25.46) (-90.51,26.16) (-89.80,26.87) (-89.80,27.58) (-89.80,28.28) (-89.80,28.99) (-89.80,29.70) (-89.80,30.41) (-89.80,31.11) (-89.80,31.82) (-89.80,32.53) (-89.80,33.23) (-89.80,33.94) (-89.80,34.65) (-89.10,35.36) (-89.10,36.06) (-89.10,36.77) (-89.10,37.48) (-89.10,38.18) (-89.10,38.89) (-89.10,39.60) (-89.10,40.31) (-89.10,41.01) (-89.10,41.72) (-89.10,42.43) (-89.10,43.13) (-88.39,43.84) (-88.39,44.55) (-88.39,45.25) (-88.39,45.96) (-88.39,46.67) (-88.39,47.38) (-88.39,48.08) (-88.39,48.79) (-88.39,49.50) (-88.39,50.20) (-88.39,50.91) (-88.39,51.62) (-87.68,52.33) (-87.68,53.03) (-87.68,53.74) (-87.68,54.45) (-87.68,55.15) (-87.68,55.86) (-87.68,55.86) (-86.97,55.15) (-86.27,54.45) (-85.56,53.74) (-84.85,53.74) (-84.15,53.03) (-83.44,52.33) (-82.73,51.62) (-82.02,50.91) (-81.32,50.20) (-80.61,50.20) (-79.90,49.50) (-79.20,48.79) (-78.49,48.08) (-77.78,47.38) (-77.07,46.67) (-76.37,46.67) (-75.66,45.96) (-74.95,45.25) (-74.25,44.55) (-73.54,43.84) (-72.83,43.13) (-72.12,43.13) (-71.42,42.43) (-70.71,41.72) (-70.00,41.01) (-69.30,40.31) (-68.59,39.60) (-67.88,39.60) (-67.18,38.89) (-66.47,38.18) (-65.76,37.48) (-65.05,36.77) (-64.35,36.06) (-63.64,36.06) (-62.93,35.36) (-62.23,34.65) (-61.52,33.94) (-61.52,33.94) (-60.81,33.23) (-60.10,33.23) (-59.40,32.53) (-58.69,31.82) (-57.98,31.82) (-57.28,31.11) (-56.57,30.41) (-55.86,30.41) (-55.15,29.70) (-54.45,28.99) (-53.74,28.99) (-53.03,28.28) (-52.33,27.58) (-51.62,27.58) (-50.91,26.87) (-50.20,26.16) (-49.50,26.16) (-48.79,25.46) (-48.08,24.75) (-47.38,24.75) (-46.67,24.04) (-45.96,24.04) (-45.25,23.33) (-44.55,22.63) (-43.84,22.63) (-43.13,21.92) (-42.43,21.21) (-41.72,21.21) (-41.01,20.51) (-40.31,19.80) (-39.60,19.80) (-38.89,19.09) (-38.18,18.38) (-37.48,18.38) (-36.77,17.68) (-36.06,16.97) (-35.36,16.97) (-34.65,16.26) (-33.94,15.56) (-33.23,15.56) (-32.53,14.85) (-32.53,14.85) (-31.82,14.85) (-31.11,14.14) (-30.41,14.14) (-29.70,13.44) (-28.99,13.44) (-28.28,12.73) (-27.58,12.73) (-26.87,12.02) (-26.16,12.02) (-25.46,11.31) (-24.75,11.31) (-24.04,10.61) (-23.33,10.61) (-22.63, 9.90) (-21.92, 9.90) (-21.21, 9.19) (-20.51, 9.19) (-19.80, 8.49) (-19.09, 8.49) (-18.38, 7.78) (-17.68, 7.78) (-16.97, 7.07) (-16.26, 7.07) (-15.56, 6.36) (-14.85, 6.36) (-14.14, 5.66) (-13.44, 5.66) (-12.73, 4.95) (-12.02, 4.95) (-11.31, 4.24) (-10.61, 4.24) (-9.90, 3.54) (-9.19, 3.54) (-8.49, 2.83) (-7.78, 2.83) (-7.07, 2.12) (-6.36, 2.12) (-5.66, 1.41) (-4.95, 1.41) (-4.24, 0.71) (-3.54, 0.71) (-2.83, 0.00) (-2.83, 0.00) (-2.12, 0.00) (-1.41, 0.00) (-0.71,-0.71) ( 0.00,-0.71) ( 0.71,-0.71) ( 1.41,-0.71) ( 2.12,-0.71) ( 2.83,-0.71) ( 3.54,-1.41) ( 4.24,-1.41) ( 4.95,-1.41) ( 5.66,-1.41) ( 6.36,-1.41) ( 7.07,-1.41) ( 7.78,-2.12) ( 8.49,-2.12) ( 9.19,-2.12) ( 9.90,-2.12) (10.61,-2.12) (11.31,-2.12) (12.02,-2.83) (12.73,-2.83) (13.44,-2.83) (14.14,-2.83) (14.85,-2.83) (15.56,-3.54) (16.26,-3.54) (16.97,-3.54) (17.68,-3.54) (18.38,-3.54) (19.09,-3.54) (19.80,-4.24) (20.51,-4.24) (21.21,-4.24) (21.92,-4.24) (22.63,-4.24) (23.33,-4.24) (24.04,-4.95) (24.75,-4.95) (25.46,-4.95) (26.16,-4.95) (26.87,-4.95) (27.58,-4.95) (28.28,-5.66) (28.99,-5.66) (29.70,-5.66) (29.70,-5.66) (30.41,-4.95) (31.11,-4.24) (31.82,-3.54) (32.53,-2.83) (33.23,-2.12) (33.23,-1.41) (33.94,-0.71) (34.65, 0.00) (35.36, 0.71) (36.06, 1.41) (36.77, 2.12) (37.48, 2.83) (38.18, 3.54) (38.89, 4.24) (39.60, 4.95) (40.31, 5.66) (40.31, 6.36) (41.01, 7.07) (41.72, 7.78) (42.43, 8.49) (43.13, 9.19) (43.84, 9.90) (44.55,10.61) (45.25,11.31) (45.96,12.02) (46.67,12.73) (47.38,13.44) (47.38,14.14) (48.08,14.85) (48.79,15.56) (49.50,16.26) (50.20,16.97) (50.91,17.68) (17.68, 5.66) (18.38, 5.66) (19.09, 6.36) (19.80, 6.36) (20.51, 6.36) (21.21, 7.07) (21.92, 7.07) (22.63, 7.78) (23.33, 7.78) (24.04, 7.78) (24.75, 8.49) (25.46, 8.49) (26.16, 8.49) (26.87, 9.19) (27.58, 9.19) (28.28, 9.19) (28.99, 9.90) (29.70, 9.90) (30.41,10.61) (31.11,10.61) (31.82,10.61) (32.53,11.31) (33.23,11.31) (33.94,11.31) (34.65,12.02) (35.36,12.02) (36.06,12.02) (36.77,12.73) (37.48,12.73) (38.18,12.73) (38.89,13.44) (39.60,13.44) (40.31,14.14) (41.01,14.14) (41.72,14.14) (42.43,14.85) (43.13,14.85) (43.84,14.85) (44.55,15.56) (45.25,15.56) (45.96,15.56) (46.67,16.26) (47.38,16.26) (48.08,16.97) (48.79,16.97) (49.50,16.97) (50.20,17.68) (50.91,17.68) (17.68,-26.87) (17.68,-26.16) (17.68,-25.46) (17.68,-24.75) (17.68,-24.04) (17.68,-23.33) (17.68,-22.63) (17.68,-21.92) (17.68,-21.21) (17.68,-20.51) (17.68,-19.80) (17.68,-19.09) (17.68,-18.38) (17.68,-17.68) (17.68,-16.97) (17.68,-16.26) (17.68,-15.56) (17.68,-14.85) (17.68,-14.14) (17.68,-13.44) (17.68,-12.73) (17.68,-12.02) (17.68,-11.31) (17.68,-10.61) (17.68,-9.90) (17.68,-9.19) (17.68,-8.49) (17.68,-7.78) (17.68,-7.07) (17.68,-6.36) (17.68,-5.66) (17.68,-4.95) (17.68,-4.24) (17.68,-3.54) (17.68,-2.83) (17.68,-2.12) (17.68,-1.41) (17.68,-0.71) (17.68, 0.00) (17.68, 0.71) (17.68, 1.41) (17.68, 2.12) (17.68, 2.83) (17.68, 3.54) (17.68, 4.24) (17.68, 4.95) (17.68, 5.66) (-17.68,-15.56) (-16.97,-15.56) (-16.26,-16.26) (-15.56,-16.26) (-14.85,-16.26) (-14.14,-16.97) (-13.44,-16.97) (-12.73,-16.97) (-12.02,-17.68) (-11.31,-17.68) (-10.61,-17.68) (-9.90,-18.38) (-9.19,-18.38) (-8.49,-18.38) (-7.78,-18.38) (-7.07,-19.09) (-6.36,-19.09) (-5.66,-19.09) (-4.95,-19.80) (-4.24,-19.80) (-3.54,-19.80) (-2.83,-20.51) (-2.12,-20.51) (-1.41,-20.51) (-0.71,-21.21) ( 0.00,-21.21) ( 0.71,-21.21) ( 1.41,-21.92) ( 2.12,-21.92) ( 2.83,-21.92) ( 3.54,-22.63) ( 4.24,-22.63) ( 4.95,-22.63) ( 5.66,-23.33) ( 6.36,-23.33) ( 7.07,-23.33) ( 7.78,-24.04) ( 8.49,-24.04) ( 9.19,-24.04) ( 9.90,-24.04) (10.61,-24.75) (11.31,-24.75) (12.02,-24.75) (12.73,-25.46) (13.44,-25.46) (14.14,-25.46) (14.85,-26.16) (15.56,-26.16) (16.26,-26.16) (16.97,-26.87) (17.68,-26.87) (-47.38,-1.41) (-46.67,-1.41) (-45.96,-2.12) (-45.25,-2.12) (-44.55,-2.83) (-43.84,-2.83) (-43.13,-3.54) (-42.43,-3.54) (-41.72,-4.24) (-41.01,-4.24) (-40.31,-4.95) (-39.60,-4.95) (-38.89,-5.66) (-38.18,-5.66) (-37.48,-6.36) (-36.77,-6.36) (-36.06,-7.07) (-35.36,-7.07) (-34.65,-7.78) (-33.94,-7.78) (-33.23,-8.49) (-32.53,-8.49) (-31.82,-8.49) (-31.11,-9.19) (-30.41,-9.19) (-29.70,-9.90) (-28.99,-9.90) (-28.28,-10.61) (-27.58,-10.61) (-26.87,-11.31) (-26.16,-11.31) (-25.46,-12.02) (-24.75,-12.02) (-24.04,-12.73) (-23.33,-12.73) (-22.63,-13.44) (-21.92,-13.44) (-21.21,-14.14) (-20.51,-14.14) (-19.80,-14.85) (-19.09,-14.85) (-18.38,-15.56) (-17.68,-15.56) (-80.61, 3.54) (-79.90, 3.54) (-79.20, 3.54) (-78.49, 3.54) (-77.78, 2.83) (-77.07, 2.83) (-76.37, 2.83) (-75.66, 2.83) (-74.95, 2.83) (-74.25, 2.83) (-73.54, 2.83) (-72.83, 2.12) (-72.12, 2.12) (-71.42, 2.12) (-70.71, 2.12) (-70.00, 2.12) (-69.30, 2.12) (-68.59, 1.41) (-67.88, 1.41) (-67.18, 1.41) (-66.47, 1.41) (-65.76, 1.41) (-65.05, 1.41) (-64.35, 1.41) (-63.64, 0.71) (-62.93, 0.71) (-62.23, 0.71) (-61.52, 0.71) (-60.81, 0.71) (-60.10, 0.71) (-59.40, 0.71) (-58.69, 0.00) (-57.98, 0.00) (-57.28, 0.00) (-56.57, 0.00) (-55.86, 0.00) (-55.15, 0.00) (-54.45,-0.71) (-53.74,-0.71) (-53.03,-0.71) (-52.33,-0.71) (-51.62,-0.71) (-50.91,-0.71) (-50.20,-0.71) (-49.50,-1.41) (-48.79,-1.41) (-48.08,-1.41) (-47.38,-1.41) (-110.31,-5.66) (-109.60,-5.66) (-108.89,-4.95) (-108.19,-4.95) (-107.48,-4.95) (-106.77,-4.24) (-106.07,-4.24) (-105.36,-4.24) (-104.65,-4.24) (-103.94,-3.54) (-103.24,-3.54) (-102.53,-3.54) (-101.82,-2.83) (-101.12,-2.83) (-100.41,-2.83) (-99.70,-2.12) (-98.99,-2.12) (-98.29,-2.12) (-97.58,-1.41) (-96.87,-1.41) (-96.17,-1.41) (-95.46,-1.41) (-94.75,-0.71) (-94.05,-0.71) (-93.34,-0.71) (-92.63, 0.00) (-91.92, 0.00) (-91.22, 0.00) (-90.51, 0.71) (-89.80, 0.71) (-89.10, 0.71) (-88.39, 1.41) (-87.68, 1.41) (-86.97, 1.41) (-86.27, 2.12) (-85.56, 2.12) (-84.85, 2.12) (-84.15, 2.12) (-83.44, 2.83) (-82.73, 2.83) (-82.02, 2.83) (-81.32, 3.54) (-80.61, 3.54) (-100.41,-37.48) (-100.41,-36.77) (-101.12,-36.06) (-101.12,-35.36) (-101.12,-34.65) (-101.82,-33.94) (-101.82,-33.23) (-101.82,-32.53) (-101.82,-31.82) (-102.53,-31.11) (-102.53,-30.41) (-102.53,-29.70) (-103.24,-28.99) (-103.24,-28.28) (-103.24,-27.58) (-103.94,-26.87) (-103.94,-26.16) (-103.94,-25.46) (-104.65,-24.75) (-104.65,-24.04) (-104.65,-23.33) (-105.36,-22.63) (-105.36,-21.92) (-105.36,-21.21) (-105.36,-20.51) (-106.07,-19.80) (-106.07,-19.09) (-106.07,-18.38) (-106.77,-17.68) (-106.77,-16.97) (-106.77,-16.26) (-107.48,-15.56) (-107.48,-14.85) (-107.48,-14.14) (-108.19,-13.44) (-108.19,-12.73) (-108.19,-12.02) (-108.89,-11.31) (-108.89,-10.61) (-108.89,-9.90) (-108.89,-9.19) (-109.60,-8.49) (-109.60,-7.78) (-109.60,-7.07) (-110.31,-6.36) (-110.31,-5.66) (-134.35,-43.84) (-133.64,-43.84) (-132.94,-43.84) (-132.23,-43.13) (-131.52,-43.13) (-130.81,-43.13) (-130.11,-43.13) (-129.40,-43.13) (-128.69,-43.13) (-127.99,-42.43) (-127.28,-42.43) (-126.57,-42.43) (-125.87,-42.43) (-125.16,-42.43) (-124.45,-41.72) (-123.74,-41.72) (-123.04,-41.72) (-122.33,-41.72) (-121.62,-41.72) (-120.92,-41.01) (-120.21,-41.01) (-119.50,-41.01) (-118.79,-41.01) (-118.09,-41.01) (-117.38,-41.01) (-116.67,-40.31) (-115.97,-40.31) (-115.26,-40.31) (-114.55,-40.31) (-113.84,-40.31) (-113.14,-39.60) (-112.43,-39.60) (-111.72,-39.60) (-111.02,-39.60) (-110.31,-39.60) (-109.60,-38.89) (-108.89,-38.89) (-108.19,-38.89) (-107.48,-38.89) (-106.77,-38.89) (-106.07,-38.89) (-105.36,-38.18) (-104.65,-38.18) (-103.94,-38.18) (-103.24,-38.18) (-102.53,-38.18) (-101.82,-37.48) (-101.12,-37.48) (-100.41,-37.48) (-153.44,-71.42) (-152.74,-70.71) (-152.74,-70.00) (-152.03,-69.30) (-151.32,-68.59) (-151.32,-67.88) (-150.61,-67.18) (-149.91,-66.47) (-149.20,-65.76) (-149.20,-65.05) (-148.49,-64.35) (-147.79,-63.64) (-147.79,-62.93) (-147.08,-62.23) (-146.37,-61.52) (-146.37,-60.81) (-145.66,-60.10) (-144.96,-59.40) (-144.96,-58.69) (-144.25,-57.98) (-143.54,-57.28) (-142.84,-56.57) (-142.84,-55.86) (-142.13,-55.15) (-141.42,-54.45) (-141.42,-53.74) (-140.71,-53.03) (-140.01,-52.33) (-140.01,-51.62) (-139.30,-50.91) (-138.59,-50.20) (-138.59,-49.50) (-137.89,-48.79) (-137.18,-48.08) (-136.47,-47.38) (-136.47,-46.67) (-135.76,-45.96) (-135.06,-45.25) (-135.06,-44.55) (-134.35,-43.84) (-153.44,-71.42) (-152.74,-70.71) (-152.03,-70.71) (-151.32,-70.00) (-150.61,-70.00) (-149.91,-69.30) (-149.20,-68.59) (-148.49,-68.59) (-147.79,-67.88) (-147.08,-67.88) (-146.37,-67.18) (-145.66,-66.47) (-144.96,-66.47) (-144.25,-65.76) (-143.54,-65.76) (-142.84,-65.05) (-142.13,-64.35) (-141.42,-64.35) (-140.71,-63.64) (-140.01,-63.64) (-139.30,-62.93) (-138.59,-62.93) (-137.89,-62.23) (-137.18,-61.52) (-136.47,-61.52) (-135.76,-60.81) (-135.06,-60.81) (-134.35,-60.10) (-133.64,-59.40) (-132.94,-59.40) (-132.23,-58.69) (-131.52,-58.69) (-130.81,-57.98) (-130.11,-57.28) (-129.40,-57.28) (-128.69,-56.57) (-127.99,-56.57) (-127.28,-55.86) (-126.57,-55.15) (-125.87,-55.15) (-125.16,-54.45) (-124.45,-54.45) (-123.74,-53.74) (-123.74,-53.74) (-123.74,-53.03) (-124.45,-52.33) (-124.45,-51.62) (-124.45,-50.91) (-125.16,-50.20) (-125.16,-49.50) (-125.16,-48.79) (-125.87,-48.08) (-125.87,-47.38) (-125.87,-46.67) (-125.87,-45.96) (-126.57,-45.25) (-126.57,-44.55) (-126.57,-43.84) (-127.28,-43.13) (-127.28,-42.43) (-127.28,-41.72) (-127.99,-41.01) (-127.99,-40.31) (-127.99,-39.60) (-128.69,-38.89) (-128.69,-38.18) (-128.69,-37.48) (-129.40,-36.77) (-129.40,-36.06) (-129.40,-35.36) (-130.11,-34.65) (-130.11,-33.94) (-130.11,-33.23) (-130.81,-32.53) (-130.81,-31.82) (-130.81,-31.11) (-130.81,-30.41) (-131.52,-29.70) (-131.52,-28.99) (-131.52,-28.28) (-132.23,-27.58) (-132.23,-26.87) (-132.23,-26.16) (-132.94,-25.46) (-132.94,-24.75) (-132.94,-24.04) (-133.64,-23.33) (-133.64,-22.63) (-133.64,-22.63) (-132.94,-22.63) (-132.23,-22.63) (-131.52,-23.33) (-130.81,-23.33) (-130.11,-23.33) (-129.40,-23.33) (-128.69,-23.33) (-127.99,-24.04) (-127.28,-24.04) (-126.57,-24.04) (-125.87,-24.04) (-125.16,-24.04) (-124.45,-24.04) (-123.74,-24.75) (-123.04,-24.75) (-122.33,-24.75) (-121.62,-24.75) (-120.92,-24.75) (-120.21,-25.46) (-119.50,-25.46) (-118.79,-25.46) (-118.09,-25.46) (-117.38,-25.46) (-116.67,-26.16) (-115.97,-26.16) (-115.26,-26.16) (-114.55,-26.16) (-113.84,-26.16) (-113.14,-26.87) (-112.43,-26.87) (-111.72,-26.87) (-111.02,-26.87) (-110.31,-26.87) (-109.60,-27.58) (-108.89,-27.58) (-108.19,-27.58) (-107.48,-27.58) (-106.77,-27.58) (-106.07,-27.58) (-105.36,-28.28) (-104.65,-28.28) (-103.94,-28.28) (-103.24,-28.28) (-102.53,-28.28) (-101.82,-28.99) (-101.12,-28.99) (-100.41,-28.99) (-100.41,-28.99) (-99.70,-28.28) (-98.99,-28.28) (-98.29,-27.58) (-97.58,-26.87) (-96.87,-26.87) (-96.17,-26.16) (-95.46,-25.46) (-94.75,-24.75) (-94.05,-24.75) (-93.34,-24.04) (-92.63,-23.33) (-91.92,-23.33) (-91.22,-22.63) (-90.51,-21.92) (-89.80,-21.92) (-89.10,-21.21) (-88.39,-20.51) (-87.68,-20.51) (-86.97,-19.80) (-86.27,-19.09) (-85.56,-18.38) (-84.85,-18.38) (-84.15,-17.68) (-83.44,-16.97) (-82.73,-16.97) (-82.02,-16.26) (-81.32,-15.56) (-80.61,-15.56) (-79.90,-14.85) (-79.20,-14.14) (-78.49,-14.14) (-77.78,-13.44) (-77.07,-12.73) (-76.37,-12.02) (-75.66,-12.02) (-74.95,-11.31) (-74.25,-10.61) (-73.54,-10.61) (-72.83,-9.90) (-101.82, 4.95) (-101.12, 4.24) (-100.41, 4.24) (-99.70, 3.54) (-98.99, 3.54) (-98.29, 2.83) (-97.58, 2.83) (-96.87, 2.12) (-96.17, 2.12) (-95.46, 1.41) (-94.75, 1.41) (-94.05, 0.71) (-93.34, 0.71) (-92.63, 0.00) (-91.92, 0.00) (-91.22,-0.71) (-90.51,-0.71) (-89.80,-1.41) (-89.10,-1.41) (-88.39,-2.12) (-87.68,-2.12) (-86.97,-2.83) (-86.27,-2.83) (-85.56,-3.54) (-84.85,-3.54) (-84.15,-4.24) (-83.44,-4.24) (-82.73,-4.95) (-82.02,-4.95) (-81.32,-5.66) (-80.61,-5.66) (-79.90,-6.36) (-79.20,-6.36) (-78.49,-7.07) (-77.78,-7.07) (-77.07,-7.78) (-76.37,-7.78) (-75.66,-8.49) (-74.95,-8.49) (-74.25,-9.19) (-73.54,-9.19) (-72.83,-9.90) (-133.64,14.85) (-132.94,14.85) (-132.23,14.14) (-131.52,14.14) (-130.81,14.14) (-130.11,13.44) (-129.40,13.44) (-128.69,13.44) (-127.99,13.44) (-127.28,12.73) (-126.57,12.73) (-125.87,12.73) (-125.16,12.02) (-124.45,12.02) (-123.74,12.02) (-123.04,11.31) (-122.33,11.31) (-121.62,11.31) (-120.92,10.61) (-120.21,10.61) (-119.50,10.61) (-118.79, 9.90) (-118.09, 9.90) (-117.38, 9.90) (-116.67, 9.90) (-115.97, 9.19) (-115.26, 9.19) (-114.55, 9.19) (-113.84, 8.49) (-113.14, 8.49) (-112.43, 8.49) (-111.72, 7.78) (-111.02, 7.78) (-110.31, 7.78) (-109.60, 7.07) (-108.89, 7.07) (-108.19, 7.07) (-107.48, 6.36) (-106.77, 6.36) (-106.07, 6.36) (-105.36, 6.36) (-104.65, 5.66) (-103.94, 5.66) (-103.24, 5.66) (-102.53, 4.95) (-101.82, 4.95) (-163.34, 1.41) (-162.63, 1.41) (-161.93, 2.12) (-161.22, 2.12) (-160.51, 2.83) (-159.81, 2.83) (-159.10, 3.54) (-158.39, 3.54) (-157.68, 4.24) (-156.98, 4.24) (-156.27, 4.95) (-155.56, 4.95) (-154.86, 4.95) (-154.15, 5.66) (-153.44, 5.66) (-152.74, 6.36) (-152.03, 6.36) (-151.32, 7.07) (-150.61, 7.07) (-149.91, 7.78) (-149.20, 7.78) (-148.49, 7.78) (-147.79, 8.49) (-147.08, 8.49) (-146.37, 9.19) (-145.66, 9.19) (-144.96, 9.90) (-144.25, 9.90) (-143.54,10.61) (-142.84,10.61) (-142.13,11.31) (-141.42,11.31) (-140.71,11.31) (-140.01,12.02) (-139.30,12.02) (-138.59,12.73) (-137.89,12.73) (-137.18,13.44) (-136.47,13.44) (-135.76,14.14) (-135.06,14.14) (-134.35,14.85) (-133.64,14.85) (-146.37,-25.46) (-147.08,-24.75) (-147.08,-24.04) (-147.79,-23.33) (-148.49,-22.63) (-148.49,-21.92) (-149.20,-21.21) (-149.20,-20.51) (-149.91,-19.80) (-150.61,-19.09) (-150.61,-18.38) (-151.32,-17.68) (-152.03,-16.97) (-152.03,-16.26) (-152.74,-15.56) (-152.74,-14.85) (-153.44,-14.14) (-154.15,-13.44) (-154.15,-12.73) (-154.86,-12.02) (-155.56,-11.31) (-155.56,-10.61) (-156.27,-9.90) (-156.98,-9.19) (-156.98,-8.49) (-157.68,-7.78) (-157.68,-7.07) (-158.39,-6.36) (-159.10,-5.66) (-159.10,-4.95) (-159.81,-4.24) (-160.51,-3.54) (-160.51,-2.83) (-161.22,-2.12) (-161.22,-1.41) (-161.93,-0.71) (-162.63, 0.00) (-162.63, 0.71) (-163.34, 1.41) (-169.00,-51.62) (-168.29,-50.91) (-167.58,-50.20) (-166.88,-49.50) (-166.88,-48.79) (-166.17,-48.08) (-165.46,-47.38) (-164.76,-46.67) (-164.05,-45.96) (-163.34,-45.25) (-162.63,-44.55) (-161.93,-43.84) (-161.93,-43.13) (-161.22,-42.43) (-160.51,-41.72) (-159.81,-41.01) (-159.10,-40.31) (-158.39,-39.60) (-157.68,-38.89) (-157.68,-38.18) (-156.98,-37.48) (-156.27,-36.77) (-155.56,-36.06) (-154.86,-35.36) (-154.15,-34.65) (-153.44,-33.94) (-153.44,-33.23) (-152.74,-32.53) (-152.03,-31.82) (-151.32,-31.11) (-150.61,-30.41) (-149.91,-29.70) (-149.20,-28.99) (-148.49,-28.28) (-148.49,-27.58) (-147.79,-26.87) (-147.08,-26.16) (-146.37,-25.46) (-169.00,-51.62) (-168.29,-50.91) (-168.29,-50.20) (-167.58,-49.50) (-166.88,-48.79) (-166.17,-48.08) (-166.17,-47.38) (-165.46,-46.67) (-164.76,-45.96) (-164.05,-45.25) (-164.05,-44.55) (-163.34,-43.84) (-162.63,-43.13) (-161.93,-42.43) (-161.93,-41.72) (-161.22,-41.01) (-160.51,-40.31) (-159.81,-39.60) (-159.81,-38.89) (-159.10,-38.18) (-158.39,-37.48) (-157.68,-36.77) (-157.68,-36.06) (-156.98,-35.36) (-156.27,-34.65) (-155.56,-33.94) (-155.56,-33.23) (-154.86,-32.53) (-154.15,-31.82) (-153.44,-31.11) (-153.44,-30.41) (-152.74,-29.70) (-152.03,-28.99) (-151.32,-28.28) (-151.32,-27.58) (-150.61,-26.87) (-149.91,-26.16) (-149.20,-25.46) (-149.20,-24.75) (-148.49,-24.04) (-147.79,-23.33) (-179.61,-14.85) (-178.90,-14.85) (-178.19,-15.56) (-177.48,-15.56) (-176.78,-15.56) (-176.07,-15.56) (-175.36,-16.26) (-174.66,-16.26) (-173.95,-16.26) (-173.24,-16.26) (-172.53,-16.97) (-171.83,-16.97) (-171.12,-16.97) (-170.41,-16.97) (-169.71,-17.68) (-169.00,-17.68) (-168.29,-17.68) (-167.58,-18.38) (-166.88,-18.38) (-166.17,-18.38) (-165.46,-18.38) (-164.76,-19.09) (-164.05,-19.09) (-163.34,-19.09) (-162.63,-19.09) (-161.93,-19.80) (-161.22,-19.80) (-160.51,-19.80) (-159.81,-19.80) (-159.10,-20.51) (-158.39,-20.51) (-157.68,-20.51) (-156.98,-21.21) (-156.27,-21.21) (-155.56,-21.21) (-154.86,-21.21) (-154.15,-21.92) (-153.44,-21.92) (-152.74,-21.92) (-152.03,-21.92) (-151.32,-22.63) (-150.61,-22.63) (-149.91,-22.63) (-149.20,-22.63) (-148.49,-23.33) (-147.79,-23.33) (-209.30,-30.41) (-208.60,-29.70) (-207.89,-29.70) (-207.18,-28.99) (-206.48,-28.99) (-205.77,-28.28) (-205.06,-28.28) (-204.35,-27.58) (-203.65,-27.58) (-202.94,-26.87) (-202.23,-26.87) (-201.53,-26.16) (-200.82,-26.16) (-200.11,-25.46) (-199.40,-25.46) (-198.70,-24.75) (-197.99,-24.75) (-197.28,-24.04) (-196.58,-24.04) (-195.87,-23.33) (-195.16,-23.33) (-194.45,-22.63) (-193.75,-21.92) (-193.04,-21.92) (-192.33,-21.21) (-191.63,-21.21) (-190.92,-20.51) (-190.21,-20.51) (-189.50,-19.80) (-188.80,-19.80) (-188.09,-19.09) (-187.38,-19.09) (-186.68,-18.38) (-185.97,-18.38) (-185.26,-17.68) (-184.55,-17.68) (-183.85,-16.97) (-183.14,-16.97) (-182.43,-16.26) (-181.73,-16.26) (-181.02,-15.56) (-180.31,-15.56) (-179.61,-14.85) (-194.45,-61.52) (-194.45,-60.81) (-195.16,-60.10) (-195.16,-59.40) (-195.87,-58.69) (-195.87,-57.98) (-196.58,-57.28) (-196.58,-56.57) (-197.28,-55.86) (-197.28,-55.15) (-197.99,-54.45) (-197.99,-53.74) (-198.70,-53.03) (-198.70,-52.33) (-199.40,-51.62) (-199.40,-50.91) (-200.11,-50.20) (-200.11,-49.50) (-200.82,-48.79) (-200.82,-48.08) (-201.53,-47.38) (-201.53,-46.67) (-201.53,-45.96) (-202.23,-45.25) (-202.23,-44.55) (-202.94,-43.84) (-202.94,-43.13) (-203.65,-42.43) (-203.65,-41.72) (-204.35,-41.01) (-204.35,-40.31) (-205.06,-39.60) (-205.06,-38.89) (-205.77,-38.18) (-205.77,-37.48) (-206.48,-36.77) (-206.48,-36.06) (-207.18,-35.36) (-207.18,-34.65) (-207.89,-33.94) (-207.89,-33.23) (-208.60,-32.53) (-208.60,-31.82) (-209.30,-31.11) (-209.30,-30.41) (-217.79,-85.56) (-217.08,-84.85) (-216.37,-84.15) (-215.67,-83.44) (-214.96,-82.73) (-214.25,-82.02) (-213.55,-81.32) (-212.84,-80.61) (-212.13,-79.90) (-211.42,-79.20) (-210.72,-78.49) (-210.01,-77.78) (-209.30,-77.07) (-208.60,-76.37) (-207.89,-75.66) (-207.18,-74.95) (-206.48,-74.25) (-206.48,-73.54) (-205.77,-72.83) (-205.06,-72.12) (-204.35,-71.42) (-203.65,-70.71) (-202.94,-70.00) (-202.23,-69.30) (-201.53,-68.59) (-200.82,-67.88) (-200.11,-67.18) (-199.40,-66.47) (-198.70,-65.76) (-197.99,-65.05) (-197.28,-64.35) (-196.58,-63.64) (-195.87,-62.93) (-195.16,-62.23) (-194.45,-61.52) (-217.79,-85.56) (-217.08,-85.56) (-216.37,-85.56) (-215.67,-86.27) (-214.96,-86.27) (-214.25,-86.27) (-213.55,-86.27) (-212.84,-86.27) (-212.13,-86.27) (-211.42,-86.97) (-210.72,-86.97) (-210.01,-86.97) (-209.30,-86.97) (-208.60,-86.97) (-207.89,-87.68) (-207.18,-87.68) (-206.48,-87.68) (-205.77,-87.68) (-205.06,-87.68) (-204.35,-87.68) (-203.65,-88.39) (-202.94,-88.39) (-202.23,-88.39) (-201.53,-88.39) (-200.82,-88.39) (-200.11,-88.39) (-199.40,-89.10) (-198.70,-89.10) (-197.99,-89.10) (-197.28,-89.10) (-196.58,-89.10) (-195.87,-89.80) (-195.16,-89.80) (-194.45,-89.80) (-193.75,-89.80) (-193.04,-89.80) (-192.33,-89.80) (-191.63,-90.51) (-190.92,-90.51) (-190.21,-90.51) (-189.50,-90.51) (-188.80,-90.51) (-188.09,-91.22) (-187.38,-91.22) (-186.68,-91.22) (-185.97,-91.22) (-185.26,-91.22) (-184.55,-91.22) (-183.85,-91.92) (-183.14,-91.92) (-182.43,-91.92) (-182.43,-91.92) (-181.73,-91.22) (-181.73,-90.51) (-181.02,-89.80) (-181.02,-89.10) (-180.31,-88.39) (-180.31,-87.68) (-179.61,-86.97) (-179.61,-86.27) (-178.90,-85.56) (-178.90,-84.85) (-178.19,-84.15) (-178.19,-83.44) (-177.48,-82.73) (-177.48,-82.02) (-176.78,-81.32) (-176.78,-80.61) (-176.07,-79.90) (-176.07,-79.20) (-175.36,-78.49) (-175.36,-77.78) (-174.66,-77.07) (-173.95,-76.37) (-173.95,-75.66) (-173.24,-74.95) (-173.24,-74.25) (-172.53,-73.54) (-172.53,-72.83) (-171.83,-72.12) (-171.83,-71.42) (-171.12,-70.71) (-171.12,-70.00) (-170.41,-69.30) (-170.41,-68.59) (-169.71,-67.88) (-169.71,-67.18) (-169.00,-66.47) (-169.00,-65.76) (-168.29,-65.05) (-168.29,-64.35) (-167.58,-63.64) (-182.43,-95.46) (-182.43,-94.75) (-181.73,-94.05) (-181.73,-93.34) (-181.02,-92.63) (-181.02,-91.92) (-180.31,-91.22) (-180.31,-90.51) (-179.61,-89.80) (-179.61,-89.10) (-178.90,-88.39) (-178.90,-87.68) (-178.19,-86.97) (-178.19,-86.27) (-177.48,-85.56) (-177.48,-84.85) (-177.48,-84.15) (-176.78,-83.44) (-176.78,-82.73) (-176.07,-82.02) (-176.07,-81.32) (-175.36,-80.61) (-175.36,-79.90) (-174.66,-79.20) (-174.66,-78.49) (-173.95,-77.78) (-173.95,-77.07) (-173.24,-76.37) (-173.24,-75.66) (-172.53,-74.95) (-172.53,-74.25) (-172.53,-73.54) (-171.83,-72.83) (-171.83,-72.12) (-171.12,-71.42) (-171.12,-70.71) (-170.41,-70.00) (-170.41,-69.30) (-169.71,-68.59) (-169.71,-67.88) (-169.00,-67.18) (-169.00,-66.47) (-168.29,-65.76) (-168.29,-65.05) (-167.58,-64.35) (-167.58,-63.64) (-182.43,-95.46) (-182.43,-94.75) (-182.43,-94.05) (-182.43,-93.34) (-182.43,-92.63) (-182.43,-91.92) (-181.73,-91.22) (-181.73,-90.51) (-181.73,-89.80) (-181.73,-89.10) (-181.73,-88.39) (-181.73,-87.68) (-181.73,-86.97) (-181.73,-86.27) (-181.73,-85.56) (-181.73,-84.85) (-181.73,-84.15) (-181.73,-83.44) (-181.02,-82.73) (-181.02,-82.02) (-181.02,-81.32) (-181.02,-80.61) (-181.02,-79.90) (-181.02,-79.20) (-181.02,-78.49) (-181.02,-77.78) (-181.02,-77.07) (-181.02,-76.37) (-181.02,-75.66) (-180.31,-74.95) (-180.31,-74.25) (-180.31,-73.54) (-180.31,-72.83) (-180.31,-72.12) (-180.31,-71.42) (-180.31,-70.71) (-180.31,-70.00) (-180.31,-69.30) (-180.31,-68.59) (-180.31,-67.88) (-180.31,-67.18) (-179.61,-66.47) (-179.61,-65.76) (-179.61,-65.05) (-179.61,-64.35) (-179.61,-63.64) (-179.61,-62.93) (-173.95,-93.34) (-173.95,-92.63) (-173.95,-91.92) (-174.66,-91.22) (-174.66,-90.51) (-174.66,-89.80) (-174.66,-89.10) (-174.66,-88.39) (-174.66,-87.68) (-175.36,-86.97) (-175.36,-86.27) (-175.36,-85.56) (-175.36,-84.85) (-175.36,-84.15) (-176.07,-83.44) (-176.07,-82.73) (-176.07,-82.02) (-176.07,-81.32) (-176.07,-80.61) (-176.78,-79.90) (-176.78,-79.20) (-176.78,-78.49) (-176.78,-77.78) (-176.78,-77.07) (-176.78,-76.37) (-177.48,-75.66) (-177.48,-74.95) (-177.48,-74.25) (-177.48,-73.54) (-177.48,-72.83) (-178.19,-72.12) (-178.19,-71.42) (-178.19,-70.71) (-178.19,-70.00) (-178.19,-69.30) (-178.90,-68.59) (-178.90,-67.88) (-178.90,-67.18) (-178.90,-66.47) (-178.90,-65.76) (-178.90,-65.05) (-179.61,-64.35) (-179.61,-63.64) (-179.61,-62.93) (-209.30,-100.41) (-208.60,-100.41) (-207.89,-100.41) (-207.18,-99.70) (-206.48,-99.70) (-205.77,-99.70) (-205.06,-99.70) (-204.35,-99.70) (-203.65,-98.99) (-202.94,-98.99) (-202.23,-98.99) (-201.53,-98.99) (-200.82,-98.99) (-200.11,-98.29) (-199.40,-98.29) (-198.70,-98.29) (-197.99,-98.29) (-197.28,-98.29) (-196.58,-97.58) (-195.87,-97.58) (-195.16,-97.58) (-194.45,-97.58) (-193.75,-97.58) (-193.04,-96.87) (-192.33,-96.87) (-191.63,-96.87) (-190.92,-96.87) (-190.21,-96.87) (-189.50,-96.17) (-188.80,-96.17) (-188.09,-96.17) (-187.38,-96.17) (-186.68,-96.17) (-185.97,-95.46) (-185.26,-95.46) (-184.55,-95.46) (-183.85,-95.46) (-183.14,-95.46) (-182.43,-94.75) (-181.73,-94.75) (-181.02,-94.75) (-180.31,-94.75) (-179.61,-94.75) (-178.90,-94.05) (-178.19,-94.05) (-177.48,-94.05) (-176.78,-94.05) (-176.07,-94.05) (-175.36,-93.34) (-174.66,-93.34) (-173.95,-93.34) (-211.42,-133.64) (-211.42,-132.94) (-211.42,-132.23) (-211.42,-131.52) (-211.42,-130.81) (-211.42,-130.11) (-211.42,-129.40) (-211.42,-128.69) (-210.72,-127.99) (-210.72,-127.28) (-210.72,-126.57) (-210.72,-125.87) (-210.72,-125.16) (-210.72,-124.45) (-210.72,-123.74) (-210.72,-123.04) (-210.72,-122.33) (-210.72,-121.62) (-210.72,-120.92) (-210.72,-120.21) (-210.72,-119.50) (-210.72,-118.79) (-210.72,-118.09) (-210.72,-117.38) (-210.01,-116.67) (-210.01,-115.97) (-210.01,-115.26) (-210.01,-114.55) (-210.01,-113.84) (-210.01,-113.14) (-210.01,-112.43) (-210.01,-111.72) (-210.01,-111.02) (-210.01,-110.31) (-210.01,-109.60) (-210.01,-108.89) (-210.01,-108.19) (-210.01,-107.48) (-210.01,-106.77) (-210.01,-106.07) (-209.30,-105.36) (-209.30,-104.65) (-209.30,-103.94) (-209.30,-103.24) (-209.30,-102.53) (-209.30,-101.82) (-209.30,-101.12) (-209.30,-100.41) (-241.83,-124.45) (-241.12,-124.45) (-240.42,-125.16) (-239.71,-125.16) (-239.00,-125.16) (-238.29,-125.87) (-237.59,-125.87) (-236.88,-125.87) (-236.17,-125.87) (-235.47,-126.57) (-234.76,-126.57) (-234.05,-126.57) (-233.35,-127.28) (-232.64,-127.28) (-231.93,-127.28) (-231.22,-127.99) (-230.52,-127.99) (-229.81,-127.99) (-229.10,-127.99) (-228.40,-128.69) (-227.69,-128.69) (-226.98,-128.69) (-226.27,-129.40) (-225.57,-129.40) (-224.86,-129.40) (-224.15,-130.11) (-223.45,-130.11) (-222.74,-130.11) (-222.03,-130.11) (-221.32,-130.81) (-220.62,-130.81) (-219.91,-130.81) (-219.20,-131.52) (-218.50,-131.52) (-217.79,-131.52) (-217.08,-132.23) (-216.37,-132.23) (-215.67,-132.23) (-214.96,-132.23) (-214.25,-132.94) (-213.55,-132.94) (-212.84,-132.94) (-212.13,-133.64) (-211.42,-133.64) (-271.53,-136.47) (-270.82,-136.47) (-270.11,-135.76) (-269.41,-135.76) (-268.70,-135.06) (-267.99,-135.06) (-267.29,-135.06) (-266.58,-134.35) (-265.87,-134.35) (-265.17,-133.64) (-264.46,-133.64) (-263.75,-133.64) (-263.04,-132.94) (-262.34,-132.94) (-261.63,-132.23) (-260.92,-132.23) (-260.22,-132.23) (-259.51,-131.52) (-258.80,-131.52) (-258.09,-130.81) (-257.39,-130.81) (-256.68,-130.81) (-255.97,-130.11) (-255.27,-130.11) (-254.56,-129.40) (-253.85,-129.40) (-253.14,-128.69) (-252.44,-128.69) (-251.73,-128.69) (-251.02,-127.99) (-250.32,-127.99) (-249.61,-127.28) (-248.90,-127.28) (-248.19,-127.28) (-247.49,-126.57) (-246.78,-126.57) (-246.07,-125.87) (-245.37,-125.87) (-244.66,-125.87) (-243.95,-125.16) (-243.24,-125.16) (-242.54,-124.45) (-241.83,-124.45) (-271.53,-136.47) (-270.82,-137.18) (-270.11,-137.18) (-269.41,-137.89) (-268.70,-138.59) (-267.99,-138.59) (-267.29,-139.30) (-266.58,-140.01) (-265.87,-140.01) (-265.17,-140.71) (-264.46,-141.42) (-263.75,-141.42) (-263.04,-142.13) (-262.34,-142.84) (-261.63,-142.84) (-260.92,-143.54) (-260.22,-144.25) (-259.51,-144.25) (-258.80,-144.96) (-258.09,-145.66) (-257.39,-145.66) (-256.68,-146.37) (-255.97,-146.37) (-255.27,-147.08) (-254.56,-147.79) (-253.85,-147.79) (-253.14,-148.49) (-252.44,-149.20) (-251.73,-149.20) (-251.02,-149.91) (-250.32,-150.61) (-249.61,-150.61) (-248.90,-151.32) (-248.19,-152.03) (-247.49,-152.03) (-246.78,-152.74) (-246.07,-153.44) (-245.37,-153.44) (-244.66,-154.15) (-243.95,-154.86) (-243.24,-154.86) (-242.54,-155.56) (-242.54,-155.56) (-241.83,-156.27) (-241.12,-156.27) (-240.42,-156.98) (-239.71,-157.68) (-239.00,-157.68) (-238.29,-158.39) (-237.59,-158.39) (-236.88,-159.10) (-236.17,-159.81) (-235.47,-159.81) (-234.76,-160.51) (-234.05,-161.22) (-233.35,-161.22) (-232.64,-161.93) (-231.93,-162.63) (-231.22,-162.63) (-230.52,-163.34) (-229.81,-163.34) (-229.10,-164.05) (-228.40,-164.76) (-227.69,-164.76) (-226.98,-165.46) (-226.27,-166.17) (-225.57,-166.17) (-224.86,-166.88) (-224.15,-166.88) (-223.45,-167.58) (-222.74,-168.29) (-222.03,-168.29) (-221.32,-169.00) (-220.62,-169.71) (-219.91,-169.71) (-219.20,-170.41) (-218.50,-171.12) (-217.79,-171.12) (-217.08,-171.83) (-216.37,-171.83) (-215.67,-172.53) (-214.96,-173.24) (-214.25,-173.24) (-213.55,-173.95) (-213.55,-173.95) (-212.84,-174.66) (-212.13,-174.66) (-211.42,-175.36) (-210.72,-176.07) (-210.01,-176.07) (-209.30,-176.78) (-208.60,-177.48) (-207.89,-177.48) (-207.18,-178.19) (-206.48,-178.90) (-205.77,-179.61) (-205.06,-179.61) (-204.35,-180.31) (-203.65,-181.02) (-202.94,-181.02) (-202.23,-181.73) (-201.53,-182.43) (-200.82,-182.43) (-200.11,-183.14) (-199.40,-183.85) (-198.70,-183.85) (-197.99,-184.55) (-197.28,-185.26) (-196.58,-185.26) (-195.87,-185.97) (-195.16,-186.68) (-194.45,-186.68) (-193.75,-187.38) (-193.04,-188.09) (-192.33,-188.09) (-191.63,-188.80) (-190.92,-189.50) (-190.21,-190.21) (-189.50,-190.21) (-188.80,-190.92) (-188.09,-191.63) (-187.38,-191.63) (-186.68,-192.33) (-185.97,-193.04) (-185.26,-193.04) (-184.55,-193.75) (-184.55,-193.75) (-183.85,-194.45) (-183.14,-195.16) (-182.43,-195.16) (-181.73,-195.87) (-181.02,-196.58) (-180.31,-197.28) (-179.61,-197.99) (-178.90,-198.70) (-178.19,-198.70) (-177.48,-199.40) (-176.78,-200.11) (-176.07,-200.82) (-175.36,-201.53) (-174.66,-201.53) (-173.95,-202.23) (-173.24,-202.94) (-172.53,-203.65) (-171.83,-204.35) (-171.12,-204.35) (-170.41,-205.06) (-169.71,-205.77) (-169.00,-206.48) (-168.29,-207.18) (-167.58,-207.89) (-166.88,-207.89) (-166.17,-208.60) (-165.46,-209.30) (-164.76,-210.01) (-164.05,-210.72) (-163.34,-210.72) (-162.63,-211.42) (-161.93,-212.13) (-161.22,-212.84) (-160.51,-213.55) (-159.81,-214.25) (-159.10,-214.25) (-158.39,-214.96) (-157.68,-215.67) (-157.68,-215.67) (-156.98,-216.37) (-156.27,-216.37) (-155.56,-217.08) (-154.86,-217.08) (-154.15,-217.79) (-153.44,-218.50) (-152.74,-218.50) (-152.03,-219.20) (-151.32,-219.20) (-150.61,-219.91) (-149.91,-219.91) (-149.20,-220.62) (-148.49,-221.32) (-147.79,-221.32) (-147.08,-222.03) (-146.37,-222.03) (-145.66,-222.74) (-144.96,-223.45) (-144.25,-223.45) (-143.54,-224.15) (-142.84,-224.15) (-142.13,-224.86) (-141.42,-224.86) (-140.71,-225.57) (-140.01,-226.27) (-139.30,-226.27) (-138.59,-226.98) (-137.89,-226.98) (-137.18,-227.69) (-136.47,-228.40) (-135.76,-228.40) (-135.06,-229.10) (-134.35,-229.10) (-133.64,-229.81) (-132.94,-229.81) (-132.23,-230.52) (-131.52,-231.22) (-130.81,-231.22) (-130.11,-231.93) (-129.40,-231.93) (-128.69,-232.64) (-128.69,-232.64) (-127.99,-232.64) (-127.28,-233.35) (-126.57,-233.35) (-125.87,-234.05) (-125.16,-234.05) (-124.45,-234.76) (-123.74,-234.76) (-123.04,-235.47) (-122.33,-235.47) (-121.62,-236.17) (-120.92,-236.17) (-120.21,-236.88) (-119.50,-236.88) (-118.79,-237.59) (-118.09,-237.59) (-117.38,-238.29) (-116.67,-238.29) (-115.97,-239.00) (-115.26,-239.00) (-114.55,-239.71) (-113.84,-239.71) (-113.14,-239.71) (-112.43,-240.42) (-111.72,-240.42) (-111.02,-241.12) (-110.31,-241.12) (-109.60,-241.83) (-108.89,-241.83) (-108.19,-242.54) (-107.48,-242.54) (-106.77,-243.24) (-106.07,-243.24) (-105.36,-243.95) (-104.65,-243.95) (-103.94,-244.66) (-103.24,-244.66) (-102.53,-245.37) (-101.82,-245.37) (-101.12,-246.07) (-100.41,-246.07) (-99.70,-246.78) (-98.99,-246.78) (-98.29,-247.49) (-97.58,-247.49) (-97.58,-247.49) (-96.87,-247.49) (-96.17,-248.19) (-95.46,-248.19) (-94.75,-248.90) (-94.05,-248.90) (-93.34,-248.90) (-92.63,-249.61) (-91.92,-249.61) (-91.22,-250.32) (-90.51,-250.32) (-89.80,-251.02) (-89.10,-251.02) (-88.39,-251.02) (-87.68,-251.73) (-86.97,-251.73) (-86.27,-252.44) (-85.56,-252.44) (-84.85,-252.44) (-84.15,-253.14) (-83.44,-253.14) (-82.73,-253.85) (-82.02,-253.85) (-81.32,-254.56) (-80.61,-254.56) (-79.90,-254.56) (-79.20,-255.27) (-78.49,-255.27) (-77.78,-255.97) (-77.07,-255.97) (-76.37,-255.97) (-75.66,-256.68) (-74.95,-256.68) (-74.25,-257.39) (-73.54,-257.39) (-72.83,-258.09) (-72.12,-258.09) (-71.42,-258.09) (-70.71,-258.80) (-70.00,-258.80) (-69.30,-259.51) (-68.59,-259.51) (-67.88,-259.51) (-67.18,-260.22) (-66.47,-260.22) (-65.76,-260.92) (-65.05,-260.92) (-64.35,-261.63) (-63.64,-261.63) (-63.64,-261.63) (-62.93,-261.63) (-62.23,-262.34) (-61.52,-262.34) (-60.81,-262.34) (-60.10,-263.04) (-59.40,-263.04) (-58.69,-263.04) (-57.98,-263.75) (-57.28,-263.75) (-56.57,-264.46) (-55.86,-264.46) (-55.15,-264.46) (-54.45,-265.17) (-53.74,-265.17) (-53.03,-265.17) (-52.33,-265.87) (-51.62,-265.87) (-50.91,-265.87) (-50.20,-266.58) (-49.50,-266.58) (-48.79,-266.58) (-48.08,-267.29) (-47.38,-267.29) (-46.67,-267.99) (-45.96,-267.99) (-45.25,-267.99) (-44.55,-268.70) (-43.84,-268.70) (-43.13,-268.70) (-42.43,-269.41) (-41.72,-269.41) (-41.01,-269.41) (-40.31,-270.11) (-39.60,-270.11) (-38.89,-270.11) (-38.18,-270.82) (-37.48,-270.82) (-36.77,-271.53) (-36.06,-271.53) (-35.36,-271.53) (-34.65,-272.24) (-33.94,-272.24) (-33.23,-272.24) (-32.53,-272.94) (-31.82,-272.94) (-31.82,-272.94) (-31.11,-272.24) (-30.41,-272.24) (-29.70,-271.53) (-28.99,-270.82) (-28.28,-270.11) (-27.58,-270.11) (-26.87,-269.41) (-26.16,-268.70) (-25.46,-267.99) (-24.75,-267.99) (-24.04,-267.29) (-23.33,-266.58) (-22.63,-266.58) (-21.92,-265.87) (-21.21,-265.17) (-20.51,-264.46) (-19.80,-264.46) (-19.09,-263.75) (-18.38,-263.04) (-17.68,-262.34) (-16.97,-262.34) (-16.26,-261.63) (-15.56,-260.92) (-14.85,-260.22) (-14.14,-260.22) (-13.44,-259.51) (-12.73,-258.80) (-12.02,-258.80) (-11.31,-258.09) (-10.61,-257.39) (-9.90,-256.68) (-9.19,-256.68) (-8.49,-255.97) (-7.78,-255.27) (-7.07,-254.56) (-6.36,-254.56) (-5.66,-253.85) (-33.94,-234.76) (-33.23,-235.47) (-32.53,-235.47) (-31.82,-236.17) (-31.11,-236.88) (-30.41,-236.88) (-29.70,-237.59) (-28.99,-238.29) (-28.28,-238.29) (-27.58,-239.00) (-26.87,-239.71) (-26.16,-239.71) (-25.46,-240.42) (-24.75,-241.12) (-24.04,-241.12) (-23.33,-241.83) (-22.63,-242.54) (-21.92,-242.54) (-21.21,-243.24) (-20.51,-243.95) (-19.80,-243.95) (-19.09,-244.66) (-18.38,-245.37) (-17.68,-246.07) (-16.97,-246.07) (-16.26,-246.78) (-15.56,-247.49) (-14.85,-247.49) (-14.14,-248.19) (-13.44,-248.90) (-12.73,-248.90) (-12.02,-249.61) (-11.31,-250.32) (-10.61,-250.32) (-9.90,-251.02) (-9.19,-251.73) (-8.49,-251.73) (-7.78,-252.44) (-7.07,-253.14) (-6.36,-253.14) (-5.66,-253.85) (-63.64,-213.55) (-62.93,-214.25) (-62.23,-214.25) (-61.52,-214.96) (-60.81,-215.67) (-60.10,-216.37) (-59.40,-216.37) (-58.69,-217.08) (-57.98,-217.79) (-57.28,-217.79) (-56.57,-218.50) (-55.86,-219.20) (-55.15,-219.91) (-54.45,-219.91) (-53.74,-220.62) (-53.03,-221.32) (-52.33,-221.32) (-51.62,-222.03) (-50.91,-222.74) (-50.20,-223.45) (-49.50,-223.45) (-48.79,-224.15) (-48.08,-224.86) (-47.38,-224.86) (-46.67,-225.57) (-45.96,-226.27) (-45.25,-226.98) (-44.55,-226.98) (-43.84,-227.69) (-43.13,-228.40) (-42.43,-228.40) (-41.72,-229.10) (-41.01,-229.81) (-40.31,-230.52) (-39.60,-230.52) (-38.89,-231.22) (-38.18,-231.93) (-37.48,-231.93) (-36.77,-232.64) (-36.06,-233.35) (-35.36,-234.05) (-34.65,-234.05) (-33.94,-234.76) (-63.64,-213.55) (-63.64,-212.84) (-63.64,-212.13) (-63.64,-211.42) (-64.35,-210.72) (-64.35,-210.01) (-64.35,-209.30) (-64.35,-208.60) (-64.35,-207.89) (-64.35,-207.18) (-64.35,-206.48) (-65.05,-205.77) (-65.05,-205.06) (-65.05,-204.35) (-65.05,-203.65) (-65.05,-202.94) (-65.05,-202.23) (-65.05,-201.53) (-65.76,-200.82) (-65.76,-200.11) (-65.76,-199.40) (-65.76,-198.70) (-65.76,-197.99) (-65.76,-197.28) (-65.76,-196.58) (-66.47,-195.87) (-66.47,-195.16) (-66.47,-194.45) (-66.47,-193.75) (-66.47,-193.04) (-66.47,-192.33) (-67.18,-191.63) (-67.18,-190.92) (-67.18,-190.21) (-67.18,-189.50) (-67.18,-188.80) (-67.18,-188.09) (-67.18,-187.38) (-67.88,-186.68) (-67.88,-185.97) (-67.88,-185.26) (-67.88,-184.55) (-67.88,-183.85) (-67.88,-183.14) (-67.88,-182.43) (-68.59,-181.73) (-68.59,-181.02) (-68.59,-180.31) (-68.59,-179.61) (-68.59,-179.61) (-67.88,-179.61) (-67.18,-180.31) (-66.47,-180.31) (-65.76,-181.02) (-65.05,-181.02) (-64.35,-181.02) (-63.64,-181.73) (-62.93,-181.73) (-62.23,-181.73) (-61.52,-182.43) (-60.81,-182.43) (-60.10,-183.14) (-59.40,-183.14) (-58.69,-183.14) (-57.98,-183.85) (-57.28,-183.85) (-56.57,-183.85) (-55.86,-184.55) (-55.15,-184.55) (-54.45,-185.26) (-53.74,-185.26) (-53.03,-185.26) (-52.33,-185.97) (-51.62,-185.97) (-50.91,-185.97) (-50.20,-186.68) (-49.50,-186.68) (-48.79,-187.38) (-48.08,-187.38) (-47.38,-187.38) (-46.67,-188.09) (-45.96,-188.09) (-45.25,-188.09) (-44.55,-188.80) (-43.84,-188.80) (-43.13,-189.50) (-42.43,-189.50) (-41.72,-189.50) (-41.01,-190.21) (-40.31,-190.21) (-39.60,-190.21) (-38.89,-190.92) (-38.18,-190.92) (-37.48,-191.63) (-36.77,-191.63) (-36.77,-191.63) (-36.06,-191.63) (-35.36,-191.63) (-34.65,-191.63) (-33.94,-191.63) (-33.23,-191.63) (-32.53,-191.63) (-31.82,-191.63) (-31.11,-191.63) (-30.41,-191.63) (-29.70,-191.63) (-28.99,-191.63) (-28.28,-191.63) (-27.58,-191.63) (-26.87,-191.63) (-26.16,-191.63) (-25.46,-191.63) (-24.75,-191.63) (-24.04,-191.63) (-23.33,-191.63) (-22.63,-191.63) (-21.92,-191.63) (-21.21,-191.63) (-20.51,-190.92) (-19.80,-190.92) (-19.09,-190.92) (-18.38,-190.92) (-17.68,-190.92) (-16.97,-190.92) (-16.26,-190.92) (-15.56,-190.92) (-14.85,-190.92) (-14.14,-190.92) (-13.44,-190.92) (-12.73,-190.92) (-12.02,-190.92) (-11.31,-190.92) (-10.61,-190.92) (-9.90,-190.92) (-9.19,-190.92) (-8.49,-190.92) (-7.78,-190.92) (-7.07,-190.92) (-6.36,-190.92) (-5.66,-190.92) (-4.95,-190.92) (-4.95,-190.92) (-4.24,-190.92) (-3.54,-190.92) (-2.83,-190.21) (-2.12,-190.21) (-1.41,-190.21) (-0.71,-190.21) ( 0.00,-189.50) ( 0.71,-189.50) ( 1.41,-189.50) ( 2.12,-189.50) ( 2.83,-189.50) ( 3.54,-188.80) ( 4.24,-188.80) ( 4.95,-188.80) ( 5.66,-188.80) ( 6.36,-188.09) ( 7.07,-188.09) ( 7.78,-188.09) ( 8.49,-188.09) ( 9.19,-188.09) ( 9.90,-187.38) (10.61,-187.38) (11.31,-187.38) (12.02,-187.38) (12.73,-186.68) (13.44,-186.68) (14.14,-186.68) (14.85,-186.68) (15.56,-186.68) (16.26,-185.97) (16.97,-185.97) (17.68,-185.97) (18.38,-185.97) (19.09,-185.26) (19.80,-185.26) (20.51,-185.26) (21.21,-185.26) (21.92,-185.26) (22.63,-184.55) (23.33,-184.55) (24.04,-184.55) (24.75,-184.55) (25.46,-183.85) (26.16,-183.85) (26.87,-183.85) (26.87,-183.85) (27.58,-183.14) (28.28,-182.43) (28.99,-181.73) (28.99,-181.02) (29.70,-180.31) (30.41,-179.61) (31.11,-178.90) (31.82,-178.19) (32.53,-177.48) (33.23,-176.78) (33.23,-176.07) (33.94,-175.36) (34.65,-174.66) (35.36,-173.95) (36.06,-173.24) (36.77,-172.53) (37.48,-171.83) (37.48,-171.12) (38.18,-170.41) (38.89,-169.71) (39.60,-169.00) (40.31,-168.29) (41.01,-167.58) (41.72,-166.88) (42.43,-166.17) (42.43,-165.46) (43.13,-164.76) (43.84,-164.05) (44.55,-163.34) (45.25,-162.63) (45.96,-161.93) (46.67,-161.22) (46.67,-160.51) (47.38,-159.81) (48.08,-159.10) (48.79,-158.39) (14.85,-167.58) (15.56,-167.58) (16.26,-166.88) (16.97,-166.88) (17.68,-166.88) (18.38,-166.88) (19.09,-166.17) (19.80,-166.17) (20.51,-166.17) (21.21,-166.17) (21.92,-165.46) (22.63,-165.46) (23.33,-165.46) (24.04,-164.76) (24.75,-164.76) (25.46,-164.76) (26.16,-164.76) (26.87,-164.05) (27.58,-164.05) (28.28,-164.05) (28.99,-164.05) (29.70,-163.34) (30.41,-163.34) (31.11,-163.34) (31.82,-163.34) (32.53,-162.63) (33.23,-162.63) (33.94,-162.63) (34.65,-161.93) (35.36,-161.93) (36.06,-161.93) (36.77,-161.93) (37.48,-161.22) (38.18,-161.22) (38.89,-161.22) (39.60,-161.22) (40.31,-160.51) (41.01,-160.51) (41.72,-160.51) (42.43,-159.81) (43.13,-159.81) (43.84,-159.81) (44.55,-159.81) (45.25,-159.10) (45.96,-159.10) (46.67,-159.10) (47.38,-159.10) (48.08,-158.39) (48.79,-158.39) (21.92,-202.94) (21.92,-202.23) (21.92,-201.53) (21.21,-200.82) (21.21,-200.11) (21.21,-199.40) (21.21,-198.70) (21.21,-197.99) (20.51,-197.28) (20.51,-196.58) (20.51,-195.87) (20.51,-195.16) (20.51,-194.45) (19.80,-193.75) (19.80,-193.04) (19.80,-192.33) (19.80,-191.63) (19.80,-190.92) (19.09,-190.21) (19.09,-189.50) (19.09,-188.80) (19.09,-188.09) (19.09,-187.38) (18.38,-186.68) (18.38,-185.97) (18.38,-185.26) (18.38,-184.55) (18.38,-183.85) (17.68,-183.14) (17.68,-182.43) (17.68,-181.73) (17.68,-181.02) (17.68,-180.31) (16.97,-179.61) (16.97,-178.90) (16.97,-178.19) (16.97,-177.48) (16.97,-176.78) (16.26,-176.07) (16.26,-175.36) (16.26,-174.66) (16.26,-173.95) (16.26,-173.24) (15.56,-172.53) (15.56,-171.83) (15.56,-171.12) (15.56,-170.41) (15.56,-169.71) (14.85,-169.00) (14.85,-168.29) (14.85,-167.58) (-9.90,-211.42) (-9.19,-211.42) (-8.49,-210.72) (-7.78,-210.72) (-7.07,-210.72) (-6.36,-210.72) (-5.66,-210.01) (-4.95,-210.01) (-4.24,-210.01) (-3.54,-210.01) (-2.83,-209.30) (-2.12,-209.30) (-1.41,-209.30) (-0.71,-209.30) ( 0.00,-208.60) ( 0.71,-208.60) ( 1.41,-208.60) ( 2.12,-207.89) ( 2.83,-207.89) ( 3.54,-207.89) ( 4.24,-207.89) ( 4.95,-207.18) ( 5.66,-207.18) ( 6.36,-207.18) ( 7.07,-207.18) ( 7.78,-206.48) ( 8.49,-206.48) ( 9.19,-206.48) ( 9.90,-206.48) (10.61,-205.77) (11.31,-205.77) (12.02,-205.77) (12.73,-205.06) (13.44,-205.06) (14.14,-205.06) (14.85,-205.06) (15.56,-204.35) (16.26,-204.35) (16.97,-204.35) (17.68,-204.35) (18.38,-203.65) (19.09,-203.65) (19.80,-203.65) (20.51,-203.65) (21.21,-202.94) (21.92,-202.94) (10.61,-238.29) ( 9.90,-237.59) ( 9.19,-236.88) ( 9.19,-236.17) ( 8.49,-235.47) ( 7.78,-234.76) ( 7.07,-234.05) ( 7.07,-233.35) ( 6.36,-232.64) ( 5.66,-231.93) ( 4.95,-231.22) ( 4.95,-230.52) ( 4.24,-229.81) ( 3.54,-229.10) ( 2.83,-228.40) ( 2.83,-227.69) ( 2.12,-226.98) ( 1.41,-226.27) ( 0.71,-225.57) ( 0.71,-224.86) ( 0.00,-224.15) (-0.71,-223.45) (-1.41,-222.74) (-2.12,-222.03) (-2.12,-221.32) (-2.83,-220.62) (-3.54,-219.91) (-4.24,-219.20) (-4.24,-218.50) (-4.95,-217.79) (-5.66,-217.08) (-6.36,-216.37) (-6.36,-215.67) (-7.07,-214.96) (-7.78,-214.25) (-8.49,-213.55) (-8.49,-212.84) (-9.19,-212.13) (-9.90,-211.42) (10.61,-269.41) (10.61,-268.70) (10.61,-267.99) (10.61,-267.29) (10.61,-266.58) (10.61,-265.87) (10.61,-265.17) (10.61,-264.46) (10.61,-263.75) (10.61,-263.04) (10.61,-262.34) (10.61,-261.63) (10.61,-260.92) (10.61,-260.22) (10.61,-259.51) (10.61,-258.80) (10.61,-258.09) (10.61,-257.39) (10.61,-256.68) (10.61,-255.97) (10.61,-255.27) (10.61,-254.56) (10.61,-253.85) (10.61,-253.14) (10.61,-252.44) (10.61,-251.73) (10.61,-251.02) (10.61,-250.32) (10.61,-249.61) (10.61,-248.90) (10.61,-248.19) (10.61,-247.49) (10.61,-246.78) (10.61,-246.07) (10.61,-245.37) (10.61,-244.66) (10.61,-243.95) (10.61,-243.24) (10.61,-242.54) (10.61,-241.83) (10.61,-241.12) (10.61,-240.42) (10.61,-239.71) (10.61,-239.00) (10.61,-238.29) (-13.44,-246.78) (-12.73,-247.49) (-12.02,-248.19) (-11.31,-248.90) (-10.61,-249.61) (-9.90,-250.32) (-9.19,-251.02) (-8.49,-251.73) (-7.78,-252.44) (-7.07,-252.44) (-6.36,-253.14) (-5.66,-253.85) (-4.95,-254.56) (-4.24,-255.27) (-3.54,-255.97) (-2.83,-256.68) (-2.12,-257.39) (-1.41,-258.09) (-0.71,-258.80) ( 0.00,-259.51) ( 0.71,-260.22) ( 1.41,-260.92) ( 2.12,-261.63) ( 2.83,-262.34) ( 3.54,-263.04) ( 4.24,-263.75) ( 4.95,-263.75) ( 5.66,-264.46) ( 6.36,-265.17) ( 7.07,-265.87) ( 7.78,-266.58) ( 8.49,-267.29) ( 9.19,-267.99) ( 9.90,-268.70) (10.61,-269.41) (-13.44,-246.78) (-13.44,-246.07) (-14.14,-245.37) (-14.14,-244.66) (-14.14,-243.95) (-14.14,-243.24) (-14.85,-242.54) (-14.85,-241.83) (-14.85,-241.12) (-14.85,-240.42) (-15.56,-239.71) (-15.56,-239.00) (-15.56,-238.29) (-16.26,-237.59) (-16.26,-236.88) (-16.26,-236.17) (-16.26,-235.47) (-16.97,-234.76) (-16.97,-234.05) (-16.97,-233.35) (-17.68,-232.64) (-17.68,-231.93) (-17.68,-231.22) (-17.68,-230.52) (-18.38,-229.81) (-18.38,-229.10) (-18.38,-228.40) (-18.38,-227.69) (-19.09,-226.98) (-19.09,-226.27) (-19.09,-225.57) (-19.80,-224.86) (-19.80,-224.15) (-19.80,-223.45) (-19.80,-222.74) (-20.51,-222.03) (-20.51,-221.32) (-20.51,-220.62) (-21.21,-219.91) (-21.21,-219.20) (-21.21,-218.50) (-21.21,-217.79) (-21.92,-217.08) (-21.92,-216.37) (-21.92,-215.67) (-21.92,-214.96) (-22.63,-214.25) (-22.63,-213.55) (-22.63,-213.55) (-21.92,-213.55) (-21.21,-213.55) (-20.51,-213.55) (-19.80,-214.25) (-19.09,-214.25) (-18.38,-214.25) (-17.68,-214.25) (-16.97,-214.25) (-16.26,-214.25) (-15.56,-214.25) (-14.85,-214.96) (-14.14,-214.96) (-13.44,-214.96) (-12.73,-214.96) (-12.02,-214.96) (-11.31,-214.96) (-10.61,-215.67) (-9.90,-215.67) (-9.19,-215.67) (-8.49,-215.67) (-7.78,-215.67) (-7.07,-215.67) (-6.36,-215.67) (-5.66,-216.37) (-4.95,-216.37) (-4.24,-216.37) (-3.54,-216.37) (-2.83,-216.37) (-2.12,-216.37) (-1.41,-216.37) (-0.71,-217.08) ( 0.00,-217.08) ( 0.71,-217.08) ( 1.41,-217.08) ( 2.12,-217.08) ( 2.83,-217.08) ( 3.54,-217.79) ( 4.24,-217.79) ( 4.95,-217.79) ( 5.66,-217.79) ( 6.36,-217.79) ( 7.07,-217.79) ( 7.78,-217.79) ( 8.49,-218.50) ( 9.19,-218.50) ( 9.90,-218.50) (10.61,-218.50) (10.61,-218.50) (11.31,-217.79) (12.02,-217.08) (12.73,-216.37) (13.44,-215.67) (14.14,-214.96) (14.85,-214.25) (15.56,-213.55) (16.26,-212.84) (16.97,-212.13) (17.68,-211.42) (18.38,-210.72) (19.09,-210.01) (19.80,-209.30) (20.51,-208.60) (21.21,-207.89) (21.92,-207.18) (21.92,-206.48) (22.63,-205.77) (23.33,-205.06) (24.04,-204.35) (24.75,-203.65) (25.46,-202.94) (26.16,-202.23) (26.87,-201.53) (27.58,-200.82) (28.28,-200.11) (28.99,-199.40) (29.70,-198.70) (30.41,-197.99) (31.11,-197.28) (31.82,-196.58) (32.53,-195.87) (33.23,-195.16) (33.94,-194.45) ( 2.83,-175.36) ( 3.54,-176.07) ( 4.24,-176.07) ( 4.95,-176.78) ( 5.66,-176.78) ( 6.36,-177.48) ( 7.07,-178.19) ( 7.78,-178.19) ( 8.49,-178.90) ( 9.19,-179.61) ( 9.90,-179.61) (10.61,-180.31) (11.31,-180.31) (12.02,-181.02) (12.73,-181.73) (13.44,-181.73) (14.14,-182.43) (14.85,-182.43) (15.56,-183.14) (16.26,-183.85) (16.97,-183.85) (17.68,-184.55) (18.38,-184.55) (19.09,-185.26) (19.80,-185.97) (20.51,-185.97) (21.21,-186.68) (21.92,-187.38) (22.63,-187.38) (23.33,-188.09) (24.04,-188.09) (24.75,-188.80) (25.46,-189.50) (26.16,-189.50) (26.87,-190.21) (27.58,-190.21) (28.28,-190.92) (28.99,-191.63) (29.70,-191.63) (30.41,-192.33) (31.11,-193.04) (31.82,-193.04) (32.53,-193.75) (33.23,-193.75) (33.94,-194.45) (-26.87,-161.93) (-26.16,-161.93) (-25.46,-162.63) (-24.75,-162.63) (-24.04,-163.34) (-23.33,-163.34) (-22.63,-164.05) (-21.92,-164.05) (-21.21,-164.76) (-20.51,-164.76) (-19.80,-165.46) (-19.09,-165.46) (-18.38,-165.46) (-17.68,-166.17) (-16.97,-166.17) (-16.26,-166.88) (-15.56,-166.88) (-14.85,-167.58) (-14.14,-167.58) (-13.44,-168.29) (-12.73,-168.29) (-12.02,-168.29) (-11.31,-169.00) (-10.61,-169.00) (-9.90,-169.71) (-9.19,-169.71) (-8.49,-170.41) (-7.78,-170.41) (-7.07,-171.12) (-6.36,-171.12) (-5.66,-171.83) (-4.95,-171.83) (-4.24,-171.83) (-3.54,-172.53) (-2.83,-172.53) (-2.12,-173.24) (-1.41,-173.24) (-0.71,-173.95) ( 0.00,-173.95) ( 0.71,-174.66) ( 1.41,-174.66) ( 2.12,-175.36) ( 2.83,-175.36) (-59.40,-150.61) (-58.69,-150.61) (-57.98,-151.32) (-57.28,-151.32) (-56.57,-151.32) (-55.86,-152.03) (-55.15,-152.03) (-54.45,-152.03) (-53.74,-152.74) (-53.03,-152.74) (-52.33,-152.74) (-51.62,-153.44) (-50.91,-153.44) (-50.20,-154.15) (-49.50,-154.15) (-48.79,-154.15) (-48.08,-154.86) (-47.38,-154.86) (-46.67,-154.86) (-45.96,-155.56) (-45.25,-155.56) (-44.55,-155.56) (-43.84,-156.27) (-43.13,-156.27) (-42.43,-156.27) (-41.72,-156.98) (-41.01,-156.98) (-40.31,-156.98) (-39.60,-157.68) (-38.89,-157.68) (-38.18,-157.68) (-37.48,-158.39) (-36.77,-158.39) (-36.06,-158.39) (-35.36,-159.10) (-34.65,-159.10) (-33.94,-159.81) (-33.23,-159.81) (-32.53,-159.81) (-31.82,-160.51) (-31.11,-160.51) (-30.41,-160.51) (-29.70,-161.22) (-28.99,-161.22) (-28.28,-161.22) (-27.58,-161.93) (-26.87,-161.93) (-92.63,-149.91) (-91.92,-149.91) (-91.22,-149.91) (-90.51,-149.91) (-89.80,-149.91) (-89.10,-149.91) (-88.39,-149.91) (-87.68,-149.91) (-86.97,-149.91) (-86.27,-149.91) (-85.56,-149.91) (-84.85,-149.91) (-84.15,-149.91) (-83.44,-149.91) (-82.73,-149.91) (-82.02,-149.91) (-81.32,-149.91) (-80.61,-149.91) (-79.90,-149.91) (-79.20,-149.91) (-78.49,-149.91) (-77.78,-149.91) (-77.07,-149.91) (-76.37,-149.91) (-75.66,-150.61) (-74.95,-150.61) (-74.25,-150.61) (-73.54,-150.61) (-72.83,-150.61) (-72.12,-150.61) (-71.42,-150.61) (-70.71,-150.61) (-70.00,-150.61) (-69.30,-150.61) (-68.59,-150.61) (-67.88,-150.61) (-67.18,-150.61) (-66.47,-150.61) (-65.76,-150.61) (-65.05,-150.61) (-64.35,-150.61) (-63.64,-150.61) (-62.93,-150.61) (-62.23,-150.61) (-61.52,-150.61) (-60.81,-150.61) (-60.10,-150.61) (-59.40,-150.61) (-105.36,-178.90) (-105.36,-178.19) (-104.65,-177.48) (-104.65,-176.78) (-103.94,-176.07) (-103.94,-175.36) (-103.24,-174.66) (-103.24,-173.95) (-102.53,-173.24) (-102.53,-172.53) (-102.53,-171.83) (-101.82,-171.12) (-101.82,-170.41) (-101.12,-169.71) (-101.12,-169.00) (-100.41,-168.29) (-100.41,-167.58) (-100.41,-166.88) (-99.70,-166.17) (-99.70,-165.46) (-98.99,-164.76) (-98.99,-164.05) (-98.29,-163.34) (-98.29,-162.63) (-97.58,-161.93) (-97.58,-161.22) (-97.58,-160.51) (-96.87,-159.81) (-96.87,-159.10) (-96.17,-158.39) (-96.17,-157.68) (-95.46,-156.98) (-95.46,-156.27) (-95.46,-155.56) (-94.75,-154.86) (-94.75,-154.15) (-94.05,-153.44) (-94.05,-152.74) (-93.34,-152.03) (-93.34,-151.32) (-92.63,-150.61) (-92.63,-149.91) (-105.36,-178.90) (-105.36,-178.19) (-105.36,-177.48) (-104.65,-176.78) (-104.65,-176.07) (-104.65,-175.36) (-104.65,-174.66) (-103.94,-173.95) (-103.94,-173.24) (-103.94,-172.53) (-103.94,-171.83) (-103.94,-171.12) (-103.24,-170.41) (-103.24,-169.71) (-103.24,-169.00) (-103.24,-168.29) (-102.53,-167.58) (-102.53,-166.88) (-102.53,-166.17) (-102.53,-165.46) (-102.53,-164.76) (-101.82,-164.05) (-101.82,-163.34) (-101.82,-162.63) (-101.82,-161.93) (-101.82,-161.22) (-101.12,-160.51) (-101.12,-159.81) (-101.12,-159.10) (-101.12,-158.39) (-100.41,-157.68) (-100.41,-156.98) (-100.41,-156.27) (-100.41,-155.56) (-100.41,-154.86) (-99.70,-154.15) (-99.70,-153.44) (-99.70,-152.74) (-99.70,-152.03) (-98.99,-151.32) (-98.99,-150.61) (-98.99,-149.91) (-98.99,-149.20) (-98.99,-148.49) (-98.29,-147.79) (-98.29,-147.08) (-98.29,-146.37) (-98.29,-145.66) (-97.58,-144.96) (-97.58,-144.25) (-97.58,-143.54) (-83.44,-173.95) (-83.44,-173.24) (-84.15,-172.53) (-84.15,-171.83) (-84.85,-171.12) (-84.85,-170.41) (-85.56,-169.71) (-85.56,-169.00) (-86.27,-168.29) (-86.27,-167.58) (-86.97,-166.88) (-86.97,-166.17) (-87.68,-165.46) (-87.68,-164.76) (-88.39,-164.05) (-88.39,-163.34) (-88.39,-162.63) (-89.10,-161.93) (-89.10,-161.22) (-89.80,-160.51) (-89.80,-159.81) (-90.51,-159.10) (-90.51,-158.39) (-91.22,-157.68) (-91.22,-156.98) (-91.92,-156.27) (-91.92,-155.56) (-92.63,-154.86) (-92.63,-154.15) (-92.63,-153.44) (-93.34,-152.74) (-93.34,-152.03) (-94.05,-151.32) (-94.05,-150.61) (-94.75,-149.91) (-94.75,-149.20) (-95.46,-148.49) (-95.46,-147.79) (-96.17,-147.08) (-96.17,-146.37) (-96.87,-145.66) (-96.87,-144.96) (-97.58,-144.25) (-97.58,-143.54) (-112.43,-190.92) (-111.72,-190.21) (-111.02,-190.21) (-110.31,-189.50) (-109.60,-189.50) (-108.89,-188.80) (-108.19,-188.09) (-107.48,-188.09) (-106.77,-187.38) (-106.07,-187.38) (-105.36,-186.68) (-104.65,-186.68) (-103.94,-185.97) (-103.24,-185.26) (-102.53,-185.26) (-101.82,-184.55) (-101.12,-184.55) (-100.41,-183.85) (-99.70,-183.14) (-98.99,-183.14) (-98.29,-182.43) (-97.58,-182.43) (-96.87,-181.73) (-96.17,-181.73) (-95.46,-181.02) (-94.75,-180.31) (-94.05,-180.31) (-93.34,-179.61) (-92.63,-179.61) (-91.92,-178.90) (-91.22,-178.19) (-90.51,-178.19) (-89.80,-177.48) (-89.10,-177.48) (-88.39,-176.78) (-87.68,-176.78) (-86.97,-176.07) (-86.27,-175.36) (-85.56,-175.36) (-84.85,-174.66) (-84.15,-174.66) (-83.44,-173.95) (-91.92,-217.79) (-92.63,-217.08) (-93.34,-216.37) (-93.34,-215.67) (-94.05,-214.96) (-94.75,-214.25) (-95.46,-213.55) (-95.46,-212.84) (-96.17,-212.13) (-96.87,-211.42) (-97.58,-210.72) (-97.58,-210.01) (-98.29,-209.30) (-98.99,-208.60) (-99.70,-207.89) (-99.70,-207.18) (-100.41,-206.48) (-101.12,-205.77) (-101.82,-205.06) (-101.82,-204.35) (-102.53,-203.65) (-103.24,-202.94) (-103.94,-202.23) (-104.65,-201.53) (-104.65,-200.82) (-105.36,-200.11) (-106.07,-199.40) (-106.77,-198.70) (-106.77,-197.99) (-107.48,-197.28) (-108.19,-196.58) (-108.89,-195.87) (-108.89,-195.16) (-109.60,-194.45) (-110.31,-193.75) (-111.02,-193.04) (-111.02,-192.33) (-111.72,-191.63) (-112.43,-190.92) (-84.85,-251.73) (-84.85,-251.02) (-84.85,-250.32) (-85.56,-249.61) (-85.56,-248.90) (-85.56,-248.19) (-85.56,-247.49) (-85.56,-246.78) (-86.27,-246.07) (-86.27,-245.37) (-86.27,-244.66) (-86.27,-243.95) (-86.27,-243.24) (-86.97,-242.54) (-86.97,-241.83) (-86.97,-241.12) (-86.97,-240.42) (-87.68,-239.71) (-87.68,-239.00) (-87.68,-238.29) (-87.68,-237.59) (-87.68,-236.88) (-88.39,-236.17) (-88.39,-235.47) (-88.39,-234.76) (-88.39,-234.05) (-88.39,-233.35) (-89.10,-232.64) (-89.10,-231.93) (-89.10,-231.22) (-89.10,-230.52) (-89.10,-229.81) (-89.80,-229.10) (-89.80,-228.40) (-89.80,-227.69) (-89.80,-226.98) (-89.80,-226.27) (-90.51,-225.57) (-90.51,-224.86) (-90.51,-224.15) (-90.51,-223.45) (-91.22,-222.74) (-91.22,-222.03) (-91.22,-221.32) (-91.22,-220.62) (-91.22,-219.91) (-91.92,-219.20) (-91.92,-218.50) (-91.92,-217.79) (-116.67,-255.97) (-115.97,-255.97) (-115.26,-255.97) (-114.55,-255.97) (-113.84,-255.27) (-113.14,-255.27) (-112.43,-255.27) (-111.72,-255.27) (-111.02,-255.27) (-110.31,-255.27) (-109.60,-255.27) (-108.89,-255.27) (-108.19,-254.56) (-107.48,-254.56) (-106.77,-254.56) (-106.07,-254.56) (-105.36,-254.56) (-104.65,-254.56) (-103.94,-254.56) (-103.24,-253.85) (-102.53,-253.85) (-101.82,-253.85) (-101.12,-253.85) (-100.41,-253.85) (-99.70,-253.85) (-98.99,-253.85) (-98.29,-253.85) (-97.58,-253.14) (-96.87,-253.14) (-96.17,-253.14) (-95.46,-253.14) (-94.75,-253.14) (-94.05,-253.14) (-93.34,-253.14) (-92.63,-252.44) (-91.92,-252.44) (-91.22,-252.44) (-90.51,-252.44) (-89.80,-252.44) (-89.10,-252.44) (-88.39,-252.44) (-87.68,-252.44) (-86.97,-251.73) (-86.27,-251.73) (-85.56,-251.73) (-84.85,-251.73) (-136.47,-281.43) (-135.76,-280.72) (-135.06,-280.01) (-135.06,-279.31) (-134.35,-278.60) (-133.64,-277.89) (-132.94,-277.19) (-132.94,-276.48) (-132.23,-275.77) (-131.52,-275.06) (-130.81,-274.36) (-130.11,-273.65) (-130.11,-272.94) (-129.40,-272.24) (-128.69,-271.53) (-127.99,-270.82) (-127.99,-270.11) (-127.28,-269.41) (-126.57,-268.70) (-125.87,-267.99) (-125.16,-267.29) (-125.16,-266.58) (-124.45,-265.87) (-123.74,-265.17) (-123.04,-264.46) (-123.04,-263.75) (-122.33,-263.04) (-121.62,-262.34) (-120.92,-261.63) (-120.21,-260.92) (-120.21,-260.22) (-119.50,-259.51) (-118.79,-258.80) (-118.09,-258.09) (-118.09,-257.39) (-117.38,-256.68) (-116.67,-255.97) (-136.47,-281.43) (-135.76,-280.72) (-135.76,-280.01) (-135.06,-279.31) (-135.06,-278.60) (-134.35,-277.89) (-134.35,-277.19) (-133.64,-276.48) (-133.64,-275.77) (-132.94,-275.06) (-132.23,-274.36) (-132.23,-273.65) (-131.52,-272.94) (-131.52,-272.24) (-130.81,-271.53) (-130.81,-270.82) (-130.11,-270.11) (-130.11,-269.41) (-129.40,-268.70) (-128.69,-267.99) (-128.69,-267.29) (-127.99,-266.58) (-127.99,-265.87) (-127.28,-265.17) (-127.28,-264.46) (-126.57,-263.75) (-126.57,-263.04) (-125.87,-262.34) (-125.16,-261.63) (-125.16,-260.92) (-124.45,-260.22) (-124.45,-259.51) (-123.74,-258.80) (-123.74,-258.09) (-123.04,-257.39) (-123.04,-256.68) (-122.33,-255.97) (-121.62,-255.27) (-121.62,-254.56) (-120.92,-253.85) (-120.92,-253.14) (-120.21,-252.44) (-120.21,-251.73) (-119.50,-251.02) (-119.50,-250.32) (-118.79,-249.61) (-152.74,-255.97) (-152.03,-255.97) (-151.32,-255.97) (-150.61,-255.27) (-149.91,-255.27) (-149.20,-255.27) (-148.49,-255.27) (-147.79,-255.27) (-147.08,-255.27) (-146.37,-254.56) (-145.66,-254.56) (-144.96,-254.56) (-144.25,-254.56) (-143.54,-254.56) (-142.84,-253.85) (-142.13,-253.85) (-141.42,-253.85) (-140.71,-253.85) (-140.01,-253.85) (-139.30,-253.14) (-138.59,-253.14) (-137.89,-253.14) (-137.18,-253.14) (-136.47,-253.14) (-135.76,-253.14) (-135.06,-252.44) (-134.35,-252.44) (-133.64,-252.44) (-132.94,-252.44) (-132.23,-252.44) (-131.52,-251.73) (-130.81,-251.73) (-130.11,-251.73) (-129.40,-251.73) (-128.69,-251.73) (-127.99,-251.02) (-127.28,-251.02) (-126.57,-251.02) (-125.87,-251.02) (-125.16,-251.02) (-124.45,-251.02) (-123.74,-250.32) (-123.04,-250.32) (-122.33,-250.32) (-121.62,-250.32) (-120.92,-250.32) (-120.21,-249.61) (-119.50,-249.61) (-118.79,-249.61) (-166.88,-286.38) (-166.88,-285.67) (-166.17,-284.96) (-166.17,-284.26) (-165.46,-283.55) (-165.46,-282.84) (-164.76,-282.14) (-164.76,-281.43) (-164.05,-280.72) (-164.05,-280.01) (-163.34,-279.31) (-163.34,-278.60) (-162.63,-277.89) (-162.63,-277.19) (-161.93,-276.48) (-161.93,-275.77) (-161.93,-275.06) (-161.22,-274.36) (-161.22,-273.65) (-160.51,-272.94) (-160.51,-272.24) (-159.81,-271.53) (-159.81,-270.82) (-159.10,-270.11) (-159.10,-269.41) (-158.39,-268.70) (-158.39,-267.99) (-157.68,-267.29) (-157.68,-266.58) (-157.68,-265.87) (-156.98,-265.17) (-156.98,-264.46) (-156.27,-263.75) (-156.27,-263.04) (-155.56,-262.34) (-155.56,-261.63) (-154.86,-260.92) (-154.86,-260.22) (-154.15,-259.51) (-154.15,-258.80) (-153.44,-258.09) (-153.44,-257.39) (-152.74,-256.68) (-152.74,-255.97) (-166.88,-286.38) (-166.88,-285.67) (-166.88,-284.96) (-166.88,-284.26) (-167.58,-283.55) (-167.58,-282.84) (-167.58,-282.14) (-167.58,-281.43) (-167.58,-280.72) (-167.58,-280.01) (-168.29,-279.31) (-168.29,-278.60) (-168.29,-277.89) (-168.29,-277.19) (-168.29,-276.48) (-168.29,-275.77) (-168.29,-275.06) (-169.00,-274.36) (-169.00,-273.65) (-169.00,-272.94) (-169.00,-272.24) (-169.00,-271.53) (-169.00,-270.82) (-169.00,-270.11) (-169.71,-269.41) (-169.71,-268.70) (-169.71,-267.99) (-169.71,-267.29) (-169.71,-266.58) (-169.71,-265.87) (-170.41,-265.17) (-170.41,-264.46) (-170.41,-263.75) (-170.41,-263.04) (-170.41,-262.34) (-170.41,-261.63) (-170.41,-260.92) (-171.12,-260.22) (-171.12,-259.51) (-171.12,-258.80) (-171.12,-258.09) (-171.12,-257.39) (-171.12,-256.68) (-171.83,-255.97) (-171.83,-255.27) (-171.83,-254.56) (-171.83,-253.85) (-171.83,-253.85) (-171.12,-253.85) (-170.41,-254.56) (-169.71,-254.56) (-169.00,-255.27) (-168.29,-255.27) (-167.58,-255.27) (-166.88,-255.97) (-166.17,-255.97) (-165.46,-255.97) (-164.76,-256.68) (-164.05,-256.68) (-163.34,-257.39) (-162.63,-257.39) (-161.93,-257.39) (-161.22,-258.09) (-160.51,-258.09) (-159.81,-258.80) (-159.10,-258.80) (-158.39,-258.80) (-157.68,-259.51) (-156.98,-259.51) (-156.27,-259.51) (-155.56,-260.22) (-154.86,-260.22) (-154.15,-260.92) (-153.44,-260.92) (-152.74,-260.92) (-152.03,-261.63) (-151.32,-261.63) (-150.61,-262.34) (-149.91,-262.34) (-149.20,-262.34) (-148.49,-263.04) (-147.79,-263.04) (-147.08,-263.75) (-146.37,-263.75) (-145.66,-263.75) (-144.96,-264.46) (-144.25,-264.46) (-143.54,-264.46) (-142.84,-265.17) (-142.13,-265.17) (-141.42,-265.87) (-140.71,-265.87) (-140.71,-265.87) (-140.01,-265.87) (-139.30,-265.17) (-138.59,-265.17) (-137.89,-265.17) (-137.18,-265.17) (-136.47,-264.46) (-135.76,-264.46) (-135.06,-264.46) (-134.35,-263.75) (-133.64,-263.75) (-132.94,-263.75) (-132.23,-263.75) (-131.52,-263.04) (-130.81,-263.04) (-130.11,-263.04) (-129.40,-262.34) (-128.69,-262.34) (-127.99,-262.34) (-127.28,-262.34) (-126.57,-261.63) (-125.87,-261.63) (-125.16,-261.63) (-124.45,-260.92) (-123.74,-260.92) (-123.04,-260.92) (-122.33,-260.22) (-121.62,-260.22) (-120.92,-260.22) (-120.21,-260.22) (-119.50,-259.51) (-118.79,-259.51) (-118.09,-259.51) (-117.38,-258.80) (-116.67,-258.80) (-115.97,-258.80) (-115.26,-258.80) (-114.55,-258.09) (-113.84,-258.09) (-113.14,-258.09) (-112.43,-257.39) (-111.72,-257.39) (-111.02,-257.39) (-110.31,-257.39) (-109.60,-256.68) (-108.89,-256.68) (-108.89,-256.68) (-108.19,-255.97) (-108.19,-255.27) (-107.48,-254.56) (-106.77,-253.85) (-106.07,-253.14) (-106.07,-252.44) (-105.36,-251.73) (-104.65,-251.02) (-104.65,-250.32) (-103.94,-249.61) (-103.24,-248.90) (-103.24,-248.19) (-102.53,-247.49) (-101.82,-246.78) (-101.12,-246.07) (-101.12,-245.37) (-100.41,-244.66) (-99.70,-243.95) (-99.70,-243.24) (-98.99,-242.54) (-98.29,-241.83) (-97.58,-241.12) (-97.58,-240.42) (-96.87,-239.71) (-96.17,-239.00) (-96.17,-238.29) (-95.46,-237.59) (-94.75,-236.88) (-94.05,-236.17) (-94.05,-235.47) (-93.34,-234.76) (-92.63,-234.05) (-92.63,-233.35) (-91.92,-232.64) (-91.22,-231.93) (-91.22,-231.22) (-90.51,-230.52) (-89.80,-229.81) (-89.10,-229.10) (-89.10,-228.40) (-88.39,-227.69) (-111.72,-251.73) (-111.02,-251.02) (-110.31,-250.32) (-109.60,-249.61) (-108.89,-248.90) (-108.19,-248.19) (-107.48,-247.49) (-106.77,-246.78) (-106.07,-246.07) (-105.36,-245.37) (-104.65,-244.66) (-103.94,-243.95) (-103.24,-243.24) (-102.53,-242.54) (-101.82,-241.83) (-101.12,-241.12) (-100.41,-240.42) (-100.41,-239.71) (-99.70,-239.00) (-98.99,-238.29) (-98.29,-237.59) (-97.58,-236.88) (-96.87,-236.17) (-96.17,-235.47) (-95.46,-234.76) (-94.75,-234.05) (-94.05,-233.35) (-93.34,-232.64) (-92.63,-231.93) (-91.92,-231.22) (-91.22,-230.52) (-90.51,-229.81) (-89.80,-229.10) (-89.10,-228.40) (-88.39,-227.69) (-111.72,-251.73) (-111.02,-251.73) (-110.31,-251.73) (-109.60,-251.73) (-108.89,-251.73) (-108.19,-251.02) (-107.48,-251.02) (-106.77,-251.02) (-106.07,-251.02) (-105.36,-251.02) (-104.65,-251.02) (-103.94,-251.02) (-103.24,-251.02) (-102.53,-250.32) (-101.82,-250.32) (-101.12,-250.32) (-100.41,-250.32) (-99.70,-250.32) (-98.99,-250.32) (-98.29,-250.32) (-97.58,-250.32) (-96.87,-250.32) (-96.17,-249.61) (-95.46,-249.61) (-94.75,-249.61) (-94.05,-249.61) (-93.34,-249.61) (-92.63,-249.61) (-91.92,-249.61) (-91.22,-249.61) (-90.51,-248.90) (-89.80,-248.90) (-89.10,-248.90) (-88.39,-248.90) (-87.68,-248.90) (-86.97,-248.90) (-86.27,-248.90) (-85.56,-248.90) (-84.85,-248.90) (-84.15,-248.19) (-83.44,-248.19) (-82.73,-248.19) (-82.02,-248.19) (-81.32,-248.19) (-80.61,-248.19) (-79.90,-248.19) (-79.20,-248.19) (-78.49,-247.49) (-77.78,-247.49) (-77.07,-247.49) (-76.37,-247.49) (-75.66,-247.49) (-75.66,-247.49) (-74.95,-246.78) (-74.25,-246.07) (-73.54,-245.37) (-72.83,-244.66) (-72.12,-243.95) (-71.42,-243.24) (-70.71,-242.54) (-70.00,-241.83) (-70.00,-241.12) (-69.30,-240.42) (-68.59,-239.71) (-67.88,-239.00) (-67.18,-238.29) (-66.47,-237.59) (-65.76,-236.88) (-65.05,-236.17) (-64.35,-235.47) (-63.64,-234.76) (-62.93,-234.05) (-62.23,-233.35) (-61.52,-232.64) (-60.81,-231.93) (-60.10,-231.22) (-59.40,-230.52) (-58.69,-229.81) (-57.98,-229.10) (-57.98,-228.40) (-57.28,-227.69) (-56.57,-226.98) (-55.86,-226.27) (-55.15,-225.57) (-54.45,-224.86) (-53.74,-224.15) (-53.03,-223.45) (-52.33,-222.74) (-84.85,-211.42) (-84.15,-211.42) (-83.44,-212.13) (-82.73,-212.13) (-82.02,-212.13) (-81.32,-212.84) (-80.61,-212.84) (-79.90,-212.84) (-79.20,-213.55) (-78.49,-213.55) (-77.78,-213.55) (-77.07,-214.25) (-76.37,-214.25) (-75.66,-214.96) (-74.95,-214.96) (-74.25,-214.96) (-73.54,-215.67) (-72.83,-215.67) (-72.12,-215.67) (-71.42,-216.37) (-70.71,-216.37) (-70.00,-216.37) (-69.30,-217.08) (-68.59,-217.08) (-67.88,-217.08) (-67.18,-217.79) (-66.47,-217.79) (-65.76,-217.79) (-65.05,-218.50) (-64.35,-218.50) (-63.64,-218.50) (-62.93,-219.20) (-62.23,-219.20) (-61.52,-219.20) (-60.81,-219.91) (-60.10,-219.91) (-59.40,-220.62) (-58.69,-220.62) (-57.98,-220.62) (-57.28,-221.32) (-56.57,-221.32) (-55.86,-221.32) (-55.15,-222.03) (-54.45,-222.03) (-53.74,-222.03) (-53.03,-222.74) (-52.33,-222.74) (-116.67,-201.53) (-115.97,-201.53) (-115.26,-202.23) (-114.55,-202.23) (-113.84,-202.23) (-113.14,-202.94) (-112.43,-202.94) (-111.72,-202.94) (-111.02,-202.94) (-110.31,-203.65) (-109.60,-203.65) (-108.89,-203.65) (-108.19,-204.35) (-107.48,-204.35) (-106.77,-204.35) (-106.07,-205.06) (-105.36,-205.06) (-104.65,-205.06) (-103.94,-205.77) (-103.24,-205.77) (-102.53,-205.77) (-101.82,-206.48) (-101.12,-206.48) (-100.41,-206.48) (-99.70,-206.48) (-98.99,-207.18) (-98.29,-207.18) (-97.58,-207.18) (-96.87,-207.89) (-96.17,-207.89) (-95.46,-207.89) (-94.75,-208.60) (-94.05,-208.60) (-93.34,-208.60) (-92.63,-209.30) (-91.92,-209.30) (-91.22,-209.30) (-90.51,-210.01) (-89.80,-210.01) (-89.10,-210.01) (-88.39,-210.01) (-87.68,-210.72) (-86.97,-210.72) (-86.27,-210.72) (-85.56,-211.42) (-84.85,-211.42) (-148.49,-191.63) (-147.79,-191.63) (-147.08,-192.33) (-146.37,-192.33) (-145.66,-192.33) (-144.96,-193.04) (-144.25,-193.04) (-143.54,-193.04) (-142.84,-193.04) (-142.13,-193.75) (-141.42,-193.75) (-140.71,-193.75) (-140.01,-194.45) (-139.30,-194.45) (-138.59,-194.45) (-137.89,-195.16) (-137.18,-195.16) (-136.47,-195.16) (-135.76,-195.87) (-135.06,-195.87) (-134.35,-195.87) (-133.64,-196.58) (-132.94,-196.58) (-132.23,-196.58) (-131.52,-196.58) (-130.81,-197.28) (-130.11,-197.28) (-129.40,-197.28) (-128.69,-197.99) (-127.99,-197.99) (-127.28,-197.99) (-126.57,-198.70) (-125.87,-198.70) (-125.16,-198.70) (-124.45,-199.40) (-123.74,-199.40) (-123.04,-199.40) (-122.33,-200.11) (-121.62,-200.11) (-120.92,-200.11) (-120.21,-200.11) (-119.50,-200.82) (-118.79,-200.82) (-118.09,-200.82) (-117.38,-201.53) (-116.67,-201.53) (-181.73,-175.36) (-181.02,-175.36) (-180.31,-176.07) (-179.61,-176.07) (-178.90,-176.78) (-178.19,-176.78) (-177.48,-177.48) (-176.78,-177.48) (-176.07,-178.19) (-175.36,-178.19) (-174.66,-178.90) (-173.95,-178.90) (-173.24,-179.61) (-172.53,-179.61) (-171.83,-180.31) (-171.12,-180.31) (-170.41,-181.02) (-169.71,-181.02) (-169.00,-181.73) (-168.29,-181.73) (-167.58,-182.43) (-166.88,-182.43) (-166.17,-183.14) (-165.46,-183.14) (-164.76,-183.85) (-164.05,-183.85) (-163.34,-184.55) (-162.63,-184.55) (-161.93,-185.26) (-161.22,-185.26) (-160.51,-185.97) (-159.81,-185.97) (-159.10,-186.68) (-158.39,-186.68) (-157.68,-187.38) (-156.98,-187.38) (-156.27,-188.09) (-155.56,-188.09) (-154.86,-188.80) (-154.15,-188.80) (-153.44,-189.50) (-152.74,-189.50) (-152.03,-190.21) (-151.32,-190.21) (-150.61,-190.92) (-149.91,-190.92) (-149.20,-191.63) (-148.49,-191.63) (-181.73,-175.36) (-182.43,-174.66) (-183.14,-173.95) (-183.85,-173.24) (-184.55,-172.53) (-185.26,-171.83) (-185.97,-171.12) (-186.68,-170.41) (-187.38,-169.71) (-187.38,-169.00) (-188.09,-168.29) (-188.80,-167.58) (-189.50,-166.88) (-190.21,-166.17) (-190.92,-165.46) (-191.63,-164.76) (-192.33,-164.05) (-193.04,-163.34) (-193.75,-162.63) (-194.45,-161.93) (-195.16,-161.22) (-195.87,-160.51) (-196.58,-159.81) (-197.28,-159.10) (-197.99,-158.39) (-198.70,-157.68) (-199.40,-156.98) (-199.40,-156.27) (-200.11,-155.56) (-200.82,-154.86) (-201.53,-154.15) (-202.23,-153.44) (-202.94,-152.74) (-203.65,-152.03) (-204.35,-151.32) (-205.06,-150.61) (-205.77,-149.91) (-205.77,-149.91) (-205.77,-149.20) (-205.77,-148.49) (-205.06,-147.79) (-205.06,-147.08) (-205.06,-146.37) (-205.06,-145.66) (-205.06,-144.96) (-204.35,-144.25) (-204.35,-143.54) (-204.35,-142.84) (-204.35,-142.13) (-203.65,-141.42) (-203.65,-140.71) (-203.65,-140.01) (-203.65,-139.30) (-203.65,-138.59) (-202.94,-137.89) (-202.94,-137.18) (-202.94,-136.47) (-202.94,-135.76) (-202.94,-135.06) (-202.23,-134.35) (-202.23,-133.64) (-202.23,-132.94) (-202.23,-132.23) (-201.53,-131.52) (-201.53,-130.81) (-201.53,-130.11) (-201.53,-129.40) (-201.53,-128.69) (-200.82,-127.99) (-200.82,-127.28) (-200.82,-126.57) (-200.82,-125.87) (-200.82,-125.16) (-200.11,-124.45) (-200.11,-123.74) (-200.11,-123.04) (-200.11,-122.33) (-199.40,-121.62) (-199.40,-120.92) (-199.40,-120.21) (-199.40,-119.50) (-199.40,-118.79) (-198.70,-118.09) (-198.70,-117.38) (-198.70,-116.67) (-208.60,-150.61) (-208.60,-149.91) (-207.89,-149.20) (-207.89,-148.49) (-207.89,-147.79) (-207.89,-147.08) (-207.18,-146.37) (-207.18,-145.66) (-207.18,-144.96) (-206.48,-144.25) (-206.48,-143.54) (-206.48,-142.84) (-206.48,-142.13) (-205.77,-141.42) (-205.77,-140.71) (-205.77,-140.01) (-205.06,-139.30) (-205.06,-138.59) (-205.06,-137.89) (-204.35,-137.18) (-204.35,-136.47) (-204.35,-135.76) (-204.35,-135.06) (-203.65,-134.35) (-203.65,-133.64) (-203.65,-132.94) (-202.94,-132.23) (-202.94,-131.52) (-202.94,-130.81) (-202.94,-130.11) (-202.23,-129.40) (-202.23,-128.69) (-202.23,-127.99) (-201.53,-127.28) (-201.53,-126.57) (-201.53,-125.87) (-201.53,-125.16) (-200.82,-124.45) (-200.82,-123.74) (-200.82,-123.04) (-200.11,-122.33) (-200.11,-121.62) (-200.11,-120.92) (-199.40,-120.21) (-199.40,-119.50) (-199.40,-118.79) (-199.40,-118.09) (-198.70,-117.38) (-198.70,-116.67) (-208.60,-150.61) (-208.60,-149.91) (-209.30,-149.20) (-209.30,-148.49) (-209.30,-147.79) (-210.01,-147.08) (-210.01,-146.37) (-210.01,-145.66) (-210.72,-144.96) (-210.72,-144.25) (-210.72,-143.54) (-211.42,-142.84) (-211.42,-142.13) (-211.42,-141.42) (-212.13,-140.71) (-212.13,-140.01) (-212.13,-139.30) (-212.84,-138.59) (-212.84,-137.89) (-212.84,-137.18) (-213.55,-136.47) (-213.55,-135.76) (-213.55,-135.06) (-214.25,-134.35) (-214.25,-133.64) (-214.25,-132.94) (-214.25,-132.23) (-214.96,-131.52) (-214.96,-130.81) (-214.96,-130.11) (-215.67,-129.40) (-215.67,-128.69) (-215.67,-127.99) (-216.37,-127.28) (-216.37,-126.57) (-216.37,-125.87) (-217.08,-125.16) (-217.08,-124.45) (-217.08,-123.74) (-217.79,-123.04) (-217.79,-122.33) (-217.79,-121.62) (-218.50,-120.92) (-218.50,-120.21) (-218.50,-119.50) (-219.20,-118.79) (-219.20,-118.09) (-219.20,-117.38) (-219.91,-116.67) (-219.91,-115.97) (-219.91,-115.97) (-219.20,-115.26) (-218.50,-115.26) (-217.79,-114.55) (-217.08,-113.84) (-216.37,-113.84) (-215.67,-113.14) (-214.96,-113.14) (-214.25,-112.43) (-213.55,-111.72) (-212.84,-111.72) (-212.13,-111.02) (-211.42,-110.31) (-210.72,-110.31) (-210.01,-109.60) (-209.30,-108.89) (-208.60,-108.89) (-207.89,-108.19) (-207.18,-108.19) (-206.48,-107.48) (-205.77,-106.77) (-205.06,-106.77) (-204.35,-106.07) (-203.65,-105.36) (-202.94,-105.36) (-202.23,-104.65) (-201.53,-104.65) (-200.82,-103.94) (-200.11,-103.24) (-199.40,-103.24) (-198.70,-102.53) (-197.99,-101.82) (-197.28,-101.82) (-196.58,-101.12) (-195.87,-100.41) (-195.16,-100.41) (-194.45,-99.70) (-193.75,-99.70) (-193.04,-98.99) (-192.33,-98.29) (-191.63,-98.29) (-190.92,-97.58) (-190.92,-97.58) (-190.92,-96.87) (-191.63,-96.17) (-191.63,-95.46) (-192.33,-94.75) (-192.33,-94.05) (-192.33,-93.34) (-193.04,-92.63) (-193.04,-91.92) (-193.75,-91.22) (-193.75,-90.51) (-193.75,-89.80) (-194.45,-89.10) (-194.45,-88.39) (-195.16,-87.68) (-195.16,-86.97) (-195.16,-86.27) (-195.87,-85.56) (-195.87,-84.85) (-196.58,-84.15) (-196.58,-83.44) (-196.58,-82.73) (-197.28,-82.02) (-197.28,-81.32) (-197.99,-80.61) (-197.99,-79.90) (-198.70,-79.20) (-198.70,-78.49) (-198.70,-77.78) (-199.40,-77.07) (-199.40,-76.37) (-200.11,-75.66) (-200.11,-74.95) (-200.11,-74.25) (-200.82,-73.54) (-200.82,-72.83) (-201.53,-72.12) (-201.53,-71.42) (-201.53,-70.71) (-202.23,-70.00) (-202.23,-69.30) (-202.94,-68.59) (-202.94,-67.88) (-202.94,-67.18) (-203.65,-66.47) (-203.65,-65.76) (-204.35,-65.05) (-204.35,-64.35) (-204.35,-64.35) (-203.65,-64.35) (-202.94,-63.64) (-202.23,-63.64) (-201.53,-62.93) (-200.82,-62.93) (-200.11,-62.23) (-199.40,-62.23) (-198.70,-61.52) (-197.99,-61.52) (-197.28,-60.81) (-196.58,-60.81) (-195.87,-60.10) (-195.16,-60.10) (-194.45,-59.40) (-193.75,-59.40) (-193.04,-58.69) (-192.33,-58.69) (-191.63,-57.98) (-190.92,-57.98) (-190.21,-57.28) (-189.50,-57.28) (-188.80,-56.57) (-188.09,-56.57) (-187.38,-55.86) (-186.68,-55.86) (-185.97,-55.15) (-185.26,-55.15) (-184.55,-54.45) (-183.85,-54.45) (-183.14,-53.74) (-182.43,-53.74) (-181.73,-53.03) (-181.02,-53.03) (-180.31,-52.33) (-179.61,-52.33) (-178.90,-51.62) (-178.19,-51.62) (-177.48,-50.91) (-176.78,-50.91) (-176.07,-50.20) (-175.36,-50.20) (-174.66,-49.50) (-173.95,-49.50) (-173.95,-49.50) (-174.66,-48.79) (-175.36,-48.08) (-176.07,-47.38) (-176.07,-46.67) (-176.78,-45.96) (-177.48,-45.25) (-178.19,-44.55) (-178.90,-43.84) (-179.61,-43.13) (-179.61,-42.43) (-180.31,-41.72) (-181.02,-41.01) (-181.73,-40.31) (-182.43,-39.60) (-183.14,-38.89) (-183.14,-38.18) (-183.85,-37.48) (-184.55,-36.77) (-185.26,-36.06) (-185.97,-35.36) (-186.68,-34.65) (-187.38,-33.94) (-187.38,-33.23) (-188.09,-32.53) (-188.80,-31.82) (-189.50,-31.11) (-190.21,-30.41) (-190.92,-29.70) (-190.92,-28.99) (-191.63,-28.28) (-192.33,-27.58) (-193.04,-26.87) (-193.75,-26.16) (-194.45,-25.46) (-194.45,-24.75) (-195.16,-24.04) (-195.87,-23.33) (-196.58,-22.63) (-196.58,-22.63) (-195.87,-21.92) (-195.87,-21.21) (-195.16,-20.51) (-194.45,-19.80) (-194.45,-19.09) (-193.75,-18.38) (-193.04,-17.68) (-192.33,-16.97) (-192.33,-16.26) (-191.63,-15.56) (-190.92,-14.85) (-190.92,-14.14) (-190.21,-13.44) (-189.50,-12.73) (-189.50,-12.02) (-188.80,-11.31) (-188.09,-10.61) (-188.09,-9.90) (-187.38,-9.19) (-186.68,-8.49) (-185.97,-7.78) (-185.97,-7.07) (-185.26,-6.36) (-184.55,-5.66) (-184.55,-4.95) (-183.85,-4.24) (-183.14,-3.54) (-183.14,-2.83) (-182.43,-2.12) (-181.73,-1.41) (-181.73,-0.71) (-181.02, 0.00) (-180.31, 0.71) (-179.61, 1.41) (-179.61, 2.12) (-178.90, 2.83) (-178.19, 3.54) (-178.19, 4.24) (-177.48, 4.95) (-209.30,-10.61) (-208.60,-10.61) (-207.89,-9.90) (-207.18,-9.90) (-206.48,-9.19) (-205.77,-9.19) (-205.06,-8.49) (-204.35,-8.49) (-203.65,-7.78) (-202.94,-7.78) (-202.23,-7.07) (-201.53,-7.07) (-200.82,-6.36) (-200.11,-6.36) (-199.40,-5.66) (-198.70,-5.66) (-197.99,-4.95) (-197.28,-4.95) (-196.58,-4.24) (-195.87,-4.24) (-195.16,-3.54) (-194.45,-3.54) (-193.75,-2.83) (-193.04,-2.83) (-192.33,-2.12) (-191.63,-2.12) (-190.92,-1.41) (-190.21,-1.41) (-189.50,-0.71) (-188.80,-0.71) (-188.09, 0.00) (-187.38, 0.00) (-186.68, 0.71) (-185.97, 0.71) (-185.26, 1.41) (-184.55, 1.41) (-183.85, 2.12) (-183.14, 2.12) (-182.43, 2.83) (-181.73, 2.83) (-181.02, 3.54) (-180.31, 3.54) (-179.61, 4.24) (-178.90, 4.24) (-178.19, 4.95) (-177.48, 4.95) (-193.75,-37.48) (-194.45,-36.77) (-194.45,-36.06) (-195.16,-35.36) (-195.16,-34.65) (-195.87,-33.94) (-195.87,-33.23) (-196.58,-32.53) (-197.28,-31.82) (-197.28,-31.11) (-197.99,-30.41) (-197.99,-29.70) (-198.70,-28.99) (-199.40,-28.28) (-199.40,-27.58) (-200.11,-26.87) (-200.11,-26.16) (-200.82,-25.46) (-200.82,-24.75) (-201.53,-24.04) (-202.23,-23.33) (-202.23,-22.63) (-202.94,-21.92) (-202.94,-21.21) (-203.65,-20.51) (-203.65,-19.80) (-204.35,-19.09) (-205.06,-18.38) (-205.06,-17.68) (-205.77,-16.97) (-205.77,-16.26) (-206.48,-15.56) (-207.18,-14.85) (-207.18,-14.14) (-207.89,-13.44) (-207.89,-12.73) (-208.60,-12.02) (-208.60,-11.31) (-209.30,-10.61) (-211.42,-66.47) (-210.72,-65.76) (-210.72,-65.05) (-210.01,-64.35) (-210.01,-63.64) (-209.30,-62.93) (-208.60,-62.23) (-208.60,-61.52) (-207.89,-60.81) (-207.89,-60.10) (-207.18,-59.40) (-206.48,-58.69) (-206.48,-57.98) (-205.77,-57.28) (-205.06,-56.57) (-205.06,-55.86) (-204.35,-55.15) (-204.35,-54.45) (-203.65,-53.74) (-202.94,-53.03) (-202.94,-52.33) (-202.23,-51.62) (-202.23,-50.91) (-201.53,-50.20) (-200.82,-49.50) (-200.82,-48.79) (-200.11,-48.08) (-200.11,-47.38) (-199.40,-46.67) (-198.70,-45.96) (-198.70,-45.25) (-197.99,-44.55) (-197.28,-43.84) (-197.28,-43.13) (-196.58,-42.43) (-196.58,-41.72) (-195.87,-41.01) (-195.16,-40.31) (-195.16,-39.60) (-194.45,-38.89) (-194.45,-38.18) (-193.75,-37.48) (-211.42,-66.47) (-210.72,-65.76) (-210.01,-65.05) (-209.30,-65.05) (-208.60,-64.35) (-207.89,-63.64) (-207.18,-62.93) (-206.48,-62.23) (-205.77,-61.52) (-205.06,-61.52) (-204.35,-60.81) (-203.65,-60.10) (-202.94,-59.40) (-202.23,-58.69) (-201.53,-58.69) (-200.82,-57.98) (-200.11,-57.28) (-199.40,-56.57) (-198.70,-55.86) (-197.99,-55.86) (-197.28,-55.15) (-196.58,-54.45) (-195.87,-53.74) (-195.16,-53.03) (-194.45,-52.33) (-193.75,-52.33) (-193.04,-51.62) (-192.33,-50.91) (-191.63,-50.20) (-190.92,-49.50) (-190.21,-49.50) (-189.50,-48.79) (-188.80,-48.08) (-188.09,-47.38) (-187.38,-46.67) (-186.68,-45.96) (-185.97,-45.96) (-185.26,-45.25) (-184.55,-44.55) (-184.55,-44.55) (-185.26,-43.84) (-185.26,-43.13) (-185.97,-42.43) (-186.68,-41.72) (-186.68,-41.01) (-187.38,-40.31) (-188.09,-39.60) (-188.09,-38.89) (-188.80,-38.18) (-189.50,-37.48) (-189.50,-36.77) (-190.21,-36.06) (-190.92,-35.36) (-190.92,-34.65) (-191.63,-33.94) (-192.33,-33.23) (-192.33,-32.53) (-193.04,-31.82) (-193.75,-31.11) (-193.75,-30.41) (-194.45,-29.70) (-194.45,-28.99) (-195.16,-28.28) (-195.87,-27.58) (-195.87,-26.87) (-196.58,-26.16) (-197.28,-25.46) (-197.28,-24.75) (-197.99,-24.04) (-198.70,-23.33) (-198.70,-22.63) (-199.40,-21.92) (-200.11,-21.21) (-200.11,-20.51) (-200.82,-19.80) (-201.53,-19.09) (-201.53,-18.38) (-202.23,-17.68) (-202.94,-16.97) (-202.94,-16.26) (-203.65,-15.56) (-203.65,-15.56) (-202.94,-14.85) (-202.94,-14.14) (-202.23,-13.44) (-201.53,-12.73) (-201.53,-12.02) (-200.82,-11.31) (-200.11,-10.61) (-199.40,-9.90) (-199.40,-9.19) (-198.70,-8.49) (-197.99,-7.78) (-197.99,-7.07) (-197.28,-6.36) (-196.58,-5.66) (-196.58,-4.95) (-195.87,-4.24) (-195.16,-3.54) (-195.16,-2.83) (-194.45,-2.12) (-193.75,-1.41) (-193.04,-0.71) (-193.04, 0.00) (-192.33, 0.71) (-191.63, 1.41) (-191.63, 2.12) (-190.92, 2.83) (-190.21, 3.54) (-190.21, 4.24) (-189.50, 4.95) (-188.80, 5.66) (-188.09, 6.36) (-188.09, 7.07) (-187.38, 7.78) (-186.68, 8.49) (-186.68, 9.19) (-185.97, 9.90) (-213.55,-4.95) (-212.84,-4.24) (-212.13,-4.24) (-211.42,-3.54) (-210.72,-3.54) (-210.01,-2.83) (-209.30,-2.83) (-208.60,-2.12) (-207.89,-2.12) (-207.18,-1.41) (-206.48,-1.41) (-205.77,-0.71) (-205.06,-0.71) (-204.35, 0.00) (-203.65, 0.71) (-202.94, 0.71) (-202.23, 1.41) (-201.53, 1.41) (-200.82, 2.12) (-200.11, 2.12) (-199.40, 2.83) (-198.70, 2.83) (-197.99, 3.54) (-197.28, 3.54) (-196.58, 4.24) (-195.87, 4.24) (-195.16, 4.95) (-194.45, 5.66) (-193.75, 5.66) (-193.04, 6.36) (-192.33, 6.36) (-191.63, 7.07) (-190.92, 7.07) (-190.21, 7.78) (-189.50, 7.78) (-188.80, 8.49) (-188.09, 8.49) (-187.38, 9.19) (-186.68, 9.19) (-185.97, 9.90) (-213.55,-4.95) (-212.84,-4.95) (-212.13,-5.66) (-211.42,-5.66) (-210.72,-5.66) (-210.01,-5.66) (-209.30,-6.36) (-208.60,-6.36) (-207.89,-6.36) (-207.18,-6.36) (-206.48,-7.07) (-205.77,-7.07) (-205.06,-7.07) (-204.35,-7.07) (-203.65,-7.78) (-202.94,-7.78) (-202.23,-7.78) (-201.53,-8.49) (-200.82,-8.49) (-200.11,-8.49) (-199.40,-8.49) (-198.70,-9.19) (-197.99,-9.19) (-197.28,-9.19) (-196.58,-9.19) (-195.87,-9.90) (-195.16,-9.90) (-194.45,-9.90) (-193.75,-9.90) (-193.04,-10.61) (-192.33,-10.61) (-191.63,-10.61) (-190.92,-11.31) (-190.21,-11.31) (-189.50,-11.31) (-188.80,-11.31) (-188.09,-12.02) (-187.38,-12.02) (-186.68,-12.02) (-185.97,-12.02) (-185.26,-12.73) (-184.55,-12.73) (-183.85,-12.73) (-183.14,-12.73) (-182.43,-13.44) (-181.73,-13.44) (-181.73,-13.44) (-181.02,-13.44) (-180.31,-12.73) (-179.61,-12.73) (-178.90,-12.73) (-178.19,-12.73) (-177.48,-12.02) (-176.78,-12.02) (-176.07,-12.02) (-175.36,-12.02) (-174.66,-11.31) (-173.95,-11.31) (-173.24,-11.31) (-172.53,-11.31) (-171.83,-10.61) (-171.12,-10.61) (-170.41,-10.61) (-169.71,-10.61) (-169.00,-9.90) (-168.29,-9.90) (-167.58,-9.90) (-166.88,-9.90) (-166.17,-9.19) (-165.46,-9.19) (-164.76,-9.19) (-164.05,-9.19) (-163.34,-8.49) (-162.63,-8.49) (-161.93,-8.49) (-161.22,-8.49) (-160.51,-7.78) (-159.81,-7.78) (-159.10,-7.78) (-158.39,-7.78) (-157.68,-7.07) (-156.98,-7.07) (-156.27,-7.07) (-155.56,-7.07) (-154.86,-6.36) (-154.15,-6.36) (-153.44,-6.36) (-152.74,-6.36) (-152.03,-5.66) (-151.32,-5.66) (-150.61,-5.66) (-149.91,-5.66) (-149.20,-4.95) (-148.49,-4.95) (-148.49,-4.95) (-147.79,-4.24) (-147.79,-3.54) (-147.08,-2.83) (-147.08,-2.12) (-146.37,-1.41) (-145.66,-0.71) (-145.66, 0.00) (-144.96, 0.71) (-144.25, 1.41) (-144.25, 2.12) (-143.54, 2.83) (-143.54, 3.54) (-142.84, 4.24) (-142.13, 4.95) (-142.13, 5.66) (-141.42, 6.36) (-140.71, 7.07) (-140.71, 7.78) (-140.01, 8.49) (-140.01, 9.19) (-139.30, 9.90) (-138.59,10.61) (-138.59,11.31) (-137.89,12.02) (-137.18,12.73) (-137.18,13.44) (-136.47,14.14) (-136.47,14.85) (-135.76,15.56) (-135.06,16.26) (-135.06,16.97) (-134.35,17.68) (-133.64,18.38) (-133.64,19.09) (-132.94,19.80) (-132.94,20.51) (-132.23,21.21) (-131.52,21.92) (-131.52,22.63) (-130.81,23.33) (-148.49,-2.83) (-147.79,-2.12) (-147.79,-1.41) (-147.08,-0.71) (-146.37, 0.00) (-146.37, 0.71) (-145.66, 1.41) (-144.96, 2.12) (-144.96, 2.83) (-144.25, 3.54) (-143.54, 4.24) (-143.54, 4.95) (-142.84, 5.66) (-142.13, 6.36) (-142.13, 7.07) (-141.42, 7.78) (-140.71, 8.49) (-140.71, 9.19) (-140.01, 9.90) (-139.30,10.61) (-138.59,11.31) (-138.59,12.02) (-137.89,12.73) (-137.18,13.44) (-137.18,14.14) (-136.47,14.85) (-135.76,15.56) (-135.76,16.26) (-135.06,16.97) (-134.35,17.68) (-134.35,18.38) (-133.64,19.09) (-132.94,19.80) (-132.94,20.51) (-132.23,21.21) (-131.52,21.92) (-131.52,22.63) (-130.81,23.33) (-148.49,-2.83) (-147.79,-2.12) (-147.08,-1.41) (-146.37,-0.71) (-145.66, 0.00) (-144.96, 0.00) (-144.25, 0.71) (-143.54, 1.41) (-142.84, 2.12) (-142.13, 2.83) (-141.42, 3.54) (-140.71, 4.24) (-140.01, 4.95) (-139.30, 4.95) (-138.59, 5.66) (-137.89, 6.36) (-137.18, 7.07) (-136.47, 7.78) (-135.76, 8.49) (-135.06, 9.19) (-134.35, 9.90) (-133.64,10.61) (-132.94,10.61) (-132.23,11.31) (-131.52,12.02) (-130.81,12.73) (-130.11,13.44) (-129.40,14.14) (-128.69,14.85) (-127.99,15.56) (-127.28,15.56) (-126.57,16.26) (-125.87,16.97) (-125.16,17.68) (-124.45,18.38) (-158.39,22.63) (-157.68,22.63) (-156.98,22.63) (-156.27,22.63) (-155.56,22.63) (-154.86,21.92) (-154.15,21.92) (-153.44,21.92) (-152.74,21.92) (-152.03,21.92) (-151.32,21.92) (-150.61,21.92) (-149.91,21.92) (-149.20,21.21) (-148.49,21.21) (-147.79,21.21) (-147.08,21.21) (-146.37,21.21) (-145.66,21.21) (-144.96,21.21) (-144.25,21.21) (-143.54,20.51) (-142.84,20.51) (-142.13,20.51) (-141.42,20.51) (-140.71,20.51) (-140.01,20.51) (-139.30,20.51) (-138.59,20.51) (-137.89,19.80) (-137.18,19.80) (-136.47,19.80) (-135.76,19.80) (-135.06,19.80) (-134.35,19.80) (-133.64,19.80) (-132.94,19.80) (-132.23,19.09) (-131.52,19.09) (-130.81,19.09) (-130.11,19.09) (-129.40,19.09) (-128.69,19.09) (-127.99,19.09) (-127.28,19.09) (-126.57,18.38) (-125.87,18.38) (-125.16,18.38) (-124.45,18.38) (-191.63,17.68) (-190.92,17.68) (-190.21,17.68) (-189.50,17.68) (-188.80,18.38) (-188.09,18.38) (-187.38,18.38) (-186.68,18.38) (-185.97,18.38) (-185.26,18.38) (-184.55,18.38) (-183.85,19.09) (-183.14,19.09) (-182.43,19.09) (-181.73,19.09) (-181.02,19.09) (-180.31,19.09) (-179.61,19.80) (-178.90,19.80) (-178.19,19.80) (-177.48,19.80) (-176.78,19.80) (-176.07,19.80) (-175.36,19.80) (-174.66,20.51) (-173.95,20.51) (-173.24,20.51) (-172.53,20.51) (-171.83,20.51) (-171.12,20.51) (-170.41,20.51) (-169.71,21.21) (-169.00,21.21) (-168.29,21.21) (-167.58,21.21) (-166.88,21.21) (-166.17,21.21) (-165.46,21.92) (-164.76,21.92) (-164.05,21.92) (-163.34,21.92) (-162.63,21.92) (-161.93,21.92) (-161.22,21.92) (-160.51,22.63) (-159.81,22.63) (-159.10,22.63) (-158.39,22.63) (-196.58,-15.56) (-196.58,-14.85) (-196.58,-14.14) (-196.58,-13.44) (-195.87,-12.73) (-195.87,-12.02) (-195.87,-11.31) (-195.87,-10.61) (-195.87,-9.90) (-195.87,-9.19) (-195.87,-8.49) (-195.16,-7.78) (-195.16,-7.07) (-195.16,-6.36) (-195.16,-5.66) (-195.16,-4.95) (-195.16,-4.24) (-194.45,-3.54) (-194.45,-2.83) (-194.45,-2.12) (-194.45,-1.41) (-194.45,-0.71) (-194.45, 0.00) (-194.45, 0.71) (-193.75, 1.41) (-193.75, 2.12) (-193.75, 2.83) (-193.75, 3.54) (-193.75, 4.24) (-193.75, 4.95) (-193.75, 5.66) (-193.04, 6.36) (-193.04, 7.07) (-193.04, 7.78) (-193.04, 8.49) (-193.04, 9.19) (-193.04, 9.90) (-192.33,10.61) (-192.33,11.31) (-192.33,12.02) (-192.33,12.73) (-192.33,13.44) (-192.33,14.14) (-192.33,14.85) (-191.63,15.56) (-191.63,16.26) (-191.63,16.97) (-191.63,17.68) (-223.45, 0.00) (-222.74,-0.71) (-222.03,-0.71) (-221.32,-1.41) (-220.62,-1.41) (-219.91,-2.12) (-219.20,-2.12) (-218.50,-2.83) (-217.79,-3.54) (-217.08,-3.54) (-216.37,-4.24) (-215.67,-4.24) (-214.96,-4.95) (-214.25,-5.66) (-213.55,-5.66) (-212.84,-6.36) (-212.13,-6.36) (-211.42,-7.07) (-210.72,-7.07) (-210.01,-7.78) (-209.30,-8.49) (-208.60,-8.49) (-207.89,-9.19) (-207.18,-9.19) (-206.48,-9.90) (-205.77,-9.90) (-205.06,-10.61) (-204.35,-11.31) (-203.65,-11.31) (-202.94,-12.02) (-202.23,-12.02) (-201.53,-12.73) (-200.82,-13.44) (-200.11,-13.44) (-199.40,-14.14) (-198.70,-14.14) (-197.99,-14.85) (-197.28,-14.85) (-196.58,-15.56) (-251.73,22.63) (-251.02,21.92) (-250.32,21.21) (-249.61,21.21) (-248.90,20.51) (-248.19,19.80) (-247.49,19.09) (-246.78,18.38) (-246.07,18.38) (-245.37,17.68) (-244.66,16.97) (-243.95,16.26) (-243.24,15.56) (-242.54,15.56) (-241.83,14.85) (-241.12,14.14) (-240.42,13.44) (-239.71,12.73) (-239.00,12.73) (-238.29,12.02) (-237.59,11.31) (-236.88,10.61) (-236.17, 9.90) (-235.47, 9.90) (-234.76, 9.19) (-234.05, 8.49) (-233.35, 7.78) (-232.64, 7.07) (-231.93, 7.07) (-231.22, 6.36) (-230.52, 5.66) (-229.81, 4.95) (-229.10, 4.24) (-228.40, 4.24) (-227.69, 3.54) (-226.98, 2.83) (-226.27, 2.12) (-225.57, 1.41) (-224.86, 1.41) (-224.15, 0.71) (-223.45, 0.00) (-251.73,22.63) (-252.44,23.33) (-253.14,24.04) (-253.85,24.75) (-254.56,25.46) (-255.27,26.16) (-255.97,26.87) (-256.68,27.58) (-257.39,28.28) (-257.39,28.99) (-258.09,29.70) (-258.80,30.41) (-259.51,31.11) (-260.22,31.82) (-260.92,32.53) (-261.63,33.23) (-262.34,33.94) (-263.04,34.65) (-263.75,35.36) (-264.46,36.06) (-265.17,36.77) (-265.87,37.48) (-266.58,38.18) (-267.29,38.89) (-267.99,39.60) (-268.70,40.31) (-268.70,41.01) (-269.41,41.72) (-270.11,42.43) (-270.82,43.13) (-271.53,43.84) (-272.24,44.55) (-272.94,45.25) (-273.65,45.96) (-274.36,46.67) (-274.36,46.67) (-275.06,47.38) (-275.06,48.08) (-275.77,48.79) (-275.77,49.50) (-276.48,50.20) (-276.48,50.91) (-277.19,51.62) (-277.19,52.33) (-277.89,53.03) (-277.89,53.74) (-278.60,54.45) (-279.31,55.15) (-279.31,55.86) (-280.01,56.57) (-280.01,57.28) (-280.72,57.98) (-280.72,58.69) (-281.43,59.40) (-281.43,60.10) (-282.14,60.81) (-282.14,61.52) (-282.84,62.23) (-283.55,62.93) (-283.55,63.64) (-284.26,64.35) (-284.26,65.05) (-284.96,65.76) (-284.96,66.47) (-285.67,67.18) (-285.67,67.88) (-286.38,68.59) (-287.09,69.30) (-287.09,70.00) (-287.79,70.71) (-287.79,71.42) (-288.50,72.12) (-288.50,72.83) (-289.21,73.54) (-289.21,74.25) (-289.91,74.95) (-289.91,75.66) (-290.62,76.37) (-290.62,76.37) (-289.91,77.07) (-289.21,77.07) (-288.50,77.78) (-287.79,78.49) (-287.09,79.20) (-286.38,79.20) (-285.67,79.90) (-284.96,80.61) (-284.26,80.61) (-283.55,81.32) (-282.84,82.02) (-282.14,82.02) (-281.43,82.73) (-280.72,83.44) (-280.01,84.15) (-279.31,84.15) (-278.60,84.85) (-277.89,85.56) (-277.19,85.56) (-276.48,86.27) (-275.77,86.97) (-275.06,86.97) (-274.36,87.68) (-273.65,88.39) (-272.94,89.10) (-272.24,89.10) (-271.53,89.80) (-270.82,90.51) (-270.11,90.51) (-269.41,91.22) (-268.70,91.92) (-267.99,91.92) (-267.29,92.63) (-266.58,93.34) (-265.87,94.05) (-265.17,94.05) (-264.46,94.75) (-289.91,115.97) (-289.21,115.26) (-288.50,114.55) (-287.79,114.55) (-287.09,113.84) (-286.38,113.14) (-285.67,112.43) (-284.96,111.72) (-284.26,111.02) (-283.55,111.02) (-282.84,110.31) (-282.14,109.60) (-281.43,108.89) (-280.72,108.19) (-280.01,107.48) (-279.31,107.48) (-278.60,106.77) (-277.89,106.07) (-277.19,105.36) (-276.48,104.65) (-275.77,103.94) (-275.06,103.94) (-274.36,103.24) (-273.65,102.53) (-272.94,101.82) (-272.24,101.12) (-271.53,100.41) (-270.82,100.41) (-270.11,99.70) (-269.41,98.99) (-268.70,98.29) (-267.99,97.58) (-267.29,96.87) (-266.58,96.87) (-265.87,96.17) (-265.17,95.46) (-264.46,94.75) (-289.91,115.97) (-290.62,116.67) (-290.62,117.38) (-291.33,118.09) (-292.04,118.79) (-292.04,119.50) (-292.74,120.21) (-292.74,120.92) (-293.45,121.62) (-294.16,122.33) (-294.16,123.04) (-294.86,123.74) (-295.57,124.45) (-295.57,125.16) (-296.28,125.87) (-296.28,126.57) (-296.98,127.28) (-297.69,127.99) (-297.69,128.69) (-298.40,129.40) (-299.11,130.11) (-299.11,130.81) (-299.81,131.52) (-300.52,132.23) (-300.52,132.94) (-301.23,133.64) (-301.23,134.35) (-301.93,135.06) (-302.64,135.76) (-302.64,136.47) (-303.35,137.18) (-304.06,137.89) (-304.06,138.59) (-304.76,139.30) (-304.76,140.01) (-305.47,140.71) (-306.18,141.42) (-306.18,142.13) (-306.88,142.84) (-306.88,142.84) (-306.88,143.54) (-306.18,144.25) (-306.18,144.96) (-305.47,145.66) (-305.47,146.37) (-304.76,147.08) (-304.76,147.79) (-304.06,148.49) (-304.06,149.20) (-304.06,149.91) (-303.35,150.61) (-303.35,151.32) (-302.64,152.03) (-302.64,152.74) (-301.93,153.44) (-301.93,154.15) (-301.23,154.86) (-301.23,155.56) (-301.23,156.27) (-300.52,156.98) (-300.52,157.68) (-299.81,158.39) (-299.81,159.10) (-299.11,159.81) (-299.11,160.51) (-298.40,161.22) (-298.40,161.93) (-298.40,162.63) (-297.69,163.34) (-297.69,164.05) (-296.98,164.76) (-296.98,165.46) (-296.28,166.17) (-296.28,166.88) (-295.57,167.58) (-295.57,168.29) (-295.57,169.00) (-294.86,169.71) (-294.86,170.41) (-294.16,171.12) (-294.16,171.83) (-293.45,172.53) (-293.45,173.24) (-292.74,173.95) (-292.74,174.66) (-309.71,145.66) (-309.01,146.37) (-309.01,147.08) (-308.30,147.79) (-308.30,148.49) (-307.59,149.20) (-306.88,149.91) (-306.88,150.61) (-306.18,151.32) (-306.18,152.03) (-305.47,152.74) (-305.47,153.44) (-304.76,154.15) (-304.06,154.86) (-304.06,155.56) (-303.35,156.27) (-303.35,156.98) (-302.64,157.68) (-301.93,158.39) (-301.93,159.10) (-301.23,159.81) (-301.23,160.51) (-300.52,161.22) (-300.52,161.93) (-299.81,162.63) (-299.11,163.34) (-299.11,164.05) (-298.40,164.76) (-298.40,165.46) (-297.69,166.17) (-296.98,166.88) (-296.98,167.58) (-296.28,168.29) (-296.28,169.00) (-295.57,169.71) (-295.57,170.41) (-294.86,171.12) (-294.16,171.83) (-294.16,172.53) (-293.45,173.24) (-293.45,173.95) (-292.74,174.66) (-309.71,145.66) (-309.01,146.37) (-308.30,146.37) (-307.59,147.08) (-306.88,147.08) (-306.18,147.79) (-305.47,147.79) (-304.76,148.49) (-304.06,148.49) (-303.35,149.20) (-302.64,149.91) (-301.93,149.91) (-301.23,150.61) (-300.52,150.61) (-299.81,151.32) (-299.11,151.32) (-298.40,152.03) (-297.69,152.03) (-296.98,152.74) (-296.28,153.44) (-295.57,153.44) (-294.86,154.15) (-294.16,154.15) (-293.45,154.86) (-292.74,154.86) (-292.04,155.56) (-291.33,156.27) (-290.62,156.27) (-289.91,156.98) (-289.21,156.98) (-288.50,157.68) (-287.79,157.68) (-287.09,158.39) (-286.38,158.39) (-285.67,159.10) (-284.96,159.81) (-284.26,159.81) (-283.55,160.51) (-282.84,160.51) (-282.14,161.22) (-281.43,161.22) (-280.72,161.93) (-280.01,161.93) (-279.31,162.63) (-304.76,186.68) (-304.06,185.97) (-303.35,185.26) (-302.64,184.55) (-301.93,183.85) (-301.23,183.14) (-300.52,182.43) (-299.81,181.73) (-299.11,181.02) (-298.40,181.02) (-297.69,180.31) (-296.98,179.61) (-296.28,178.90) (-295.57,178.19) (-294.86,177.48) (-294.16,176.78) (-293.45,176.07) (-292.74,175.36) (-292.04,174.66) (-291.33,173.95) (-290.62,173.24) (-289.91,172.53) (-289.21,171.83) (-288.50,171.12) (-287.79,170.41) (-287.09,169.71) (-286.38,169.00) (-285.67,169.00) (-284.96,168.29) (-284.26,167.58) (-283.55,166.88) (-282.84,166.17) (-282.14,165.46) (-281.43,164.76) (-280.72,164.05) (-280.01,163.34) (-279.31,162.63) (-304.76,186.68) (-304.76,187.38) (-305.47,188.09) (-305.47,188.80) (-306.18,189.50) (-306.18,190.21) (-306.88,190.92) (-306.88,191.63) (-307.59,192.33) (-307.59,193.04) (-308.30,193.75) (-308.30,194.45) (-309.01,195.16) (-309.01,195.87) (-309.71,196.58) (-309.71,197.28) (-310.42,197.99) (-310.42,198.70) (-311.13,199.40) (-311.13,200.11) (-311.83,200.82) (-311.83,201.53) (-312.54,202.23) (-312.54,202.94) (-313.25,203.65) (-313.25,204.35) (-313.96,205.06) (-313.96,205.77) (-314.66,206.48) (-314.66,207.18) (-315.37,207.89) (-315.37,208.60) (-316.08,209.30) (-316.08,210.01) (-316.78,210.72) (-316.78,211.42) (-317.49,212.13) (-317.49,212.84) (-318.20,213.55) (-318.20,214.25) (-318.91,214.96) (-318.91,215.67) (-318.91,215.67) (-318.20,216.37) (-317.49,217.08) (-316.78,217.08) (-316.08,217.79) (-315.37,218.50) (-314.66,219.20) (-313.96,219.20) (-313.25,219.91) (-312.54,220.62) (-311.83,221.32) (-311.13,221.32) (-310.42,222.03) (-309.71,222.74) (-309.01,223.45) (-308.30,223.45) (-307.59,224.15) (-306.88,224.86) (-306.18,225.57) (-305.47,225.57) (-304.76,226.27) (-304.06,226.98) (-303.35,227.69) (-302.64,227.69) (-301.93,228.40) (-301.23,229.10) (-300.52,229.81) (-299.81,229.81) (-299.11,230.52) (-298.40,231.22) (-297.69,231.93) (-296.98,231.93) (-296.28,232.64) (-295.57,233.35) (-294.86,234.05) (-294.16,234.05) (-293.45,234.76) (-292.74,235.47) (-326.68,241.83) (-325.98,241.83) (-325.27,241.83) (-324.56,241.12) (-323.85,241.12) (-323.15,241.12) (-322.44,241.12) (-321.73,241.12) (-321.03,241.12) (-320.32,240.42) (-319.61,240.42) (-318.91,240.42) (-318.20,240.42) (-317.49,240.42) (-316.78,239.71) (-316.08,239.71) (-315.37,239.71) (-314.66,239.71) (-313.96,239.71) (-313.25,239.00) (-312.54,239.00) (-311.83,239.00) (-311.13,239.00) (-310.42,239.00) (-309.71,239.00) (-309.01,238.29) (-308.30,238.29) (-307.59,238.29) (-306.88,238.29) (-306.18,238.29) (-305.47,237.59) (-304.76,237.59) (-304.06,237.59) (-303.35,237.59) (-302.64,237.59) (-301.93,236.88) (-301.23,236.88) (-300.52,236.88) (-299.81,236.88) (-299.11,236.88) (-298.40,236.88) (-297.69,236.17) (-296.98,236.17) (-296.28,236.17) (-295.57,236.17) (-294.86,236.17) (-294.16,235.47) (-293.45,235.47) (-292.74,235.47) (-353.55,220.62) (-352.85,221.32) (-352.14,222.03) (-351.43,222.03) (-350.72,222.74) (-350.02,223.45) (-349.31,224.15) (-348.60,224.86) (-347.90,224.86) (-347.19,225.57) (-346.48,226.27) (-345.78,226.98) (-345.07,226.98) (-344.36,227.69) (-343.65,228.40) (-342.95,229.10) (-342.24,229.81) (-341.53,229.81) (-340.83,230.52) (-340.12,231.22) (-339.41,231.93) (-338.70,232.64) (-338.00,232.64) (-337.29,233.35) (-336.58,234.05) (-335.88,234.76) (-335.17,235.47) (-334.46,235.47) (-333.75,236.17) (-333.05,236.88) (-332.34,237.59) (-331.63,237.59) (-330.93,238.29) (-330.22,239.00) (-329.51,239.71) (-328.80,240.42) (-328.10,240.42) (-327.39,241.12) (-326.68,241.83) (-353.55,220.62) (-352.85,220.62) (-352.14,219.91) (-351.43,219.91) (-350.72,219.20) (-350.02,219.20) (-349.31,218.50) (-348.60,218.50) (-347.90,217.79) (-347.19,217.79) (-346.48,217.08) (-345.78,217.08) (-345.07,216.37) (-344.36,216.37) (-343.65,215.67) (-342.95,215.67) (-342.24,214.96) (-341.53,214.96) (-340.83,214.25) (-340.12,214.25) (-339.41,213.55) (-338.70,213.55) (-338.00,212.84) (-337.29,212.84) (-336.58,212.13) (-335.88,212.13) (-335.17,211.42) (-334.46,211.42) (-333.75,210.72) (-333.05,210.72) (-332.34,210.01) (-331.63,210.01) (-330.93,209.30) (-330.22,209.30) (-329.51,208.60) (-328.80,208.60) (-328.10,207.89) (-327.39,207.89) (-326.68,207.18) (-325.98,207.18) (-325.27,206.48) (-324.56,206.48) (-323.85,205.77) (-323.85,205.77) (-323.15,205.77) (-322.44,205.77) (-321.73,206.48) (-321.03,206.48) (-320.32,206.48) (-319.61,206.48) (-318.91,206.48) (-318.20,207.18) (-317.49,207.18) (-316.78,207.18) (-316.08,207.18) (-315.37,207.18) (-314.66,207.89) (-313.96,207.89) (-313.25,207.89) (-312.54,207.89) (-311.83,208.60) (-311.13,208.60) (-310.42,208.60) (-309.71,208.60) (-309.01,208.60) (-308.30,209.30) (-307.59,209.30) (-306.88,209.30) (-306.18,209.30) (-305.47,209.30) (-304.76,210.01) (-304.06,210.01) (-303.35,210.01) (-302.64,210.01) (-301.93,210.01) (-301.23,210.72) (-300.52,210.72) (-299.81,210.72) (-299.11,210.72) (-298.40,210.72) (-297.69,211.42) (-296.98,211.42) (-296.28,211.42) (-295.57,211.42) (-294.86,212.13) (-294.16,212.13) (-293.45,212.13) (-292.74,212.13) (-292.04,212.13) (-291.33,212.84) (-290.62,212.84) (-289.91,212.84) (-289.91,212.84) (-289.21,213.55) (-288.50,214.25) (-287.79,214.96) (-287.09,214.96) (-286.38,215.67) (-285.67,216.37) (-284.96,217.08) (-284.26,217.79) (-283.55,218.50) (-282.84,219.20) (-282.14,219.91) (-281.43,219.91) (-280.72,220.62) (-280.01,221.32) (-279.31,222.03) (-278.60,222.74) (-277.89,223.45) (-277.19,224.15) (-276.48,224.86) (-275.77,224.86) (-275.06,225.57) (-274.36,226.27) (-273.65,226.98) (-272.94,227.69) (-272.24,228.40) (-271.53,229.10) (-270.82,229.81) (-270.11,229.81) (-269.41,230.52) (-268.70,231.22) (-267.99,231.93) (-267.29,232.64) (-266.58,233.35) (-265.87,234.05) (-265.17,234.76) (-264.46,234.76) (-263.75,235.47) (-263.04,236.17) (-262.34,236.88) (-288.50,255.97) (-287.79,255.27) (-287.09,255.27) (-286.38,254.56) (-285.67,253.85) (-284.96,253.14) (-284.26,253.14) (-283.55,252.44) (-282.84,251.73) (-282.14,251.02) (-281.43,251.02) (-280.72,250.32) (-280.01,249.61) (-279.31,249.61) (-278.60,248.90) (-277.89,248.19) (-277.19,247.49) (-276.48,247.49) (-275.77,246.78) (-275.06,246.07) (-274.36,245.37) (-273.65,245.37) (-272.94,244.66) (-272.24,243.95) (-271.53,243.24) (-270.82,243.24) (-270.11,242.54) (-269.41,241.83) (-268.70,241.83) (-267.99,241.12) (-267.29,240.42) (-266.58,239.71) (-265.87,239.71) (-265.17,239.00) (-264.46,238.29) (-263.75,237.59) (-263.04,237.59) (-262.34,236.88) (-288.50,255.97) (-288.50,256.68) (-289.21,257.39) (-289.21,258.09) (-289.91,258.80) (-289.91,259.51) (-290.62,260.22) (-290.62,260.92) (-291.33,261.63) (-291.33,262.34) (-292.04,263.04) (-292.04,263.75) (-292.74,264.46) (-292.74,265.17) (-293.45,265.87) (-293.45,266.58) (-294.16,267.29) (-294.16,267.99) (-294.86,268.70) (-294.86,269.41) (-295.57,270.11) (-295.57,270.82) (-296.28,271.53) (-296.28,272.24) (-296.98,272.94) (-296.98,273.65) (-297.69,274.36) (-297.69,275.06) (-298.40,275.77) (-298.40,276.48) (-299.11,277.19) (-299.11,277.89) (-299.81,278.60) (-299.81,279.31) (-300.52,280.01) (-300.52,280.72) (-301.23,281.43) (-301.23,282.14) (-301.93,282.84) (-301.93,283.55) (-301.93,283.55) (-301.23,284.26) (-300.52,284.96) (-299.81,285.67) (-299.11,285.67) (-298.40,286.38) (-297.69,287.09) (-296.98,287.79) (-296.28,288.50) (-295.57,289.21) (-294.86,289.91) (-294.16,289.91) (-293.45,290.62) (-292.74,291.33) (-292.04,292.04) (-291.33,292.74) (-290.62,293.45) (-289.91,294.16) (-289.21,294.16) (-288.50,294.86) (-287.79,295.57) (-287.09,296.28) (-286.38,296.98) (-285.67,297.69) (-284.96,297.69) (-284.26,298.40) (-283.55,299.11) (-282.84,299.81) (-282.14,300.52) (-281.43,301.23) (-280.72,301.93) (-280.01,301.93) (-279.31,302.64) (-278.60,303.35) (-277.89,304.06) (-277.19,304.76) (-276.48,305.47) (-275.77,306.18) (-275.06,306.18) (-274.36,306.88) (-273.65,307.59) (-272.94,308.30) (-303.35,324.56) (-302.64,323.85) (-301.93,323.85) (-301.23,323.15) (-300.52,323.15) (-299.81,322.44) (-299.11,322.44) (-298.40,321.73) (-297.69,321.73) (-296.98,321.03) (-296.28,321.03) (-295.57,320.32) (-294.86,320.32) (-294.16,319.61) (-293.45,319.61) (-292.74,318.91) (-292.04,318.20) (-291.33,318.20) (-290.62,317.49) (-289.91,317.49) (-289.21,316.78) (-288.50,316.78) (-287.79,316.08) (-287.09,316.08) (-286.38,315.37) (-285.67,315.37) (-284.96,314.66) (-284.26,314.66) (-283.55,313.96) (-282.84,313.25) (-282.14,313.25) (-281.43,312.54) (-280.72,312.54) (-280.01,311.83) (-279.31,311.83) (-278.60,311.13) (-277.89,311.13) (-277.19,310.42) (-276.48,310.42) (-275.77,309.71) (-275.06,309.71) (-274.36,309.01) (-273.65,309.01) (-272.94,308.30) (-331.63,341.53) (-330.93,340.83) (-330.22,340.83) (-329.51,340.12) (-328.80,340.12) (-328.10,339.41) (-327.39,338.70) (-326.68,338.70) (-325.98,338.00) (-325.27,338.00) (-324.56,337.29) (-323.85,336.58) (-323.15,336.58) (-322.44,335.88) (-321.73,335.88) (-321.03,335.17) (-320.32,334.46) (-319.61,334.46) (-318.91,333.75) (-318.20,333.75) (-317.49,333.05) (-316.78,332.34) (-316.08,332.34) (-315.37,331.63) (-314.66,331.63) (-313.96,330.93) (-313.25,330.22) (-312.54,330.22) (-311.83,329.51) (-311.13,329.51) (-310.42,328.80) (-309.71,328.10) (-309.01,328.10) (-308.30,327.39) (-307.59,327.39) (-306.88,326.68) (-306.18,325.98) (-305.47,325.98) (-304.76,325.27) (-304.06,325.27) (-303.35,324.56) (-361.33,359.92) (-360.62,359.21) (-359.92,359.21) (-359.21,358.50) (-358.50,358.50) (-357.80,357.80) (-357.09,357.09) (-356.38,357.09) (-355.67,356.38) (-354.97,355.67) (-354.26,355.67) (-353.55,354.97) (-352.85,354.97) (-352.14,354.26) (-351.43,353.55) (-350.72,353.55) (-350.02,352.85) (-349.31,352.14) (-348.60,352.14) (-347.90,351.43) (-347.19,351.43) (-346.48,350.72) (-345.78,350.02) (-345.07,350.02) (-344.36,349.31) (-343.65,349.31) (-342.95,348.60) (-342.24,347.90) (-341.53,347.90) (-340.83,347.19) (-340.12,346.48) (-339.41,346.48) (-338.70,345.78) (-338.00,345.78) (-337.29,345.07) (-336.58,344.36) (-335.88,344.36) (-335.17,343.65) (-334.46,342.95) (-333.75,342.95) (-333.05,342.24) (-332.34,342.24) (-331.63,341.53) (-384.67,381.84) (-383.96,381.13) (-383.25,380.42) (-382.54,379.72) (-381.84,379.01) (-381.13,378.30) (-380.42,377.60) (-379.72,376.89) (-379.01,376.18) (-378.30,376.18) (-377.60,375.47) (-376.89,374.77) (-376.18,374.06) (-375.47,373.35) (-374.77,372.65) (-374.06,371.94) (-373.35,371.23) (-372.65,370.52) (-371.94,369.82) (-371.23,369.11) (-370.52,368.40) (-369.82,367.70) (-369.11,366.99) (-368.40,366.28) (-367.70,365.57) (-366.99,365.57) (-366.28,364.87) (-365.57,364.16) (-364.87,363.45) (-364.16,362.75) (-363.45,362.04) (-362.75,361.33) (-362.04,360.62) (-361.33,359.92) (-384.67,381.84) (-384.67,382.54) (-385.37,383.25) (-385.37,383.96) (-386.08,384.67) (-386.08,385.37) (-386.79,386.08) (-386.79,386.79) (-387.49,387.49) (-387.49,388.20) (-387.49,388.91) (-388.20,389.62) (-388.20,390.32) (-388.91,391.03) (-388.91,391.74) (-389.62,392.44) (-389.62,393.15) (-390.32,393.86) (-390.32,394.57) (-390.32,395.27) (-391.03,395.98) (-391.03,396.69) (-391.74,397.39) (-391.74,398.10) (-392.44,398.81) (-392.44,399.52) (-393.15,400.22) (-393.15,400.93) (-393.15,401.64) (-393.86,402.34) (-393.86,403.05) (-394.57,403.76) (-394.57,404.47) (-395.27,405.17) (-395.27,405.88) (-395.98,406.59) (-395.98,407.29) (-395.98,408.00) (-396.69,408.71) (-396.69,409.41) (-397.39,410.12) (-397.39,410.83) (-398.10,411.54) (-398.10,412.24) (-398.81,412.95) (-398.81,413.66) (-398.81,413.66) (-398.10,413.66) (-397.39,413.66) (-396.69,414.36) (-395.98,414.36) (-395.27,414.36) (-394.57,414.36) (-393.86,415.07) (-393.15,415.07) (-392.44,415.07) (-391.74,415.07) (-391.03,415.07) (-390.32,415.78) (-389.62,415.78) (-388.91,415.78) (-388.20,415.78) (-387.49,416.49) (-386.79,416.49) (-386.08,416.49) (-385.37,416.49) (-384.67,416.49) (-383.96,417.19) (-383.25,417.19) (-382.54,417.19) (-381.84,417.19) (-381.13,417.19) (-380.42,417.90) (-379.72,417.90) (-379.01,417.90) (-378.30,417.90) (-377.60,418.61) (-376.89,418.61) (-376.18,418.61) (-375.47,418.61) (-374.77,418.61) (-374.06,419.31) (-373.35,419.31) (-372.65,419.31) (-371.94,419.31) (-371.23,420.02) (-370.52,420.02) (-369.82,420.02) (-369.11,420.02) (-368.40,420.02) (-367.70,420.73) (-366.99,420.73) (-366.28,420.73) (-365.57,420.73) (-364.87,421.44) (-364.16,421.44) (-363.45,421.44) (-363.45,421.44) (-363.45,422.14) (-363.45,422.85) (-362.75,423.56) (-362.75,424.26) (-362.75,424.97) (-362.75,425.68) (-362.75,426.39) (-362.04,427.09) (-362.04,427.80) (-362.04,428.51) (-362.04,429.21) (-362.04,429.92) (-361.33,430.63) (-361.33,431.34) (-361.33,432.04) (-361.33,432.75) (-360.62,433.46) (-360.62,434.16) (-360.62,434.87) (-360.62,435.58) (-360.62,436.28) (-359.92,436.99) (-359.92,437.70) (-359.92,438.41) (-359.92,439.11) (-359.92,439.82) (-359.21,440.53) (-359.21,441.23) (-359.21,441.94) (-359.21,442.65) (-359.21,443.36) (-358.50,444.06) (-358.50,444.77) (-358.50,445.48) (-358.50,446.18) (-358.50,446.89) (-357.80,447.60) (-357.80,448.31) (-357.80,449.01) (-357.80,449.72) (-357.09,450.43) (-357.09,451.13) (-357.09,451.84) (-357.09,452.55) (-357.09,453.26) (-356.38,453.96) (-356.38,454.67) (-356.38,455.38) (-344.36,421.44) (-344.36,422.14) (-345.07,422.85) (-345.07,423.56) (-345.07,424.26) (-345.78,424.97) (-345.78,425.68) (-345.78,426.39) (-346.48,427.09) (-346.48,427.80) (-347.19,428.51) (-347.19,429.21) (-347.19,429.92) (-347.90,430.63) (-347.90,431.34) (-347.90,432.04) (-348.60,432.75) (-348.60,433.46) (-348.60,434.16) (-349.31,434.87) (-349.31,435.58) (-349.31,436.28) (-350.02,436.99) (-350.02,437.70) (-350.02,438.41) (-350.72,439.11) (-350.72,439.82) (-351.43,440.53) (-351.43,441.23) (-351.43,441.94) (-352.14,442.65) (-352.14,443.36) (-352.14,444.06) (-352.85,444.77) (-352.85,445.48) (-352.85,446.18) (-353.55,446.89) (-353.55,447.60) (-353.55,448.31) (-354.26,449.01) (-354.26,449.72) (-354.97,450.43) (-354.97,451.13) (-354.97,451.84) (-355.67,452.55) (-355.67,453.26) (-355.67,453.96) (-356.38,454.67) (-356.38,455.38) (-379.72,416.49) (-379.01,416.49) (-378.30,416.49) (-377.60,416.49) (-376.89,417.19) (-376.18,417.19) (-375.47,417.19) (-374.77,417.19) (-374.06,417.19) (-373.35,417.19) (-372.65,417.19) (-371.94,417.90) (-371.23,417.90) (-370.52,417.90) (-369.82,417.90) (-369.11,417.90) (-368.40,417.90) (-367.70,417.90) (-366.99,418.61) (-366.28,418.61) (-365.57,418.61) (-364.87,418.61) (-364.16,418.61) (-363.45,418.61) (-362.75,418.61) (-362.04,418.61) (-361.33,419.31) (-360.62,419.31) (-359.92,419.31) (-359.21,419.31) (-358.50,419.31) (-357.80,419.31) (-357.09,419.31) (-356.38,420.02) (-355.67,420.02) (-354.97,420.02) (-354.26,420.02) (-353.55,420.02) (-352.85,420.02) (-352.14,420.02) (-351.43,420.73) (-350.72,420.73) (-350.02,420.73) (-349.31,420.73) (-348.60,420.73) (-347.90,420.73) (-347.19,420.73) (-346.48,421.44) (-345.78,421.44) (-345.07,421.44) (-344.36,421.44) (-382.54,383.96) (-382.54,384.67) (-382.54,385.37) (-382.54,386.08) (-382.54,386.79) (-382.54,387.49) (-381.84,388.20) (-381.84,388.91) (-381.84,389.62) (-381.84,390.32) (-381.84,391.03) (-381.84,391.74) (-381.84,392.44) (-381.84,393.15) (-381.84,393.86) (-381.84,394.57) (-381.84,395.27) (-381.84,395.98) (-381.13,396.69) (-381.13,397.39) (-381.13,398.10) (-381.13,398.81) (-381.13,399.52) (-381.13,400.22) (-381.13,400.93) (-381.13,401.64) (-381.13,402.34) (-381.13,403.05) (-381.13,403.76) (-380.42,404.47) (-380.42,405.17) (-380.42,405.88) (-380.42,406.59) (-380.42,407.29) (-380.42,408.00) (-380.42,408.71) (-380.42,409.41) (-380.42,410.12) (-380.42,410.83) (-380.42,411.54) (-380.42,412.24) (-379.72,412.95) (-379.72,413.66) (-379.72,414.36) (-379.72,415.07) (-379.72,415.78) (-379.72,416.49) (-410.83,404.47) (-410.12,403.76) (-409.41,403.76) (-408.71,403.05) (-408.00,402.34) (-407.29,401.64) (-406.59,401.64) (-405.88,400.93) (-405.17,400.22) (-404.47,399.52) (-403.76,399.52) (-403.05,398.81) (-402.34,398.10) (-401.64,398.10) (-400.93,397.39) (-400.22,396.69) (-399.52,395.98) (-398.81,395.98) (-398.10,395.27) (-397.39,394.57) (-396.69,394.57) (-395.98,393.86) (-395.27,393.15) (-394.57,392.44) (-393.86,392.44) (-393.15,391.74) (-392.44,391.03) (-391.74,390.32) (-391.03,390.32) (-390.32,389.62) (-389.62,388.91) (-388.91,388.91) (-388.20,388.20) (-387.49,387.49) (-386.79,386.79) (-386.08,386.79) (-385.37,386.08) (-384.67,385.37) (-383.96,384.67) (-383.25,384.67) (-382.54,383.96) (-410.83,404.47) (-411.54,405.17) (-411.54,405.88) (-412.24,406.59) (-412.24,407.29) (-412.95,408.00) (-413.66,408.71) (-413.66,409.41) (-414.36,410.12) (-414.36,410.83) (-415.07,411.54) (-415.78,412.24) (-415.78,412.95) (-416.49,413.66) (-416.49,414.36) (-417.19,415.07) (-417.90,415.78) (-417.90,416.49) (-418.61,417.19) (-418.61,417.90) (-419.31,418.61) (-420.02,419.31) (-420.02,420.02) (-420.73,420.73) (-420.73,421.44) (-421.44,422.14) (-422.14,422.85) (-422.14,423.56) (-422.85,424.26) (-422.85,424.97) (-423.56,425.68) (-424.26,426.39) (-424.26,427.09) (-424.97,427.80) (-424.97,428.51) (-425.68,429.21) (-426.39,429.92) (-426.39,430.63) (-427.09,431.34) (-427.09,432.04) (-427.80,432.75) (-427.80,432.75) (-427.09,433.46) (-427.09,434.16) (-426.39,434.87) (-425.68,435.58) (-425.68,436.28) (-424.97,436.99) (-424.26,437.70) (-424.26,438.41) (-423.56,439.11) (-422.85,439.82) (-422.85,440.53) (-422.14,441.23) (-421.44,441.94) (-421.44,442.65) (-420.73,443.36) (-420.02,444.06) (-420.02,444.77) (-419.31,445.48) (-418.61,446.18) (-418.61,446.89) (-417.90,447.60) (-417.90,448.31) (-417.19,449.01) (-416.49,449.72) (-416.49,450.43) (-415.78,451.13) (-415.07,451.84) (-415.07,452.55) (-414.36,453.26) (-413.66,453.96) (-413.66,454.67) (-412.95,455.38) (-412.24,456.08) (-412.24,456.79) (-411.54,457.50) (-410.83,458.21) (-410.83,458.91) (-410.12,459.62) (-409.41,460.33) (-409.41,461.03) (-408.71,461.74) (-429.92,432.75) (-429.21,433.46) (-429.21,434.16) (-428.51,434.87) (-427.80,435.58) (-427.09,436.28) (-427.09,436.99) (-426.39,437.70) (-425.68,438.41) (-424.97,439.11) (-424.97,439.82) (-424.26,440.53) (-423.56,441.23) (-422.85,441.94) (-422.85,442.65) (-422.14,443.36) (-421.44,444.06) (-421.44,444.77) (-420.73,445.48) (-420.02,446.18) (-419.31,446.89) (-419.31,447.60) (-418.61,448.31) (-417.90,449.01) (-417.19,449.72) (-417.19,450.43) (-416.49,451.13) (-415.78,451.84) (-415.78,452.55) (-415.07,453.26) (-414.36,453.96) (-413.66,454.67) (-413.66,455.38) (-412.95,456.08) (-412.24,456.79) (-411.54,457.50) (-411.54,458.21) (-410.83,458.91) (-410.12,459.62) (-409.41,460.33) (-409.41,461.03) (-408.71,461.74) (-429.92,432.75) (-429.21,433.46) (-429.21,434.16) (-428.51,434.87) (-428.51,435.58) (-427.80,436.28) (-427.80,436.99) (-427.09,437.70) (-427.09,438.41) (-426.39,439.11) (-426.39,439.82) (-425.68,440.53) (-425.68,441.23) (-424.97,441.94) (-424.97,442.65) (-424.26,443.36) (-424.26,444.06) (-423.56,444.77) (-423.56,445.48) (-422.85,446.18) (-422.85,446.89) (-422.14,447.60) (-421.44,448.31) (-421.44,449.01) (-420.73,449.72) (-420.73,450.43) (-420.02,451.13) (-420.02,451.84) (-419.31,452.55) (-419.31,453.26) (-418.61,453.96) (-418.61,454.67) (-417.90,455.38) (-417.90,456.08) (-417.19,456.79) (-417.19,457.50) (-416.49,458.21) (-416.49,458.91) (-415.78,459.62) (-415.78,460.33) (-415.07,461.03) (-415.07,461.74) (-414.36,462.45) (-429.92,430.63) (-429.92,431.34) (-429.21,432.04) (-429.21,432.75) (-428.51,433.46) (-428.51,434.16) (-427.80,434.87) (-427.80,435.58) (-427.09,436.28) (-427.09,436.99) (-426.39,437.70) (-426.39,438.41) (-425.68,439.11) (-425.68,439.82) (-424.97,440.53) (-424.97,441.23) (-424.26,441.94) (-424.26,442.65) (-423.56,443.36) (-423.56,444.06) (-422.85,444.77) (-422.85,445.48) (-422.14,446.18) (-422.14,446.89) (-421.44,447.60) (-421.44,448.31) (-420.73,449.01) (-420.73,449.72) (-420.02,450.43) (-420.02,451.13) (-419.31,451.84) (-419.31,452.55) (-418.61,453.26) (-418.61,453.96) (-417.90,454.67) (-417.90,455.38) (-417.19,456.08) (-417.19,456.79) (-416.49,457.50) (-416.49,458.21) (-415.78,458.91) (-415.78,459.62) (-415.07,460.33) (-415.07,461.03) (-414.36,461.74) (-414.36,462.45) (-429.92,430.63) (-429.92,431.34) (-429.92,432.04) (-429.92,432.75) (-429.21,433.46) (-429.21,434.16) (-429.21,434.87) (-429.21,435.58) (-429.21,436.28) (-429.21,436.99) (-429.21,437.70) (-428.51,438.41) (-428.51,439.11) (-428.51,439.82) (-428.51,440.53) (-428.51,441.23) (-428.51,441.94) (-428.51,442.65) (-427.80,443.36) (-427.80,444.06) (-427.80,444.77) (-427.80,445.48) (-427.80,446.18) (-427.80,446.89) (-427.80,447.60) (-427.80,448.31) (-427.09,449.01) (-427.09,449.72) (-427.09,450.43) (-427.09,451.13) (-427.09,451.84) (-427.09,452.55) (-427.09,453.26) (-426.39,453.96) (-426.39,454.67) (-426.39,455.38) (-426.39,456.08) (-426.39,456.79) (-426.39,457.50) (-426.39,458.21) (-425.68,458.91) (-425.68,459.62) (-425.68,460.33) (-425.68,461.03) (-425.68,461.74) (-425.68,462.45) (-425.68,463.15) (-424.97,463.86) (-424.97,464.57) (-424.97,465.28) (-424.97,465.98) (-427.80,428.51) (-427.80,429.21) (-427.80,429.92) (-427.80,430.63) (-427.80,431.34) (-427.80,432.04) (-427.80,432.75) (-427.09,433.46) (-427.09,434.16) (-427.09,434.87) (-427.09,435.58) (-427.09,436.28) (-427.09,436.99) (-427.09,437.70) (-427.09,438.41) (-427.09,439.11) (-427.09,439.82) (-427.09,440.53) (-427.09,441.23) (-427.09,441.94) (-426.39,442.65) (-426.39,443.36) (-426.39,444.06) (-426.39,444.77) (-426.39,445.48) (-426.39,446.18) (-426.39,446.89) (-426.39,447.60) (-426.39,448.31) (-426.39,449.01) (-426.39,449.72) (-426.39,450.43) (-426.39,451.13) (-426.39,451.84) (-425.68,452.55) (-425.68,453.26) (-425.68,453.96) (-425.68,454.67) (-425.68,455.38) (-425.68,456.08) (-425.68,456.79) (-425.68,457.50) (-425.68,458.21) (-425.68,458.91) (-425.68,459.62) (-425.68,460.33) (-425.68,461.03) (-424.97,461.74) (-424.97,462.45) (-424.97,463.15) (-424.97,463.86) (-424.97,464.57) (-424.97,465.28) (-424.97,465.98) (-427.80,428.51) (-427.80,429.21) (-428.51,429.92) (-428.51,430.63) (-428.51,431.34) (-429.21,432.04) (-429.21,432.75) (-429.92,433.46) (-429.92,434.16) (-429.92,434.87) (-430.63,435.58) (-430.63,436.28) (-430.63,436.99) (-431.34,437.70) (-431.34,438.41) (-431.34,439.11) (-432.04,439.82) (-432.04,440.53) (-432.75,441.23) (-432.75,441.94) (-432.75,442.65) (-433.46,443.36) (-433.46,444.06) (-433.46,444.77) (-434.16,445.48) (-434.16,446.18) (-434.16,446.89) (-434.87,447.60) (-434.87,448.31) (-434.87,449.01) (-435.58,449.72) (-435.58,450.43) (-436.28,451.13) (-436.28,451.84) (-436.28,452.55) (-436.99,453.26) (-436.99,453.96) (-436.99,454.67) (-437.70,455.38) (-437.70,456.08) (-437.70,456.79) (-438.41,457.50) (-438.41,458.21) (-439.11,458.91) (-439.11,459.62) (-439.11,460.33) (-439.82,461.03) (-439.82,461.74) (-439.82,461.74) (-439.11,461.74) (-438.41,462.45) (-437.70,462.45) (-436.99,462.45) (-436.28,463.15) (-435.58,463.15) (-434.87,463.15) (-434.16,463.86) (-433.46,463.86) (-432.75,463.86) (-432.04,464.57) (-431.34,464.57) (-430.63,464.57) (-429.92,465.28) (-429.21,465.28) (-428.51,465.28) (-427.80,465.98) (-427.09,465.98) (-426.39,465.98) (-425.68,466.69) (-424.97,466.69) (-424.26,466.69) (-423.56,467.40) (-422.85,467.40) (-422.14,468.10) (-421.44,468.10) (-420.73,468.10) (-420.02,468.81) (-419.31,468.81) (-418.61,468.81) (-417.90,469.52) (-417.19,469.52) (-416.49,469.52) (-415.78,470.23) (-415.07,470.23) (-414.36,470.23) (-413.66,470.93) (-412.95,470.93) (-412.24,470.93) (-411.54,471.64) (-410.83,471.64) (-410.12,471.64) (-409.41,472.35) (-408.71,472.35) (-408.71,472.35) (-408.71,473.05) (-408.71,473.76) (-408.71,474.47) (-408.71,475.18) (-408.71,475.88) (-408.71,476.59) (-408.71,477.30) (-408.71,478.00) (-408.71,478.71) (-408.71,479.42) (-408.71,480.13) (-408.00,480.83) (-408.00,481.54) (-408.00,482.25) (-408.00,482.95) (-408.00,483.66) (-408.00,484.37) (-408.00,485.08) (-408.00,485.78) (-408.00,486.49) (-408.00,487.20) (-408.00,487.90) (-408.00,488.61) (-408.00,489.32) (-408.00,490.02) (-408.00,490.73) (-408.00,491.44) (-408.00,492.15) (-408.00,492.85) (-408.00,493.56) (-408.00,494.27) (-408.00,494.97) (-408.00,495.68) (-408.00,496.39) (-408.00,497.10) (-407.29,497.80) (-407.29,498.51) (-407.29,499.22) (-407.29,499.92) (-407.29,500.63) (-407.29,501.34) (-407.29,502.05) (-407.29,502.75) (-407.29,503.46) (-407.29,504.17) (-407.29,504.87) (-407.29,505.58) (-383.96,479.42) (-384.67,480.13) (-385.37,480.83) (-386.08,481.54) (-386.79,482.25) (-386.79,482.95) (-387.49,483.66) (-388.20,484.37) (-388.91,485.08) (-389.62,485.78) (-390.32,486.49) (-391.03,487.20) (-391.74,487.90) (-392.44,488.61) (-392.44,489.32) (-393.15,490.02) (-393.86,490.73) (-394.57,491.44) (-395.27,492.15) (-395.98,492.85) (-396.69,493.56) (-397.39,494.27) (-398.10,494.97) (-398.81,495.68) (-398.81,496.39) (-399.52,497.10) (-400.22,497.80) (-400.93,498.51) (-401.64,499.22) (-402.34,499.92) (-403.05,500.63) (-403.76,501.34) (-404.47,502.05) (-404.47,502.75) (-405.17,503.46) (-405.88,504.17) (-406.59,504.87) (-407.29,505.58) (-376.89,445.48) (-376.89,446.18) (-376.89,446.89) (-377.60,447.60) (-377.60,448.31) (-377.60,449.01) (-377.60,449.72) (-377.60,450.43) (-378.30,451.13) (-378.30,451.84) (-378.30,452.55) (-378.30,453.26) (-378.30,453.96) (-379.01,454.67) (-379.01,455.38) (-379.01,456.08) (-379.01,456.79) (-379.72,457.50) (-379.72,458.21) (-379.72,458.91) (-379.72,459.62) (-379.72,460.33) (-380.42,461.03) (-380.42,461.74) (-380.42,462.45) (-380.42,463.15) (-380.42,463.86) (-381.13,464.57) (-381.13,465.28) (-381.13,465.98) (-381.13,466.69) (-381.13,467.40) (-381.84,468.10) (-381.84,468.81) (-381.84,469.52) (-381.84,470.23) (-381.84,470.93) (-382.54,471.64) (-382.54,472.35) (-382.54,473.05) (-382.54,473.76) (-383.25,474.47) (-383.25,475.18) (-383.25,475.88) (-383.25,476.59) (-383.25,477.30) (-383.96,478.00) (-383.96,478.71) (-383.96,479.42) (-408.71,450.43) (-408.00,450.43) (-407.29,450.43) (-406.59,450.43) (-405.88,449.72) (-405.17,449.72) (-404.47,449.72) (-403.76,449.72) (-403.05,449.72) (-402.34,449.72) (-401.64,449.01) (-400.93,449.01) (-400.22,449.01) (-399.52,449.01) (-398.81,449.01) (-398.10,449.01) (-397.39,449.01) (-396.69,448.31) (-395.98,448.31) (-395.27,448.31) (-394.57,448.31) (-393.86,448.31) (-393.15,448.31) (-392.44,447.60) (-391.74,447.60) (-391.03,447.60) (-390.32,447.60) (-389.62,447.60) (-388.91,447.60) (-388.20,446.89) (-387.49,446.89) (-386.79,446.89) (-386.08,446.89) (-385.37,446.89) (-384.67,446.89) (-383.96,446.89) (-383.25,446.18) (-382.54,446.18) (-381.84,446.18) (-381.13,446.18) (-380.42,446.18) (-379.72,446.18) (-379.01,445.48) (-378.30,445.48) (-377.60,445.48) (-376.89,445.48) (-443.36,454.67) (-442.65,454.67) (-441.94,454.67) (-441.23,454.67) (-440.53,454.67) (-439.82,453.96) (-439.11,453.96) (-438.41,453.96) (-437.70,453.96) (-436.99,453.96) (-436.28,453.96) (-435.58,453.96) (-434.87,453.96) (-434.16,453.26) (-433.46,453.26) (-432.75,453.26) (-432.04,453.26) (-431.34,453.26) (-430.63,453.26) (-429.92,453.26) (-429.21,453.26) (-428.51,452.55) (-427.80,452.55) (-427.09,452.55) (-426.39,452.55) (-425.68,452.55) (-424.97,452.55) (-424.26,452.55) (-423.56,452.55) (-422.85,451.84) (-422.14,451.84) (-421.44,451.84) (-420.73,451.84) (-420.02,451.84) (-419.31,451.84) (-418.61,451.84) (-417.90,451.84) (-417.19,451.13) (-416.49,451.13) (-415.78,451.13) (-415.07,451.13) (-414.36,451.13) (-413.66,451.13) (-412.95,451.13) (-412.24,451.13) (-411.54,450.43) (-410.83,450.43) (-410.12,450.43) (-409.41,450.43) (-408.71,450.43) (-471.64,435.58) (-470.93,436.28) (-470.23,436.28) (-469.52,436.99) (-468.81,437.70) (-468.10,437.70) (-467.40,438.41) (-466.69,439.11) (-465.98,439.11) (-465.28,439.82) (-464.57,440.53) (-463.86,440.53) (-463.15,441.23) (-462.45,441.94) (-461.74,441.94) (-461.03,442.65) (-460.33,443.36) (-459.62,443.36) (-458.91,444.06) (-458.21,444.77) (-457.50,444.77) (-456.79,445.48) (-456.08,446.18) (-455.38,446.89) (-454.67,446.89) (-453.96,447.60) (-453.26,448.31) (-452.55,448.31) (-451.84,449.01) (-451.13,449.72) (-450.43,449.72) (-449.72,450.43) (-449.01,451.13) (-448.31,451.13) (-447.60,451.84) (-446.89,452.55) (-446.18,452.55) (-445.48,453.26) (-444.77,453.96) (-444.06,453.96) (-443.36,454.67) (-471.64,435.58) (-470.93,435.58) (-470.23,435.58) (-469.52,434.87) (-468.81,434.87) (-468.10,434.87) (-467.40,434.87) (-466.69,434.87) (-465.98,434.87) (-465.28,434.16) (-464.57,434.16) (-463.86,434.16) (-463.15,434.16) (-462.45,434.16) (-461.74,434.16) (-461.03,433.46) (-460.33,433.46) (-459.62,433.46) (-458.91,433.46) (-458.21,433.46) (-457.50,432.75) (-456.79,432.75) (-456.08,432.75) (-455.38,432.75) (-454.67,432.75) (-453.96,432.75) (-453.26,432.04) (-452.55,432.04) (-451.84,432.04) (-451.13,432.04) (-450.43,432.04) (-449.72,431.34) (-449.01,431.34) (-448.31,431.34) (-447.60,431.34) (-446.89,431.34) (-446.18,431.34) (-445.48,430.63) (-444.77,430.63) (-444.06,430.63) (-443.36,430.63) (-442.65,430.63) (-441.94,430.63) (-441.23,429.92) (-440.53,429.92) (-439.82,429.92) (-439.82,429.92) (-439.82,430.63) (-439.11,431.34) (-439.11,432.04) (-438.41,432.75) (-438.41,433.46) (-437.70,434.16) (-437.70,434.87) (-436.99,435.58) (-436.99,436.28) (-436.28,436.99) (-436.28,437.70) (-435.58,438.41) (-435.58,439.11) (-434.87,439.82) (-434.87,440.53) (-434.16,441.23) (-434.16,441.94) (-433.46,442.65) (-433.46,443.36) (-432.75,444.06) (-432.75,444.77) (-432.04,445.48) (-432.04,446.18) (-431.34,446.89) (-431.34,447.60) (-430.63,448.31) (-430.63,449.01) (-429.92,449.72) (-429.92,450.43) (-429.21,451.13) (-429.21,451.84) (-428.51,452.55) (-428.51,453.26) (-427.80,453.96) (-427.80,454.67) (-427.09,455.38) (-427.09,456.08) (-426.39,456.79) (-426.39,457.50) (-425.68,458.21) (-425.68,458.91) (-424.97,459.62) (-446.89,431.34) (-446.18,432.04) (-445.48,432.75) (-445.48,433.46) (-444.77,434.16) (-444.06,434.87) (-443.36,435.58) (-443.36,436.28) (-442.65,436.99) (-441.94,437.70) (-441.23,438.41) (-440.53,439.11) (-440.53,439.82) (-439.82,440.53) (-439.11,441.23) (-438.41,441.94) (-438.41,442.65) (-437.70,443.36) (-436.99,444.06) (-436.28,444.77) (-436.28,445.48) (-435.58,446.18) (-434.87,446.89) (-434.16,447.60) (-433.46,448.31) (-433.46,449.01) (-432.75,449.72) (-432.04,450.43) (-431.34,451.13) (-431.34,451.84) (-430.63,452.55) (-429.92,453.26) (-429.21,453.96) (-428.51,454.67) (-428.51,455.38) (-427.80,456.08) (-427.09,456.79) (-426.39,457.50) (-426.39,458.21) (-425.68,458.91) (-424.97,459.62) (-446.89,431.34) (-446.18,432.04) (-445.48,432.75) (-444.77,433.46) (-444.06,434.16) (-443.36,434.87) (-442.65,435.58) (-442.65,436.28) (-441.94,436.99) (-441.23,437.70) (-440.53,438.41) (-439.82,439.11) (-439.11,439.82) (-438.41,440.53) (-437.70,441.23) (-436.99,441.94) (-436.28,442.65) (-435.58,443.36) (-434.87,444.06) (-434.87,444.77) (-434.16,445.48) (-433.46,446.18) (-432.75,446.89) (-432.04,447.60) (-431.34,448.31) (-430.63,449.01) (-429.92,449.72) (-429.21,450.43) (-428.51,451.13) (-427.80,451.84) (-427.09,452.55) (-427.09,453.26) (-426.39,453.96) (-425.68,454.67) (-424.97,455.38) (-424.26,456.08) (-423.56,456.79) (-422.85,457.50) (-449.72,444.77) (-449.01,444.77) (-448.31,445.48) (-447.60,445.48) (-446.89,446.18) (-446.18,446.18) (-445.48,446.89) (-444.77,446.89) (-444.06,447.60) (-443.36,447.60) (-442.65,448.31) (-441.94,448.31) (-441.23,449.01) (-440.53,449.01) (-439.82,449.72) (-439.11,449.72) (-438.41,450.43) (-437.70,450.43) (-436.99,451.13) (-436.28,451.13) (-435.58,451.13) (-434.87,451.84) (-434.16,451.84) (-433.46,452.55) (-432.75,452.55) (-432.04,453.26) (-431.34,453.26) (-430.63,453.96) (-429.92,453.96) (-429.21,454.67) (-428.51,454.67) (-427.80,455.38) (-427.09,455.38) (-426.39,456.08) (-425.68,456.08) (-424.97,456.79) (-424.26,456.79) (-423.56,457.50) (-422.85,457.50) (-425.68,419.31) (-426.39,420.02) (-427.09,420.73) (-427.80,421.44) (-428.51,422.14) (-429.21,422.85) (-429.92,423.56) (-430.63,424.26) (-431.34,424.97) (-431.34,425.68) (-432.04,426.39) (-432.75,427.09) (-433.46,427.80) (-434.16,428.51) (-434.87,429.21) (-435.58,429.92) (-436.28,430.63) (-436.99,431.34) (-437.70,432.04) (-438.41,432.75) (-439.11,433.46) (-439.82,434.16) (-440.53,434.87) (-441.23,435.58) (-441.94,436.28) (-442.65,436.99) (-443.36,437.70) (-443.36,438.41) (-444.06,439.11) (-444.77,439.82) (-445.48,440.53) (-446.18,441.23) (-446.89,441.94) (-447.60,442.65) (-448.31,443.36) (-449.01,444.06) (-449.72,444.77) (-403.76,393.86) (-404.47,394.57) (-405.17,395.27) (-405.88,395.98) (-405.88,396.69) (-406.59,397.39) (-407.29,398.10) (-408.00,398.81) (-408.71,399.52) (-409.41,400.22) (-410.12,400.93) (-410.12,401.64) (-410.83,402.34) (-411.54,403.05) (-412.24,403.76) (-412.95,404.47) (-413.66,405.17) (-414.36,405.88) (-414.36,406.59) (-415.07,407.29) (-415.78,408.00) (-416.49,408.71) (-417.19,409.41) (-417.90,410.12) (-418.61,410.83) (-419.31,411.54) (-419.31,412.24) (-420.02,412.95) (-420.73,413.66) (-421.44,414.36) (-422.14,415.07) (-422.85,415.78) (-423.56,416.49) (-423.56,417.19) (-424.26,417.90) (-424.97,418.61) (-425.68,419.31) (-403.76,359.92) (-403.76,360.62) (-403.76,361.33) (-403.76,362.04) (-403.76,362.75) (-403.76,363.45) (-403.76,364.16) (-403.76,364.87) (-403.76,365.57) (-403.76,366.28) (-403.76,366.99) (-403.76,367.70) (-403.76,368.40) (-403.76,369.11) (-403.76,369.82) (-403.76,370.52) (-403.76,371.23) (-403.76,371.94) (-403.76,372.65) (-403.76,373.35) (-403.76,374.06) (-403.76,374.77) (-403.76,375.47) (-403.76,376.18) (-403.76,376.89) (-403.76,377.60) (-403.76,378.30) (-403.76,379.01) (-403.76,379.72) (-403.76,380.42) (-403.76,381.13) (-403.76,381.84) (-403.76,382.54) (-403.76,383.25) (-403.76,383.96) (-403.76,384.67) (-403.76,385.37) (-403.76,386.08) (-403.76,386.79) (-403.76,387.49) (-403.76,388.20) (-403.76,388.91) (-403.76,389.62) (-403.76,390.32) (-403.76,391.03) (-403.76,391.74) (-403.76,392.44) (-403.76,393.15) (-403.76,393.86) (-439.82,363.45) (-439.11,363.45) (-438.41,363.45) (-437.70,363.45) (-436.99,363.45) (-436.28,363.45) (-435.58,362.75) (-434.87,362.75) (-434.16,362.75) (-433.46,362.75) (-432.75,362.75) (-432.04,362.75) (-431.34,362.75) (-430.63,362.75) (-429.92,362.75) (-429.21,362.75) (-428.51,362.04) (-427.80,362.04) (-427.09,362.04) (-426.39,362.04) (-425.68,362.04) (-424.97,362.04) (-424.26,362.04) (-423.56,362.04) (-422.85,362.04) (-422.14,362.04) (-421.44,361.33) (-420.73,361.33) (-420.02,361.33) (-419.31,361.33) (-418.61,361.33) (-417.90,361.33) (-417.19,361.33) (-416.49,361.33) (-415.78,361.33) (-415.07,361.33) (-414.36,360.62) (-413.66,360.62) (-412.95,360.62) (-412.24,360.62) (-411.54,360.62) (-410.83,360.62) (-410.12,360.62) (-409.41,360.62) (-408.71,360.62) (-408.00,360.62) (-407.29,359.92) (-406.59,359.92) (-405.88,359.92) (-405.17,359.92) (-404.47,359.92) (-403.76,359.92) (-464.57,344.36) (-463.86,345.07) (-463.15,345.78) (-462.45,345.78) (-461.74,346.48) (-461.03,347.19) (-460.33,347.90) (-459.62,347.90) (-458.91,348.60) (-458.21,349.31) (-457.50,350.02) (-456.79,350.02) (-456.08,350.72) (-455.38,351.43) (-454.67,352.14) (-453.96,352.85) (-453.26,352.85) (-452.55,353.55) (-451.84,354.26) (-451.13,354.97) (-450.43,354.97) (-449.72,355.67) (-449.01,356.38) (-448.31,357.09) (-447.60,357.80) (-446.89,357.80) (-446.18,358.50) (-445.48,359.21) (-444.77,359.92) (-444.06,359.92) (-443.36,360.62) (-442.65,361.33) (-441.94,362.04) (-441.23,362.04) (-440.53,362.75) (-439.82,363.45) (-464.57,344.36) (-463.86,343.65) (-463.15,343.65) (-462.45,342.95) (-461.74,342.24) (-461.03,342.24) (-460.33,341.53) (-459.62,341.53) (-458.91,340.83) (-458.21,340.12) (-457.50,340.12) (-456.79,339.41) (-456.08,338.70) (-455.38,338.70) (-454.67,338.00) (-453.96,337.29) (-453.26,337.29) (-452.55,336.58) (-451.84,336.58) (-451.13,335.88) (-450.43,335.17) (-449.72,335.17) (-449.01,334.46) (-448.31,333.75) (-447.60,333.75) (-446.89,333.05) (-446.18,333.05) (-445.48,332.34) (-444.77,331.63) (-444.06,331.63) (-443.36,330.93) (-442.65,330.22) (-441.94,330.22) (-441.23,329.51) (-440.53,328.80) (-439.82,328.80) (-439.11,328.10) (-438.41,328.10) (-437.70,327.39) (-436.99,326.68) (-436.28,326.68) (-435.58,325.98) (-413.66,297.69) (-414.36,298.40) (-415.07,299.11) (-415.07,299.81) (-415.78,300.52) (-416.49,301.23) (-417.19,301.93) (-417.19,302.64) (-417.90,303.35) (-418.61,304.06) (-419.31,304.76) (-420.02,305.47) (-420.02,306.18) (-420.73,306.88) (-421.44,307.59) (-422.14,308.30) (-422.14,309.01) (-422.85,309.71) (-423.56,310.42) (-424.26,311.13) (-424.26,311.83) (-424.97,312.54) (-425.68,313.25) (-426.39,313.96) (-427.09,314.66) (-427.09,315.37) (-427.80,316.08) (-428.51,316.78) (-429.21,317.49) (-429.21,318.20) (-429.92,318.91) (-430.63,319.61) (-431.34,320.32) (-432.04,321.03) (-432.04,321.73) (-432.75,322.44) (-433.46,323.15) (-434.16,323.85) (-434.16,324.56) (-434.87,325.27) (-435.58,325.98) (-395.98,266.58) (-396.69,267.29) (-396.69,267.99) (-397.39,268.70) (-397.39,269.41) (-398.10,270.11) (-398.10,270.82) (-398.81,271.53) (-399.52,272.24) (-399.52,272.94) (-400.22,273.65) (-400.22,274.36) (-400.93,275.06) (-400.93,275.77) (-401.64,276.48) (-402.34,277.19) (-402.34,277.89) (-403.05,278.60) (-403.05,279.31) (-403.76,280.01) (-403.76,280.72) (-404.47,281.43) (-404.47,282.14) (-405.17,282.84) (-405.88,283.55) (-405.88,284.26) (-406.59,284.96) (-406.59,285.67) (-407.29,286.38) (-407.29,287.09) (-408.00,287.79) (-408.71,288.50) (-408.71,289.21) (-409.41,289.91) (-409.41,290.62) (-410.12,291.33) (-410.12,292.04) (-410.83,292.74) (-411.54,293.45) (-411.54,294.16) (-412.24,294.86) (-412.24,295.57) (-412.95,296.28) (-412.95,296.98) (-413.66,297.69) (-399.52,232.64) (-399.52,233.35) (-399.52,234.05) (-399.52,234.76) (-399.52,235.47) (-398.81,236.17) (-398.81,236.88) (-398.81,237.59) (-398.81,238.29) (-398.81,239.00) (-398.81,239.71) (-398.81,240.42) (-398.81,241.12) (-398.81,241.83) (-398.81,242.54) (-398.10,243.24) (-398.10,243.95) (-398.10,244.66) (-398.10,245.37) (-398.10,246.07) (-398.10,246.78) (-398.10,247.49) (-398.10,248.19) (-398.10,248.90) (-398.10,249.61) (-397.39,250.32) (-397.39,251.02) (-397.39,251.73) (-397.39,252.44) (-397.39,253.14) (-397.39,253.85) (-397.39,254.56) (-397.39,255.27) (-397.39,255.97) (-396.69,256.68) (-396.69,257.39) (-396.69,258.09) (-396.69,258.80) (-396.69,259.51) (-396.69,260.22) (-396.69,260.92) (-396.69,261.63) (-396.69,262.34) (-396.69,263.04) (-395.98,263.75) (-395.98,264.46) (-395.98,265.17) (-395.98,265.87) (-395.98,266.58) (-399.52,232.64) (-400.22,233.35) (-400.93,234.05) (-401.64,234.76) (-402.34,235.47) (-402.34,236.17) (-403.05,236.88) (-403.76,237.59) (-404.47,238.29) (-405.17,239.00) (-405.88,239.71) (-406.59,240.42) (-407.29,241.12) (-408.00,241.83) (-408.00,242.54) (-408.71,243.24) (-409.41,243.95) (-410.12,244.66) (-410.83,245.37) (-411.54,246.07) (-412.24,246.78) (-412.95,247.49) (-412.95,248.19) (-413.66,248.90) (-414.36,249.61) (-415.07,250.32) (-415.78,251.02) (-416.49,251.73) (-417.19,252.44) (-417.90,253.14) (-418.61,253.85) (-418.61,254.56) (-419.31,255.27) (-420.02,255.97) (-420.73,256.68) (-421.44,257.39) (-421.44,257.39) (-421.44,258.09) (-422.14,258.80) (-422.14,259.51) (-422.14,260.22) (-422.14,260.92) (-422.85,261.63) (-422.85,262.34) (-422.85,263.04) (-422.85,263.75) (-423.56,264.46) (-423.56,265.17) (-423.56,265.87) (-424.26,266.58) (-424.26,267.29) (-424.26,267.99) (-424.26,268.70) (-424.97,269.41) (-424.97,270.11) (-424.97,270.82) (-424.97,271.53) (-425.68,272.24) (-425.68,272.94) (-425.68,273.65) (-425.68,274.36) (-426.39,275.06) (-426.39,275.77) (-426.39,276.48) (-427.09,277.19) (-427.09,277.89) (-427.09,278.60) (-427.09,279.31) (-427.80,280.01) (-427.80,280.72) (-427.80,281.43) (-427.80,282.14) (-428.51,282.84) (-428.51,283.55) (-428.51,284.26) (-429.21,284.96) (-429.21,285.67) (-429.21,286.38) (-429.21,287.09) (-429.92,287.79) (-429.92,288.50) (-429.92,289.21) (-429.92,289.91) (-430.63,290.62) (-430.63,291.33) (-430.63,291.33) (-429.92,291.33) (-429.21,292.04) (-428.51,292.04) (-427.80,292.74) (-427.09,292.74) (-426.39,292.74) (-425.68,293.45) (-424.97,293.45) (-424.26,294.16) (-423.56,294.16) (-422.85,294.16) (-422.14,294.86) (-421.44,294.86) (-420.73,295.57) (-420.02,295.57) (-419.31,295.57) (-418.61,296.28) (-417.90,296.28) (-417.19,296.98) (-416.49,296.98) (-415.78,296.98) (-415.07,297.69) (-414.36,297.69) (-413.66,298.40) (-412.95,298.40) (-412.24,299.11) (-411.54,299.11) (-410.83,299.11) (-410.12,299.81) (-409.41,299.81) (-408.71,300.52) (-408.00,300.52) (-407.29,300.52) (-406.59,301.23) (-405.88,301.23) (-405.17,301.93) (-404.47,301.93) (-403.76,301.93) (-403.05,302.64) (-402.34,302.64) (-401.64,303.35) (-400.93,303.35) (-400.22,303.35) (-399.52,304.06) (-398.81,304.06) (-398.10,304.76) (-397.39,304.76) (-397.39,304.76) (-397.39,305.47) (-397.39,306.18) (-397.39,306.88) (-397.39,307.59) (-397.39,308.30) (-397.39,309.01) (-397.39,309.71) (-397.39,310.42) (-397.39,311.13) (-397.39,311.83) (-397.39,312.54) (-397.39,313.25) (-398.10,313.96) (-398.10,314.66) (-398.10,315.37) (-398.10,316.08) (-398.10,316.78) (-398.10,317.49) (-398.10,318.20) (-398.10,318.91) (-398.10,319.61) (-398.10,320.32) (-398.10,321.03) (-398.10,321.73) (-398.10,322.44) (-398.10,323.15) (-398.10,323.85) (-398.10,324.56) (-398.10,325.27) (-398.10,325.98) (-398.10,326.68) (-398.10,327.39) (-398.10,328.10) (-398.10,328.80) (-398.10,329.51) (-398.10,330.22) (-398.81,330.93) (-398.81,331.63) (-398.81,332.34) (-398.81,333.05) (-398.81,333.75) (-398.81,334.46) (-398.81,335.17) (-398.81,335.88) (-398.81,336.58) (-398.81,337.29) (-398.81,338.00) (-398.81,338.70) (-398.81,338.70) (-398.10,338.00) (-397.39,338.00) (-396.69,337.29) (-395.98,336.58) (-395.27,336.58) (-394.57,335.88) (-393.86,335.88) (-393.15,335.17) (-392.44,334.46) (-391.74,334.46) (-391.03,333.75) (-390.32,333.05) (-389.62,333.05) (-388.91,332.34) (-388.20,331.63) (-387.49,331.63) (-386.79,330.93) (-386.08,330.93) (-385.37,330.22) (-384.67,329.51) (-383.96,329.51) (-383.25,328.80) (-382.54,328.10) (-381.84,328.10) (-381.13,327.39) (-380.42,327.39) (-379.72,326.68) (-379.01,325.98) (-378.30,325.98) (-377.60,325.27) (-376.89,324.56) (-376.18,324.56) (-375.47,323.85) (-374.77,323.15) (-374.06,323.15) (-373.35,322.44) (-372.65,322.44) (-371.94,321.73) (-371.23,321.03) (-370.52,321.03) (-369.82,320.32) (-369.82,320.32) (-369.11,319.61) (-368.40,319.61) (-367.70,318.91) (-366.99,318.91) (-366.28,318.20) (-365.57,318.20) (-364.87,317.49) (-364.16,317.49) (-363.45,316.78) (-362.75,316.78) (-362.04,316.08) (-361.33,315.37) (-360.62,315.37) (-359.92,314.66) (-359.21,314.66) (-358.50,313.96) (-357.80,313.96) (-357.09,313.25) (-356.38,313.25) (-355.67,312.54) (-354.97,311.83) (-354.26,311.83) (-353.55,311.13) (-352.85,311.13) (-352.14,310.42) (-351.43,310.42) (-350.72,309.71) (-350.02,309.71) (-349.31,309.01) (-348.60,309.01) (-347.90,308.30) (-347.19,307.59) (-346.48,307.59) (-345.78,306.88) (-345.07,306.88) (-344.36,306.18) (-343.65,306.18) (-342.95,305.47) (-342.24,305.47) (-341.53,304.76) (-341.53,304.76) (-340.83,304.06) (-340.12,303.35) (-339.41,302.64) (-338.70,301.93) (-338.00,301.23) (-337.29,300.52) (-336.58,299.81) (-335.88,299.11) (-335.17,299.11) (-334.46,298.40) (-333.75,297.69) (-333.05,296.98) (-332.34,296.28) (-331.63,295.57) (-330.93,294.86) (-330.22,294.16) (-329.51,293.45) (-328.80,292.74) (-328.10,292.04) (-327.39,291.33) (-326.68,290.62) (-325.98,289.91) (-325.27,289.21) (-324.56,288.50) (-323.85,287.79) (-323.15,287.09) (-322.44,287.09) (-321.73,286.38) (-321.03,285.67) (-320.32,284.96) (-319.61,284.26) (-318.91,283.55) (-318.20,282.84) (-317.49,282.14) (-316.78,281.43) (-294.86,254.56) (-295.57,255.27) (-296.28,255.97) (-296.28,256.68) (-296.98,257.39) (-297.69,258.09) (-298.40,258.80) (-299.11,259.51) (-299.81,260.22) (-299.81,260.92) (-300.52,261.63) (-301.23,262.34) (-301.93,263.04) (-302.64,263.75) (-302.64,264.46) (-303.35,265.17) (-304.06,265.87) (-304.76,266.58) (-305.47,267.29) (-305.47,267.99) (-306.18,268.70) (-306.88,269.41) (-307.59,270.11) (-308.30,270.82) (-309.01,271.53) (-309.01,272.24) (-309.71,272.94) (-310.42,273.65) (-311.13,274.36) (-311.83,275.06) (-311.83,275.77) (-312.54,276.48) (-313.25,277.19) (-313.96,277.89) (-314.66,278.60) (-315.37,279.31) (-315.37,280.01) (-316.08,280.72) (-316.78,281.43) (-279.31,225.57) (-280.01,226.27) (-280.01,226.98) (-280.72,227.69) (-280.72,228.40) (-281.43,229.10) (-281.43,229.81) (-282.14,230.52) (-282.14,231.22) (-282.84,231.93) (-282.84,232.64) (-283.55,233.35) (-283.55,234.05) (-284.26,234.76) (-284.96,235.47) (-284.96,236.17) (-285.67,236.88) (-285.67,237.59) (-286.38,238.29) (-286.38,239.00) (-287.09,239.71) (-287.09,240.42) (-287.79,241.12) (-287.79,241.83) (-288.50,242.54) (-288.50,243.24) (-289.21,243.95) (-289.21,244.66) (-289.91,245.37) (-290.62,246.07) (-290.62,246.78) (-291.33,247.49) (-291.33,248.19) (-292.04,248.90) (-292.04,249.61) (-292.74,250.32) (-292.74,251.02) (-293.45,251.73) (-293.45,252.44) (-294.16,253.14) (-294.16,253.85) (-294.86,254.56) (-310.42,206.48) (-309.71,207.18) (-309.01,207.18) (-308.30,207.89) (-307.59,207.89) (-306.88,208.60) (-306.18,209.30) (-305.47,209.30) (-304.76,210.01) (-304.06,210.72) (-303.35,210.72) (-302.64,211.42) (-301.93,211.42) (-301.23,212.13) (-300.52,212.84) (-299.81,212.84) (-299.11,213.55) (-298.40,213.55) (-297.69,214.25) (-296.98,214.96) (-296.28,214.96) (-295.57,215.67) (-294.86,215.67) (-294.16,216.37) (-293.45,217.08) (-292.74,217.08) (-292.04,217.79) (-291.33,218.50) (-290.62,218.50) (-289.91,219.20) (-289.21,219.20) (-288.50,219.91) (-287.79,220.62) (-287.09,220.62) (-286.38,221.32) (-285.67,221.32) (-284.96,222.03) (-284.26,222.74) (-283.55,222.74) (-282.84,223.45) (-282.14,224.15) (-281.43,224.15) (-280.72,224.86) (-280.01,224.86) (-279.31,225.57) (-290.62,178.90) (-291.33,179.61) (-291.33,180.31) (-292.04,181.02) (-292.74,181.73) (-293.45,182.43) (-293.45,183.14) (-294.16,183.85) (-294.86,184.55) (-294.86,185.26) (-295.57,185.97) (-296.28,186.68) (-296.98,187.38) (-296.98,188.09) (-297.69,188.80) (-298.40,189.50) (-298.40,190.21) (-299.11,190.92) (-299.81,191.63) (-300.52,192.33) (-300.52,193.04) (-301.23,193.75) (-301.93,194.45) (-302.64,195.16) (-302.64,195.87) (-303.35,196.58) (-304.06,197.28) (-304.06,197.99) (-304.76,198.70) (-305.47,199.40) (-306.18,200.11) (-306.18,200.82) (-306.88,201.53) (-307.59,202.23) (-307.59,202.94) (-308.30,203.65) (-309.01,204.35) (-309.71,205.06) (-309.71,205.77) (-310.42,206.48) (-307.59,149.91) (-306.88,150.61) (-306.88,151.32) (-306.18,152.03) (-306.18,152.74) (-305.47,153.44) (-304.76,154.15) (-304.76,154.86) (-304.06,155.56) (-304.06,156.27) (-303.35,156.98) (-303.35,157.68) (-302.64,158.39) (-301.93,159.10) (-301.93,159.81) (-301.23,160.51) (-301.23,161.22) (-300.52,161.93) (-299.81,162.63) (-299.81,163.34) (-299.11,164.05) (-299.11,164.76) (-298.40,165.46) (-298.40,166.17) (-297.69,166.88) (-296.98,167.58) (-296.98,168.29) (-296.28,169.00) (-296.28,169.71) (-295.57,170.41) (-294.86,171.12) (-294.86,171.83) (-294.16,172.53) (-294.16,173.24) (-293.45,173.95) (-293.45,174.66) (-292.74,175.36) (-292.04,176.07) (-292.04,176.78) (-291.33,177.48) (-291.33,178.19) (-290.62,178.90) (-307.59,149.91) (-307.59,150.61) (-307.59,151.32) (-306.88,152.03) (-306.88,152.74) (-306.88,153.44) (-306.88,154.15) (-306.18,154.86) (-306.18,155.56) (-306.18,156.27) (-306.18,156.98) (-306.18,157.68) (-305.47,158.39) (-305.47,159.10) (-305.47,159.81) (-305.47,160.51) (-305.47,161.22) (-304.76,161.93) (-304.76,162.63) (-304.76,163.34) (-304.76,164.05) (-304.06,164.76) (-304.06,165.46) (-304.06,166.17) (-304.06,166.88) (-304.06,167.58) (-303.35,168.29) (-303.35,169.00) (-303.35,169.71) (-303.35,170.41) (-302.64,171.12) (-302.64,171.83) (-302.64,172.53) (-302.64,173.24) (-302.64,173.95) (-301.93,174.66) (-301.93,175.36) (-301.93,176.07) (-301.93,176.78) (-301.93,177.48) (-301.23,178.19) (-301.23,178.90) (-301.23,179.61) (-301.23,180.31) (-300.52,181.02) (-300.52,181.73) (-300.52,182.43) (-300.52,146.37) (-300.52,147.08) (-300.52,147.79) (-300.52,148.49) (-300.52,149.20) (-300.52,149.91) (-300.52,150.61) (-300.52,151.32) (-300.52,152.03) (-300.52,152.74) (-300.52,153.44) (-300.52,154.15) (-300.52,154.86) (-300.52,155.56) (-300.52,156.27) (-300.52,156.98) (-300.52,157.68) (-300.52,158.39) (-300.52,159.10) (-300.52,159.81) (-300.52,160.51) (-300.52,161.22) (-300.52,161.93) (-300.52,162.63) (-300.52,163.34) (-300.52,164.05) (-300.52,164.76) (-300.52,165.46) (-300.52,166.17) (-300.52,166.88) (-300.52,167.58) (-300.52,168.29) (-300.52,169.00) (-300.52,169.71) (-300.52,170.41) (-300.52,171.12) (-300.52,171.83) (-300.52,172.53) (-300.52,173.24) (-300.52,173.95) (-300.52,174.66) (-300.52,175.36) (-300.52,176.07) (-300.52,176.78) (-300.52,177.48) (-300.52,178.19) (-300.52,178.90) (-300.52,179.61) (-300.52,180.31) (-300.52,181.02) (-300.52,181.73) (-300.52,182.43) (-325.98,170.41) (-325.27,169.71) (-324.56,169.00) (-323.85,168.29) (-323.15,167.58) (-322.44,166.88) (-321.73,166.17) (-321.03,165.46) (-320.32,164.76) (-319.61,164.76) (-318.91,164.05) (-318.20,163.34) (-317.49,162.63) (-316.78,161.93) (-316.08,161.22) (-315.37,160.51) (-314.66,159.81) (-313.96,159.10) (-313.25,158.39) (-312.54,157.68) (-311.83,156.98) (-311.13,156.27) (-310.42,155.56) (-309.71,154.86) (-309.01,154.15) (-308.30,153.44) (-307.59,152.74) (-306.88,152.74) (-306.18,152.03) (-305.47,151.32) (-304.76,150.61) (-304.06,149.91) (-303.35,149.20) (-302.64,148.49) (-301.93,147.79) (-301.23,147.08) (-300.52,146.37) (-325.98,170.41) (-325.98,171.12) (-325.27,171.83) (-325.27,172.53) (-325.27,173.24) (-325.27,173.95) (-324.56,174.66) (-324.56,175.36) (-324.56,176.07) (-324.56,176.78) (-323.85,177.48) (-323.85,178.19) (-323.85,178.90) (-323.15,179.61) (-323.15,180.31) (-323.15,181.02) (-323.15,181.73) (-322.44,182.43) (-322.44,183.14) (-322.44,183.85) (-322.44,184.55) (-321.73,185.26) (-321.73,185.97) (-321.73,186.68) (-321.73,187.38) (-321.03,188.09) (-321.03,188.80) (-321.03,189.50) (-320.32,190.21) (-320.32,190.92) (-320.32,191.63) (-320.32,192.33) (-319.61,193.04) (-319.61,193.75) (-319.61,194.45) (-319.61,195.16) (-318.91,195.87) (-318.91,196.58) (-318.91,197.28) (-318.20,197.99) (-318.20,198.70) (-318.20,199.40) (-318.20,200.11) (-317.49,200.82) (-317.49,201.53) (-317.49,202.23) (-317.49,202.94) (-316.78,203.65) (-316.78,204.35) (-332.34,174.66) (-331.63,175.36) (-331.63,176.07) (-330.93,176.78) (-330.93,177.48) (-330.22,178.19) (-330.22,178.90) (-329.51,179.61) (-329.51,180.31) (-328.80,181.02) (-328.80,181.73) (-328.10,182.43) (-328.10,183.14) (-327.39,183.85) (-327.39,184.55) (-326.68,185.26) (-326.68,185.97) (-325.98,186.68) (-325.98,187.38) (-325.27,188.09) (-325.27,188.80) (-324.56,189.50) (-323.85,190.21) (-323.85,190.92) (-323.15,191.63) (-323.15,192.33) (-322.44,193.04) (-322.44,193.75) (-321.73,194.45) (-321.73,195.16) (-321.03,195.87) (-321.03,196.58) (-320.32,197.28) (-320.32,197.99) (-319.61,198.70) (-319.61,199.40) (-318.91,200.11) (-318.91,200.82) (-318.20,201.53) (-318.20,202.23) (-317.49,202.94) (-317.49,203.65) (-316.78,204.35) (-332.34,174.66) (-331.63,175.36) (-330.93,176.07) (-330.22,176.07) (-329.51,176.78) (-328.80,177.48) (-328.10,178.19) (-327.39,178.90) (-326.68,178.90) (-325.98,179.61) (-325.27,180.31) (-324.56,181.02) (-323.85,181.02) (-323.15,181.73) (-322.44,182.43) (-321.73,183.14) (-321.03,183.85) (-320.32,183.85) (-319.61,184.55) (-318.91,185.26) (-318.20,185.97) (-317.49,186.68) (-316.78,186.68) (-316.08,187.38) (-315.37,188.09) (-314.66,188.80) (-313.96,189.50) (-313.25,189.50) (-312.54,190.21) (-311.83,190.92) (-311.13,191.63) (-310.42,191.63) (-309.71,192.33) (-309.01,193.04) (-308.30,193.75) (-307.59,194.45) (-306.88,194.45) (-306.18,195.16) (-305.47,195.87) (-339.41,198.70) (-338.70,198.70) (-338.00,198.70) (-337.29,198.70) (-336.58,198.70) (-335.88,198.70) (-335.17,198.70) (-334.46,197.99) (-333.75,197.99) (-333.05,197.99) (-332.34,197.99) (-331.63,197.99) (-330.93,197.99) (-330.22,197.99) (-329.51,197.99) (-328.80,197.99) (-328.10,197.99) (-327.39,197.99) (-326.68,197.99) (-325.98,197.28) (-325.27,197.28) (-324.56,197.28) (-323.85,197.28) (-323.15,197.28) (-322.44,197.28) (-321.73,197.28) (-321.03,197.28) (-320.32,197.28) (-319.61,197.28) (-318.91,197.28) (-318.20,197.28) (-317.49,196.58) (-316.78,196.58) (-316.08,196.58) (-315.37,196.58) (-314.66,196.58) (-313.96,196.58) (-313.25,196.58) (-312.54,196.58) (-311.83,196.58) (-311.13,196.58) (-310.42,196.58) (-309.71,196.58) (-309.01,195.87) (-308.30,195.87) (-307.59,195.87) (-306.88,195.87) (-306.18,195.87) (-305.47,195.87) (-371.94,192.33) (-371.23,192.33) (-370.52,192.33) (-369.82,193.04) (-369.11,193.04) (-368.40,193.04) (-367.70,193.04) (-366.99,193.04) (-366.28,193.75) (-365.57,193.75) (-364.87,193.75) (-364.16,193.75) (-363.45,193.75) (-362.75,194.45) (-362.04,194.45) (-361.33,194.45) (-360.62,194.45) (-359.92,194.45) (-359.21,195.16) (-358.50,195.16) (-357.80,195.16) (-357.09,195.16) (-356.38,195.16) (-355.67,195.16) (-354.97,195.87) (-354.26,195.87) (-353.55,195.87) (-352.85,195.87) (-352.14,195.87) (-351.43,196.58) (-350.72,196.58) (-350.02,196.58) (-349.31,196.58) (-348.60,196.58) (-347.90,197.28) (-347.19,197.28) (-346.48,197.28) (-345.78,197.28) (-345.07,197.28) (-344.36,197.99) (-343.65,197.99) (-342.95,197.99) (-342.24,197.99) (-341.53,197.99) (-340.83,198.70) (-340.12,198.70) (-339.41,198.70) (-387.49,162.63) (-386.79,163.34) (-386.79,164.05) (-386.08,164.76) (-386.08,165.46) (-385.37,166.17) (-385.37,166.88) (-384.67,167.58) (-384.67,168.29) (-383.96,169.00) (-383.96,169.71) (-383.25,170.41) (-383.25,171.12) (-382.54,171.83) (-382.54,172.53) (-381.84,173.24) (-381.84,173.95) (-381.13,174.66) (-381.13,175.36) (-380.42,176.07) (-380.42,176.78) (-379.72,177.48) (-379.01,178.19) (-379.01,178.90) (-378.30,179.61) (-378.30,180.31) (-377.60,181.02) (-377.60,181.73) (-376.89,182.43) (-376.89,183.14) (-376.18,183.85) (-376.18,184.55) (-375.47,185.26) (-375.47,185.97) (-374.77,186.68) (-374.77,187.38) (-374.06,188.09) (-374.06,188.80) (-373.35,189.50) (-373.35,190.21) (-372.65,190.92) (-372.65,191.63) (-371.94,192.33) (-387.49,162.63) (-386.79,163.34) (-386.08,164.05) (-385.37,164.05) (-384.67,164.76) (-383.96,165.46) (-383.25,166.17) (-382.54,166.17) (-381.84,166.88) (-381.13,167.58) (-380.42,168.29) (-379.72,168.29) (-379.01,169.00) (-378.30,169.71) (-377.60,170.41) (-376.89,170.41) (-376.18,171.12) (-375.47,171.83) (-374.77,172.53) (-374.06,172.53) (-373.35,173.24) (-372.65,173.95) (-371.94,174.66) (-371.23,174.66) (-370.52,175.36) (-369.82,176.07) (-369.11,176.78) (-368.40,176.78) (-367.70,177.48) (-366.99,178.19) (-366.28,178.90) (-365.57,178.90) (-364.87,179.61) (-364.16,180.31) (-363.45,181.02) (-362.75,181.02) (-362.04,181.73) (-361.33,182.43) (-393.86,192.33) (-393.15,192.33) (-392.44,191.63) (-391.74,191.63) (-391.03,191.63) (-390.32,190.92) (-389.62,190.92) (-388.91,190.92) (-388.20,190.92) (-387.49,190.21) (-386.79,190.21) (-386.08,190.21) (-385.37,189.50) (-384.67,189.50) (-383.96,189.50) (-383.25,188.80) (-382.54,188.80) (-381.84,188.80) (-381.13,188.80) (-380.42,188.09) (-379.72,188.09) (-379.01,188.09) (-378.30,187.38) (-377.60,187.38) (-376.89,187.38) (-376.18,186.68) (-375.47,186.68) (-374.77,186.68) (-374.06,185.97) (-373.35,185.97) (-372.65,185.97) (-371.94,185.97) (-371.23,185.26) (-370.52,185.26) (-369.82,185.26) (-369.11,184.55) (-368.40,184.55) (-367.70,184.55) (-366.99,183.85) (-366.28,183.85) (-365.57,183.85) (-364.87,183.85) (-364.16,183.14) (-363.45,183.14) (-362.75,183.14) (-362.04,182.43) (-361.33,182.43) (-426.39,199.40) (-425.68,199.40) (-424.97,199.40) (-424.26,198.70) (-423.56,198.70) (-422.85,198.70) (-422.14,198.70) (-421.44,197.99) (-420.73,197.99) (-420.02,197.99) (-419.31,197.99) (-418.61,197.99) (-417.90,197.28) (-417.19,197.28) (-416.49,197.28) (-415.78,197.28) (-415.07,197.28) (-414.36,196.58) (-413.66,196.58) (-412.95,196.58) (-412.24,196.58) (-411.54,195.87) (-410.83,195.87) (-410.12,195.87) (-409.41,195.87) (-408.71,195.87) (-408.00,195.16) (-407.29,195.16) (-406.59,195.16) (-405.88,195.16) (-405.17,194.45) (-404.47,194.45) (-403.76,194.45) (-403.05,194.45) (-402.34,194.45) (-401.64,193.75) (-400.93,193.75) (-400.22,193.75) (-399.52,193.75) (-398.81,193.75) (-398.10,193.04) (-397.39,193.04) (-396.69,193.04) (-395.98,193.04) (-395.27,192.33) (-394.57,192.33) (-393.86,192.33) (-455.38,181.73) (-454.67,182.43) (-453.96,182.43) (-453.26,183.14) (-452.55,183.14) (-451.84,183.85) (-451.13,184.55) (-450.43,184.55) (-449.72,185.26) (-449.01,185.26) (-448.31,185.97) (-447.60,186.68) (-446.89,186.68) (-446.18,187.38) (-445.48,188.09) (-444.77,188.09) (-444.06,188.80) (-443.36,188.80) (-442.65,189.50) (-441.94,190.21) (-441.23,190.21) (-440.53,190.92) (-439.82,190.92) (-439.11,191.63) (-438.41,192.33) (-437.70,192.33) (-436.99,193.04) (-436.28,193.04) (-435.58,193.75) (-434.87,194.45) (-434.16,194.45) (-433.46,195.16) (-432.75,195.87) (-432.04,195.87) (-431.34,196.58) (-430.63,196.58) (-429.92,197.28) (-429.21,197.99) (-428.51,197.99) (-427.80,198.70) (-427.09,198.70) (-426.39,199.40) (-455.38,181.73) (-454.67,181.73) (-453.96,181.73) (-453.26,181.73) (-452.55,181.73) (-451.84,181.73) (-451.13,181.73) (-450.43,181.73) (-449.72,181.73) (-449.01,181.02) (-448.31,181.02) (-447.60,181.02) (-446.89,181.02) (-446.18,181.02) (-445.48,181.02) (-444.77,181.02) (-444.06,181.02) (-443.36,181.02) (-442.65,181.02) (-441.94,181.02) (-441.23,181.02) (-440.53,181.02) (-439.82,181.02) (-439.11,181.02) (-438.41,181.02) (-437.70,180.31) (-436.99,180.31) (-436.28,180.31) (-435.58,180.31) (-434.87,180.31) (-434.16,180.31) (-433.46,180.31) (-432.75,180.31) (-432.04,180.31) (-431.34,180.31) (-430.63,180.31) (-429.92,180.31) (-429.21,180.31) (-428.51,180.31) (-427.80,180.31) (-427.09,180.31) (-426.39,179.61) (-425.68,179.61) (-424.97,179.61) (-424.26,179.61) (-423.56,179.61) (-422.85,179.61) (-422.14,179.61) (-421.44,179.61) (-421.44,179.61) (-420.73,180.31) (-420.73,181.02) (-420.02,181.73) (-419.31,182.43) (-419.31,183.14) (-418.61,183.85) (-417.90,184.55) (-417.19,185.26) (-417.19,185.97) (-416.49,186.68) (-415.78,187.38) (-415.78,188.09) (-415.07,188.80) (-414.36,189.50) (-414.36,190.21) (-413.66,190.92) (-412.95,191.63) (-412.24,192.33) (-412.24,193.04) (-411.54,193.75) (-410.83,194.45) (-410.83,195.16) (-410.12,195.87) (-409.41,196.58) (-409.41,197.28) (-408.71,197.99) (-408.00,198.70) (-407.29,199.40) (-407.29,200.11) (-406.59,200.82) (-405.88,201.53) (-405.88,202.23) (-405.17,202.94) (-404.47,203.65) (-404.47,204.35) (-403.76,205.06) (-403.05,205.77) (-402.34,206.48) (-402.34,207.18) (-401.64,207.89) (-432.75,189.50) (-432.04,190.21) (-431.34,190.21) (-430.63,190.92) (-429.92,190.92) (-429.21,191.63) (-428.51,192.33) (-427.80,192.33) (-427.09,193.04) (-426.39,193.04) (-425.68,193.75) (-424.97,193.75) (-424.26,194.45) (-423.56,195.16) (-422.85,195.16) (-422.14,195.87) (-421.44,195.87) (-420.73,196.58) (-420.02,197.28) (-419.31,197.28) (-418.61,197.99) (-417.90,197.99) (-417.19,198.70) (-416.49,199.40) (-415.78,199.40) (-415.07,200.11) (-414.36,200.11) (-413.66,200.82) (-412.95,201.53) (-412.24,201.53) (-411.54,202.23) (-410.83,202.23) (-410.12,202.94) (-409.41,202.94) (-408.71,203.65) (-408.00,204.35) (-407.29,204.35) (-406.59,205.06) (-405.88,205.06) (-405.17,205.77) (-404.47,206.48) (-403.76,206.48) (-403.05,207.18) (-402.34,207.18) (-401.64,207.89) (-432.75,189.50) (-432.04,188.80) (-431.34,188.09) (-430.63,188.09) (-429.92,187.38) (-429.21,186.68) (-428.51,185.97) (-427.80,185.97) (-427.09,185.26) (-426.39,184.55) (-425.68,183.85) (-424.97,183.14) (-424.26,183.14) (-423.56,182.43) (-422.85,181.73) (-422.14,181.02) (-421.44,181.02) (-420.73,180.31) (-420.02,179.61) (-419.31,178.90) (-418.61,178.90) (-417.90,178.19) (-417.19,177.48) (-416.49,176.78) (-415.78,176.07) (-415.07,176.07) (-414.36,175.36) (-413.66,174.66) (-412.95,173.95) (-412.24,173.95) (-411.54,173.24) (-410.83,172.53) (-410.12,171.83) (-409.41,171.12) (-408.71,171.12) (-408.00,170.41) (-407.29,169.71) (-406.59,169.00) (-405.88,169.00) (-405.17,168.29) (-404.47,167.58) (-381.84,142.84) (-382.54,143.54) (-383.25,144.25) (-383.96,144.96) (-384.67,145.66) (-385.37,146.37) (-385.37,147.08) (-386.08,147.79) (-386.79,148.49) (-387.49,149.20) (-388.20,149.91) (-388.91,150.61) (-389.62,151.32) (-390.32,152.03) (-391.03,152.74) (-391.74,153.44) (-392.44,154.15) (-393.15,154.86) (-393.15,155.56) (-393.86,156.27) (-394.57,156.98) (-395.27,157.68) (-395.98,158.39) (-396.69,159.10) (-397.39,159.81) (-398.10,160.51) (-398.81,161.22) (-399.52,161.93) (-400.22,162.63) (-400.93,163.34) (-400.93,164.05) (-401.64,164.76) (-402.34,165.46) (-403.05,166.17) (-403.76,166.88) (-404.47,167.58) (-381.84,108.89) (-381.84,109.60) (-381.84,110.31) (-381.84,111.02) (-381.84,111.72) (-381.84,112.43) (-381.84,113.14) (-381.84,113.84) (-381.84,114.55) (-381.84,115.26) (-381.84,115.97) (-381.84,116.67) (-381.84,117.38) (-381.84,118.09) (-381.84,118.79) (-381.84,119.50) (-381.84,120.21) (-381.84,120.92) (-381.84,121.62) (-381.84,122.33) (-381.84,123.04) (-381.84,123.74) (-381.84,124.45) (-381.84,125.16) (-381.84,125.87) (-381.84,126.57) (-381.84,127.28) (-381.84,127.99) (-381.84,128.69) (-381.84,129.40) (-381.84,130.11) (-381.84,130.81) (-381.84,131.52) (-381.84,132.23) (-381.84,132.94) (-381.84,133.64) (-381.84,134.35) (-381.84,135.06) (-381.84,135.76) (-381.84,136.47) (-381.84,137.18) (-381.84,137.89) (-381.84,138.59) (-381.84,139.30) (-381.84,140.01) (-381.84,140.71) (-381.84,141.42) (-381.84,142.13) (-381.84,142.84) (-412.95,105.36) (-412.24,105.36) (-411.54,105.36) (-410.83,105.36) (-410.12,105.36) (-409.41,106.07) (-408.71,106.07) (-408.00,106.07) (-407.29,106.07) (-406.59,106.07) (-405.88,106.07) (-405.17,106.07) (-404.47,106.07) (-403.76,106.07) (-403.05,106.77) (-402.34,106.77) (-401.64,106.77) (-400.93,106.77) (-400.22,106.77) (-399.52,106.77) (-398.81,106.77) (-398.10,106.77) (-397.39,106.77) (-396.69,107.48) (-395.98,107.48) (-395.27,107.48) (-394.57,107.48) (-393.86,107.48) (-393.15,107.48) (-392.44,107.48) (-391.74,107.48) (-391.03,108.19) (-390.32,108.19) (-389.62,108.19) (-388.91,108.19) (-388.20,108.19) (-387.49,108.19) (-386.79,108.19) (-386.08,108.19) (-385.37,108.19) (-384.67,108.89) (-383.96,108.89) (-383.25,108.89) (-382.54,108.89) (-381.84,108.89) (-431.34,77.78) (-430.63,78.49) (-430.63,79.20) (-429.92,79.90) (-429.21,80.61) (-429.21,81.32) (-428.51,82.02) (-427.80,82.73) (-427.80,83.44) (-427.09,84.15) (-426.39,84.85) (-426.39,85.56) (-425.68,86.27) (-424.97,86.97) (-424.97,87.68) (-424.26,88.39) (-423.56,89.10) (-423.56,89.80) (-422.85,90.51) (-422.14,91.22) (-422.14,91.92) (-421.44,92.63) (-420.73,93.34) (-420.73,94.05) (-420.02,94.75) (-419.31,95.46) (-419.31,96.17) (-418.61,96.87) (-417.90,97.58) (-417.90,98.29) (-417.19,98.99) (-416.49,99.70) (-416.49,100.41) (-415.78,101.12) (-415.07,101.82) (-415.07,102.53) (-414.36,103.24) (-413.66,103.94) (-413.66,104.65) (-412.95,105.36) (-431.34,77.78) (-430.63,77.78) (-429.92,78.49) (-429.21,78.49) (-428.51,78.49) (-427.80,78.49) (-427.09,79.20) (-426.39,79.20) (-425.68,79.20) (-424.97,79.90) (-424.26,79.90) (-423.56,79.90) (-422.85,79.90) (-422.14,80.61) (-421.44,80.61) (-420.73,80.61) (-420.02,81.32) (-419.31,81.32) (-418.61,81.32) (-417.90,82.02) (-417.19,82.02) (-416.49,82.02) (-415.78,82.02) (-415.07,82.73) (-414.36,82.73) (-413.66,82.73) (-412.95,83.44) (-412.24,83.44) (-411.54,83.44) (-410.83,83.44) (-410.12,84.15) (-409.41,84.15) (-408.71,84.15) (-408.00,84.85) (-407.29,84.85) (-406.59,84.85) (-405.88,84.85) (-405.17,85.56) (-404.47,85.56) (-403.76,85.56) (-403.05,86.27) (-402.34,86.27) (-401.64,86.27) (-400.93,86.97) (-400.22,86.97) (-399.52,86.97) (-398.81,86.97) (-398.10,87.68) (-397.39,87.68) (-397.39,87.68) (-396.69,88.39) (-396.69,89.10) (-395.98,89.80) (-395.98,90.51) (-395.27,91.22) (-395.27,91.92) (-394.57,92.63) (-394.57,93.34) (-393.86,94.05) (-393.86,94.75) (-393.15,95.46) (-393.15,96.17) (-392.44,96.87) (-392.44,97.58) (-391.74,98.29) (-391.74,98.99) (-391.03,99.70) (-391.03,100.41) (-390.32,101.12) (-390.32,101.82) (-389.62,102.53) (-389.62,103.24) (-388.91,103.94) (-388.91,104.65) (-388.20,105.36) (-388.20,106.07) (-387.49,106.77) (-387.49,107.48) (-386.79,108.19) (-386.79,108.89) (-386.08,109.60) (-386.08,110.31) (-385.37,111.02) (-385.37,111.72) (-384.67,112.43) (-384.67,113.14) (-383.96,113.84) (-383.96,114.55) (-383.25,115.26) (-383.25,115.97) (-382.54,116.67) (-397.39,84.85) (-397.39,85.56) (-396.69,86.27) (-396.69,86.97) (-395.98,87.68) (-395.98,88.39) (-395.27,89.10) (-395.27,89.80) (-394.57,90.51) (-394.57,91.22) (-393.86,91.92) (-393.86,92.63) (-393.15,93.34) (-393.15,94.05) (-392.44,94.75) (-392.44,95.46) (-392.44,96.17) (-391.74,96.87) (-391.74,97.58) (-391.03,98.29) (-391.03,98.99) (-390.32,99.70) (-390.32,100.41) (-389.62,101.12) (-389.62,101.82) (-388.91,102.53) (-388.91,103.24) (-388.20,103.94) (-388.20,104.65) (-387.49,105.36) (-387.49,106.07) (-387.49,106.77) (-386.79,107.48) (-386.79,108.19) (-386.08,108.89) (-386.08,109.60) (-385.37,110.31) (-385.37,111.02) (-384.67,111.72) (-384.67,112.43) (-383.96,113.14) (-383.96,113.84) (-383.25,114.55) (-383.25,115.26) (-382.54,115.97) (-382.54,116.67) (-397.39,84.85) (-397.39,85.56) (-397.39,86.27) (-396.69,86.97) (-396.69,87.68) (-396.69,88.39) (-396.69,89.10) (-395.98,89.80) (-395.98,90.51) (-395.98,91.22) (-395.98,91.92) (-395.27,92.63) (-395.27,93.34) (-395.27,94.05) (-395.27,94.75) (-394.57,95.46) (-394.57,96.17) (-394.57,96.87) (-394.57,97.58) (-393.86,98.29) (-393.86,98.99) (-393.86,99.70) (-393.86,100.41) (-393.15,101.12) (-393.15,101.82) (-393.15,102.53) (-393.15,103.24) (-392.44,103.94) (-392.44,104.65) (-392.44,105.36) (-392.44,106.07) (-391.74,106.77) (-391.74,107.48) (-391.74,108.19) (-391.74,108.89) (-391.03,109.60) (-391.03,110.31) (-391.03,111.02) (-391.03,111.72) (-390.32,112.43) (-390.32,113.14) (-390.32,113.84) (-390.32,114.55) (-389.62,115.26) (-389.62,115.97) (-389.62,116.67) (-387.49,82.73) (-387.49,83.44) (-387.49,84.15) (-387.49,84.85) (-387.49,85.56) (-387.49,86.27) (-387.49,86.97) (-387.49,87.68) (-387.49,88.39) (-388.20,89.10) (-388.20,89.80) (-388.20,90.51) (-388.20,91.22) (-388.20,91.92) (-388.20,92.63) (-388.20,93.34) (-388.20,94.05) (-388.20,94.75) (-388.20,95.46) (-388.20,96.17) (-388.20,96.87) (-388.20,97.58) (-388.20,98.29) (-388.20,98.99) (-388.20,99.70) (-388.91,100.41) (-388.91,101.12) (-388.91,101.82) (-388.91,102.53) (-388.91,103.24) (-388.91,103.94) (-388.91,104.65) (-388.91,105.36) (-388.91,106.07) (-388.91,106.77) (-388.91,107.48) (-388.91,108.19) (-388.91,108.89) (-388.91,109.60) (-388.91,110.31) (-388.91,111.02) (-389.62,111.72) (-389.62,112.43) (-389.62,113.14) (-389.62,113.84) (-389.62,114.55) (-389.62,115.26) (-389.62,115.97) (-389.62,116.67) (-415.78,100.41) (-415.07,99.70) (-414.36,99.70) (-413.66,98.99) (-412.95,98.99) (-412.24,98.29) (-411.54,97.58) (-410.83,97.58) (-410.12,96.87) (-409.41,96.17) (-408.71,96.17) (-408.00,95.46) (-407.29,95.46) (-406.59,94.75) (-405.88,94.05) (-405.17,94.05) (-404.47,93.34) (-403.76,92.63) (-403.05,92.63) (-402.34,91.92) (-401.64,91.92) (-400.93,91.22) (-400.22,90.51) (-399.52,90.51) (-398.81,89.80) (-398.10,89.10) (-397.39,89.10) (-396.69,88.39) (-395.98,88.39) (-395.27,87.68) (-394.57,86.97) (-393.86,86.97) (-393.15,86.27) (-392.44,85.56) (-391.74,85.56) (-391.03,84.85) (-390.32,84.85) (-389.62,84.15) (-388.91,83.44) (-388.20,83.44) (-387.49,82.73) (-442.65,116.67) (-441.94,115.97) (-441.23,115.97) (-440.53,115.26) (-439.82,115.26) (-439.11,114.55) (-438.41,113.84) (-437.70,113.84) (-436.99,113.14) (-436.28,113.14) (-435.58,112.43) (-434.87,111.72) (-434.16,111.72) (-433.46,111.02) (-432.75,111.02) (-432.04,110.31) (-431.34,109.60) (-430.63,109.60) (-429.92,108.89) (-429.21,108.89) (-428.51,108.19) (-427.80,107.48) (-427.09,107.48) (-426.39,106.77) (-425.68,106.07) (-424.97,106.07) (-424.26,105.36) (-423.56,105.36) (-422.85,104.65) (-422.14,103.94) (-421.44,103.94) (-420.73,103.24) (-420.02,103.24) (-419.31,102.53) (-418.61,101.82) (-417.90,101.82) (-417.19,101.12) (-416.49,101.12) (-415.78,100.41) (-467.40,137.89) (-466.69,137.18) (-465.98,136.47) (-465.28,135.76) (-464.57,135.76) (-463.86,135.06) (-463.15,134.35) (-462.45,133.64) (-461.74,132.94) (-461.03,132.23) (-460.33,131.52) (-459.62,131.52) (-458.91,130.81) (-458.21,130.11) (-457.50,129.40) (-456.79,128.69) (-456.08,127.99) (-455.38,127.28) (-454.67,127.28) (-453.96,126.57) (-453.26,125.87) (-452.55,125.16) (-451.84,124.45) (-451.13,123.74) (-450.43,123.04) (-449.72,123.04) (-449.01,122.33) (-448.31,121.62) (-447.60,120.92) (-446.89,120.21) (-446.18,119.50) (-445.48,118.79) (-444.77,118.79) (-444.06,118.09) (-443.36,117.38) (-442.65,116.67) (-467.40,137.89) (-468.10,138.59) (-468.81,139.30) (-468.81,140.01) (-469.52,140.71) (-470.23,141.42) (-470.93,142.13) (-471.64,142.84) (-472.35,143.54) (-472.35,144.25) (-473.05,144.96) (-473.76,145.66) (-474.47,146.37) (-475.18,147.08) (-475.88,147.79) (-475.88,148.49) (-476.59,149.20) (-477.30,149.91) (-478.00,150.61) (-478.71,151.32) (-479.42,152.03) (-479.42,152.74) (-480.13,153.44) (-480.83,154.15) (-481.54,154.86) (-482.25,155.56) (-482.95,156.27) (-482.95,156.98) (-483.66,157.68) (-484.37,158.39) (-485.08,159.10) (-485.78,159.81) (-486.49,160.51) (-486.49,161.22) (-487.20,161.93) (-487.90,162.63) (-488.61,163.34) (-488.61,163.34) (-487.90,164.05) (-487.90,164.76) (-487.20,165.46) (-487.20,166.17) (-486.49,166.88) (-485.78,167.58) (-485.78,168.29) (-485.08,169.00) (-485.08,169.71) (-484.37,170.41) (-483.66,171.12) (-483.66,171.83) (-482.95,172.53) (-482.25,173.24) (-482.25,173.95) (-481.54,174.66) (-481.54,175.36) (-480.83,176.07) (-480.13,176.78) (-480.13,177.48) (-479.42,178.19) (-479.42,178.90) (-478.71,179.61) (-478.00,180.31) (-478.00,181.02) (-477.30,181.73) (-477.30,182.43) (-476.59,183.14) (-475.88,183.85) (-475.88,184.55) (-475.18,185.26) (-474.47,185.97) (-474.47,186.68) (-473.76,187.38) (-473.76,188.09) (-473.05,188.80) (-472.35,189.50) (-472.35,190.21) (-471.64,190.92) (-471.64,191.63) (-470.93,192.33) (-503.46,182.43) (-502.75,182.43) (-502.05,183.14) (-501.34,183.14) (-500.63,183.14) (-499.92,183.85) (-499.22,183.85) (-498.51,183.85) (-497.80,183.85) (-497.10,184.55) (-496.39,184.55) (-495.68,184.55) (-494.97,185.26) (-494.27,185.26) (-493.56,185.26) (-492.85,185.97) (-492.15,185.97) (-491.44,185.97) (-490.73,185.97) (-490.02,186.68) (-489.32,186.68) (-488.61,186.68) (-487.90,187.38) (-487.20,187.38) (-486.49,187.38) (-485.78,188.09) (-485.08,188.09) (-484.37,188.09) (-483.66,188.80) (-482.95,188.80) (-482.25,188.80) (-481.54,188.80) (-480.83,189.50) (-480.13,189.50) (-479.42,189.50) (-478.71,190.21) (-478.00,190.21) (-477.30,190.21) (-476.59,190.92) (-475.88,190.92) (-475.18,190.92) (-474.47,190.92) (-473.76,191.63) (-473.05,191.63) (-472.35,191.63) (-471.64,192.33) (-470.93,192.33) (-509.82,147.79) (-509.82,148.49) (-509.82,149.20) (-509.12,149.91) (-509.12,150.61) (-509.12,151.32) (-509.12,152.03) (-509.12,152.74) (-509.12,153.44) (-508.41,154.15) (-508.41,154.86) (-508.41,155.56) (-508.41,156.27) (-508.41,156.98) (-507.70,157.68) (-507.70,158.39) (-507.70,159.10) (-507.70,159.81) (-507.70,160.51) (-507.70,161.22) (-507.00,161.93) (-507.00,162.63) (-507.00,163.34) (-507.00,164.05) (-507.00,164.76) (-506.29,165.46) (-506.29,166.17) (-506.29,166.88) (-506.29,167.58) (-506.29,168.29) (-505.58,169.00) (-505.58,169.71) (-505.58,170.41) (-505.58,171.12) (-505.58,171.83) (-505.58,172.53) (-504.87,173.24) (-504.87,173.95) (-504.87,174.66) (-504.87,175.36) (-504.87,176.07) (-504.17,176.78) (-504.17,177.48) (-504.17,178.19) (-504.17,178.90) (-504.17,179.61) (-504.17,180.31) (-503.46,181.02) (-503.46,181.73) (-503.46,182.43) (-509.82,147.79) (-510.53,148.49) (-510.53,149.20) (-511.24,149.91) (-511.95,150.61) (-512.65,151.32) (-512.65,152.03) (-513.36,152.74) (-514.07,153.44) (-514.07,154.15) (-514.77,154.86) (-515.48,155.56) (-516.19,156.27) (-516.19,156.98) (-516.90,157.68) (-517.60,158.39) (-517.60,159.10) (-518.31,159.81) (-519.02,160.51) (-519.02,161.22) (-519.72,161.93) (-520.43,162.63) (-521.14,163.34) (-521.14,164.05) (-521.84,164.76) (-522.55,165.46) (-522.55,166.17) (-523.26,166.88) (-523.97,167.58) (-524.67,168.29) (-524.67,169.00) (-525.38,169.71) (-526.09,170.41) (-526.09,171.12) (-526.79,171.83) (-527.50,172.53) (-528.21,173.24) (-528.21,173.95) (-528.92,174.66) (-528.92,174.66) (-528.21,175.36) (-527.50,175.36) (-526.79,176.07) (-526.09,176.78) (-525.38,176.78) (-524.67,177.48) (-523.97,178.19) (-523.26,178.19) (-522.55,178.90) (-521.84,179.61) (-521.14,179.61) (-520.43,180.31) (-519.72,181.02) (-519.02,181.02) (-518.31,181.73) (-517.60,182.43) (-516.90,182.43) (-516.19,183.14) (-515.48,183.85) (-514.77,183.85) (-514.07,184.55) (-513.36,185.26) (-512.65,185.97) (-511.95,185.97) (-511.24,186.68) (-510.53,187.38) (-509.82,187.38) (-509.12,188.09) (-508.41,188.80) (-507.70,188.80) (-507.00,189.50) (-506.29,190.21) (-505.58,190.21) (-504.87,190.92) (-504.17,191.63) (-503.46,191.63) (-502.75,192.33) (-502.05,193.04) (-501.34,193.04) (-500.63,193.75) (-500.63,193.75) (-500.63,194.45) (-501.34,195.16) (-501.34,195.87) (-501.34,196.58) (-502.05,197.28) (-502.05,197.99) (-502.75,198.70) (-502.75,199.40) (-502.75,200.11) (-503.46,200.82) (-503.46,201.53) (-503.46,202.23) (-504.17,202.94) (-504.17,203.65) (-504.17,204.35) (-504.87,205.06) (-504.87,205.77) (-505.58,206.48) (-505.58,207.18) (-505.58,207.89) (-506.29,208.60) (-506.29,209.30) (-506.29,210.01) (-507.00,210.72) (-507.00,211.42) (-507.00,212.13) (-507.70,212.84) (-507.70,213.55) (-508.41,214.25) (-508.41,214.96) (-508.41,215.67) (-509.12,216.37) (-509.12,217.08) (-509.12,217.79) (-509.82,218.50) (-509.82,219.20) (-509.82,219.91) (-510.53,220.62) (-510.53,221.32) (-511.24,222.03) (-511.24,222.74) (-511.24,223.45) (-511.95,224.15) (-511.95,224.86) (-511.95,224.86) (-511.24,224.86) (-510.53,224.86) (-509.82,225.57) (-509.12,225.57) (-508.41,225.57) (-507.70,225.57) (-507.00,225.57) (-506.29,226.27) (-505.58,226.27) (-504.87,226.27) (-504.17,226.27) (-503.46,226.98) (-502.75,226.98) (-502.05,226.98) (-501.34,226.98) (-500.63,226.98) (-499.92,227.69) (-499.22,227.69) (-498.51,227.69) (-497.80,227.69) (-497.10,227.69) (-496.39,228.40) (-495.68,228.40) (-494.97,228.40) (-494.27,228.40) (-493.56,229.10) (-492.85,229.10) (-492.15,229.10) (-491.44,229.10) (-490.73,229.10) (-490.02,229.81) (-489.32,229.81) (-488.61,229.81) (-487.90,229.81) (-487.20,229.81) (-486.49,230.52) (-485.78,230.52) (-485.08,230.52) (-484.37,230.52) (-483.66,231.22) (-482.95,231.22) (-482.25,231.22) (-481.54,231.22) (-480.83,231.22) (-480.13,231.93) (-479.42,231.93) (-478.71,231.93) (-478.71,231.93) (-478.00,232.64) (-477.30,233.35) (-476.59,234.05) (-475.88,234.76) (-475.18,235.47) (-474.47,236.17) (-473.76,236.88) (-473.05,237.59) (-472.35,238.29) (-471.64,239.00) (-470.93,239.71) (-470.23,240.42) (-469.52,241.12) (-468.81,241.83) (-468.10,242.54) (-467.40,243.24) (-467.40,243.95) (-466.69,244.66) (-465.98,245.37) (-465.28,246.07) (-464.57,246.78) (-463.86,247.49) (-463.15,248.19) (-462.45,248.90) (-461.74,249.61) (-461.03,250.32) (-460.33,251.02) (-459.62,251.73) (-458.91,252.44) (-458.21,253.14) (-457.50,253.85) (-456.79,254.56) (-456.08,255.27) (-455.38,255.97) (-484.37,273.65) (-483.66,272.94) (-482.95,272.94) (-482.25,272.24) (-481.54,272.24) (-480.83,271.53) (-480.13,270.82) (-479.42,270.82) (-478.71,270.11) (-478.00,270.11) (-477.30,269.41) (-476.59,268.70) (-475.88,268.70) (-475.18,267.99) (-474.47,267.29) (-473.76,267.29) (-473.05,266.58) (-472.35,266.58) (-471.64,265.87) (-470.93,265.17) (-470.23,265.17) (-469.52,264.46) (-468.81,264.46) (-468.10,263.75) (-467.40,263.04) (-466.69,263.04) (-465.98,262.34) (-465.28,262.34) (-464.57,261.63) (-463.86,260.92) (-463.15,260.92) (-462.45,260.22) (-461.74,259.51) (-461.03,259.51) (-460.33,258.80) (-459.62,258.80) (-458.91,258.09) (-458.21,257.39) (-457.50,257.39) (-456.79,256.68) (-456.08,256.68) (-455.38,255.97) (-484.37,273.65) (-484.37,274.36) (-485.08,275.06) (-485.08,275.77) (-485.78,276.48) (-485.78,277.19) (-486.49,277.89) (-486.49,278.60) (-486.49,279.31) (-487.20,280.01) (-487.20,280.72) (-487.90,281.43) (-487.90,282.14) (-487.90,282.84) (-488.61,283.55) (-488.61,284.26) (-489.32,284.96) (-489.32,285.67) (-490.02,286.38) (-490.02,287.09) (-490.02,287.79) (-490.73,288.50) (-490.73,289.21) (-491.44,289.91) (-491.44,290.62) (-492.15,291.33) (-492.15,292.04) (-492.15,292.74) (-492.85,293.45) (-492.85,294.16) (-493.56,294.86) (-493.56,295.57) (-494.27,296.28) (-494.27,296.98) (-494.27,297.69) (-494.97,298.40) (-494.97,299.11) (-495.68,299.81) (-495.68,300.52) (-495.68,301.23) (-496.39,301.93) (-496.39,302.64) (-497.10,303.35) (-497.10,304.06) (-497.80,304.76) (-497.80,305.47) (-497.80,305.47) (-497.10,305.47) (-496.39,306.18) (-495.68,306.18) (-494.97,306.18) (-494.27,306.18) (-493.56,306.88) (-492.85,306.88) (-492.15,306.88) (-491.44,306.88) (-490.73,307.59) (-490.02,307.59) (-489.32,307.59) (-488.61,308.30) (-487.90,308.30) (-487.20,308.30) (-486.49,308.30) (-485.78,309.01) (-485.08,309.01) (-484.37,309.01) (-483.66,309.71) (-482.95,309.71) (-482.25,309.71) (-481.54,309.71) (-480.83,310.42) (-480.13,310.42) (-479.42,310.42) (-478.71,310.42) (-478.00,311.13) (-477.30,311.13) (-476.59,311.13) (-475.88,311.83) (-475.18,311.83) (-474.47,311.83) (-473.76,311.83) (-473.05,312.54) (-472.35,312.54) (-471.64,312.54) (-470.93,313.25) (-470.23,313.25) (-469.52,313.25) (-468.81,313.25) (-468.10,313.96) (-467.40,313.96) (-466.69,313.96) (-465.98,313.96) (-465.28,314.66) (-464.57,314.66) (-464.57,314.66) (-464.57,315.37) (-463.86,316.08) (-463.86,316.78) (-463.86,317.49) (-463.15,318.20) (-463.15,318.91) (-463.15,319.61) (-462.45,320.32) (-462.45,321.03) (-461.74,321.73) (-461.74,322.44) (-461.74,323.15) (-461.03,323.85) (-461.03,324.56) (-461.03,325.27) (-460.33,325.98) (-460.33,326.68) (-460.33,327.39) (-459.62,328.10) (-459.62,328.80) (-459.62,329.51) (-458.91,330.22) (-458.91,330.93) (-458.21,331.63) (-458.21,332.34) (-458.21,333.05) (-457.50,333.75) (-457.50,334.46) (-457.50,335.17) (-456.79,335.88) (-456.79,336.58) (-456.79,337.29) (-456.08,338.00) (-456.08,338.70) (-456.08,339.41) (-455.38,340.12) (-455.38,340.83) (-454.67,341.53) (-454.67,342.24) (-454.67,342.95) (-453.96,343.65) (-453.96,344.36) (-475.88,315.37) (-475.18,316.08) (-474.47,316.78) (-474.47,317.49) (-473.76,318.20) (-473.05,318.91) (-472.35,319.61) (-472.35,320.32) (-471.64,321.03) (-470.93,321.73) (-470.23,322.44) (-470.23,323.15) (-469.52,323.85) (-468.81,324.56) (-468.10,325.27) (-468.10,325.98) (-467.40,326.68) (-466.69,327.39) (-465.98,328.10) (-465.98,328.80) (-465.28,329.51) (-464.57,330.22) (-463.86,330.93) (-463.86,331.63) (-463.15,332.34) (-462.45,333.05) (-461.74,333.75) (-461.74,334.46) (-461.03,335.17) (-460.33,335.88) (-459.62,336.58) (-459.62,337.29) (-458.91,338.00) (-458.21,338.70) (-457.50,339.41) (-457.50,340.12) (-456.79,340.83) (-456.08,341.53) (-455.38,342.24) (-455.38,342.95) (-454.67,343.65) (-453.96,344.36) (-475.88,315.37) (-475.18,316.08) (-474.47,316.78) (-474.47,317.49) (-473.76,318.20) (-473.05,318.91) (-472.35,319.61) (-472.35,320.32) (-471.64,321.03) (-470.93,321.73) (-470.23,322.44) (-469.52,323.15) (-469.52,323.85) (-468.81,324.56) (-468.10,325.27) (-467.40,325.98) (-466.69,326.68) (-466.69,327.39) (-465.98,328.10) (-465.28,328.80) (-464.57,329.51) (-464.57,330.22) (-463.86,330.93) (-463.15,331.63) (-462.45,332.34) (-461.74,333.05) (-461.74,333.75) (-461.03,334.46) (-460.33,335.17) (-459.62,335.88) (-458.91,336.58) (-458.91,337.29) (-458.21,338.00) (-457.50,338.70) (-456.79,339.41) (-456.79,340.12) (-456.08,340.83) (-455.38,341.53) (-481.54,319.61) (-480.83,320.32) (-480.13,321.03) (-479.42,321.73) (-478.71,321.73) (-478.00,322.44) (-477.30,323.15) (-476.59,323.85) (-475.88,324.56) (-475.18,325.27) (-474.47,325.27) (-473.76,325.98) (-473.05,326.68) (-472.35,327.39) (-471.64,328.10) (-470.93,328.80) (-470.23,328.80) (-469.52,329.51) (-468.81,330.22) (-468.10,330.93) (-467.40,331.63) (-466.69,332.34) (-465.98,332.34) (-465.28,333.05) (-464.57,333.75) (-463.86,334.46) (-463.15,335.17) (-462.45,335.88) (-461.74,335.88) (-461.03,336.58) (-460.33,337.29) (-459.62,338.00) (-458.91,338.70) (-458.21,339.41) (-457.50,339.41) (-456.79,340.12) (-456.08,340.83) (-455.38,341.53) (-481.54,319.61) (-480.83,319.61) (-480.13,319.61) (-479.42,319.61) (-478.71,319.61) (-478.00,320.32) (-477.30,320.32) (-476.59,320.32) (-475.88,320.32) (-475.18,320.32) (-474.47,320.32) (-473.76,320.32) (-473.05,320.32) (-472.35,321.03) (-471.64,321.03) (-470.93,321.03) (-470.23,321.03) (-469.52,321.03) (-468.81,321.03) (-468.10,321.03) (-467.40,321.03) (-466.69,321.73) (-465.98,321.73) (-465.28,321.73) (-464.57,321.73) (-463.86,321.73) (-463.15,321.73) (-462.45,321.73) (-461.74,321.73) (-461.03,322.44) (-460.33,322.44) (-459.62,322.44) (-458.91,322.44) (-458.21,322.44) (-457.50,322.44) (-456.79,322.44) (-456.08,322.44) (-455.38,323.15) (-454.67,323.15) (-453.96,323.15) (-453.26,323.15) (-452.55,323.15) (-451.84,323.15) (-451.13,323.15) (-450.43,323.15) (-449.72,323.85) (-449.01,323.85) (-448.31,323.85) (-447.60,323.85) (-447.60,323.85) (-446.89,324.56) (-446.18,325.27) (-445.48,325.98) (-445.48,326.68) (-444.77,327.39) (-444.06,328.10) (-443.36,328.80) (-442.65,329.51) (-441.94,330.22) (-441.23,330.93) (-441.23,331.63) (-440.53,332.34) (-439.82,333.05) (-439.11,333.75) (-438.41,334.46) (-437.70,335.17) (-436.99,335.88) (-436.99,336.58) (-436.28,337.29) (-435.58,338.00) (-434.87,338.70) (-434.16,339.41) (-433.46,340.12) (-432.75,340.83) (-432.04,341.53) (-432.04,342.24) (-431.34,342.95) (-430.63,343.65) (-429.92,344.36) (-429.21,345.07) (-428.51,345.78) (-427.80,346.48) (-427.80,347.19) (-427.09,347.90) (-426.39,348.60) (-425.68,349.31) (-459.62,345.78) (-458.91,345.78) (-458.21,345.78) (-457.50,345.78) (-456.79,345.78) (-456.08,346.48) (-455.38,346.48) (-454.67,346.48) (-453.96,346.48) (-453.26,346.48) (-452.55,346.48) (-451.84,346.48) (-451.13,346.48) (-450.43,346.48) (-449.72,346.48) (-449.01,347.19) (-448.31,347.19) (-447.60,347.19) (-446.89,347.19) (-446.18,347.19) (-445.48,347.19) (-444.77,347.19) (-444.06,347.19) (-443.36,347.19) (-442.65,347.19) (-441.94,347.90) (-441.23,347.90) (-440.53,347.90) (-439.82,347.90) (-439.11,347.90) (-438.41,347.90) (-437.70,347.90) (-436.99,347.90) (-436.28,347.90) (-435.58,348.60) (-434.87,348.60) (-434.16,348.60) (-433.46,348.60) (-432.75,348.60) (-432.04,348.60) (-431.34,348.60) (-430.63,348.60) (-429.92,348.60) (-429.21,348.60) (-428.51,349.31) (-427.80,349.31) (-427.09,349.31) (-426.39,349.31) (-425.68,349.31) (-480.83,319.61) (-480.13,320.32) (-479.42,321.03) (-479.42,321.73) (-478.71,322.44) (-478.00,323.15) (-477.30,323.85) (-476.59,324.56) (-476.59,325.27) (-475.88,325.98) (-475.18,326.68) (-474.47,327.39) (-473.76,328.10) (-473.05,328.80) (-473.05,329.51) (-472.35,330.22) (-471.64,330.93) (-470.93,331.63) (-470.23,332.34) (-470.23,333.05) (-469.52,333.75) (-468.81,334.46) (-468.10,335.17) (-467.40,335.88) (-467.40,336.58) (-466.69,337.29) (-465.98,338.00) (-465.28,338.70) (-464.57,339.41) (-463.86,340.12) (-463.86,340.83) (-463.15,341.53) (-462.45,342.24) (-461.74,342.95) (-461.03,343.65) (-461.03,344.36) (-460.33,345.07) (-459.62,345.78) (-480.83,319.61) (-480.13,319.61) (-479.42,319.61) (-478.71,319.61) (-478.00,319.61) (-477.30,319.61) (-476.59,319.61) (-475.88,319.61) (-475.18,319.61) (-474.47,319.61) (-473.76,319.61) (-473.05,319.61) (-472.35,319.61) (-471.64,319.61) (-470.93,319.61) (-470.23,319.61) (-469.52,319.61) (-468.81,319.61) (-468.10,319.61) (-467.40,319.61) (-466.69,319.61) (-465.98,319.61) (-465.28,319.61) (-464.57,319.61) (-463.86,319.61) (-463.15,319.61) (-462.45,319.61) (-461.74,319.61) (-461.03,319.61) (-460.33,319.61) (-459.62,319.61) (-458.91,319.61) (-458.21,319.61) (-457.50,319.61) (-456.79,319.61) (-456.08,319.61) (-455.38,319.61) (-454.67,319.61) (-453.96,319.61) (-453.26,319.61) (-452.55,319.61) (-451.84,319.61) (-451.13,319.61) (-450.43,319.61) (-449.72,319.61) (-449.01,319.61) (-448.31,319.61) (-447.60,319.61) (-446.89,319.61) (-446.18,319.61) (-445.48,319.61) (-444.77,319.61) (-444.06,319.61) (-443.36,319.61) (-443.36,319.61) (-442.65,320.32) (-441.94,321.03) (-441.23,321.03) (-440.53,321.73) (-439.82,322.44) (-439.11,323.15) (-438.41,323.85) (-437.70,323.85) (-436.99,324.56) (-436.28,325.27) (-435.58,325.98) (-434.87,326.68) (-434.16,326.68) (-433.46,327.39) (-432.75,328.10) (-432.04,328.80) (-431.34,329.51) (-430.63,329.51) (-429.92,330.22) (-429.21,330.93) (-428.51,331.63) (-427.80,331.63) (-427.09,332.34) (-426.39,333.05) (-425.68,333.75) (-424.97,334.46) (-424.26,334.46) (-423.56,335.17) (-422.85,335.88) (-422.14,336.58) (-421.44,337.29) (-420.73,337.29) (-420.02,338.00) (-419.31,338.70) (-418.61,339.41) (-417.90,340.12) (-417.19,340.12) (-416.49,340.83) (-415.78,341.53) (-415.78,341.53) (-416.49,342.24) (-416.49,342.95) (-417.19,343.65) (-417.90,344.36) (-417.90,345.07) (-418.61,345.78) (-419.31,346.48) (-420.02,347.19) (-420.02,347.90) (-420.73,348.60) (-421.44,349.31) (-421.44,350.02) (-422.14,350.72) (-422.85,351.43) (-422.85,352.14) (-423.56,352.85) (-424.26,353.55) (-424.97,354.26) (-424.97,354.97) (-425.68,355.67) (-426.39,356.38) (-426.39,357.09) (-427.09,357.80) (-427.80,358.50) (-427.80,359.21) (-428.51,359.92) (-429.21,360.62) (-429.92,361.33) (-429.92,362.04) (-430.63,362.75) (-431.34,363.45) (-431.34,364.16) (-432.04,364.87) (-432.75,365.57) (-432.75,366.28) (-433.46,366.99) (-434.16,367.70) (-434.87,368.40) (-434.87,369.11) (-435.58,369.82) (-435.58,369.82) (-434.87,370.52) (-434.16,371.23) (-433.46,371.94) (-432.75,371.94) (-432.04,372.65) (-431.34,373.35) (-430.63,374.06) (-429.92,374.77) (-429.21,375.47) (-428.51,375.47) (-427.80,376.18) (-427.09,376.89) (-426.39,377.60) (-425.68,378.30) (-424.97,379.01) (-424.26,379.01) (-423.56,379.72) (-422.85,380.42) (-422.14,381.13) (-421.44,381.84) (-420.73,382.54) (-420.02,382.54) (-419.31,383.25) (-418.61,383.96) (-417.90,384.67) (-417.19,385.37) (-416.49,386.08) (-415.78,386.08) (-415.07,386.79) (-414.36,387.49) (-413.66,388.20) (-412.95,388.91) (-412.24,389.62) (-411.54,389.62) (-410.83,390.32) (-410.12,391.03) (-409.41,391.74) (-438.41,375.47) (-437.70,376.18) (-436.99,376.18) (-436.28,376.89) (-435.58,376.89) (-434.87,377.60) (-434.16,377.60) (-433.46,378.30) (-432.75,378.30) (-432.04,379.01) (-431.34,379.72) (-430.63,379.72) (-429.92,380.42) (-429.21,380.42) (-428.51,381.13) (-427.80,381.13) (-427.09,381.84) (-426.39,382.54) (-425.68,382.54) (-424.97,383.25) (-424.26,383.25) (-423.56,383.96) (-422.85,383.96) (-422.14,384.67) (-421.44,384.67) (-420.73,385.37) (-420.02,386.08) (-419.31,386.08) (-418.61,386.79) (-417.90,386.79) (-417.19,387.49) (-416.49,387.49) (-415.78,388.20) (-415.07,388.91) (-414.36,388.91) (-413.66,389.62) (-412.95,389.62) (-412.24,390.32) (-411.54,390.32) (-410.83,391.03) (-410.12,391.03) (-409.41,391.74) (-418.61,349.31) (-419.31,350.02) (-420.02,350.72) (-420.02,351.43) (-420.73,352.14) (-421.44,352.85) (-422.14,353.55) (-422.14,354.26) (-422.85,354.97) (-423.56,355.67) (-424.26,356.38) (-424.26,357.09) (-424.97,357.80) (-425.68,358.50) (-426.39,359.21) (-426.39,359.92) (-427.09,360.62) (-427.80,361.33) (-428.51,362.04) (-428.51,362.75) (-429.21,363.45) (-429.92,364.16) (-430.63,364.87) (-430.63,365.57) (-431.34,366.28) (-432.04,366.99) (-432.75,367.70) (-432.75,368.40) (-433.46,369.11) (-434.16,369.82) (-434.87,370.52) (-434.87,371.23) (-435.58,371.94) (-436.28,372.65) (-436.99,373.35) (-436.99,374.06) (-437.70,374.77) (-438.41,375.47) (-426.39,316.78) (-426.39,317.49) (-426.39,318.20) (-425.68,318.91) (-425.68,319.61) (-425.68,320.32) (-425.68,321.03) (-424.97,321.73) (-424.97,322.44) (-424.97,323.15) (-424.97,323.85) (-424.26,324.56) (-424.26,325.27) (-424.26,325.98) (-424.26,326.68) (-423.56,327.39) (-423.56,328.10) (-423.56,328.80) (-423.56,329.51) (-422.85,330.22) (-422.85,330.93) (-422.85,331.63) (-422.85,332.34) (-422.85,333.05) (-422.14,333.75) (-422.14,334.46) (-422.14,335.17) (-422.14,335.88) (-421.44,336.58) (-421.44,337.29) (-421.44,338.00) (-421.44,338.70) (-420.73,339.41) (-420.73,340.12) (-420.73,340.83) (-420.73,341.53) (-420.02,342.24) (-420.02,342.95) (-420.02,343.65) (-420.02,344.36) (-419.31,345.07) (-419.31,345.78) (-419.31,346.48) (-419.31,347.19) (-418.61,347.90) (-418.61,348.60) (-418.61,349.31) (-426.39,316.78) (-427.09,317.49) (-427.80,318.20) (-428.51,318.91) (-429.21,319.61) (-429.92,320.32) (-430.63,321.03) (-431.34,321.73) (-431.34,322.44) (-432.04,323.15) (-432.75,323.85) (-433.46,324.56) (-434.16,325.27) (-434.87,325.98) (-435.58,326.68) (-436.28,327.39) (-436.99,328.10) (-437.70,328.80) (-438.41,329.51) (-439.11,330.22) (-439.82,330.93) (-440.53,331.63) (-441.23,332.34) (-441.94,333.05) (-441.94,333.75) (-442.65,334.46) (-443.36,335.17) (-444.06,335.88) (-444.77,336.58) (-445.48,337.29) (-446.18,338.00) (-446.89,338.70) (-447.60,339.41) (-447.60,339.41) (-447.60,340.12) (-448.31,340.83) (-448.31,341.53) (-448.31,342.24) (-449.01,342.95) (-449.01,343.65) (-449.01,344.36) (-449.72,345.07) (-449.72,345.78) (-449.72,346.48) (-450.43,347.19) (-450.43,347.90) (-450.43,348.60) (-451.13,349.31) (-451.13,350.02) (-451.13,350.72) (-451.84,351.43) (-451.84,352.14) (-451.84,352.85) (-452.55,353.55) (-452.55,354.26) (-452.55,354.97) (-453.26,355.67) (-453.26,356.38) (-453.96,357.09) (-453.96,357.80) (-453.96,358.50) (-454.67,359.21) (-454.67,359.92) (-454.67,360.62) (-455.38,361.33) (-455.38,362.04) (-455.38,362.75) (-456.08,363.45) (-456.08,364.16) (-456.08,364.87) (-456.79,365.57) (-456.79,366.28) (-456.79,366.99) (-457.50,367.70) (-457.50,368.40) (-457.50,369.11) (-458.21,369.82) (-458.21,370.52) (-458.21,371.23) (-458.91,371.94) (-458.91,372.65) (-458.91,372.65) (-458.21,372.65) (-457.50,372.65) (-456.79,372.65) (-456.08,372.65) (-455.38,372.65) (-454.67,372.65) (-453.96,372.65) (-453.26,372.65) (-452.55,372.65) (-451.84,372.65) (-451.13,372.65) (-450.43,372.65) (-449.72,372.65) (-449.01,372.65) (-448.31,372.65) (-447.60,372.65) (-446.89,372.65) (-446.18,372.65) (-445.48,372.65) (-444.77,372.65) (-444.06,372.65) (-443.36,372.65) (-442.65,372.65) (-441.94,372.65) (-441.23,372.65) (-440.53,373.35) (-439.82,373.35) (-439.11,373.35) (-438.41,373.35) (-437.70,373.35) (-436.99,373.35) (-436.28,373.35) (-435.58,373.35) (-434.87,373.35) (-434.16,373.35) (-433.46,373.35) (-432.75,373.35) (-432.04,373.35) (-431.34,373.35) (-430.63,373.35) (-429.92,373.35) (-429.21,373.35) (-428.51,373.35) (-427.80,373.35) (-427.09,373.35) (-426.39,373.35) (-425.68,373.35) (-424.97,373.35) (-424.26,373.35) (-423.56,373.35) (-423.56,373.35) (-422.85,373.35) (-422.14,373.35) (-421.44,374.06) (-420.73,374.06) (-420.02,374.06) (-419.31,374.06) (-418.61,374.77) (-417.90,374.77) (-417.19,374.77) (-416.49,374.77) (-415.78,375.47) (-415.07,375.47) (-414.36,375.47) (-413.66,375.47) (-412.95,376.18) (-412.24,376.18) (-411.54,376.18) (-410.83,376.18) (-410.12,376.89) (-409.41,376.89) (-408.71,376.89) (-408.00,376.89) (-407.29,377.60) (-406.59,377.60) (-405.88,377.60) (-405.17,377.60) (-404.47,378.30) (-403.76,378.30) (-403.05,378.30) (-402.34,378.30) (-401.64,379.01) (-400.93,379.01) (-400.22,379.01) (-399.52,379.01) (-398.81,379.72) (-398.10,379.72) (-397.39,379.72) (-396.69,379.72) (-395.98,380.42) (-395.27,380.42) (-394.57,380.42) (-393.86,380.42) (-393.15,381.13) (-392.44,381.13) (-391.74,381.13) (-391.03,381.13) (-390.32,381.84) (-389.62,381.84) (-388.91,381.84) (-388.91,381.84) (-388.91,382.54) (-388.20,383.25) (-388.20,383.96) (-388.20,384.67) (-388.20,385.37) (-387.49,386.08) (-387.49,386.79) (-387.49,387.49) (-387.49,388.20) (-386.79,388.91) (-386.79,389.62) (-386.79,390.32) (-386.79,391.03) (-386.08,391.74) (-386.08,392.44) (-386.08,393.15) (-385.37,393.86) (-385.37,394.57) (-385.37,395.27) (-385.37,395.98) (-384.67,396.69) (-384.67,397.39) (-384.67,398.10) (-384.67,398.81) (-383.96,399.52) (-383.96,400.22) (-383.96,400.93) (-383.96,401.64) (-383.25,402.34) (-383.25,403.05) (-383.25,403.76) (-382.54,404.47) (-382.54,405.17) (-382.54,405.88) (-382.54,406.59) (-381.84,407.29) (-381.84,408.00) (-381.84,408.71) (-381.84,409.41) (-381.13,410.12) (-381.13,410.83) (-381.13,411.54) (-381.13,412.24) (-380.42,412.95) (-380.42,413.66) (-385.37,381.84) (-385.37,382.54) (-385.37,383.25) (-385.37,383.96) (-384.67,384.67) (-384.67,385.37) (-384.67,386.08) (-384.67,386.79) (-384.67,387.49) (-384.67,388.20) (-383.96,388.91) (-383.96,389.62) (-383.96,390.32) (-383.96,391.03) (-383.96,391.74) (-383.96,392.44) (-383.96,393.15) (-383.25,393.86) (-383.25,394.57) (-383.25,395.27) (-383.25,395.98) (-383.25,396.69) (-383.25,397.39) (-382.54,398.10) (-382.54,398.81) (-382.54,399.52) (-382.54,400.22) (-382.54,400.93) (-382.54,401.64) (-381.84,402.34) (-381.84,403.05) (-381.84,403.76) (-381.84,404.47) (-381.84,405.17) (-381.84,405.88) (-381.84,406.59) (-381.13,407.29) (-381.13,408.00) (-381.13,408.71) (-381.13,409.41) (-381.13,410.12) (-381.13,410.83) (-380.42,411.54) (-380.42,412.24) (-380.42,412.95) (-380.42,413.66) (-385.37,381.84) (-385.37,382.54) (-385.37,383.25) (-386.08,383.96) (-386.08,384.67) (-386.08,385.37) (-386.08,386.08) (-386.79,386.79) (-386.79,387.49) (-386.79,388.20) (-386.79,388.91) (-386.79,389.62) (-387.49,390.32) (-387.49,391.03) (-387.49,391.74) (-387.49,392.44) (-387.49,393.15) (-388.20,393.86) (-388.20,394.57) (-388.20,395.27) (-388.20,395.98) (-388.91,396.69) (-388.91,397.39) (-388.91,398.10) (-388.91,398.81) (-388.91,399.52) (-389.62,400.22) (-389.62,400.93) (-389.62,401.64) (-389.62,402.34) (-390.32,403.05) (-390.32,403.76) (-390.32,404.47) (-390.32,405.17) (-390.32,405.88) (-391.03,406.59) (-391.03,407.29) (-391.03,408.00) (-391.03,408.71) (-391.03,409.41) (-391.74,410.12) (-391.74,410.83) (-391.74,411.54) (-391.74,412.24) (-392.44,412.95) (-392.44,413.66) (-392.44,414.36) (-392.44,414.36) (-391.74,413.66) (-391.03,412.95) (-390.32,412.95) (-389.62,412.24) (-388.91,411.54) (-388.20,410.83) (-387.49,410.83) (-386.79,410.12) (-386.08,409.41) (-385.37,408.71) (-384.67,408.71) (-383.96,408.00) (-383.25,407.29) (-382.54,406.59) (-381.84,406.59) (-381.13,405.88) (-380.42,405.17) (-379.72,404.47) (-379.01,404.47) (-378.30,403.76) (-377.60,403.05) (-376.89,402.34) (-376.18,402.34) (-375.47,401.64) (-374.77,400.93) (-374.06,400.22) (-373.35,400.22) (-372.65,399.52) (-371.94,398.81) (-371.23,398.10) (-370.52,398.10) (-369.82,397.39) (-369.11,396.69) (-368.40,395.98) (-367.70,395.98) (-366.99,395.27) (-366.28,394.57) (-365.57,393.86) (-364.87,393.86) (-364.16,393.15) (-363.45,392.44) (-363.45,392.44) (-362.75,391.74) (-362.04,391.74) (-361.33,391.03) (-360.62,391.03) (-359.92,390.32) (-359.21,389.62) (-358.50,389.62) (-357.80,388.91) (-357.09,388.91) (-356.38,388.20) (-355.67,387.49) (-354.97,387.49) (-354.26,386.79) (-353.55,386.08) (-352.85,386.08) (-352.14,385.37) (-351.43,385.37) (-350.72,384.67) (-350.02,383.96) (-349.31,383.96) (-348.60,383.25) (-347.90,383.25) (-347.19,382.54) (-346.48,381.84) (-345.78,381.84) (-345.07,381.13) (-344.36,381.13) (-343.65,380.42) (-342.95,379.72) (-342.24,379.72) (-341.53,379.01) (-340.83,378.30) (-340.12,378.30) (-339.41,377.60) (-338.70,377.60) (-338.00,376.89) (-337.29,376.18) (-336.58,376.18) (-335.88,375.47) (-335.17,375.47) (-334.46,374.77) (-334.46,374.77) (-333.75,374.77) (-333.05,374.77) (-332.34,374.77) (-331.63,374.77) (-330.93,374.77) (-330.22,374.77) (-329.51,374.77) (-328.80,374.77) (-328.10,374.77) (-327.39,374.77) (-326.68,374.77) (-325.98,374.77) (-325.27,374.77) (-324.56,374.77) (-323.85,374.77) (-323.15,374.77) (-322.44,374.77) (-321.73,374.77) (-321.03,374.77) (-320.32,374.77) (-319.61,374.77) (-318.91,374.77) (-318.20,374.77) (-317.49,374.77) (-316.78,374.77) (-316.08,374.77) (-315.37,374.77) (-314.66,374.77) (-313.96,374.77) (-313.25,374.77) (-312.54,374.77) (-311.83,374.77) (-311.13,374.77) (-310.42,374.77) (-309.71,374.77) (-309.01,374.77) (-308.30,374.77) (-307.59,374.77) (-306.88,374.77) (-306.18,374.77) (-305.47,374.77) (-304.76,374.77) (-304.06,374.77) (-303.35,374.77) (-302.64,374.77) (-301.93,374.77) (-301.93,374.77) (-301.23,374.77) (-300.52,374.77) (-299.81,375.47) (-299.11,375.47) (-298.40,375.47) (-297.69,375.47) (-296.98,375.47) (-296.28,376.18) (-295.57,376.18) (-294.86,376.18) (-294.16,376.18) (-293.45,376.18) (-292.74,376.89) (-292.04,376.89) (-291.33,376.89) (-290.62,376.89) (-289.91,377.60) (-289.21,377.60) (-288.50,377.60) (-287.79,377.60) (-287.09,377.60) (-286.38,378.30) (-285.67,378.30) (-284.96,378.30) (-284.26,378.30) (-283.55,378.30) (-282.84,379.01) (-282.14,379.01) (-281.43,379.01) (-280.72,379.01) (-280.01,379.01) (-279.31,379.72) (-278.60,379.72) (-277.89,379.72) (-277.19,379.72) (-276.48,379.72) (-275.77,380.42) (-275.06,380.42) (-274.36,380.42) (-273.65,380.42) (-272.94,381.13) (-272.24,381.13) (-271.53,381.13) (-270.82,381.13) (-270.11,381.13) (-269.41,381.84) (-268.70,381.84) (-267.99,381.84) (-267.99,381.84) (-267.99,382.54) (-267.99,383.25) (-267.29,383.96) (-267.29,384.67) (-267.29,385.37) (-267.29,386.08) (-267.29,386.79) (-267.29,387.49) (-266.58,388.20) (-266.58,388.91) (-266.58,389.62) (-266.58,390.32) (-266.58,391.03) (-266.58,391.74) (-265.87,392.44) (-265.87,393.15) (-265.87,393.86) (-265.87,394.57) (-265.87,395.27) (-265.17,395.98) (-265.17,396.69) (-265.17,397.39) (-265.17,398.10) (-265.17,398.81) (-265.17,399.52) (-264.46,400.22) (-264.46,400.93) (-264.46,401.64) (-264.46,402.34) (-264.46,403.05) (-263.75,403.76) (-263.75,404.47) (-263.75,405.17) (-263.75,405.88) (-263.75,406.59) (-263.75,407.29) (-263.04,408.00) (-263.04,408.71) (-263.04,409.41) (-263.04,410.12) (-263.04,410.83) (-263.04,411.54) (-262.34,412.24) (-262.34,412.95) (-262.34,413.66) (-245.37,383.96) (-246.07,384.67) (-246.07,385.37) (-246.78,386.08) (-246.78,386.79) (-247.49,387.49) (-247.49,388.20) (-248.19,388.91) (-248.90,389.62) (-248.90,390.32) (-249.61,391.03) (-249.61,391.74) (-250.32,392.44) (-250.32,393.15) (-251.02,393.86) (-251.73,394.57) (-251.73,395.27) (-252.44,395.98) (-252.44,396.69) (-253.14,397.39) (-253.14,398.10) (-253.85,398.81) (-254.56,399.52) (-254.56,400.22) (-255.27,400.93) (-255.27,401.64) (-255.97,402.34) (-255.97,403.05) (-256.68,403.76) (-257.39,404.47) (-257.39,405.17) (-258.09,405.88) (-258.09,406.59) (-258.80,407.29) (-258.80,408.00) (-259.51,408.71) (-260.22,409.41) (-260.22,410.12) (-260.92,410.83) (-260.92,411.54) (-261.63,412.24) (-261.63,412.95) (-262.34,413.66) (-263.75,357.80) (-263.04,358.50) (-263.04,359.21) (-262.34,359.92) (-261.63,360.62) (-260.92,361.33) (-260.92,362.04) (-260.22,362.75) (-259.51,363.45) (-259.51,364.16) (-258.80,364.87) (-258.09,365.57) (-258.09,366.28) (-257.39,366.99) (-256.68,367.70) (-255.97,368.40) (-255.97,369.11) (-255.27,369.82) (-254.56,370.52) (-254.56,371.23) (-253.85,371.94) (-253.14,372.65) (-253.14,373.35) (-252.44,374.06) (-251.73,374.77) (-251.02,375.47) (-251.02,376.18) (-250.32,376.89) (-249.61,377.60) (-249.61,378.30) (-248.90,379.01) (-248.19,379.72) (-248.19,380.42) (-247.49,381.13) (-246.78,381.84) (-246.07,382.54) (-246.07,383.25) (-245.37,383.96) (-263.75,357.80) (-263.04,357.80) (-262.34,358.50) (-261.63,358.50) (-260.92,358.50) (-260.22,359.21) (-259.51,359.21) (-258.80,359.92) (-258.09,359.92) (-257.39,359.92) (-256.68,360.62) (-255.97,360.62) (-255.27,360.62) (-254.56,361.33) (-253.85,361.33) (-253.14,361.33) (-252.44,362.04) (-251.73,362.04) (-251.02,362.75) (-250.32,362.75) (-249.61,362.75) (-248.90,363.45) (-248.19,363.45) (-247.49,363.45) (-246.78,364.16) (-246.07,364.16) (-245.37,364.16) (-244.66,364.87) (-243.95,364.87) (-243.24,364.87) (-242.54,365.57) (-241.83,365.57) (-241.12,366.28) (-240.42,366.28) (-239.71,366.28) (-239.00,366.99) (-238.29,366.99) (-237.59,366.99) (-236.88,367.70) (-236.17,367.70) (-235.47,367.70) (-234.76,368.40) (-234.05,368.40) (-233.35,369.11) (-232.64,369.11) (-231.93,369.11) (-231.22,369.82) (-230.52,369.82) (-230.52,369.82) (-230.52,370.52) (-231.22,371.23) (-231.22,371.94) (-231.22,372.65) (-231.93,373.35) (-231.93,374.06) (-231.93,374.77) (-232.64,375.47) (-232.64,376.18) (-232.64,376.89) (-233.35,377.60) (-233.35,378.30) (-233.35,379.01) (-234.05,379.72) (-234.05,380.42) (-234.05,381.13) (-234.76,381.84) (-234.76,382.54) (-234.76,383.25) (-235.47,383.96) (-235.47,384.67) (-235.47,385.37) (-236.17,386.08) (-236.17,386.79) (-236.17,387.49) (-236.88,388.20) (-236.88,388.91) (-236.88,389.62) (-237.59,390.32) (-237.59,391.03) (-237.59,391.74) (-238.29,392.44) (-238.29,393.15) (-238.29,393.86) (-239.00,394.57) (-239.00,395.27) (-239.00,395.98) (-239.71,396.69) (-239.71,397.39) (-239.71,398.10) (-240.42,398.81) (-240.42,399.52) (-240.42,399.52) (-239.71,399.52) (-239.00,399.52) (-238.29,399.52) (-237.59,399.52) (-236.88,399.52) (-236.17,399.52) (-235.47,398.81) (-234.76,398.81) (-234.05,398.81) (-233.35,398.81) (-232.64,398.81) (-231.93,398.81) (-231.22,398.81) (-230.52,398.81) (-229.81,398.81) (-229.10,398.81) (-228.40,398.81) (-227.69,398.81) (-226.98,398.10) (-226.27,398.10) (-225.57,398.10) (-224.86,398.10) (-224.15,398.10) (-223.45,398.10) (-222.74,398.10) (-222.03,398.10) (-221.32,398.10) (-220.62,398.10) (-219.91,398.10) (-219.20,398.10) (-218.50,397.39) (-217.79,397.39) (-217.08,397.39) (-216.37,397.39) (-215.67,397.39) (-214.96,397.39) (-214.25,397.39) (-213.55,397.39) (-212.84,397.39) (-212.13,397.39) (-211.42,397.39) (-210.72,397.39) (-210.01,396.69) (-209.30,396.69) (-208.60,396.69) (-207.89,396.69) (-207.18,396.69) (-206.48,396.69) (-206.48,396.69) (-206.48,397.39) (-206.48,398.10) (-206.48,398.81) (-206.48,399.52) (-206.48,400.22) (-205.77,400.93) (-205.77,401.64) (-205.77,402.34) (-205.77,403.05) (-205.77,403.76) (-205.77,404.47) (-205.77,405.17) (-205.77,405.88) (-205.77,406.59) (-205.77,407.29) (-205.77,408.00) (-205.06,408.71) (-205.06,409.41) (-205.06,410.12) (-205.06,410.83) (-205.06,411.54) (-205.06,412.24) (-205.06,412.95) (-205.06,413.66) (-205.06,414.36) (-205.06,415.07) (-205.06,415.78) (-205.06,416.49) (-204.35,417.19) (-204.35,417.90) (-204.35,418.61) (-204.35,419.31) (-204.35,420.02) (-204.35,420.73) (-204.35,421.44) (-204.35,422.14) (-204.35,422.85) (-204.35,423.56) (-204.35,424.26) (-203.65,424.97) (-203.65,425.68) (-203.65,426.39) (-203.65,427.09) (-203.65,427.80) (-203.65,428.51) (-191.63,397.39) (-191.63,398.10) (-192.33,398.81) (-192.33,399.52) (-193.04,400.22) (-193.04,400.93) (-193.04,401.64) (-193.75,402.34) (-193.75,403.05) (-193.75,403.76) (-194.45,404.47) (-194.45,405.17) (-195.16,405.88) (-195.16,406.59) (-195.16,407.29) (-195.87,408.00) (-195.87,408.71) (-196.58,409.41) (-196.58,410.12) (-196.58,410.83) (-197.28,411.54) (-197.28,412.24) (-197.28,412.95) (-197.99,413.66) (-197.99,414.36) (-198.70,415.07) (-198.70,415.78) (-198.70,416.49) (-199.40,417.19) (-199.40,417.90) (-200.11,418.61) (-200.11,419.31) (-200.11,420.02) (-200.82,420.73) (-200.82,421.44) (-201.53,422.14) (-201.53,422.85) (-201.53,423.56) (-202.23,424.26) (-202.23,424.97) (-202.23,425.68) (-202.94,426.39) (-202.94,427.09) (-203.65,427.80) (-203.65,428.51) (-223.45,387.49) (-222.74,387.49) (-222.03,388.20) (-221.32,388.20) (-220.62,388.20) (-219.91,388.91) (-219.20,388.91) (-218.50,388.91) (-217.79,388.91) (-217.08,389.62) (-216.37,389.62) (-215.67,389.62) (-214.96,390.32) (-214.25,390.32) (-213.55,390.32) (-212.84,391.03) (-212.13,391.03) (-211.42,391.03) (-210.72,391.74) (-210.01,391.74) (-209.30,391.74) (-208.60,392.44) (-207.89,392.44) (-207.18,392.44) (-206.48,392.44) (-205.77,393.15) (-205.06,393.15) (-204.35,393.15) (-203.65,393.86) (-202.94,393.86) (-202.23,393.86) (-201.53,394.57) (-200.82,394.57) (-200.11,394.57) (-199.40,395.27) (-198.70,395.27) (-197.99,395.27) (-197.28,395.98) (-196.58,395.98) (-195.87,395.98) (-195.16,395.98) (-194.45,396.69) (-193.75,396.69) (-193.04,396.69) (-192.33,397.39) (-191.63,397.39) (-223.45,353.55) (-223.45,354.26) (-223.45,354.97) (-223.45,355.67) (-223.45,356.38) (-223.45,357.09) (-223.45,357.80) (-223.45,358.50) (-223.45,359.21) (-223.45,359.92) (-223.45,360.62) (-223.45,361.33) (-223.45,362.04) (-223.45,362.75) (-223.45,363.45) (-223.45,364.16) (-223.45,364.87) (-223.45,365.57) (-223.45,366.28) (-223.45,366.99) (-223.45,367.70) (-223.45,368.40) (-223.45,369.11) (-223.45,369.82) (-223.45,370.52) (-223.45,371.23) (-223.45,371.94) (-223.45,372.65) (-223.45,373.35) (-223.45,374.06) (-223.45,374.77) (-223.45,375.47) (-223.45,376.18) (-223.45,376.89) (-223.45,377.60) (-223.45,378.30) (-223.45,379.01) (-223.45,379.72) (-223.45,380.42) (-223.45,381.13) (-223.45,381.84) (-223.45,382.54) (-223.45,383.25) (-223.45,383.96) (-223.45,384.67) (-223.45,385.37) (-223.45,386.08) (-223.45,386.79) (-223.45,387.49) (-248.90,373.35) (-248.19,372.65) (-247.49,371.94) (-246.78,371.94) (-246.07,371.23) (-245.37,370.52) (-244.66,369.82) (-243.95,369.82) (-243.24,369.11) (-242.54,368.40) (-241.83,367.70) (-241.12,366.99) (-240.42,366.99) (-239.71,366.28) (-239.00,365.57) (-238.29,364.87) (-237.59,364.87) (-236.88,364.16) (-236.17,363.45) (-235.47,362.75) (-234.76,362.04) (-234.05,362.04) (-233.35,361.33) (-232.64,360.62) (-231.93,359.92) (-231.22,359.92) (-230.52,359.21) (-229.81,358.50) (-229.10,357.80) (-228.40,357.09) (-227.69,357.09) (-226.98,356.38) (-226.27,355.67) (-225.57,354.97) (-224.86,354.97) (-224.15,354.26) (-223.45,353.55) (-248.90,373.35) (-249.61,374.06) (-250.32,374.77) (-250.32,375.47) (-251.02,376.18) (-251.73,376.89) (-252.44,377.60) (-253.14,378.30) (-253.14,379.01) (-253.85,379.72) (-254.56,380.42) (-255.27,381.13) (-255.97,381.84) (-255.97,382.54) (-256.68,383.25) (-257.39,383.96) (-258.09,384.67) (-258.80,385.37) (-258.80,386.08) (-259.51,386.79) (-260.22,387.49) (-260.92,388.20) (-261.63,388.91) (-262.34,389.62) (-262.34,390.32) (-263.04,391.03) (-263.75,391.74) (-264.46,392.44) (-265.17,393.15) (-265.17,393.86) (-265.87,394.57) (-266.58,395.27) (-267.29,395.98) (-267.99,396.69) (-267.99,397.39) (-268.70,398.10) (-269.41,398.81) (-269.41,398.81) (-269.41,399.52) (-269.41,400.22) (-269.41,400.93) (-269.41,401.64) (-269.41,402.34) (-269.41,403.05) (-269.41,403.76) (-269.41,404.47) (-269.41,405.17) (-269.41,405.88) (-269.41,406.59) (-269.41,407.29) (-269.41,408.00) (-269.41,408.71) (-269.41,409.41) (-269.41,410.12) (-269.41,410.83) (-269.41,411.54) (-269.41,412.24) (-269.41,412.95) (-269.41,413.66) (-269.41,414.36) (-269.41,415.07) (-269.41,415.78) (-269.41,416.49) (-269.41,417.19) (-269.41,417.90) (-269.41,418.61) (-269.41,419.31) (-269.41,420.02) (-269.41,420.73) (-269.41,421.44) (-269.41,422.14) (-269.41,422.85) (-269.41,423.56) (-269.41,424.26) (-269.41,424.97) (-269.41,425.68) (-269.41,426.39) (-269.41,427.09) (-269.41,427.80) (-269.41,428.51) (-269.41,429.21) (-269.41,429.92) (-269.41,430.63) (-269.41,431.34) (-269.41,432.04) (-269.41,432.75) (-269.41,432.75) (-268.70,432.75) (-267.99,432.04) (-267.29,432.04) (-266.58,431.34) (-265.87,431.34) (-265.17,430.63) (-264.46,430.63) (-263.75,430.63) (-263.04,429.92) (-262.34,429.92) (-261.63,429.21) (-260.92,429.21) (-260.22,428.51) (-259.51,428.51) (-258.80,427.80) (-258.09,427.80) (-257.39,427.80) (-256.68,427.09) (-255.97,427.09) (-255.27,426.39) (-254.56,426.39) (-253.85,425.68) (-253.14,425.68) (-252.44,425.68) (-251.73,424.97) (-251.02,424.97) (-250.32,424.26) (-249.61,424.26) (-248.90,423.56) (-248.19,423.56) (-247.49,423.56) (-246.78,422.85) (-246.07,422.85) (-245.37,422.14) (-244.66,422.14) (-243.95,421.44) (-243.24,421.44) (-242.54,420.73) (-241.83,420.73) (-241.12,420.73) (-240.42,420.02) (-239.71,420.02) (-239.00,419.31) (-238.29,419.31) (-237.59,418.61) (-236.88,418.61) (-236.88,418.61) (-236.17,418.61) (-235.47,418.61) (-234.76,418.61) (-234.05,418.61) (-233.35,418.61) (-232.64,418.61) (-231.93,418.61) (-231.22,418.61) (-230.52,418.61) (-229.81,418.61) (-229.10,418.61) (-228.40,418.61) (-227.69,418.61) (-226.98,418.61) (-226.27,418.61) (-225.57,418.61) (-224.86,418.61) (-224.15,418.61) (-223.45,418.61) (-222.74,418.61) (-222.03,418.61) (-221.32,418.61) (-220.62,418.61) (-219.91,418.61) (-219.20,418.61) (-218.50,417.90) (-217.79,417.90) (-217.08,417.90) (-216.37,417.90) (-215.67,417.90) (-214.96,417.90) (-214.25,417.90) (-213.55,417.90) (-212.84,417.90) (-212.13,417.90) (-211.42,417.90) (-210.72,417.90) (-210.01,417.90) (-209.30,417.90) (-208.60,417.90) (-207.89,417.90) (-207.18,417.90) (-206.48,417.90) (-205.77,417.90) (-205.06,417.90) (-204.35,417.90) (-203.65,417.90) (-202.94,417.90) (-202.23,417.90) (-201.53,417.90) (-201.53,417.90) (-200.82,418.61) (-200.11,419.31) (-199.40,420.02) (-198.70,420.73) (-197.99,421.44) (-197.28,422.14) (-196.58,422.85) (-195.87,423.56) (-195.87,424.26) (-195.16,424.97) (-194.45,425.68) (-193.75,426.39) (-193.04,427.09) (-192.33,427.80) (-191.63,428.51) (-190.92,429.21) (-190.21,429.92) (-189.50,430.63) (-188.80,431.34) (-188.09,432.04) (-187.38,432.75) (-186.68,433.46) (-185.97,434.16) (-185.26,434.87) (-184.55,435.58) (-183.85,436.28) (-183.85,436.99) (-183.14,437.70) (-182.43,438.41) (-181.73,439.11) (-181.02,439.82) (-180.31,440.53) (-179.61,441.23) (-178.90,441.94) (-178.19,442.65) (-177.48,443.36) (-209.30,437.70) (-208.60,437.70) (-207.89,437.70) (-207.18,438.41) (-206.48,438.41) (-205.77,438.41) (-205.06,438.41) (-204.35,438.41) (-203.65,438.41) (-202.94,439.11) (-202.23,439.11) (-201.53,439.11) (-200.82,439.11) (-200.11,439.11) (-199.40,439.11) (-198.70,439.82) (-197.99,439.82) (-197.28,439.82) (-196.58,439.82) (-195.87,439.82) (-195.16,440.53) (-194.45,440.53) (-193.75,440.53) (-193.04,440.53) (-192.33,440.53) (-191.63,440.53) (-190.92,441.23) (-190.21,441.23) (-189.50,441.23) (-188.80,441.23) (-188.09,441.23) (-187.38,441.94) (-186.68,441.94) (-185.97,441.94) (-185.26,441.94) (-184.55,441.94) (-183.85,441.94) (-183.14,442.65) (-182.43,442.65) (-181.73,442.65) (-181.02,442.65) (-180.31,442.65) (-179.61,442.65) (-178.90,443.36) (-178.19,443.36) (-177.48,443.36) (-217.79,403.76) (-217.79,404.47) (-217.79,405.17) (-217.08,405.88) (-217.08,406.59) (-217.08,407.29) (-217.08,408.00) (-216.37,408.71) (-216.37,409.41) (-216.37,410.12) (-216.37,410.83) (-215.67,411.54) (-215.67,412.24) (-215.67,412.95) (-215.67,413.66) (-214.96,414.36) (-214.96,415.07) (-214.96,415.78) (-214.96,416.49) (-214.25,417.19) (-214.25,417.90) (-214.25,418.61) (-214.25,419.31) (-213.55,420.02) (-213.55,420.73) (-213.55,421.44) (-213.55,422.14) (-212.84,422.85) (-212.84,423.56) (-212.84,424.26) (-212.84,424.97) (-212.13,425.68) (-212.13,426.39) (-212.13,427.09) (-212.13,427.80) (-211.42,428.51) (-211.42,429.21) (-211.42,429.92) (-211.42,430.63) (-210.72,431.34) (-210.72,432.04) (-210.72,432.75) (-210.72,433.46) (-210.01,434.16) (-210.01,434.87) (-210.01,435.58) (-210.01,436.28) (-209.30,436.99) (-209.30,437.70) (-217.79,403.76) (-217.79,404.47) (-218.50,405.17) (-218.50,405.88) (-218.50,406.59) (-219.20,407.29) (-219.20,408.00) (-219.20,408.71) (-219.91,409.41) (-219.91,410.12) (-219.91,410.83) (-219.91,411.54) (-220.62,412.24) (-220.62,412.95) (-220.62,413.66) (-221.32,414.36) (-221.32,415.07) (-221.32,415.78) (-222.03,416.49) (-222.03,417.19) (-222.03,417.90) (-222.74,418.61) (-222.74,419.31) (-222.74,420.02) (-223.45,420.73) (-223.45,421.44) (-223.45,422.14) (-224.15,422.85) (-224.15,423.56) (-224.15,424.26) (-224.86,424.97) (-224.86,425.68) (-224.86,426.39) (-224.86,427.09) (-225.57,427.80) (-225.57,428.51) (-225.57,429.21) (-226.27,429.92) (-226.27,430.63) (-226.27,431.34) (-226.98,432.04) (-226.98,432.75) (-226.98,433.46) (-227.69,434.16) (-227.69,434.87) (-227.69,434.87) (-226.98,434.87) (-226.27,435.58) (-225.57,435.58) (-224.86,435.58) (-224.15,435.58) (-223.45,436.28) (-222.74,436.28) (-222.03,436.28) (-221.32,436.99) (-220.62,436.99) (-219.91,436.99) (-219.20,437.70) (-218.50,437.70) (-217.79,437.70) (-217.08,437.70) (-216.37,438.41) (-215.67,438.41) (-214.96,438.41) (-214.25,439.11) (-213.55,439.11) (-212.84,439.11) (-212.13,439.82) (-211.42,439.82) (-210.72,439.82) (-210.01,439.82) (-209.30,440.53) (-208.60,440.53) (-207.89,440.53) (-207.18,441.23) (-206.48,441.23) (-205.77,441.23) (-205.06,441.94) (-204.35,441.94) (-203.65,441.94) (-202.94,441.94) (-202.23,442.65) (-201.53,442.65) (-200.82,442.65) (-200.11,443.36) (-199.40,443.36) (-198.70,443.36) (-197.99,444.06) (-197.28,444.06) (-196.58,444.06) (-195.87,444.06) (-195.16,444.77) (-194.45,444.77) (-194.45,444.77) (-194.45,445.48) (-195.16,446.18) (-195.16,446.89) (-195.16,447.60) (-195.16,448.31) (-195.87,449.01) (-195.87,449.72) (-195.87,450.43) (-195.87,451.13) (-196.58,451.84) (-196.58,452.55) (-196.58,453.26) (-196.58,453.96) (-197.28,454.67) (-197.28,455.38) (-197.28,456.08) (-197.99,456.79) (-197.99,457.50) (-197.99,458.21) (-197.99,458.91) (-198.70,459.62) (-198.70,460.33) (-198.70,461.03) (-198.70,461.74) (-199.40,462.45) (-199.40,463.15) (-199.40,463.86) (-199.40,464.57) (-200.11,465.28) (-200.11,465.98) (-200.11,466.69) (-200.82,467.40) (-200.82,468.10) (-200.82,468.81) (-200.82,469.52) (-201.53,470.23) (-201.53,470.93) (-201.53,471.64) (-201.53,472.35) (-202.23,473.05) (-202.23,473.76) (-202.23,474.47) (-202.23,475.18) (-202.94,475.88) (-202.94,476.59) (-202.94,476.59) (-202.23,476.59) (-201.53,476.59) (-200.82,476.59) (-200.11,477.30) (-199.40,477.30) (-198.70,477.30) (-197.99,477.30) (-197.28,477.30) (-196.58,477.30) (-195.87,477.30) (-195.16,478.00) (-194.45,478.00) (-193.75,478.00) (-193.04,478.00) (-192.33,478.00) (-191.63,478.00) (-190.92,478.00) (-190.21,478.00) (-189.50,478.71) (-188.80,478.71) (-188.09,478.71) (-187.38,478.71) (-186.68,478.71) (-185.97,478.71) (-185.26,478.71) (-184.55,479.42) (-183.85,479.42) (-183.14,479.42) (-182.43,479.42) (-181.73,479.42) (-181.02,479.42) (-180.31,479.42) (-179.61,480.13) (-178.90,480.13) (-178.19,480.13) (-177.48,480.13) (-176.78,480.13) (-176.07,480.13) (-175.36,480.13) (-174.66,480.13) (-173.95,480.83) (-173.24,480.83) (-172.53,480.83) (-171.83,480.83) (-171.12,480.83) (-170.41,480.83) (-169.71,480.83) (-169.00,481.54) (-168.29,481.54) (-167.58,481.54) (-166.88,481.54) (-166.88,481.54) (-166.17,481.54) (-165.46,482.25) (-164.76,482.25) (-164.05,482.95) (-163.34,482.95) (-162.63,483.66) (-161.93,483.66) (-161.22,484.37) (-160.51,484.37) (-159.81,485.08) (-159.10,485.08) (-158.39,485.78) (-157.68,485.78) (-156.98,486.49) (-156.27,486.49) (-155.56,487.20) (-154.86,487.20) (-154.15,487.90) (-153.44,487.90) (-152.74,488.61) (-152.03,488.61) (-151.32,489.32) (-150.61,489.32) (-149.91,490.02) (-149.20,490.02) (-148.49,490.73) (-147.79,490.73) (-147.08,491.44) (-146.37,491.44) (-145.66,492.15) (-144.96,492.15) (-144.25,492.85) (-143.54,492.85) (-142.84,493.56) (-142.13,493.56) (-141.42,494.27) (-140.71,494.27) (-140.01,494.97) (-139.30,494.97) (-138.59,495.68) (-137.89,495.68) (-137.89,495.68) (-138.59,496.39) (-138.59,497.10) (-139.30,497.80) (-140.01,498.51) (-140.01,499.22) (-140.71,499.92) (-141.42,500.63) (-141.42,501.34) (-142.13,502.05) (-142.84,502.75) (-142.84,503.46) (-143.54,504.17) (-144.25,504.87) (-144.25,505.58) (-144.96,506.29) (-145.66,507.00) (-145.66,507.70) (-146.37,508.41) (-147.08,509.12) (-147.79,509.82) (-147.79,510.53) (-148.49,511.24) (-149.20,511.95) (-149.20,512.65) (-149.91,513.36) (-150.61,514.07) (-150.61,514.77) (-151.32,515.48) (-152.03,516.19) (-152.03,516.90) (-152.74,517.60) (-153.44,518.31) (-153.44,519.02) (-154.15,519.72) (-154.86,520.43) (-154.86,521.14) (-155.56,521.84) (-155.56,521.84) (-155.56,522.55) (-154.86,523.26) (-154.86,523.97) (-154.86,524.67) (-154.15,525.38) (-154.15,526.09) (-154.15,526.79) (-153.44,527.50) (-153.44,528.21) (-153.44,528.92) (-152.74,529.62) (-152.74,530.33) (-152.74,531.04) (-152.03,531.74) (-152.03,532.45) (-152.03,533.16) (-151.32,533.87) (-151.32,534.57) (-151.32,535.28) (-150.61,535.99) (-150.61,536.69) (-150.61,537.40) (-149.91,538.11) (-149.91,538.82) (-149.91,539.52) (-149.20,540.23) (-149.20,540.94) (-149.20,541.64) (-148.49,542.35) (-148.49,543.06) (-148.49,543.77) (-147.79,544.47) (-147.79,545.18) (-147.79,545.89) (-147.08,546.59) (-147.08,547.30) (-147.08,548.01) (-146.37,548.71) (-146.37,549.42) (-146.37,550.13) (-145.66,550.84) (-145.66,551.54) (-145.66,552.25) (-144.96,552.96) (-144.96,553.66) (-166.88,527.50) (-166.17,528.21) (-165.46,528.92) (-164.76,529.62) (-164.76,530.33) (-164.05,531.04) (-163.34,531.74) (-162.63,532.45) (-161.93,533.16) (-161.22,533.87) (-161.22,534.57) (-160.51,535.28) (-159.81,535.99) (-159.10,536.69) (-158.39,537.40) (-157.68,538.11) (-157.68,538.82) (-156.98,539.52) (-156.27,540.23) (-155.56,540.94) (-154.86,541.64) (-154.15,542.35) (-154.15,543.06) (-153.44,543.77) (-152.74,544.47) (-152.03,545.18) (-151.32,545.89) (-150.61,546.59) (-150.61,547.30) (-149.91,548.01) (-149.20,548.71) (-148.49,549.42) (-147.79,550.13) (-147.08,550.84) (-147.08,551.54) (-146.37,552.25) (-145.66,552.96) (-144.96,553.66) (-166.88,527.50) (-166.17,527.50) (-165.46,527.50) (-164.76,527.50) (-164.05,527.50) (-163.34,527.50) (-162.63,527.50) (-161.93,527.50) (-161.22,527.50) (-160.51,527.50) (-159.81,527.50) (-159.10,527.50) (-158.39,527.50) (-157.68,527.50) (-156.98,527.50) (-156.27,527.50) (-155.56,527.50) (-154.86,527.50) (-154.15,527.50) (-153.44,527.50) (-152.74,527.50) (-152.03,527.50) (-151.32,527.50) (-150.61,527.50) (-149.91,526.79) (-149.20,526.79) (-148.49,526.79) (-147.79,526.79) (-147.08,526.79) (-146.37,526.79) (-145.66,526.79) (-144.96,526.79) (-144.25,526.79) (-143.54,526.79) (-142.84,526.79) (-142.13,526.79) (-141.42,526.79) (-140.71,526.79) (-140.01,526.79) (-139.30,526.79) (-138.59,526.79) (-137.89,526.79) (-137.18,526.79) (-136.47,526.79) (-135.76,526.79) (-135.06,526.79) (-134.35,526.79) (-133.64,526.79) (-544,-293)[(0,0)\[lt\][$\rho: 0.95$]{}]{}
$\delta$ $\alpha_f$ $l_f$ $\tau$ $\alpha_n$ $l_n$
---------- ------------ -------- -------- ------------ -------
0.95 0.00 136.94 57 34.25 33.50
0.5 199.06 15.53 0 2.00 60.50
0.1 309.18 1.75 18 285.18 58.00
0.01 227.29 1.50 37 345.88 45.50
: The values of $A_F,L_F,T,A_N,L_F$ for the best individual for different values of the food density. The values are normalized as in (\[normal\]) to obtain the walk parameters $(\alpha_f,l_f,\tau,\alpha_n,l_n)$.[]{data-label="params"}
[cc]{}
(1050,1050)(0,0) (172.0,131.0)
------------------------------------------------------------------------
(152,131)[(0,0)\[r\][ 0]{}]{} (947.0,131.0)
------------------------------------------------------------------------
(172.0,230.0)
------------------------------------------------------------------------
(152,230)[(0,0)\[r\][ 500]{}]{} (947.0,230.0)
------------------------------------------------------------------------
(172.0,330.0)
------------------------------------------------------------------------
(152,330)[(0,0)\[r\][ 1000]{}]{} (947.0,330.0)
------------------------------------------------------------------------
(172.0,429.0)
------------------------------------------------------------------------
(152,429)[(0,0)\[r\][ 1500]{}]{} (947.0,429.0)
------------------------------------------------------------------------
(172.0,529.0)
------------------------------------------------------------------------
(152,529)[(0,0)\[r\][ 2000]{}]{} (947.0,529.0)
------------------------------------------------------------------------
(172.0,628.0)
------------------------------------------------------------------------
(152,628)[(0,0)\[r\][ 2500]{}]{} (947.0,628.0)
------------------------------------------------------------------------
(172.0,727.0)
------------------------------------------------------------------------
(152,727)[(0,0)\[r\][ 3000]{}]{} (947.0,727.0)
------------------------------------------------------------------------
(172.0,827.0)
------------------------------------------------------------------------
(152,827)[(0,0)\[r\][ 3500]{}]{} (947.0,827.0)
------------------------------------------------------------------------
(172.0,926.0)
------------------------------------------------------------------------
(152,926)[(0,0)\[r\][ 4000]{}]{} (947.0,926.0)
------------------------------------------------------------------------
(172.0,131.0)
------------------------------------------------------------------------
(172,90)[(0,0)[ 0]{}]{} (172.0,906.0)
------------------------------------------------------------------------
(253.0,131.0)
------------------------------------------------------------------------
(253,90)[(0,0)[ 5]{}]{} (253.0,906.0)
------------------------------------------------------------------------
(334.0,131.0)
------------------------------------------------------------------------
(334,90)[(0,0)[ 10]{}]{} (334.0,906.0)
------------------------------------------------------------------------
(415.0,131.0)
------------------------------------------------------------------------
(415,90)[(0,0)[ 15]{}]{} (415.0,906.0)
------------------------------------------------------------------------
(496.0,131.0)
------------------------------------------------------------------------
(496,90)[(0,0)[ 20]{}]{} (496.0,906.0)
------------------------------------------------------------------------
(578.0,131.0)
------------------------------------------------------------------------
(578,90)[(0,0)[ 25]{}]{} (578.0,906.0)
------------------------------------------------------------------------
(659.0,131.0)
------------------------------------------------------------------------
(659,90)[(0,0)[ 30]{}]{} (659.0,906.0)
------------------------------------------------------------------------
(740.0,131.0)
------------------------------------------------------------------------
(740,90)[(0,0)[ 35]{}]{} (740.0,906.0)
------------------------------------------------------------------------
(821.0,131.0)
------------------------------------------------------------------------
(821,90)[(0,0)[ 40]{}]{} (821.0,906.0)
------------------------------------------------------------------------
(902.0,131.0)
------------------------------------------------------------------------
(902,90)[(0,0)[ 45]{}]{} (902.0,906.0)
------------------------------------------------------------------------
(172.0,131.0)
------------------------------------------------------------------------
(172.0,131.0)
------------------------------------------------------------------------
(967.0,131.0)
------------------------------------------------------------------------
(172.0,926.0)
------------------------------------------------------------------------
(569,29)[(0,0)[$t$]{}]{} (569,988)[(0,0)[density: $\rho=0.01$]{}]{} (434,807)[(0,0)\[r\][walk]{}]{} (172,131)[(0,0)[$+$]{}]{} (204,132)[(0,0)[$+$]{}]{} (237,135)[(0,0)[$+$]{}]{} (269,135)[(0,0)[$+$]{}]{} (302,138)[(0,0)[$+$]{}]{} (334,139)[(0,0)[$+$]{}]{} (367,141)[(0,0)[$+$]{}]{} (399,144)[(0,0)[$+$]{}]{} (432,145)[(0,0)[$+$]{}]{} (464,150)[(0,0)[$+$]{}]{} (496,158)[(0,0)[$+$]{}]{} (529,169)[(0,0)[$+$]{}]{} (561,183)[(0,0)[$+$]{}]{} (594,201)[(0,0)[$+$]{}]{} (626,223)[(0,0)[$+$]{}]{} (659,250)[(0,0)[$+$]{}]{} (691,302)[(0,0)[$+$]{}]{} (724,324)[(0,0)[$+$]{}]{} (756,391)[(0,0)[$+$]{}]{} (789,453)[(0,0)[$+$]{}]{} (821,514)[(0,0)[$+$]{}]{} (853,616)[(0,0)[$+$]{}]{} (886,702)[(0,0)[$+$]{}]{} (918,812)[(0,0)[$+$]{}]{} (951,907)[(0,0)[$+$]{}]{} (504,807)[(0,0)[$+$]{}]{} (434,766)[(0,0)\[r\][$x^{3.78}$]{}]{} (454.0,766.0)
------------------------------------------------------------------------
(180,131) (276,130.67)
------------------------------------------------------------------------
(276.00,130.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(180.0,131.0)
------------------------------------------------------------------------
(317,131.67)
------------------------------------------------------------------------
(317.00,131.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(284.0,132.0)
------------------------------------------------------------------------
(341,132.67)
------------------------------------------------------------------------
(341.00,132.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(325.0,133.0)
------------------------------------------------------------------------
(357,133.67)
------------------------------------------------------------------------
(357.00,133.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(365,134.67)
------------------------------------------------------------------------
(365.00,134.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(349.0,134.0)
------------------------------------------------------------------------
(381,135.67)
------------------------------------------------------------------------
(381.00,135.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(389,136.67)
------------------------------------------------------------------------
(389.00,136.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(397,137.67)
------------------------------------------------------------------------
(397.00,137.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(405,138.67)
------------------------------------------------------------------------
(405.00,138.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(413,139.67)
------------------------------------------------------------------------
(413.00,139.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(421,141.17)
------------------------------------------------------------------------
(421.00,140.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(429,142.67)
------------------------------------------------------------------------
(429.00,142.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(437,144.17)
------------------------------------------------------------------------
(437.00,143.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(445,146.17)
------------------------------------------------------------------------
(445.00,145.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(453,147.67)
------------------------------------------------------------------------
(453.00,147.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(461,149.17)
------------------------------------------------------------------------
(461.00,148.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(469.00,151.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(469.00,150.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(477,154.17)
------------------------------------------------------------------------
(477.00,153.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(485,156.17)
------------------------------------------------------------------------
(485.00,155.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(493.00,158.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(493.00,157.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(501.00,161.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(501.00,160.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(509.00,164.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(509.00,163.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(517.00,167.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(517.00,166.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(525.00,170.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(525.00,169.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(533.00,174.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(533.00,173.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(541.00,177.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(541.00,176.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(549.00,181.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(549.00,180.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(557.00,185.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(557.00,184.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(565.00,190.59)(0.933,0.477)[7]{}
------------------------------------------------------------------------
(565.00,189.17)(7.298,5.000)[2]{}
------------------------------------------------------------------------
(574.00,195.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(574.00,194.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(582.00,199.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(582.00,198.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(590.00,205.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(590.00,204.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(598.00,210.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(598.00,209.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(606.00,216.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(606.00,215.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(614.00,222.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(614.00,221.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(622.00,228.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(622.00,227.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(630.00,235.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(630.00,234.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(638.00,242.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(638.00,241.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(646.00,250.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(646.00,249.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(654.00,257.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(654.00,256.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(662.59,265.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(661.17,265.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(670.59,274.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(669.17,274.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(678.59,283.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(677.17,283.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(686.59,292.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(685.17,292.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(694.59,302.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(693.17,302.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(702.59,312.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(701.17,312.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(710.59,322.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(709.17,322.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(718.59,333.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(717.17,333.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(726.59,345.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(725.17,345.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(734.59,357.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(733.17,357.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(742.59,369.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(741.17,369.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(750.59,382.00)(0.488,0.890)[13]{}
------------------------------------------------------------------------
(749.17,382.00)(8.000,12.340)[2]{}
------------------------------------------------------------------------
(758.59,396.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(757.17,396.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(766.59,409.00)(0.488,0.956)[13]{}
------------------------------------------------------------------------
(765.17,409.00)(8.000,13.236)[2]{}
------------------------------------------------------------------------
(774.59,424.00)(0.488,0.956)[13]{}
------------------------------------------------------------------------
(773.17,424.00)(8.000,13.236)[2]{}
------------------------------------------------------------------------
(782.59,439.00)(0.488,0.956)[13]{}
------------------------------------------------------------------------
(781.17,439.00)(8.000,13.236)[2]{}
------------------------------------------------------------------------
(790.59,454.00)(0.488,1.088)[13]{}
------------------------------------------------------------------------
(789.17,454.00)(8.000,15.028)[2]{}
------------------------------------------------------------------------
(798.59,471.00)(0.488,1.022)[13]{}
------------------------------------------------------------------------
(797.17,471.00)(8.000,14.132)[2]{}
------------------------------------------------------------------------
(806.59,487.00)(0.488,1.154)[13]{}
------------------------------------------------------------------------
(805.17,487.00)(8.000,15.924)[2]{}
------------------------------------------------------------------------
(814.59,505.00)(0.488,1.088)[13]{}
------------------------------------------------------------------------
(813.17,505.00)(8.000,15.028)[2]{}
------------------------------------------------------------------------
(822.59,522.00)(0.488,1.220)[13]{}
------------------------------------------------------------------------
(821.17,522.00)(8.000,16.821)[2]{}
------------------------------------------------------------------------
(830.59,541.00)(0.489,1.077)[15]{}
------------------------------------------------------------------------
(829.17,541.00)(9.000,17.040)[2]{}
------------------------------------------------------------------------
(839.59,560.00)(0.488,1.286)[13]{}
------------------------------------------------------------------------
(838.17,560.00)(8.000,17.717)[2]{}
------------------------------------------------------------------------
(847.59,580.00)(0.488,1.286)[13]{}
------------------------------------------------------------------------
(846.17,580.00)(8.000,17.717)[2]{}
------------------------------------------------------------------------
(855.59,600.00)(0.488,1.418)[13]{}
------------------------------------------------------------------------
(854.17,600.00)(8.000,19.509)[2]{}
------------------------------------------------------------------------
(863.59,622.00)(0.488,1.352)[13]{}
------------------------------------------------------------------------
(862.17,622.00)(8.000,18.613)[2]{}
------------------------------------------------------------------------
(871.59,643.00)(0.488,1.484)[13]{}
------------------------------------------------------------------------
(870.17,643.00)(8.000,20.406)[2]{}
------------------------------------------------------------------------
(879.59,666.00)(0.488,1.484)[13]{}
------------------------------------------------------------------------
(878.17,666.00)(8.000,20.406)[2]{}
------------------------------------------------------------------------
(887.59,689.00)(0.488,1.550)[13]{}
------------------------------------------------------------------------
(886.17,689.00)(8.000,21.302)[2]{}
------------------------------------------------------------------------
(895.59,713.00)(0.488,1.616)[13]{}
------------------------------------------------------------------------
(894.17,713.00)(8.000,22.198)[2]{}
------------------------------------------------------------------------
(903.59,738.00)(0.488,1.682)[13]{}
------------------------------------------------------------------------
(902.17,738.00)(8.000,23.094)[2]{}
------------------------------------------------------------------------
(911.59,764.00)(0.488,1.682)[13]{}
------------------------------------------------------------------------
(910.17,764.00)(8.000,23.094)[2]{}
------------------------------------------------------------------------
(919.59,790.00)(0.488,1.748)[13]{}
------------------------------------------------------------------------
(918.17,790.00)(8.000,23.990)[2]{}
------------------------------------------------------------------------
(927.59,817.00)(0.488,1.814)[13]{}
------------------------------------------------------------------------
(926.17,817.00)(8.000,24.887)[2]{}
------------------------------------------------------------------------
(935.59,845.00)(0.488,1.880)[13]{}
------------------------------------------------------------------------
(934.17,845.00)(8.000,25.783)[2]{}
------------------------------------------------------------------------
(943.59,874.00)(0.488,1.880)[13]{}
------------------------------------------------------------------------
(942.17,874.00)(8.000,25.783)[2]{}
------------------------------------------------------------------------
(951.59,903.00)(0.482,2.027)[9]{}
------------------------------------------------------------------------
(950.17,903.00)(6.000,19.610)[2]{}
------------------------------------------------------------------------
(373.0,136.0)
------------------------------------------------------------------------
(172.0,131.0)
------------------------------------------------------------------------
(172.0,131.0)
------------------------------------------------------------------------
(967.0,131.0)
------------------------------------------------------------------------
(172.0,926.0)
------------------------------------------------------------------------
(1050,1050)(0,0) (182.0,131.0)
------------------------------------------------------------------------
(162,131)[(0,0)\[r\][ 0]{}]{} (957.0,131.0)
------------------------------------------------------------------------
(182.0,330.0)
------------------------------------------------------------------------
(162,330)[(0,0)\[r\][ 5000]{}]{} (957.0,330.0)
------------------------------------------------------------------------
(182.0,529.0)
------------------------------------------------------------------------
(162,529)[(0,0)\[r\][ 10000]{}]{} (957.0,529.0)
------------------------------------------------------------------------
(182.0,727.0)
------------------------------------------------------------------------
(162,727)[(0,0)\[r\][ 15000]{}]{} (957.0,727.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
(162,926)[(0,0)\[r\][ 20000]{}]{} (957.0,926.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182,90)[(0,0)[ 0]{}]{} (182.0,906.0)
------------------------------------------------------------------------
(263.0,131.0)
------------------------------------------------------------------------
(263,90)[(0,0)[ 5]{}]{} (263.0,906.0)
------------------------------------------------------------------------
(344.0,131.0)
------------------------------------------------------------------------
(344,90)[(0,0)[ 10]{}]{} (344.0,906.0)
------------------------------------------------------------------------
(425.0,131.0)
------------------------------------------------------------------------
(425,90)[(0,0)[ 15]{}]{} (425.0,906.0)
------------------------------------------------------------------------
(506.0,131.0)
------------------------------------------------------------------------
(506,90)[(0,0)[ 20]{}]{} (506.0,906.0)
------------------------------------------------------------------------
(588.0,131.0)
------------------------------------------------------------------------
(588,90)[(0,0)[ 25]{}]{} (588.0,906.0)
------------------------------------------------------------------------
(669.0,131.0)
------------------------------------------------------------------------
(669,90)[(0,0)[ 30]{}]{} (669.0,906.0)
------------------------------------------------------------------------
(750.0,131.0)
------------------------------------------------------------------------
(750,90)[(0,0)[ 35]{}]{} (750.0,906.0)
------------------------------------------------------------------------
(831.0,131.0)
------------------------------------------------------------------------
(831,90)[(0,0)[ 40]{}]{} (831.0,906.0)
------------------------------------------------------------------------
(912.0,131.0)
------------------------------------------------------------------------
(912,90)[(0,0)[ 45]{}]{} (912.0,906.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(977.0,131.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
(579,29)[(0,0)[$t$]{}]{} (579,988)[(0,0)[density: $\rho=0.1$]{}]{} (424,827)[(0,0)\[r\][walk]{}]{} (198,131)[(0,0)[$+$]{}]{} (231,132)[(0,0)[$+$]{}]{} (263,134)[(0,0)[$+$]{}]{} (296,138)[(0,0)[$+$]{}]{} (328,145)[(0,0)[$+$]{}]{} (360,157)[(0,0)[$+$]{}]{} (393,167)[(0,0)[$+$]{}]{} (425,185)[(0,0)[$+$]{}]{} (458,208)[(0,0)[$+$]{}]{} (490,240)[(0,0)[$+$]{}]{} (523,276)[(0,0)[$+$]{}]{} (555,316)[(0,0)[$+$]{}]{} (588,350)[(0,0)[$+$]{}]{} (620,396)[(0,0)[$+$]{}]{} (653,436)[(0,0)[$+$]{}]{} (685,480)[(0,0)[$+$]{}]{} (717,533)[(0,0)[$+$]{}]{} (750,567)[(0,0)[$+$]{}]{} (782,622)[(0,0)[$+$]{}]{} (815,658)[(0,0)[$+$]{}]{} (847,704)[(0,0)[$+$]{}]{} (880,747)[(0,0)[$+$]{}]{} (912,790)[(0,0)[$+$]{}]{} (945,838)[(0,0)[$+$]{}]{} (977,893)[(0,0)[$+$]{}]{} (494,827)[(0,0)[$+$]{}]{} (424,786)[(0,0)\[r\][$x^{1.8}$]{}]{} (444.0,786.0)
------------------------------------------------------------------------
(190,131) (190,130.67)
------------------------------------------------------------------------
(190.00,130.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(206,131.67)
------------------------------------------------------------------------
(206.00,131.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(214,133.17)
------------------------------------------------------------------------
(214.00,132.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(222,134.67)
------------------------------------------------------------------------
(222.00,134.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(230,135.67)
------------------------------------------------------------------------
(230.00,135.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(238,137.17)
------------------------------------------------------------------------
(238.00,136.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(246,139.17)
------------------------------------------------------------------------
(246.00,138.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(254,141.17)
------------------------------------------------------------------------
(254.00,140.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(262.00,143.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(262.00,142.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(270,146.17)
------------------------------------------------------------------------
(270.00,145.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(278.00,148.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(278.00,147.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(286.00,151.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(286.00,150.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(294.00,154.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(294.00,153.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(302.00,157.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(302.00,156.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(310.00,160.61)(1.802,0.447)[3]{}
------------------------------------------------------------------------
(310.00,159.17)(6.302,3.000)[2]{}
------------------------------------------------------------------------
(319.00,163.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(319.00,162.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(327.00,166.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(327.00,165.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(335.00,170.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(335.00,169.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(343.00,174.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(343.00,173.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(351.00,178.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(351.00,177.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(359.00,182.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(359.00,181.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(367.00,186.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(367.00,185.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(375.00,190.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(375.00,189.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(383.00,195.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(383.00,194.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(391.00,200.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(391.00,199.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(399.00,204.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(399.00,203.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(407.00,209.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(407.00,208.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(415.00,215.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(415.00,214.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(423.00,220.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(423.00,219.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(431.00,225.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(431.00,224.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(439.00,231.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(439.00,230.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(447.00,236.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(447.00,235.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(455.00,242.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(455.00,241.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(463.00,248.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(463.00,247.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(471.00,254.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(471.00,253.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(479.00,261.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(479.00,260.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(487.00,267.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(487.00,266.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(495.00,273.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(495.00,272.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(503.00,280.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(503.00,279.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(511.00,287.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(511.00,286.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(519.00,294.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(519.00,293.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(527.00,301.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(527.00,300.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(535.00,308.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(535.00,307.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(543.00,315.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(543.00,314.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(551.00,323.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(551.00,322.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(559.00,330.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(559.00,329.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(567.00,338.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(567.00,337.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(575.00,346.59)(0.560,0.488)[13]{}
------------------------------------------------------------------------
(575.00,345.17)(7.858,8.000)[2]{}
------------------------------------------------------------------------
(584.00,354.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(584.00,353.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(592.00,362.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(592.00,361.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(600.00,370.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(600.00,369.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(608.59,378.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(607.17,378.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(616.00,387.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(616.00,386.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(624.59,395.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(623.17,395.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(632.59,404.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(631.17,404.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(640.59,413.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(639.17,413.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(648.59,422.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(647.17,422.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(656.59,431.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(655.17,431.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(664.59,440.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(663.17,440.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(672.59,450.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(671.17,450.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(680.59,459.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(679.17,459.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(688.59,469.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(687.17,469.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(696.59,478.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(695.17,478.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(704.59,488.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(703.17,488.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(712.59,498.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(711.17,498.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(720.59,508.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(719.17,508.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(728.59,518.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(727.17,518.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(736.59,529.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(735.17,529.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(744.59,539.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(743.17,539.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(752.59,550.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(751.17,550.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(760.59,560.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(759.17,560.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(768.59,571.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(767.17,571.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(776.59,582.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(775.17,582.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(784.59,593.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(783.17,593.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(792.59,604.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(791.17,604.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(800.59,616.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(799.17,616.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(808.59,627.00)(0.488,0.692)[13]{}
------------------------------------------------------------------------
(807.17,627.00)(8.000,9.651)[2]{}
------------------------------------------------------------------------
(816.59,638.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(815.17,638.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(824.59,650.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(823.17,650.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(832.59,662.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(831.17,662.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(840.59,674.00)(0.489,0.669)[15]{}
------------------------------------------------------------------------
(839.17,674.00)(9.000,10.685)[2]{}
------------------------------------------------------------------------
(849.59,686.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(848.17,686.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(857.59,698.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(856.17,698.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(865.59,710.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(864.17,710.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(873.59,722.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(872.17,722.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(881.59,735.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(880.17,735.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(889.59,747.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(888.17,747.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(897.59,760.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(896.17,760.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(905.59,773.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(904.17,773.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(913.59,785.00)(0.488,0.890)[13]{}
------------------------------------------------------------------------
(912.17,785.00)(8.000,12.340)[2]{}
------------------------------------------------------------------------
(921.59,799.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(920.17,799.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(929.59,812.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(928.17,812.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(937.59,825.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(936.17,825.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(945.59,838.00)(0.488,0.890)[13]{}
------------------------------------------------------------------------
(944.17,838.00)(8.000,12.340)[2]{}
------------------------------------------------------------------------
(953.59,852.00)(0.488,0.824)[13]{}
------------------------------------------------------------------------
(952.17,852.00)(8.000,11.443)[2]{}
------------------------------------------------------------------------
(961.59,865.00)(0.488,0.890)[13]{}
------------------------------------------------------------------------
(960.17,865.00)(8.000,12.340)[2]{}
------------------------------------------------------------------------
(969.59,879.00)(0.488,0.890)[13]{}
------------------------------------------------------------------------
(968.17,879.00)(8.000,12.340)[2]{}
------------------------------------------------------------------------
(198.0,132.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(977.0,131.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
\
(1050,1050)(0,0) (182.0,131.0)
------------------------------------------------------------------------
(162,131)[(0,0)\[r\][ 0]{}]{} (957.0,131.0)
------------------------------------------------------------------------
(182.0,219.0)
------------------------------------------------------------------------
(162,219)[(0,0)\[r\][ 5000]{}]{} (957.0,219.0)
------------------------------------------------------------------------
(182.0,308.0)
------------------------------------------------------------------------
(162,308)[(0,0)\[r\][ 10000]{}]{} (957.0,308.0)
------------------------------------------------------------------------
(182.0,396.0)
------------------------------------------------------------------------
(162,396)[(0,0)\[r\][ 15000]{}]{} (957.0,396.0)
------------------------------------------------------------------------
(182.0,484.0)
------------------------------------------------------------------------
(162,484)[(0,0)\[r\][ 20000]{}]{} (957.0,484.0)
------------------------------------------------------------------------
(182.0,573.0)
------------------------------------------------------------------------
(162,573)[(0,0)\[r\][ 25000]{}]{} (957.0,573.0)
------------------------------------------------------------------------
(182.0,661.0)
------------------------------------------------------------------------
(162,661)[(0,0)\[r\][ 30000]{}]{} (957.0,661.0)
------------------------------------------------------------------------
(182.0,749.0)
------------------------------------------------------------------------
(162,749)[(0,0)\[r\][ 35000]{}]{} (957.0,749.0)
------------------------------------------------------------------------
(182.0,838.0)
------------------------------------------------------------------------
(162,838)[(0,0)\[r\][ 40000]{}]{} (957.0,838.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
(162,926)[(0,0)\[r\][ 45000]{}]{} (957.0,926.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182,90)[(0,0)[ 0]{}]{} (182.0,906.0)
------------------------------------------------------------------------
(263.0,131.0)
------------------------------------------------------------------------
(263,90)[(0,0)[ 5]{}]{} (263.0,906.0)
------------------------------------------------------------------------
(344.0,131.0)
------------------------------------------------------------------------
(344,90)[(0,0)[ 10]{}]{} (344.0,906.0)
------------------------------------------------------------------------
(425.0,131.0)
------------------------------------------------------------------------
(425,90)[(0,0)[ 15]{}]{} (425.0,906.0)
------------------------------------------------------------------------
(506.0,131.0)
------------------------------------------------------------------------
(506,90)[(0,0)[ 20]{}]{} (506.0,906.0)
------------------------------------------------------------------------
(588.0,131.0)
------------------------------------------------------------------------
(588,90)[(0,0)[ 25]{}]{} (588.0,906.0)
------------------------------------------------------------------------
(669.0,131.0)
------------------------------------------------------------------------
(669,90)[(0,0)[ 30]{}]{} (669.0,906.0)
------------------------------------------------------------------------
(750.0,131.0)
------------------------------------------------------------------------
(750,90)[(0,0)[ 35]{}]{} (750.0,906.0)
------------------------------------------------------------------------
(831.0,131.0)
------------------------------------------------------------------------
(831,90)[(0,0)[ 40]{}]{} (831.0,906.0)
------------------------------------------------------------------------
(912.0,131.0)
------------------------------------------------------------------------
(912,90)[(0,0)[ 45]{}]{} (912.0,906.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(977.0,131.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
(579,29)[(0,0)[$t$]{}]{} (579,988)[(0,0)[density: $\rho=0.5$]{}]{} (424,729)[(0,0)\[r\][walk]{}]{} (198,135)[(0,0)[$+$]{}]{} (231,159)[(0,0)[$+$]{}]{} (263,191)[(0,0)[$+$]{}]{} (296,219)[(0,0)[$+$]{}]{} (328,253)[(0,0)[$+$]{}]{} (360,289)[(0,0)[$+$]{}]{} (393,313)[(0,0)[$+$]{}]{} (425,347)[(0,0)[$+$]{}]{} (458,378)[(0,0)[$+$]{}]{} (490,412)[(0,0)[$+$]{}]{} (523,442)[(0,0)[$+$]{}]{} (555,472)[(0,0)[$+$]{}]{} (588,502)[(0,0)[$+$]{}]{} (620,530)[(0,0)[$+$]{}]{} (653,562)[(0,0)[$+$]{}]{} (685,599)[(0,0)[$+$]{}]{} (717,621)[(0,0)[$+$]{}]{} (750,661)[(0,0)[$+$]{}]{} (782,696)[(0,0)[$+$]{}]{} (815,731)[(0,0)[$+$]{}]{} (847,770)[(0,0)[$+$]{}]{} (880,795)[(0,0)[$+$]{}]{} (912,831)[(0,0)[$+$]{}]{} (945,864)[(0,0)[$+$]{}]{} (977,908)[(0,0)[$+$]{}]{} (494,729)[(0,0)[$+$]{}]{} (424,688)[(0,0)\[r\][$x^{1.1}$]{}]{} (444.0,688.0)
------------------------------------------------------------------------
(190,136) (190.00,136.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(190.00,135.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(198.00,142.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(198.00,141.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(206.00,148.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(206.00,147.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(214.00,154.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(214.00,153.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(222.00,160.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(222.00,159.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(230.00,167.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(230.00,166.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(238.00,173.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(238.00,172.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(246.00,180.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(246.00,179.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(254.00,187.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(254.00,186.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(262.00,193.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(262.00,192.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(270.00,200.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(270.00,199.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(278.00,207.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(278.00,206.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(286.00,214.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(286.00,213.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(294.00,221.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(294.00,220.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(302.00,228.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(302.00,227.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(310.00,236.59)(0.645,0.485)[11]{}
------------------------------------------------------------------------
(310.00,235.17)(7.725,7.000)[2]{}
------------------------------------------------------------------------
(319.00,243.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(319.00,242.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(327.00,250.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(327.00,249.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(335.00,257.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(335.00,256.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(343.00,265.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(343.00,264.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(351.00,272.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(351.00,271.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(359.00,280.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(359.00,279.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(367.00,287.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(367.00,286.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(375.00,294.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(375.00,293.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(383.00,302.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(383.00,301.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(391.00,309.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(391.00,308.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(399.00,317.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(399.00,316.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(407.00,325.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(407.00,324.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(415.00,332.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(415.00,331.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(423.00,340.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(423.00,339.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(431.00,348.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(431.00,347.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(439.00,355.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(439.00,354.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(447.00,363.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(447.00,362.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(455.00,371.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(455.00,370.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(463.00,379.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(463.00,378.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(471.00,386.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(471.00,385.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(479.00,394.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(479.00,393.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(487.00,402.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(487.00,401.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(495.00,410.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(495.00,409.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(503.00,418.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(503.00,417.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(511.00,426.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(511.00,425.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(519.00,434.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(519.00,433.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(527.00,441.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(527.00,440.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(535.00,449.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(535.00,448.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(543.00,457.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(543.00,456.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(551.00,465.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(551.00,464.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(559.00,473.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(559.00,472.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(567.00,481.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(567.00,480.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(575.00,489.59)(0.560,0.488)[13]{}
------------------------------------------------------------------------
(575.00,488.17)(7.858,8.000)[2]{}
------------------------------------------------------------------------
(584.59,497.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(583.17,497.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(592.00,506.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(592.00,505.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(600.00,514.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(600.00,513.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(608.00,522.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(608.00,521.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(616.00,530.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(616.00,529.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(624.00,538.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(624.00,537.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(632.00,546.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(632.00,545.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(640.00,554.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(640.00,553.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(648.59,562.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(647.17,562.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(656.00,571.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(656.00,570.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(664.00,579.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(664.00,578.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(672.00,587.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(672.00,586.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(680.59,595.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(679.17,595.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(688.00,604.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(688.00,603.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(696.00,612.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(696.00,611.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(704.00,620.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(704.00,619.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(712.59,628.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(711.17,628.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(720.00,637.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(720.00,636.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(728.00,645.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(728.00,644.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(736.59,653.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(735.17,653.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(744.00,662.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(744.00,661.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(752.00,670.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(752.00,669.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(760.59,678.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(759.17,678.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(768.00,687.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(768.00,686.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(776.00,695.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(776.00,694.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(784.59,703.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(783.17,703.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(792.00,712.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(792.00,711.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(800.59,720.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(799.17,720.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(808.00,729.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(808.00,728.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(816.59,737.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(815.17,737.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(824.00,746.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(824.00,745.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(832.00,754.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(832.00,753.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(840.00,762.59)(0.495,0.489)[15]{}
------------------------------------------------------------------------
(840.00,761.17)(7.962,9.000)[2]{}
------------------------------------------------------------------------
(849.00,771.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(849.00,770.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(857.59,779.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(856.17,779.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(865.00,788.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(865.00,787.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(873.59,796.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(872.17,796.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(881.00,805.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(881.00,804.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(889.59,813.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(888.17,813.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(897.59,822.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(896.17,822.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(905.00,831.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(905.00,830.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(913.59,839.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(912.17,839.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(921.00,848.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(921.00,847.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(929.59,856.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(928.17,856.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(937.00,865.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(937.00,864.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(945.59,873.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(944.17,873.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(953.59,882.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(952.17,882.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(961.00,891.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(961.00,890.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(969.59,899.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(968.17,899.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(977.0,131.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
(1050,1050)(0,0) (182.0,131.0)
------------------------------------------------------------------------
(162,131)[(0,0)\[r\][ 0]{}]{} (957.0,131.0)
------------------------------------------------------------------------
(182.0,212.0)
------------------------------------------------------------------------
(162,212)[(0,0)\[r\][ 5000]{}]{} (957.0,212.0)
------------------------------------------------------------------------
(182.0,293.0)
------------------------------------------------------------------------
(162,293)[(0,0)\[r\][ 10000]{}]{} (957.0,293.0)
------------------------------------------------------------------------
(182.0,374.0)
------------------------------------------------------------------------
(162,374)[(0,0)\[r\][ 15000]{}]{} (957.0,374.0)
------------------------------------------------------------------------
(182.0,455.0)
------------------------------------------------------------------------
(162,455)[(0,0)\[r\][ 20000]{}]{} (957.0,455.0)
------------------------------------------------------------------------
(182.0,537.0)
------------------------------------------------------------------------
(162,537)[(0,0)\[r\][ 25000]{}]{} (957.0,537.0)
------------------------------------------------------------------------
(182.0,618.0)
------------------------------------------------------------------------
(162,618)[(0,0)\[r\][ 30000]{}]{} (957.0,618.0)
------------------------------------------------------------------------
(182.0,699.0)
------------------------------------------------------------------------
(162,699)[(0,0)\[r\][ 35000]{}]{} (957.0,699.0)
------------------------------------------------------------------------
(182.0,780.0)
------------------------------------------------------------------------
(162,780)[(0,0)\[r\][ 40000]{}]{} (957.0,780.0)
------------------------------------------------------------------------
(182.0,861.0)
------------------------------------------------------------------------
(162,861)[(0,0)\[r\][ 45000]{}]{} (957.0,861.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182,90)[(0,0)[ 0]{}]{} (182.0,906.0)
------------------------------------------------------------------------
(263.0,131.0)
------------------------------------------------------------------------
(263,90)[(0,0)[ 5]{}]{} (263.0,906.0)
------------------------------------------------------------------------
(344.0,131.0)
------------------------------------------------------------------------
(344,90)[(0,0)[ 10]{}]{} (344.0,906.0)
------------------------------------------------------------------------
(425.0,131.0)
------------------------------------------------------------------------
(425,90)[(0,0)[ 15]{}]{} (425.0,906.0)
------------------------------------------------------------------------
(506.0,131.0)
------------------------------------------------------------------------
(506,90)[(0,0)[ 20]{}]{} (506.0,906.0)
------------------------------------------------------------------------
(588.0,131.0)
------------------------------------------------------------------------
(588,90)[(0,0)[ 25]{}]{} (588.0,906.0)
------------------------------------------------------------------------
(669.0,131.0)
------------------------------------------------------------------------
(669,90)[(0,0)[ 30]{}]{} (669.0,906.0)
------------------------------------------------------------------------
(750.0,131.0)
------------------------------------------------------------------------
(750,90)[(0,0)[ 35]{}]{} (750.0,906.0)
------------------------------------------------------------------------
(831.0,131.0)
------------------------------------------------------------------------
(831,90)[(0,0)[ 40]{}]{} (831.0,906.0)
------------------------------------------------------------------------
(912.0,131.0)
------------------------------------------------------------------------
(912,90)[(0,0)[ 45]{}]{} (912.0,906.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(977.0,131.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
(579,29)[(0,0)[$t$]{}]{} (579,988)[(0,0)[density: $\rho=0.95$]{}]{} (444,679)[(0,0)\[r\][walk]{}]{} (198,149)[(0,0)[$+$]{}]{} (231,186)[(0,0)[$+$]{}]{} (263,224)[(0,0)[$+$]{}]{} (296,261)[(0,0)[$+$]{}]{} (328,290)[(0,0)[$+$]{}]{} (360,326)[(0,0)[$+$]{}]{} (393,368)[(0,0)[$+$]{}]{} (425,405)[(0,0)[$+$]{}]{} (458,427)[(0,0)[$+$]{}]{} (490,458)[(0,0)[$+$]{}]{} (523,490)[(0,0)[$+$]{}]{} (555,517)[(0,0)[$+$]{}]{} (588,537)[(0,0)[$+$]{}]{} (620,564)[(0,0)[$+$]{}]{} (653,594)[(0,0)[$+$]{}]{} (685,627)[(0,0)[$+$]{}]{} (717,662)[(0,0)[$+$]{}]{} (750,696)[(0,0)[$+$]{}]{} (782,735)[(0,0)[$+$]{}]{} (815,766)[(0,0)[$+$]{}]{} (847,782)[(0,0)[$+$]{}]{} (880,813)[(0,0)[$+$]{}]{} (912,844)[(0,0)[$+$]{}]{} (945,888)[(0,0)[$+$]{}]{} (977,917)[(0,0)[$+$]{}]{} (514,679)[(0,0)[$+$]{}]{} (444,638)[(0,0)\[r\][$x^{0.95}$]{}]{} (464.0,638.0)
------------------------------------------------------------------------
(190,141) (190.59,141.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(189.17,141.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(198.59,150.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(197.17,150.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(206.59,159.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(205.17,159.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(214.59,168.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(213.17,168.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(222.59,177.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(221.17,177.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(230.00,186.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(230.00,185.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(238.59,194.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(237.17,194.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(246.59,203.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(245.17,203.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(254.00,212.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(254.00,211.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(262.59,220.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(261.17,220.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(270.00,229.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(270.00,228.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(278.00,237.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(278.00,236.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(286.59,245.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(285.17,245.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(294.00,254.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(294.00,253.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(302.00,262.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(302.00,261.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(310.00,270.59)(0.560,0.488)[13]{}
------------------------------------------------------------------------
(310.00,269.17)(7.858,8.000)[2]{}
------------------------------------------------------------------------
(319.59,278.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(318.17,278.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(327.00,287.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(327.00,286.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(335.00,295.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(335.00,294.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(343.00,303.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(343.00,302.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(351.00,311.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(351.00,310.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(359.00,319.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(359.00,318.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(367.59,327.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(366.17,327.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(375.00,336.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(375.00,335.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(383.00,344.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(383.00,343.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(391.00,352.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(391.00,351.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(399.00,360.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(399.00,359.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(407.00,368.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(407.00,367.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(415.00,376.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(415.00,375.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(423.00,384.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(423.00,383.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(431.00,392.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(431.00,391.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(439.00,400.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(439.00,399.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(447.00,408.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(447.00,407.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(455.00,416.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(455.00,415.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(463.00,424.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(463.00,423.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(471.00,432.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(471.00,431.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(479.00,440.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(479.00,439.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(487.00,448.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(487.00,447.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(495.00,455.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(495.00,454.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(503.00,463.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(503.00,462.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(511.00,471.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(511.00,470.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(519.00,479.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(519.00,478.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(527.00,487.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(527.00,486.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(535.00,495.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(535.00,494.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(543.00,503.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(543.00,502.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(551.00,511.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(551.00,510.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(559.00,518.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(559.00,517.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(567.00,526.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(567.00,525.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(575.00,534.59)(0.560,0.488)[13]{}
------------------------------------------------------------------------
(575.00,533.17)(7.858,8.000)[2]{}
------------------------------------------------------------------------
(584.00,542.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(584.00,541.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(592.00,550.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(592.00,549.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(600.00,557.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(600.00,556.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(608.00,565.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(608.00,564.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(616.00,573.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(616.00,572.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(624.00,581.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(624.00,580.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(632.00,589.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(632.00,588.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(640.00,596.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(640.00,595.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(648.00,604.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(648.00,603.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(656.00,612.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(656.00,611.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(664.00,620.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(664.00,619.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(672.00,627.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(672.00,626.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(680.00,635.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(680.00,634.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(688.00,643.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(688.00,642.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(696.00,650.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(696.00,649.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(704.00,658.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(704.00,657.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(712.00,666.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(712.00,665.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(720.00,674.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(720.00,673.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(728.00,681.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(728.00,680.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(736.00,689.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(736.00,688.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(744.00,697.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(744.00,696.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(752.00,704.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(752.00,703.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(760.00,712.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(760.00,711.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(768.00,720.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(768.00,719.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(776.00,727.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(776.00,726.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(784.00,735.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(784.00,734.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(792.00,743.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(792.00,742.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(800.00,750.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(800.00,749.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(808.00,758.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(808.00,757.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(816.00,766.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(816.00,765.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(824.00,773.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(824.00,772.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(832.00,781.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(832.00,780.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(840.00,788.59)(0.560,0.488)[13]{}
------------------------------------------------------------------------
(840.00,787.17)(7.858,8.000)[2]{}
------------------------------------------------------------------------
(849.00,796.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(849.00,795.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(857.00,804.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(857.00,803.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(865.00,811.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(865.00,810.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(873.00,819.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(873.00,818.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(881.00,826.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(881.00,825.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(889.00,834.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(889.00,833.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(897.00,842.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(897.00,841.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(905.00,849.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(905.00,848.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(913.00,857.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(913.00,856.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(921.00,864.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(921.00,863.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(929.00,872.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(929.00,871.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(937.00,879.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(937.00,878.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(945.00,887.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(945.00,886.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(953.00,895.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(953.00,894.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(961.00,902.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(961.00,901.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(969.00,910.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(969.00,909.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(182.0,131.0)
------------------------------------------------------------------------
(977.0,131.0)
------------------------------------------------------------------------
(182.0,926.0)
------------------------------------------------------------------------
We note that in the case of high food density the random walk does not exhibit the characteristics of ARS (this observation will be made more formal later on), simply because the individual is always, or almost always, on food. As the density decreases, the behavior becomes more characteristic of ARS.
In order to study the characteristics of these walks, we consider the square average of the displacement from the initial position: $$\langle X^2 \rangle \dfeq \langle (x-x_0)^2 + (y-y_0)^2 \rangle.$$ Let the individual be fixed (viz. let it be the best performing individual). We execute, with this individual, $N$ random walks on the environment with the prescribed density, each of length $T$. Let $$w^i = [p_0^i,\ldots,p_t^i,\ldots,p_T^i]$$ be the $i$th random walk, where $p_t^i=(x_t^i-x_0,y_t^i-y_0)$. We are interested in knowing how far the individual has gone, on average, from its initial position, at time $t$. That is, we are interested in studying the function $$\langle X^2(t) \rangle = \frac{1}{N} \sum_{i=1}^N (p_t^i)^2$$ Figure \[variances\] shows the behavior of $\langle{X^2(t)}\rangle$ as a function of $t$ for various values of the density $\rho$, together with the best approximation of the form $\langle{X^2(t)}\rangle\sim{x^\nu}$, where the exponent $\nu$ depends on $\rho$:
$\rho$ 0.01 0.1 0.5 0.9
-------- ------ ----- ----- ------
$\nu$ 3.78 1.8 1.1 0.95
Figure \[varplot\] shows the behavior of the exponent $\nu$ as a function of $\rho$. As density approaches 1 or 0, that is, as the environment becomes more homogeneous (either with a lot of food or very little food), the exponent approaches 1, that is, we approach a situation in which $\langle{X^2(t)}\rangle\sim{x^1}$. This, as we shall see, is the behavior characteristic of Brownian motion, as well as of several other types of random walks. This should not come as a surprise: when food can be found everywhere, the individuals have no particular reason to modify their behavior, and will simply move to and fro in an haphazard manner: they will do a random walk. The same is true if there is very little food. The patches are so small and far apart that the in-patch behavior will last for a very short time and will not change significantly the characteristics of the walk, which will be a random walk from patch to patch looking for resources.
When the food is patchy, on the other hand, the random walk of the individual doesn’t follow the standard Brownian model. In the next section I shall consider the foraging walk more closely from a mathematical point of view. We shall begin, in the next section by considering the switching from the on-patch behavior to the off-patch, without taking onto account the spatial characteristics of the environment. Then, in the following section, we shall study random walks in search of a model that fits the characteristics of a forager on patchy resources. We shall see that such a model is given by the so-called *Levy walks*.
(1050,1050)(0,0) (162.0,169.0)
------------------------------------------------------------------------
(142,169)[(0,0)\[r\][ 1]{}]{} (937.0,169.0)
------------------------------------------------------------------------
(162.0,358.0)
------------------------------------------------------------------------
(142,358)[(0,0)\[r\][ 1.5]{}]{} (937.0,358.0)
------------------------------------------------------------------------
(162.0,547.0)
------------------------------------------------------------------------
(142,547)[(0,0)\[r\][ 2]{}]{} (937.0,547.0)
------------------------------------------------------------------------
(162.0,737.0)
------------------------------------------------------------------------
(142,737)[(0,0)\[r\][ 2.5]{}]{} (937.0,737.0)
------------------------------------------------------------------------
(162.0,926.0)
------------------------------------------------------------------------
(142,926)[(0,0)\[r\][ 3]{}]{} (937.0,926.0)
------------------------------------------------------------------------
(162.0,131.0)
------------------------------------------------------------------------
(162,90)[(0,0)[ 0]{}]{} (162.0,906.0)
------------------------------------------------------------------------
(261.0,131.0)
------------------------------------------------------------------------
(261,90)[(0,0)[ 0.1]{}]{} (261.0,906.0)
------------------------------------------------------------------------
(361.0,131.0)
------------------------------------------------------------------------
(361,90)[(0,0)[ 0.2]{}]{} (361.0,906.0)
------------------------------------------------------------------------
(460.0,131.0)
------------------------------------------------------------------------
(460,90)[(0,0)[ 0.3]{}]{} (460.0,906.0)
------------------------------------------------------------------------
(560.0,131.0)
------------------------------------------------------------------------
(560,90)[(0,0)[ 0.4]{}]{} (560.0,906.0)
------------------------------------------------------------------------
(659.0,131.0)
------------------------------------------------------------------------
(659,90)[(0,0)[ 0.5]{}]{} (659.0,906.0)
------------------------------------------------------------------------
(758.0,131.0)
------------------------------------------------------------------------
(758,90)[(0,0)[ 0.6]{}]{} (758.0,906.0)
------------------------------------------------------------------------
(858.0,131.0)
------------------------------------------------------------------------
(858,90)[(0,0)[ 0.7]{}]{} (858.0,906.0)
------------------------------------------------------------------------
(957.0,131.0)
------------------------------------------------------------------------
(957,90)[(0,0)[ 0.8]{}]{} (957.0,906.0)
------------------------------------------------------------------------
(162.0,131.0)
------------------------------------------------------------------------
(162.0,131.0)
------------------------------------------------------------------------
(957.0,131.0)
------------------------------------------------------------------------
(162.0,926.0)
------------------------------------------------------------------------
(559,29)[(0,0)[$\rho$]{}]{} (559,988)[(0,0)[Exponent $\nu$ of $\langle{X^2(t)}\rangle\sim{t^\nu}$]{}]{} (172,250)[(0,0)[$\blacksquare$]{}]{} (182,877)[(0,0)[$\blacksquare$]{}]{} (212,832)[(0,0)[$\blacksquare$]{}]{} (242,750)[(0,0)[$\blacksquare$]{}]{} (261,297)[(0,0)[$\blacksquare$]{}]{} (361,213)[(0,0)[$\blacksquare$]{}]{} (659,133)[(0,0)[$\blacksquare$]{}]{} (957,182)[(0,0)[$\blacksquare$]{}]{} (170.59,131.00)(0.488,39.654)[13]{}
------------------------------------------------------------------------
(169.17,131.00)(8.000,538.422)[2]{}
------------------------------------------------------------------------
(178.59,732.00)(0.488,9.475)[13]{}
------------------------------------------------------------------------
(177.17,732.00)(8.000,128.848)[2]{}
------------------------------------------------------------------------
(186.59,873.72)(0.488,-0.560)[13]{}
------------------------------------------------------------------------
(185.17,874.86)(8.000,-7.858)[2]{}
------------------------------------------------------------------------
(194.59,863.47)(0.488,-0.956)[13]{}
------------------------------------------------------------------------
(193.17,865.24)(8.000,-13.236)[2]{}
------------------------------------------------------------------------
(202.59,848.26)(0.488,-1.022)[13]{}
------------------------------------------------------------------------
(201.17,850.13)(8.000,-14.132)[2]{}
------------------------------------------------------------------------
(210.59,832.47)(0.488,-0.956)[13]{}
------------------------------------------------------------------------
(209.17,834.24)(8.000,-13.236)[2]{}
------------------------------------------------------------------------
(218.59,817.26)(0.488,-1.022)[13]{}
------------------------------------------------------------------------
(217.17,819.13)(8.000,-14.132)[2]{}
------------------------------------------------------------------------
(226.59,800.23)(0.488,-1.352)[13]{}
------------------------------------------------------------------------
(225.17,802.61)(8.000,-18.613)[2]{}
------------------------------------------------------------------------
(234.59,776.53)(0.488,-2.211)[13]{}
------------------------------------------------------------------------
(233.17,780.26)(8.000,-30.264)[2]{}
------------------------------------------------------------------------
(242.59,709.02)(0.485,-13.002)[11]{}
------------------------------------------------------------------------
(241.17,729.51)(7.000,-150.511)[2]{}
------------------------------------------------------------------------
(249.59,530.22)(0.488,-15.352)[13]{}
------------------------------------------------------------------------
(248.17,554.61)(8.000,-208.612)[2]{}
------------------------------------------------------------------------
(257.59,333.75)(0.488,-3.730)[13]{}
------------------------------------------------------------------------
(256.17,339.88)(8.000,-50.877)[2]{}
------------------------------------------------------------------------
(265.59,285.68)(0.488,-0.890)[13]{}
------------------------------------------------------------------------
(264.17,287.34)(8.000,-12.340)[2]{}
------------------------------------------------------------------------
(273.59,272.09)(0.488,-0.758)[13]{}
------------------------------------------------------------------------
(272.17,273.55)(8.000,-10.547)[2]{}
------------------------------------------------------------------------
(281.59,260.51)(0.488,-0.626)[13]{}
------------------------------------------------------------------------
(280.17,261.75)(8.000,-8.755)[2]{}
------------------------------------------------------------------------
(289.00,251.93)(0.494,-0.488)[13]{}
------------------------------------------------------------------------
(289.00,252.17)(6.962,-8.000)[2]{}
------------------------------------------------------------------------
(297.00,243.93)(0.671,-0.482)[9]{}
------------------------------------------------------------------------
(297.00,244.17)(6.685,-6.000)[2]{}
------------------------------------------------------------------------
(305.00,237.93)(0.671,-0.482)[9]{}
------------------------------------------------------------------------
(305.00,238.17)(6.685,-6.000)[2]{}
------------------------------------------------------------------------
(313.00,231.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(313.00,232.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(321.00,227.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(321.00,228.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(329.00,223.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(329.00,224.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(337.00,220.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(337.00,221.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(345.00,217.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(345.00,218.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(353.00,214.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(353.00,215.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(361.00,211.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(361.00,212.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(369.00,207.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(369.00,208.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(377.00,204.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(377.00,205.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(385.00,200.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(385.00,201.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(393.00,197.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(393.00,198.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(400.00,194.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(400.00,195.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(408.00,190.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(408.00,191.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(416.00,187.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(416.00,188.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(424.00,184.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(424.00,185.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(432.00,181.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(432.00,182.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(440.00,178.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(440.00,179.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(448.00,175.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(448.00,176.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(456.00,172.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(456.00,173.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(464.00,169.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(464.00,170.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(472,166.17)
------------------------------------------------------------------------
(472.00,167.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(480.00,164.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(480.00,165.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(488,161.17)
------------------------------------------------------------------------
(488.00,162.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(496.00,159.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(496.00,160.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(504,156.17)
------------------------------------------------------------------------
(504.00,157.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(512,154.17)
------------------------------------------------------------------------
(512.00,155.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(520,152.17)
------------------------------------------------------------------------
(520.00,153.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(528,150.17)
------------------------------------------------------------------------
(528.00,151.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(536,148.17)
------------------------------------------------------------------------
(536.00,149.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(544,146.17)
------------------------------------------------------------------------
(544.00,147.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(552,144.17)
------------------------------------------------------------------------
(552.00,145.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(560,142.67)
------------------------------------------------------------------------
(560.00,143.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(567,141.17)
------------------------------------------------------------------------
(567.00,142.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(575,139.67)
------------------------------------------------------------------------
(575.00,140.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(583,138.67)
------------------------------------------------------------------------
(583.00,139.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(591,137.17)
------------------------------------------------------------------------
(591.00,138.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(599,135.67)
------------------------------------------------------------------------
(599.00,136.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(615,134.67)
------------------------------------------------------------------------
(615.00,135.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(623,133.67)
------------------------------------------------------------------------
(623.00,134.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(607.0,136.0)
------------------------------------------------------------------------
(647,132.67)
------------------------------------------------------------------------
(647.00,133.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(631.0,134.0)
------------------------------------------------------------------------
(671,132.67)
------------------------------------------------------------------------
(671.00,132.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(655.0,133.0)
------------------------------------------------------------------------
(703,133.67)
------------------------------------------------------------------------
(703.00,133.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(679.0,134.0)
------------------------------------------------------------------------
(726,134.67)
------------------------------------------------------------------------
(726.00,134.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(711.0,135.0)
------------------------------------------------------------------------
(742,135.67)
------------------------------------------------------------------------
(742.00,135.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(750,136.67)
------------------------------------------------------------------------
(750.00,136.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(758,137.67)
------------------------------------------------------------------------
(758.00,137.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(734.0,136.0)
------------------------------------------------------------------------
(774,138.67)
------------------------------------------------------------------------
(774.00,138.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(782,139.67)
------------------------------------------------------------------------
(782.00,139.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(790,140.67)
------------------------------------------------------------------------
(790.00,140.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(798,141.67)
------------------------------------------------------------------------
(798.00,141.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(806,143.17)
------------------------------------------------------------------------
(806.00,142.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(814,144.67)
------------------------------------------------------------------------
(814.00,144.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(822,145.67)
------------------------------------------------------------------------
(822.00,145.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(830,147.17)
------------------------------------------------------------------------
(830.00,146.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(838,148.67)
------------------------------------------------------------------------
(838.00,148.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(846,150.17)
------------------------------------------------------------------------
(846.00,149.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(854,152.17)
------------------------------------------------------------------------
(854.00,151.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(862,154.17)
------------------------------------------------------------------------
(862.00,153.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(870,156.17)
------------------------------------------------------------------------
(870.00,155.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(877,158.17)
------------------------------------------------------------------------
(877.00,157.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(885,160.17)
------------------------------------------------------------------------
(885.00,159.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(893,162.17)
------------------------------------------------------------------------
(893.00,161.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(901,164.17)
------------------------------------------------------------------------
(901.00,163.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(909.00,166.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(909.00,165.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(917,169.17)
------------------------------------------------------------------------
(917.00,168.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(925.00,171.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(925.00,170.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(933.00,174.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(933.00,173.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(941,177.17)
------------------------------------------------------------------------
(941.00,176.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(949.00,179.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(949.00,178.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(766.0,139.0)
------------------------------------------------------------------------
(162.0,131.0)
------------------------------------------------------------------------
(162.0,131.0)
------------------------------------------------------------------------
(957.0,131.0)
------------------------------------------------------------------------
(162.0,926.0)
------------------------------------------------------------------------
Should I stay of Should I go?
=============================
In the next section, I shall analyze the global characteristics of ARS exploration considering it as a random walk such as those that emerge from our genetic experiment. Before that, in this section I shall consider a more basic problem. Suppose that you are in a patch. For a while, you stay there happily eating, as there is plenty of food. After a while, the food begins to dwindle, the resources of the patch begin to be exhausted. When is it a good time to leave? You are confronted with two contrasting criteria. On the one hand, staying implies that you can continue eating without having to make a possibly long journey without food before you find another patch. In the long run, you want to spend more time in a patch and less between patches. On the other hand, the new patch that you will find has lots and lots of food so it might be a good idea for you to move now to greener pastures instead of half starving in this half barren patch. When is it a good time to leave? This is the question I want to answer in this section. I shall consider a very simple model: I analyze only the time that an individual spends on a patch ($t_p$) and the time that it spends between patches ($t_b$), and how to optimize them for maximum foraging efficiency. In this, I follow essentially the techniques developed for *optimal foraging theory* [@stephens:86].
Suppose that a forager searches for food for a certain (long) amount of time. It spends a total time $T_b$ looking for the next patch, and a total time $T_p$ staying on a patch and eating (all the symbols used in this section are shown in Table \[forasymbols\])
----------- ----------------------------------------------------------------------------------
$T_b$ Total time spent looking for a patch,
$T_p$ total time spent on a patch,
$t_b$ average time spent looking for a patch,
$t_p$ average time spent on a patch,
$G$ total resource gain,
$g$ average gain per patch,
$g(t)$ gain per patch as a function of the time spent on a patch,
$R$ rate of reward: average gain per time unit,
$\pi$ ${=g/t_p}$, profitability of a patch (resource per unit time when on the patch),
$\lambda$ average number of patches found per unit time,
$P$ number of types of patches,
$p_i$ probability of using a resource of type $i$.
----------- ----------------------------------------------------------------------------------
: Symbols used in this section.[]{data-label="forasymbols"}
If the total gain of the activity is $G$, then the rate of reward (reward per unit time), is $$\label{totalgain}
R = \frac{G}{T_b+T_p}$$ This equation is inconvenient as it depends on the total times $T_p$ and $T_b$ and on the total gain $G$ (if the time spent goes to infinity, both the numerator and the denominator go to infinity). One can derive a more convenient equation, independent on the actual foraging time, by considering the average on-patch time $t_p$, the average time between patches $t_b$ and the average gain per patch $g$. The rate at which patches are discovered, that is, the number of patches discovered per unit time, is $\lambda=1/t_b$. The total number of patches discovered during foraging is therefore $\lambda{T_b}$. The total gain and the total time spent on patches depend on this number, that is: $$\begin{aligned}
G &= \lambda T_b g \\
T_p &= \lambda T_b t_p
\end{aligned}$$ Introducing these values in (\[totalgain\]) we have $$R = \frac{\lambda T_b g}{T_b + \lambda T_b t_p} = \frac{\lambda g}{1 + \lambda t_p}$$ or, in terms of average times $$R = \frac{g/t_b}{1 + t_p/t_b} = \frac{g}{t_b+t_p}$$ This equation is known as the *Holling disk* equation [@holling:59][^2]. Define the profitability of a patch as $\pi=g/t_p$, that is, the gain per unit of time spent on the patch. With this definition we have: $$R = \frac{\pi}{{\displaystyle 1+\frac{t_b}{t_p}}}$$ When the patches become more and more dense, then $t_b\rightarrow{0}$, and $$\label{pilimit}
\lim_{t_b\rightarrow{0}} \frac{\pi}{{\displaystyle 1+\frac{t_b}{t_p}}} = \pi$$ This simple model can be extended in several ways. One very useful one is to consider that a patch has *diminishing returns*: as the forager spends time on a path, its resources become depleted, so it becomes harder to get rewards, and the profitability of the patch is reduced.
That is, the reward $g$ is a function of $t$, $g(t)$, that tells us how much reward one accumulates while foraging on a patch for a time equal to $t$. The profitability is also a function of time: $\pi(t)=g(t)/t$. Physical considerations place certain constraints on these functions. The gain is positive and cumulative (you never lose what you have gained), which implies $g(t)\ge{0}$ and $g'(t)\ge{0}$. The first inequality also entails $\pi(t)\ge{0}$. On the other hand, it is reasonable to assume that, as time goes by and the resources become depleted, it will take longer and longer to amass the same amount of reward; this entails $\pi'(t)\le{0}$. These two relations imply $\lim_{t\rightarrow\infty}\pi(t)=C\ge{0}$. The condition $\pi'(t)\le{0}$ imposes conditions on $g'(t)$. From $\pi(t)=g(t)/t$, we have $$\pi'(t) = \frac{1}{t^2}\Bigl[g'(t)t-g(t)\Bigr] \le 0$$ that is $$\label{gprime}
g'(t)\le \frac{g(t)}{t} = \pi(t)$$ I assume certain regularity conditions. In particular, that $\pi(t)$ decreases without [“[bumps]{}”]{}, that is, that $\pi'$ is monotonically increasing, which entails that $\pi''\ge{0}$. Similarly, I assume $g''(t)\le{0}$. Note that in the most common case the patch will become depleted, that is, $$\lim_{t\rightarrow\infty} g(t) = g_\infty > 0$$ but the analysis applies to the more general case in which $g(t)$ goes to infinity slower than a linear function.
Given the average between-patches time $t_b$, we are interested in finding the optimal time that the forager should spend on a patch ($t_p$) to maximize the reward $R$. The idea is that if you spend too little time on a patch, then you don’t take full advantage of its resources, and spend comparatively too much time without patches, in an area where you have no reward: your rate of gain will be reduced.
On the other hand, since the resources get depleted as we stay on a patch, if you spend too much time there you shall waste your time on a depleted patch that won’t yield too much, while it would be more convenient to invest some time ($t_b$) to find a new patch with better yield. Given the equality $$R(t) = \frac{g(t)}{t_b+t}$$ compute the derivative $$\frac{\partial R}{\partial t} = \frac{g'(t)(t_b+t)-g(t)}{(t_b+t)^2}$$ It is $\partial{R}/\partial{t}=0$ if $g'(t)(t_b+t)-g(t)=0$, that is, if $$g'(t)=\frac{g(t)}{t_b+t}=R$$ This result is known as the *Charnov’s Marginal Value Theorem* [@charnov:76].
This equation has a simple geometric interpretation. The average gain $R$ results in a straight line in a $t$-$g$ diagram (Figure \[patchdiag\]).
(17.746531,10.000000)(-6.760408,0) (-6.760408,0)[(1,0)[17.746531]{}]{} (0,0)[(0,1)[10.000000]{}]{} (0.000000,0.000000) (0.036620,0.072973) (0.073241,0.145414) (0.109861,0.217326) (0.146482,0.288714) (0.183102,0.359580) (0.219722,0.429929) (0.256343,0.499765) (0.292963,0.569091) (0.329584,0.637912) (0.366204,0.706230) (0.402825,0.774050) (0.439445,0.841375) (0.476065,0.908208) (0.512686,0.974554) (0.549306,1.040415) (0.585927,1.105796) (0.622547,1.170700) (0.659167,1.235131) (0.695788,1.299091) (0.732408,1.362584) (0.769029,1.425614) (0.805649,1.488184) (0.842269,1.550298) (0.878890,1.611958) (0.915510,1.673168) (0.952131,1.733932) (0.988751,1.794252) (1.025371,1.854132) (1.061992,1.913575) (1.098612,1.972584) (1.135233,2.031163) (1.171853,2.089314) (1.208474,2.147041) (1.245094,2.204347) (1.281714,2.261234) (1.318335,2.317706) (1.354955,2.373767) (1.391576,2.429418) (1.428196,2.484663) (1.464816,2.539505) (1.501437,2.593946) (1.538057,2.647991) (1.574678,2.701641) (1.611298,2.754899) (1.647918,2.807769) (1.684539,2.860253) (1.721159,2.912354) (1.757780,2.964075) (1.794400,3.015418) (1.831020,3.066387) (1.867641,3.116984) (1.904261,3.167212) (1.940882,3.217073) (1.977502,3.266570) (2.014123,3.315706) (2.050743,3.364484) (2.087363,3.412905) (2.123984,3.460973) (2.160604,3.508691) (2.197225,3.556060) (2.233845,3.603083) (2.270465,3.649764) (2.307086,3.696104) (2.343706,3.742105) (2.380327,3.787771) (2.416947,3.833104) (2.453567,3.878106) (2.490188,3.922779) (2.526808,3.967126) (2.563429,4.011150) (2.600049,4.054853) (2.636669,4.098237) (2.673290,4.141304) (2.709910,4.184056) (2.746531,4.226497) (2.783151,4.268628) (2.819772,4.310452) (2.856392,4.351971) (2.893012,4.393186) (2.929633,4.434101) (2.966253,4.474717) (3.002874,4.515037) (3.039494,4.555062) (3.076114,4.594796) (3.112735,4.634239) (3.149355,4.673395) (3.185976,4.712265) (3.222596,4.750851) (3.259216,4.789156) (3.295837,4.827181) (3.332457,4.864929) (3.369078,4.902401) (3.405698,4.939600) (3.442319,4.976528) (3.478939,5.013186) (3.515559,5.049576) (3.552180,5.085701) (3.588800,5.121562) (3.625421,5.157162) (3.662041,5.192501) (3.698661,5.227583) (3.735282,5.262409) (3.771902,5.296981) (3.808523,5.331300) (3.845143,5.365369) (3.881763,5.399190) (3.918384,5.432763) (3.955004,5.466092) (3.991625,5.499177) (4.028245,5.532021) (4.064865,5.564626) (4.101486,5.596992) (4.138106,5.629122) (4.174727,5.661018) (4.211347,5.692681) (4.247968,5.724113) (4.284588,5.755315) (4.321208,5.786290) (4.357829,5.817039) (4.394449,5.847564) (4.431070,5.877865) (4.467690,5.907946) (4.504310,5.937807) (4.540931,5.967450) (4.577551,5.996877) (4.614172,6.026089) (4.650792,6.055088) (4.687412,6.083875) (4.724033,6.112452) (4.760653,6.140821) (4.797274,6.168983) (4.833894,6.196939) (4.870514,6.224691) (4.907135,6.252241) (4.943755,6.279589) (4.980376,6.306738) (5.016996,6.333689) (5.053617,6.360444) (5.090237,6.387003) (5.126857,6.413368) (5.163478,6.439541) (5.200098,6.465523) (5.236719,6.491315) (5.273339,6.516919) (5.309959,6.542336) (5.346580,6.567568) (5.383200,6.592615) (5.419821,6.617480) (5.456441,6.642163) (5.493061,6.666667) (5.529682,6.690991) (5.566302,6.715138) (5.602923,6.739109) (5.639543,6.762905) (5.676163,6.786527) (5.712784,6.809976) (5.749404,6.833255) (5.786025,6.856364) (5.822645,6.879304) (5.859266,6.902077) (5.895886,6.924683) (5.932506,6.947125) (5.969127,6.969403) (6.005747,6.991518) (6.042368,7.013472) (6.078988,7.035265) (6.115608,7.056900) (6.152229,7.078377) (6.188849,7.099697) (6.225470,7.120861) (6.262090,7.141871) (6.298710,7.162728) (6.335331,7.183433) (6.371951,7.203986) (6.408572,7.224389) (6.445192,7.244644) (6.481813,7.264751) (6.518433,7.284711) (6.555053,7.304525) (6.591674,7.324195) (6.628294,7.343721) (6.664915,7.363105) (6.701535,7.382347) (6.738155,7.401449) (6.774776,7.420411) (6.811396,7.439235) (6.848017,7.457922) (6.884637,7.476473) (6.921257,7.494888) (6.957878,7.513168) (6.994498,7.531315) (7.031119,7.549330) (7.067739,7.567214) (7.104359,7.584966) (7.140980,7.602590) (7.177600,7.620084) (7.214221,7.637451) (7.250841,7.654692) (7.287462,7.671806) (7.324082,7.688796) (7.360702,7.705661) (7.397323,7.722404) (7.433943,7.739024) (7.470564,7.755523) (7.507184,7.771902) (7.543804,7.788161) (7.580425,7.804302) (7.617045,7.820324) (7.653666,7.836230) (7.690286,7.852020) (7.726906,7.867694) (7.763527,7.883255) (7.800147,7.898701) (7.836768,7.914035) (7.873388,7.929257) (7.910008,7.944368) (7.946629,7.959369) (7.983249,7.974260) (8.019870,7.989042) (8.056490,8.003717) (8.093111,8.018284) (8.129731,8.032746) (8.166351,8.047101) (8.202972,8.061352) (8.239592,8.075499) (8.276213,8.089543) (8.312833,8.103484) (8.349453,8.117324) (8.386074,8.131062) (8.422694,8.144700) (8.459315,8.158239) (8.495935,8.171679) (8.532555,8.185021) (8.569176,8.198265) (8.605796,8.211413) (8.642417,8.224465) (8.679037,8.237422) (8.715657,8.250284) (8.752278,8.263052) (8.788898,8.275727) (8.825519,8.288310) (8.862139,8.300800) (8.898760,8.313200) (8.935380,8.325509) (8.972000,8.337729) (9.008621,8.349859) (9.045241,8.361900) (9.081862,8.373854) (9.118482,8.385721) (9.155102,8.397500) (9.191723,8.409194) (9.228343,8.420803) (9.264964,8.432327) (9.301584,8.443767) (9.338204,8.455123) (9.374825,8.466397) (9.411445,8.477588) (9.448066,8.488697) (9.484686,8.499726) (9.521307,8.510674) (9.557927,8.521542) (9.594547,8.532331) (9.631168,8.543041) (9.667788,8.553673) (9.704409,8.564227) (9.741029,8.574704) (9.777649,8.585105) (9.814270,8.595430) (9.850890,8.605680) (9.887511,8.615855) (9.924131,8.625955) (9.960751,8.635982) (9.997372,8.645936) (10.033992,8.655817) (10.070613,8.665626) (10.107233,8.675363) (10.143853,8.685029) (10.180474,8.694625) (10.217094,8.704151) (10.253715,8.713607) (10.290335,8.722994) (10.326956,8.732313) (10.363576,8.741564) (10.400196,8.750747) (10.436817,8.759863) (10.473437,8.768913) (10.510058,8.777896) (10.546678,8.786815) (10.583298,8.795668) (10.619919,8.804456) (10.656539,8.813180) (10.693160,8.821841) (10.729780,8.830438) (10.766400,8.838973) (10.803021,8.847445) (10.839641,8.855856) (10.876262,8.864205) (10.912882,8.872493) (10.949502,8.880721) (-4.506939,0)[(3,2)[15.493061]{}]{} (5.493061,-1)(0,0.766667)[12]{}[(0,1)[0.383333]{}]{} (5.493061,-0.8)[(0,0)\[t\][$t_p$]{}]{} (-4.506939,-0.8)[(0,0)\[t\][$-t_b$]{}]{} (10.986123,8.444444)[(0,0)\[tl\][$g(t)$]{}]{} (10.986123,10.328708)[(0,0)\[tl\][$R$]{}]{} (-3.605551,0.600925)[(0,1)[1]{}]{} (-3.605551,1.600925)[(1,0)[1.500000]{}]{} (-3.805551,0.800925)[(0,0)\[br\][$R$]{}]{}
The patch gain is a curve that stays at zero for a time $t_b$ and then grows as $g(t)$; the optimal $t_p$ occurs when the slope of the curve $g(t)$ is equal to the average rate of gain. Note that there are two ways in which the environment can change so that $R$ increases. First, the profitability of the patch may increase, that is, the curve $g(t)$ can be pulled upward (Figure \[increase\_patch\].a). Second, the patches may become dense, reducing $t_b$ (Figure \[increase\_patch\].b). As $t_b\rightarrow{0}$ the rate $R$ approaches $\pi$ (the profitability of the patch), as per (\[pilimit\]).
[ccc]{}
(17.746531,17.000000)(-6.760408,-2) (-6.760408,0)[(1,0)[17.746531]{}]{} (0,0)[(0,1)[10.000000]{}]{} (0.000000,0.000000) (0.036620,0.072973) (0.073241,0.145414) (0.109861,0.217326) (0.146482,0.288714) (0.183102,0.359580) (0.219722,0.429929) (0.256343,0.499765) (0.292963,0.569091) (0.329584,0.637912) (0.366204,0.706230) (0.402825,0.774050) (0.439445,0.841375) (0.476065,0.908208) (0.512686,0.974554) (0.549306,1.040415) (0.585927,1.105796) (0.622547,1.170700) (0.659167,1.235131) (0.695788,1.299091) (0.732408,1.362584) (0.769029,1.425614) (0.805649,1.488184) (0.842269,1.550298) (0.878890,1.611958) (0.915510,1.673168) (0.952131,1.733932) (0.988751,1.794252) (1.025371,1.854132) (1.061992,1.913575) (1.098612,1.972584) (1.135233,2.031163) (1.171853,2.089314) (1.208474,2.147041) (1.245094,2.204347) (1.281714,2.261234) (1.318335,2.317706) (1.354955,2.373767) (1.391576,2.429418) (1.428196,2.484663) (1.464816,2.539505) (1.501437,2.593946) (1.538057,2.647991) (1.574678,2.701641) (1.611298,2.754899) (1.647918,2.807769) (1.684539,2.860253) (1.721159,2.912354) (1.757780,2.964075) (1.794400,3.015418) (1.831020,3.066387) (1.867641,3.116984) (1.904261,3.167212) (1.940882,3.217073) (1.977502,3.266570) (2.014123,3.315706) (2.050743,3.364484) (2.087363,3.412905) (2.123984,3.460973) (2.160604,3.508691) (2.197225,3.556060) (2.233845,3.603083) (2.270465,3.649764) (2.307086,3.696104) (2.343706,3.742105) (2.380327,3.787771) (2.416947,3.833104) (2.453567,3.878106) (2.490188,3.922779) (2.526808,3.967126) (2.563429,4.011150) (2.600049,4.054853) (2.636669,4.098237) (2.673290,4.141304) (2.709910,4.184056) (2.746531,4.226497) (2.783151,4.268628) (2.819772,4.310452) (2.856392,4.351971) (2.893012,4.393186) (2.929633,4.434101) (2.966253,4.474717) (3.002874,4.515037) (3.039494,4.555062) (3.076114,4.594796) (3.112735,4.634239) (3.149355,4.673395) (3.185976,4.712265) (3.222596,4.750851) (3.259216,4.789156) (3.295837,4.827181) (3.332457,4.864929) (3.369078,4.902401) (3.405698,4.939600) (3.442319,4.976528) (3.478939,5.013186) (3.515559,5.049576) (3.552180,5.085701) (3.588800,5.121562) (3.625421,5.157162) (3.662041,5.192501) (3.698661,5.227583) (3.735282,5.262409) (3.771902,5.296981) (3.808523,5.331300) (3.845143,5.365369) (3.881763,5.399190) (3.918384,5.432763) (3.955004,5.466092) (3.991625,5.499177) (4.028245,5.532021) (4.064865,5.564626) (4.101486,5.596992) (4.138106,5.629122) (4.174727,5.661018) (4.211347,5.692681) (4.247968,5.724113) (4.284588,5.755315) (4.321208,5.786290) (4.357829,5.817039) (4.394449,5.847564) (4.431070,5.877865) (4.467690,5.907946) (4.504310,5.937807) (4.540931,5.967450) (4.577551,5.996877) (4.614172,6.026089) (4.650792,6.055088) (4.687412,6.083875) (4.724033,6.112452) (4.760653,6.140821) (4.797274,6.168983) (4.833894,6.196939) (4.870514,6.224691) (4.907135,6.252241) (4.943755,6.279589) (4.980376,6.306738) (5.016996,6.333689) (5.053617,6.360444) (5.090237,6.387003) (5.126857,6.413368) (5.163478,6.439541) (5.200098,6.465523) (5.236719,6.491315) (5.273339,6.516919) (5.309959,6.542336) (5.346580,6.567568) (5.383200,6.592615) (5.419821,6.617480) (5.456441,6.642163) (5.493061,6.666667) (5.529682,6.690991) (5.566302,6.715138) (5.602923,6.739109) (5.639543,6.762905) (5.676163,6.786527) (5.712784,6.809976) (5.749404,6.833255) (5.786025,6.856364) (5.822645,6.879304) (5.859266,6.902077) (5.895886,6.924683) (5.932506,6.947125) (5.969127,6.969403) (6.005747,6.991518) (6.042368,7.013472) (6.078988,7.035265) (6.115608,7.056900) (6.152229,7.078377) (6.188849,7.099697) (6.225470,7.120861) (6.262090,7.141871) (6.298710,7.162728) (6.335331,7.183433) (6.371951,7.203986) (6.408572,7.224389) (6.445192,7.244644) (6.481813,7.264751) (6.518433,7.284711) (6.555053,7.304525) (6.591674,7.324195) (6.628294,7.343721) (6.664915,7.363105) (6.701535,7.382347) (6.738155,7.401449) (6.774776,7.420411) (6.811396,7.439235) (6.848017,7.457922) (6.884637,7.476473) (6.921257,7.494888) (6.957878,7.513168) (6.994498,7.531315) (7.031119,7.549330) (7.067739,7.567214) (7.104359,7.584966) (7.140980,7.602590) (7.177600,7.620084) (7.214221,7.637451) (7.250841,7.654692) (7.287462,7.671806) (7.324082,7.688796) (7.360702,7.705661) (7.397323,7.722404) (7.433943,7.739024) (7.470564,7.755523) (7.507184,7.771902) (7.543804,7.788161) (7.580425,7.804302) (7.617045,7.820324) (7.653666,7.836230) (7.690286,7.852020) (7.726906,7.867694) (7.763527,7.883255) (7.800147,7.898701) (7.836768,7.914035) (7.873388,7.929257) (7.910008,7.944368) (7.946629,7.959369) (7.983249,7.974260) (8.019870,7.989042) (8.056490,8.003717) (8.093111,8.018284) (8.129731,8.032746) (8.166351,8.047101) (8.202972,8.061352) (8.239592,8.075499) (8.276213,8.089543) (8.312833,8.103484) (8.349453,8.117324) (8.386074,8.131062) (8.422694,8.144700) (8.459315,8.158239) (8.495935,8.171679) (8.532555,8.185021) (8.569176,8.198265) (8.605796,8.211413) (8.642417,8.224465) (8.679037,8.237422) (8.715657,8.250284) (8.752278,8.263052) (8.788898,8.275727) (8.825519,8.288310) (8.862139,8.300800) (8.898760,8.313200) (8.935380,8.325509) (8.972000,8.337729) (9.008621,8.349859) (9.045241,8.361900) (9.081862,8.373854) (9.118482,8.385721) (9.155102,8.397500) (9.191723,8.409194) (9.228343,8.420803) (9.264964,8.432327) (9.301584,8.443767) (9.338204,8.455123) (9.374825,8.466397) (9.411445,8.477588) (9.448066,8.488697) (9.484686,8.499726) (9.521307,8.510674) (9.557927,8.521542) (9.594547,8.532331) (9.631168,8.543041) (9.667788,8.553673) (9.704409,8.564227) (9.741029,8.574704) (9.777649,8.585105) (9.814270,8.595430) (9.850890,8.605680) (9.887511,8.615855) (9.924131,8.625955) (9.960751,8.635982) (9.997372,8.645936) (10.033992,8.655817) (10.070613,8.665626) (10.107233,8.675363) (10.143853,8.685029) (10.180474,8.694625) (10.217094,8.704151) (10.253715,8.713607) (10.290335,8.722994) (10.326956,8.732313) (10.363576,8.741564) (10.400196,8.750747) (10.436817,8.759863) (10.473437,8.768913) (10.510058,8.777896) (10.546678,8.786815) (10.583298,8.795668) (10.619919,8.804456) (10.656539,8.813180) (10.693160,8.821841) (10.729780,8.830438) (10.766400,8.838973) (10.803021,8.847445) (10.839641,8.855856) (10.876262,8.864205) (10.912882,8.872493) (10.949502,8.880721) (-4.506939,0)[(3,2)[15.493061]{}]{} (5.493061,-0.8)[(0,0)\[t\][$t_w$]{}]{} (-4.506939,-0.8)[(0,0)\[t\][$-t_s$]{}]{} (10.986123,8.444444)[(0,0)\[tl\][$g(t)$]{}]{} (10.986123,10.328708)[(0,0)\[tl\][$R$]{}]{} (0.000000,0.000000) (0.036620,0.109460) (0.073241,0.218121) (0.109861,0.325989) (0.146482,0.433070) (0.183102,0.539370) (0.219722,0.644894) (0.256343,0.749648) (0.292963,0.853637) (0.329584,0.956868) (0.366204,1.059345) (0.402825,1.161075) (0.439445,1.262062) (0.476065,1.362312) (0.512686,1.461831) (0.549306,1.560623) (0.585927,1.658695) (0.622547,1.756050) (0.659167,1.852696) (0.695788,1.948636) (0.732408,2.043876) (0.769029,2.138421) (0.805649,2.232276) (0.842269,2.325447) (0.878890,2.417937) (0.915510,2.509752) (0.952131,2.600898) (0.988751,2.691378) (1.025371,2.781198) (1.061992,2.870363) (1.098612,2.958877) (1.135233,3.046745) (1.171853,3.133971) (1.208474,3.220562) (1.245094,3.306520) (1.281714,3.391851) (1.318335,3.476560) (1.354955,3.560650) (1.391576,3.644127) (1.428196,3.726994) (1.464816,3.809257) (1.501437,3.890919) (1.538057,3.971986) (1.574678,4.052461) (1.611298,4.132349) (1.647918,4.211654) (1.684539,4.290380) (1.721159,4.368531) (1.757780,4.446113) (1.794400,4.523128) (1.831020,4.599581) (1.867641,4.675476) (1.904261,4.750818) (1.940882,4.825609) (1.977502,4.899855) (2.014123,4.973559) (2.050743,5.046725) (2.087363,5.119358) (2.123984,5.191460) (2.160604,5.263036) (2.197225,5.334090) (2.233845,5.404625) (2.270465,5.474646) (2.307086,5.544155) (2.343706,5.613158) (2.380327,5.681657) (2.416947,5.749655) (2.453567,5.817158) (2.490188,5.884168) (2.526808,5.950690) (2.563429,6.016725) (2.600049,6.082279) (2.636669,6.147355) (2.673290,6.211955) (2.709910,6.276085) (2.746531,6.339746) (2.783151,6.402943) (2.819772,6.465678) (2.856392,6.527956) (2.893012,6.589779) (2.929633,6.651151) (2.966253,6.712076) (3.002874,6.772555) (3.039494,6.832594) (3.076114,6.892194) (3.112735,6.951359) (3.149355,7.010093) (3.185976,7.068398) (3.222596,7.126277) (3.259216,7.183734) (3.295837,7.240772) (3.332457,7.297394) (3.369078,7.353602) (3.405698,7.409400) (3.442319,7.464792) (3.478939,7.519778) (3.515559,7.574364) (3.552180,7.628551) (3.588800,7.682343) (3.625421,7.735742) (3.662041,7.788752) (3.698661,7.841375) (3.735282,7.893614) (3.771902,7.945471) (3.808523,7.996951) (3.845143,8.048054) (3.881763,8.098785) (3.918384,8.149145) (3.955004,8.199138) (3.991625,8.248766) (4.028245,8.298032) (4.064865,8.346939) (4.101486,8.395488) (4.138106,8.443683) (4.174727,8.491527) (4.211347,8.539021) (4.247968,8.586169) (4.284588,8.632973) (4.321208,8.679435) (4.357829,8.725559) (4.394449,8.771345) (4.431070,8.816798) (4.467690,8.861919) (4.504310,8.906710) (4.540931,8.951175) (4.577551,8.995315) (4.614172,9.039133) (4.650792,9.082632) (4.687412,9.125813) (4.724033,9.168679) (4.760653,9.211232) (4.797274,9.253474) (4.833894,9.295408) (4.870514,9.337037) (4.907135,9.378361) (4.943755,9.419384) (4.980376,9.460108) (5.016996,9.500534) (5.053617,9.540666) (5.090237,9.580504) (5.126857,9.620052) (5.163478,9.659311) (5.200098,9.698284) (5.236719,9.736972) (5.273339,9.775378) (5.309959,9.813504) (5.346580,9.851352) (5.383200,9.888923) (5.419821,9.926220) (5.456441,9.963245) (5.493061,10.000000) (5.529682,10.036487) (5.566302,10.072707) (5.602923,10.108663) (5.639543,10.144357) (5.676163,10.179790) (5.712784,10.214965) (5.749404,10.249883) (5.786025,10.284546) (5.822645,10.318956) (5.859266,10.353115) (5.895886,10.387025) (5.932506,10.420687) (5.969127,10.454104) (6.005747,10.487277) (6.042368,10.520208) (6.078988,10.552898) (6.115608,10.585350) (6.152229,10.617565) (6.188849,10.649545) (6.225470,10.681292) (6.262090,10.712807) (6.298710,10.744092) (6.335331,10.775149) (6.371951,10.805979) (6.408572,10.836584) (6.445192,10.866966) (6.481813,10.897126) (6.518433,10.927066) (6.555053,10.956788) (6.591674,10.986292) (6.628294,11.015582) (6.664915,11.044657) (6.701535,11.073521) (6.738155,11.102173) (6.774776,11.130617) (6.811396,11.158853) (6.848017,11.186883) (6.884637,11.214709) (6.921257,11.242331) (6.957878,11.269752) (6.994498,11.296973) (7.031119,11.323995) (7.067739,11.350820) (7.104359,11.377450) (7.140980,11.403885) (7.177600,11.430127) (7.214221,11.456177) (7.250841,11.482038) (7.287462,11.507709) (7.324082,11.533194) (7.360702,11.558492) (7.397323,11.583606) (7.433943,11.608536) (7.470564,11.633285) (7.507184,11.657853) (7.543804,11.682242) (7.580425,11.706453) (7.617045,11.730487) (7.653666,11.754345) (7.690286,11.778030) (7.726906,11.801542) (7.763527,11.824882) (7.800147,11.848052) (7.836768,11.871053) (7.873388,11.893886) (7.910008,11.916552) (7.946629,11.939053) (7.983249,11.961389) (8.019870,11.983563) (8.056490,12.005575) (8.093111,12.027426) (8.129731,12.049118) (8.166351,12.070652) (8.202972,12.092028) (8.239592,12.113249) (8.276213,12.134314) (8.312833,12.155226) (8.349453,12.175985) (8.386074,12.196593) (8.422694,12.217050) (8.459315,12.237359) (8.495935,12.257518) (8.532555,12.277531) (8.569176,12.297398) (8.605796,12.317120) (8.642417,12.336698) (8.679037,12.356133) (8.715657,12.375426) (8.752278,12.394578) (8.788898,12.413591) (8.825519,12.432465) (8.862139,12.451201) (8.898760,12.469800) (8.935380,12.488264) (8.972000,12.506593) (9.008621,12.524788) (9.045241,12.542850) (9.081862,12.560781) (9.118482,12.578581) (9.155102,12.596251) (9.191723,12.613792) (9.228343,12.631205) (9.264964,12.648490) (9.301584,12.665650) (9.338204,12.682685) (9.374825,12.699595) (9.411445,12.716382) (9.448066,12.733046) (9.484686,12.749589) (9.521307,12.766011) (9.557927,12.782313) (9.594547,12.798496) (9.631168,12.814561) (9.667788,12.830509) (9.704409,12.846340) (9.741029,12.862056) (9.777649,12.877658) (9.814270,12.893145) (9.850890,12.908520) (9.887511,12.923782) (9.924131,12.938933) (9.960751,12.953973) (9.997372,12.968903) (10.033992,12.983725) (10.070613,12.998438) (10.107233,13.013044) (10.143853,13.027544) (10.180474,13.041938) (10.217094,13.056226) (10.253715,13.070411) (10.290335,13.084491) (10.326956,13.098469) (10.363576,13.112346) (10.400196,13.126120) (10.436817,13.139795) (10.473437,13.153369) (10.510058,13.166845) (10.546678,13.180222) (10.583298,13.193501) (10.619919,13.206684) (10.656539,13.219770) (10.693160,13.232761) (10.729780,13.245657) (10.766400,13.258459) (10.803021,13.271168) (10.839641,13.283784) (10.876262,13.296308) (10.912882,13.308740) (10.949502,13.321082) (-4.506939,0)[(1,1)[15.493061]{}]{} (5.493061,-1)(0,1.100000)[12]{}[(0,1)[0.550000]{}]{} (10.986123,15.493061)[(0,0)\[tl\][$R'>R$]{}]{} (10.986123,12.666667)[(0,0)\[tl\][$g(t)$]{}]{}
(17.746531,17.000000)(-6.760408,-2) (-6.760408,0)[(1,0)[17.746531]{}]{} (0,0)[(0,1)[10.000000]{}]{} (0.000000,0.000000) (0.036620,0.072973) (0.073241,0.145414) (0.109861,0.217326) (0.146482,0.288714) (0.183102,0.359580) (0.219722,0.429929) (0.256343,0.499765) (0.292963,0.569091) (0.329584,0.637912) (0.366204,0.706230) (0.402825,0.774050) (0.439445,0.841375) (0.476065,0.908208) (0.512686,0.974554) (0.549306,1.040415) (0.585927,1.105796) (0.622547,1.170700) (0.659167,1.235131) (0.695788,1.299091) (0.732408,1.362584) (0.769029,1.425614) (0.805649,1.488184) (0.842269,1.550298) (0.878890,1.611958) (0.915510,1.673168) (0.952131,1.733932) (0.988751,1.794252) (1.025371,1.854132) (1.061992,1.913575) (1.098612,1.972584) (1.135233,2.031163) (1.171853,2.089314) (1.208474,2.147041) (1.245094,2.204347) (1.281714,2.261234) (1.318335,2.317706) (1.354955,2.373767) (1.391576,2.429418) (1.428196,2.484663) (1.464816,2.539505) (1.501437,2.593946) (1.538057,2.647991) (1.574678,2.701641) (1.611298,2.754899) (1.647918,2.807769) (1.684539,2.860253) (1.721159,2.912354) (1.757780,2.964075) (1.794400,3.015418) (1.831020,3.066387) (1.867641,3.116984) (1.904261,3.167212) (1.940882,3.217073) (1.977502,3.266570) (2.014123,3.315706) (2.050743,3.364484) (2.087363,3.412905) (2.123984,3.460973) (2.160604,3.508691) (2.197225,3.556060) (2.233845,3.603083) (2.270465,3.649764) (2.307086,3.696104) (2.343706,3.742105) (2.380327,3.787771) (2.416947,3.833104) (2.453567,3.878106) (2.490188,3.922779) (2.526808,3.967126) (2.563429,4.011150) (2.600049,4.054853) (2.636669,4.098237) (2.673290,4.141304) (2.709910,4.184056) (2.746531,4.226497) (2.783151,4.268628) (2.819772,4.310452) (2.856392,4.351971) (2.893012,4.393186) (2.929633,4.434101) (2.966253,4.474717) (3.002874,4.515037) (3.039494,4.555062) (3.076114,4.594796) (3.112735,4.634239) (3.149355,4.673395) (3.185976,4.712265) (3.222596,4.750851) (3.259216,4.789156) (3.295837,4.827181) (3.332457,4.864929) (3.369078,4.902401) (3.405698,4.939600) (3.442319,4.976528) (3.478939,5.013186) (3.515559,5.049576) (3.552180,5.085701) (3.588800,5.121562) (3.625421,5.157162) (3.662041,5.192501) (3.698661,5.227583) (3.735282,5.262409) (3.771902,5.296981) (3.808523,5.331300) (3.845143,5.365369) (3.881763,5.399190) (3.918384,5.432763) (3.955004,5.466092) (3.991625,5.499177) (4.028245,5.532021) (4.064865,5.564626) (4.101486,5.596992) (4.138106,5.629122) (4.174727,5.661018) (4.211347,5.692681) (4.247968,5.724113) (4.284588,5.755315) (4.321208,5.786290) (4.357829,5.817039) (4.394449,5.847564) (4.431070,5.877865) (4.467690,5.907946) (4.504310,5.937807) (4.540931,5.967450) (4.577551,5.996877) (4.614172,6.026089) (4.650792,6.055088) (4.687412,6.083875) (4.724033,6.112452) (4.760653,6.140821) (4.797274,6.168983) (4.833894,6.196939) (4.870514,6.224691) (4.907135,6.252241) (4.943755,6.279589) (4.980376,6.306738) (5.016996,6.333689) (5.053617,6.360444) (5.090237,6.387003) (5.126857,6.413368) (5.163478,6.439541) (5.200098,6.465523) (5.236719,6.491315) (5.273339,6.516919) (5.309959,6.542336) (5.346580,6.567568) (5.383200,6.592615) (5.419821,6.617480) (5.456441,6.642163) (5.493061,6.666667) (5.529682,6.690991) (5.566302,6.715138) (5.602923,6.739109) (5.639543,6.762905) (5.676163,6.786527) (5.712784,6.809976) (5.749404,6.833255) (5.786025,6.856364) (5.822645,6.879304) (5.859266,6.902077) (5.895886,6.924683) (5.932506,6.947125) (5.969127,6.969403) (6.005747,6.991518) (6.042368,7.013472) (6.078988,7.035265) (6.115608,7.056900) (6.152229,7.078377) (6.188849,7.099697) (6.225470,7.120861) (6.262090,7.141871) (6.298710,7.162728) (6.335331,7.183433) (6.371951,7.203986) (6.408572,7.224389) (6.445192,7.244644) (6.481813,7.264751) (6.518433,7.284711) (6.555053,7.304525) (6.591674,7.324195) (6.628294,7.343721) (6.664915,7.363105) (6.701535,7.382347) (6.738155,7.401449) (6.774776,7.420411) (6.811396,7.439235) (6.848017,7.457922) (6.884637,7.476473) (6.921257,7.494888) (6.957878,7.513168) (6.994498,7.531315) (7.031119,7.549330) (7.067739,7.567214) (7.104359,7.584966) (7.140980,7.602590) (7.177600,7.620084) (7.214221,7.637451) (7.250841,7.654692) (7.287462,7.671806) (7.324082,7.688796) (7.360702,7.705661) (7.397323,7.722404) (7.433943,7.739024) (7.470564,7.755523) (7.507184,7.771902) (7.543804,7.788161) (7.580425,7.804302) (7.617045,7.820324) (7.653666,7.836230) (7.690286,7.852020) (7.726906,7.867694) (7.763527,7.883255) (7.800147,7.898701) (7.836768,7.914035) (7.873388,7.929257) (7.910008,7.944368) (7.946629,7.959369) (7.983249,7.974260) (8.019870,7.989042) (8.056490,8.003717) (8.093111,8.018284) (8.129731,8.032746) (8.166351,8.047101) (8.202972,8.061352) (8.239592,8.075499) (8.276213,8.089543) (8.312833,8.103484) (8.349453,8.117324) (8.386074,8.131062) (8.422694,8.144700) (8.459315,8.158239) (8.495935,8.171679) (8.532555,8.185021) (8.569176,8.198265) (8.605796,8.211413) (8.642417,8.224465) (8.679037,8.237422) (8.715657,8.250284) (8.752278,8.263052) (8.788898,8.275727) (8.825519,8.288310) (8.862139,8.300800) (8.898760,8.313200) (8.935380,8.325509) (8.972000,8.337729) (9.008621,8.349859) (9.045241,8.361900) (9.081862,8.373854) (9.118482,8.385721) (9.155102,8.397500) (9.191723,8.409194) (9.228343,8.420803) (9.264964,8.432327) (9.301584,8.443767) (9.338204,8.455123) (9.374825,8.466397) (9.411445,8.477588) (9.448066,8.488697) (9.484686,8.499726) (9.521307,8.510674) (9.557927,8.521542) (9.594547,8.532331) (9.631168,8.543041) (9.667788,8.553673) (9.704409,8.564227) (9.741029,8.574704) (9.777649,8.585105) (9.814270,8.595430) (9.850890,8.605680) (9.887511,8.615855) (9.924131,8.625955) (9.960751,8.635982) (9.997372,8.645936) (10.033992,8.655817) (10.070613,8.665626) (10.107233,8.675363) (10.143853,8.685029) (10.180474,8.694625) (10.217094,8.704151) (10.253715,8.713607) (10.290335,8.722994) (10.326956,8.732313) (10.363576,8.741564) (10.400196,8.750747) (10.436817,8.759863) (10.473437,8.768913) (10.510058,8.777896) (10.546678,8.786815) (10.583298,8.795668) (10.619919,8.804456) (10.656539,8.813180) (10.693160,8.821841) (10.729780,8.830438) (10.766400,8.838973) (10.803021,8.847445) (10.839641,8.855856) (10.876262,8.864205) (10.912882,8.872493) (10.949502,8.880721) (-4.506939,0)[(3,2)[15.493061]{}]{} (5.493061,-1)(0,0.766667)[12]{}[(0,1)[0.383333]{}]{} (5.493061,-0.8)[(0,0)\[t\][$t_w$]{}]{} (-4.506939,-0.8)[(0,0)\[t\][$-t_s$]{}]{} (10.986123,8.444444)[(0,0)\[tl\][$g(t)$]{}]{} (10.986123,10.328708)[(0,0)\[tl\][$R$]{}]{} (-1.534264,0)[(1,1)[12.520387]{}]{} (3.465736,-1)(0,0.600000)[12]{}[(0,1)[0.300000]{}]{} (3.465736,-0.8)[(0,0)\[t\][$t_w'$]{}]{} (-1.534264,-0.8)[(0,0)\[t\][$-t_s'$]{}]{} (10.986123,12.520387)[(0,0)\[tl\][$R'>R$]{}]{}
\(a) & & (b)
Suppose we are looking for a low-priced hotel in Paris. We have several web sites available, and our strategy is to log in to one, start checking prices looking for the cheapest price for a while, then move to another one looking for a new price or, simply, stop looking and accept the lowest price we have found. The question is: how long should we stay on the site and keep looking before we move on?
When we are on a page, the important events are the prices that we look at so, for the sake of convenience, we take the time that it takes to move from one hotel to the next one on the same page as the time unit so that at time $t=n$ we have looked at $n+1$ prices (we look at the first price at $t=0$). The actual length depends of what we are looking for: if we look just for the best price, the interval is very small; if we look for price subject to certain constraint (hotel with a bar, with a sauna, etc.), it will take longer. In any case, we look at a new hotel (one the same page) per unit of time. In these units, let $t_b$ be the time that it takes to get set on a page (including typing the address, logging in, etc.)
We begin by determinig the expected value of the minimum price that we have observed if we have observed $n$ prices. In this simple example we shall favor simplicity over plausibility and we shall assume that the prices of teh hotels are uniformly distributed in the interval $[\mu-a,\mu+a]$.
To begin with, we shall answer an even simpler question: given $n$ observations $X=\{x_1,\ldots,x_n\}$ of $n$ random variables independent and uniformly distributed in $[-1,1]$, which is the expected value of $\min(X)$? The cumulative distribution and the density for each of the $x_i$ are $$\Phi(x) =
\begin{cases}
0 & x \le 1 \\
\frac{x+1}{2} & -1<x<1 \\
1 & x\ge 1
\end{cases}$$ and $$\phi(x) =
\begin{cases}
\frac{1}{2} & -1\le{x}\le{1} \\
0 & \mbox{otherwise}
\end{cases}$$ respectively. The probability density for the minimum is given by (\[mindense\]): $$\phi_{\min}(x) = n[1 - \Phi(x)]^{n-1}\phi(x) =
\begin{cases}
\frac{n}{2^n}[1-x]^{n-1} & (-1\le{x}\le{1}) \\
0 & \mbox{otherwise}
\end{cases}$$ and its expected value is $$\begin{aligned}
m(n) &= \int_{-1}^{1} u \phi_{\min}(u) du = \frac{n}{2^n} \int_{-1}^{1} u(1-u)^{n-1} du \\
&= \frac{n}{2^n} \left[ \int_{-1}^{1} (1-u)^{n-1}du - \int_{-1}^{1}(1-u)^ndu\right] \\
&= \frac{n}{2^n} \left[ - \int_{2}^{0} u^{n-1}du + \int_{2}^{0}u^n du\right] \\
&= \frac{n}{2^n} \left[ \frac{2^n}{n} - \frac{2^{n+1}}{n+1} \right] \\
&= \frac{1-n}{1+n}
\end{aligned}$$ Scaling and shifting one obtains, for variables distributed in $[\mu-a,\mu+a]$: $$m(n) = \mu - a \frac{1-n}{1+n}$$ Note that if we only observe one price, the expected value of the minimum is $\mu$. We we observe more prices, the expected value decreases, with $$\lim_{t\rightarrow\infty} m(t) = \mu-a$$ Let us consider that we start exploring at the time when we observe the first price, and that everything that comes before that is preparation that is included in the time $t_b$. At time $t$, we have looked at $t+1$ prices, and the expected value for the minimum price is $m(t+1)$. The gain that we have obtained is the difference between the first price that we saw (expected value equal to $\mu$) and the current one: $$\label{gdef}
g(t) = m(1) = m(t+1) = a \frac{t}{t+2}$$ Up to now, I have considered $t$ as a discrete variable (the number of prices observed); from now on we regard it as a continuous variable, so I can take the derivative: $$g'(t) = \frac{2a}{(t+2)^2}$$ I now apply Charnov’s Marginal value theorem: the optimal value to spend on the site is given by $$g'(t) = \frac{2a}{(t+2)^2} = \frac{at}{t+2} \frac{1}{t+t_b} = \frac{g(t)}{t+t_b}$$ which has solution $\tau_1=\sqrt{2t_b}$[^3]. The corresponding rate of reward is $$\label{t1bum}
R_1 = g'(\tau_1) = \frac{2a}{(\sqrt{2t_b}+2)^2} = \frac{a}{(\sqrt{t_b}+\sqrt{2})^2}.$$ Note that the time that we spend on a site grown sub-linearly with the time we spend looking for the site: if we spend twice as long looking for the web page, the time we should spend on the web page grows like $\sqrt{2}$, that is, we should stay about 40% longer.
The considerations of the previous example are valid for the first [“[patch]{}”]{}, that is, for the first site that we visit. Suppose now that we want to visit a second site (the time necessary to do the switch is assumed to be $t_b$). Now we already have a minimum, the one that we found in the first patch, namely $$m_1 \dfeq \mu - a\frac{\tau_1}{\tau_1+2} = \mu - a \frac{\sqrt{t_b}}{\sqrt{t_b}+\sqrt{2}} \dfeq \mu - a\alpha$$ with $$\alpha \dfeq \frac{\sqrt{t_b}}{\sqrt{t_b}+\sqrt{2}} < 1$$ While we explore the second site, as long as the minimum price that we find there is greater than $m_1$, our gain is zero. Assume that the second patch has the same average price as the first, but a larger spread, that is, in the second patch the prices are uniformly distributed in $[\mu-b,\mu+b]$, with $b>a$. Also, define $\gamma=b/a>1$ (I shall need it later).
The current minimum after we have explored the second site for a time $t$ is $$\tilde{m}(t) = \mu - b\frac{t}{t+2}$$ The expected minimum in the second site is the same as the optimal price in the first at a time $t_0$ such that $\tilde{m}(t_0)=m_1$, that is $$t_0 = \frac{2a\alpha}{b-a\alpha} = \frac{2\alpha}{\gamma-\alpha}$$ The gain in the second site is given by the savings we obtain over the previous minimum, that is $$g_2(t) =
\begin{cases}
0 & t<t_0 \\
b\frac{t}{t+2} - a\alpha & t \ge t_0
\end{cases}$$ and $$g_2^\prime(t) =
\begin{cases}
0 & t<t_0 \\
b\frac{2b}{(t+2)^2} - a\alpha & t \ge t_0
\end{cases}$$ The situation is depicted schematically in figure \[patchdiag2\].
(24.722816,6.630996)(0,0) (0,0)[(1,0)[24.722816]{}]{} (5.000000,0.000000) (5.037947,0.093102) (5.075895,0.182800) (5.113842,0.269277) (5.151789,0.352705) (5.189737,0.433241) (5.227684,0.511033) (5.265631,0.586219) (5.303579,0.658928) (5.341526,0.729281) (5.379473,0.797389) (5.417421,0.863360) (5.455368,0.927291) (5.493315,0.989276) (5.531263,1.049402) (5.569210,1.107753) (5.607157,1.164405) (5.645105,1.219431) (5.683052,1.272901) (5.720999,1.324880) (5.758947,1.375428) (5.796894,1.424605) (5.834841,1.472466) (5.872789,1.519062) (5.910736,1.564443) (5.948683,1.608656) (5.986631,1.651745) (6.024578,1.693754) (6.062525,1.734721) (6.100473,1.774685) (6.138420,1.813683) (6.176367,1.851749) (6.214315,1.888917) (6.252262,1.925217) (6.290209,1.960680) (6.328157,1.995334) (6.366104,2.029206) (6.404051,2.062324) (6.441999,2.094711) (6.479946,2.126392) (6.517893,2.157390) (6.555841,2.187725) (6.593788,2.217421) (6.631735,2.246495) (6.669683,2.274969) (6.707630,2.302859) (6.745577,2.330185) (6.783525,2.356962) (6.821472,2.383207) (6.859419,2.408937) (6.897367,2.434165) (6.935314,2.458907) (6.973261,2.483176) (7.011209,2.506986) (7.049156,2.530349) (7.087103,2.553279) (7.125051,2.575787) (7.162998,2.597885) (7.200945,2.619583) (7.238893,2.640893) (7.276840,2.661825) (7.314787,2.682389) (7.352735,2.702594) (7.390682,2.722449) (7.428629,2.741965) (7.466577,2.761149) (7.504524,2.780010) (7.542471,2.798555) (7.580419,2.816793) (7.618366,2.834732) (7.656313,2.852378) (7.694261,2.869739) (7.732208,2.886821) (7.770155,2.903632) (7.808103,2.920177) (7.846050,2.936464) (7.883997,2.952497) (7.921945,2.968283) (7.959892,2.983827) (7.997839,2.999135) (8.035787,3.014213) (8.073734,3.029065) (8.111681,3.043696) (8.149629,3.058112) (8.187576,3.072317) (8.225523,3.086316) (8.263471,3.100113) (8.301418,3.113712) (8.339365,3.127118) (8.377313,3.140335) (8.415260,3.153367) (8.453207,3.166217) (8.491155,3.178889) (8.529102,3.191388) (8.567049,3.203716) (8.604997,3.215878) (8.642944,3.227875) (8.680891,3.239713) (8.718839,3.251393) (8.756786,3.262920) (6.264911,1.937129)[(0,0)\[lt\][$g_1(t)$]{}]{} (0.000000,0.000000) (0.030303,0.000000) (0.060606,0.030303) (0.090909,0.030303) (0.121212,0.060606) (0.151515,0.060606) (0.181818,0.060606) (0.212121,0.090909) (0.242424,0.090909) (0.272727,0.090909) (0.303030,0.121212) (0.333333,0.121212) (0.363636,0.151515) (0.393939,0.151515) (0.424242,0.151515) (0.454545,0.181818) (0.484848,0.181818) (0.515152,0.181818) (0.545455,0.212121) (0.575758,0.212121) (0.606061,0.242424) (0.636364,0.242424) (0.666667,0.242424) (0.696970,0.272727) (0.727273,0.272727) (0.757576,0.272727) (0.787879,0.303030) (0.818182,0.303030) (0.848485,0.333333) (0.878788,0.333333) (0.909091,0.333333) (0.939394,0.363636) (0.969697,0.363636) (1.000000,0.363636) (1.030303,0.393939) (1.060606,0.393939) (1.090909,0.424242) (1.121212,0.424242) (1.151515,0.424242) (1.181818,0.454545) (1.212121,0.454545) (1.242424,0.454545) (1.272727,0.484848) (1.303030,0.484848) (1.333333,0.515152) (1.363636,0.515152) (1.393939,0.515152) (1.424242,0.545455) (1.454545,0.545455) (1.484848,0.545455) (1.515152,0.575758) (1.545455,0.575758) (1.575758,0.606061) (1.606061,0.606061) (1.636364,0.606061) (1.666667,0.636364) (1.696970,0.636364) (1.727273,0.636364) (1.757576,0.666667) (1.787879,0.666667) (1.818182,0.696970) (1.848485,0.696970) (1.878788,0.696970) (1.909091,0.727273) (1.939394,0.727273) (1.969697,0.727273) (2.000000,0.757576) (2.030303,0.757576) (2.060606,0.787879) (2.090909,0.787879) (2.121212,0.787879) (2.151515,0.818182) (2.181818,0.818182) (2.212121,0.818182) (2.242424,0.848485) (2.272727,0.848485) (2.303030,0.878788) (2.333333,0.878788) (2.363636,0.878788) (2.393939,0.909091) (2.424242,0.909091) (2.454545,0.909091) (2.484848,0.939394) (2.515152,0.939394) (2.545455,0.969697) (2.575758,0.969697) (2.606061,0.969697) (2.636364,1.000000) (2.666667,1.000000) (2.696970,1.000000) (2.727273,1.030303) (2.757576,1.030303) (2.787879,1.060606) (2.818182,1.060606) (2.848485,1.060606) (2.878788,1.090909) (2.909091,1.090909) (2.939394,1.090909) (2.969697,1.121212) (3.000000,1.121212) (3.030303,1.151515) (3.060606,1.151515) (3.090909,1.151515) (3.121212,1.181818) (3.151515,1.181818) (3.181818,1.181818) (3.212121,1.212121) (3.242424,1.212121) (3.272727,1.242424) (3.303030,1.242424) (3.333333,1.242424) (3.363636,1.272727) (3.393939,1.272727) (3.424242,1.272727) (3.454545,1.303030) (3.484848,1.303030) (3.515152,1.333333) (3.545455,1.333333) (3.575758,1.333333) (3.606061,1.363636) (3.636364,1.363636) (3.666667,1.363636) (3.696970,1.393939) (3.727273,1.393939) (3.757576,1.424242) (3.787879,1.424242) (3.818182,1.424242) (3.848485,1.454545) (3.878788,1.454545) (3.909091,1.454545) (3.939394,1.484848) (3.969697,1.484848) (4.000000,1.515152) (4.030303,1.515152) (4.060606,1.515152) (4.090909,1.545455) (4.121212,1.545455) (4.151515,1.545455) (4.181818,1.575758) (4.212121,1.575758) (4.242424,1.606061) (4.272727,1.606061) (4.303030,1.606061) (4.333333,1.636364) (4.363636,1.636364) (4.393939,1.636364) (4.424242,1.666667) (4.454545,1.666667) (4.484848,1.696970) (4.515152,1.696970) (4.545455,1.696970) (4.575758,1.727273) (4.606061,1.727273) (4.636364,1.727273) (4.666667,1.757576) (4.696970,1.757576) (4.727273,1.787879) (4.757576,1.787879) (4.787879,1.787879) (4.818182,1.818182) (4.848485,1.818182) (4.878788,1.818182) (4.909091,1.848485) (4.939394,1.848485) (4.969697,1.878788) (5.000000,1.878788) (5.030303,1.878788) (5.060606,1.909091) (5.090909,1.909091) (5.121212,1.909091) (5.151515,1.939394) (5.181818,1.939394) (5.212121,1.969697) (5.242424,1.969697) (5.272727,1.969697) (5.303030,2.000000) (5.333333,2.000000) (5.363636,2.000000) (5.393939,2.030303) (5.424242,2.030303) (5.454545,2.060606) (5.484848,2.060606) (5.515152,2.060606) (5.545455,2.090909) (5.575758,2.090909) (5.606061,2.090909) (5.636364,2.121212) (5.666667,2.121212) (5.696970,2.151515) (5.727273,2.151515) (5.757576,2.151515) (5.787879,2.181818) (5.818182,2.181818) (5.848485,2.181818) (5.878788,2.212121) (5.909091,2.212121) (5.939394,2.242424) (5.969697,2.242424) (6.000000,2.242424) (6.030303,2.272727) (6.060606,2.272727) (6.090909,2.272727) (6.121212,2.303030) (6.151515,2.303030) (6.181818,2.333333) (6.212121,2.333333) (6.242424,2.333333) (6.272727,2.363636) (6.303030,2.363636) (6.333333,2.363636) (6.363636,2.393939) (6.393939,2.393939) (6.424242,2.424242) (6.454545,2.424242) (6.484848,2.424242) (6.515152,2.454545) (6.545455,2.454545) (6.575758,2.454545) (6.606061,2.484848) (6.636364,2.484848) (6.666667,2.515152) (6.696970,2.515152) (6.727273,2.515152) (6.757576,2.545455) (6.787879,2.545455) (6.818182,2.545455) (6.848485,2.575758) (6.878788,2.575758) (6.909091,2.606061) (6.939394,2.606061) (6.969697,2.606061) (7.000000,2.636364) (7.030303,2.636364) (7.060606,2.636364) (7.090909,2.666667) (7.121212,2.666667) (7.151515,2.696970) (7.181818,2.696970) (7.212121,2.696970) (7.242424,2.727273) (7.272727,2.727273) (7.303030,2.727273) (7.333333,2.757576) (7.363636,2.757576) (7.393939,2.787879) (7.424242,2.787879) (7.454545,2.787879) (7.484848,2.818182) (7.515152,2.818182) (7.545455,2.818182) (7.575758,2.848485) (7.606061,2.848485) (7.636364,2.878788) (7.666667,2.878788) (7.696970,2.878788) (7.727273,2.909091) (7.757576,2.909091) (7.787879,2.909091) (7.818182,2.939394) (7.848485,2.939394) (7.878788,2.969697) (7.909091,2.969697) (7.939394,2.969697) (7.969697,3.000000) (8.000000,3.000000) (8.030303,3.000000) (8.060606,3.030303) (8.090909,3.030303) (8.121212,3.060606) (8.151515,3.060606) (4.500000,1.788612)[(0,0)\[rb\][$R_1$]{}]{} (8.162278,0)[(0,1)[3.369158]{}]{} (8.162278,3.062871)[(1,0)[6.240750]{}]{} (0,0)[(0,-1)[0.500000]{}]{} (5.000000,0)[(0,-1)[0.500000]{}]{} (8.162278,0)[(0,-1)[0.500000]{}]{} (13.162278,0)[(0,-1)[0.500000]{}]{} (2.500000,-0.300000)[(0,0)[$t_b$]{}]{} (6.581139,-0.300000)[(0,0)[$\tau_1$]{}]{} (10.662278,-0.300000)[(0,0)[$t_b$]{}]{} (8.262278,2.144009)[(0,0)\[l\][$m_1$]{}]{} (13.162278,0.000000) (13.259151,0.369592) (13.356024,0.706542) (13.452898,1.014992) (13.549771,1.298411) (13.646644,1.559727) (13.743518,1.801429) (13.840391,2.025645) (13.937264,2.234206) (14.034138,2.428698) (14.131011,2.610496) (14.227884,2.780805) (14.324758,2.940679) (14.421631,3.091051) (14.518504,3.232741) (14.615378,3.366482) (14.712251,3.492924) (14.809124,3.612648) (14.905998,3.726176) (15.002871,3.833977) (15.099744,3.936473) (15.196618,4.034048) (15.293491,4.127046) (15.390364,4.215782) (15.487238,4.300544) (15.584111,4.381591) (15.680984,4.459164) (15.777858,4.533480) (15.874731,4.604741) (15.971604,4.673131) (16.068478,4.738820) (16.165351,4.801966) (16.262224,4.862712) (16.359097,4.921194) (16.455971,4.977535) (16.552844,5.031852) (16.649717,5.084250) (16.746591,5.134831) (16.843464,5.183687) (16.940337,5.230904) (17.037211,5.276565) (17.134084,5.320744) (17.230957,5.363512) (17.327831,5.404937) (17.424704,5.445080) (17.521577,5.484000) (17.618451,5.521752) (17.715324,5.558388) (17.812197,5.593956) (17.909071,5.628503) (18.005944,5.662072) (18.102817,5.694704) (18.199691,5.726437) (18.296564,5.757309) (18.393437,5.787354) (18.490311,5.816604) (18.587184,5.845091) (18.684057,5.872844) (18.780931,5.899891) (18.877804,5.926259) (18.974677,5.951974) (19.071551,5.977058) (19.168424,6.001535) (19.265297,6.025427) (19.362171,6.048755) (19.459044,6.071538) (19.555917,6.093795) (19.652791,6.115544) (19.749664,6.136802) (19.846537,6.157586) (19.943411,6.177912) (20.040284,6.197794) (20.137157,6.217246) (20.234031,6.236284) (20.330904,6.254919) (20.427777,6.273164) (20.524651,6.291032) (20.621524,6.308533) (20.718397,6.325680) (20.815271,6.342483) (20.912144,6.358952) (21.009017,6.375097) (21.105891,6.390927) (21.202764,6.406452) (21.299637,6.421680) (21.396511,6.436620) (21.493384,6.451279) (21.590257,6.465666) (21.687131,6.479789) (21.784004,6.493654) (21.880877,6.507268) (21.977751,6.520638) (22.074624,6.533771) (22.171497,6.546673) (22.268371,6.559349) (22.365244,6.571807) (22.462117,6.584051) (22.558991,6.596086) (22.655864,6.607919) (22.752737,6.619554) (14.403027,0)[(0,1)[3.062871]{}]{} (13.782652,-0.300000)[(0,0)[$t_0$]{}]{} (14.403027,0)[(0,-1)[0.500000]{}]{} (13.162278,-0.750000)[(1,0)[8.072777]{}]{} (17.198666,-0.800000)[(0,0)\[t\][$\tau_2$]{}]{} (15.584111,1.318721)[(0,0)\[lt\][$g_2(t)$]{}]{} (17.198666,2.912166)[(0,0)\[rb\][$R_2$]{}]{} (14.403027,0.000000) (14.445260,0.063512) (14.487493,0.125411) (14.529726,0.185757) (14.571959,0.244609) (14.614192,0.302020) (14.656425,0.358043) (14.698658,0.412729) (14.740891,0.466123) (14.783123,0.518272) (14.825356,0.569219) (14.867589,0.619004) (14.909822,0.667667) (14.952055,0.715246) (14.994288,0.761775) (15.036521,0.807291) (15.078754,0.851824) (15.120987,0.895408) (15.163220,0.938071) (15.205453,0.979843) (15.247686,1.020752) (15.289918,1.060823) (15.332151,1.100083) (15.374384,1.138555) (15.416617,1.176264) (15.458850,1.213231) (15.501083,1.249478) (15.543316,1.285027) (15.585549,1.319897) (15.627782,1.354107) (15.670015,1.387677) (15.712248,1.420623) (15.754481,1.452963) (15.796713,1.484713) (15.838946,1.515891) (15.881179,1.546510) (15.923412,1.576586) (15.965645,1.606133) (16.007878,1.635165) (16.050111,1.663695) (16.092344,1.691737) (16.134577,1.719302) (16.176810,1.746403) (16.219043,1.773051) (16.261276,1.799258) (16.303508,1.825034) (16.345741,1.850390) (16.387974,1.875337) (16.430207,1.899883) (16.472440,1.924039) (16.514673,1.947814) (16.556906,1.971216) (16.599139,1.994255) (16.641372,2.016939) (16.683605,2.039275) (16.725838,2.061273) (16.768071,2.082939) (16.810303,2.104281) (16.852536,2.125306) (16.894769,2.146022) (16.937002,2.166435) (16.979235,2.186551) (17.021468,2.206377) (17.063701,2.225919) (17.105934,2.245184) (17.148167,2.264176) (17.190400,2.282903) (17.232633,2.301369) (17.274866,2.319580) (17.317098,2.337541) (17.359331,2.355257) (17.401564,2.372734) (17.443797,2.389975) (17.486030,2.406986) (17.528263,2.423771) (17.570496,2.440336) (17.612729,2.456683) (17.654962,2.472817) (17.697195,2.488743) (17.739428,2.504465) (17.781661,2.519986) (17.823893,2.535310) (17.866126,2.550441) (17.908359,2.565382) (17.950592,2.580138) (17.992825,2.594711) (18.035058,2.609105) (18.077291,2.623323) (18.119524,2.637369) (18.161757,2.651245) (18.203990,2.664955) (18.246223,2.678501) (18.288456,2.691887) (18.330688,2.705114) (18.372921,2.718187) (18.415154,2.731108) (18.457387,2.743879) (18.499620,2.756503) (18.541853,2.768983) (18.584086,2.781321) (18.626319,2.793519) (18.668552,2.805579) (18.710785,2.817505) (18.753018,2.829298) (18.795251,2.840961) (18.837483,2.852495) (18.879716,2.863903) (18.921949,2.875187) (18.964182,2.886348) (19.006415,2.897390) (19.048648,2.908313) (19.090881,2.919120) (19.133114,2.929812) (19.175347,2.940391) (19.217580,2.950860) (19.259813,2.961219) (19.302046,2.971471) (19.344278,2.981618) (19.386511,2.991659) (19.428744,3.001599) (19.470977,3.011437) (19.513210,3.021176) (19.555443,3.030816) (19.597676,3.040361) (19.639909,3.049810) (19.682142,3.059165) (19.724375,3.068428) (19.766608,3.077601) (19.808841,3.086683) (19.851073,3.095677) (19.893306,3.104585) (19.935539,3.113406) (19.977772,3.122143) (20.020005,3.130797) (20.062238,3.139369) (20.104471,3.147859) (20.146704,3.156270) (20.188937,3.164602) (20.231170,3.172857) (20.273403,3.181035) (20.315636,3.189137) (20.357868,3.197165) (20.400101,3.205120) (20.442334,3.213002) (20.484567,3.220813) (20.526800,3.228553) (20.569033,3.236224) (20.611266,3.243826) (20.653499,3.251361) (20.695732,3.258829) (20.737965,3.266231) (20.780198,3.273568) (20.822431,3.280841) (20.864663,3.288050) (20.906896,3.295198) (20.949129,3.302283) (20.991362,3.309307) (21.033595,3.316272) (21.075828,3.323177) (21.118061,3.330023) (21.160294,3.336812) (21.202527,3.343543) (21.244760,3.350219) (21.286993,3.356838) (21.329226,3.363402) (21.371458,3.369913) (21.413691,3.376369) (21.455924,3.382773) (21.498157,3.389124) (21.540390,3.395423) (21.582623,3.401672) (21.624856,3.407870) (21.667089,3.414018) (21.709322,3.420117) (21.751555,3.426167) (21.793788,3.432169) (21.836020,3.438124) (21.878253,3.444032) (21.920486,3.449893) (21.962719,3.455709) (22.004952,3.461479) (22.047185,3.467204) (22.089418,3.472885) (22.131651,3.478523) (22.173884,3.484117) (22.216117,3.489669) (22.258350,3.495178) (22.300583,3.500645) (22.342815,3.506071) (22.385048,3.511457) (22.427281,3.516801) (22.469514,3.522106) (22.511747,3.527372) (22.553980,3.532598) (22.596213,3.537786) (22.638446,3.542936) (22.680679,3.548048) (22.722912,3.553122) (22.765145,3.558160) (22.807378,3.563161) (0.000000,0.000000) (0.030303,0.000000) (0.060606,0.000000) (0.090909,0.000000) (0.121212,0.030303) (0.151515,0.030303) (0.181818,0.030303) (0.212121,0.030303) (0.242424,0.030303) (0.272727,0.030303) (0.303030,0.060606) (0.333333,0.060606) (0.363636,0.060606) (0.393939,0.060606) (0.424242,0.060606) (0.454545,0.060606) (0.484848,0.090909) (0.515152,0.090909) (0.545455,0.090909) (0.575758,0.090909) (0.606061,0.090909) (0.636364,0.090909) (0.666667,0.090909) (0.696970,0.121212) (0.727273,0.121212) (0.757576,0.121212) (0.787879,0.121212) (0.818182,0.121212) (0.848485,0.121212) (0.878788,0.151515) (0.909091,0.151515) (0.939394,0.151515) (0.969697,0.151515) (1.000000,0.151515) (1.030303,0.151515) (1.060606,0.151515) (1.090909,0.181818) (1.121212,0.181818) (1.151515,0.181818) (1.181818,0.181818) (1.212121,0.181818) (1.242424,0.181818) (1.272727,0.212121) (1.303030,0.212121) (1.333333,0.212121) (1.363636,0.212121) (1.393939,0.212121) (1.424242,0.212121) (1.454545,0.242424) (1.484848,0.242424) (1.515152,0.242424) (1.545455,0.242424) (1.575758,0.242424) (1.606061,0.242424) (1.636364,0.242424) (1.666667,0.272727) (1.696970,0.272727) (1.727273,0.272727) (1.757576,0.272727) (1.787879,0.272727) (1.818182,0.272727) (1.848485,0.303030) (1.878788,0.303030) (1.909091,0.303030) (1.939394,0.303030) (1.969697,0.303030) (2.000000,0.303030) (2.030303,0.333333) (2.060606,0.333333) (2.090909,0.333333) (2.121212,0.333333) (2.151515,0.333333) (2.181818,0.333333) (2.212121,0.333333) (2.242424,0.363636) (2.272727,0.363636) (2.303030,0.363636) (2.333333,0.363636) (2.363636,0.363636) (2.393939,0.363636) (2.424242,0.393939) (2.454545,0.393939) (2.484848,0.393939) (2.515152,0.393939) (2.545455,0.393939) (2.575758,0.393939) (2.606061,0.424242) (2.636364,0.424242) (2.666667,0.424242) (2.696970,0.424242) (2.727273,0.424242) (2.757576,0.424242) (2.787879,0.424242) (2.818182,0.454545) (2.848485,0.454545) (2.878788,0.454545) (2.909091,0.454545) (2.939394,0.454545) (2.969697,0.454545) (3.000000,0.484848) (3.030303,0.484848) (3.060606,0.484848) (3.090909,0.484848) (3.121212,0.484848) (3.151515,0.484848) (3.181818,0.484848) (3.212121,0.515152) (3.242424,0.515152) (3.272727,0.515152) (3.303030,0.515152) (3.333333,0.515152) (3.363636,0.515152) (3.393939,0.545455) (3.424242,0.545455) (3.454545,0.545455) (3.484848,0.545455) (3.515152,0.545455) (3.545455,0.545455) (3.575758,0.575758) (3.606061,0.575758) (3.636364,0.575758) (3.666667,0.575758) (3.696970,0.575758) (3.727273,0.575758) (3.757576,0.575758) (3.787879,0.606061) (3.818182,0.606061) (3.848485,0.606061) (3.878788,0.606061) (3.909091,0.606061) (3.939394,0.606061) (3.969697,0.636364) (4.000000,0.636364) (4.030303,0.636364) (4.060606,0.636364) (4.090909,0.636364) (4.121212,0.636364) (4.151515,0.666667) (4.181818,0.666667) (4.212121,0.666667) (4.242424,0.666667) (4.272727,0.666667) (4.303030,0.666667) (4.333333,0.666667) (4.363636,0.696970) (4.393939,0.696970) (4.424242,0.696970) (4.454545,0.696970) (4.484848,0.696970) (4.515152,0.696970) (4.545455,0.727273) (4.575758,0.727273) (4.606061,0.727273) (4.636364,0.727273) (4.666667,0.727273) (4.696970,0.727273) (4.727273,0.757576) (4.757576,0.757576) (4.787879,0.757576) (4.818182,0.757576) (4.848485,0.757576) (4.878788,0.757576) (4.909091,0.757576) (4.939394,0.787879) (4.969697,0.787879) (5.000000,0.787879) (5.030303,0.787879) (5.060606,0.787879) (5.090909,0.787879) (5.121212,0.818182) (5.151515,0.818182) (5.181818,0.818182) (5.212121,0.818182) (5.242424,0.818182) (5.272727,0.818182) (5.303030,0.818182) (5.333333,0.848485) (5.363636,0.848485) (5.393939,0.848485) (5.424242,0.848485) (5.454545,0.848485) (5.484848,0.848485) (5.515152,0.878788) (5.545455,0.878788) (5.575758,0.878788) (5.606061,0.878788) (5.636364,0.878788) (5.666667,0.878788) (5.696970,0.909091) (5.727273,0.909091) (5.757576,0.909091) (5.787879,0.909091) (5.818182,0.909091) (5.848485,0.909091) (5.878788,0.909091) (5.909091,0.939394) (5.939394,0.939394) (5.969697,0.939394) (6.000000,0.939394) (6.030303,0.939394) (6.060606,0.939394) (6.090909,0.969697) (6.121212,0.969697) (6.151515,0.969697) (6.181818,0.969697) (6.212121,0.969697) (6.242424,0.969697) (6.272727,1.000000) (6.303030,1.000000) (6.333333,1.000000) (6.363636,1.000000) (6.393939,1.000000) (6.424242,1.000000) (6.454545,1.000000) (6.484848,1.030303) (6.515152,1.030303) (6.545455,1.030303) (6.575758,1.030303) (6.606061,1.030303) (6.636364,1.030303) (6.666667,1.060606) (6.696970,1.060606) (6.727273,1.060606) (6.757576,1.060606) (6.787879,1.060606) (6.818182,1.060606) (6.848485,1.090909) (6.878788,1.090909) (6.909091,1.090909) (6.939394,1.090909) (6.969697,1.090909) (7.000000,1.090909) (7.030303,1.090909) (7.060606,1.121212) (7.090909,1.121212) (7.121212,1.121212) (7.151515,1.121212) (7.181818,1.121212) (7.212121,1.121212) (7.242424,1.151515) (7.272727,1.151515) (7.303030,1.151515) (7.333333,1.151515) (7.363636,1.151515) (7.393939,1.151515) (7.424242,1.151515) (7.454545,1.181818) (7.484848,1.181818) (7.515152,1.181818) (7.545455,1.181818) (7.575758,1.181818) (7.606061,1.181818) (7.636364,1.212121) (7.666667,1.212121) (7.696970,1.212121) (7.727273,1.212121) (7.757576,1.212121) (7.787879,1.212121) (7.818182,1.242424) (7.848485,1.242424) (7.878788,1.242424) (7.909091,1.242424) (7.939394,1.242424) (7.969697,1.242424) (8.000000,1.242424) (8.030303,1.272727) (8.060606,1.272727) (8.090909,1.272727) (8.121212,1.272727) (8.151515,1.272727) (8.181818,1.272727) (8.212121,1.303030) (8.242424,1.303030) (8.272727,1.303030) (8.303030,1.303030) (8.333333,1.303030) (8.363636,1.303030) (8.393939,1.333333) (8.424242,1.333333) (8.454545,1.333333) (8.484848,1.333333) (8.515152,1.333333) (8.545455,1.333333) (8.575758,1.333333) (8.606061,1.363636) (8.636364,1.363636) (8.666667,1.363636) (8.696970,1.363636) (8.727273,1.363636) (8.757576,1.363636) (8.787879,1.393939) (8.818182,1.393939) (8.848485,1.393939) (8.878788,1.393939) (8.909091,1.393939) (8.939394,1.393939) (8.969697,1.424242) (9.000000,1.424242) (9.030303,1.424242) (9.060606,1.424242) (9.090909,1.424242) (9.121212,1.424242) (9.151515,1.424242) (9.181818,1.454545) (9.212121,1.454545) (9.242424,1.454545) (9.272727,1.454545) (9.303030,1.454545) (9.333333,1.454545) (9.363636,1.484848) (9.393939,1.484848) (9.424242,1.484848) (9.454545,1.484848) (9.484848,1.484848) (9.515152,1.484848) (9.545455,1.484848) (9.575758,1.515152) (9.606061,1.515152) (9.636364,1.515152) (9.666667,1.515152) (9.696970,1.515152) (9.727273,1.515152) (9.757576,1.545455) (9.787879,1.545455) (9.818182,1.545455) (9.848485,1.545455) (9.878788,1.545455) (9.909091,1.545455) (9.939394,1.575758) (9.969697,1.575758) (10.000000,1.575758) (10.030303,1.575758) (10.060606,1.575758) (10.090909,1.575758) (10.121212,1.575758) (10.151515,1.606061) (10.181818,1.606061) (10.212121,1.606061) (10.242424,1.606061) (10.272727,1.606061) (10.303030,1.606061) (10.333333,1.636364) (10.363636,1.636364) (10.393939,1.636364) (10.424242,1.636364) (10.454545,1.636364) (10.484848,1.636364) (10.515152,1.666667) (10.545455,1.666667) (10.575758,1.666667) (10.606061,1.666667) (10.636364,1.666667) (10.666667,1.666667) (10.696970,1.666667) (10.727273,1.696970) (10.757576,1.696970) (10.787879,1.696970) (10.818182,1.696970) (10.848485,1.696970) (10.878788,1.696970) (10.909091,1.727273) (10.939394,1.727273) (10.969697,1.727273) (11.000000,1.727273) (11.030303,1.727273) (11.060606,1.727273) (11.090909,1.757576) (11.121212,1.757576) (11.151515,1.757576) (11.181818,1.757576) (11.212121,1.757576) (11.242424,1.757576) (11.272727,1.757576) (11.303030,1.787879) (11.333333,1.787879) (11.363636,1.787879) (11.393939,1.787879) (11.424242,1.787879) (11.454545,1.787879) (11.484848,1.818182) (11.515152,1.818182) (11.545455,1.818182) (11.575758,1.818182) (11.606061,1.818182) (11.636364,1.818182) (11.666667,1.818182) (11.696970,1.848485) (11.727273,1.848485) (11.757576,1.848485) (11.787879,1.848485) (11.818182,1.848485) (11.848485,1.848485) (11.878788,1.878788) (11.909091,1.878788) (11.939394,1.878788) (11.969697,1.878788) (12.000000,1.878788) (12.030303,1.878788) (12.060606,1.909091) (12.090909,1.909091) (12.121212,1.909091) (12.151515,1.909091) (12.181818,1.909091) (12.212121,1.909091) (12.242424,1.909091) (12.272727,1.939394) (12.303030,1.939394) (12.333333,1.939394) (12.363636,1.939394) (12.393939,1.939394) (12.424242,1.939394) (12.454545,1.969697) (12.484848,1.969697) (12.515152,1.969697) (12.545455,1.969697) (12.575758,1.969697) (12.606061,1.969697) (12.636364,2.000000) (12.666667,2.000000) (12.696970,2.000000) (12.727273,2.000000) (12.757576,2.000000) (12.787879,2.000000) (12.818182,2.000000) (12.848485,2.030303) (12.878788,2.030303) (12.909091,2.030303) (12.939394,2.030303) (12.969697,2.030303) (13.000000,2.030303) (13.030303,2.060606) (13.060606,2.060606) (13.090909,2.060606) (13.121212,2.060606) (13.151515,2.060606) (13.181818,2.060606) (13.212121,2.090909) (13.242424,2.090909) (13.272727,2.090909) (13.303030,2.090909) (13.333333,2.090909) (13.363636,2.090909) (13.393939,2.090909) (13.424242,2.121212) (13.454545,2.121212) (13.484848,2.121212) (13.515152,2.121212) (13.545455,2.121212) (13.575758,2.121212) (13.606061,2.151515) (13.636364,2.151515) (13.666667,2.151515) (13.696970,2.151515) (13.727273,2.151515) (13.757576,2.151515) (13.787879,2.151515) (13.818182,2.181818) (13.848485,2.181818) (13.878788,2.181818) (13.909091,2.181818) (13.939394,2.181818) (13.969697,2.181818) (14.000000,2.212121) (14.030303,2.212121) (14.060606,2.212121) (14.090909,2.212121) (14.121212,2.212121) (14.151515,2.212121) (14.181818,2.242424) (14.212121,2.242424) (14.242424,2.242424) (14.272727,2.242424) (14.303030,2.242424) (14.333333,2.242424) (14.363636,2.242424) (14.393939,2.272727) (14.424242,2.272727) (14.454545,2.272727) (14.484848,2.272727) (14.515152,2.272727) (14.545455,2.272727) (14.575758,2.303030) (14.606061,2.303030) (14.636364,2.303030) (14.666667,2.303030) (14.696970,2.303030) (14.727273,2.303030) (14.757576,2.333333) (14.787879,2.333333) (14.818182,2.333333) (14.848485,2.333333) (14.878788,2.333333) (14.909091,2.333333) (14.939394,2.333333) (14.969697,2.363636) (15.000000,2.363636) (15.030303,2.363636) (15.060606,2.363636) (15.090909,2.363636) (15.121212,2.363636) (15.151515,2.393939) (15.181818,2.393939) (15.212121,2.393939) (15.242424,2.393939) (15.272727,2.393939) (15.303030,2.393939) (15.333333,2.424242) (15.363636,2.424242) (15.393939,2.424242) (15.424242,2.424242) (15.454545,2.424242) (15.484848,2.424242) (15.515152,2.424242) (15.545455,2.454545) (15.575758,2.454545) (15.606061,2.454545) (15.636364,2.454545) (15.666667,2.454545) (15.696970,2.454545) (15.727273,2.484848) (15.757576,2.484848) (15.787879,2.484848) (15.818182,2.484848) (15.848485,2.484848) (15.878788,2.484848) (15.909091,2.484848) (15.939394,2.515152) (15.969697,2.515152) (16.000000,2.515152) (16.030303,2.515152) (16.060606,2.515152) (16.090909,2.515152) (16.121212,2.545455) (16.151515,2.545455) (16.181818,2.545455) (16.212121,2.545455) (16.242424,2.545455) (16.272727,2.545455) (16.303030,2.575758) (16.333333,2.575758) (16.363636,2.575758) (16.393939,2.575758) (16.424242,2.575758) (16.454545,2.575758) (16.484848,2.575758) (16.515152,2.606061) (16.545455,2.606061) (16.575758,2.606061) (16.606061,2.606061) (16.636364,2.606061) (16.666667,2.606061) (16.696970,2.636364) (16.727273,2.636364) (16.757576,2.636364) (16.787879,2.636364) (16.818182,2.636364) (16.848485,2.636364) (16.878788,2.666667) (16.909091,2.666667) (16.939394,2.666667) (16.969697,2.666667) (17.000000,2.666667) (17.030303,2.666667) (17.060606,2.666667) (17.090909,2.696970) (17.121212,2.696970) (17.151515,2.696970) (17.181818,2.696970) (17.212121,2.696970) (17.242424,2.696970) (17.272727,2.727273) (17.303030,2.727273) (17.333333,2.727273) (17.363636,2.727273) (17.393939,2.727273) (17.424242,2.727273) (17.454545,2.757576) (17.484848,2.757576) (17.515152,2.757576) (17.545455,2.757576) (17.575758,2.757576) (17.606061,2.757576) (17.636364,2.757576) (17.666667,2.787879) (17.696970,2.787879) (17.727273,2.787879) (17.757576,2.787879) (17.787879,2.787879) (17.818182,2.787879) (17.848485,2.818182) (17.878788,2.818182) (17.909091,2.818182) (17.939394,2.818182) (17.969697,2.818182) (18.000000,2.818182) (18.030303,2.818182) (18.060606,2.848485) (18.090909,2.848485) (18.121212,2.848485) (18.151515,2.848485) (18.181818,2.848485) (18.212121,2.848485) (18.242424,2.878788) (18.272727,2.878788) (18.303030,2.878788) (18.333333,2.878788) (18.363636,2.878788) (18.393939,2.878788) (18.424242,2.909091) (18.454545,2.909091) (18.484848,2.909091) (18.515152,2.909091) (18.545455,2.909091) (18.575758,2.909091) (18.606061,2.909091) (18.636364,2.939394) (18.666667,2.939394) (18.696970,2.939394) (18.727273,2.939394) (18.757576,2.939394) (18.787879,2.939394) (18.818182,2.969697) (18.848485,2.969697) (18.878788,2.969697) (18.909091,2.969697) (18.939394,2.969697) (18.969697,2.969697) (19.000000,3.000000) (19.030303,3.000000) (19.060606,3.000000) (19.090909,3.000000) (19.121212,3.000000) (19.151515,3.000000) (19.181818,3.000000) (19.212121,3.030303) (19.242424,3.030303) (19.272727,3.030303) (19.303030,3.030303) (19.333333,3.030303) (19.363636,3.030303) (19.393939,3.060606) (19.424242,3.060606) (19.454545,3.060606) (19.484848,3.060606) (19.515152,3.060606) (19.545455,3.060606) (19.575758,3.090909) (19.606061,3.090909) (19.636364,3.090909) (19.666667,3.090909) (19.696970,3.090909) (19.727273,3.090909) (19.757576,3.090909) (19.787879,3.121212) (19.818182,3.121212) (19.848485,3.121212) (19.878788,3.121212) (19.909091,3.121212) (19.939394,3.121212) (19.969697,3.151515) (20.000000,3.151515) (20.030303,3.151515) (20.060606,3.151515) (20.090909,3.151515) (20.121212,3.151515) (20.151515,3.151515) (20.181818,3.181818) (20.212121,3.181818) (20.242424,3.181818) (20.272727,3.181818) (20.303030,3.181818) (20.333333,3.181818) (20.363636,3.212121) (20.393939,3.212121) (20.424242,3.212121) (20.454545,3.212121) (20.484848,3.212121) (20.515152,3.212121) (20.545455,3.242424) (20.575758,3.242424) (20.606061,3.242424) (20.636364,3.242424) (20.666667,3.242424) (20.696970,3.242424) (20.727273,3.242424) (20.757576,3.272727) (20.787879,3.272727) (20.818182,3.272727) (20.848485,3.272727) (20.878788,3.272727) (20.909091,3.272727) (20.939394,3.303030) (20.969697,3.303030) (21.000000,3.303030) (21.030303,3.303030) (21.060606,3.303030) (21.090909,3.303030) (21.121212,3.333333) (21.151515,3.333333) (21.181818,3.333333) (21.212121,3.333333) (21.235055,0)[(0,1)[3.348690]{}]{}
When we arrive at the second site, a time $T\dfeq{2t_b}+\tau_1$ has already elapsed, so if we stay on the site a time $\tau$, the rate of gain is: $$R_2(\tau) = \frac{g_2(\tau)}{T+\tau} = \frac{g_2(\tau)}{2t_b+\tau_1+\tau}$$ To find the maximum of $R_2(\tau)$ we proceed as in the previous example, setting $$g_2^\prime(\tau) = \frac{g_2(\tau)}{T+\tau}$$ that is $$\frac{2b}{(\tau+2)^2} = \frac{1}{T+\tau} \Bigl[\frac{\tau b}{\tau+2} - a\alpha\Bigr]$$ or $$\frac{2}{(\tau+2)^2} = \frac{1}{T+\tau} \Bigl[\frac{\tau}{\tau+2} - \frac{\alpha}{\gamma}\Bigr]$$ that is $$2(T+\tau) = (\tau+2)\Bigl[\tau - \frac{\alpha}{\gamma}(\tau+2)\Bigr]$$ Rearranging the terms, we obtain the equation $$(\gamma-\alpha)\tau^2-4\alpha\tau-(4\alpha+2\gamma{T})=0$$ whose only positive solution is given by $$\begin{cases}
\displaystyle \Delta = 8\gamma\Bigl[2\alpha+(\gamma-\alpha)T\Bigr] \\
\displaystyle \tau_2 = \frac{4\alpha+\sqrt{\Delta}}{2(\gamma-\alpha)}
\end{cases}$$ The corresponding maximum rate of gain is $$R_2 = g_2^\prime(\tau_2) = \frac{2b}{(\tau_2+2)^2}$$ Using the second patch is convenient if we improve our gain rate, that is, if $R_2>R_1$, where $R_1$ is as in (\[t1bum\]). This imposes a condition on $b/a$, that is, on $\gamma$: it is convenient to use the second patch if $\gamma$ is large enough that we can find a better price that offsets the extra time spent in searching. The limit condition $R_2=R_1$ yields $$\gamma = \left[ \frac{\tau_2+2}{\sqrt{2t_b}+2}\right]^2$$ The value of $\tau_2$ depends on $\gamma$ in such a way that we can’t find a closed form solution to this equation. We can, however, determine its limits. For $t_b\rightarrow{0}$, we have $\alpha\rightarrow{0}$, $T\rightarrow{0}$, $\Delta\rightarrow{0}$, and $\tau_2\rightarrow{0}$, therefore $$\lim_{t_b\rightarrow{0}} \gamma = 1$$ For $t_b\rightarrow\infty$, $\alpha\rightarrow{1}$, $T\sum2t_b$, and $$\begin{aligned}
\Delta &\sim 16\gamma(\gamma-1)t_b \\
\tau &\sim 2\frac{\sqrt{\gamma(\gamma-1)}}{\gamma-1}\sqrt{t_b}
\end{aligned}$$ The equation $$\label{gamma}
\gamma = \left[ \frac{2\sqrt{\gamma(\gamma-1)}}{\sqrt{2}(\gamma-1)} \right]^2 = \frac{2\gamma}{\gamma-1}$$ has solution $\gamma=3$, therefore $$\lim_{t_b\rightarrow\infty} \gamma = 3$$ The behavior of $\gamma$ can be found solving (\[gamma\]) numerically, which, given the stability of the equation, can be done with a simple iteration. The result is shown in figure \[gammafig\]
(839,839)(0,0) (151.0,151.0)
------------------------------------------------------------------------
(131,151)[(0,0)\[r\][$1.2$]{}]{} (758.0,151.0)
------------------------------------------------------------------------
(151.0,221.0)
------------------------------------------------------------------------
(131,221)[(0,0)\[r\][$1.4$]{}]{} (758.0,221.0)
------------------------------------------------------------------------
(151.0,290.0)
------------------------------------------------------------------------
(131,290)[(0,0)\[r\][$1.6$]{}]{} (758.0,290.0)
------------------------------------------------------------------------
(151.0,360.0)
------------------------------------------------------------------------
(131,360)[(0,0)\[r\][$1.8$]{}]{} (758.0,360.0)
------------------------------------------------------------------------
(151.0,430.0)
------------------------------------------------------------------------
(131,430)[(0,0)\[r\][$2$]{}]{} (758.0,430.0)
------------------------------------------------------------------------
(151.0,499.0)
------------------------------------------------------------------------
(131,499)[(0,0)\[r\][$2.2$]{}]{} (758.0,499.0)
------------------------------------------------------------------------
(151.0,569.0)
------------------------------------------------------------------------
(131,569)[(0,0)\[r\][$2.4$]{}]{} (758.0,569.0)
------------------------------------------------------------------------
(151.0,639.0)
------------------------------------------------------------------------
(131,639)[(0,0)\[r\][$2.6$]{}]{} (758.0,639.0)
------------------------------------------------------------------------
(151.0,708.0)
------------------------------------------------------------------------
(131,708)[(0,0)\[r\][$2.8$]{}]{} (758.0,708.0)
------------------------------------------------------------------------
(151.0,778.0)
------------------------------------------------------------------------
(131,778)[(0,0)\[r\][$3$]{}]{} (758.0,778.0)
------------------------------------------------------------------------
(151.0,151.0)
------------------------------------------------------------------------
(151,110)[(0,0)[$10^{-4}$]{}]{} (151.0,758.0)
------------------------------------------------------------------------
(182.0,151.0)
------------------------------------------------------------------------
(182.0,768.0)
------------------------------------------------------------------------
(224.0,151.0)
------------------------------------------------------------------------
(224.0,768.0)
------------------------------------------------------------------------
(245.0,151.0)
------------------------------------------------------------------------
(245.0,768.0)
------------------------------------------------------------------------
(255.0,151.0)
------------------------------------------------------------------------
(255,110)[(0,0)[$10^{-3}$]{}]{} (255.0,758.0)
------------------------------------------------------------------------
(287.0,151.0)
------------------------------------------------------------------------
(287.0,768.0)
------------------------------------------------------------------------
(329.0,151.0)
------------------------------------------------------------------------
(329.0,768.0)
------------------------------------------------------------------------
(350.0,151.0)
------------------------------------------------------------------------
(350.0,768.0)
------------------------------------------------------------------------
(360.0,151.0)
------------------------------------------------------------------------
(360,110)[(0,0)[$10^{-2}$]{}]{} (360.0,758.0)
------------------------------------------------------------------------
(391.0,151.0)
------------------------------------------------------------------------
(391.0,768.0)
------------------------------------------------------------------------
(433.0,151.0)
------------------------------------------------------------------------
(433.0,768.0)
------------------------------------------------------------------------
(454.0,151.0)
------------------------------------------------------------------------
(454.0,768.0)
------------------------------------------------------------------------
(464.0,151.0)
------------------------------------------------------------------------
(464,110)[(0,0)[$10^{-1}$]{}]{} (464.0,758.0)
------------------------------------------------------------------------
(496.0,151.0)
------------------------------------------------------------------------
(496.0,768.0)
------------------------------------------------------------------------
(538.0,151.0)
------------------------------------------------------------------------
(538.0,768.0)
------------------------------------------------------------------------
(559.0,151.0)
------------------------------------------------------------------------
(559.0,768.0)
------------------------------------------------------------------------
(569.0,151.0)
------------------------------------------------------------------------
(569,110)[(0,0)[$1$]{}]{} (569.0,758.0)
------------------------------------------------------------------------
(600.0,151.0)
------------------------------------------------------------------------
(600.0,768.0)
------------------------------------------------------------------------
(642.0,151.0)
------------------------------------------------------------------------
(642.0,768.0)
------------------------------------------------------------------------
(663.0,151.0)
------------------------------------------------------------------------
(663.0,768.0)
------------------------------------------------------------------------
(674.0,151.0)
------------------------------------------------------------------------
(674,110)[(0,0)[$10$]{}]{} (674.0,758.0)
------------------------------------------------------------------------
(705.0,151.0)
------------------------------------------------------------------------
(705.0,768.0)
------------------------------------------------------------------------
(747.0,151.0)
------------------------------------------------------------------------
(747.0,768.0)
------------------------------------------------------------------------
(768.0,151.0)
------------------------------------------------------------------------
(768.0,768.0)
------------------------------------------------------------------------
(778.0,151.0)
------------------------------------------------------------------------
(778,110)[(0,0)[$10^2$]{}]{} (778.0,758.0)
------------------------------------------------------------------------
(151.0,151.0)
------------------------------------------------------------------------
(151.0,151.0)
------------------------------------------------------------------------
(778.0,151.0)
------------------------------------------------------------------------
(151.0,778.0)
------------------------------------------------------------------------
(30,464)[(0,0)[$\gamma$]{}]{} (464,49)[(0,0)[$t_b$]{}]{} (151,163) (151,162.67)
------------------------------------------------------------------------
(151.00,162.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(153,163.67)
------------------------------------------------------------------------
(153.00,163.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(155,164.67)
------------------------------------------------------------------------
(155.00,164.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(158,165.67)
------------------------------------------------------------------------
(158.00,165.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(160,166.67)
------------------------------------------------------------------------
(160.00,166.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(162,167.67)
------------------------------------------------------------------------
(162.00,167.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(164,169.17)
------------------------------------------------------------------------
(164.00,168.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(166,170.67)
------------------------------------------------------------------------
(166.00,170.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(168,171.67)
------------------------------------------------------------------------
(168.00,171.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(171,172.67)
------------------------------------------------------------------------
(171.00,172.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(173,173.67)
------------------------------------------------------------------------
(173.00,173.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(175,174.67)
------------------------------------------------------------------------
(175.00,174.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(177,175.67)
------------------------------------------------------------------------
(177.00,175.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(180,176.67)
------------------------------------------------------------------------
(180.00,176.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(182,177.67)
------------------------------------------------------------------------
(182.00,177.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(184,179.17)
------------------------------------------------------------------------
(184.00,178.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(186,180.67)
------------------------------------------------------------------------
(186.00,180.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(188,181.67)
------------------------------------------------------------------------
(188.00,181.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(191,182.67)
------------------------------------------------------------------------
(191.00,182.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(193,183.67)
------------------------------------------------------------------------
(193.00,183.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(195,185.17)
------------------------------------------------------------------------
(195.00,184.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(197,186.67)
------------------------------------------------------------------------
(197.00,186.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(199,187.67)
------------------------------------------------------------------------
(199.00,187.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(202,188.67)
------------------------------------------------------------------------
(202.00,188.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(204,190.17)
------------------------------------------------------------------------
(204.00,189.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(206,191.67)
------------------------------------------------------------------------
(206.00,191.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(208,192.67)
------------------------------------------------------------------------
(208.00,192.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(210,194.17)
------------------------------------------------------------------------
(210.00,193.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(213,195.67)
------------------------------------------------------------------------
(213.00,195.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(215,196.67)
------------------------------------------------------------------------
(215.00,196.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(217,198.17)
------------------------------------------------------------------------
(217.00,197.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(219,199.67)
------------------------------------------------------------------------
(219.00,199.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(221,200.67)
------------------------------------------------------------------------
(221.00,200.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(224,202.17)
------------------------------------------------------------------------
(224.00,201.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(226,203.67)
------------------------------------------------------------------------
(226.00,203.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(228,205.17)
------------------------------------------------------------------------
(228.00,204.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(230,207.17)
------------------------------------------------------------------------
(230.00,206.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(232,208.67)
------------------------------------------------------------------------
(232.00,208.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(235,210.17)
------------------------------------------------------------------------
(235.00,209.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(237,211.67)
------------------------------------------------------------------------
(237.00,211.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(239,213.17)
------------------------------------------------------------------------
(239.00,212.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(241,214.67)
------------------------------------------------------------------------
(241.00,214.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(244,216.17)
------------------------------------------------------------------------
(244.00,215.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(246,217.67)
------------------------------------------------------------------------
(246.00,217.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(248,219.17)
------------------------------------------------------------------------
(248.00,218.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(250,221.17)
------------------------------------------------------------------------
(250.00,220.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(252,222.67)
------------------------------------------------------------------------
(252.00,222.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(254,224.17)
------------------------------------------------------------------------
(254.00,223.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(257,226.17)
------------------------------------------------------------------------
(257.00,225.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(259,227.67)
------------------------------------------------------------------------
(259.00,227.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(261,229.17)
------------------------------------------------------------------------
(261.00,228.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(263,231.17)
------------------------------------------------------------------------
(263.00,230.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(266,232.67)
------------------------------------------------------------------------
(266.00,232.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(268,234.17)
------------------------------------------------------------------------
(268.00,233.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(270,236.17)
------------------------------------------------------------------------
(270.00,235.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(272,238.17)
------------------------------------------------------------------------
(272.00,237.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(274,240.17)
------------------------------------------------------------------------
(274.00,239.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(277,242.17)
------------------------------------------------------------------------
(277.00,241.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(279,243.67)
------------------------------------------------------------------------
(279.00,243.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(281,245.17)
------------------------------------------------------------------------
(281.00,244.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(283,247.17)
------------------------------------------------------------------------
(283.00,246.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(285,249.17)
------------------------------------------------------------------------
(285.00,248.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(288,251.17)
------------------------------------------------------------------------
(288.00,250.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(290,253.17)
------------------------------------------------------------------------
(290.00,252.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(292,255.17)
------------------------------------------------------------------------
(292.00,254.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(294,257.17)
------------------------------------------------------------------------
(294.00,256.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(296,259.17)
------------------------------------------------------------------------
(296.00,258.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(299,261.17)
------------------------------------------------------------------------
(299.00,260.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(301,263.17)
------------------------------------------------------------------------
(301.00,262.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(303,265.17)
------------------------------------------------------------------------
(303.00,264.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(305,267.17)
------------------------------------------------------------------------
(305.00,266.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(307,269.17)
------------------------------------------------------------------------
(307.00,268.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(310,271.17)
------------------------------------------------------------------------
(310.00,270.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(312.17,273)
------------------------------------------------------------------------
(311.17,273.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(314,276.17)
------------------------------------------------------------------------
(314.00,275.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(316,278.17)
------------------------------------------------------------------------
(316.00,277.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(318,280.17)
------------------------------------------------------------------------
(318.00,279.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(321,282.17)
------------------------------------------------------------------------
(321.00,281.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(323.17,284)
------------------------------------------------------------------------
(322.17,284.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(325,287.17)
------------------------------------------------------------------------
(325.00,286.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(327,289.17)
------------------------------------------------------------------------
(327.00,288.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(329,291.17)
------------------------------------------------------------------------
(329.00,290.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(332.17,293)
------------------------------------------------------------------------
(331.17,293.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(334,296.17)
------------------------------------------------------------------------
(334.00,295.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(336.17,298)
------------------------------------------------------------------------
(335.17,298.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(338,301.17)
------------------------------------------------------------------------
(338.00,300.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(340,303.17)
------------------------------------------------------------------------
(340.00,302.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(343.17,305)
------------------------------------------------------------------------
(342.17,305.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(345,308.17)
------------------------------------------------------------------------
(345.00,307.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(347.17,310)
------------------------------------------------------------------------
(346.17,310.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(349,313.17)
------------------------------------------------------------------------
(349.00,312.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(351.00,315.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(351.00,314.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(354,318.17)
------------------------------------------------------------------------
(354.00,317.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(356.17,320)
------------------------------------------------------------------------
(355.17,320.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(358,323.17)
------------------------------------------------------------------------
(358.00,322.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(360.17,325)
------------------------------------------------------------------------
(359.17,325.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(362.00,328.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(362.00,327.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(365,331.17)
------------------------------------------------------------------------
(365.00,330.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(367.17,333)
------------------------------------------------------------------------
(366.17,333.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(369,336.17)
------------------------------------------------------------------------
(369.00,335.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(371.17,338)
------------------------------------------------------------------------
(370.17,338.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(373.00,341.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(373.00,340.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(376.17,344)
------------------------------------------------------------------------
(375.17,344.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(378,347.17)
------------------------------------------------------------------------
(378.00,346.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(380,349.17)
------------------------------------------------------------------------
(380.00,348.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(382.17,351)
------------------------------------------------------------------------
(381.17,351.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(384.00,354.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(384.00,353.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(387.17,357)
------------------------------------------------------------------------
(386.17,357.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(389.17,360)
------------------------------------------------------------------------
(388.17,360.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(391,363.17)
------------------------------------------------------------------------
(391.00,362.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(393.17,365)
------------------------------------------------------------------------
(392.17,365.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(395.00,368.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(395.00,367.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(398.17,371)
------------------------------------------------------------------------
(397.17,371.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(400.17,374)
------------------------------------------------------------------------
(399.17,374.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(402.17,377)
------------------------------------------------------------------------
(401.17,377.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(404.17,380)
------------------------------------------------------------------------
(403.17,380.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(406.00,383.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(406.00,382.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(409.17,386)
------------------------------------------------------------------------
(408.17,386.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(411,389.17)
------------------------------------------------------------------------
(411.00,388.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(413.17,391)
------------------------------------------------------------------------
(412.17,391.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(415.17,394)
------------------------------------------------------------------------
(414.17,394.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(417.00,397.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(417.00,396.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(420.17,400)
------------------------------------------------------------------------
(419.17,400.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(422.17,403)
------------------------------------------------------------------------
(421.17,403.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(424.17,406)
------------------------------------------------------------------------
(423.17,406.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(426.17,410)
------------------------------------------------------------------------
(425.17,410.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(428.00,413.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(428.00,412.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(431.17,416)
------------------------------------------------------------------------
(430.17,416.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(433.17,419)
------------------------------------------------------------------------
(432.17,419.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(435.17,422)
------------------------------------------------------------------------
(434.17,422.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(437.17,425)
------------------------------------------------------------------------
(436.17,425.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(439.00,428.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(439.00,427.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(442.17,431)
------------------------------------------------------------------------
(441.17,431.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(444.17,434)
------------------------------------------------------------------------
(443.17,434.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(446.17,437)
------------------------------------------------------------------------
(445.17,437.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(448.17,440)
------------------------------------------------------------------------
(447.17,440.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(450.00,444.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(450.00,443.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(453.17,447)
------------------------------------------------------------------------
(452.17,447.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(455.17,450)
------------------------------------------------------------------------
(454.17,450.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(457.17,453)
------------------------------------------------------------------------
(456.17,453.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(459.00,456.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(459.00,455.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(462.17,459)
------------------------------------------------------------------------
(461.17,459.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(464.17,462)
------------------------------------------------------------------------
(463.17,462.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(466.17,466)
------------------------------------------------------------------------
(465.17,466.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(468.17,469)
------------------------------------------------------------------------
(467.17,469.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(470.00,472.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(470.00,471.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(473.17,475)
------------------------------------------------------------------------
(472.17,475.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(475.17,479)
------------------------------------------------------------------------
(474.17,479.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(477.17,482)
------------------------------------------------------------------------
(476.17,482.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(479.17,486)
------------------------------------------------------------------------
(478.17,486.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(481.00,489.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(481.00,488.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(484.17,492)
------------------------------------------------------------------------
(483.17,492.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(486.17,495)
------------------------------------------------------------------------
(485.17,495.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(488.17,498)
------------------------------------------------------------------------
(487.17,498.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(490.17,502)
------------------------------------------------------------------------
(489.17,502.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(492.00,505.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(492.00,504.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(495.17,508)
------------------------------------------------------------------------
(494.17,508.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(497.17,511)
------------------------------------------------------------------------
(496.17,511.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(499.17,515)
------------------------------------------------------------------------
(498.17,515.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(501.17,518)
------------------------------------------------------------------------
(500.17,518.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(503.00,521.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(503.00,520.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(506.17,524)
------------------------------------------------------------------------
(505.17,524.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(508.17,527)
------------------------------------------------------------------------
(507.17,527.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(510.17,530)
------------------------------------------------------------------------
(509.17,530.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(512.17,534)
------------------------------------------------------------------------
(511.17,534.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(514.00,537.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(514.00,536.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(517.17,540)
------------------------------------------------------------------------
(516.17,540.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(519.17,543)
------------------------------------------------------------------------
(518.17,543.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(521.17,546)
------------------------------------------------------------------------
(520.17,546.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(523.17,549)
------------------------------------------------------------------------
(522.17,549.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(525.00,552.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(525.00,551.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(528.17,555)
------------------------------------------------------------------------
(527.17,555.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(530.17,558)
------------------------------------------------------------------------
(529.17,558.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(532.17,561)
------------------------------------------------------------------------
(531.17,561.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(534.17,564)
------------------------------------------------------------------------
(533.17,564.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(536.00,568.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(536.00,567.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(539.17,571)
------------------------------------------------------------------------
(538.17,571.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(541,574.17)
------------------------------------------------------------------------
(541.00,573.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(543,576.17)
------------------------------------------------------------------------
(543.00,575.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(545.17,578)
------------------------------------------------------------------------
(544.17,578.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(547.00,581.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(547.00,580.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(550.17,584)
------------------------------------------------------------------------
(549.17,584.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(552.17,587)
------------------------------------------------------------------------
(551.17,587.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(554.17,590)
------------------------------------------------------------------------
(553.17,590.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(556,593.17)
------------------------------------------------------------------------
(556.00,592.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(558.00,595.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(558.00,594.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(561.17,598)
------------------------------------------------------------------------
(560.17,598.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(563.17,601)
------------------------------------------------------------------------
(562.17,601.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(565,604.17)
------------------------------------------------------------------------
(565.00,603.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(567.17,606)
------------------------------------------------------------------------
(566.17,606.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(569.00,609.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(569.00,608.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(572,612.17)
------------------------------------------------------------------------
(572.00,611.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(574.17,614)
------------------------------------------------------------------------
(573.17,614.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(576,617.17)
------------------------------------------------------------------------
(576.00,616.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(578.17,619)
------------------------------------------------------------------------
(577.17,619.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(580.00,622.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(580.00,621.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(583,625.17)
------------------------------------------------------------------------
(583.00,624.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(585.17,627)
------------------------------------------------------------------------
(584.17,627.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(587,630.17)
------------------------------------------------------------------------
(587.00,629.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(589,632.17)
------------------------------------------------------------------------
(589.00,631.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(591.00,634.61)(0.462,0.447)[3]{}
------------------------------------------------------------------------
(591.00,633.17)(1.962,3.000)[2]{}
------------------------------------------------------------------------
(594,637.17)
------------------------------------------------------------------------
(594.00,636.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(596.17,639)
------------------------------------------------------------------------
(595.17,639.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(598,642.17)
------------------------------------------------------------------------
(598.00,641.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(600,644.17)
------------------------------------------------------------------------
(600.00,643.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(602,646.17)
------------------------------------------------------------------------
(602.00,645.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(605.17,648)
------------------------------------------------------------------------
(604.17,648.00)(2.000,1.547)[2]{}
------------------------------------------------------------------------
(607,651.17)
------------------------------------------------------------------------
(607.00,650.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(609,653.17)
------------------------------------------------------------------------
(609.00,652.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(611.17,655)
------------------------------------------------------------------------
(610.17,655.00)(2.000,2.132)[2]{}
------------------------------------------------------------------------
(613,659.17)
------------------------------------------------------------------------
(613.00,658.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(616,661.17)
------------------------------------------------------------------------
(616.00,660.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(618,663.17)
------------------------------------------------------------------------
(618.00,662.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(620,665.17)
------------------------------------------------------------------------
(620.00,664.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(622,667.17)
------------------------------------------------------------------------
(622.00,666.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(624,669.17)
------------------------------------------------------------------------
(624.00,668.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(627,671.17)
------------------------------------------------------------------------
(627.00,670.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(629,673.17)
------------------------------------------------------------------------
(629.00,672.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(631,675.17)
------------------------------------------------------------------------
(631.00,674.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(633,677.17)
------------------------------------------------------------------------
(633.00,676.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(635,679.17)
------------------------------------------------------------------------
(635.00,678.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(638,681.17)
------------------------------------------------------------------------
(638.00,680.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(640,683.17)
------------------------------------------------------------------------
(640.00,682.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(642,685.17)
------------------------------------------------------------------------
(642.00,684.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(644,686.67)
------------------------------------------------------------------------
(644.00,686.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(646,688.17)
------------------------------------------------------------------------
(646.00,687.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(649,690.17)
------------------------------------------------------------------------
(649.00,689.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(651,692.17)
------------------------------------------------------------------------
(651.00,691.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(653,693.67)
------------------------------------------------------------------------
(653.00,693.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(655,695.17)
------------------------------------------------------------------------
(655.00,694.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(658,697.17)
------------------------------------------------------------------------
(658.00,696.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(660,698.67)
------------------------------------------------------------------------
(660.00,698.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(662,700.17)
------------------------------------------------------------------------
(662.00,699.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(664,701.67)
------------------------------------------------------------------------
(664.00,701.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(666,703.17)
------------------------------------------------------------------------
(666.00,702.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(669,705.17)
------------------------------------------------------------------------
(669.00,704.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(671,706.67)
------------------------------------------------------------------------
(671.00,706.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(673,708.17)
------------------------------------------------------------------------
(673.00,707.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(675,709.67)
------------------------------------------------------------------------
(675.00,709.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(677,710.67)
------------------------------------------------------------------------
(677.00,710.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(680,712.17)
------------------------------------------------------------------------
(680.00,711.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(682,713.67)
------------------------------------------------------------------------
(682.00,713.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(684,714.67)
------------------------------------------------------------------------
(684.00,714.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(686,716.17)
------------------------------------------------------------------------
(686.00,715.17)(1.000,2.000)[2]{}
------------------------------------------------------------------------
(691,717.67)
------------------------------------------------------------------------
(691.00,717.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(693,718.67)
------------------------------------------------------------------------
(693.00,718.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(695,719.67)
------------------------------------------------------------------------
(695.00,719.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(697,720.67)
------------------------------------------------------------------------
(697.00,720.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(699,722.17)
------------------------------------------------------------------------
(699.00,721.17)(1.547,2.000)[2]{}
------------------------------------------------------------------------
(702,723.67)
------------------------------------------------------------------------
(702.00,723.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(704,724.67)
------------------------------------------------------------------------
(704.00,724.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(706,725.67)
------------------------------------------------------------------------
(706.00,725.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(708,726.67)
------------------------------------------------------------------------
(708.00,726.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(710,727.67)
------------------------------------------------------------------------
(710.00,727.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(713,728.67)
------------------------------------------------------------------------
(713.00,728.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(715,729.67)
------------------------------------------------------------------------
(715.00,729.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(717,730.67)
------------------------------------------------------------------------
(717.00,730.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(719,731.67)
------------------------------------------------------------------------
(719.00,731.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(721,732.67)
------------------------------------------------------------------------
(721.00,732.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(724,733.67)
------------------------------------------------------------------------
(724.00,733.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(726,734.67)
------------------------------------------------------------------------
(726.00,734.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(728,735.67)
------------------------------------------------------------------------
(728.00,735.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(730,736.67)
------------------------------------------------------------------------
(730.00,736.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(688.0,718.0)
------------------------------------------------------------------------
(735,737.67)
------------------------------------------------------------------------
(735.00,737.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(737,738.67)
------------------------------------------------------------------------
(737.00,738.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(739,739.67)
------------------------------------------------------------------------
(739.00,739.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(741,740.67)
------------------------------------------------------------------------
(741.00,740.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(732.0,738.0)
------------------------------------------------------------------------
(746,741.67)
------------------------------------------------------------------------
(746.00,741.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(748,742.67)
------------------------------------------------------------------------
(748.00,742.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(750,743.67)
------------------------------------------------------------------------
(750.00,743.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(743.0,742.0)
------------------------------------------------------------------------
(754,744.67)
------------------------------------------------------------------------
(754.00,744.17)(1.500,1.000)[2]{}
------------------------------------------------------------------------
(757,745.67)
------------------------------------------------------------------------
(757.00,745.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(752.0,745.0)
------------------------------------------------------------------------
(761,746.67)
------------------------------------------------------------------------
(761.00,746.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(763,747.67)
------------------------------------------------------------------------
(763.00,747.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(759.0,747.0)
------------------------------------------------------------------------
(768,748.67)
------------------------------------------------------------------------
(768.00,748.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(770,749.67)
------------------------------------------------------------------------
(770.00,749.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(765.0,749.0)
------------------------------------------------------------------------
(774,750.67)
------------------------------------------------------------------------
(774.00,750.17)(1.000,1.000)[2]{}
------------------------------------------------------------------------
(772.0,751.0)
------------------------------------------------------------------------
(776.0,752.0)
------------------------------------------------------------------------
(151.0,151.0)
------------------------------------------------------------------------
(151.0,151.0)
------------------------------------------------------------------------
(778.0,151.0)
------------------------------------------------------------------------
(151.0,778.0)
------------------------------------------------------------------------
For $\frac{b}{a}>\gamma(t_b)$ and switch time $t_b$, it is convenient to explore the second patch.
Walking in Continuous Space
===========================
We have seen, in the first section, the neurological basis of ARS, and its widespread presence to solve many apparently unrelated problems in the animal kingdom. All these problems have a common abstract structure: that of foraging in a patchy environment, that is, in an environment in which the resources are distributed in clumps: there are concentrated areas in which the resource is present, separated by stretches in which no (or little) resource is found.
Some fiddling with genetic algorithms convinced us that ARS is indeed an optimal strategy for this kind of problems. The problem that I should like to consider now is how to characterize this behavior at a large scale. I have shown in the previous section how to decide when to leave a patch and venture in search of another, now I am interested in analyzing the global behavior that results from these decisions. One result, which we have already glimpsed at the end of section \[genetic\] is to see ARS as a type of *random walk*. In this section, we shall study the paths produced by ARS as random walks in a continuum (viz. in ${\mathbb{R}}^n$, usually in ${\mathbb{R}}^2$). As we shall see, an important parameter in this exploration is the exponent $\nu$ of the curves in Figure \[variances\], that is, in the relation $\langle{X^2(t)}\rangle\sim{t^\nu}$; the fact that $\nu>1$ makes ARS walk of a peculiar kind, known as *Levy walks*.
Random Walks and Diffusion Processes
------------------------------------
A random walk is the description of the motion of a point (sometimes called, in reference to the physics in which random walks were first studied, a *particle* or, in reference to ecology, an *individual*) subject to forces that can be modeled as a stochastic process. Random walks can be described at three levels of detail, given by the schema of Figure \[walkonsunshine\]).
Input Description Level Output Math, formalism
------- ------------------- --------------- -------------------------
trajectories Langevin equations
prob. density Master equation
prob. density Fokker-Planck equations
(0,0)(0,0) (0,-4)[(0,1)[8]{}]{} (0.2,-2)[(0,0)\[l\][complexity]{}]{}
We shall consider these levels of description one by one, especially as they apply to the best known model of random walk: *Brownian motion*.
Microscopic description–Langevin equations
------------------------------------------
Langevin’s equations were originally studied for what become the prototype of diffusive random walks, namely *Brownian motion* [@hida:80; @karatzas:12]. In 1827, botanist R. Brown discovered, during microscopic observations, that particles of pollen suspended in water exhibited an incessant and irregular motion [@powles:78]. Vitalist explanations were soon discarded since mineral (viz. non-living) particles exhibited the same phenomenon. Brownian motion occurs when the mass of the particle of pollen is larger than the mass of the molecules of the liquid, so that the continuous collisions drive the particles in a chaotic way (Figure \[brownian\]). The first theoretical explanation of this phenomenon was given in 1905 by Albert Einstein at a macroscopic level [@einstein:05], and we shall consider his approach shortly. In 1906, Paul Langevin offered a microscopic model of Brownian motion based on stochastic differential equations. Langevin’s equations for a one-dimensional Brownian motion (the case that is commonly studied) are: $$\begin{aligned}
\frac{dx}{dt} &= v \\
m\frac{dv}{dt} &= -\gamma v + \sigma \xi(t)
\end{aligned}$$ where $m$ is the particle mass, $\gamma$ is the friction coefficient, and $\xi(t)$ is the force resulting from the impact with the molecules. Langevin assumed that $\xi(t)$ is a Gaussian, uncorrelated stochastic process with zero mean, that is $\langle\xi(t)\rangle=0$, $\langle\xi(t)\xi(t')\rangle=\delta(t-t')$. He also considered the limit of strong friction $m|dv/dt|\ll|\gamma{v}|$ (more on this hypothesis later) so the equations become $$\gamma \frac{dx}{dt} = \sigma\xi(t)$$ and, defining the *diffusion coefficient* $D=\frac{\sigma^2}{2\gamma^2}$, $$\frac{dx}{dt} = \sqrt{2D} \xi(t)$$ or $$\label{intme}
dx = \sqrt{2D}\xi(t) dt = \sqrt{2D}dW(t)$$ where $W(t)$ is a Wiener process (see section \[gausswie\]). Integrating (\[intme\]) we get $$x(t) = x(0) + \sqrt{2D}w(t)$$ So, Brownian motion is the motion of a particle whose displacement is a Wiener process and whose velocity is an uncorrelated Gaussian process. The hypothesis of strong friction is key to obtain velocity as an uncorrelated stochastic process: if inertial phenomena are present, then the velocities at two different instants in time are correlated.
From the solution of this equation we can determine macroscopic quantities such as the mean position $\langle{X}\rangle$ and the mean square displacement $\langle{X^2}\rangle-\langle{X}\rangle^2$: $$\label{lookieloopsie}
\begin{aligned}
\langle X \rangle &= \langle X(0) \rangle + \sqrt{2D} \langle W(t) \rangle = \langle X(0) \rangle \\
\langle{X^2}\rangle-\langle{X}\rangle^2 &= 2D\langle W^2(t) \rangle = 2Dt
\end{aligned}$$ From the second equation we see that $\langle{X^2}\rangle\sim{t}$. This is an equation that we have already encountered: it characterizes the ARS walk for very low $\rho$ and for $\rho\sim{1}$ (figure \[variances\]). It is a behavior typical of *diffusive processes*, and we shall meet it quite a few times in the following.
(15,14)(0,0) (4.5,5.5) (5.5,2.5) (8.5,8.5) (9.5,6.5) (6.5,7.5) (7.5,4.5) (13,4) (10.5,2.5) (12.5,6) (10,11.5) (6.5,10.5) (8.5,12.5) (0.5,1)[(1,1)[4]{}]{} (4.5,5)[(1,-2)[1]{}]{} (5.5,3)[(3,5)[3]{}]{} (8.5,8)[(1,-2)[0.5]{}]{} (9,7)[(-4,1)[2]{}]{} (7,7.5)[(1,-5)[0.5]{}]{} (7.5,5)[(5,-1)[5]{}]{} (12.5,4)[(-3,-2)[1.5]{}]{} (11,3)[(1,3)[1]{}]{} (12,6)[(-2,5)[2]{}]{} (10,11)[(-6,-1)[3]{}]{} (7,10.5)[(1,2)[1]{}]{} (8,12.5)[(-1,2)[0.5]{}]{} (3.5,1.5) (7.5,0.5) (8.5,2.5) (2,7) (2.5,4.5) (4,9) (2.5,11.5) (4.5,12.5) (12.5,10.5) (13.5,8)
Mesoscopic description–Master equation
--------------------------------------
The *master equation* is an ensemble equation that expresses the probability $P(x,t)$ that a particle be at position $x$ at time $t$. It is an integro-differential equation, expressing the time derivative of $P(x,t)$ as a balance of the probability of arriving at $x$ and the probability of leaving the position $x$ once we are there.
Consider a stationary Markov process. For this kind of process, the probability $P(x,t|z,t')$ depends only on $t-t'$, so we can define $P_\tau(x,z)=P(x,t+\tau|z,t)=P(x,\tau|z,0)$.
Consider now $P(x,t+\tau)$, that is, the probability that the walking particle will be in position $x$ at time $t+\tau$. The particle is in $x$ at time $t+\tau$ if there is a position $z$ such that the particle was in $z$ at time $t$ and has moved from $z$ to $x$ in the time interval $\tau$ (if $z=x$, this is the probability that the particle were already in $x$ and hasn’t moved). This event, for a specific $z$, has a probability $P(z,t)P_\tau(x,z)$. Integrating over all possible $z$, we obtain $$P(x,t+\tau) = \int_{-\infty}^\infty\!\!\!\!P(z,t)P_\tau(x,z)\,dz$$ If $\tau\ll{t}$, we can approximate $P(x,t+\tau)$ as $$P(x,t+\tau) = P(x,t) + \frac{\partial P}{\partial t} \tau + O(\tau^2)$$ Let $\omega(x|z)$ be the transition probability per unit time from $z$ to $x$, that is, $\omega(x|z)\tau$ is the probability that the particle go from $z$ to $x$ in a time $\tau$. If the particle is in $x$ at time $t$, then $$\int_{-\infty}^{\infty}\!\!\!\!\omega(z|x)\tau\,dz$$ is the probability that it will move somewhere else, and $$1 - \int_{-\infty}^{\infty}\!\!\!\!\omega(z|x)\tau\,dz$$ is the probability that it will stay in $x$. Balancing the probability of arriving at $x$ and that of not moving if we are already there, we obtain $$P(x,t+\tau) = P(x,t)\left( 1 - \int_{-\infty}^{\infty}\!\!\!\!\omega(z|x)\tau\,dz \right) + \int_{-\infty}^{\infty}\!\!\!\!\omega(x|z)P(z,t)\tau\,dz$$ The first terms gives us the probability that the particle were in $x$ at time $t$ and did not move in the interval $[t,t+\tau]$, while the second is the probability that the particle were in a different position at $t$ and that it moved to $x$ in the interval $[t,t+\tau]$. Rearranging and taking the limit for $\tau\rightarrow{0}$, we obtain the *master equation* $$\frac{\partial}{\partial t} P(x,t) = \int_{-\infty}^{\infty}\!\!\!\!\omega(x|z)P(z,t)\,dz - \int_{-\infty}^{\infty}\!\!\!\!\omega(z|x)P(x,t)\tau\,dz$$ If $X$ is a discrete stochastic process, then, calling $\omega_{nm}$ the prbability of moving from position $x_n$ to position $x_m$ in unit time, the equation becomes $$\frac{\partial}{\partial t} P(n,t) = \sum_m \omega_{mn}P(m,t) - \sum_m \omega_{nm}P(n,t)$$
As an example, consider a counting process that transitions from $n$ to $n+1$ with probability $\lambda$ at each instant, that is, $\omega_{n,n+1}=\lambda$ and $\omega_{nm}=0$ for $m\ne{n+1}$. Then the master equation reads $$\frac{\partial}{\partial t} P(n,t) = \lambda\bigr[ P(n-1,t) - P(n,t) \bigr]$$
This type of equation can be solved through the use of the $z$-transform of the sequence $P(n,t)$, defined as $$\label{this}
F(z,t) = {\mathcal{Z}}[P(n,t)] = \sum_{n=0}^\infty z^n P(n,t)$$ Then $$\sum_{n=0}^\infty z^n \frac{\partial}{\partial t} P(n,t) = \sum_{n=0}^\infty \frac{\partial}{\partial t} (z^n P(n,t)) =
\frac{\partial}{\partial t} F(z,t)$$ and $$\sum_{n=0}^\infty z^n P(n-1,t) = z \sum_{n=1}^\infty z^{n-1} P(n-1,t) = z \sum_{n=0}^\infty z^n P(n,t) = z F(z,t)$$ so that $$\sum_{n=0}^\infty z^n \lambda\bigr[ P(n-1,t) - P(n,t) \bigr] =
\lambda \Bigl[ \sum_{n=0}^\infty z^n P(n-1,t) - \sum_{n=0}^\infty z^n P(n,t) \Bigr] =
\lambda(z-1)F(z,t)$$ resulting in $$\label{ohohoh}
\frac{\partial}{\partial t} F(z,t) = \lambda(z-1)F(z,t)$$ If $P(n,0)=\delta_{n,0}$, it is easy to check that $F(z,0)=1$. With this initial condition, (\[ohohoh\]) can be easily integrated yielding $$F(z,t) = \exp(\lambda(z-1)t) = \sum_{n=0}^\infty z^n \frac{(\lambda t)^n}{n!} e^{-\lambda t}$$ Comparing with (\[this\]) we have that $P(n,t)$ follows a Poisson distribution $$P(n,t) = \frac{(\lambda t)^n}{n!} e^{-\lambda t}$$
Macroscopic level–Fokker-Planck equations
-----------------------------------------
### Diffusion
I shall introduce the macroscopic level of description in a slightly more general setting that needed here before seeing how it related to Brownian motion: as *diffusion*. As I mentioned, the macroscopic level consists in considering that the characteristic magnitudes of the walk (time and distance between collisions) are much smaller than the magnitudes we are considering. This means that we can characterize the problem using a continuous *population density* $\rho({\mathbf{{x}}},t)$: the number of particles in a unit volume around ${\mathbf{{x}}}$ at time $t$ (Figure \[diffmodel\]).
(6,10)(0,0) (0,0)(0,4)[2]{}[(1,0)[4]{}]{} (0,0)(4,0)[2]{}[(0,1)[4]{}]{} (4,4)[(1,1)[2]{}]{} (4,0)[(1,1)[2]{}]{} (0,4)[(1,1)[2]{}]{} (2,6)[(1,0)[4]{}]{} (6,6)[(0,-1)[4]{}]{} (0,0)(0.125,0.125)[16]{} (2,2)(0.125,0)[32]{} (2,2)(0,0.125)[32]{} (3,5)[(0,1)[2]{}]{} (3,5)[(1,2)[2]{}]{} (2.9,7)[(0,0)\[r\][$\mathbf{n}$]{}]{} (5.1,9.1)[(0,0)\[l\][$\mathbf{J}$]{}]{} (3,3)[(0,0)[$\mathbf{\rho}(x)$]{}]{} (0,-0.1)[(0,0)\[t\][$x$]{}]{} (4,-0.1)[(0,0)\[t\][$x+dx$]{}]{} (3,5)[(0,1)[5]{}]{} (5,9)(-0.125,-0.0675)[17]{}
We can approximate the local density of particles with a continuous field, and take the limit for space and time going to zero. In addition to the density, we define the population flow ${\mathbf{{J}}}({\mathbf{{x}}},t)$, which is a vector pointing in the direction of movement and indicating how many particles move per unit time in a surface patch of unitary area. Considering a closed volume $V$ with a closed surface $S$, if there is no generation or annihilation of particles, the variation in density is due to the particles that enter and leave through the surface. So, we have: $$\label{booze}
\frac{\partial}{\partial t}\int_V\!\!\!\rho({\mathbf{{x}}},t)\,dV =
- \oint_S {\mathbf{{J}}}({\mathbf{{x}}},t)\cdot{\mathbf{{n}}}\,dS$$ where ${\mathbf{{n}}}$ is the normal to the surface at ${\mathbf{{x}}}$. By the divergence theorem: $$\oint_S {\mathbf{{J}}}({\mathbf{{x}}},t)\cdot{\mathbf{{n}}}\,dS = \int_V\!\!\!\nabla\cdot{\mathbf{{J}}}\,dV$$ Applying this theorem to (\[booze\]) we have $$\int_V \Bigl[ \frac{\partial \rho({\mathbf{{x}}},t)}{\partial t} + \nabla\cdot{\mathbf{{J}}}({\mathbf{{x}}},t) \Bigr]\,dV = 0$$ Since the volume $V$ is arbitrary, we get the continuity equation $$\label{fock}
\frac{\partial \rho({\mathbf{{x}}},t)}{\partial t} + \nabla\cdot{\mathbf{{J}}}({\mathbf{{x}}},t) = 0$$ In order to get a solvable equation in $\rho$, we need to determine how ${\mathbf{{J}}}$ emerges as a consequence of variations of the population density, that is, how ${\mathbf{{J}}}$ relates to $\rho$. Such an expression is called a *constitutive equation*. One common constitutive equation, known as *Fick’s law*, assume that the flow is proportional to the local population gradient, that is $$\label{fick}
{\mathbf{{J}}}({\mathbf{{x}}},t) = - D \nabla\rho({\mathbf{{x}}},t)$$ The minus sign takes into account the fact that the flow goes from regions of high density to regions of low density. Introducing (\[fick\]) into (\[fock\]) one gets the diffusion equation $$\label{fuck}
\frac{\partial\rho}{\partial t} = \nabla \cdot (D\nabla\rho)$$ If $D$ is a constant independent of ${\mathbf{{x}}}$ then (\[fuck\]) turns into $$\label{diff}
\frac{\partial\rho}{\partial t} = D \nabla^2\rho$$ and, in the one-dimensional case $$\label{diffone}
\frac{\partial\rho}{\partial t} = D \frac{\partial^2 \rho}{\partial x^2}$$ This *diffusion equation* has, in principle, nothing to do with Brownian motion or random walks: it has been derived considering a completely different problem, namely the diffusion of a fluid into space under the action of the gradient of its density. Yet, surprisingly, it turns out that this equation does indeed describe Brownian motion. In particular, it describes the evolution of the probability of finding a Brownian walker in $x$ at time $t$.
### Fokker-Planck equation
The *Fokker-Planck equation* is a partial differential equation in time and space that describes Brownian motion at a macroscopic level. This makes it a macroscopic equation, since the use of differential operators entails that we are considering times and distances much greater than the time and space between changes in direction of a particle. In this section we present the Einstein derivation of the Fokker-Planck equations considering, for the sake of simplicity, the one-dimensional case [@einstein:05].
The motion of a particle in a Brownian motion can be interpreted as a series of jumps that can have an arbitrary length $z$. Let the jump lengths be distributed according to a PDF $\phi(z)$, and let them be i.i.d. The density of individuals at position $x$ at time $t+\tau$ is given by those individuals that were at a position $z$ at time $t$ and that have jumped to $x$ after waiting a time $\tau$. Since $z$ is arbitrary, we integrate over all possible $z$, obtaining a form of non-Markovian Chapman-Kolmogorov equation $$\label{albert}
\rho(x,t+\tau) = \int_{-\infty}^\infty\!\!\!\!\!\rho(x-z,t)\phi(z)\,dz$$ Note that this equation is continuous in space and discrete in time. In particular, the PDF $\phi(z)$, which in ecology is called the *dispersion kernel* is continuous, meaning that we are making an implicit assumption of a large number of individuals/particles. If we now take the macroscopic limit, that is, is we consider that $\tau$ and $z$ are both small with respect to the scale of interest, then we can use a Taylor expansion in $t$ and $z$: $$\begin{aligned}
\rho(x,t+\tau) &= \sum_{n=0}^\infty \frac{\tau^n}{n!} \frac{\partial^n \rho}{\partial \tau^n} \\
\rho(x-z,t) &= \sum_{n=0}^\infty \frac{(-z)^n}{n!} \frac{\partial^n \rho}{\partial z^n}
\end{aligned}$$ Inserting into (\[albert\]), one gets: $$\rho(x,t) + \tau \frac{\partial \rho}{\partial \tau} + \cdots =
\rho(x,t)\int_{-\infty}^\infty\!\!\!\!\!\phi(z)\,dz - \frac{\partial \rho}{\partial x}\int_{-\infty}^\infty\!\!\!\!\!z\phi(z)\,dz
+ \frac{\partial^2 \rho}{\partial x^2}\int_{-\infty}^\infty\!\frac{z^2}{2!}\phi(z)\,dz + \cdots$$ The kernel $\phi(z)$ is a PDF, therefore $\int_{-\infty}^\infty\phi(z)\,dz=1$. Moreover, if the movements are isotropic, that is, there is no preferential direction of movement, then $\phi(z)=\phi(-z)$, and $\int_{-\infty}^\infty{z^n}\phi(z)\,dz=0$ for $n$ odd. So, we have: $$\rho(x,t) + \tau \frac{\partial \rho}{\partial \tau} + O(\tau^2) =
\rho(x,t) + \frac{\partial^2 \rho}{\partial x^2}\int_{-\infty}^\infty\!\frac{z^2}{2!}\phi(z)\,dz + O(z^4)$$ Or, simplifying the common term and dividing by $\tau$, $$\frac{\partial \rho}{\partial \tau} =
\frac{\partial^2 \rho}{\partial x^2}\int_{-\infty}^\infty\!\frac{z^2}{2\tau}\phi(z)\,dz + O(z^4/\tau^2)$$ We now take the macroscopic limit $z,\tau\rightarrow{0}$, but in such a way that $\lim z^2/\tau=C\ne{0}$, that is, keeping $z^2$ and $\tau$ of the same order of magnitude. Then we obtain, as a Fokker-Planck equation, a diffusion equation like (\[diff\]): $$\label{einfock}
\frac{\partial \rho}{\partial t} = D \frac{\partial^2 \rho}{\partial x^2}$$ where $$D = \frac{1}{2\tau} \int_{-\infty}^\infty\!\!\!\!\!z^2\phi(z)\,dz = \frac{\langle{z^2}\rangle}{2\tau}$$ Note that (\[einfock\]) depends only on the second moment of the diffusion kernel $\phi(z)$. In no place have we made the hypothesis that $\phi(z)$ is Gaussian so very different diffusion kernels with the same second moment will generate the same Fokker-Planck equation. We have lost information with respect to the distribution $\phi(z)$: in particular, *given any distribution, a Gaussian distribution with the same second moment will generate the same macroscopic distribution*.
In this sense, we also notice that the Langevin equation does indeed make the hypothesis that $\xi(t)$ is Gaussian. It would therefore seem that the Einstein derivation is more general than the Langevin equation. In reality, it is not so, and the reason is the Central Limit Theorem[^4]: the hypothesis that $z^2$ and $\tau$ be of the same magnitude entails that $D$ is finite and therefore, by (\[einfock\]), that $\langle{Z^2}\rangle$ is finite. We are in the hypotheses of the Central Limit Theorem, so the sum of all the jumps of the Einstein’s derivation, with distribution $\phi(z)$, will end up being Gaussian regardless of the exact shape of $\phi(z)$.
### Solution of the Diffusion Equation
One simple way to solve the diffusion equation is through the use of the Characteristic function of the distribution $\rho$, viz. its Fourier Transform. Taking the Fourier transform of (\[diffone\]) we get the ordinary differential equation $$\frac{d\tilde{\rho}}{dt}(\omega,t) = -D \omega^2 \tilde{\rho}(\omega,t)$$ which has solution $$\tilde{\rho}(\omega,t) = \tilde{\rho}(\omega,0)\exp\bigl[-D\omega^2t\bigr]$$ In the simplest case, at the beginning all the individuals are concentrated at $x_0$, that is, $\rho(x,0)=\delta(x-x_0)$. By the formula for the characteristic function of the Dirac distribition (\[deltachar\]), we have $\tilde{\rho}(\omega,0)=\exp\bigl[-i\omega{x_0}\bigr]$. That is $$\tilde{\rho}(\omega,t) = \exp\bigl[-i\omega{x_0}-D\omega^2t\bigr]$$ The inverse Fourier transform gives us $$\label{pussy}
\rho(x,t) = \frac{1}{2\pi} \int_{-\infty}^\infty\!\!\!e^{i\omega{x}}\tilde{\rho}(\omega,t)\,d\omega =
\frac{1}{2\pi} \int_{-\infty}^\infty\!\!\!\exp\bigl[ i\omega(x-x_0)-D\omega^2{t}\bigr]\,d\omega =
\frac{1}{\sqrt{4\pi{Dt}}} \exp\Bigl[-\frac{(x-x_0)^2}{4Dt}\Bigr]$$ From this solution we can obtain the general solution for $\rho(x,0)=g(x)$. Writing $$g(y) = \int_{-\infty}^\infty\!\!\!g(x)\delta(x-y)\,dy$$ and applying superposition we have $$\rho(x,t) = \frac{1}{\sqrt{4\pi{Dt}}} \int_{-\infty}^\infty\!\!\!\!\!g(y)\exp\Bigl[-\frac{(y-x_0)^2}{4Dt}\Bigr]\,dy$$ With reference to the simple solution (\[pussy\]), Figure \[difsol\] shows $\rho(x,t)$ as a function of $x$ for several values of $t$.
(839,839)(0,0) (56.0,82.0)
------------------------------------------------------------------------
(56,41)[(0,0)[$-8$]{}]{} (56.0,778.0)
------------------------------------------------------------------------
(146.0,82.0)
------------------------------------------------------------------------
(146,41)[(0,0)[$-6$]{}]{} (146.0,778.0)
------------------------------------------------------------------------
(235.0,82.0)
------------------------------------------------------------------------
(235,41)[(0,0)[$-4$]{}]{} (235.0,778.0)
------------------------------------------------------------------------
(325.0,82.0)
------------------------------------------------------------------------
(325,41)[(0,0)[$-2$]{}]{} (325.0,778.0)
------------------------------------------------------------------------
(414.0,82.0)
------------------------------------------------------------------------
(414,41)[(0,0)[$0$]{}]{} (414.0,778.0)
------------------------------------------------------------------------
(504.0,82.0)
------------------------------------------------------------------------
(504,41)[(0,0)[$2$]{}]{} (504.0,778.0)
------------------------------------------------------------------------
(593.0,82.0)
------------------------------------------------------------------------
(593,41)[(0,0)[$4$]{}]{} (593.0,778.0)
------------------------------------------------------------------------
(683.0,82.0)
------------------------------------------------------------------------
(683,41)[(0,0)[$6$]{}]{} (683.0,778.0)
------------------------------------------------------------------------
(772.0,82.0)
------------------------------------------------------------------------
(772,41)[(0,0)[$8$]{}]{} (772.0,778.0)
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(772.0,82.0)
------------------------------------------------------------------------
(56.0,798.0)
------------------------------------------------------------------------
(450,735)[(0,0)\[l\][t=0.1]{}]{} (463,286)[(0,0)\[l\][t=0.5]{}]{} (597,123)[(0,0)\[l\][t=2]{}]{} (56,82) (331,81.67)
------------------------------------------------------------------------
(331.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(338,82.67)
------------------------------------------------------------------------
(338.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(345.00,84.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(345.00,83.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(353.59,89.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(352.17,89.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(360.59,100.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(359.17,100.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(367.59,128.00)(0.485,4.154)[11]{}
------------------------------------------------------------------------
(366.17,128.00)(7.000,48.269)[2]{}
------------------------------------------------------------------------
(374.59,183.00)(0.485,7.052)[11]{}
------------------------------------------------------------------------
(373.17,183.00)(7.000,81.762)[2]{}
------------------------------------------------------------------------
(381.59,276.00)(0.488,8.748)[13]{}
------------------------------------------------------------------------
(380.17,276.00)(8.000,118.990)[2]{}
------------------------------------------------------------------------
(389.59,409.00)(0.485,11.934)[11]{}
------------------------------------------------------------------------
(388.17,409.00)(7.000,138.172)[2]{}
------------------------------------------------------------------------
(396.59,566.00)(0.485,10.943)[11]{}
------------------------------------------------------------------------
(395.17,566.00)(7.000,126.714)[2]{}
------------------------------------------------------------------------
(403.59,710.00)(0.485,6.671)[11]{}
------------------------------------------------------------------------
(402.17,710.00)(7.000,77.355)[2]{}
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(418.59,776.71)(0.485,-6.671)[11]{}
------------------------------------------------------------------------
(417.17,787.36)(7.000,-77.355)[2]{}
------------------------------------------------------------------------
(425.59,675.43)(0.485,-10.943)[11]{}
------------------------------------------------------------------------
(424.17,692.71)(7.000,-126.714)[2]{}
------------------------------------------------------------------------
(432.59,528.34)(0.485,-11.934)[11]{}
------------------------------------------------------------------------
(431.17,547.17)(7.000,-138.172)[2]{}
------------------------------------------------------------------------
(439.59,380.98)(0.488,-8.748)[13]{}
------------------------------------------------------------------------
(438.17,394.99)(8.000,-118.990)[2]{}
------------------------------------------------------------------------
(447.59,253.52)(0.485,-7.052)[11]{}
------------------------------------------------------------------------
(446.17,264.76)(7.000,-81.762)[2]{}
------------------------------------------------------------------------
(454.59,169.54)(0.485,-4.154)[11]{}
------------------------------------------------------------------------
(453.17,176.27)(7.000,-48.269)[2]{}
------------------------------------------------------------------------
(461.59,120.94)(0.485,-2.094)[11]{}
------------------------------------------------------------------------
(460.17,124.47)(7.000,-24.472)[2]{}
------------------------------------------------------------------------
(468.59,96.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(467.17,98.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(475.00,87.93)(0.821,-0.477)[7]{}
------------------------------------------------------------------------
(475.00,88.17)(6.464,-5.000)[2]{}
------------------------------------------------------------------------
(483,82.67)
------------------------------------------------------------------------
(483.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(490,81.67)
------------------------------------------------------------------------
(490.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(410.0,798.0)
------------------------------------------------------------------------
(497.0,82.0)
------------------------------------------------------------------------
(56,82) (251,81.67)
------------------------------------------------------------------------
(251.00,81.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(266,82.67)
------------------------------------------------------------------------
(266.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(273,84.17)
------------------------------------------------------------------------
(273.00,83.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(280,86.17)
------------------------------------------------------------------------
(280.00,85.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(287.00,88.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(287.00,87.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(295.00,91.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(295.00,90.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(302.00,96.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(302.00,95.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(309.59,103.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(308.17,103.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(316.59,112.00)(0.488,0.758)[13]{}
------------------------------------------------------------------------
(315.17,112.00)(8.000,10.547)[2]{}
------------------------------------------------------------------------
(324.59,124.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(323.17,124.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(331.59,140.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(330.17,140.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(338.59,159.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(337.17,159.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(345.59,182.00)(0.488,1.748)[13]{}
------------------------------------------------------------------------
(344.17,182.00)(8.000,23.990)[2]{}
------------------------------------------------------------------------
(353.59,209.00)(0.485,2.171)[11]{}
------------------------------------------------------------------------
(352.17,209.00)(7.000,25.353)[2]{}
------------------------------------------------------------------------
(360.59,238.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(359.17,238.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(367.59,269.00)(0.485,2.399)[11]{}
------------------------------------------------------------------------
(366.17,269.00)(7.000,27.997)[2]{}
------------------------------------------------------------------------
(374.59,301.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(373.17,301.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(381.59,332.00)(0.488,1.748)[13]{}
------------------------------------------------------------------------
(380.17,332.00)(8.000,23.990)[2]{}
------------------------------------------------------------------------
(389.59,359.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(388.17,359.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(396.59,382.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(395.17,382.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(403.59,398.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(402.17,398.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(259.0,83.0)
------------------------------------------------------------------------
(418.59,403.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(417.17,404.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(425.59,393.79)(0.485,-1.179)[11]{}
------------------------------------------------------------------------
(424.17,395.89)(7.000,-13.895)[2]{}
------------------------------------------------------------------------
(432.59,376.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(431.17,379.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(439.59,352.98)(0.488,-1.748)[13]{}
------------------------------------------------------------------------
(438.17,355.99)(8.000,-23.990)[2]{}
------------------------------------------------------------------------
(447.59,324.23)(0.485,-2.323)[11]{}
------------------------------------------------------------------------
(446.17,328.12)(7.000,-27.116)[2]{}
------------------------------------------------------------------------
(454.59,292.99)(0.485,-2.399)[11]{}
------------------------------------------------------------------------
(453.17,297.00)(7.000,-27.997)[2]{}
------------------------------------------------------------------------
(461.59,261.23)(0.485,-2.323)[11]{}
------------------------------------------------------------------------
(460.17,265.12)(7.000,-27.116)[2]{}
------------------------------------------------------------------------
(468.59,230.71)(0.485,-2.171)[11]{}
------------------------------------------------------------------------
(467.17,234.35)(7.000,-25.353)[2]{}
------------------------------------------------------------------------
(475.59,202.98)(0.488,-1.748)[13]{}
------------------------------------------------------------------------
(474.17,205.99)(8.000,-23.990)[2]{}
------------------------------------------------------------------------
(483.59,176.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(482.17,179.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(490.59,154.08)(0.485,-1.408)[11]{}
------------------------------------------------------------------------
(489.17,156.54)(7.000,-16.539)[2]{}
------------------------------------------------------------------------
(497.59,135.79)(0.485,-1.179)[11]{}
------------------------------------------------------------------------
(496.17,137.89)(7.000,-13.895)[2]{}
------------------------------------------------------------------------
(504.59,121.09)(0.488,-0.758)[13]{}
------------------------------------------------------------------------
(503.17,122.55)(8.000,-10.547)[2]{}
------------------------------------------------------------------------
(512.59,109.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(511.17,110.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(519.00,101.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(519.00,102.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(526.00,94.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(526.00,95.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(533.00,89.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(533.00,90.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(541,86.17)
------------------------------------------------------------------------
(541.00,87.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(548,84.17)
------------------------------------------------------------------------
(548.00,85.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(555,82.67)
------------------------------------------------------------------------
(555.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(410.0,406.0)
------------------------------------------------------------------------
(569,81.67)
------------------------------------------------------------------------
(569.00,82.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(562.0,83.0)
------------------------------------------------------------------------
(577.0,82.0)
------------------------------------------------------------------------
(56,82) (107,81.67)
------------------------------------------------------------------------
(107.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(136,82.67)
------------------------------------------------------------------------
(136.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(114.0,83.0)
------------------------------------------------------------------------
(150,83.67)
------------------------------------------------------------------------
(150.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(143.0,84.0)
------------------------------------------------------------------------
(164,84.67)
------------------------------------------------------------------------
(164.00,84.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(172,85.67)
------------------------------------------------------------------------
(172.00,85.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(179,86.67)
------------------------------------------------------------------------
(179.00,86.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(186,88.17)
------------------------------------------------------------------------
(186.00,87.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(193,89.67)
------------------------------------------------------------------------
(193.00,89.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(201,91.17)
------------------------------------------------------------------------
(201.00,90.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(208.00,93.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(208.00,92.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(215,96.17)
------------------------------------------------------------------------
(215.00,95.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(222.00,98.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(222.00,97.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(230.00,101.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(230.00,100.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(237.00,105.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(237.00,104.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(244.00,109.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(244.00,108.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(251.00,113.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(251.00,112.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(259.00,118.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(259.00,117.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(266.00,123.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(266.00,122.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(273.00,129.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(273.00,128.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(280.00,135.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(280.00,134.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(287.00,142.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(287.00,141.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(295.00,149.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(295.00,148.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(302.59,156.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(301.17,156.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(309.59,164.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(308.17,164.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(316.00,172.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(316.00,171.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(324.59,180.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(323.17,180.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(331.59,188.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(330.17,188.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(338.00,196.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(338.00,195.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(345.00,203.59)(0.494,0.488)[13]{}
------------------------------------------------------------------------
(345.00,202.17)(6.962,8.000)[2]{}
------------------------------------------------------------------------
(353.00,211.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(353.00,210.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(360.00,217.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(360.00,216.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(367.00,224.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(367.00,223.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(374.00,229.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(374.00,228.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(381.00,234.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(381.00,233.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(389.00,238.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(389.00,237.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(396.00,241.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(396.00,240.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(403,243.67)
------------------------------------------------------------------------
(403.00,243.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(157.0,85.0)
------------------------------------------------------------------------
(418,243.67)
------------------------------------------------------------------------
(418.00,244.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(425.00,242.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(425.00,243.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(432.00,239.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(432.00,240.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(439.00,236.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(439.00,237.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(447.00,232.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(447.00,233.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(454.00,227.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(454.00,228.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(461.00,222.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(461.00,223.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(468.00,215.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(468.00,216.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(475.00,209.93)(0.494,-0.488)[13]{}
------------------------------------------------------------------------
(475.00,210.17)(6.962,-8.000)[2]{}
------------------------------------------------------------------------
(483.00,201.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(483.00,202.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(490.59,193.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(489.17,194.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(497.59,185.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(496.17,186.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(504.00,178.93)(0.494,-0.488)[13]{}
------------------------------------------------------------------------
(504.00,179.17)(6.962,-8.000)[2]{}
------------------------------------------------------------------------
(512.59,169.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(511.17,170.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(519.59,161.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(518.17,162.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(526.00,154.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(526.00,155.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(533.00,147.93)(0.569,-0.485)[11]{}
------------------------------------------------------------------------
(533.00,148.17)(6.844,-7.000)[2]{}
------------------------------------------------------------------------
(541.00,140.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(541.00,141.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(548.00,133.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(548.00,134.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(555.00,127.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(555.00,128.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(562.00,121.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(562.00,122.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(569.00,116.93)(0.821,-0.477)[7]{}
------------------------------------------------------------------------
(569.00,117.17)(6.464,-5.000)[2]{}
------------------------------------------------------------------------
(577.00,111.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(577.00,112.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(584.00,107.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(584.00,108.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(591.00,103.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(591.00,104.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(598.00,99.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(598.00,100.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(606,96.17)
------------------------------------------------------------------------
(606.00,97.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(613.00,94.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(613.00,95.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(620,91.17)
------------------------------------------------------------------------
(620.00,92.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(627,89.67)
------------------------------------------------------------------------
(627.00,90.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(635,88.17)
------------------------------------------------------------------------
(635.00,89.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(642,86.67)
------------------------------------------------------------------------
(642.00,87.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(649,85.67)
------------------------------------------------------------------------
(649.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(656,84.67)
------------------------------------------------------------------------
(656.00,85.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(410.0,245.0)
------------------------------------------------------------------------
(671,83.67)
------------------------------------------------------------------------
(671.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(664.0,85.0)
------------------------------------------------------------------------
(685,82.67)
------------------------------------------------------------------------
(685.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(678.0,84.0)
------------------------------------------------------------------------
(714,81.67)
------------------------------------------------------------------------
(714.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(692.0,83.0)
------------------------------------------------------------------------
(721.0,82.0)
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(56.0,82.0)
------------------------------------------------------------------------
(772.0,82.0)
------------------------------------------------------------------------
(56.0,798.0)
------------------------------------------------------------------------
The result is typical of diffusion processes in which the population, initially concentrated at $x=0$ ($\rho(x,0)=\delta(x)$) spreads over larger and larger areas. The [“[speed]{}”]{} of this diffusion is determined by the increasing variance $\sigma(t)=\langle{X^2}\rangle=2Dt$ which, as predicted by (\[lookie\]) and (\[einfock\]), grows linearly with $t$.
Anomalous Diffusion
-------------------
The standard diffusion process, which we have considered so far, is characterized by the relation $$\label{difflaw}
\langle X^2 \rangle \sim t$$ where the notation is shorthand for $$\lim_{t\rightarrow\infty} \frac{\langle X^2 \rangle}{t} = C \ne 0$$ The reason for this boils down to the fact that the diffusion equation has first derivatives in time and second derivatives in space, so that one obtain homologous quantities starting with a constant and integrating once in time and twice in space.
To see a different kind of behavior, consider *ballistic displacements*, that is, the motion of a particle that moves at a constant speed $v$ and never changes direction. For a movement along the $x$ axis, this can be modeled as a stochastic process with PDF $P(x,t)=\delta(x-vt)$ so that $$\label{ball}
\langle X^2 \rangle = \int_{-\infty}^\infty\!\!\!x^2\delta(x-vt)\,dx = v^2t^2\sim t^2$$ That is, in this case $$\lim_{t\rightarrow\infty} \frac{\langle X^2 \rangle}{t^2} = C \ne 0$$ We can see ballistic movement as a type of random walk, albeit a not-quite-so-random one, and one that moves from the origin much faster than the Brownian motion. Small wonder: ballistic movement moves purposely in a fixed direction, while Brownian motion is bounced to and fro. Note, however, that this means that ballistic movement will explore around much less than Brownian motion: it will stick to a trajectory and not look around at all. Just like a traveler in a rush: you may go very far, but you miss the view.
Processes that don’t follow the standard diffusion law (\[difflaw\]) are called *anomalous* . Ballistic movement is our first example of anomalous diffusion (albeit a rather pathological one). The asymptotic relation between $\langle{X^2}\rangle$ and $t$ is normally defined using the *Hurst exponent* $H$ defined by $$\label{hurst}
\langle X^2 \rangle \sim t^{2H}$$ Diffusion corresponds to $H=1/2$; if $H<1/2$ we have *subdiffusion*, while for $1/2<H<1$ we have *superdiffusion*. The ballistic limit (\[ball\]) is achieved for $H=1$ (see Figure \[hurstfig\]). Note that, because of the central limit theorem, normal diffusion ($H=1/2$) is obtained under a wide family of displacements distributions. If the displacements:
i)
: are independent,
ii)
: are identically distributed, and
iii)
: follow a PDF with finite mean and variance,
then we can apply the CLT in its standard form, and the total distance covered (that is, the sum of all these displacements) is a Gaussian $\exp(-x^2/\sigma^2)$, where $\sigma^2$ is proportional to the number of displacements, that is, $\sigma^2\sim{t}$. Then (\[difflaw\]) follow directly from the equality $\langle{X^2}\rangle=\sigma^2$ valid for a Gaussian.
The hypotheses i)–iii) hint at three possible ways in which they can be violated, resulting in three different mechanisms that can generate anomalous diffusion.
i)
: the displacements are not independent due to long range correlations: once a particle moves, it will tend to remain in motion (leading to superdiffusion—ballistic motion is an example of this kind of process) or, contrariwise [^5] , once it stops it will tend to remain at rest (leading to subdiffusion);
ii)
: the distribution of the displacements is not identical, either because they become shorter with time (leading to subdiffusion) or because they become longer (leading to superdiffusion);
iii)
: the displacements are distributed according to a PDF with infinite variance, so that arbitrary large displacements are relatively likely.
(20,10.5)(0,-0.5) (0,0)(0,10)[2]{}[(1,0)[20]{}]{} (0,0)(20,0)[2]{}[(0,1)[10]{}]{} (0,0)(1,0)[20]{}[(0,1)[0.2]{}]{} (0,0)(0,1)[10]{}[(1,0)[0.2]{}]{} (0,0)[(2,1)[20]{}]{} (0,0)[(1,1)[10]{}]{} (1,9)[(0,0)\[lt\]]{} (5,5.5)
(0,0)
[45]{}$H=1$; ballistic limit
(8,4.2)
(0,0)
[28]{}$H=1/2$; normal diffusion
(11,9)[(0,0)\[lt\]]{} (15,5)[(0,0)\[lt\]]{} (10,0.4)[(0,0)\[lb\]]{} (10,-0.2)[(0,0)\[t\][$\log\,\,{t}$]{}]{} (-0.3,5)
(0,0)\[t\]
[90]{}$\log\,\,\langle{X^2}\rangle$
In the following, we shall consider mostly the third case but, before digging into it, I shall give a brief example of how the first two work.
For the case of long-term correlations, divide the trajectory into intervals of fixed duration $\Delta{t}$. With this division, the correlations between displacements are the same as those between velocities. In this case, we can determine the derivative of $\langle{X^2}\rangle$ as $$\label{kubo}
\frac{d}{dt}\langle{X^2}\rangle = \frac{d}{dt} \int_0^t\!\!\!\int_0^t \langle{v(s)v(\tau)}\rangle\,d\tau\,ds
= 2 \int_0^t\!\!\!\langle{v(t)v(\tau)}\rangle\,d\tau$$ This result is known as the Taylor’s formula (also known as the Green-Kubo formula). If $\langle{v(t)v(\tau)}\rangle$ is integrable, then the limit for $t\rightarrow\infty$ of the integral exists, so the right-hand side of (\[kubo\]) is asymptotically a constant, that is, for $t\rightarrow\infty$, $$\frac{d}{dt}\langle{X^2}\rangle \sim C \mbox{~~~or~~~} \langle{X^2}\rangle\sim t$$ and we find again a diffusive behavior. If, on the other hand, the correlation decays slowly enough that the integral diverges, then the CLT doesn’t hold, and we observe anomalous diffusion. If, for example, $\langle{v(t)v(\tau)}\rangle\sim(t-\tau)^{-\eta}$, with $0<\eta<1$, then $\langle{X^2}\rangle\sim{t}^{2-\eta}$, that is, we have superdiffusion.
Non-identical displacements ocurr when displacements become either longer or shorter with time or, equivalently, as the particle gets farther from its initial position. If we take a macroscopic point of view—that is, if we write a diffusion-like Fokker-Planck equation—then we can model this as a time and/or space varying coefficient $D$. This is tantamount to saying that the obstacles to motions become gradually larger or smaller as we get away from the initial position. Consider a space-dependent diffusion coefficient that varies as a power law: $D=D_0x^\theta$. This leads to a diffusion equation: $$\frac{\partial\rho}{\partial t} = \nabla\cdot(D_0 x^\theta \nabla\rho)$$ A rigorous derivation of the behavior of $\langle{X^2}\rangle$ under this equation can be found in ; here we shall do a simple informal derivation using dimensional analysis. The density $\rho$ is a number of particles per unit of $x$ that is, dimensionally, $[\rho]=[x]^{-1}$ and, consequently $$\left[ \frac{\partial \rho}{\partial t} \right] = [x]^{-1}[T]^{-1}$$ where $T$ is the dimension of time. Similarly $[\nabla\rho]=[x]^{-2}$, $[x^\theta\nabla\rho]=[x]^{\theta-2}$ and $[\nabla\cdot(x^\theta\nabla\rho)]=[x]^{\theta-3}$. This equality gives us $$[x]^{-1}[T]^{-1} = \left[ \frac{\partial \rho}{\partial t} \right] = [\nabla\cdot(x^\theta\nabla\rho)]=[x]^{\theta-3}$$ that is, $[T]^{-1}=[x]^{\theta-2}$, $[T]=[x]^{2-\theta}$, or $[x]=[T]^{1/(2-\theta)}$, whch leads to $$[x^2] = [x]^2 = [T]^{\frac{2}{2-\theta}} = [X^2]$$ This dimensional equality indicates that, asymptotically, $$\langle X^2 \rangle \sim t^{\frac{2}{2-\theta}}$$ leading, again, to anomalous diffusion.
The case of divergent moments that I shall consider closely is that of *Lévy flights*. If the individual displacements are i.i.d., then we are in the conditions of the generalized Central Limit Theorem: no matter what the individual PDF are, the sum of a large number of them will converge to a Lévy stable distribution. So, just like in the finite moment case we could assume that the displacements followed a Gaussian distribution[^6], we can now assume that they follow a Lévy distribution which, as seen in (\[levypow\]), behaves like $x^{-(1+\alpha)}$ for $t\rightarrow\infty$. For $\alpha<2$, $\langle{X^2}\rangle$ diverges due to the [“[long tail]{}”]{} of the distribution, which makes arbitrarily large displacements relatively frequent.
In order to frame these ideas properly, it is first necessary to study random walks from a slightly more general point of view, that of Continuous Time Random Walks.
Continuous Time Random Walks
----------------------------
The random walks that we have considered so far were limits of what we can consider a discrete time scenario: we considered that jumps take place at regular time intervals, and we take the limit of $\Delta{t}\rightarrow{0}$, corresponding to a continuum of jumps of length zero (this is enforced by the fact that we require $\langle{x^2}\rangle/\tau$ to stay finite). In a *Continuous Time Random Walk* (CTRW) we assume that the waiting time between jumps is a random process as well, that is, that the particle will intersperse jumps of random length with pauses of random duration. I shall introduce the analysis of CTRW in two steps: first I shall consider the PDF of the position of the particle after $n$ jumps, without considering *when* did these jumps occur, then the probability of doing $n$ jumps in time $t$. This corresponds to a specific type of CTRW, one in which the jump length is independent of the waiting time. The result can easily be extended to the case in which waiting time and jump length are correlated.
Let $Z_n$ be the length of the $n$th jump. The position of a particle after $n$ jumps is $$X_n = \sum_{k=1}^n Z_k = X_{n-1} + Z_n$$ This equation shows that the walk is a Markov chain. Let $Z_k$ be i.i.d. with PDF $\phi(z)$; the function $\phi$ (the dispersal kernel) represents the transition probability of the Markov chain. Adapting the Chapman-Kolmogorov equation (\[albert\]) to this discrete-time scenario, we obtain an equation for $\rho_n(x)$, the density of individuals after $n$ jumps: $$\rho_n(x) = \int_{-\infty}^{\infty}\!\!\!\rho_{n-1}(x-z)\phi(z)\,dz = \rho_{n-1}*\phi$$ where $*$ denotes spatial convolution. If $\rho_0$ is the initial density, then: $$\begin{aligned}
\rho_1 &= \rho_0 * \phi \\
\rho_2 &= \rho_1 * \phi = \rho_0 * \phi * \phi \\
\vdots \\
\rho_n &= \rho_{n-1} * \phi = \rho_0 * \overbrace{\phi * \cdots * \phi}^{n}
\end{aligned}$$ Considering, for the sake of simplicity, the one-dimensional case, we can take the Fourier transform and apply (\[pluck\]) to obtain $$\label{palooza}
\tilde{\rho}_n(\omega) = \tilde{\rho}_0(\omega) \tilde{\phi}^n(\omega)$$
Consider now the jump times. Let $\theta_n$ be the waiting time between jump $n-1$ and jump $n$, and $\psi(t)$ its PDF. The time at which the $n$th jump is taken is then $$T_n = \sum_{k=1}^n \theta_n$$ Let $\psi^0(t)$ be the probability that no jump has ocurred by time $t$, viz. $$\psi^0(t) = \int_t^\infty\!\!\!\psi(u)\,du = 1 - \int_0^t\!\!\!\psi(u)\,du$$ Let $P_n(t)$ be the probability of performing $n$ jumps by time $t$. Then, clearly, $P_0(t)=\psi^0(t)$. The probability that there is a jump at a time $u<t$ and then no further jumps until time $t$ is $\psi(u)\psi^0(t-u)$. Integrating over all $u<t$ we have[^7]$$P_1(t) = \int_0^t\!\!\!\psi(u)\psi^0(t-u)\,du = \psi*\psi$$ Iterating this, we have $$P_n(t) = \psi^0*\overbrace{\psi* \cdots *\psi}^{n}$$ In this case, since we have different limits and a different convolution, one must use the Laplace transform in lieu of the Fourier: $$\psi(s)=\int_0^\infty\!\!\!e^{-st}\psi(t)\,dt$$ where $s\in{\mathbb{C}}$.[^8] This leads to $$\tilde{P}_n(s) = \tilde{\psi^0}(s)\tilde{\psi}^{n}(s) = \frac{1-\tilde{\psi}(s)}{s}(\tilde{\psi}(s))^{n}$$
\* \* \*
Consider now the combination of the two processes. The position of an individual at time $t$ (assume $x(0)=0$) is $$x(t)=\sum_{k=0}^{N(t)} z_k$$ where $N(t)$ is the number of jumps taken before time $t$, itself a random variable. We are interested in finding an expression for $\rho(t)$, the density of individuals at time $t$. If by time $t$ $n$ jumps have been made, then $$\rho(x,t | N(t)=n) = \rho_n(x)$$ The value of $\rho(x,t)$ is then given by the value of $\rho_n$ for all possible $n$, weighted by their probability: $$\rho(x,t) = \sum_{n=0}^\infty \rho_n(x)P_n(t)$$ that is, taking the Fourier and Laplace transforms: $$\label{ooohh}
\begin{aligned}
\tilde{\rho}(\omega,s) &= \sum_{n=0}^\infty \tilde{\rho}_n(\omega)\tilde{P}_n(t) \\
&= \tilde{\rho}(\omega,0) \frac{1-\tilde{\psi}(s)}{s} \sum_{n=0}^\infty \bigl[\phi(\omega)\psi(s)\bigr]^n \\
&= \tilde{\rho}(\omega,0) \frac{1-\tilde{\psi}(s)}{s} \frac{1}{1-\phi(\omega)\psi(s)}
\end{aligned}$$ This is known as the *Montroll-Wei$\beta$* equation. I made here the assumption that the waiting times and the jump length are independent, hence the product $\phi(\omega)\psi(s)$. If they are not, then their joint probability would be expressed by a distribution $\phi(\omega,s)$, and (\[ooohh\]) becomes $$\tilde{\rho}(\omega,s) = \tilde{\rho}(\omega,0) \frac{1-\tilde{\psi}(s)}{s} \frac{1}{1-\phi(\omega,s)}$$ For any distribution $\phi(\omega)$ and $\psi(s)$, (\[ooohh\]) allows us to determine the evolution of the density of individuals by taking the inverse Fourier/Laplace transform.
### Finite moments: diffusion
The generality of the CTRW notwithstanding, if we assume that $\phi$ and $\psi$ have finite moment we still revert to the normal diffusive behavior. In order to see this, we first rearrange (\[ooohh\]) in a more useful form. From (\[ooohh\]), we write $$\label{auhh}
s\tilde{\rho}(\omega,s) - \tilde{\rho}(\omega,0) = \tilde{\rho}(\omega,0)
\Bigl[ \frac{1 - \psi(s)}{1 - \phi(\omega)\psi(s)} - 1 \Bigr]$$ Express, from the same equation $$\tilde{\rho}(\omega,0) = \frac{s}{1-\psi(s)} \bigl[1 - \phi(\omega)\psi(s)\bigr]\tilde{\rho}(\omega,s)$$ Replacing in the right-hand side of (\[auhh\]) and simplifying we get $$\label{bebop}
s\tilde{\rho}(\omega,s) - \tilde{\rho}(\omega,0)
\frac{s\psi(s)}{1-\psi(s)} \bigl[1 - \phi(\omega)\psi(s)\bigr]\tilde{\rho}(\omega,s)$$ The quantity $$M(s) = \frac{s\psi(s)}{1 - \psi(s)}$$ is called the *memory kernel* of the CTRW. Equation (\[bebop\]) can in turn be rewritten in a way that separates the spatial and temporal variables: $$\label{alula}
\frac{1-\psi(s)}{s\psi(s)}\bigl[ s\tilde{\rho}(\omega,s) - \tilde{\rho}(\omega,o)\bigr] =
\bigl[1 - \phi(\omega)\psi(s)\bigr]\tilde{\rho}(\omega,s)$$ We now consider the macroscopic limit in space, which entails assuming that the microscopic scale of the process is very small compared to the scale of $x$. This means that we shall consider the limit for $\omega\rightarrow{0}$ in the Fourier space. Similarly, the macroscopic limit in time consists in taking the limit $s\rightarrow{0}$ in the complex plane of the Laplace transform. If $\phi$ is symmetric and has finite moments, then it has an expansion $\phi(\omega)=1-\langle\phi^2\rangle\omega^2/2+o(\omega^4)$, where $\langle\phi\rangle$ is the average displacement $\langle{X^2}\rangle$ when $X$ has PDF $\phi$. Similarly, $\psi(s)=1-\langle\psi\rangle{s}+o(s^2)$, where $\langle\psi\rangle$ is the mean waiting time. From this we get $$\frac{1-\psi^0(s)}{s\psi(s)} \sim \frac{\langle\psi\rangle}{1-\langle\psi\rangle{s}} = \langle\psi\rangle + o(s)$$ while $$\bigl[\phi(\omega)-1\bigr] = - \frac{\langle\phi^2\rangle}{2}\omega^2 + o(\omega^2)$$ Putting these in (\[alula\]) we have $$\langle\psi\rangle[s\rho(\omega,s)-\rho(\omega,0)] = \frac{\langle\phi^2\rangle}{2}\omega^2 \rho(\omega,s)$$ that is, taking the inverse Fourier and Laplace transforms $$\frac{\partial}{\partial t}\rho(x,t) = \frac{\langle\phi^2\rangle}{2\langle\psi\rangle} \frac{\partial^2}{\partial x^2}\rho(x,t)$$ that is, we are back to a diffusion equation that behaves like (\[hurst\]), with $H=1/2$.
We obtain anomalous diffusion in two ways: we can either make long pauses (viz. pauses with a distribution with diverging variance) or we can make long jumps.
### Long pauses
Let us assume that $\phi(z)$ has a Gaussian distribution[^9], while $\psi(t)$ has a Lévy distribution with a long tail $$\psi(t) \sim A_\alpha \left( \frac{\tau}{t} \right)^\alpha$$ with $0<\alpha<1$. We are interested in the long term behavior of the walk, that is, in terms of characteristic functions, in the limit $\omega\rightarrow{0}$, $|s|\rightarrow{0}$ so we can write $$\begin{aligned}
\phi(\omega) &= \exp\bigl(- \frac{\omega^2\sigma^2}{2} \bigr) \sim 1 - \sigma^2\omega^2 \\
\psi(s) &= \exp\bigl( -\tau^\alpha|s|^\alpha \bigr) \sim 1 - (\tau{s})^\alpha
\end{aligned}$$ Introducing into (\[ooohh\]), we get $$\tilde{\rho(\omega,s)} = \frac{1}{s} \frac{\rho(\omega,0)}{1 + K_\alpha\omega^2s^{-\alpha}}$$ with $K_\alpha = \sigma^2/\tau^{\alpha}$. The long term behavior of $\langle{X^2}\rangle$ can be determined using the relation $$\langle X^2 \rangle = \lim_{\omega\rightarrow{0}} - \frac{\partial \tilde{\rho}}{\partial \omega}$$ For $\rho(x,0)=\delta(x)$, i.e. $\tilde{\rho}(\omega,0)=1$, we have $$\begin{aligned}
\langle X^2 \rangle &= \lim_{\omega\rightarrow{0}} \Bigl[
- \frac{2}{s} K_\alpha s^{-\alpha} (1+K_\alpha\omega^2s^{-\alpha})^{-2} - \frac{8}{s}(1+K_\alpha\omega^2s^{-\alpha})^{-3}(K_\alpha s^{-\alpha}\omega^2)^2
\\
&= 2K_\alpha s^{-(\alpha+1)}
\end{aligned}$$ which, inverting the Laplace transform, gives $$\langle X^2 \rangle = \frac{2K_\alpha}{\Gamma(\alpha+1)}t^{\alpha}$$ Since $\alpha<1$, we are in the presence of subdiffusion ($H=\alpha/2<1/2$), as could be expected given that we have arbitrarily long pauses with relative high frequency.
### Long jumps
We consider now the opposite situation: assume that $\psi(t)$ has a distribution with finite moments (exponential, in this case, since $t>0$) and that $\phi(z)$ has a Lévy distribution with Lévy parameter $\mu$: $$\begin{aligned}
\psi(t) &= \tau\exp\bigl(-\frac{t}{\tau}\bigr) \\
\phi(z) &= A_\mu \left( \frac{z_0}{z} \right)^{1+\mu}
\end{aligned}$$ Note that in this case we consider $1<\mu<2$ for the sake of simplicity: the results are similar for $0<\mu<1$. As before, in the limit $\omega\rightarrow{0}$, $|s|\rightarrow{0}$, we can approximate them as $$\begin{aligned}
\psi(s) &\sim 1 - s\tau \\
\phi(\omega) &\sim 1 - \sigma^\mu\omega^\mu
\end{aligned}$$
(8,16)(0,0) (0,8)(0,8)[2]{}[(1,0)[8]{}]{} (0,8)(8,0)[2]{}[(0,1)[8]{}]{} (0,0)(0.5,0)[16]{}[(1,0)[0.25]{}]{} (0,0)(8,0)[2]{}[ (0,0)(0,0.5)[16]{}[(0,1)[0.25]{}]{} ]{} (0,0)[(1,2)[8]{}]{} (9,12)[(-1,0)[2.8]{}]{} (9.1,12)[(0,0)\[l\][$H=1/2$; diffusion]{}]{} (-0.1,0)[(0,0)\[r\][0]{}]{} (-0.1,8)[(0,0)\[r\][1]{}]{} (-0.1,16)[(0,0)\[r\][2]{}]{} (-0.3,12)[(0,0)\[r\][$\mu$]{}]{} (0,-0.2)[(0,0)\[t\][0]{}]{} (8,-0.2)[(0,0)\[t\][1]{}]{} (4,-0.2)[(0,0)\[t\][$\alpha$]{}]{} (1,15)[(0,0)\[l\][$H<1/2$]{}]{} (1,14)[(0,0)\[l\][subdiffusion]{}]{} (3,4)[(0,0)\[l\][$H>1/2$]{}]{} (3,3)[(0,0)\[l\][superdiffusion]{}]{}
Inserting these approximations into (\[ooohh\]) we have $$\tilde{\rho}(\omega,s) = \tau \frac{\tilde{\rho}(\omega,0)}{s + K^\mu\omega^\mu}$$ where $K^\mu=\sigma^\mu/\tau$ or, for $\rho(x,0)=\delta(x)$, $$\tilde{\rho}(\omega,s) = \frac{\tau}{s + K^\mu\omega^\mu}$$ Taking the inverse Laplace transform, we have $$\tilde{\rho}(\omega,t) = \exp(-K^\mu \omega^\mu)$$ that is, we obtain a Lévy stable distribution, as expected from the generalized Central Limit Theorem. Note that in this case $\langle{X^2}\rangle\rightarrow\infty$, so we can’t directly compare this distribution with the standard diffusion. There are however several ways to arrive at a result. The first is to use a truncated Lévy distribution, which is closer to real applications as in the physical world one doesn’t have arbitrarily long jumps. The second is to extrapolate from fractional moments $\langle{X^q}\rangle$ with $q<\mu$, which can be shown to converge and, in this case: $$\langle{X^q}\rangle \sim t^{q/\mu}$$ which leads to a Hurst exponent $H=1/\mu>1/2$, that is, to superdiffusion[^10].
### Long waits and long jumps
The case in which both jumps and waiting times have Lévy distribution can be trated similarly, leading to $$\tilde{\rho}(\omega,s) = \frac{1}{s} \frac{1}{1 + K_\alpha^\mu \omega^\mu s^{-\alpha}}$$ By analogy with the previous cases, we see that $$\langle X^2 \rangle \sim t^{2\alpha/\mu}$$ which entails $H=\alpha/\mu$. If $\mu>2\alpha$, then $H<1/2$, and we have subdiffusion, if $\mu>2\alpha$ we have superdiffusion (see figure \[diffme\]).
[10]{}
M. J. Acerbo, P. A. Gargiulo, I. Krug, and J. D. Delius. Behavioural consequences of nucleus accumbens dopaminergic stimulation and glutamatergic blocking in pigeons. , 136:171–7, 2002.
R.J. Bainton, L.T.Y. Tsai, C.M. Singh, M.S. Moore, W.S. Neckameyer, and U. Heberlein. Dopamine modulates response to cocaine, nocotine, and ethanol in *Drosophila*. , 10:187–94, 2000.
S.L. Barrett, R. Bell, D. Watson, and D.J. King. Effects of amisulpride, risperidone and chrorpromazine on auditory and visual latent inhibition, prepulse inhibition, executive function and eye movements in healthy volunteers. , 18:156–72, 2004.
G.S. Berns, S.M. McClure, G. Pagnoni, and P.R. Montague. Predictability modulates human brain response to reward. , 21:2793–8, 2001.
M Bertolucci-D’Angiò, A. Serrano, and B. Scatton. Differential effects of forced locomotion, tail-pinch, immobilization, and methyl-carboline carboxylate on extracellular 3,4-dihydroxyphenylacecetic acid levels in the rat striatum, nucleus accumbens, and prefrontal cortex: an in vivo voltammetric study. , 55:1208–15, 1990.
T.S. Braver and J.D. Cohen. On the control of control: the role of dopamine in regulating prefrontal function and working memory. In S. Monsell and J. Driver, editors, [*Control of Cognitive Processes*]{}, pages 713–37. Cambridge, MA:MIT Press, 2000.
H. Buxbaum-Conradi and J.P. Ewert. Responses of single neurons in the toad’s caudal ventral striatum to moving visual stimuli and test of their different projection by extracellular antidromic stimulation/recording techniques. , 54:338–54, 1999.
E. L. Charnov. Optimal foraging: the marginal value theorem. , 9:129–36, 1976.
T. A. Cleland and A. I. Silverston. Dopaminergic modulation of inhibitory glutamate receptors in the lobster stomatogastric ganglion. , 78:3450–2, 1997.
KJ.A. Dani and F. Zhou. Selective dopamine filter of glutamate striatal afferents. , 42:522–3, 2004.
M.R. DeLong. Primate models of movement disorders of basal ganglia origin. , 13:281–5, 1990.
M.R. Due, J. Jing, and R.W. Klaudiusz. Dopaminergic contributions to modulatory functions of a dual-transmitter interneuron in *Aplysia*. , 358:53–7, 2004.
S.M. Dursun, N. Wright, and M.A. Reveley. Effects of amphetamine on saccadic eye movements in man: Possible relevance to schizophrenia? , 13:245–7, 1999.
H. Eichenbaum and N.J. Cohen. . New York:Oxford University Press, 2001.
Albert Einstein. "uber die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen. , 17:549–560, 1905.
J.L. Evenden, M. Turpin, L. Oliver, and C. Jennings. Caffeine and nicotine improve visual tracking by ratsl a comparison with amphetamine, cocaine and apomorphine. , 110:169–76, 1993.
S.B. Floresco, D.N. Braaksma, and A.G. Phillips. Involvement of the ventral pallidum in working memory tasks with or without a delay. , 877:711–6, 1999.
S.B. Floresco, J.K. Seamans, and A.G. Phillips. A selective role for dopamine in the nucleus accumbens of the rat in random foraging but not delayed spatial win-shift-based foraging. , 80:161–8, 1996.
Crispin W Gardiner. . Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1985.
R.M. Harris-Warrick, L.M. Coniglio, N. Barazangi, J. Guckenheimer, and S. Gueron. Dopamine modulation of transient potassium current evokes phase shifts in a central pattern generator network. , 15:342–58, 1995.
S. Havlin and D. Benavraham. Diffusion in dosordered media. , 36:695–798, 1987.
Takeyuki Hida. Brownian motion. In [*Brownian Motion*]{}, pages 44–113. Springer, 1980.
T. Hills, P. J. Brockie, and A. V. Maricq. Dopamine and glutammate control area-restricted search behavior in *Caenorhabditis elegans*. , 24:1217–25, 2004.
Thomas T. Hills. Animal foraging and the evolution of goal-directed cognition. , 30:3–41, 2006.
C.S. Holling. Some characteristics of simple types of predation and parasitism. , 91:385–98, 1959.
H.E. Hurst, R.P. Black, and Y.M. Simaika. . London:Constable, 1965.
Ioannis Karatzas and Steven Shreve. , volume 113. Springer Science & Business Media, 2012.
Vicenç Méndez, Daniel Campos, and Frederic Bartumeus. . Springer Series in Synergetics. Berlin:Springer-Verlag, 2014.
Melanie Mitchell. . Cambridge, MA:MIT press, 1998.
D.R. Nassel. Neuropeptides, amines and amino acids in an elementary insect ganglion: Functional chemical anatomy of the unfised abdominal ganglion. , 48:325–420, 1996.
P. O’Donnel and A. Grace. Synaptic interactions among excitatory afferents to nucleus accumbens neurons: Hippocampal gating of prefrontal cortical input. , 15:3622–39, 1995.
B O’Shaughnessy and I. Procaccia. Analytical solutions for diffusion on fractal objects. , 54(5):455–8, 1985.
J. G. Powles. Brownian motion–june 1827. , 13:310–2, 1978.
William H. Press, Brian P. Flannery, Saul A. Teulolsky, and William T. Vetterling. . Cambridge University Press, 1986.
A. Reiner. Catecholaminergic innervation of the basal ganglia in mammals: Anatomy and function. In A.J. Smeets and A. Reiner, editors, [*Phylogeny and development of catechoamine systems in the [CNS]{} of vertebrates*]{}. Cambridge, England:Cambridge University Press, 1994.
A. Reiner, L. Medina, and C.L. Veenman. Structural and functional evolution of the basal ganglia in vertebrates. , 28:235–85, 1998.
C. Salas, C. Broglio, and F. Rodriguez. Evolution of firebrain and spatial cognition in vertebrates: Conservation across diversity. , 62:72–82, 2003.
W. Schultz. Neural coding of basic reward terms of animal learning theory, game theory, microeconomics and behavioural ecology. , 14:139–44, 2004.
W. Schultz, R. Romo, T. Ljungberg, J. Mirenowicz, J.R. Hoolerman, and A. Dickinson. Reward-related signals carried by dopamine neurons. In J.C. Houk, J.L. Davis, and D.G. Beiser, editors, [*Models of information processing in the basal ganglia*]{}, pages 233–48. Cambridge, MA:MIT Press, 1995.
J.K. Seamans, S.B. Floresco, and A.G. Phillips. receptor modulation of hippocampal-prefrontal cortical circuits integrating spatial memory with executive functions in the rat. , 18:1613–21, 1998.
D. W. Stephens. . Princeton:Princeton University Press, 1986.
G.I. Taylor. Diffusion by continuous movements. , 20:196–212, 1921.
S.P. Tipper, B. Weaver, L.M. Jerreat, and A.L. Burak. Object-based and environment-based inhibition of return of visual attention. , 20:478–99, 1994.
M. Wang, S. Vijayraghavan, and P.S. Goldman-Rakic. Selective [D2]{} receptor actions on the functional circuit of working memory. , 303:853–6, 2004.
J.G. White, E. Southgate, J.N. Thomson, and S. Brenner. The structure of the nervous system in the neumatode *Caenorhabditis Elegans*. , 314:1–340, 1986.
Random variables
----------------
A *random variable* is a mathematical object characterized by a set $\Omega$ (the *range* of the variable), which contains all possible outcomes of the variable and a function $P_X(x)$[^11] that assigns, to each $x\in\Omega$ a value $P_X(x)\in[0,1]$ called its *probability*. The function $P_X$ is not arbitrary, but must meet some minimal conditions. If the set $\Omega$ is finite or countable, these conditions can be expressed simply as
i)
: $\displaystyle \forall{x}\in\Omega.\ P_X(x)\ge0$ (positivity)
ii)
: $\displaystyle \sum_{x\in\Omega} P_X(x) = 1$ (normalization)
If $\Omega$ is uncountable, the conditions are technically more complex. In this case, $X$ is a continuous random variable, and $P_X$ is referred to as the *probability density* function (PDF) of $X$. The function $P_X$ in this case represents a probability only if it is integrated over a subset of $\Omega$ of non-zero measure. In this case, the normalization condition is $$\int_\Omega P_X(x) dx = 1$$ If $\Omega={\mathbb{R}}$ (as we shall often assume) we have $$\int_{-\infty}^\infty P_X(x) dx = 1$$ For continuous variables on ${\mathbb{R}}$ one can define the *cumulative probability function*, that is, the probability that $X$ be at most $x$: $${\mathcal{P}}(x) = {\mathbb{P}}[X\le{x}] = \int_{-\infty}^x P_X(u) du$$ Note that $$P_X(x) = \frac{\partial}{\partial x}{\mathcal{P}}(x).$$ From this and the positivity condition we can derive that ${\mathcal{P}}$ is monotonically non-decreasing and that $$\lim_{x\rightarrow-\infty} {\mathcal{P}}(x)=0\ \ \ \ \lim_{x\rightarrow\infty} {\mathcal{P}}(x)=1$$ In some cases, a whole function might be difficult to work with; it is easier to work with an enumerable set of numbers that characterizes the function completely. *Statistical moments* are such quantities. The moment of order $n$ of the variable $X$ is defined as $$\label{mom}
\langle X^n \rangle = \int_\Omega x^n P_X(x) dx$$ In general, given a function $f$ defined on $\Omega$, we define $$\langle f(X) \rangle = \int_\Omega f(x) P_X(x) dx$$ The $n$th moment is obtained for $f(x)=x^n$.
The first order moment $\langle{X}\rangle$ is called the *mean*, the *average*, or the *expected value* of $X$, while $$\sigma^2 = \langle X^2 \rangle - \langle X \rangle^2$$ is the *variance*; its square root $\sigma$ is the *standard deviation* of $X$.
Not all distributions have finite moments, that is, the integral (\[mom\]) may fail to converge. If the moments are finite, then they completely characterize the PDF. To show this, we introduce the *characteristic function* $\tilde{P}_X(\omega)$ of a PDF $P_X$: $$\label{pooh}
\tilde{P}_X(\omega) = \langle e^{i\omega{x}}\rangle = \int_\Omega e^{i\omega{x}}P_X(x)\,dx$$ This is simply the Fourier transform of $P_X$, so the PDF can be recovered from its characteristic function as $$P_X(x) = \frac{1}{2\pi} \int e^{-i\omega{x}}\tilde{P}_X(\omega)\,d\omega$$ The relation with the moments becomes evident by taking the Taylor expansion of the exponential: $$e^{i\omega{x}} = \sum_n^\infty \frac{(i\omega{x})^n}{n!}$$ Introducing this into (\[pooh\]) we get $$\tilde{P}_X(\omega) = \sum_n \frac{(i\omega)^n}{n!} \int x^nP_X(x)\,dx = \sum_n \frac{(i\omega)^n}{n!} \langle X^n \rangle$$ As a consequence, the moments of $P_X$ can be obtained by differentiating $\tilde{P}_X$: $$\langle X^n \rangle = \lim_{\omega\rightarrow{0}} (-i)^n \frac{\partial^n}{\partial \omega^n} \tilde{P}_X(\omega)$$
\* \* \*
The *joint probability* of two random variables $X_1$ and $X_2$, indicated as $P_{X_1\cap{X_2}}(x_1,x_2)$ measures the simultaneous probability that $X_1$ and $X_2$ take the values $x_1$ and $x_2$, respectively. The *conditional probability* $P_{X_1|X_2}(x_1|x_2)$ denotes the probability that $X_1$ take value $x_1$ conditioned to the fact that $X_2$ takes value $x_2$. Two variables are *independent* if for all $x_1$, $x_2$ $P_{X_1|X_2}(x_1x_2)=P_{X_1}(x_1)$, that is, knowing the value of $X_2$ does not change the distribution of $X_1$. Joint and conditional probabilities are related through Bayes’s theorem: $$P_{X_1\cap{X_2}}(x_1,x_2) = P_{X_1|X_2}(x_1|x_2)P_{X_2}(x_2) = P_{X_2|X_2}(x_2|x_1)P_{X_1}(x_1)$$
### Useful Probability Distributions
A variable $X$ follows a **Gaussian** (or *normal*) distribution if $$P_X(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\Bigl( -\frac{(x-\mu)^2}{\sigma^2} \Bigr)$$ (Figure \[gaussian\]) or, equivalently, it has characteristic function $$\label{lookie}
\tilde{P}_X(\omega) = \int_{-\infty}^{\infty} e^{i\omega{x}} P_X(x)\,dx = \exp\Bigl( i\omega\mu - \frac{\omega^2\sigma^2}{2} \Bigr)$$
[ccc]{}
(900,900)(0,0) (150.0,84.0)
------------------------------------------------------------------------
(130,84)[(0,0)\[r\][ 0]{}]{} (819.0,84.0)
------------------------------------------------------------------------
(150.0,170.0)
------------------------------------------------------------------------
(130,170)[(0,0)\[r\][ 0.05]{}]{} (819.0,170.0)
------------------------------------------------------------------------
(150.0,257.0)
------------------------------------------------------------------------
(130,257)[(0,0)\[r\][ 0.1]{}]{} (819.0,257.0)
------------------------------------------------------------------------
(150.0,343.0)
------------------------------------------------------------------------
(130,343)[(0,0)\[r\][ 0.15]{}]{} (819.0,343.0)
------------------------------------------------------------------------
(150.0,429.0)
------------------------------------------------------------------------
(130,429)[(0,0)\[r\][ 0.2]{}]{} (819.0,429.0)
------------------------------------------------------------------------
(150.0,515.0)
------------------------------------------------------------------------
(130,515)[(0,0)\[r\][ 0.25]{}]{} (819.0,515.0)
------------------------------------------------------------------------
(150.0,601.0)
------------------------------------------------------------------------
(130,601)[(0,0)\[r\][ 0.3]{}]{} (819.0,601.0)
------------------------------------------------------------------------
(150.0,688.0)
------------------------------------------------------------------------
(130,688)[(0,0)\[r\][ 0.35]{}]{} (819.0,688.0)
------------------------------------------------------------------------
(150.0,774.0)
------------------------------------------------------------------------
(130,774)[(0,0)\[r\][ 0.4]{}]{} (819.0,774.0)
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(150,43)[(0,0)[-8]{}]{} (150.0,754.0)
------------------------------------------------------------------------
(236.0,84.0)
------------------------------------------------------------------------
(236,43)[(0,0)[-6]{}]{} (236.0,754.0)
------------------------------------------------------------------------
(322.0,84.0)
------------------------------------------------------------------------
(322,43)[(0,0)[-4]{}]{} (322.0,754.0)
------------------------------------------------------------------------
(408.0,84.0)
------------------------------------------------------------------------
(408,43)[(0,0)[-2]{}]{} (408.0,754.0)
------------------------------------------------------------------------
(495.0,84.0)
------------------------------------------------------------------------
(495,43)[(0,0)[ 0]{}]{} (495.0,754.0)
------------------------------------------------------------------------
(581.0,84.0)
------------------------------------------------------------------------
(581,43)[(0,0)[ 2]{}]{} (581.0,754.0)
------------------------------------------------------------------------
(667.0,84.0)
------------------------------------------------------------------------
(667,43)[(0,0)[ 4]{}]{} (667.0,754.0)
------------------------------------------------------------------------
(753.0,84.0)
------------------------------------------------------------------------
(753,43)[(0,0)[ 6]{}]{} (753.0,754.0)
------------------------------------------------------------------------
(839.0,84.0)
------------------------------------------------------------------------
(839,43)[(0,0)[ 8]{}]{} (839.0,754.0)
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(839.0,84.0)
------------------------------------------------------------------------
(150.0,774.0)
------------------------------------------------------------------------
(494,836)[(0,0)[Gaussian PDF ($\mu$=0)]{}]{} (529,601)[(0,0)\[l\][$\sigma$=1]{}]{} (550,343)[(0,0)\[l\][$\sigma$=2]{}]{} (667,170)[(0,0)\[l\][$\sigma$=3]{}]{} (150,84) (373,83.67)
------------------------------------------------------------------------
(373.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(387,85.17)
------------------------------------------------------------------------
(387.00,84.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(394.00,87.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(394.00,86.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(401.00,90.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(401.00,89.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(408.59,96.00)(0.482,0.852)[9]{}
------------------------------------------------------------------------
(407.17,96.00)(6.000,8.409)[2]{}
------------------------------------------------------------------------
(414.59,106.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(413.17,106.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(421.59,123.00)(0.485,1.942)[11]{}
------------------------------------------------------------------------
(420.17,123.00)(7.000,22.709)[2]{}
------------------------------------------------------------------------
(428.59,149.00)(0.485,2.933)[11]{}
------------------------------------------------------------------------
(427.17,149.00)(7.000,34.167)[2]{}
------------------------------------------------------------------------
(435.59,188.00)(0.485,4.078)[11]{}
------------------------------------------------------------------------
(434.17,188.00)(7.000,47.388)[2]{}
------------------------------------------------------------------------
(442.59,242.00)(0.485,5.298)[11]{}
------------------------------------------------------------------------
(441.17,242.00)(7.000,61.490)[2]{}
------------------------------------------------------------------------
(449.59,312.00)(0.485,6.366)[11]{}
------------------------------------------------------------------------
(448.17,312.00)(7.000,73.830)[2]{}
------------------------------------------------------------------------
(456.59,396.00)(0.485,7.052)[11]{}
------------------------------------------------------------------------
(455.17,396.00)(7.000,81.762)[2]{}
------------------------------------------------------------------------
(463.59,489.00)(0.485,7.205)[11]{}
------------------------------------------------------------------------
(462.17,489.00)(7.000,83.525)[2]{}
------------------------------------------------------------------------
(470.59,584.00)(0.485,6.442)[11]{}
------------------------------------------------------------------------
(469.17,584.00)(7.000,74.711)[2]{}
------------------------------------------------------------------------
(477.59,669.00)(0.485,4.840)[11]{}
------------------------------------------------------------------------
(476.17,669.00)(7.000,56.202)[2]{}
------------------------------------------------------------------------
(484.59,733.00)(0.485,2.628)[11]{}
------------------------------------------------------------------------
(483.17,733.00)(7.000,30.641)[2]{}
------------------------------------------------------------------------
(380.0,85.0)
------------------------------------------------------------------------
(498.59,759.28)(0.485,-2.628)[11]{}
------------------------------------------------------------------------
(497.17,763.64)(7.000,-30.641)[2]{}
------------------------------------------------------------------------
(505.59,717.40)(0.485,-4.840)[11]{}
------------------------------------------------------------------------
(504.17,725.20)(7.000,-56.202)[2]{}
------------------------------------------------------------------------
(512.59,648.42)(0.485,-6.442)[11]{}
------------------------------------------------------------------------
(511.17,658.71)(7.000,-74.711)[2]{}
------------------------------------------------------------------------
(519.59,561.05)(0.485,-7.205)[11]{}
------------------------------------------------------------------------
(518.17,572.53)(7.000,-83.525)[2]{}
------------------------------------------------------------------------
(526.59,466.52)(0.485,-7.052)[11]{}
------------------------------------------------------------------------
(525.17,477.76)(7.000,-81.762)[2]{}
------------------------------------------------------------------------
(533.59,375.66)(0.485,-6.366)[11]{}
------------------------------------------------------------------------
(532.17,385.83)(7.000,-73.830)[2]{}
------------------------------------------------------------------------
(540.59,294.98)(0.485,-5.298)[11]{}
------------------------------------------------------------------------
(539.17,303.49)(7.000,-61.490)[2]{}
------------------------------------------------------------------------
(547.59,228.78)(0.485,-4.078)[11]{}
------------------------------------------------------------------------
(546.17,235.39)(7.000,-47.388)[2]{}
------------------------------------------------------------------------
(554.59,178.33)(0.485,-2.933)[11]{}
------------------------------------------------------------------------
(553.17,183.17)(7.000,-34.167)[2]{}
------------------------------------------------------------------------
(561.59,142.42)(0.485,-1.942)[11]{}
------------------------------------------------------------------------
(560.17,145.71)(7.000,-22.709)[2]{}
------------------------------------------------------------------------
(568.59,118.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(567.17,120.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(575.59,102.82)(0.482,-0.852)[9]{}
------------------------------------------------------------------------
(574.17,104.41)(6.000,-8.409)[2]{}
------------------------------------------------------------------------
(581.00,94.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(581.00,95.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(588.00,88.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(588.00,89.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(595,85.17)
------------------------------------------------------------------------
(595.00,86.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(491.0,768.0)
------------------------------------------------------------------------
(609,83.67)
------------------------------------------------------------------------
(609.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(602.0,85.0)
------------------------------------------------------------------------
(616.0,84.0)
------------------------------------------------------------------------
(150,84) (268,83.67)
------------------------------------------------------------------------
(268.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(289,84.67)
------------------------------------------------------------------------
(289.00,84.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(275.0,85.0)
------------------------------------------------------------------------
(303,86.17)
------------------------------------------------------------------------
(303.00,85.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(310,87.67)
------------------------------------------------------------------------
(310.00,87.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(317,89.17)
------------------------------------------------------------------------
(317.00,88.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(324,91.17)
------------------------------------------------------------------------
(324.00,90.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(331.00,93.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(331.00,92.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(338.00,97.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(338.00,96.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(345.00,101.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(345.00,100.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(352.00,106.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(352.00,105.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(359.59,113.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(358.17,113.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(366.59,121.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(365.17,121.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(373.59,131.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(372.17,131.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(380.59,142.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(379.17,142.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(387.59,156.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(386.17,156.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(394.59,171.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(393.17,171.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(401.59,189.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(400.17,189.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(408.59,208.00)(0.482,1.847)[9]{}
------------------------------------------------------------------------
(407.17,208.00)(6.000,17.887)[2]{}
------------------------------------------------------------------------
(414.59,229.00)(0.485,1.637)[11]{}
------------------------------------------------------------------------
(413.17,229.00)(7.000,19.183)[2]{}
------------------------------------------------------------------------
(421.59,251.00)(0.485,1.789)[11]{}
------------------------------------------------------------------------
(420.17,251.00)(7.000,20.946)[2]{}
------------------------------------------------------------------------
(428.59,275.00)(0.485,1.789)[11]{}
------------------------------------------------------------------------
(427.17,275.00)(7.000,20.946)[2]{}
------------------------------------------------------------------------
(435.59,299.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(434.17,299.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(442.59,322.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(441.17,322.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(449.59,345.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(448.17,345.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(456.59,366.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(455.17,366.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(463.59,385.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(462.17,385.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(470.59,402.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(469.17,402.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(477.59,414.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(476.17,414.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(484.00,423.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(484.00,422.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(296.0,86.0)
------------------------------------------------------------------------
(498.00,426.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(498.00,427.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(505.59,420.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(504.17,421.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(512.59,410.74)(0.485,-0.874)[11]{}
------------------------------------------------------------------------
(511.17,412.37)(7.000,-10.369)[2]{}
------------------------------------------------------------------------
(519.59,397.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(518.17,399.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(526.59,380.08)(0.485,-1.408)[11]{}
------------------------------------------------------------------------
(525.17,382.54)(7.000,-16.539)[2]{}
------------------------------------------------------------------------
(533.59,360.60)(0.485,-1.560)[11]{}
------------------------------------------------------------------------
(532.17,363.30)(7.000,-18.302)[2]{}
------------------------------------------------------------------------
(540.59,339.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(539.17,342.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(547.59,316.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(546.17,319.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(554.59,292.89)(0.485,-1.789)[11]{}
------------------------------------------------------------------------
(553.17,295.95)(7.000,-20.946)[2]{}
------------------------------------------------------------------------
(561.59,268.89)(0.485,-1.789)[11]{}
------------------------------------------------------------------------
(560.17,271.95)(7.000,-20.946)[2]{}
------------------------------------------------------------------------
(568.59,245.37)(0.485,-1.637)[11]{}
------------------------------------------------------------------------
(567.17,248.18)(7.000,-19.183)[2]{}
------------------------------------------------------------------------
(575.59,222.77)(0.482,-1.847)[9]{}
------------------------------------------------------------------------
(574.17,225.89)(6.000,-17.887)[2]{}
------------------------------------------------------------------------
(581.59,203.08)(0.485,-1.408)[11]{}
------------------------------------------------------------------------
(580.17,205.54)(7.000,-16.539)[2]{}
------------------------------------------------------------------------
(588.59,184.32)(0.485,-1.332)[11]{}
------------------------------------------------------------------------
(587.17,186.66)(7.000,-15.658)[2]{}
------------------------------------------------------------------------
(595.59,167.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(594.17,169.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(602.59,152.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(601.17,154.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(609.59,138.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(608.17,140.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(616.59,128.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(615.17,129.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(623.59,118.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(622.17,119.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(630.00,111.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(630.00,112.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(637.00,104.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(637.00,105.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(644.00,99.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(644.00,100.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(651.00,95.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(651.00,96.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(658,91.17)
------------------------------------------------------------------------
(658.00,92.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(665,89.17)
------------------------------------------------------------------------
(665.00,90.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(672,87.67)
------------------------------------------------------------------------
(672.00,88.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(679,86.17)
------------------------------------------------------------------------
(679.00,87.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(491.0,428.0)
------------------------------------------------------------------------
(693,84.67)
------------------------------------------------------------------------
(693.00,85.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(686.0,86.0)
------------------------------------------------------------------------
(714,83.67)
------------------------------------------------------------------------
(714.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(700.0,85.0)
------------------------------------------------------------------------
(721.0,84.0)
------------------------------------------------------------------------
(150,84) (171,83.67)
------------------------------------------------------------------------
(171.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(199,84.67)
------------------------------------------------------------------------
(199.00,84.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(178.0,85.0)
------------------------------------------------------------------------
(220,85.67)
------------------------------------------------------------------------
(220.00,85.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(227,86.67)
------------------------------------------------------------------------
(227.00,86.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(234,87.67)
------------------------------------------------------------------------
(234.00,87.17)(3.000,1.000)[2]{}
------------------------------------------------------------------------
(240,88.67)
------------------------------------------------------------------------
(240.00,88.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(247,89.67)
------------------------------------------------------------------------
(247.00,89.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(254,91.17)
------------------------------------------------------------------------
(254.00,90.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(261,93.17)
------------------------------------------------------------------------
(261.00,92.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(268,95.17)
------------------------------------------------------------------------
(268.00,94.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(275,97.17)
------------------------------------------------------------------------
(275.00,96.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(282.00,99.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(282.00,98.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(289.00,102.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(289.00,101.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(296.00,106.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(296.00,105.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(303.00,110.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(303.00,109.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(310.00,114.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(310.00,113.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(317.00,119.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(317.00,118.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(324.00,124.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(324.00,123.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(331.00,130.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(331.00,129.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(338.00,137.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(338.00,136.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(345.59,144.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(344.17,144.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(352.59,152.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(351.17,152.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(359.59,160.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(358.17,160.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(366.59,169.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(365.17,169.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(373.59,178.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(372.17,178.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(380.59,188.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(379.17,188.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(387.59,198.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(386.17,198.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(394.59,209.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(393.17,209.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(401.59,219.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(400.17,219.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(408.59,230.00)(0.482,0.852)[9]{}
------------------------------------------------------------------------
(407.17,230.00)(6.000,8.409)[2]{}
------------------------------------------------------------------------
(414.59,240.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(413.17,240.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(421.59,251.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(420.17,251.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(428.59,261.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(427.17,261.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(435.59,270.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(434.17,270.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(442.59,279.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(441.17,279.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(449.00,287.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(449.00,286.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(456.00,294.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(456.00,293.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(463.00,300.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(463.00,299.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(470.00,305.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(470.00,304.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(477.00,309.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(477.00,308.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(484,311.67)
------------------------------------------------------------------------
(484.00,311.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(206.0,86.0)
------------------------------------------------------------------------
(498,311.67)
------------------------------------------------------------------------
(498.00,312.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(505.00,310.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(505.00,311.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(512.00,307.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(512.00,308.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(519.00,303.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(519.00,304.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(526.00,298.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(526.00,299.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(533.00,292.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(533.00,293.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(540.59,284.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(539.17,285.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(547.59,276.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(546.17,277.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(554.59,267.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(553.17,268.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(561.59,258.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(560.17,259.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(568.59,247.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(567.17,249.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(575.59,236.82)(0.482,-0.852)[9]{}
------------------------------------------------------------------------
(574.17,238.41)(6.000,-8.409)[2]{}
------------------------------------------------------------------------
(581.59,226.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(580.17,228.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(588.59,216.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(587.17,217.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(595.59,205.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(594.17,207.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(602.59,195.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(601.17,196.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(609.59,185.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(608.17,186.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(616.59,175.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(615.17,176.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(623.59,166.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(622.17,167.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(630.59,157.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(629.17,158.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(637.59,149.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(636.17,150.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(644.00,142.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(644.00,143.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(651.00,135.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(651.00,136.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(658.00,128.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(658.00,129.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(665.00,122.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(665.00,123.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(672.00,117.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(672.00,118.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(679.00,112.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(679.00,113.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(686.00,108.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(686.00,109.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(693.00,104.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(693.00,105.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(700.00,100.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(700.00,101.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(707,97.17)
------------------------------------------------------------------------
(707.00,98.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(714,95.17)
------------------------------------------------------------------------
(714.00,96.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(721,93.17)
------------------------------------------------------------------------
(721.00,94.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(728,91.17)
------------------------------------------------------------------------
(728.00,92.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(735,89.67)
------------------------------------------------------------------------
(735.00,90.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(742,88.67)
------------------------------------------------------------------------
(742.00,89.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(749,87.67)
------------------------------------------------------------------------
(749.00,88.17)(3.000,-1.000)[2]{}
------------------------------------------------------------------------
(755,86.67)
------------------------------------------------------------------------
(755.00,87.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(762,85.67)
------------------------------------------------------------------------
(762.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(491.0,313.0)
------------------------------------------------------------------------
(783,84.67)
------------------------------------------------------------------------
(783.00,85.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(769.0,86.0)
------------------------------------------------------------------------
(811,83.67)
------------------------------------------------------------------------
(811.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(790.0,85.0)
------------------------------------------------------------------------
(818.0,84.0)
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(150.0,84.0)
------------------------------------------------------------------------
(839.0,84.0)
------------------------------------------------------------------------
(150.0,774.0)
------------------------------------------------------------------------
(900,900)(0,0) (137.0,82.0)
------------------------------------------------------------------------
(117,82)[(0,0)\[r\][ 0]{}]{} (812.0,82.0)
------------------------------------------------------------------------
(137.0,221.0)
------------------------------------------------------------------------
(117,221)[(0,0)\[r\][ 0.2]{}]{} (812.0,221.0)
------------------------------------------------------------------------
(137.0,360.0)
------------------------------------------------------------------------
(117,360)[(0,0)\[r\][ 0.4]{}]{} (812.0,360.0)
------------------------------------------------------------------------
(137.0,498.0)
------------------------------------------------------------------------
(117,498)[(0,0)\[r\][ 0.6]{}]{} (812.0,498.0)
------------------------------------------------------------------------
(137.0,637.0)
------------------------------------------------------------------------
(117,637)[(0,0)\[r\][ 0.8]{}]{} (812.0,637.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(117,776)[(0,0)\[r\][ 1]{}]{} (812.0,776.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137,41)[(0,0)[-8]{}]{} (137.0,756.0)
------------------------------------------------------------------------
(224.0,82.0)
------------------------------------------------------------------------
(224,41)[(0,0)[-6]{}]{} (224.0,756.0)
------------------------------------------------------------------------
(311.0,82.0)
------------------------------------------------------------------------
(311,41)[(0,0)[-4]{}]{} (311.0,756.0)
------------------------------------------------------------------------
(398.0,82.0)
------------------------------------------------------------------------
(398,41)[(0,0)[-2]{}]{} (398.0,756.0)
------------------------------------------------------------------------
(485.0,82.0)
------------------------------------------------------------------------
(485,41)[(0,0)[ 0]{}]{} (485.0,756.0)
------------------------------------------------------------------------
(571.0,82.0)
------------------------------------------------------------------------
(571,41)[(0,0)[ 2]{}]{} (571.0,756.0)
------------------------------------------------------------------------
(658.0,82.0)
------------------------------------------------------------------------
(658,41)[(0,0)[ 4]{}]{} (658.0,756.0)
------------------------------------------------------------------------
(745.0,82.0)
------------------------------------------------------------------------
(745,41)[(0,0)[ 6]{}]{} (745.0,756.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(832,41)[(0,0)[ 8]{}]{} (832.0,756.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(484,838)[(0,0)[Gaussian Cumulative Distribution ($\mu$=0)]{}]{} (480,221)[(0,0)\[l\][$\sigma$=1]{}]{} (571,637)[(0,0)\[l\][$\sigma$=3]{}]{} (309,290)[(0,0)\[r\][$\sigma$=2]{}]{} (311.58,287.05)(0.499,-0.764)[179]{}
------------------------------------------------------------------------
(310.17,288.52)(91.000,-137.524)[2]{}
------------------------------------------------------------------------
(402,151)[(2,-3)[0]{}]{} (137,82) (383,81.67)
------------------------------------------------------------------------
(383.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(397,83.17)
------------------------------------------------------------------------
(397.00,82.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(404.00,85.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(404.00,84.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(411.00,88.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(411.00,87.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(418.59,92.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(417.17,92.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(425.59,100.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(424.17,100.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(432.59,112.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(431.17,112.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(439.59,130.00)(0.485,1.789)[11]{}
------------------------------------------------------------------------
(438.17,130.00)(7.000,20.946)[2]{}
------------------------------------------------------------------------
(446.59,154.00)(0.485,2.476)[11]{}
------------------------------------------------------------------------
(445.17,154.00)(7.000,28.879)[2]{}
------------------------------------------------------------------------
(453.59,187.00)(0.485,3.162)[11]{}
------------------------------------------------------------------------
(452.17,187.00)(7.000,36.811)[2]{}
------------------------------------------------------------------------
(460.59,229.00)(0.485,3.772)[11]{}
------------------------------------------------------------------------
(459.17,229.00)(7.000,43.862)[2]{}
------------------------------------------------------------------------
(467.59,279.00)(0.485,4.306)[11]{}
------------------------------------------------------------------------
(466.17,279.00)(7.000,50.032)[2]{}
------------------------------------------------------------------------
(474.59,336.00)(0.485,4.612)[11]{}
------------------------------------------------------------------------
(473.17,336.00)(7.000,53.558)[2]{}
------------------------------------------------------------------------
(481.59,397.00)(0.485,4.840)[11]{}
------------------------------------------------------------------------
(480.17,397.00)(7.000,56.202)[2]{}
------------------------------------------------------------------------
(488.59,461.00)(0.485,4.612)[11]{}
------------------------------------------------------------------------
(487.17,461.00)(7.000,53.558)[2]{}
------------------------------------------------------------------------
(495.59,522.00)(0.485,4.306)[11]{}
------------------------------------------------------------------------
(494.17,522.00)(7.000,50.032)[2]{}
------------------------------------------------------------------------
(502.59,579.00)(0.485,3.772)[11]{}
------------------------------------------------------------------------
(501.17,579.00)(7.000,43.862)[2]{}
------------------------------------------------------------------------
(509.59,629.00)(0.485,3.162)[11]{}
------------------------------------------------------------------------
(508.17,629.00)(7.000,36.811)[2]{}
------------------------------------------------------------------------
(516.59,671.00)(0.485,2.476)[11]{}
------------------------------------------------------------------------
(515.17,671.00)(7.000,28.879)[2]{}
------------------------------------------------------------------------
(523.59,704.00)(0.485,1.789)[11]{}
------------------------------------------------------------------------
(522.17,704.00)(7.000,20.946)[2]{}
------------------------------------------------------------------------
(530.59,728.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(529.17,728.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(537.59,746.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(536.17,746.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(544.59,758.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(543.17,758.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(551.00,766.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(551.00,765.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(558.00,770.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(558.00,769.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(565,773.17)
------------------------------------------------------------------------
(565.00,772.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(390.0,83.0)
------------------------------------------------------------------------
(579,774.67)
------------------------------------------------------------------------
(579.00,774.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(572.0,775.0)
------------------------------------------------------------------------
(586.0,776.0)
------------------------------------------------------------------------
(137,82) (284,81.67)
------------------------------------------------------------------------
(284.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(305,82.67)
------------------------------------------------------------------------
(305.00,82.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(313,83.67)
------------------------------------------------------------------------
(313.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(320,84.67)
------------------------------------------------------------------------
(320.00,84.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(327,85.67)
------------------------------------------------------------------------
(327.00,85.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(334,87.17)
------------------------------------------------------------------------
(334.00,86.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(341,89.17)
------------------------------------------------------------------------
(341.00,88.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(348.00,91.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(348.00,90.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(355.00,94.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(355.00,93.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(362.00,98.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(362.00,97.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(369.00,103.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(369.00,102.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(376.00,109.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(376.00,108.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(383.59,116.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(382.17,116.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(390.59,125.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(389.17,125.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(397.59,135.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(396.17,135.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(404.59,148.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(403.17,148.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(411.59,162.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(410.17,162.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(418.59,178.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(417.17,178.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(425.59,197.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(424.17,197.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(432.59,218.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(431.17,218.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(439.59,241.00)(0.485,1.865)[11]{}
------------------------------------------------------------------------
(438.17,241.00)(7.000,21.827)[2]{}
------------------------------------------------------------------------
(446.59,266.00)(0.485,2.018)[11]{}
------------------------------------------------------------------------
(445.17,266.00)(7.000,23.590)[2]{}
------------------------------------------------------------------------
(453.59,293.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(452.17,293.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(460.59,321.00)(0.485,2.247)[11]{}
------------------------------------------------------------------------
(459.17,321.00)(7.000,26.234)[2]{}
------------------------------------------------------------------------
(467.59,351.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(466.17,351.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(474.59,382.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(473.17,382.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(481.59,413.00)(0.485,2.399)[11]{}
------------------------------------------------------------------------
(480.17,413.00)(7.000,27.997)[2]{}
------------------------------------------------------------------------
(488.59,445.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(487.17,445.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(495.59,476.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(494.17,476.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(502.59,507.00)(0.485,2.247)[11]{}
------------------------------------------------------------------------
(501.17,507.00)(7.000,26.234)[2]{}
------------------------------------------------------------------------
(509.59,537.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(508.17,537.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(516.59,565.00)(0.485,2.018)[11]{}
------------------------------------------------------------------------
(515.17,565.00)(7.000,23.590)[2]{}
------------------------------------------------------------------------
(523.59,592.00)(0.485,1.865)[11]{}
------------------------------------------------------------------------
(522.17,592.00)(7.000,21.827)[2]{}
------------------------------------------------------------------------
(530.59,617.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(529.17,617.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(537.59,640.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(536.17,640.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(544.59,661.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(543.17,661.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(551.59,680.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(550.17,680.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(558.59,696.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(557.17,696.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(565.59,710.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(564.17,710.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(572.59,723.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(571.17,723.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(579.59,733.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(578.17,733.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(586.00,742.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(586.00,741.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(593.00,749.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(593.00,748.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(600.00,755.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(600.00,754.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(607.00,760.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(607.00,759.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(614.00,764.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(614.00,763.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(621,767.17)
------------------------------------------------------------------------
(621.00,766.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(628,769.17)
------------------------------------------------------------------------
(628.00,768.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(635,770.67)
------------------------------------------------------------------------
(635.00,770.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(642,771.67)
------------------------------------------------------------------------
(642.00,771.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(649,772.67)
------------------------------------------------------------------------
(649.00,772.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(656,773.67)
------------------------------------------------------------------------
(656.00,773.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(291.0,83.0)
------------------------------------------------------------------------
(678,774.67)
------------------------------------------------------------------------
(678.00,774.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(664.0,775.0)
------------------------------------------------------------------------
(685.0,776.0)
------------------------------------------------------------------------
(137,82) (186,81.67)
------------------------------------------------------------------------
(186.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(221,82.67)
------------------------------------------------------------------------
(221.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(193.0,83.0)
------------------------------------------------------------------------
(235,83.67)
------------------------------------------------------------------------
(235.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(242,84.67)
------------------------------------------------------------------------
(242.00,84.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(249,85.67)
------------------------------------------------------------------------
(249.00,85.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(256,86.67)
------------------------------------------------------------------------
(256.00,86.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(263,87.67)
------------------------------------------------------------------------
(263.00,87.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(270,89.17)
------------------------------------------------------------------------
(270.00,88.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(277,90.67)
------------------------------------------------------------------------
(277.00,90.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(284.00,92.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(284.00,91.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(291,95.17)
------------------------------------------------------------------------
(291.00,94.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(298.00,97.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(298.00,96.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(305.00,100.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(305.00,99.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(313.00,104.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(313.00,103.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(320.00,107.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(320.00,106.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(327.00,112.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(327.00,111.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(334.00,117.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(334.00,116.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(341.00,123.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(341.00,122.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(348.00,130.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(348.00,129.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(355.59,137.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(354.17,137.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(362.59,145.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(361.17,145.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(369.59,154.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(368.17,154.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(376.59,164.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(375.17,164.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(383.59,175.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(382.17,175.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(390.59,187.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(389.17,187.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(397.59,200.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(396.17,200.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(404.59,214.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(403.17,214.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(411.59,229.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(410.17,229.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(418.59,245.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(417.17,245.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(425.59,261.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(424.17,261.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(432.59,279.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(431.17,279.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(439.59,297.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(438.17,297.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(446.59,316.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(445.17,316.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(453.59,336.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(452.17,336.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(460.59,356.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(459.17,356.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(467.59,377.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(466.17,377.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(474.59,397.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(473.17,397.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(481.59,418.00)(0.485,1.637)[11]{}
------------------------------------------------------------------------
(480.17,418.00)(7.000,19.183)[2]{}
------------------------------------------------------------------------
(488.59,440.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(487.17,440.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(495.59,461.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(494.17,461.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(502.59,481.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(501.17,481.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(509.59,502.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(508.17,502.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(516.59,522.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(515.17,522.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(523.59,542.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(522.17,542.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(530.59,561.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(529.17,561.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(537.59,579.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(536.17,579.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(544.59,597.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(543.17,597.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(551.59,613.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(550.17,613.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(558.59,629.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(557.17,629.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(565.59,644.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(564.17,644.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(572.59,658.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(571.17,658.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(579.59,671.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(578.17,671.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(586.59,683.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(585.17,683.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(593.59,694.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(592.17,694.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(600.59,704.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(599.17,704.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(607.59,713.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(606.17,713.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(614.00,721.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(614.00,720.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(621.00,728.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(621.00,727.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(628.00,735.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(628.00,734.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(635.00,741.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(635.00,740.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(642.00,746.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(642.00,745.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(649.00,751.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(649.00,750.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(656.00,754.60)(1.066,0.468)[5]{}
------------------------------------------------------------------------
(656.00,753.17)(6.132,4.000)[2]{}
------------------------------------------------------------------------
(664.00,758.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(664.00,757.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(671,761.17)
------------------------------------------------------------------------
(671.00,760.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(678.00,763.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(678.00,762.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(685,765.67)
------------------------------------------------------------------------
(685.00,765.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(692,767.17)
------------------------------------------------------------------------
(692.00,766.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(699,768.67)
------------------------------------------------------------------------
(699.00,768.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(706,769.67)
------------------------------------------------------------------------
(706.00,769.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(713,770.67)
------------------------------------------------------------------------
(713.00,770.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(720,771.67)
------------------------------------------------------------------------
(720.00,771.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(727,772.67)
------------------------------------------------------------------------
(727.00,772.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(228.0,84.0)
------------------------------------------------------------------------
(741,773.67)
------------------------------------------------------------------------
(741.00,773.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(734.0,774.0)
------------------------------------------------------------------------
(776,774.67)
------------------------------------------------------------------------
(776.00,774.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(748.0,775.0)
------------------------------------------------------------------------
(783.0,776.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
\(a) & & (b)
The mean of the distribution is $\langle{X}\rangle=\mu$. Note that, for $\mu=0$, the characteristic function has also the functional form of a Gaussian, a fact that will have important consequences. In this special case ($\mu=0$) the moments are given by $$\langle X^n \rangle = \int_{-\infty}^\infty x^n \frac{1}{\sigma \sqrt{2\pi}} \exp\Bigl( -\frac{(x-\mu)^2}{\sigma^2} \Bigr)\,dx =
\begin{cases}
\frac{2^{\frac{n}{2}}\sigma^n}{\sqrt{\pi}} \Gamma\Bigl( \frac{n+1}{2} \Bigr) & \mbox{$n$ even} \\
0 & \mbox{$n$ odd}
\end{cases}$$ where $\Gamma$ is Euler’s Gamma function. An important moment is $$\langle X^2 \rangle = \sigma^2 + \langle X \rangle.$$ One important property of the Gaussian distribution, vis-à-vis the Central Limit Theorem (which we shall consider in the following), is that it is *stable*: if $X,Y$ are Gaussians, and $a,b\in{\mathbb{R}}$, then $aX+bY$ is also Gaussian.
Let $X$ and $Y$ be two Gaussian-distributed variables with zero mean and variance $\sigma_1^2$ and $\sigma_2^2$, respectively. Then $$\begin{aligned}
\tilde{P}_X(\omega) &= \exp\bigl( - \frac{\omega^2\sigma_1^2}{2} \bigr) \\
\tilde{P}_Y(\omega) &= \exp\bigl( - \frac{\omega^2\sigma_2^2}{2} \bigr)
\end{aligned}$$ and consequently $$\label{ggood}
\tilde{P}_X(\omega)\tilde{P}_Y(\omega) = \exp\bigl( - \frac{\omega^2(\sigma_1^2+\sigma_2^2)}{2} \bigr)$$ that is, the product of the characteristic function of two Gaussian distributions is still the characteristic function of a Gaussian distribution.
\* \* \*
The Gaussian distribution is defined for all $x\in{\mathbb{R}}$, but many variables that one might be interested in modeling assume only positive values in such a way that the probability that $x=0$ is $0$ and, after reaching a maximum, decreases rapidly for high values of $x$. The most different things can be observed to have this distribution, from the length of messages in internet fori to the prices of hotels, or the size of particles in a collision.
All these phenomena can be modeled as following a **logonormal distribution**. A variable $X$ has logonormal distribution if $\log{X}$ has normal (viz. Gaussian) distribution. Let $\Phi$ and $\phi$ be the cumulative distribution and the density of a normally distributed variable with $0$ mean and unit variance (${\mathcal{N}}(0,1)$), and assume $\log{X}\sim{\mathcal{N}}(\mu,\sigma)$, i.e. $\log{X}$ has a normal distribution with mean $\mu$ and variance $\sigma^2$. Then $$\begin{aligned}
P_X(x) &= \frac{d}{dx} {\mathcal{P}}_X(x) = \frac{d}{dx} {\mathbb{P}}[X\le{x}] \\
&= \frac{d}{dx} {\mathbb{P}}[\log{X}\le\log{x}] \\
&= \frac{d}{dx} \Phi\bigl[ \frac{\log{x}-\mu}{\sigma}\bigr] \\
&= \phi\bigl[ \frac{\log{x}-\mu}{\sigma}\bigr] \frac{d}{dx} \bigl[ \frac{\log{x}-\mu}{\sigma}\bigr] \\
&= \frac{1}{\sigma{x}} \phi\bigl[ \frac{\log{x}-\mu}{\sigma}\bigr] \\
&= \frac{1}{\sqrt{2\pi}\sigma{x}} \exp\Big[- \frac{(\log{X}-\mu)^2}{2\sigma^2}\Bigr]
\end{aligned}$$ Figure \[logogauss\] shows the behavior of the logonormal PDF for various values of $\sigma$ and $\mu=0$
[ccc]{}
(900,900)(0,0) (137.0,82.0)
------------------------------------------------------------------------
(117,82)[(0,0)\[r\][ 0]{}]{} (812.0,82.0)
------------------------------------------------------------------------
(137.0,159.0)
------------------------------------------------------------------------
(117,159)[(0,0)\[r\][ 0.5]{}]{} (812.0,159.0)
------------------------------------------------------------------------
(137.0,236.0)
------------------------------------------------------------------------
(117,236)[(0,0)\[r\][ 1]{}]{} (812.0,236.0)
------------------------------------------------------------------------
(137.0,313.0)
------------------------------------------------------------------------
(117,313)[(0,0)\[r\][ 1.5]{}]{} (812.0,313.0)
------------------------------------------------------------------------
(137.0,390.0)
------------------------------------------------------------------------
(117,390)[(0,0)\[r\][ 2]{}]{} (812.0,390.0)
------------------------------------------------------------------------
(137.0,468.0)
------------------------------------------------------------------------
(117,468)[(0,0)\[r\][ 2.5]{}]{} (812.0,468.0)
------------------------------------------------------------------------
(137.0,545.0)
------------------------------------------------------------------------
(117,545)[(0,0)\[r\][ 3]{}]{} (812.0,545.0)
------------------------------------------------------------------------
(137.0,622.0)
------------------------------------------------------------------------
(117,622)[(0,0)\[r\][ 3.5]{}]{} (812.0,622.0)
------------------------------------------------------------------------
(137.0,699.0)
------------------------------------------------------------------------
(117,699)[(0,0)\[r\][ 4]{}]{} (812.0,699.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(117,776)[(0,0)\[r\][ 4.5]{}]{} (812.0,776.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137,41)[(0,0)[ 0]{}]{} (137.0,756.0)
------------------------------------------------------------------------
(206.0,82.0)
------------------------------------------------------------------------
(206,41)[(0,0)[ 0.2]{}]{} (206.0,756.0)
------------------------------------------------------------------------
(276.0,82.0)
------------------------------------------------------------------------
(276,41)[(0,0)[ 0.4]{}]{} (276.0,756.0)
------------------------------------------------------------------------
(345.0,82.0)
------------------------------------------------------------------------
(345,41)[(0,0)[ 0.6]{}]{} (345.0,756.0)
------------------------------------------------------------------------
(415.0,82.0)
------------------------------------------------------------------------
(415,41)[(0,0)[ 0.8]{}]{} (415.0,756.0)
------------------------------------------------------------------------
(484.0,82.0)
------------------------------------------------------------------------
(484,41)[(0,0)[ 1]{}]{} (484.0,756.0)
------------------------------------------------------------------------
(554.0,82.0)
------------------------------------------------------------------------
(554,41)[(0,0)[ 1.2]{}]{} (554.0,756.0)
------------------------------------------------------------------------
(623.0,82.0)
------------------------------------------------------------------------
(623,41)[(0,0)[ 1.4]{}]{} (623.0,756.0)
------------------------------------------------------------------------
(693.0,82.0)
------------------------------------------------------------------------
(693,41)[(0,0)[ 1.6]{}]{} (693.0,756.0)
------------------------------------------------------------------------
(762.0,82.0)
------------------------------------------------------------------------
(762,41)[(0,0)[ 1.8]{}]{} (762.0,756.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(832,41)[(0,0)[ 2]{}]{} (832.0,756.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(484,838)[(0,0)[Logonormal PDF ($\mu$=0)]{}]{} (519,545)[(0,0)\[l\][$\sigma$=1]{}]{} (589,236)[(0,0)\[l\][$\sigma$=2]{}]{} (728,144)[(0,0)\[l\][$\sigma$=3]{}]{} (137,82) (369,81.67)
------------------------------------------------------------------------
(369.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(376,82.67)
------------------------------------------------------------------------
(376.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(383.00,84.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(383.00,83.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(390.00,87.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(390.00,86.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(397.59,94.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(396.17,94.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(404.59,107.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(403.17,107.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(411.59,128.00)(0.485,2.552)[11]{}
------------------------------------------------------------------------
(410.17,128.00)(7.000,29.760)[2]{}
------------------------------------------------------------------------
(418.59,162.00)(0.485,3.620)[11]{}
------------------------------------------------------------------------
(417.17,162.00)(7.000,42.100)[2]{}
------------------------------------------------------------------------
(425.59,210.00)(0.485,4.688)[11]{}
------------------------------------------------------------------------
(424.17,210.00)(7.000,54.439)[2]{}
------------------------------------------------------------------------
(432.59,272.00)(0.485,5.679)[11]{}
------------------------------------------------------------------------
(431.17,272.00)(7.000,65.897)[2]{}
------------------------------------------------------------------------
(439.59,347.00)(0.485,6.290)[11]{}
------------------------------------------------------------------------
(438.17,347.00)(7.000,72.948)[2]{}
------------------------------------------------------------------------
(446.59,430.00)(0.485,6.366)[11]{}
------------------------------------------------------------------------
(445.17,430.00)(7.000,73.830)[2]{}
------------------------------------------------------------------------
(453.59,514.00)(0.485,5.756)[11]{}
------------------------------------------------------------------------
(452.17,514.00)(7.000,66.779)[2]{}
------------------------------------------------------------------------
(460.59,590.00)(0.485,4.535)[11]{}
------------------------------------------------------------------------
(459.17,590.00)(7.000,52.676)[2]{}
------------------------------------------------------------------------
(467.59,650.00)(0.485,2.857)[11]{}
------------------------------------------------------------------------
(466.17,650.00)(7.000,33.286)[2]{}
------------------------------------------------------------------------
(474.59,688.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(473.17,688.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(481.59,696.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(480.17,698.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(488.59,678.28)(0.485,-2.628)[11]{}
------------------------------------------------------------------------
(487.17,682.64)(7.000,-30.641)[2]{}
------------------------------------------------------------------------
(495.59,639.01)(0.485,-4.001)[11]{}
------------------------------------------------------------------------
(494.17,645.51)(7.000,-46.506)[2]{}
------------------------------------------------------------------------
(502.59,583.40)(0.485,-4.840)[11]{}
------------------------------------------------------------------------
(501.17,591.20)(7.000,-56.202)[2]{}
------------------------------------------------------------------------
(509.59,518.45)(0.485,-5.145)[11]{}
------------------------------------------------------------------------
(508.17,526.73)(7.000,-59.727)[2]{}
------------------------------------------------------------------------
(516.59,450.22)(0.485,-5.222)[11]{}
------------------------------------------------------------------------
(515.17,458.61)(7.000,-60.609)[2]{}
------------------------------------------------------------------------
(523.59,382.64)(0.485,-4.764)[11]{}
------------------------------------------------------------------------
(522.17,390.32)(7.000,-55.320)[2]{}
------------------------------------------------------------------------
(530.59,321.06)(0.485,-4.306)[11]{}
------------------------------------------------------------------------
(529.17,328.03)(7.000,-50.032)[2]{}
------------------------------------------------------------------------
(537.59,266.44)(0.485,-3.544)[11]{}
------------------------------------------------------------------------
(536.17,272.22)(7.000,-41.218)[2]{}
------------------------------------------------------------------------
(544.59,221.33)(0.485,-2.933)[11]{}
------------------------------------------------------------------------
(543.17,226.17)(7.000,-34.167)[2]{}
------------------------------------------------------------------------
(551.59,184.23)(0.485,-2.323)[11]{}
------------------------------------------------------------------------
(550.17,188.12)(7.000,-27.116)[2]{}
------------------------------------------------------------------------
(558.59,155.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(557.17,158.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(565.59,133.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(564.17,135.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(572.59,117.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(571.17,119.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(579.59,105.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(578.17,106.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(586.00,97.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(586.00,98.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(593.00,91.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(593.00,92.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(600,87.17)
------------------------------------------------------------------------
(600.00,88.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(607,85.17)
------------------------------------------------------------------------
(607.00,86.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(614,83.67)
------------------------------------------------------------------------
(614.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(621,82.67)
------------------------------------------------------------------------
(621.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(635,81.67)
------------------------------------------------------------------------
(635.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(628.0,83.0)
------------------------------------------------------------------------
(642.0,82.0)
------------------------------------------------------------------------
(137,82) (298,81.67)
------------------------------------------------------------------------
(298.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(305,82.67)
------------------------------------------------------------------------
(305.00,82.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(313,83.67)
------------------------------------------------------------------------
(313.00,83.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(320.00,85.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(320.00,84.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(327.00,88.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(327.00,87.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(334.00,92.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(334.00,91.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(341.00,97.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(341.00,96.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(348.59,104.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(347.17,104.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(355.59,114.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(354.17,114.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(362.59,127.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(361.17,127.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(369.59,142.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(368.17,142.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(376.59,159.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(375.17,159.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(383.59,180.00)(0.485,1.637)[11]{}
------------------------------------------------------------------------
(382.17,180.00)(7.000,19.183)[2]{}
------------------------------------------------------------------------
(390.59,202.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(389.17,202.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(397.59,225.00)(0.485,1.865)[11]{}
------------------------------------------------------------------------
(396.17,225.00)(7.000,21.827)[2]{}
------------------------------------------------------------------------
(404.59,250.00)(0.485,1.865)[11]{}
------------------------------------------------------------------------
(403.17,250.00)(7.000,21.827)[2]{}
------------------------------------------------------------------------
(411.59,275.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(410.17,275.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(418.59,298.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(417.17,298.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(425.59,321.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(424.17,321.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(432.59,341.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(431.17,341.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(439.59,359.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(438.17,359.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(446.59,373.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(445.17,373.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(453.59,384.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(452.17,384.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(460.00,392.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(460.00,391.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(474.00,393.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(474.00,394.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(481.00,390.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(481.00,391.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(488.59,383.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(487.17,384.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(495.59,373.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(494.17,375.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(502.59,362.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(501.17,364.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(509.59,349.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(508.17,351.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(516.59,334.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(515.17,336.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(523.59,318.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(522.17,320.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(530.59,301.79)(0.485,-1.179)[11]{}
------------------------------------------------------------------------
(529.17,303.89)(7.000,-13.895)[2]{}
------------------------------------------------------------------------
(537.59,285.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(536.17,287.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(544.59,268.79)(0.485,-1.179)[11]{}
------------------------------------------------------------------------
(543.17,270.89)(7.000,-13.895)[2]{}
------------------------------------------------------------------------
(551.59,253.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(550.17,255.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(558.59,237.79)(0.485,-1.179)[11]{}
------------------------------------------------------------------------
(557.17,239.89)(7.000,-13.895)[2]{}
------------------------------------------------------------------------
(565.59,222.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(564.17,224.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(572.59,208.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(571.17,210.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(579.59,195.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(578.17,197.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(586.59,182.74)(0.485,-0.874)[11]{}
------------------------------------------------------------------------
(585.17,184.37)(7.000,-10.369)[2]{}
------------------------------------------------------------------------
(593.59,171.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(592.17,172.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(600.59,161.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(599.17,162.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(607.59,151.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(606.17,152.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(614.59,142.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(613.17,143.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(621.00,135.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(621.00,136.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(628.00,128.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(628.00,129.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(635.00,122.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(635.00,123.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(642.00,117.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(642.00,118.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(649.00,112.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(649.00,113.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(656.00,107.94)(1.066,-0.468)[5]{}
------------------------------------------------------------------------
(656.00,108.17)(6.132,-4.000)[2]{}
------------------------------------------------------------------------
(664.00,103.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(664.00,104.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(671.00,100.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(671.00,101.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(678,97.17)
------------------------------------------------------------------------
(678.00,98.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(685,95.17)
------------------------------------------------------------------------
(685.00,96.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(692,93.17)
------------------------------------------------------------------------
(692.00,94.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(699,91.17)
------------------------------------------------------------------------
(699.00,92.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(706,89.67)
------------------------------------------------------------------------
(706.00,90.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(713,88.17)
------------------------------------------------------------------------
(713.00,89.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(720,86.67)
------------------------------------------------------------------------
(720.00,87.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(467.0,395.0)
------------------------------------------------------------------------
(734,85.67)
------------------------------------------------------------------------
(734.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(741,84.67)
------------------------------------------------------------------------
(741.00,85.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(727.0,87.0)
------------------------------------------------------------------------
(755,83.67)
------------------------------------------------------------------------
(755.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(748.0,85.0)
------------------------------------------------------------------------
(776,82.67)
------------------------------------------------------------------------
(776.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(762.0,84.0)
------------------------------------------------------------------------
(818,81.67)
------------------------------------------------------------------------
(818.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(783.0,83.0)
------------------------------------------------------------------------
(825.0,82.0)
------------------------------------------------------------------------
(137,82) (186,81.67)
------------------------------------------------------------------------
(186.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(193,82.67)
------------------------------------------------------------------------
(193.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(200,84.17)
------------------------------------------------------------------------
(200.00,83.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(207,86.17)
------------------------------------------------------------------------
(207.00,85.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(214.00,88.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(214.00,87.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(221.00,91.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(221.00,90.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(228.00,95.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(228.00,94.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(235.00,100.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(235.00,99.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(242.00,106.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(242.00,105.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(249.00,112.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(249.00,111.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(256.00,119.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(256.00,118.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(263.59,126.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(262.17,126.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(270.00,134.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(270.00,133.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(277.59,141.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(276.17,141.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(284.59,149.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(283.17,149.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(291.00,157.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(291.00,156.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(298.00,164.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(298.00,163.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(305.00,171.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(305.00,170.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(313.00,178.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(313.00,177.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(320.00,184.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(320.00,183.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(327.00,190.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(327.00,189.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(334.00,196.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(334.00,195.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(341.00,201.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(341.00,200.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(348.00,205.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(348.00,204.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(355.00,209.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(355.00,208.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(362.00,212.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(362.00,211.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(369,215.17)
------------------------------------------------------------------------
(369.00,214.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(376,217.17)
------------------------------------------------------------------------
(376.00,216.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(383,218.67)
------------------------------------------------------------------------
(383.00,218.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(390,219.67)
------------------------------------------------------------------------
(390.00,219.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(418,219.67)
------------------------------------------------------------------------
(418.00,220.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(425,218.67)
------------------------------------------------------------------------
(425.00,219.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(432,217.67)
------------------------------------------------------------------------
(432.00,218.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(439,216.67)
------------------------------------------------------------------------
(439.00,217.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(446,215.17)
------------------------------------------------------------------------
(446.00,216.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(453,213.17)
------------------------------------------------------------------------
(453.00,214.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(460,211.17)
------------------------------------------------------------------------
(460.00,212.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(467,209.17)
------------------------------------------------------------------------
(467.00,210.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(474.00,207.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(474.00,208.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(481,204.17)
------------------------------------------------------------------------
(481.00,205.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(488.00,202.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(488.00,203.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(495,199.17)
------------------------------------------------------------------------
(495.00,200.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(502.00,197.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(502.00,198.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(509.00,194.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(509.00,195.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(516.00,191.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(516.00,192.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(523.00,188.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(523.00,189.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(530,185.17)
------------------------------------------------------------------------
(530.00,186.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(537.00,183.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(537.00,184.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(544.00,180.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(544.00,181.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(551.00,177.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(551.00,178.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(558.00,174.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(558.00,175.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(565,171.17)
------------------------------------------------------------------------
(565.00,172.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(572.00,169.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(572.00,170.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(579.00,166.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(579.00,167.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(586,163.17)
------------------------------------------------------------------------
(586.00,164.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(593.00,161.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(593.00,162.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(600,158.17)
------------------------------------------------------------------------
(600.00,159.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(607.00,156.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(607.00,157.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(614,153.17)
------------------------------------------------------------------------
(614.00,154.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(621.00,151.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(621.00,152.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(628,148.17)
------------------------------------------------------------------------
(628.00,149.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(635,146.17)
------------------------------------------------------------------------
(635.00,147.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(642,144.17)
------------------------------------------------------------------------
(642.00,145.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(649,142.17)
------------------------------------------------------------------------
(649.00,143.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(656.00,140.95)(1.579,-0.447)[3]{}
------------------------------------------------------------------------
(656.00,141.17)(5.579,-3.000)[2]{}
------------------------------------------------------------------------
(664,137.17)
------------------------------------------------------------------------
(664.00,138.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(671,135.67)
------------------------------------------------------------------------
(671.00,136.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(678,134.17)
------------------------------------------------------------------------
(678.00,135.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(685,132.17)
------------------------------------------------------------------------
(685.00,133.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(692,130.17)
------------------------------------------------------------------------
(692.00,131.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(699,128.17)
------------------------------------------------------------------------
(699.00,129.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(706,126.67)
------------------------------------------------------------------------
(706.00,127.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(713,125.17)
------------------------------------------------------------------------
(713.00,126.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(720,123.17)
------------------------------------------------------------------------
(720.00,124.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(727,121.67)
------------------------------------------------------------------------
(727.00,122.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(734,120.17)
------------------------------------------------------------------------
(734.00,121.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(741,118.67)
------------------------------------------------------------------------
(741.00,119.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(748,117.67)
------------------------------------------------------------------------
(748.00,118.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(755,116.17)
------------------------------------------------------------------------
(755.00,117.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(762,114.67)
------------------------------------------------------------------------
(762.00,115.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(769,113.67)
------------------------------------------------------------------------
(769.00,114.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(776,112.67)
------------------------------------------------------------------------
(776.00,113.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(783,111.67)
------------------------------------------------------------------------
(783.00,112.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(790,110.17)
------------------------------------------------------------------------
(790.00,111.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(797,108.67)
------------------------------------------------------------------------
(797.00,109.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(804,107.67)
------------------------------------------------------------------------
(804.00,108.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(811,106.67)
------------------------------------------------------------------------
(811.00,107.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(818,105.67)
------------------------------------------------------------------------
(818.00,106.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(397.0,221.0)
------------------------------------------------------------------------
(825.0,106.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(900,900)(0,0) (137.0,82.0)
------------------------------------------------------------------------
(117,82)[(0,0)\[r\][ 0]{}]{} (812.0,82.0)
------------------------------------------------------------------------
(137.0,221.0)
------------------------------------------------------------------------
(117,221)[(0,0)\[r\][ 0.2]{}]{} (812.0,221.0)
------------------------------------------------------------------------
(137.0,360.0)
------------------------------------------------------------------------
(117,360)[(0,0)\[r\][ 0.4]{}]{} (812.0,360.0)
------------------------------------------------------------------------
(137.0,498.0)
------------------------------------------------------------------------
(117,498)[(0,0)\[r\][ 0.6]{}]{} (812.0,498.0)
------------------------------------------------------------------------
(137.0,637.0)
------------------------------------------------------------------------
(117,637)[(0,0)\[r\][ 0.8]{}]{} (812.0,637.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(117,776)[(0,0)\[r\][ 1]{}]{} (812.0,776.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137,41)[(0,0)[ 0]{}]{} (137.0,756.0)
------------------------------------------------------------------------
(206.0,82.0)
------------------------------------------------------------------------
(206,41)[(0,0)[ 0.2]{}]{} (206.0,756.0)
------------------------------------------------------------------------
(276.0,82.0)
------------------------------------------------------------------------
(276,41)[(0,0)[ 0.4]{}]{} (276.0,756.0)
------------------------------------------------------------------------
(345.0,82.0)
------------------------------------------------------------------------
(345,41)[(0,0)[ 0.6]{}]{} (345.0,756.0)
------------------------------------------------------------------------
(415.0,82.0)
------------------------------------------------------------------------
(415,41)[(0,0)[ 0.8]{}]{} (415.0,756.0)
------------------------------------------------------------------------
(484.0,82.0)
------------------------------------------------------------------------
(484,41)[(0,0)[ 1]{}]{} (484.0,756.0)
------------------------------------------------------------------------
(554.0,82.0)
------------------------------------------------------------------------
(554,41)[(0,0)[ 1.2]{}]{} (554.0,756.0)
------------------------------------------------------------------------
(623.0,82.0)
------------------------------------------------------------------------
(623,41)[(0,0)[ 1.4]{}]{} (623.0,756.0)
------------------------------------------------------------------------
(693.0,82.0)
------------------------------------------------------------------------
(693,41)[(0,0)[ 1.6]{}]{} (693.0,756.0)
------------------------------------------------------------------------
(762.0,82.0)
------------------------------------------------------------------------
(762,41)[(0,0)[ 1.8]{}]{} (762.0,756.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(832,41)[(0,0)[ 2]{}]{} (832.0,756.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(484,838)[(0,0)[Logonormal Cumulative Distribution ($\mu$=0)]{}]{} (467,151)[(0,0)\[l\][$\sigma$=1]{}]{} (540,672)[(0,0)\[l\][$\sigma$=2]{}]{} (571,568)[(0,0)\[l\][$\sigma$=3]{}]{} (137,82) (411,81.67)
------------------------------------------------------------------------
(411.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(418,83.17)
------------------------------------------------------------------------
(418.00,82.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(425.00,85.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(425.00,84.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(432.59,89.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(431.17,89.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(439.59,98.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(438.17,98.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(446.59,116.00)(0.485,2.171)[11]{}
------------------------------------------------------------------------
(445.17,116.00)(7.000,25.353)[2]{}
------------------------------------------------------------------------
(453.59,145.00)(0.485,3.162)[11]{}
------------------------------------------------------------------------
(452.17,145.00)(7.000,36.811)[2]{}
------------------------------------------------------------------------
(460.59,187.00)(0.485,4.383)[11]{}
------------------------------------------------------------------------
(459.17,187.00)(7.000,50.913)[2]{}
------------------------------------------------------------------------
(467.59,245.00)(0.485,5.222)[11]{}
------------------------------------------------------------------------
(466.17,245.00)(7.000,60.609)[2]{}
------------------------------------------------------------------------
(474.59,314.00)(0.485,5.832)[11]{}
------------------------------------------------------------------------
(473.17,314.00)(7.000,67.660)[2]{}
------------------------------------------------------------------------
(481.59,391.00)(0.485,5.985)[11]{}
------------------------------------------------------------------------
(480.17,391.00)(7.000,69.423)[2]{}
------------------------------------------------------------------------
(488.59,470.00)(0.485,5.603)[11]{}
------------------------------------------------------------------------
(487.17,470.00)(7.000,65.016)[2]{}
------------------------------------------------------------------------
(495.59,544.00)(0.485,4.917)[11]{}
------------------------------------------------------------------------
(494.17,544.00)(7.000,57.083)[2]{}
------------------------------------------------------------------------
(502.59,609.00)(0.485,3.925)[11]{}
------------------------------------------------------------------------
(501.17,609.00)(7.000,45.625)[2]{}
------------------------------------------------------------------------
(509.59,661.00)(0.485,3.010)[11]{}
------------------------------------------------------------------------
(508.17,661.00)(7.000,35.048)[2]{}
------------------------------------------------------------------------
(516.59,701.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(515.17,701.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(523.59,729.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(522.17,729.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(530.59,748.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(529.17,748.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(537.00,760.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(537.00,759.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(544.00,767.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(544.00,766.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(551,772.17)
------------------------------------------------------------------------
(551.00,771.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(558,773.67)
------------------------------------------------------------------------
(558.00,773.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(565,774.67)
------------------------------------------------------------------------
(565.00,774.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(572.0,776.0)
------------------------------------------------------------------------
(137,82) (355,81.67)
------------------------------------------------------------------------
(355.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(369,83.17)
------------------------------------------------------------------------
(369.00,82.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(376,85.17)
------------------------------------------------------------------------
(376.00,84.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(383.00,87.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(383.00,86.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(390.00,91.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(390.00,90.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(397.59,96.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(396.17,96.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(404.59,104.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(403.17,104.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(411.59,114.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(410.17,114.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(418.59,128.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(417.17,128.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(425.59,146.00)(0.485,1.637)[11]{}
------------------------------------------------------------------------
(424.17,146.00)(7.000,19.183)[2]{}
------------------------------------------------------------------------
(432.59,168.00)(0.485,1.942)[11]{}
------------------------------------------------------------------------
(431.17,168.00)(7.000,22.709)[2]{}
------------------------------------------------------------------------
(439.59,194.00)(0.485,2.171)[11]{}
------------------------------------------------------------------------
(438.17,194.00)(7.000,25.353)[2]{}
------------------------------------------------------------------------
(446.59,223.00)(0.485,2.552)[11]{}
------------------------------------------------------------------------
(445.17,223.00)(7.000,29.760)[2]{}
------------------------------------------------------------------------
(453.59,257.00)(0.485,2.705)[11]{}
------------------------------------------------------------------------
(452.17,257.00)(7.000,31.523)[2]{}
------------------------------------------------------------------------
(460.59,293.00)(0.485,2.857)[11]{}
------------------------------------------------------------------------
(459.17,293.00)(7.000,33.286)[2]{}
------------------------------------------------------------------------
(467.59,331.00)(0.485,2.933)[11]{}
------------------------------------------------------------------------
(466.17,331.00)(7.000,34.167)[2]{}
------------------------------------------------------------------------
(474.59,370.00)(0.485,3.010)[11]{}
------------------------------------------------------------------------
(473.17,370.00)(7.000,35.048)[2]{}
------------------------------------------------------------------------
(481.59,410.00)(0.485,3.010)[11]{}
------------------------------------------------------------------------
(480.17,410.00)(7.000,35.048)[2]{}
------------------------------------------------------------------------
(488.59,450.00)(0.485,2.857)[11]{}
------------------------------------------------------------------------
(487.17,450.00)(7.000,33.286)[2]{}
------------------------------------------------------------------------
(495.59,488.00)(0.485,2.705)[11]{}
------------------------------------------------------------------------
(494.17,488.00)(7.000,31.523)[2]{}
------------------------------------------------------------------------
(502.59,524.00)(0.485,2.552)[11]{}
------------------------------------------------------------------------
(501.17,524.00)(7.000,29.760)[2]{}
------------------------------------------------------------------------
(509.59,558.00)(0.485,2.399)[11]{}
------------------------------------------------------------------------
(508.17,558.00)(7.000,27.997)[2]{}
------------------------------------------------------------------------
(516.59,590.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(515.17,590.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(523.59,618.00)(0.485,1.942)[11]{}
------------------------------------------------------------------------
(522.17,618.00)(7.000,22.709)[2]{}
------------------------------------------------------------------------
(530.59,644.00)(0.485,1.637)[11]{}
------------------------------------------------------------------------
(529.17,644.00)(7.000,19.183)[2]{}
------------------------------------------------------------------------
(537.59,666.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(536.17,666.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(544.59,685.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(543.17,685.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(551.59,702.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(550.17,702.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(558.59,716.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(557.17,716.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(565.59,728.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(564.17,728.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(572.59,738.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(571.17,738.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(579.00,746.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(579.00,745.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(586.00,752.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(586.00,751.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(593.00,757.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(593.00,756.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(600.00,762.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(600.00,761.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(607,765.17)
------------------------------------------------------------------------
(607.00,764.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(614,767.17)
------------------------------------------------------------------------
(614.00,766.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(621,769.17)
------------------------------------------------------------------------
(621.00,768.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(628,770.67)
------------------------------------------------------------------------
(628.00,770.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(635,771.67)
------------------------------------------------------------------------
(635.00,771.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(642,772.67)
------------------------------------------------------------------------
(642.00,772.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(362.0,83.0)
------------------------------------------------------------------------
(656,773.67)
------------------------------------------------------------------------
(656.00,773.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(649.0,774.0)
------------------------------------------------------------------------
(678,774.67)
------------------------------------------------------------------------
(678.00,774.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(664.0,775.0)
------------------------------------------------------------------------
(685.0,776.0)
------------------------------------------------------------------------
(137,82) (249,81.67)
------------------------------------------------------------------------
(249.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(263,82.67)
------------------------------------------------------------------------
(263.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(270,84.17)
------------------------------------------------------------------------
(270.00,83.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(277,85.67)
------------------------------------------------------------------------
(277.00,85.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(284.00,87.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(284.00,86.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(291.00,90.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(291.00,89.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(298.00,93.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(298.00,92.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(305.00,96.59)(0.821,0.477)[7]{}
------------------------------------------------------------------------
(305.00,95.17)(6.464,5.000)[2]{}
------------------------------------------------------------------------
(313.00,101.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(313.00,100.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(320.00,106.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(320.00,105.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(327.00,112.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(327.00,111.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(334.59,119.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(333.17,119.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(341.59,128.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(340.17,128.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(348.59,137.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(347.17,137.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(355.59,147.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(354.17,147.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(362.59,158.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(361.17,158.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(369.59,170.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(368.17,170.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(376.59,182.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(375.17,182.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(383.59,196.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(382.17,196.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(390.59,210.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(389.17,210.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(397.59,225.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(396.17,225.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(404.59,240.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(403.17,240.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(411.59,256.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(410.17,256.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(418.59,272.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(417.17,272.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(425.59,289.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(424.17,289.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(432.59,305.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(431.17,305.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(439.59,322.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(438.17,322.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(446.59,339.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(445.17,339.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(453.59,356.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(452.17,356.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(460.59,372.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(459.17,372.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(467.59,389.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(466.17,389.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(474.59,405.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(473.17,405.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(481.59,421.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(480.17,421.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(488.59,437.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(487.17,437.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(495.59,453.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(494.17,453.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(502.59,468.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(501.17,468.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(509.59,483.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(508.17,483.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(516.59,497.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(515.17,497.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(523.59,511.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(522.17,511.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(530.59,524.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(529.17,524.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(537.59,537.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(536.17,537.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(544.59,549.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(543.17,549.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(551.59,561.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(550.17,561.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(558.59,573.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(557.17,573.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(565.59,584.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(564.17,584.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(572.59,594.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(571.17,594.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(579.59,604.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(578.17,604.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(586.59,614.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(585.17,614.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(593.59,623.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(592.17,623.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(600.59,632.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(599.17,632.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(607.59,640.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(606.17,640.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(614.59,648.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(613.17,648.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(621.00,656.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(621.00,655.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(628.00,663.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(628.00,662.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(635.00,669.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(635.00,668.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(642.00,676.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(642.00,675.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(649.00,682.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(649.00,681.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(656.00,687.59)(0.671,0.482)[9]{}
------------------------------------------------------------------------
(656.00,686.17)(6.685,6.000)[2]{}
------------------------------------------------------------------------
(664.00,693.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(664.00,692.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(671.00,698.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(671.00,697.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(678.00,703.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(678.00,702.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(685.00,707.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(685.00,706.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(692.00,711.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(692.00,710.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(699.00,715.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(699.00,714.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(706.00,719.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(706.00,718.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(713.00,723.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(713.00,722.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(720.00,726.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(720.00,725.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(727.00,729.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(727.00,728.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(734.00,732.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(734.00,731.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(741.00,735.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(741.00,734.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(748,738.17)
------------------------------------------------------------------------
(748.00,737.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(755,740.17)
------------------------------------------------------------------------
(755.00,739.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(762,742.17)
------------------------------------------------------------------------
(762.00,741.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(769.00,744.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(769.00,743.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(776,746.67)
------------------------------------------------------------------------
(776.00,746.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(783,748.17)
------------------------------------------------------------------------
(783.00,747.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(790,750.17)
------------------------------------------------------------------------
(790.00,749.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(797,751.67)
------------------------------------------------------------------------
(797.00,751.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(804,753.17)
------------------------------------------------------------------------
(804.00,752.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(811,754.67)
------------------------------------------------------------------------
(811.00,754.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(818,755.67)
------------------------------------------------------------------------
(818.00,755.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(825,757.17)
------------------------------------------------------------------------
(825.00,756.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(256.0,83.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
\(a) & & (b)
Note that $\mu$ and $\sigma$ are the mean and variance of $\log{X}$, *not* of $X$. To distinguish them, I shall indicate the mean and the variance of $X$ as $m$ and $v$, respectively.
The moments of $X$ are given by $$\langle X^n \rangle = \int_0^\infty x^n P_X(x) dx = \exp\bigl(n\mu + \frac{n^2\sigma^2}{2}\bigr)$$ as can be verified by replacing $z=\frac{1}{\sigma}\bigl[\log{X}-(\mu+n\sigma^2)\bigr]$ in the integral. From this we have $$\begin{aligned}
m &= \langle{X}\rangle = \exp\bigl(\mu+\frac{\sigma^2}{2}\bigr) \\
\langle{X^2}\rangle &= \exp\bigl(2\mu+2\sigma^2\bigr) \\
v &= \langle{X^2}\rangle - \langle{X}\rangle^2 = \exp(2\mu+\sigma^2)(e^{\sigma^2}-1)
\end{aligned}$$ From these equality, one can derive the values of $\mu$ and $\sigma^2$ for desired $m$ and $v$: $$\mu = \log\frac{m}{\sqrt{\displaystyle 1 + \frac{v}{m^2}}}\ \ \ \sigma^2 = \log\left(1 + \frac{v}{m^2}\right)$$ The characteristic function $\langle\exp(i\omega{x})\rangle$ is defined, but if we try to extend it to complex variables, $\langle\exp(s{x})\rangle$, $s\in{\mathbb{C}}$ is not defined for any $s$ with a negative imaginary part. This entails that the characteristic function is not analytical in the origin and, consequently, it can’t be represented as an infinite convergent series. In particular, the formal Taylor series $$\sum_n \frac{(i\omega{x})^n}{n!} \langle{x^n}\rangle =
\sum_n \frac{(i\omega{x})^n}{n!} \exp\bigl(n\mu + \frac{\displaystyle n^2\sigma^2}{2}\bigr)$$ diverges
\* \* \*
Other positive variables follow a different distribution, one in which the value $0$ is the most probable, and the probability decreases sharply as $x$ increases, In these cases, the variable $x$ can be modeled using an **exponential** distribution: $$P_X(x) =
\begin{cases}
\lambda e^{-\lambda{x}} & x \ge 0 \\
0 & x < 0
\end{cases}$$ If the variable can take negative values, then $$P_X(x) = \frac{\lambda}{2} e^{-\lambda|x|}$$ (Figure \[exponential\]. Its characteristic function is
[ccc]{}
(900,900)(0,0) (127.0,82.0)
------------------------------------------------------------------------
(107,82)[(0,0)\[r\][$0$]{}]{} (802.0,82.0)
------------------------------------------------------------------------
(127.0,151.0)
------------------------------------------------------------------------
(107,151)[(0,0)\[r\][$0.2$]{}]{} (802.0,151.0)
------------------------------------------------------------------------
(127.0,221.0)
------------------------------------------------------------------------
(107,221)[(0,0)\[r\][$0.4$]{}]{} (802.0,221.0)
------------------------------------------------------------------------
(127.0,290.0)
------------------------------------------------------------------------
(107,290)[(0,0)\[r\][$0.6$]{}]{} (802.0,290.0)
------------------------------------------------------------------------
(127.0,360.0)
------------------------------------------------------------------------
(107,360)[(0,0)\[r\][$0.8$]{}]{} (802.0,360.0)
------------------------------------------------------------------------
(127.0,429.0)
------------------------------------------------------------------------
(107,429)[(0,0)\[r\][$1$]{}]{} (802.0,429.0)
------------------------------------------------------------------------
(127.0,498.0)
------------------------------------------------------------------------
(107,498)[(0,0)\[r\][$1.2$]{}]{} (802.0,498.0)
------------------------------------------------------------------------
(127.0,568.0)
------------------------------------------------------------------------
(107,568)[(0,0)\[r\][$1.4$]{}]{} (802.0,568.0)
------------------------------------------------------------------------
(127.0,637.0)
------------------------------------------------------------------------
(107,637)[(0,0)\[r\][$1.6$]{}]{} (802.0,637.0)
------------------------------------------------------------------------
(127.0,707.0)
------------------------------------------------------------------------
(107,707)[(0,0)\[r\][$1.8$]{}]{} (802.0,707.0)
------------------------------------------------------------------------
(127.0,776.0)
------------------------------------------------------------------------
(107,776)[(0,0)\[r\][$2$]{}]{} (802.0,776.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(127,41)[(0,0)[$0$]{}]{} (127.0,756.0)
------------------------------------------------------------------------
(243.0,82.0)
------------------------------------------------------------------------
(243,41)[(0,0)[$0.5$]{}]{} (243.0,756.0)
------------------------------------------------------------------------
(359.0,82.0)
------------------------------------------------------------------------
(359,41)[(0,0)[$1$]{}]{} (359.0,756.0)
------------------------------------------------------------------------
(475.0,82.0)
------------------------------------------------------------------------
(475,41)[(0,0)[$1.5$]{}]{} (475.0,756.0)
------------------------------------------------------------------------
(590.0,82.0)
------------------------------------------------------------------------
(590,41)[(0,0)[$2$]{}]{} (590.0,756.0)
------------------------------------------------------------------------
(706.0,82.0)
------------------------------------------------------------------------
(706,41)[(0,0)[$2.5$]{}]{} (706.0,756.0)
------------------------------------------------------------------------
(822.0,82.0)
------------------------------------------------------------------------
(822,41)[(0,0)[$3$]{}]{} (822.0,756.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(822.0,82.0)
------------------------------------------------------------------------
(127.0,776.0)
------------------------------------------------------------------------
(173,186)[(0,0)\[l\][$\lambda=0.5$]{}]{} (150,290)[(0,0)\[l\][$\lambda=1$]{}]{} (173,568)[(0,0)\[l\][$\lambda=2$]{}]{} (474,838)[(0,0)[Exponential PDF]{}]{} (127,256) (127.00,254.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(127.00,255.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(134.00,251.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(134.00,252.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(141,248.17)
------------------------------------------------------------------------
(141.00,249.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(148.00,246.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(148.00,247.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(155,243.17)
------------------------------------------------------------------------
(155.00,244.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(162.00,241.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(162.00,242.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(169,238.17)
------------------------------------------------------------------------
(169.00,239.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(176,236.17)
------------------------------------------------------------------------
(176.00,237.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(183.00,234.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(183.00,235.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(190,231.17)
------------------------------------------------------------------------
(190.00,232.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(197,229.17)
------------------------------------------------------------------------
(197.00,230.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(204,227.17)
------------------------------------------------------------------------
(204.00,228.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(211.00,225.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(211.00,226.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(218,222.17)
------------------------------------------------------------------------
(218.00,223.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(225,220.17)
------------------------------------------------------------------------
(225.00,221.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(232,218.17)
------------------------------------------------------------------------
(232.00,219.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(239,216.17)
------------------------------------------------------------------------
(239.00,217.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(246,214.17)
------------------------------------------------------------------------
(246.00,215.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(253,212.17)
------------------------------------------------------------------------
(253.00,213.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(260,210.17)
------------------------------------------------------------------------
(260.00,211.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(267,208.17)
------------------------------------------------------------------------
(267.00,209.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(274,206.17)
------------------------------------------------------------------------
(274.00,207.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(281,204.17)
------------------------------------------------------------------------
(281.00,205.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(288,202.67)
------------------------------------------------------------------------
(288.00,203.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(295,201.17)
------------------------------------------------------------------------
(295.00,202.17)(4.472,-2.000)[2]{}
------------------------------------------------------------------------
(303,199.17)
------------------------------------------------------------------------
(303.00,200.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(310,197.17)
------------------------------------------------------------------------
(310.00,198.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(317,195.67)
------------------------------------------------------------------------
(317.00,196.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(324,194.17)
------------------------------------------------------------------------
(324.00,195.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(331,192.17)
------------------------------------------------------------------------
(331.00,193.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(338,190.17)
------------------------------------------------------------------------
(338.00,191.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(345,188.67)
------------------------------------------------------------------------
(345.00,189.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(352,187.17)
------------------------------------------------------------------------
(352.00,188.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(359,185.67)
------------------------------------------------------------------------
(359.00,186.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(366,184.17)
------------------------------------------------------------------------
(366.00,185.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(373,182.67)
------------------------------------------------------------------------
(373.00,183.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(380,181.17)
------------------------------------------------------------------------
(380.00,182.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(387,179.67)
------------------------------------------------------------------------
(387.00,180.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(394,178.17)
------------------------------------------------------------------------
(394.00,179.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(401,176.67)
------------------------------------------------------------------------
(401.00,177.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(408,175.17)
------------------------------------------------------------------------
(408.00,176.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(415,173.67)
------------------------------------------------------------------------
(415.00,174.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(422,172.17)
------------------------------------------------------------------------
(422.00,173.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(429,170.67)
------------------------------------------------------------------------
(429.00,171.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(436,169.67)
------------------------------------------------------------------------
(436.00,170.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(443,168.17)
------------------------------------------------------------------------
(443.00,169.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(450,166.67)
------------------------------------------------------------------------
(450.00,167.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(457,165.67)
------------------------------------------------------------------------
(457.00,166.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(464,164.67)
------------------------------------------------------------------------
(464.00,165.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(471,163.17)
------------------------------------------------------------------------
(471.00,164.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(478,161.67)
------------------------------------------------------------------------
(478.00,162.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(485,160.67)
------------------------------------------------------------------------
(485.00,161.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(492,159.67)
------------------------------------------------------------------------
(492.00,160.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(499,158.67)
------------------------------------------------------------------------
(499.00,159.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(506,157.17)
------------------------------------------------------------------------
(506.00,158.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(513,155.67)
------------------------------------------------------------------------
(513.00,156.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(520,154.67)
------------------------------------------------------------------------
(520.00,155.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(527,153.67)
------------------------------------------------------------------------
(527.00,154.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(534,152.67)
------------------------------------------------------------------------
(534.00,153.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(541,151.67)
------------------------------------------------------------------------
(541.00,152.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(548,150.67)
------------------------------------------------------------------------
(548.00,151.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(555,149.67)
------------------------------------------------------------------------
(555.00,150.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(562,148.67)
------------------------------------------------------------------------
(562.00,149.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(569,147.67)
------------------------------------------------------------------------
(569.00,148.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(576,146.67)
------------------------------------------------------------------------
(576.00,147.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(583,145.67)
------------------------------------------------------------------------
(583.00,146.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(590,144.67)
------------------------------------------------------------------------
(590.00,145.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(597,143.67)
------------------------------------------------------------------------
(597.00,144.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(604,142.67)
------------------------------------------------------------------------
(604.00,143.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(611,141.67)
------------------------------------------------------------------------
(611.00,142.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(618,140.67)
------------------------------------------------------------------------
(618.00,141.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(625,139.67)
------------------------------------------------------------------------
(625.00,140.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(632,138.67)
------------------------------------------------------------------------
(632.00,139.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(646,137.67)
------------------------------------------------------------------------
(646.00,138.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(654,136.67)
------------------------------------------------------------------------
(654.00,137.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(661,135.67)
------------------------------------------------------------------------
(661.00,136.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(668,134.67)
------------------------------------------------------------------------
(668.00,135.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(675,133.67)
------------------------------------------------------------------------
(675.00,134.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(639.0,139.0)
------------------------------------------------------------------------
(689,132.67)
------------------------------------------------------------------------
(689.00,133.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(696,131.67)
------------------------------------------------------------------------
(696.00,132.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(703,130.67)
------------------------------------------------------------------------
(703.00,131.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(682.0,134.0)
------------------------------------------------------------------------
(717,129.67)
------------------------------------------------------------------------
(717.00,130.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(724,128.67)
------------------------------------------------------------------------
(724.00,129.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(731,127.67)
------------------------------------------------------------------------
(731.00,128.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(710.0,131.0)
------------------------------------------------------------------------
(745,126.67)
------------------------------------------------------------------------
(745.00,127.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(752,125.67)
------------------------------------------------------------------------
(752.00,126.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(738.0,128.0)
------------------------------------------------------------------------
(766,124.67)
------------------------------------------------------------------------
(766.00,125.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(773,123.67)
------------------------------------------------------------------------
(773.00,124.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(759.0,126.0)
------------------------------------------------------------------------
(787,122.67)
------------------------------------------------------------------------
(787.00,123.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(780.0,124.0)
------------------------------------------------------------------------
(801,121.67)
------------------------------------------------------------------------
(801.00,122.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(808,120.67)
------------------------------------------------------------------------
(808.00,121.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(794.0,123.0)
------------------------------------------------------------------------
(815.0,121.0)
------------------------------------------------------------------------
(127,429) (127.59,426.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(126.17,427.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(134.59,416.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(133.17,417.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(141.59,406.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(140.17,407.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(148.59,396.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(147.17,397.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(155.59,386.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(154.17,387.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(162.59,377.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(161.17,378.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(169.59,368.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(168.17,369.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(176.59,360.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(175.17,361.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(183.59,351.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(182.17,352.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(190.59,343.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(189.17,344.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(197.00,336.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(197.00,337.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(204.59,328.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(203.17,329.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(211.00,321.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(211.00,322.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(218.00,314.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(218.00,315.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(225.00,307.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(225.00,308.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(232.00,300.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(232.00,301.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(239.00,294.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(239.00,295.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(246.00,287.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(246.00,288.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(253.00,281.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(253.00,282.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(260.00,275.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(260.00,276.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(267.00,269.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(267.00,270.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(274.00,264.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(274.00,265.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(281.00,258.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(281.00,259.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(288.00,253.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(288.00,254.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(295.00,248.93)(0.821,-0.477)[7]{}
------------------------------------------------------------------------
(295.00,249.17)(6.464,-5.000)[2]{}
------------------------------------------------------------------------
(303.00,243.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(303.00,244.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(310.00,238.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(310.00,239.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(317.00,233.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(317.00,234.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(324.00,229.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(324.00,230.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(331.00,224.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(331.00,225.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(338.00,220.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(338.00,221.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(345.00,216.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(345.00,217.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(352.00,212.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(352.00,213.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(359.00,208.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(359.00,209.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(366.00,204.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(366.00,205.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(373.00,200.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(373.00,201.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(380.00,197.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(380.00,198.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(387.00,193.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(387.00,194.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(394.00,190.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(394.00,191.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(401.00,186.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(401.00,187.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(408.00,183.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(408.00,184.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(415.00,180.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(415.00,181.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(422.00,177.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(422.00,178.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(429.00,174.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(429.00,175.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(436,171.17)
------------------------------------------------------------------------
(436.00,172.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(443.00,169.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(443.00,170.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(450,166.17)
------------------------------------------------------------------------
(450.00,167.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(457.00,164.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(457.00,165.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(464,161.17)
------------------------------------------------------------------------
(464.00,162.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(471.00,159.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(471.00,160.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(478,156.17)
------------------------------------------------------------------------
(478.00,157.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(485,154.17)
------------------------------------------------------------------------
(485.00,155.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(492,152.17)
------------------------------------------------------------------------
(492.00,153.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(499,150.17)
------------------------------------------------------------------------
(499.00,151.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(506,148.17)
------------------------------------------------------------------------
(506.00,149.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(513,146.17)
------------------------------------------------------------------------
(513.00,147.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(520,144.17)
------------------------------------------------------------------------
(520.00,145.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(527,142.17)
------------------------------------------------------------------------
(527.00,143.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(534,140.17)
------------------------------------------------------------------------
(534.00,141.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(541,138.17)
------------------------------------------------------------------------
(541.00,139.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(548,136.67)
------------------------------------------------------------------------
(548.00,137.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(555,135.17)
------------------------------------------------------------------------
(555.00,136.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(562,133.17)
------------------------------------------------------------------------
(562.00,134.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(569,131.67)
------------------------------------------------------------------------
(569.00,132.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(576,130.17)
------------------------------------------------------------------------
(576.00,131.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(583,128.67)
------------------------------------------------------------------------
(583.00,129.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(590,127.67)
------------------------------------------------------------------------
(590.00,128.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(597,126.17)
------------------------------------------------------------------------
(597.00,127.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(604,124.67)
------------------------------------------------------------------------
(604.00,125.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(611,123.67)
------------------------------------------------------------------------
(611.00,124.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(618,122.17)
------------------------------------------------------------------------
(618.00,123.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(625,120.67)
------------------------------------------------------------------------
(625.00,121.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(632,119.67)
------------------------------------------------------------------------
(632.00,120.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(639,118.67)
------------------------------------------------------------------------
(639.00,119.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(646,117.67)
------------------------------------------------------------------------
(646.00,118.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(654,116.67)
------------------------------------------------------------------------
(654.00,117.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(661,115.67)
------------------------------------------------------------------------
(661.00,116.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(668,114.67)
------------------------------------------------------------------------
(668.00,115.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(675,113.67)
------------------------------------------------------------------------
(675.00,114.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(682,112.67)
------------------------------------------------------------------------
(682.00,113.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(689,111.67)
------------------------------------------------------------------------
(689.00,112.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(696,110.67)
------------------------------------------------------------------------
(696.00,111.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(703,109.67)
------------------------------------------------------------------------
(703.00,110.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(710,108.67)
------------------------------------------------------------------------
(710.00,109.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(717,107.67)
------------------------------------------------------------------------
(717.00,108.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(731,106.67)
------------------------------------------------------------------------
(731.00,107.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(738,105.67)
------------------------------------------------------------------------
(738.00,106.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(745,104.67)
------------------------------------------------------------------------
(745.00,105.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(724.0,108.0)
------------------------------------------------------------------------
(759,103.67)
------------------------------------------------------------------------
(759.00,104.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(766,102.67)
------------------------------------------------------------------------
(766.00,103.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(752.0,105.0)
------------------------------------------------------------------------
(780,101.67)
------------------------------------------------------------------------
(780.00,102.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(773.0,103.0)
------------------------------------------------------------------------
(794,100.67)
------------------------------------------------------------------------
(794.00,101.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(801,99.67)
------------------------------------------------------------------------
(801.00,100.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(787.0,102.0)
------------------------------------------------------------------------
(815,98.67)
------------------------------------------------------------------------
(815.00,99.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(808.0,100.0)
------------------------------------------------------------------------
(127,776) (127.59,765.86)(0.485,-3.086)[11]{}
------------------------------------------------------------------------
(126.17,770.93)(7.000,-35.930)[2]{}
------------------------------------------------------------------------
(134.59,725.57)(0.485,-2.857)[11]{}
------------------------------------------------------------------------
(133.17,730.29)(7.000,-33.286)[2]{}
------------------------------------------------------------------------
(141.59,688.05)(0.485,-2.705)[11]{}
------------------------------------------------------------------------
(140.17,692.52)(7.000,-31.523)[2]{}
------------------------------------------------------------------------
(148.59,652.52)(0.485,-2.552)[11]{}
------------------------------------------------------------------------
(147.17,656.76)(7.000,-29.760)[2]{}
------------------------------------------------------------------------
(155.59,618.99)(0.485,-2.399)[11]{}
------------------------------------------------------------------------
(154.17,623.00)(7.000,-27.997)[2]{}
------------------------------------------------------------------------
(162.59,587.23)(0.485,-2.323)[11]{}
------------------------------------------------------------------------
(161.17,591.12)(7.000,-27.116)[2]{}
------------------------------------------------------------------------
(169.59,556.94)(0.485,-2.094)[11]{}
------------------------------------------------------------------------
(168.17,560.47)(7.000,-24.472)[2]{}
------------------------------------------------------------------------
(176.59,529.18)(0.485,-2.018)[11]{}
------------------------------------------------------------------------
(175.17,532.59)(7.000,-23.590)[2]{}
------------------------------------------------------------------------
(183.59,502.65)(0.485,-1.865)[11]{}
------------------------------------------------------------------------
(182.17,505.83)(7.000,-21.827)[2]{}
------------------------------------------------------------------------
(190.59,478.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(189.17,481.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(197.59,455.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(196.17,458.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(204.59,432.60)(0.485,-1.560)[11]{}
------------------------------------------------------------------------
(203.17,435.30)(7.000,-18.302)[2]{}
------------------------------------------------------------------------
(211.59,412.08)(0.485,-1.408)[11]{}
------------------------------------------------------------------------
(210.17,414.54)(7.000,-16.539)[2]{}
------------------------------------------------------------------------
(218.59,393.08)(0.485,-1.408)[11]{}
------------------------------------------------------------------------
(217.17,395.54)(7.000,-16.539)[2]{}
------------------------------------------------------------------------
(225.59,374.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(224.17,376.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(232.59,357.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(231.17,359.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(239.59,341.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(238.17,343.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(246.59,326.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(245.17,328.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(253.59,311.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(252.17,313.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(260.59,297.74)(0.485,-0.874)[11]{}
------------------------------------------------------------------------
(259.17,299.37)(7.000,-10.369)[2]{}
------------------------------------------------------------------------
(267.59,285.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(266.17,287.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(274.59,272.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(273.17,274.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(281.59,261.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(280.17,263.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(288.59,251.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(287.17,252.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(295.59,241.72)(0.488,-0.560)[13]{}
------------------------------------------------------------------------
(294.17,242.86)(8.000,-7.858)[2]{}
------------------------------------------------------------------------
(303.59,232.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(302.17,233.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(310.59,223.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(309.17,224.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(317.59,214.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(316.17,215.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(324.00,207.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(324.00,208.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(331.00,200.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(331.00,201.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(338.00,193.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(338.00,194.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(345.00,186.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(345.00,187.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(352.00,180.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(352.00,181.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(359.00,174.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(359.00,175.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(366.00,168.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(366.00,169.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(373.00,163.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(373.00,164.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(380.00,158.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(380.00,159.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(387.00,154.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(387.00,155.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(394.00,149.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(394.00,150.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(401.00,145.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(401.00,146.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(408.00,141.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(408.00,142.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(415.00,138.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(415.00,139.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(422.00,134.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(422.00,135.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(429.00,131.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(429.00,132.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(436.00,128.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(436.00,129.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(443,125.17)
------------------------------------------------------------------------
(443.00,126.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(450.00,123.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(450.00,124.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(457,120.17)
------------------------------------------------------------------------
(457.00,121.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(464,118.17)
------------------------------------------------------------------------
(464.00,119.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(471,116.17)
------------------------------------------------------------------------
(471.00,117.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(478,114.17)
------------------------------------------------------------------------
(478.00,115.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(485,112.17)
------------------------------------------------------------------------
(485.00,113.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(492,110.17)
------------------------------------------------------------------------
(492.00,111.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(499,108.17)
------------------------------------------------------------------------
(499.00,109.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(506,106.67)
------------------------------------------------------------------------
(506.00,107.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(513,105.17)
------------------------------------------------------------------------
(513.00,106.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(520,103.67)
------------------------------------------------------------------------
(520.00,104.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(527,102.67)
------------------------------------------------------------------------
(527.00,103.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(534,101.17)
------------------------------------------------------------------------
(534.00,102.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(541,99.67)
------------------------------------------------------------------------
(541.00,100.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(548,98.67)
------------------------------------------------------------------------
(548.00,99.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(555,97.67)
------------------------------------------------------------------------
(555.00,98.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(562,96.67)
------------------------------------------------------------------------
(562.00,97.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(569,95.67)
------------------------------------------------------------------------
(569.00,96.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(583,94.67)
------------------------------------------------------------------------
(583.00,95.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(590,93.67)
------------------------------------------------------------------------
(590.00,94.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(597,92.67)
------------------------------------------------------------------------
(597.00,93.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(576.0,96.0)
------------------------------------------------------------------------
(611,91.67)
------------------------------------------------------------------------
(611.00,92.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(618,90.67)
------------------------------------------------------------------------
(618.00,91.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(604.0,93.0)
------------------------------------------------------------------------
(632,89.67)
------------------------------------------------------------------------
(632.00,90.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(625.0,91.0)
------------------------------------------------------------------------
(646,88.67)
------------------------------------------------------------------------
(646.00,89.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(639.0,90.0)
------------------------------------------------------------------------
(668,87.67)
------------------------------------------------------------------------
(668.00,88.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(654.0,89.0)
------------------------------------------------------------------------
(682,86.67)
------------------------------------------------------------------------
(682.00,87.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(675.0,88.0)
------------------------------------------------------------------------
(710,85.67)
------------------------------------------------------------------------
(710.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(689.0,87.0)
------------------------------------------------------------------------
(738,84.67)
------------------------------------------------------------------------
(738.00,85.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(717.0,86.0)
------------------------------------------------------------------------
(773,83.67)
------------------------------------------------------------------------
(773.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(745.0,85.0)
------------------------------------------------------------------------
(780.0,84.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(822.0,82.0)
------------------------------------------------------------------------
(127.0,776.0)
------------------------------------------------------------------------
(900,900)(0,0) (127.0,82.0)
------------------------------------------------------------------------
(107,82)[(0,0)\[r\][$0$]{}]{} (802.0,82.0)
------------------------------------------------------------------------
(127.0,151.0)
------------------------------------------------------------------------
(107,151)[(0,0)\[r\][$0.1$]{}]{} (802.0,151.0)
------------------------------------------------------------------------
(127.0,221.0)
------------------------------------------------------------------------
(107,221)[(0,0)\[r\][$0.2$]{}]{} (802.0,221.0)
------------------------------------------------------------------------
(127.0,290.0)
------------------------------------------------------------------------
(107,290)[(0,0)\[r\][$0.3$]{}]{} (802.0,290.0)
------------------------------------------------------------------------
(127.0,360.0)
------------------------------------------------------------------------
(107,360)[(0,0)\[r\][$0.4$]{}]{} (802.0,360.0)
------------------------------------------------------------------------
(127.0,429.0)
------------------------------------------------------------------------
(107,429)[(0,0)\[r\][$0.5$]{}]{} (802.0,429.0)
------------------------------------------------------------------------
(127.0,498.0)
------------------------------------------------------------------------
(107,498)[(0,0)\[r\][$0.6$]{}]{} (802.0,498.0)
------------------------------------------------------------------------
(127.0,568.0)
------------------------------------------------------------------------
(107,568)[(0,0)\[r\][$0.7$]{}]{} (802.0,568.0)
------------------------------------------------------------------------
(127.0,637.0)
------------------------------------------------------------------------
(107,637)[(0,0)\[r\][$0.8$]{}]{} (802.0,637.0)
------------------------------------------------------------------------
(127.0,707.0)
------------------------------------------------------------------------
(107,707)[(0,0)\[r\][$0.9$]{}]{} (802.0,707.0)
------------------------------------------------------------------------
(127.0,776.0)
------------------------------------------------------------------------
(107,776)[(0,0)\[r\][$1$]{}]{} (802.0,776.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(127,41)[(0,0)[$0$]{}]{} (127.0,756.0)
------------------------------------------------------------------------
(243.0,82.0)
------------------------------------------------------------------------
(243,41)[(0,0)[$0.5$]{}]{} (243.0,756.0)
------------------------------------------------------------------------
(359.0,82.0)
------------------------------------------------------------------------
(359,41)[(0,0)[$1$]{}]{} (359.0,756.0)
------------------------------------------------------------------------
(475.0,82.0)
------------------------------------------------------------------------
(475,41)[(0,0)[$1.5$]{}]{} (475.0,756.0)
------------------------------------------------------------------------
(590.0,82.0)
------------------------------------------------------------------------
(590,41)[(0,0)[$2$]{}]{} (590.0,756.0)
------------------------------------------------------------------------
(706.0,82.0)
------------------------------------------------------------------------
(706,41)[(0,0)[$2.5$]{}]{} (706.0,756.0)
------------------------------------------------------------------------
(822.0,82.0)
------------------------------------------------------------------------
(822,41)[(0,0)[$3$]{}]{} (822.0,756.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(822.0,82.0)
------------------------------------------------------------------------
(127.0,776.0)
------------------------------------------------------------------------
(590,498)[(0,0)\[l\][$\lambda=0.5$]{}]{} (428,568)[(0,0)\[l\][$\lambda=1$]{}]{} (197,637)[(0,0)\[l\][$\lambda=2$]{}]{} (474,838)[(0,0)[Exponential PDF (Cumulative)]{}]{} (127,82) (127.59,82.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(126.17,82.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(134.59,92.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(133.17,92.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(141.59,103.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(140.17,103.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(148.59,113.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(147.17,113.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(155.59,123.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(154.17,123.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(162.59,133.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(161.17,133.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(169.59,142.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(168.17,142.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(176.59,152.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(175.17,152.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(183.59,161.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(182.17,161.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(190.59,170.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(189.17,170.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(197.59,180.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(196.17,180.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(204.59,189.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(203.17,189.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(211.59,197.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(210.17,197.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(218.59,206.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(217.17,206.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(225.59,215.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(224.17,215.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(232.59,223.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(231.17,223.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(239.59,231.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(238.17,231.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(246.59,240.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(245.17,240.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(253.59,248.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(252.17,248.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(260.00,256.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(260.00,255.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(267.59,263.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(266.17,263.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(274.59,271.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(273.17,271.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(281.00,279.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(281.00,278.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(288.59,286.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(287.17,286.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(295.00,294.59)(0.569,0.485)[11]{}
------------------------------------------------------------------------
(295.00,293.17)(6.844,7.000)[2]{}
------------------------------------------------------------------------
(303.00,301.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(303.00,300.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(310.00,308.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(310.00,307.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(317.00,315.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(317.00,314.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(324.00,322.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(324.00,321.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(331.00,329.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(331.00,328.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(338.00,335.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(338.00,334.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(345.00,342.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(345.00,341.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(352.00,349.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(352.00,348.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(359.00,355.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(359.00,354.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(366.00,361.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(366.00,360.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(373.00,368.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(373.00,367.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(380.00,374.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(380.00,373.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(387.00,380.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(387.00,379.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(394.00,386.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(394.00,385.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(401.00,392.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(401.00,391.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(408.00,397.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(408.00,396.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(415.00,403.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(415.00,402.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(422.00,409.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(422.00,408.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(429.00,414.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(429.00,413.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(436.00,420.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(436.00,419.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(443.00,425.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(443.00,424.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(450.00,430.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(450.00,429.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(457.00,436.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(457.00,435.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(464.00,441.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(464.00,440.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(471.00,446.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(471.00,445.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(478.00,451.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(478.00,450.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(485.00,456.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(485.00,455.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(492.00,460.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(492.00,459.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(499.00,465.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(499.00,464.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(506.00,470.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(506.00,469.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(513.00,474.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(513.00,473.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(520.00,479.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(520.00,478.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(527.00,483.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(527.00,482.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(534.00,488.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(534.00,487.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(541.00,492.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(541.00,491.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(548.00,496.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(548.00,495.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(555.00,501.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(555.00,500.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(562.00,505.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(562.00,504.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(569.00,509.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(569.00,508.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(576.00,513.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(576.00,512.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(583.00,517.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(583.00,516.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(590.00,521.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(590.00,520.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(597.00,525.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(597.00,524.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(604.00,528.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(604.00,527.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(611.00,532.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(611.00,531.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(618.00,536.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(618.00,535.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(625.00,539.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(625.00,538.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(632.00,543.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(632.00,542.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(639.00,546.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(639.00,545.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(646.00,550.61)(1.579,0.447)[3]{}
------------------------------------------------------------------------
(646.00,549.17)(5.579,3.000)[2]{}
------------------------------------------------------------------------
(654.00,553.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(654.00,552.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(661.00,557.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(661.00,556.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(668.00,560.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(668.00,559.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(675.00,563.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(675.00,562.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(682.00,566.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(682.00,565.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(689.00,569.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(689.00,568.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(696.00,573.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(696.00,572.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(703.00,576.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(703.00,575.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(710.00,579.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(710.00,578.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(717.00,582.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(717.00,581.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(724,585.17)
------------------------------------------------------------------------
(724.00,584.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(731.00,587.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(731.00,586.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(738.00,590.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(738.00,589.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(745.00,593.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(745.00,592.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(752.00,596.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(752.00,595.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(759,599.17)
------------------------------------------------------------------------
(759.00,598.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(766.00,601.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(766.00,600.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(773,604.17)
------------------------------------------------------------------------
(773.00,603.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(780.00,606.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(780.00,605.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(787,609.17)
------------------------------------------------------------------------
(787.00,608.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(794.00,611.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(794.00,610.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(801,614.17)
------------------------------------------------------------------------
(801.00,613.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(808.00,616.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(808.00,615.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(815,619.17)
------------------------------------------------------------------------
(815.00,618.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(127,82) (127.59,82.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(126.17,82.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(134.59,103.00)(0.485,1.484)[11]{}
------------------------------------------------------------------------
(133.17,103.00)(7.000,17.420)[2]{}
------------------------------------------------------------------------
(141.59,123.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(140.17,123.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(148.59,142.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(147.17,142.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(155.59,161.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(154.17,161.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(162.59,180.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(161.17,180.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(169.59,197.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(168.17,197.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(176.59,215.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(175.17,215.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(183.59,231.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(182.17,231.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(190.59,248.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(189.17,248.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(197.59,263.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(196.17,263.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(204.59,279.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(203.17,279.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(211.59,294.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(210.17,294.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(218.59,308.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(217.17,308.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(225.59,322.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(224.17,322.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(232.59,335.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(231.17,335.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(239.59,349.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(238.17,349.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(246.59,361.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(245.17,361.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(253.59,374.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(252.17,374.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(260.59,386.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(259.17,386.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(267.59,397.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(266.17,397.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(274.59,409.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(273.17,409.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(281.59,420.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(280.17,420.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(288.59,430.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(287.17,430.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(295.59,441.00)(0.488,0.626)[13]{}
------------------------------------------------------------------------
(294.17,441.00)(8.000,8.755)[2]{}
------------------------------------------------------------------------
(303.59,451.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(302.17,451.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(310.59,460.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(309.17,460.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(317.59,470.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(316.17,470.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(324.59,479.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(323.17,479.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(331.59,488.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(330.17,488.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(338.59,496.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(337.17,496.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(345.59,505.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(344.17,505.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(352.59,513.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(351.17,513.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(359.00,521.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(359.00,520.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(366.59,528.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(365.17,528.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(373.00,536.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(373.00,535.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(380.00,543.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(380.00,542.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(387.00,550.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(387.00,549.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(394.00,557.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(394.00,556.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(401.00,563.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(401.00,562.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(408.00,569.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(408.00,568.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(415.00,576.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(415.00,575.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(422.00,582.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(422.00,581.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(429.00,587.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(429.00,586.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(436.00,593.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(436.00,592.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(443.00,599.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(443.00,598.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(450.00,604.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(450.00,603.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(457.00,609.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(457.00,608.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(464.00,614.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(464.00,613.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(471.00,619.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(471.00,618.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(478.00,623.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(478.00,622.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(485.00,628.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(485.00,627.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(492.00,632.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(492.00,631.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(499.00,637.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(499.00,636.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(506.00,641.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(506.00,640.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(513.00,645.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(513.00,644.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(520.00,649.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(520.00,648.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(527.00,653.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(527.00,652.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(534.00,656.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(534.00,655.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(541.00,660.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(541.00,659.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(548.00,663.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(548.00,662.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(555.00,667.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(555.00,666.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(562.00,670.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(562.00,669.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(569.00,673.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(569.00,672.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(576.00,676.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(576.00,675.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(583.00,679.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(583.00,678.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(590.00,682.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(590.00,681.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(597.00,685.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(597.00,684.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(604,688.17)
------------------------------------------------------------------------
(604.00,687.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(611.00,690.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(611.00,689.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(618,693.17)
------------------------------------------------------------------------
(618.00,692.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(625.00,695.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(625.00,694.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(632,698.17)
------------------------------------------------------------------------
(632.00,697.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(639,700.17)
------------------------------------------------------------------------
(639.00,699.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(646,702.17)
------------------------------------------------------------------------
(646.00,701.17)(4.472,2.000)[2]{}
------------------------------------------------------------------------
(654.00,704.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(654.00,703.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(661,707.17)
------------------------------------------------------------------------
(661.00,706.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(668,709.17)
------------------------------------------------------------------------
(668.00,708.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(675,711.17)
------------------------------------------------------------------------
(675.00,710.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(682,713.17)
------------------------------------------------------------------------
(682.00,712.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(689,714.67)
------------------------------------------------------------------------
(689.00,714.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(696,716.17)
------------------------------------------------------------------------
(696.00,715.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(703,718.17)
------------------------------------------------------------------------
(703.00,717.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(710,720.17)
------------------------------------------------------------------------
(710.00,719.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(717,721.67)
------------------------------------------------------------------------
(717.00,721.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(724,723.17)
------------------------------------------------------------------------
(724.00,722.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(731,724.67)
------------------------------------------------------------------------
(731.00,724.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(738,726.17)
------------------------------------------------------------------------
(738.00,725.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(745,727.67)
------------------------------------------------------------------------
(745.00,727.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(752,729.17)
------------------------------------------------------------------------
(752.00,728.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(759,730.67)
------------------------------------------------------------------------
(759.00,730.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(766,731.67)
------------------------------------------------------------------------
(766.00,731.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(773,733.17)
------------------------------------------------------------------------
(773.00,732.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(780,734.67)
------------------------------------------------------------------------
(780.00,734.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(787,735.67)
------------------------------------------------------------------------
(787.00,735.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(794,736.67)
------------------------------------------------------------------------
(794.00,736.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(801,737.67)
------------------------------------------------------------------------
(801.00,737.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(808,738.67)
------------------------------------------------------------------------
(808.00,738.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(815,739.67)
------------------------------------------------------------------------
(815.00,739.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(127,82) (127.59,82.00)(0.485,3.086)[11]{}
------------------------------------------------------------------------
(126.17,82.00)(7.000,35.930)[2]{}
------------------------------------------------------------------------
(134.59,123.00)(0.485,2.857)[11]{}
------------------------------------------------------------------------
(133.17,123.00)(7.000,33.286)[2]{}
------------------------------------------------------------------------
(141.59,161.00)(0.485,2.705)[11]{}
------------------------------------------------------------------------
(140.17,161.00)(7.000,31.523)[2]{}
------------------------------------------------------------------------
(148.59,197.00)(0.485,2.552)[11]{}
------------------------------------------------------------------------
(147.17,197.00)(7.000,29.760)[2]{}
------------------------------------------------------------------------
(155.59,231.00)(0.485,2.399)[11]{}
------------------------------------------------------------------------
(154.17,231.00)(7.000,27.997)[2]{}
------------------------------------------------------------------------
(162.59,263.00)(0.485,2.323)[11]{}
------------------------------------------------------------------------
(161.17,263.00)(7.000,27.116)[2]{}
------------------------------------------------------------------------
(169.59,294.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(168.17,294.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(176.59,322.00)(0.485,2.018)[11]{}
------------------------------------------------------------------------
(175.17,322.00)(7.000,23.590)[2]{}
------------------------------------------------------------------------
(183.59,349.00)(0.485,1.865)[11]{}
------------------------------------------------------------------------
(182.17,349.00)(7.000,21.827)[2]{}
------------------------------------------------------------------------
(190.59,374.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(189.17,374.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(197.59,397.00)(0.485,1.713)[11]{}
------------------------------------------------------------------------
(196.17,397.00)(7.000,20.065)[2]{}
------------------------------------------------------------------------
(204.59,420.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(203.17,420.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(211.59,441.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(210.17,441.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(218.59,460.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(217.17,460.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(225.59,479.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(224.17,479.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(232.59,496.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(231.17,496.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(239.59,513.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(238.17,513.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(246.59,528.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(245.17,528.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(253.59,543.00)(0.485,1.026)[11]{}
------------------------------------------------------------------------
(252.17,543.00)(7.000,12.132)[2]{}
------------------------------------------------------------------------
(260.59,557.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(259.17,557.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(267.59,569.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(266.17,569.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(274.59,582.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(273.17,582.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(281.59,593.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(280.17,593.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(288.59,604.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(287.17,604.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(295.59,614.00)(0.488,0.560)[13]{}
------------------------------------------------------------------------
(294.17,614.00)(8.000,7.858)[2]{}
------------------------------------------------------------------------
(303.59,623.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(302.17,623.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(310.59,632.00)(0.485,0.645)[11]{}
------------------------------------------------------------------------
(309.17,632.00)(7.000,7.725)[2]{}
------------------------------------------------------------------------
(317.59,641.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(316.17,641.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(324.00,649.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(324.00,648.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(331.00,656.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(331.00,655.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(338.00,663.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(338.00,662.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(345.00,670.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(345.00,669.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(352.00,676.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(352.00,675.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(359.00,682.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(359.00,681.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(366.00,688.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(366.00,687.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(373.00,693.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(373.00,692.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(380.00,698.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(380.00,697.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(387.00,702.59)(0.710,0.477)[7]{}
------------------------------------------------------------------------
(387.00,701.17)(5.630,5.000)[2]{}
------------------------------------------------------------------------
(394.00,707.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(394.00,706.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(401.00,711.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(401.00,710.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(408.00,715.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(408.00,714.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(415.00,718.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(415.00,717.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(422.00,722.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(422.00,721.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(429.00,725.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(429.00,724.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(436.00,728.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(436.00,727.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(443,731.17)
------------------------------------------------------------------------
(443.00,730.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(450.00,733.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(450.00,732.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(457,736.17)
------------------------------------------------------------------------
(457.00,735.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(464,738.17)
------------------------------------------------------------------------
(464.00,737.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(471,740.17)
------------------------------------------------------------------------
(471.00,739.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(478,742.17)
------------------------------------------------------------------------
(478.00,741.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(485,744.17)
------------------------------------------------------------------------
(485.00,743.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(492,746.17)
------------------------------------------------------------------------
(492.00,745.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(499,748.17)
------------------------------------------------------------------------
(499.00,747.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(506,749.67)
------------------------------------------------------------------------
(506.00,749.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(513,751.17)
------------------------------------------------------------------------
(513.00,750.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(520,752.67)
------------------------------------------------------------------------
(520.00,752.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(527,753.67)
------------------------------------------------------------------------
(527.00,753.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(534,755.17)
------------------------------------------------------------------------
(534.00,754.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(541,756.67)
------------------------------------------------------------------------
(541.00,756.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(548,757.67)
------------------------------------------------------------------------
(548.00,757.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(555,758.67)
------------------------------------------------------------------------
(555.00,758.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(562,759.67)
------------------------------------------------------------------------
(562.00,759.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(569,760.67)
------------------------------------------------------------------------
(569.00,760.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(583,761.67)
------------------------------------------------------------------------
(583.00,761.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(590,762.67)
------------------------------------------------------------------------
(590.00,762.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(597,763.67)
------------------------------------------------------------------------
(597.00,763.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(576.0,762.0)
------------------------------------------------------------------------
(611,764.67)
------------------------------------------------------------------------
(611.00,764.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(618,765.67)
------------------------------------------------------------------------
(618.00,765.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(604.0,765.0)
------------------------------------------------------------------------
(632,766.67)
------------------------------------------------------------------------
(632.00,766.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(625.0,767.0)
------------------------------------------------------------------------
(646,767.67)
------------------------------------------------------------------------
(646.00,767.17)(4.000,1.000)[2]{}
------------------------------------------------------------------------
(639.0,768.0)
------------------------------------------------------------------------
(668,768.67)
------------------------------------------------------------------------
(668.00,768.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(654.0,769.0)
------------------------------------------------------------------------
(682,769.67)
------------------------------------------------------------------------
(682.00,769.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(675.0,770.0)
------------------------------------------------------------------------
(710,770.67)
------------------------------------------------------------------------
(710.00,770.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(689.0,771.0)
------------------------------------------------------------------------
(738,771.67)
------------------------------------------------------------------------
(738.00,771.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(717.0,772.0)
------------------------------------------------------------------------
(773,772.67)
------------------------------------------------------------------------
(773.00,772.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(745.0,773.0)
------------------------------------------------------------------------
(780.0,774.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(127.0,82.0)
------------------------------------------------------------------------
(822.0,82.0)
------------------------------------------------------------------------
(127.0,776.0)
------------------------------------------------------------------------
\(a) & & (b)
$$\tilde{P}_X(\omega) = \frac{\lambda^2}{\lambda^2+\omega^2}$$
and its moments $$\langle X^n \rangle = \frac{1}{\lambda^n} \Gamma(n+1)$$
\* \* \*
A **uniform** or *flat* distribution assigns the same probability density to each point in $\Omega$. So, if $\Omega=[a,b]$, $$P_X(x) =
\begin{cases}
\frac{1}{b-a} & a \le x \le b \\
0 & \mbox{otherwise}
\end{cases}$$ The characteristic function of the uniform distribution is $$\tilde{P}_X(\omega) = \frac{e^{i\omega{b}}-e^{i\omega{a}}}{i\omega(b-a)}$$ and its moments $$\langle X^n \rangle = \frac{1}{n+1}\frac{b^{n+1}-a^{n+1}}{b-a}$$
\* \* \*
A **Cauchy**, or **Lorentz** distribution has PDF $$P_X(x) = \frac{1}{\pi} \frac{\gamma}{x^2 + \gamma^2}$$ where $\gamma$ is a positive parameter, and characteristic function $$\tilde{P}_X(\omega) = e^{-\gamma|\omega|}$$ If one tries to compute the moments using the definition $$\langle X^n \rangle = \frac{\gamma}{\pi} \int \frac{x^n}{x^2 + \gamma^2}\,dx$$ then, since the integrand behaves as $x^{n-2}$ for $x\rightarrow\infty$, one observes that they diverge for $n\ge{1}$. This limits the usefulness of this distribution as a model of real phenomena (which typically have finite moments), and in practice one [“[truncates]{}”]{} the distribution to a finite interval $[a,b]$.
\* \* \*
We have mentioned that one important property of the Gaussian distribution is the preservation of the functional form of their characteristic function under multiplication, as in (\[ggood\]). The Gaussian distribution is not the most general distribution with this property (although it is the only one with this property *and* finite moments): it is shared by the family of **Lévy distributions**. Lévy distributions depend on four parameters: $\alpha$ (Lévy index), $\beta$ (skew), $\mu$ (shift), and $\sigma$ (scale), and they are defined through their characteristic function: $$\tilde{P}_{\alpha,\beta}(\omega;\mu,\sigma) = \int_{-\infty}^\infty e^{i\omega{x}} P_{\alpha,\beta}(x;\mu,\sigma)\,dx
\stackrel{\triangle}{=} \exp \left[ i\mu\omega - \sigma^\alpha |\omega|^\alpha \left( 1 - i\beta\frac{\omega}{|\omega|}\Phi \right)\right]$$ where $$\Phi =
\begin{cases}
\tan \frac{\alpha\pi}{2} & \alpha \ne 1, 0<\alpha<2 \\
- \frac{2}{\pi} \ln |x| & \alpha = 1
\end{cases}$$ the four parameters determine the shape of the distribution. Of these, $\alpha$ and $\beta$ play a major rôle in this note, while $\mu$ and $\sigma$ can be eliminated through proper scale and shift transformations (much like mean and variance for the Gaussian distribution): $$P_{\alpha,\beta}(x;\mu,\sigma) = \frac{1}{\sigma} P_{\alpha,\beta}(\frac{x-\mu}{\sigma}; 0, 1)$$ From now on, I shall therefore ignore $\mu$ and $\sigma$ and refer to the distribution as $P_{\alpha,\beta}(x)$. Note the symmetry relation $$P_{\alpha,-\beta}(x) = P_{\alpha,\beta}(x)$$ The distributions with $\beta=0$ are symmetric, and these are the ones that are the most relevant in this context. The closed form of $P_{\alpha,\beta}$ is known only for a few cases. If $\alpha=2$ one obtains the Gaussian distribution ($\beta$ is irrelevant, since $\Phi=0$); if $\alpha=1,\beta=0$ one obtains the Cauchy distribution, and for $\alpha=1/2,\beta=1$, the Lévy-Smirnov distribution $$P_{1/2,1}(x) =
\begin{cases}
\frac{1}{\sqrt{2\pi}} x^{-\frac{3}{2}}\exp\bigl( -\frac{1}{2x} \bigr) & x \ge 0 \\
0 & x<0
\end{cases}$$ The most important property in this context is the asymptotic behavior of $P_{\alpha,\beta}$ which is given by the power law $$\label{levypow}
P_{\alpha,0}(x) \sim \frac{C(\alpha)}{|x|^{1+\alpha}}$$ with $$C(\alpha) = \frac{1}{\pi} \sin \bigl( \frac{\pi\alpha}{2} \bigr) \Gamma(1+\alpha)$$ This power law behavior entails that arbitrarily large values are relatively probable (compared with the exponential decay of the Gaussian). Consequently, as can be expected, $\langle{X}^2\rangle$ diverges for $\alpha<2$.
\* \* \*
The **Dirac delta distribution** is a pathological distribution useful in many contexts; for example, when dealing with certainty in a probabilistic framework, or when analyzing discrete random variables in a context created for continuous ones. The distribution is: $$P_X(x) = \delta(x-x_0)$$ where $\delta(\cdot)$ is the Dirac distribution. The characteristic function of the distribution is $$\label{deltachar}
\tilde{P}_X(\omega)=\exp(i\omega{x_0}).$$ The function $\delta(x)$ is zero everywhere except for $x=0$, and $$\int_{-\infty}^\infty \delta(x)\,dx = 1$$ This property entails $\delta(ax)=\delta(x)/a$. Also $$\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx = f(x_0)$$ from which we derive $$\langle x^n \rangle = x_0^n$$
\* \* \*
Unlike the previous distribution, the **binomial distribution** is defined for discrete variables, in particular for a variable $X$ that can take two values, the first one with probability $p$, and the second one with probability $1-p$. Suppose, for example, that we play a game in which, at each turn, I have a probability $p$ of winning and $1-p$ of losing (think of head-and-tails game with a tricked coin). If we play $N$ rounds of the game, what is the probability that I win exactly $n$ times? This turns out to be $$P(X=n) = \left(\begin{array}{c}N\\n\end{array}\right) p^n (1-p)^{N-n}=
\frac{N!}{n!(N-n)!}p^n (1-p)^{N-n}$$ which is precisely the binomial distribution. Its characteristic function is $$\tilde{P}(\omega) = (1-p+pe^{i\omega})^N$$ from which the moments can be derived. For example $$\label{binmean}
\langle X \rangle = \lim_{\omega\rightarrow{0}} \frac{d\tilde{P}}{d\omega} =
\lim_{\omega\rightarrow{0}} pN e^{i\omega} (1-p+pe^{i\omega})^{N-1} = pN$$
\* \* \*
An important and common distribution, one that appears as a limiting case of many finite processes, is the **Poisson Distribution**. Its importance will probably be more evident if we derive it as a limiting case in some examples.
Consider events that may happen at any moment in time (the events are punctual: they have no duration). Divide the time-line in small intervals of duration $\Delta{t}$, so short that the probability that two or more events will take place in the same interval is negligible. Assume that the probability that *one* event take place in $[t,t+\Delta{t})$ is constant, and proportional to the length of the interval: $$P(1;\Delta{t}) = \lambda\Delta{t}$$ and, because no two events happen in the same interval, $$P(0;\Delta{t}) = 1-\lambda\Delta{t}$$ Let $P(0;t)$ be the probability that no event has taken place up to time $t$. Then $$P(0;t+\Delta{t}) = P(0;t)(1-\lambda\Delta{t})$$ Rearranging the terms we get $$\frac{P(0;t+\Delta{t})-P(0;t)}{\Delta{t}}=-\lambda P(0;t)$$ and, taking the limit for $\Delta{t}\rightarrow{0}$ $$\frac{\partial}{\partial{t}} P(0;t) = -\lambda P(0;t)$$ that is, $P(0;t)=C\exp(-\lambda{t})$ or, considering the boundary condition $P(0,0)=1$, $$P(0;t) = e^{-\lambda{t}}$$ This takes care of the case in which no event takes place before time $t$. On to the general case. There were $n$ events by time $t+\Delta{t}$ if either (1) we had $n$ events up to time $t$ and no event occurred in $[t,t+\Delta{t}]$, or (2) there were $n-1$ events at $t$ and one event occurred in $[t,t+\Delta{t}]$. This leads to $$P(n;t+\Delta{t}) = (1-\lambda\Delta{t})P(n;t) + \lambda\Delta{t}P(n-1;t)$$ rearranging and taking the limit $\Delta{t}\rightarrow{0}$, we have $$\label{poissonpartial}
\frac{\partial}{\partial t} P(n;t) + \lambda P(n;t) = \lambda P(n-1;t)$$ In order to transform this equation into a more manageable form, we look for a function that, multiplied by the left-hand side, transforms it into the derivative of a product. That is, we look for a function $\mu(t)$ such that $$\mu(t) \left[ \frac{\partial P}{\partial t} + \lambda P \right] = \frac{\partial}{\partial t} \bigl[ \mu(t) P \bigr]$$ It is easy to verify that $\mu(t)=\exp(\lambda{t})$ fits the bill. Equation (\[poissonpartial\]) therefore becomes $$\frac{\partial}{\partial t} \Bigl[ e^{\lambda{t}}P(n;t) \Bigr] = e^{\lambda{t}}\lambda P(n-1;t)$$ For $n=1$ we have $$\frac{\partial}{\partial t} \Bigl[ e^{\lambda{t}}P(1;t) \Bigr] = e^{\lambda{t}}\lambda e^{-\lambda{t}} = \lambda$$ That is, integrating both sides and multiplying by $e^{-\lambda{t}}$ $$P(1;t) = \lambda t e^{-\lambda{t}}$$ For arbitrary $n$, I’ll show by induction that $$\label{fish}
P(n;t) = \frac{(\lambda{t})^n}{n!}e^{-\lambda{t}}$$ We have already derived the result for $n=0$ and for $n=1$. For arbitrary $n$, we have $$\begin{array}{lcll}
\displaystyle \frac{\partial}{\partial t} \Bigl[ e^{\lambda{t}}P(n+1;t) \Bigl] & = & e^{\lambda{t}} \lambda P(n;t) & \\
\displaystyle & = & e^{\lambda{t}} \lambda \frac{(\lambda{t})^n}{n!} e^{-\lambda{t}} & \mbox{(induction hypothesis)} \\
\displaystyle & = & \lambda \frac{(\lambda{t})^n}{n!}
\end{array}$$ So, integrating $$e^{\lambda{t}}P(n+1;t) = \frac{\lambda}{n!} \int (\lambda{t}^n) dt = \frac{(\lambda{t})^{n+1}}{(n+1)!} + C$$ where $C=0$ because of the initial conditions, so $$P(n+1;t) = e^{-\lambda{t}}\frac{(\lambda{t})^{n+1}}{(n+1)!}$$
The distribution that results from this example: $$\label{poisson}
P_X(x) = e^{-x} \frac{x^n}{n!}$$ is the Poisson distribution that, in the example, gives us the probability that $n$ events take place in a time $x$. Figure \[poissonfig\] shows the shape of this distribution as a function of $x$ for various values of $n$.
(900,900)(0,0) (137.0,82.0)
------------------------------------------------------------------------
(117,82)[(0,0)\[r\][$0$]{}]{} (812.0,82.0)
------------------------------------------------------------------------
(137.0,198.0)
------------------------------------------------------------------------
(117,198)[(0,0)\[r\][$0.05$]{}]{} (812.0,198.0)
------------------------------------------------------------------------
(137.0,313.0)
------------------------------------------------------------------------
(117,313)[(0,0)\[r\][$0.1$]{}]{} (812.0,313.0)
------------------------------------------------------------------------
(137.0,429.0)
------------------------------------------------------------------------
(117,429)[(0,0)\[r\][$0.15$]{}]{} (812.0,429.0)
------------------------------------------------------------------------
(137.0,545.0)
------------------------------------------------------------------------
(117,545)[(0,0)\[r\][$0.2$]{}]{} (812.0,545.0)
------------------------------------------------------------------------
(137.0,660.0)
------------------------------------------------------------------------
(117,660)[(0,0)\[r\][$0.25$]{}]{} (812.0,660.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(117,776)[(0,0)\[r\][$0.3$]{}]{} (812.0,776.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137,41)[(0,0)[$0$]{}]{} (137.0,756.0)
------------------------------------------------------------------------
(253.0,82.0)
------------------------------------------------------------------------
(253,41)[(0,0)[$5$]{}]{} (253.0,756.0)
------------------------------------------------------------------------
(369.0,82.0)
------------------------------------------------------------------------
(369,41)[(0,0)[$10$]{}]{} (369.0,756.0)
------------------------------------------------------------------------
(485.0,82.0)
------------------------------------------------------------------------
(485,41)[(0,0)[$15$]{}]{} (485.0,756.0)
------------------------------------------------------------------------
(600.0,82.0)
------------------------------------------------------------------------
(600,41)[(0,0)[$20$]{}]{} (600.0,756.0)
------------------------------------------------------------------------
(716.0,82.0)
------------------------------------------------------------------------
(716,41)[(0,0)[$25$]{}]{} (716.0,756.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(832,41)[(0,0)[$30$]{}]{} (832.0,756.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
(207,660)[(0,0)\[l\][$n=1$]{}]{} (299,429)[(0,0)\[l\][$n=5$]{}]{} (438,313)[(0,0)\[l\][$n=10$]{}]{} (484,838)[(0,0)[Poisson PDF]{}]{} (137,82) (137.59,82.00)(0.485,5.908)[11]{}
------------------------------------------------------------------------
(136.17,82.00)(7.000,68.541)[2]{}
------------------------------------------------------------------------
(144.59,160.00)(0.485,11.706)[11]{}
------------------------------------------------------------------------
(143.17,160.00)(7.000,135.528)[2]{}
------------------------------------------------------------------------
(151.59,314.00)(0.485,11.629)[11]{}
------------------------------------------------------------------------
(150.17,314.00)(7.000,134.646)[2]{}
------------------------------------------------------------------------
(158.59,467.00)(0.485,9.188)[11]{}
------------------------------------------------------------------------
(157.17,467.00)(7.000,106.441)[2]{}
------------------------------------------------------------------------
(165.59,588.00)(0.485,5.908)[11]{}
------------------------------------------------------------------------
(164.17,588.00)(7.000,68.541)[2]{}
------------------------------------------------------------------------
(172.59,666.00)(0.485,2.781)[11]{}
------------------------------------------------------------------------
(171.17,666.00)(7.000,32.404)[2]{}
------------------------------------------------------------------------
(179.00,703.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(179.00,702.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(186.59,700.37)(0.485,-1.637)[11]{}
------------------------------------------------------------------------
(185.17,703.18)(7.000,-19.183)[2]{}
------------------------------------------------------------------------
(193.59,674.33)(0.485,-2.933)[11]{}
------------------------------------------------------------------------
(192.17,679.17)(7.000,-34.167)[2]{}
------------------------------------------------------------------------
(200.59,632.72)(0.485,-3.772)[11]{}
------------------------------------------------------------------------
(199.17,638.86)(7.000,-43.862)[2]{}
------------------------------------------------------------------------
(207.59,581.54)(0.485,-4.154)[11]{}
------------------------------------------------------------------------
(206.17,588.27)(7.000,-48.269)[2]{}
------------------------------------------------------------------------
(214.59,526.54)(0.485,-4.154)[11]{}
------------------------------------------------------------------------
(213.17,533.27)(7.000,-48.269)[2]{}
------------------------------------------------------------------------
(221.59,471.78)(0.485,-4.078)[11]{}
------------------------------------------------------------------------
(220.17,478.39)(7.000,-47.388)[2]{}
------------------------------------------------------------------------
(228.59,418.72)(0.485,-3.772)[11]{}
------------------------------------------------------------------------
(227.17,424.86)(7.000,-43.862)[2]{}
------------------------------------------------------------------------
(235.59,369.91)(0.485,-3.391)[11]{}
------------------------------------------------------------------------
(234.17,375.46)(7.000,-39.455)[2]{}
------------------------------------------------------------------------
(242.59,325.86)(0.485,-3.086)[11]{}
------------------------------------------------------------------------
(241.17,330.93)(7.000,-35.930)[2]{}
------------------------------------------------------------------------
(249.59,286.28)(0.485,-2.628)[11]{}
------------------------------------------------------------------------
(248.17,290.64)(7.000,-30.641)[2]{}
------------------------------------------------------------------------
(256.59,252.23)(0.485,-2.323)[11]{}
------------------------------------------------------------------------
(255.17,256.12)(7.000,-27.116)[2]{}
------------------------------------------------------------------------
(263.59,222.42)(0.485,-1.942)[11]{}
------------------------------------------------------------------------
(262.17,225.71)(7.000,-22.709)[2]{}
------------------------------------------------------------------------
(270.59,197.37)(0.485,-1.637)[11]{}
------------------------------------------------------------------------
(269.17,200.18)(7.000,-19.183)[2]{}
------------------------------------------------------------------------
(277.59,176.32)(0.485,-1.332)[11]{}
------------------------------------------------------------------------
(276.17,178.66)(7.000,-15.658)[2]{}
------------------------------------------------------------------------
(284.59,158.79)(0.485,-1.179)[11]{}
------------------------------------------------------------------------
(283.17,160.89)(7.000,-13.895)[2]{}
------------------------------------------------------------------------
(291.59,143.74)(0.485,-0.874)[11]{}
------------------------------------------------------------------------
(290.17,145.37)(7.000,-10.369)[2]{}
------------------------------------------------------------------------
(298.59,131.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(297.17,133.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(305.00,122.93)(0.494,-0.488)[13]{}
------------------------------------------------------------------------
(305.00,123.17)(6.962,-8.000)[2]{}
------------------------------------------------------------------------
(313.00,114.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(313.00,115.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(320.00,107.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(320.00,108.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(327.00,102.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(327.00,103.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(334.00,97.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(334.00,98.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(341.00,94.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(341.00,95.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(348.00,91.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(348.00,92.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(355,88.67)
------------------------------------------------------------------------
(355.00,89.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(362,87.17)
------------------------------------------------------------------------
(362.00,88.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(369,85.67)
------------------------------------------------------------------------
(369.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(376,84.67)
------------------------------------------------------------------------
(376.00,85.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(390,83.67)
------------------------------------------------------------------------
(390.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(383.0,85.0)
------------------------------------------------------------------------
(404,82.67)
------------------------------------------------------------------------
(404.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(397.0,84.0)
------------------------------------------------------------------------
(432,81.67)
------------------------------------------------------------------------
(432.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(411.0,83.0)
------------------------------------------------------------------------
(439.0,82.0)
------------------------------------------------------------------------
(137,82) (144,81.67)
------------------------------------------------------------------------
(144.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(151.00,83.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(151.00,82.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(158.59,87.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(157.17,87.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(165.59,97.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(164.17,97.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(172.59,116.00)(0.485,2.094)[11]{}
------------------------------------------------------------------------
(171.17,116.00)(7.000,24.472)[2]{}
------------------------------------------------------------------------
(179.59,144.00)(0.485,2.781)[11]{}
------------------------------------------------------------------------
(178.17,144.00)(7.000,32.404)[2]{}
------------------------------------------------------------------------
(186.59,181.00)(0.485,3.315)[11]{}
------------------------------------------------------------------------
(185.17,181.00)(7.000,38.574)[2]{}
------------------------------------------------------------------------
(193.59,225.00)(0.485,3.544)[11]{}
------------------------------------------------------------------------
(192.17,225.00)(7.000,41.218)[2]{}
------------------------------------------------------------------------
(200.59,272.00)(0.485,3.620)[11]{}
------------------------------------------------------------------------
(199.17,272.00)(7.000,42.100)[2]{}
------------------------------------------------------------------------
(207.59,320.00)(0.485,3.391)[11]{}
------------------------------------------------------------------------
(206.17,320.00)(7.000,39.455)[2]{}
------------------------------------------------------------------------
(214.59,365.00)(0.485,3.010)[11]{}
------------------------------------------------------------------------
(213.17,365.00)(7.000,35.048)[2]{}
------------------------------------------------------------------------
(221.59,405.00)(0.485,2.476)[11]{}
------------------------------------------------------------------------
(220.17,405.00)(7.000,28.879)[2]{}
------------------------------------------------------------------------
(228.59,438.00)(0.485,1.865)[11]{}
------------------------------------------------------------------------
(227.17,438.00)(7.000,21.827)[2]{}
------------------------------------------------------------------------
(235.59,463.00)(0.485,1.179)[11]{}
------------------------------------------------------------------------
(234.17,463.00)(7.000,13.895)[2]{}
------------------------------------------------------------------------
(242.59,479.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(241.17,479.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(256.00,485.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(256.00,486.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(263.59,476.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(262.17,478.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(270.59,462.55)(0.485,-1.255)[11]{}
------------------------------------------------------------------------
(269.17,464.78)(7.000,-14.776)[2]{}
------------------------------------------------------------------------
(277.59,444.60)(0.485,-1.560)[11]{}
------------------------------------------------------------------------
(276.17,447.30)(7.000,-18.302)[2]{}
------------------------------------------------------------------------
(284.59,422.89)(0.485,-1.789)[11]{}
------------------------------------------------------------------------
(283.17,425.95)(7.000,-20.946)[2]{}
------------------------------------------------------------------------
(291.59,398.65)(0.485,-1.865)[11]{}
------------------------------------------------------------------------
(290.17,401.83)(7.000,-21.827)[2]{}
------------------------------------------------------------------------
(298.59,373.42)(0.485,-1.942)[11]{}
------------------------------------------------------------------------
(297.17,376.71)(7.000,-22.709)[2]{}
------------------------------------------------------------------------
(305.59,348.40)(0.488,-1.616)[13]{}
------------------------------------------------------------------------
(304.17,351.20)(8.000,-22.198)[2]{}
------------------------------------------------------------------------
(313.59,322.65)(0.485,-1.865)[11]{}
------------------------------------------------------------------------
(312.17,325.83)(7.000,-21.827)[2]{}
------------------------------------------------------------------------
(320.59,297.89)(0.485,-1.789)[11]{}
------------------------------------------------------------------------
(319.17,300.95)(7.000,-20.946)[2]{}
------------------------------------------------------------------------
(327.59,274.13)(0.485,-1.713)[11]{}
------------------------------------------------------------------------
(326.17,277.06)(7.000,-20.065)[2]{}
------------------------------------------------------------------------
(334.59,251.60)(0.485,-1.560)[11]{}
------------------------------------------------------------------------
(333.17,254.30)(7.000,-18.302)[2]{}
------------------------------------------------------------------------
(341.59,231.08)(0.485,-1.408)[11]{}
------------------------------------------------------------------------
(340.17,233.54)(7.000,-16.539)[2]{}
------------------------------------------------------------------------
(348.59,212.32)(0.485,-1.332)[11]{}
------------------------------------------------------------------------
(347.17,214.66)(7.000,-15.658)[2]{}
------------------------------------------------------------------------
(355.59,195.03)(0.485,-1.103)[11]{}
------------------------------------------------------------------------
(354.17,197.01)(7.000,-13.013)[2]{}
------------------------------------------------------------------------
(362.59,180.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(361.17,182.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(369.59,166.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(368.17,168.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(376.59,153.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(375.17,155.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(383.59,143.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(382.17,144.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(390.59,133.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(389.17,134.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(397.00,126.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(397.00,127.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(404.00,119.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(404.00,120.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(411.00,113.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(411.00,114.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(418.00,107.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(418.00,108.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(425.00,103.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(425.00,104.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(432.00,99.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(432.00,100.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(439.00,96.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(439.00,97.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(446,93.17)
------------------------------------------------------------------------
(446.00,94.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(453,91.17)
------------------------------------------------------------------------
(453.00,92.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(460,89.17)
------------------------------------------------------------------------
(460.00,90.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(467,87.67)
------------------------------------------------------------------------
(467.00,88.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(474,86.67)
------------------------------------------------------------------------
(474.00,87.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(481,85.67)
------------------------------------------------------------------------
(481.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(488,84.67)
------------------------------------------------------------------------
(488.00,85.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(249.0,487.0)
------------------------------------------------------------------------
(502,83.67)
------------------------------------------------------------------------
(502.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(495.0,85.0)
------------------------------------------------------------------------
(516,82.67)
------------------------------------------------------------------------
(516.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(509.0,84.0)
------------------------------------------------------------------------
(551,81.67)
------------------------------------------------------------------------
(551.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(523.0,83.0)
------------------------------------------------------------------------
(558.0,82.0)
------------------------------------------------------------------------
(137,82) (193,81.67)
------------------------------------------------------------------------
(193.00,81.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(200,82.67)
------------------------------------------------------------------------
(200.00,82.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(207,84.17)
------------------------------------------------------------------------
(207.00,83.17)(3.887,2.000)[2]{}
------------------------------------------------------------------------
(214.00,86.61)(1.355,0.447)[3]{}
------------------------------------------------------------------------
(214.00,85.17)(4.855,3.000)[2]{}
------------------------------------------------------------------------
(221.00,89.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(221.00,88.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(228.00,93.59)(0.581,0.482)[9]{}
------------------------------------------------------------------------
(228.00,92.17)(5.824,6.000)[2]{}
------------------------------------------------------------------------
(235.59,99.00)(0.485,0.569)[11]{}
------------------------------------------------------------------------
(234.17,99.00)(7.000,6.844)[2]{}
------------------------------------------------------------------------
(242.59,107.00)(0.485,0.798)[11]{}
------------------------------------------------------------------------
(241.17,107.00)(7.000,9.488)[2]{}
------------------------------------------------------------------------
(249.59,118.00)(0.485,0.950)[11]{}
------------------------------------------------------------------------
(248.17,118.00)(7.000,11.251)[2]{}
------------------------------------------------------------------------
(256.59,131.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(255.17,131.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(263.59,146.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(262.17,146.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(270.59,163.00)(0.485,1.332)[11]{}
------------------------------------------------------------------------
(269.17,163.00)(7.000,15.658)[2]{}
------------------------------------------------------------------------
(277.59,181.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(276.17,181.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(284.59,202.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(283.17,202.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(291.59,223.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(290.17,223.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(298.59,244.00)(0.485,1.560)[11]{}
------------------------------------------------------------------------
(297.17,244.00)(7.000,18.302)[2]{}
------------------------------------------------------------------------
(305.59,265.00)(0.488,1.352)[13]{}
------------------------------------------------------------------------
(304.17,265.00)(8.000,18.613)[2]{}
------------------------------------------------------------------------
(313.59,286.00)(0.485,1.408)[11]{}
------------------------------------------------------------------------
(312.17,286.00)(7.000,16.539)[2]{}
------------------------------------------------------------------------
(320.59,305.00)(0.485,1.255)[11]{}
------------------------------------------------------------------------
(319.17,305.00)(7.000,14.776)[2]{}
------------------------------------------------------------------------
(327.59,322.00)(0.485,1.103)[11]{}
------------------------------------------------------------------------
(326.17,322.00)(7.000,13.013)[2]{}
------------------------------------------------------------------------
(334.59,337.00)(0.485,0.874)[11]{}
------------------------------------------------------------------------
(333.17,337.00)(7.000,10.369)[2]{}
------------------------------------------------------------------------
(341.59,349.00)(0.485,0.721)[11]{}
------------------------------------------------------------------------
(340.17,349.00)(7.000,8.606)[2]{}
------------------------------------------------------------------------
(348.00,359.59)(0.492,0.485)[11]{}
------------------------------------------------------------------------
(348.00,358.17)(5.962,7.000)[2]{}
------------------------------------------------------------------------
(355.00,366.60)(0.920,0.468)[5]{}
------------------------------------------------------------------------
(355.00,365.17)(5.340,4.000)[2]{}
------------------------------------------------------------------------
(362,369.67)
------------------------------------------------------------------------
(362.00,369.17)(3.500,1.000)[2]{}
------------------------------------------------------------------------
(369,369.67)
------------------------------------------------------------------------
(369.00,370.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(376.00,368.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(376.00,369.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(383.00,364.93)(0.581,-0.482)[9]{}
------------------------------------------------------------------------
(383.00,365.17)(5.824,-6.000)[2]{}
------------------------------------------------------------------------
(390.59,357.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(389.17,358.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(397.59,349.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(396.17,350.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(404.59,339.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(403.17,341.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(411.59,328.74)(0.485,-0.874)[11]{}
------------------------------------------------------------------------
(410.17,330.37)(7.000,-10.369)[2]{}
------------------------------------------------------------------------
(418.59,316.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(417.17,318.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(425.59,303.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(424.17,305.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(432.59,289.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(431.17,291.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(439.59,275.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(438.17,277.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(446.59,261.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(445.17,263.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(453.59,248.26)(0.485,-1.026)[11]{}
------------------------------------------------------------------------
(452.17,250.13)(7.000,-12.132)[2]{}
------------------------------------------------------------------------
(460.59,234.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(459.17,236.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(467.59,221.50)(0.485,-0.950)[11]{}
------------------------------------------------------------------------
(466.17,223.25)(7.000,-11.251)[2]{}
------------------------------------------------------------------------
(474.59,208.74)(0.485,-0.874)[11]{}
------------------------------------------------------------------------
(473.17,210.37)(7.000,-10.369)[2]{}
------------------------------------------------------------------------
(481.59,196.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(480.17,198.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(488.59,185.98)(0.485,-0.798)[11]{}
------------------------------------------------------------------------
(487.17,187.49)(7.000,-9.488)[2]{}
------------------------------------------------------------------------
(495.59,175.21)(0.485,-0.721)[11]{}
------------------------------------------------------------------------
(494.17,176.61)(7.000,-8.606)[2]{}
------------------------------------------------------------------------
(502.59,165.45)(0.485,-0.645)[11]{}
------------------------------------------------------------------------
(501.17,166.73)(7.000,-7.725)[2]{}
------------------------------------------------------------------------
(509.59,156.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(508.17,157.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(516.59,148.69)(0.485,-0.569)[11]{}
------------------------------------------------------------------------
(515.17,149.84)(7.000,-6.844)[2]{}
------------------------------------------------------------------------
(523.00,141.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(523.00,142.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(530.00,134.93)(0.492,-0.485)[11]{}
------------------------------------------------------------------------
(530.00,135.17)(5.962,-7.000)[2]{}
------------------------------------------------------------------------
(537.00,127.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(537.00,128.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(544.00,122.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(544.00,123.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(551.00,117.93)(0.710,-0.477)[7]{}
------------------------------------------------------------------------
(551.00,118.17)(5.630,-5.000)[2]{}
------------------------------------------------------------------------
(558.00,112.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(558.00,113.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(565.00,108.94)(0.920,-0.468)[5]{}
------------------------------------------------------------------------
(565.00,109.17)(5.340,-4.000)[2]{}
------------------------------------------------------------------------
(572.00,104.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(572.00,105.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(579.00,101.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(579.00,102.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(586,98.17)
------------------------------------------------------------------------
(586.00,99.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(593.00,96.95)(1.355,-0.447)[3]{}
------------------------------------------------------------------------
(593.00,97.17)(4.855,-3.000)[2]{}
------------------------------------------------------------------------
(600,93.67)
------------------------------------------------------------------------
(600.00,94.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(607,92.17)
------------------------------------------------------------------------
(607.00,93.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(614,90.17)
------------------------------------------------------------------------
(614.00,91.17)(3.887,-2.000)[2]{}
------------------------------------------------------------------------
(621,88.67)
------------------------------------------------------------------------
(621.00,89.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(628,87.67)
------------------------------------------------------------------------
(628.00,88.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(635,86.67)
------------------------------------------------------------------------
(635.00,87.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(642,85.67)
------------------------------------------------------------------------
(642.00,86.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(656,84.67)
------------------------------------------------------------------------
(656.00,85.17)(4.000,-1.000)[2]{}
------------------------------------------------------------------------
(649.0,86.0)
------------------------------------------------------------------------
(671,83.67)
------------------------------------------------------------------------
(671.00,84.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(664.0,85.0)
------------------------------------------------------------------------
(692,82.67)
------------------------------------------------------------------------
(692.00,83.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(678.0,84.0)
------------------------------------------------------------------------
(734,81.67)
------------------------------------------------------------------------
(734.00,82.17)(3.500,-1.000)[2]{}
------------------------------------------------------------------------
(699.0,83.0)
------------------------------------------------------------------------
(741.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(137.0,82.0)
------------------------------------------------------------------------
(832.0,82.0)
------------------------------------------------------------------------
(137.0,776.0)
------------------------------------------------------------------------
The Poisson distribution can also be seen as a limiting case of the binomial distribution. If $p$ is the probability of success, then $\nu=Np$ is the expected number of successful trials, as per (\[binmean\]). This approximation is valid for large $N$. In this case, we have $$P(n;N) = \frac{N!}{n!(N-n)!} \left(\frac{\nu}{N}\right)^n \left(1-\frac{\nu}{N}\right)^{N-n}$$ Taking $N\rightarrow\infty$, we have $$\begin{aligned}
P_\nu(n) &= \lim_{N\rightarrow\infty} P(n;N) \\
&= \lim_{N\rightarrow\infty} \frac{N\cdot(N-1)\cdots(N-n+1)}{n} \frac{\nu^n}{N^n} \left(1-\frac{\nu}{N}\right)^N \left(1-\frac{\nu}{N}\right)^{-n} \\
&= \lim_{N\rightarrow\infty} \frac{N\cdot(N-1)\cdots(N-n+1)}{N^n} \frac{\nu^n}{n!} \left(1-\frac{\nu}{N}\right)^N \left(1-\frac{\nu}{N}\right)^{-n} \\
&= 1 \cdot \frac{\nu^n}{n!} e^{-\nu} \cdot 1 \\
&= \frac{\nu^n}{n!} e^{-\nu}
\end{aligned}$$ So, once again, we find that the number of successes has a Poisson distribution.
The characteristic function of the distribution (\[poisson\]) is $$\tilde{P}(\omega) = e^{\lambda(e^{i\omega}-1)}$$ from which we obtain $$\langle X \rangle = \lambda$$
### Functions of Random Variables
If $X$ is a random variable on $\Omega$, and $f:\Omega\rightarrow\Omega'$, then $Y=f(X)$ is a random variable on $\Omega'$. Here I’ll consider, for the sake of simplicity, the case $\Omega=\Omega'={\mathbb{R}}$ (all our considerations can be generalized to arbitrary continua $\Omega$ under fairly general conditions, essentially that $\Omega$ be a metric space). In order to determine the distribution of $y$, I begin with a preliminary observation. For a random variable $X$, let ${\mathbb{P}}_X[x,x+\Delta{x}]$ the probability that the value of $X$ falls in $[x,x+\Delta{x}]$. Then, for small $\Delta{x}$, $$\label{poof}
\begin{aligned}
{\mathbb{P}}_X[x,x+\Delta{x}] &= P(X\le x+\Delta{x}) - P(X\le x) \\
&= \frac{\partial}{\partial x} P(X\le x)\Delta{x} + O(\Delta{x}^2) \\
&= P_X(x)\Delta{x} + O(\Delta{x}^2)
\end{aligned}$$ Let now $f$ be invertible, and $g=f^{-1}$. Then $$\begin{aligned}
P_Y\Delta{y} &= {\mathbb{P}}_Y[y,y+\Delta{y}] \\
&= {\mathbb{P}}_X[g(y),g(y+\Delta{y})] \\
&\approx {\mathbb{P}}_X\Bigl[g(y),g(y) + \left|\frac{dg}{dy}\right|\Delta{y})\Bigr] \\
&= P_X(g(y))\left|\frac{dg}{dy}\right|\Delta{y}
\end{aligned}$$ from which we get $$P_Y(y) = P_X(g(y))\left|\frac{dg}{dy}\right|$$ Note that equivalently one could have defined $$P_Y(y) = \int \delta(y-f(x))P_X\,dx = \langle \delta(y-f(x)) \rangle_X$$ where the subscript on the average reminds us that we are taking the average with respect to the distribution of $X$. From this, we can determine the characteristic function of $Y$: $$\label{doh}
\begin{aligned}
\tilde{P}_Y(\omega) &= \int e^{i\omega{y}}P_Y(y)\,dy \\
&= \int P_X(x) \Bigl[ \int e^{i\omega{y}} \delta(y-f(x))\,dy \Bigr]\,dx \\
&= \int e^{i\omega{f(x)}}P_X(x)\,dx \\
&= \langle \exp\bigl[i\omega f(x)\bigr] \rangle_X
\end{aligned}$$ If $Y=aX$, then $$\label{quack}
\tilde{P}_Y(\omega) = \langle \exp\bigl[i\omega a X\bigr] \rangle_X - \tilde{P}_X(a\omega)$$
\* \* \*
Consider now the sum of two random variables: $Z=X+Y$. Each value of $Z$ can be obtained through an infinity of events: each time $X$ takes an arbitrary value $x$, and $y$ takes a value $z-x$, $Z$ takes the same value, namely $z$. Summing up all these possible events we obtain $$P_Z(z) = \int_{-\infty}^\infty P_X(x)P_Y(z-x)\,dx$$ This is known as the *convolution* of $P_X$ and $P_Y$, often indicated as $P_Z=P_X*P_Y$. The properties of the Fourier transform entail that the corresponding relation between characteristic functions is $$\label{pluck}
\tilde{P}_Z(\omega) = \tilde{P}_X(\omega)\tilde{P}_Y(\omega)$$
\* \* \*
Let $Y=\{y_1,\ldots,y_n\}$ be a set of independent and identically distributed (i.i.d.) variables with cumulative distribution ${\mathcal{P}}_Y$ and density $P_Y$. Consider the function $\min(Y)$: we are interested in finding its density $P_{\min}$ and cumulative distribution ${\mathcal{P}}_{\min}$. We have: $${\mathcal{P}}_Y(x) = {\mathbb{P}}\bigl[\min(Y)\le{x}\bigr] = 1 - {\mathbb{P}}\bigl[\min(Y)\ge{x}\bigr]$$ We have $\min(Y)\ge{x}$ iff we have $y_i\ge{x}$ for all $i$, that is $$\begin{aligned}
{\mathcal{P}}_{\min}(x) &= 1 - {\mathbb{P}}\Bigl[\forall y\in{Y}.y\ge{x}\Bigr] \\
&= 1 - {\mathbb{P}}\bigl[y\ge{x}\bigr]^n \\
&= 1 - \Bigl(1-{\mathbb{P}}\bigl[y\le{x}\bigr]\Bigr)^n \\
&= 1 - \Bigl(1-{\mathcal{P}}_Y(x)\Bigr)^n
\end{aligned}$$ The density is $$\begin{aligned}
P_{\min}(x) &= \frac{d}{dx} {\mathcal{P}}_{\min}(x) \\
&= n\Bigl(1-{\mathcal{P}}_Y(x)\Bigr)^{n-1} \frac{d}{dx} {\mathcal{P}}_Y(x) \\
&= n\Bigl(1-{\mathcal{P}}_Y(x)\Bigr)^{n-1} P_Y(x)
\end{aligned}
\label{mindense}$$ For the function $\max(Y)$, working in a similar way, we have $$\begin{aligned}
{\mathcal{P}}_{\max}(x) &= ({\mathcal{P}}_Y(x))^n
P_{\max}(x) &= n({\mathcal{P}}_Y(x))^{n-1}P_Y(x)
\end{aligned}$$
### The Central Limit Theorem
The Central Limit Theorem (important enough to be granted its own acronym: CLT) is one of the fundamental results in basic probability theory and the main reason why the Gaussian distribution is so important and so common in modeling natural events. In a nutshell, the theorem tells us the following: if we take a lot of random variables, independent and identically distributed (i.i.d.), and add them up, the result will be a random variable with Gaussian distribution. So, for example, if we repeat an experiment many times and take the average of the results that we obtain (the average is, normalization apart, a sum), no matter what the characteristics of the experiment are, the resulting average will have (more or less) a Gaussian distribution.
But, ay, there’s the rub! The theorem works only in the assumption that the moments of the distributions involved be finite. We shall see shortly what happens if this assumption is not satisfied.
Let $X_1,\ldots,X_n$ be a set of i.i.d. random variables with distribution $P_X$, zero mean, and (finite) variance $\sigma^2$. Note that $Y=\sum_iX_i$ has zero mean and variance $n\sigma^2$, while $Y=(\sum_iX_i)/n$ has zero mean and variance $\sigma^2/n$. It is therefore convenient to work with the variable $$Z_n = \frac{1}{\sqrt{n}} \sum_i X_i$$ which has zero mean and variance $\sigma^2$ independently of $n$.
For any distribution $P_X$ with finite mean and variance, and $X_1,\ldots,X_n$ i.i.d. with distribution $P_X$, for $n\rightarrow\infty$, we have $Z_n\rightarrow{Z_\infty}$, where $Z_\infty$ is a Gaussian random variable with zero mean and variance $\sigma^2$ equal to the variance of $P_X$.
Consider the first terms of the expansion of the characteristic function of $P_X$: $$\tilde{P}_X(\omega) = \int e^{i\omega{x}} P_X(x)\,dx = 1 - \frac{1}{2}\sigma^2\omega^2 + O(\omega^3)$$ The characteristic function of $Y=\sum_iX_i$ is given by (\[pluck\]): $$\label{bingo}
\tilde{P}_Y(\omega) = \prod_i \tilde{P}_{X_i}(\omega) = \bigl[ \tilde{P}_X(\omega) \bigr]^n$$ (the second equality holds because the $X$s have the same distribution) while (\[quack\]) with $a=1/\sqrt{n}$ gives $$\tilde{P}_Z(\omega) = P_Y\left( \frac{\omega}{\sqrt{n}} \right)
= \left[ P_Y\left( \frac{\omega}{\sqrt{n}} \right) \right]^n
\approx \left(1 - \frac{\sigma^2 \omega^2}{2 n} \right)^n
\stackrel{n\rightarrow\infty}{\longrightarrow} \exp(-\frac{1}{2}\sigma^2\omega^2)$$ Finally, from (\[lookie\]) we have the inverse transform $$P_Z(z) = \frac{1}{\sigma\sqrt{2\pi}} \exp\Bigl(-\frac{z^2}{2\sigma^2}\Bigr)$$
This theorem is true, in the form in which we have presented it, only for distributions $X$ with finite mean and variance[^12]. However, the key to the theorem is an invariance property of the characteristic function of the Gaussian. Consider the equality (\[bingo\]); we can split it up as: $$\label{pringles}
\tilde{P}_Z(\omega;n) = \bigl[ \tilde{P}_X(\omega) \bigr]^n = \bigl[ \tilde{P}_X(\omega) \bigr]^{n/2} \bigl[ \tilde{P}_X(\omega) \bigr]^{n/2} =
\tilde{P}_Z(\omega;n/2) \tilde{P}_Z(\omega;n/2)$$ Taking the limit $n\rightarrow\infty$, this gives us $P_Z(\omega)=P_Z(\omega)P_Z(\omega)$. That is: the condition for a distribution to be a central limit is that the product of two characteristic functions have the same functional form as the original distributions. As we have seen in (\[ggood\]), the Gaussian distribution does have this property. Nay: it is the *only* distribution with finite moments that has this property, hence its appearance in the theorem in the finite moments case, and hence its great importance in application as a model of many processes resulting from the sum of identical sub-processes.
If we abandon the finite moment hypothesis, however, there is a more general distribution to which (\[pringles\]) applies: the stable Levy distribution. So, a more general form of the CLT can be enunciated as:
For any distribution $P_X$, and $X_1,\ldots,X_n$ i.i.d. with distribution $P_X$, for $n\rightarrow\infty$, we have $$\lim_{n\rightarrow\infty} \frac{1}{\sqrt{n}} \sum_{i=1}^n X_i = Z_\infty$$ where $Z_\infty$ is a random variable with Levy distribution. If the variance of $P_X$ is finite and equal to $\sigma^2$, then $Z_\infty$ has a Gaussian distribution with variance $\sigma^2$.
### Stochastic Processes
A *stochastic process* is a set of random variables $X(t)$ indexed by a variable $t$ (commonly identified with time) that takes value either in ${\mathbb{N}}$ or ${\mathbb{R}}^+$ (less frequently in ${\mathbb{R}}$). We indicate with $P(x,t)$ the probability that the process take value $x$ at time $t$ (the probability density if $t$ is continuous; I shall omit the subscript $X$ to avoid complicating the notation), and with $P(x_2,t_2;x_1,t_1)$ the joint probability density for the two variables $X(t_1)$ and $X(t_2)$. The multiple joint probability density $P(x_1,t_1;\ldots,x_n,t_n)$ is defined analogously. In the following, whenever possible, I shall use the joint probability $P(x_2,t_2;x_1,t_1)$ to simplify the notation, but all considerations hold for the more general multiple joint probability.
Just as a stochastic variable is instantiated to a specific value $x\in\Omega$ with a certain probability, so a stochastic process is instantiated as a trajectory $X:{\mathbb{R}}\rightarrow\Omega$ (or with a discrete series $X:{\mathbb{N}}\rightarrow\Omega$ if the process is discrete). Each $X(t)$, for fixed $t$, is a stochastic variable with a probability distribution that, in general, depends on $t$. A stochastic process is *stationary* if all these distributions are the same, that is, $P(x,t)\equiv{P(x)}$, or, equivalently, if $$P(x_1,t_1;x_2,t_2) = P(x_1,t_1+\tau;x_2,t_2+\tau)$$ In a stochastic process, there are two ways of computing averages: one can compute the *ensemble average* $\langle{X(t)}\rangle$, that is, the average of the random variable $X(t)$, or the mean value along a trajectory $$\bar{X} = \lim_{T\rightarrow\infty} \frac{1}{T} \int_0^T x(t)\,dt$$ A process is *ergodic* if the two coincide $$\langle X \rangle = \bar{X}$$ Ergodicity is an important property for random walks: many times we are interested in the characteristics of the motion of one individual, but many of the equations that we shall use involve ensemble probabilities based on a whole population. Ergodicity allows us to switch from one to the other with impunity.
Note that in a stationary process the correlation $\langle{X(t_1)}X(t_2)\rangle$ does not depend on $t_1$ and $t_2$ individually, but only on their difference $\tau=t_2-t_1$. Joint probabilities are positive, symmetric ($P(x_1,t_1;x_2,t_2)=P(x_2,t_2;x_1,t_1)$) and normalized: $$\int\!\!\!\int_{\Omega^2} P(x_1,t_1;x_2,t_2)\,dx_1\,dx_2 = 1$$ Joint probabilities can be reduced by integration $$P(x_1,t_1) = \int_\Omega P(x_1,t_1;x_2,t_2)\,dx_2$$ and Bayes theorem can be extended to stochastic processes $$\label{extobayes}
P(x_1,t_2 = \int_\Omega P(x_2,t_2|x_1,t_1)P(x_1,t_1)\,dx_1$$ A process if *Markov* if, for all $t_1<t_2<\cdots<t_n$, $$P(x_n,t_n|x_{n-1},t_{n-1};\ldots;x_1,t_1) = P(x_n,t_n|x_{n-1},t_{n-1})$$ this entails that, at any time, the status of the process encodes all the information necessary to make predictions about its future: it is not necessary to know how the process reached that status. The Markov property can be chained: $$\begin{aligned}
P(x_1,t_1;x_1,t_2;x_3,t_3) &= P(x_2,t_2;x_3,t_3|x_1,t_1) P(x_1,t_2) \\
&= P(x_3,t_3|x_2,t_2)P(x_2,t_2|x_1,t_1)P(x_1,t_1)
\end{aligned}$$ Finally, it can be shown [@gardiner:85] that Markov processes must satisfy the *Chapman-Kolmogorov* equation: $$P(x_3,t_3|x_1,t_1) = \int_\Omega P(x_3,t_3|x_2,t_2)P(x_2,t_2|x_1,t_1)\,dx_2$$ The characteristics of the Markov process is evidenced by the fact that the probability of the transition $(x_1,t_1)\rightarrow(x_2,t_2)\rightarrow(x_3,t_3)$ is the product of the probabilities of the transitions $(x_1,t_1)\rightarrow(x_2,t_2)$ and $(x_2,t_2)\rightarrow(x_3,t_3)$, that is, the two transitions are *statistically independent*.
### Gaussian and Wiener processes {#gausswie}
I shall provide here some details on two types of processes of considerable importance for random walks and diffusion.
A stochastic process $X(t)$ is *Gaussian* with zero mean if $\langle{X(t)}\rangle=0$ and $$P(x_i,t_i) = \sqrt{\frac{A_{ii}}{2\pi}} \exp\Bigl( -\frac{1}{2} A_{ii} x^2 \Bigr)$$ ($A_{ii}>0$). The joint probability $P(x_1,t_1;\ldots;x_n,t_n)$ then follows a multivariate Gaussian distribution $$P(x_1,t_1;\ldots;x_n,t_n) = \frac{\mbox{det}(\mathbf{A})^{1/2}}{(2\pi)^{n/2}} \exp\left[ -\frac{1}{2} \sum_{i,j=1}^n x_i A_{ij} x_j \right]$$ Where $\mathbf{A}\in{\mathbb{R}}^{n\times{n}}$ is symmetric (strictly) positive definite. The matrix $\mathbf{A}$ is a measure of the covariance between two variables of the Gaussian process $$\langle X(t_i) X(t_j) \rangle = (\mathbf{A}^{-1})_{ij}$$ (this is true since we assume zero mean). A process is uncorrelated if $\langle{X(t_i)}X(t_j)\rangle=D\delta(t_i-t_j)$, in which case $A_{ij}=D^{-1}\delta_{ij}$.
A *Wiener process* $W$ is a process in which the variables $W(t)$ are real and with independent increments $W(t_2)-W(t_1)$ that follow a Gaussian distribution. That is, they define a conditional probability $$P(w_2,t_2|w_1,t_1) = \frac{1}{\sigma\sqrt{2\pi(t_2-t_1)}} \exp\left[ -\frac{(w_2-w_1)^2}{2\sigma^2(t_2-t_1)} \right]$$ from which the covariance can be computed $$\begin{aligned}
\langle (W(t_2)-\langle{W}\rangle)(W(t_1)-\langle{W}\rangle)\rangle &= \langle (W(t_2)-W(0))(W(t_1)-W(0))\rangle \\
&= \int_{-\infty}^\infty (w_2-w_0)\,dw_2 \int_{-\infty}^\infty\,dw_1 (w_1-w_0) P(w_2,t_2;w_1,t_1) \\
&= \sigma^2 \min(t_1,t_2) + w_0^2
\end{aligned}$$ From this we get $$\label{booh}
\langle W(t)^2 \rangle = \sigma^2 t + w_0^2$$ Wiener processes are related to Gaussian processes, in particular to uncorrelated (white) Gaussian processes. Let $X(t)$ be a Gaussian process with $\langle{X(t_1)}X(t_2)\rangle=\sigma^2\delta(t_2-t_1)$, and define a new stochastic process as the integral of $X(t)$: $$Y(t)=\int_0^t X(u)\,du$$ then $$\begin{aligned}
\langle Y(t_2) Y(t_1) \rangle &= \int_0^{t_2}\!\!\!\!\!du_2\int_0^{t_1}\!\!\!\!\!du_1 \langle{X(u_1)}X(u_2)\rangle \\
&= \int_0^{t_2}\!\!\!\!\!du_2\int_0^{t_1}\!\!\!\!\!du_1 \delta(u_2-u_1)
\end{aligned}$$ By the properties of the Dirac function $$\int_0^{t_1}\!\!\!\!\!du_1 \delta(u_2-u_1) =
\begin{cases}
1 & 0<u_2<t_1 \\
0 & \mbox{otherwise}
\end{cases}$$ Then $$\label{sponge}
\langle Y(t_2) Y_(t_1) \rangle = \sigma^2 \min(t_2,t_1)$$ which coincides with (\[booh\]) for $w_1=0$. That is, the integral of a Gaussian process is a Wiener process.
[^1]: Things are more complex in the case of looking at images: the actual paths that people’s gaze follow depend on what they are looking for. Given an image of an interior, the actual paths are different if the subjects are asked e.g. [“[From what epoch is the interior?]{}”]{} or if they are asked e.g. [“[What are the people in the image doing?]{}”]{} This is not important for our present considerations: all these paths, different as they may be, exhibit the features of ARS.
[^2]: The name [“[disk equation]{}”]{} has nothing to do with the properties of the equation. Holling developed his model by studying the behavior of a blindfolded researcher assistant who was given the task of picking up randomly scattered sandpaper disks.
[^3]: Dimensionally, the equation is sound: the factor $t+2$ in (\[gdef\]) entails that the term 2 has the dimensions of a time.
[^4]: We shall not do the derivation here, but from the Langevin equation one can derive the same Fokker-Planck equation as from the Einstein’s derivation. So, despite the different hypotheses the two methods describe the same macroscopic phenomenon.
[^5]: Neologism courtesy of Lewis Carroll.
[^6]: Remember that the Langevin equation, which assume Gaussian displacements, leads to the same macroscopic result as the Einstein method, which doesn’t.
[^7]: The asterisk denotes here convolution in time, which has different integration limits than convolution in space, since $\psi$ and $\psi^0$ can only take non-negative arguments. Strictly speaking, we should have used a different symbol. However, since is it usually clear what convolution is being used, I have preferred not to complicate the notation using non-standard symbols.
[^8]: The Laplace transform has similar properties as the Fourier, but is more general. If $f(t)$ is a function and ${\mathcal{L}}[f]$ is its Laplace transform (which I shall also indicate as $\tilde{f}$), then $$f(t) = \frac{1}{2\pi{i}} \lim_{T\rightarrow\infty} \int_{\gamma-iT}^{\gamma+iT}\!\!\!\!{\mathcal{L}}[f](s)e^{st}\,ds$$ where $\gamma$ is a real number that exceeds the real part of all singularities of ${\mathcal{L}}[f]$. Also: $$\begin{aligned}
{\mathcal{L}}\Bigl[ \int_0^t\!\!\!f(u)g(t-u)\,du\Bigr] &= {\mathcal{L}}[f]{\mathcal{L}}[g] \\
{\mathcal{L}}\Bigl[ \int_0^t\!\!\!f(u)\,du\Bigr] &= \frac{1}{s}{\mathcal{L}}[f] \\
{\mathcal{L}}\bigl[ e^{at}f(t)\bigr] &= {\mathcal{L}}[f](s-a) \\
{\mathcal{L}}\bigl[1\bigr] &= \frac{1}{s}
\end{aligned}$$
[^9]: As we have seen, any distribution, as long as it has finite moments, will give the same results, as we are in the hypotheses of the standard Central Limit Theorem.
[^10]: An informal way to reach the same conclusion is to note that $K^\mu=\sigma^\mu/\tau$ so that, in order to have $K^\mu$ finite, we must have $\sigma^\mu\sim\tau$, that is, $\sigma^2\sim\tau^{2/\mu}$, leading again to $H=1/\mu$.
[^11]: Following the standard notation, we shall use capital letters to indicate random variables and lowercase letters to indicate the values that they assume.
[^12]: I have assumed zero mean since, if the mean of the $X$ is non-zero, the mean of $Z$ goes to infinity; this doesn’t represent a major hurdle for the theorem, which can easily be generalized by subtracting the mean from the variables $X$ and then adding it back.
|
---
author:
- 'Benito Hernández–Bermejo$^{\; *}$'
- 'Víctor Fairén$^{\; 1,*}$'
title: Algebraic decoupling of variables for systems of ODEs of quasipolynomial form
---
[*Departamento de Física Fundamental, Universidad Nacional de Educación a Distancia. Senda del Rey S/N, 28040 Madrid, Spain.*]{}
[**Abstract**]{}
A generalization of the reduction technique for ODEs recently introduced by Gao and Liu is given. It is shown that the use of algebraic methods allows the extension of the procedure to much more general flows, as well as the derivation of simple criteria for the identification of reducible systems.
[**Keywords:**]{} Ordinary differential equations, integrability, algebraic methods, reduction techniques.
$^1$ To whom all correspondence should be addressed. E-mail: vfairen@uned.es
$^*$ Present address: Departamento de Física Matemática y Fluidos, Universidad Nacional de Educación a Distancia. Senda del Rey S/N, 28040 Madrid, Spain.
[**1. Introduction**]{}
The problem of finding first integrals and identifying integrability conditions of dynamical systems has deserved a continued effort along many decades. The relevance and interest of the problem is reflected in the number of different procedures developed for these purposes, such as the Lie symmetries method [@os1], Carleman embedding [@ks1], Prelle-Singer procedure [@ps1] or Painlevé test [@se1], to cite a sample. However, none of the presently known methods can account for the problem in its full generality, and only partial answers have been developed.
Among the different analytic tools available, the Quasipolynomial (QP) formalism for ODEs has received increasing attention in the last years. This interest was initially centered in integrability properties [@pm1]–[@at1] and canonical forms [@bv1; @bv3], but applications are also starting to reach different fields such as chemical kinetics [@vb1], theoretical biochemistry [@bv2], normal forms [@sb1] and Hamiltonian systems [@bv4].
In a recent article [@gl2], Gao and Liu have applied a changing variables method (CVM from now on) in order to find first integrals of 3D quadratic systems of Lotka-Volterra form. This is done by decoupling one of the variables of the initial 3D flow, thus reducing the effective dimension of the system in one unit. Analysis of integrability conditions and identification of first integrals is thus a much simpler task in the reduced 2D system.
It is worth noting that most transformations employed in [@gl2] find their place in a natural way within the QP formalism. In this work we explore the consequences arising from this fact. As Gao and Liu, we shall be primarily concerned with the possibility of reducing a flow into a two-dimensional one. In addition to the possibility of finding first integrals and integrability conditions already mentioned in [@gl2], it should be added that knowledge that a system of dimension three or higher can be reduced into a two-dimensional one is interesting in itself because it excludes the possibility of chaotic behavior —a problem to which a considerable effort has been devoted recently [@fh1] in the case of 3D systems.
We shall demonstrate that the CVM can be completely reformulated in terms of the QP formalism. This has four major implications:
[[*i)*]{}]{} The first is that the procedure can be made more systematic and simpler, since all manipulations can be carried out easily in terms of matrix algebra.
[[*ii)*]{}]{} The second is that the use of the QP formalism allows generalizing the scope of the procedure. Generically, the method allows a reduction of one unit in the effective dimension of an $n$-dimensional system, with arbitrary $n$. This makes the technique particularly interesting for the reduction of 3D flows into two-dimensional ones, thus precluding the existence of irregular motion. Most examples will accordingly be on standard 3D systems. However, such reduction into a two-dimensional flow may also be possible for some $n$-dimensional systems (an example is given in Subsection 3.3). Moreover, we shall demonstrate that the procedure is not limited to Lotka-Volterra quadratic systems, but is equally valid for flows with much more general nonlinearities.
[[*iii)*]{}]{} The third is that the use of matrix algebra leads to simple criteria for the identification of reducible systems.
[[*iv)*]{}]{} The fourth is that some of the CVM transformations are just particular cases of wider transformation families that we characterize. We shall see in the examples that, in some cases, different members of those families are preferable to the CVM choice.
Before describing the reduction technique, it is convenient to give a short account of some relevant features of the QP formalism.
[**2. Transformations on QP systems**]{}
We shall begin by briefly recalling those basic properties of QP equations that will be necessary in what is to come. We refer the reader to the cited literature for further details.
The starting point of the formalism are QP systems of ODE’s of the form: $$\label{eq:glv}
\dot{x}_i = x_{i} \left( \lambda _{i} + \sum_{j=1}^{m}A_{ij}\prod_{k=
1}^{n}x_{k}^{B_{jk}} \right) , \;\:\;\: i = 1 \ldots n , \;\:\; m \geq n$$ where $n$ and $m$ are positive integers, and $A$, $B$ and $\lambda$ are $n \times m$, $m \times n$ and $n \times 1$ real matrices, respectively. In what follows, $n$ will always denote the number of variables of a QP system, and $m$ the number of quasimonomials: $$\prod_{k=1}^{n}x_{k}^{B_{jk}} \; , \;\:\;\: j = 1 \ldots m$$
We will assume that $m \geq n$ and that $B$ is of maximal rank, i.e. rank($B$) $=n$. If $m<n$ and/or rank($B$) is not maximal, then it can be shown [@bv3] that the system is redundant and can always be reduced to the standard situation $m \geq n$ and rank($B$) $=n$, which is our starting assumption.
QP equations (\[eq:glv\]) are form-invariant under quasimonomial transformations (QMTs): $$x_{i} = \prod_{k=1}^{n} y _{k}^{C_{ik}} , \;\: i=1,\ldots ,n
\label{bec}$$ for any invertible real matrix $C$. After (\[bec\]), matrices $B$, $A$, and $\lambda$ change to $$B' = B \cdot C \;\:, \:\;\: A' = C^{-1} \cdot A \;\: , \:\;\:
\lambda ' = C^{-1} \cdot \lambda \;\: , \:\;\:$$ respectively, but the QP format is preserved.
Quasimonomial transformations are complemented by the new-time transformations (NTTs) of the form [@ha1]: $$\label{ntt}
d \tau \; = \; \xi(x_1, \ldots ,x_n) \; dt$$ where $\xi(x_1, \ldots ,x_n)$ is a smooth function. The most important choice for $\xi$ in the QP formalism is [@gs2]: $$\label{xi1}
\xi(x_1, \ldots ,x_n) = \prod_{i=1}^n x_i^{\, \beta _i}$$ where the $\beta _i$ are real constants. With $\xi$ given by (\[xi1\]), transformation (\[ntt\]) also preserves the QP format.
Although we will need in certain specific steps of the procedure some additional sets of transformations, it is not necessary to elaborate on them now. Therefore, we can proceed to describe the reduction method. For this, we shall distinguish three cases of increasing complexity.
[**3. Criteria and algorithms for the reduction of systems**]{}
[*3.1 Case I: $\lambda = 0$ and $m = n$*]{}
In this case we shall see that the reduction in one dimension of the system is always possible. The set of ODEs takes the form: $$\label{case1}
\dot{x}_i = x_{i} \left( \sum_{j=1}^{n}A_{ij}\prod_{k=
1}^{n}x_{k}^{B_{jk}} \right) , \;\:\;\: i = 1 \ldots n$$ We look for a QMT such that for the new QP flow we have: $$\label{bcase1}
B' = B \cdot C = \left( \begin{array}{cccc}
1 & B'_{12} & \ldots & B'_{1n} \\
1 & B'_{22} & \ldots & B'_{2n} \\
\vdots & \vdots & \mbox{} & \vdots \\
1 & B'_{n2} & \ldots & B'_{nn}
\end{array} \right) \;\: ,$$ where the columns 2 to $n$ can be chosen arbitrarily (with the obvious restriction rank($B'$) $=n$). Note that the CVM choice [@gl2] is a particular case of (\[bcase1\]) with $B'_{ij} = \delta _{ij}$, with $2 \leq j \leq n$ and symbol $\delta$ standing for Kronecker’s delta. Given $B'$, we immediately find from (\[bcase1\]) that $C$ exists and is unique: $C$ $=$ $B^{-1} \cdot B'$. Let $y_i$ denote the variables obtained after the QMT of matrix $C$. Then we arrive at the following system: $$\dot{y}_i = y_1 y_{i} \left( \sum_{j=1}^{n}A'_{ij} \prod_{k=2}^{n}
y_{k}^{B'_{jk}} \right) , \;\:\;\: i = 1 \ldots n$$ We now rescale the time variable by means of the NTT: $$\label{ntt1}
d \tau = y_1 dt \;\: ,$$ where $t$ is the old time variable and $\tau$ the new one. Let us denote from now on the derivative of any function $\chi (\tau)$ of a new time $\tau$ as $d\chi /d \tau \equiv \hat{\chi}$. Then after (\[ntt1\]) we are led to: $$\label{case13}
\hat{y}_i = y_{i} \left( \sum_{j=1}^{n}A'_{ij} \prod_{k=2}^{n}
y_{k}^{B'_{jk}} \right) , \;\:\;\: i = 1 \ldots n$$ Now notice that the only variables appearing in the r.h.s. of equations 2 to $n$ of system (\[case13\]) are precisely $\{ y_2, \ldots, y_n \}$, i.e. $y_1$ has been decoupled. Thus the equation for $y_1$ in (\[case13\]) is a quadrature, and system (\[case1\]) has been reduced to an $(n-1)$-dimensional one in the variables $\{ y_2 , \ldots , y_n \}$ and the new time $\tau$.
[*Example of Case I: Euler equations for the free rigid body*]{}
As an example of Case I we can choose Euler’s equations for the free rigid body [@as1] which are given by: $$\begin{aligned}
\dot{x}_1 & = & a _1 x_2 x_3 \nonumber \\
\dot{x}_2 & = & a _2 x_1 x_3 \label{ej1} \\
\dot{x}_3 & = & a _3 x_1 x_2 \nonumber\end{aligned}$$ The QP matrices of system (\[ej1\]) are: $$B = \left( \begin{array}{ccc}
-1 & 1 & 1 \\
1 & -1 & 1 \\
1 & 1 & -1
\end{array} \right) \; , \;\:
A = \left( \begin{array}{ccc}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & a_3
\end{array} \right) \; , \;\:
\lambda = \left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right)$$ We now apply a QMT of matrix: $$C = \left( \begin{array}{ccc}
1 & 1/2 & 1/2 \\
1 & 0 & 1/2 \\
1 & 1/2 & 0
\end{array} \right)$$ The resulting QP system is characterized by matrices: $$\label{ej12}
B' = \left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 0 & 1
\end{array} \right) \; , \;\:
A' = \left( \begin{array}{ccc}
-a_1 & a_2 & a_3 \\
2 a_1 & -2 a_2 & 0 \\
2 a_1 & 0 & -2 a_3
\end{array} \right) \; , \;\:
\lambda ' = \left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right)$$ In this case we have chosen $C$ so as to arrive to a matrix $B'$ of the form used in [@gl2]. Let $\{ y_1, y_2, y_3 \}$ be the variables of the transformed equations corresponding to matrices (\[ej12\]). If we finally perform the NTT $d \tau = y_1 dt$ we arrive to: $$\begin{aligned}
\hat{y}_1 & = & y_1(-a_1 + a_2y_2 + a_3y_3) \nonumber \\
\hat{y}_2 & = & y_2(2a_1 - 2a_2y_2) \label{ej1f} \\
\hat{y}_3 & = & y_3(2a_1 - 2a_3y_3) \nonumber \end{aligned}$$ This completes the decoupling procedure. In (\[ej1f\]) the equation of $y_1$ has been reduced to a quadrature, while the equations for $y_2$ and $y_3$ do not depend on $y_1$. Moreover, the equations for $y_2$ and $y_3$ are also decoupled from each other, and can be integrated straightforwardly. The integrability of the system is thus made manifest.
[*3.2 Case II: $\lambda = 0$ and $m > n$*]{}
We now write the system as: $$\dot{x}_i = x_{i} \left( \sum_{j=1}^{m}A_{ij}\prod_{k=1}^{n}
x_{k}^{B_{jk}} \right) , \;\:\;\: i = 1 \ldots n , \;\:\; m > n$$ We again look for a QMT of matrix $C$ such that: $$\label{bcase2}
B' = B \cdot C = \left( \begin{array}{cccc}
1 & B'_{12} & \ldots & B'_{1n} \\
1 & B'_{22} & \ldots & B'_{2n} \\
\vdots & \vdots & \mbox{} & \vdots \\
1 & B'_{m2} & \ldots & B'_{mn}
\end{array} \right)$$ As in Case I, columns 2 to $n$ of $B'$ in (\[bcase2\]) can be chosen freely in such a way that Rank($B'$) $=n$. Now note that matrices $B$ and $B'$ are not square but $m \times n$. This implies that a suitable $C$ may or may not exist, depending on the form of $B$. Let us denote by $\tilde{B}$ the following $m \times (n+1)$ matrix: $$\tilde{B} \equiv \left( \begin{array}{ccccc}
B_{11} & B_{12} & \ldots & B_{1n} & 1 \\
B_{21} & B_{22} & \ldots & B_{2n} & 1 \\
\vdots & \vdots & \mbox{} & \vdots & \vdots \\
B_{m1} & B_{m2} & \ldots & B_{mn} & 1
\end{array} \right)$$ It is not difficult to prove that $C$ exists if and only if Rank($\tilde{B}$) $=n$, and in this case it is unique.
If, according to the matrix criterion Rank($\tilde{B}$) $=n$, a suitable $C$ exists, then the rest of the procedure is completely similar to Case I: We first perform the QMT of matrix $C$. Let $\{ y_1, \ldots , y_n \}$ be the new variables of the transformed system. Then we can factor out $y_1$ in each of the equations, and eliminate it later by means of the NTT $d \tau = y_1 dt$. The result is the decoupling of $y_1$ and the reduction in one unit of the effective dimension of the vector field.
[*Example of Case II: Halphen system*]{}
We now consider the Halphen equations [@hs1; @ah1] which describe the two-monopole system. We shall write them in the form given in [@gn1]: $$\begin{aligned}
\dot{x}_1 & = & x_2 x_3 - x_1 x_2 - x_1 x_3 \nonumber \\
\do{x}_2 & = & x_1 x_3 - x_1 x_2 - x_2 x_3 \label{ej2} \\
\dot{x}_3 & = & x_1 x_2 - x_1 x_3 - x_2 x_3 \nonumber\end{aligned}$$ In QP terms we have: $$\label{mej2}
B = \left( \begin{array}{ccc}
-1 & 1 & 1 \\
1 & -1 & 1 \\
1 & 1 & -1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) \; , \;\:
A = \left( \begin{array}{cccccc}
1 & 0 & 0 & 0 & -1 & -1 \\
0 & 1 & 0 & -1 & 0 & -1 \\
0 & 0 & 1 & -1 & -1 & 0
\end{array} \right) \; , \;\:
\lambda = \mbox{\bf 0}$$ It is a simple task to check that rank($\tilde{B}$) $= 3$ when $B$ is given by (\[mej2\]). Consequently, the system can be reduced. For this purpose we may choose a QMT of matrix: $$C = \left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 0 & 1
\end{array} \right)$$ After the QMT, the first variable of the transformed system, say $y_1$, can be factored out. If we then eliminate it by means of the NTT (\[ntt1\]) the result is: $$\begin{aligned}
\hat{y}_1 & = & y_1 (y_2 y_3 - y_2 - y_3) \nonumber \\
\hat{y}_2 & = & -y_2 + y_2^2 - y_2^2 y_3 + y_3 \label{ej22} \\
\hat{y}_3 & = & -y_3 + y_3^2 - y_2 y_3^2 + y_2 \nonumber\end{aligned}$$ As expected, the first variable is reduced to quadrature and the study of system (\[ej2\]) is reduced to a 2D flow corresponding to the equations for $y_2$ and $y_3$ in (\[ej22\]).
[*3.3 Case III: $\lambda \neq 0$*]{}
We now focus on QP systems of the general form: $$\label{case3}
\dot{x}_i = x_{i} \left( \lambda _{i} + \sum_{j=1}^{m}A_{ij}\prod_{k=
1}^{n}x_{k}^{B_{jk}} \right) , \;\:\;\: i = 1 \ldots n , \;\:\; m \geq n$$ The line of action now consists in reducing the problem to either Case I or II, i.e. in suppressing the $\lambda$ terms. For this we first introduce the following new variables: $$\label{tre}
y_i = e^{- \lambda _i t} x_i \; , \;\:\;\: i = 1 \ldots n$$ Transformation (\[tre\]) is similar to the one employed in [@gl2], and was also used in [@bs1] in connection with the construction of canonical forms for ODEs. We find: $$\label{case32}
\dot{y}_i = y_{i} \left( \sum_{j=1}^{m}A_{ij} e^{\Gamma _j t}
\prod_{k=1}^{n}y_{k}^{B_{jk}} \right) , \;\:\;\: i = 1 \ldots n \; ,$$ where $\Gamma = B \cdot \lambda$. In order to reduce (\[case32\]) to Cases I or II, the following condition is sufficient: $$\label{lamb}
\Gamma _1 = \Gamma _2 = \ldots = \Gamma _m \equiv \gamma$$ Provided (\[lamb\]) holds, we can perform the transformation: $$d \tau = e^{\gamma t} dt$$ We finally arrive to the system: $$\label{case34}
\hat{y}_i = y_{i} \left( \sum_{j=1}^{m}A_{ij}
\prod_{k=1}^{n}y_{k}^{B_{jk}} \right) , \;\:\;\: i = 1 \ldots n$$ Thus, if condition (\[lamb\]) is satisfied, equations (\[case3\]) can be reduced to system (\[case34\]), which corresponds to Case I if $m=n$ and to Case II if $m>n$.
This completes our enumeration of criteria and reduction algorithms. We now give two examples corresponding to this last situation.
[*A first example of Case III: Maxwell-Bloch system*]{}
As an example we may consider the Maxwell-Bloch equations for laser systems. In the case of periodic boundary conditions, the equations are [@lm1]: $$\begin{aligned}
\dot{x}_1 & = & -a_1 x_1 + a_2 x_2 \nonumber \\
\dot{x}_2 & = & -a_3 x_2 + a_2 x_1 x_3 \label{ej3} \\
\dot{x}_3 & = & -a_4 (x_3 - x_{30}) - 4 a_2 x_1 x_2 \nonumber\end{aligned}$$ The QP matrices are: $$B = \left( \begin{array}{ccc}
-1 & 1 & 0 \\
1 & -1 & 1 \\
1 & 1 & -1 \\
0 & 0 & -1
\end{array} \right) \; , \;\:
A = \left( \begin{array}{cccc}
a_2 & 0 & 0 & 0 \\
0 & a_2 & 0 & 0 \\
0 & 0 & -4a_2 & a_4 x_{30}
\end{array} \right) \; , \;\:
\lambda = \left( \begin{array}{c}
-a_1 \\ -a_3 \\ -a_4
\end{array} \right)$$ We first compute $\Gamma$: $$\label{gam3}
\Gamma = \left( \begin{array}{c}
a_1 - a_3 \\
- a_1 + a_3 - a_4 \\
- a_1 - a_3 + a_4 \\
a_4
\end{array} \right)$$ Let us look at the compatibility condition (\[lamb\]) for $\Gamma$ in (\[gam3\]). If we impose $\Gamma _1 = \Gamma _2 = \Gamma _3$ we immediately find: $$\label{e1}
2 a_1 = a_3 = a_4$$ However, it is not possible to simultaneously verify the last requirement $\Gamma _4 = \Gamma _i$, for $i=1,2,3$. Then, system (\[ej3\]) cannot be reduced in general. However, we can follow Gümral and Nutku [@gn1] and consider the case in which $x_{30}=0$ in equations (\[ej3\]). The resulting system is given by the following QP matrices: $$\label{m32}
B = \left( \begin{array}{ccc}
-1 & 1 & 0 \\
1 & -1 & 1 \\
1 & 1 & -1
\end{array} \right) \; , \;\:
A = \left( \begin{array}{ccc}
a_2 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & -4a_2
\end{array} \right) \; , \;\:
\lambda = \left( \begin{array}{c}
-a_1 \\ -a_3 \\ -a_4
\end{array} \right)$$ Now condition (\[lamb\]) is satisfied iff (\[e1\]) holds. The parameter values that we have obtained in order to verify equation (\[lamb\]) $$2 a_1 = a_3 = a_4 \;\: , \;\:\;\: x_{30} = 0 \;\: ,$$ are precisely those found after some [*ad hoc*]{} transformations by Gümral and Nutku in [@gn1] when characterizing the values of the parameters for which the Maxwell-Bloch system (\[ej3\]) is bi-Hamiltonian. Notice that such parameter values arise here in a natural way and allow the identification of these integrable cases.
Then, in what follows we will write in (\[m32\]): $$a_1 \equiv \alpha \;\: , \;\:\;\: a_3 = a_4 \equiv 2 \alpha$$ Thus we can perform transformation (\[tre\]): $$y_1 = e^{\alpha t} x_1 \;\: , \;\:\;\:
y_2 = e^{2 \alpha t} x_2 \;\: , \;\:\;\:
y_3 = e^{2 \alpha t} x_3 \;\:\;\: ,$$ and then the change $d \tau = e^{- \alpha t} dt$. The result is: $$\begin{aligned}
\dot{y}_1 & = & a_2 y_2 \nonumber \\
\dot{y}_2 & = & a_2 y_1 y_3 \label{ej32} \\
\dot{y}_3 & = & -4 a_2 y_1 y_2 \nonumber \end{aligned}$$ The QP matrices $A'$ and $B'$ of system (\[ej32\]) coincide, respectively, with $A$ and $B$ in (\[m32\]), while now $\lambda ' = $ [**0**]{}. Since we have $m=n=3$ in (\[ej32\]), equations (\[ej3\]) have been reduced to Case I of the algorithm. As we know from Subsection 3.1, the reduction to a 2D flow is always possible in this case.
According to the procedure for Case I, we first apply to system (\[ej32\]) a QMT of matrix: $$C = \left( \begin{array}{ccc}
1 & 1/2 & 1/2 \\
2 & 1/2 & 1/2 \\
2 & 1 & 0
\end{array} \right)$$ Let $\{ z_1,z_2,z_3 \}$ be the variables of the transformed system. The last step is a NTT $d \tau = a_2 z_1 dt$. The final result is: $$\begin{aligned}
\hat{z}_1 & = & z_1 ( z_2 - 1 ) \nonumber \\
\hat{z}_2 & = & 2 z_2 ( 1 - z_2 - 2 z_3 ) \label{ej33} \\
\hat{z}_3 & = & 2 z_3 ( 1 + 2 z_3 ) \nonumber\end{aligned}$$ Then the first variable is decoupled and we obtain a reduced 2D system. Note also that the equation for $z_3$ is directly integrable, so the whole system is, in fact, reduced to a one-dimensional problem.
[*An $n$-dimensional example: Riccati projective systems*]{}
We conclude the examples with the Riccati projective equations which have recently deserved some attention in different areas, such as selection dynamics [@se2] or normal forms [@sb1]. These systems are given by: $$\label{ej4}
\dot{x}_i = \lambda _i x_i + x_i \sum_{j=1}^n a_j x_j
, \;\:\;\: i = 1 \ldots n$$ We can follow the steps given in Case III and evaluate $\Gamma$. Thus we could simplify the system provided condition (\[lamb\]) is satisfied, i.e. $ \lambda _1 $ $=$ $ \ldots $ $=$ $ \lambda _n$.
We shall not proceed according to this line of action, however. Instead, we shall demonstrate that the techniques described above allow solving equations (\[ej4\]) in general. Since $\lambda \neq$ [**0**]{} in (\[ej4\]) we start, as usual, by applying transformation (\[tre\]): $$y_i = e^{- \lambda _i t} x_i \; , \;\:\;\: i = 1 \ldots n$$ The result is: $$\label{ej42}
\dot{y}_i = y_i \left( \sum_{j=1}^n a_j e^{\lambda _j t} y_j \right)
, \;\:\;\: i = 1 \ldots n$$ In general, we cannot factor out the exponentials in equation (\[ej42\]). In other words, relations (\[lamb\]) will not be usually satisfied. However, this is not an unavoidable difficulty in the case of system (\[ej42\]): We can anyhow perform a QMT of the form (\[bcase1\]) described in Subsection 3.1. The best possibility can be easily seen to be: $$\label{qmte4}
C = \left( \begin{array}{ccccc}
1 & 0 & 0 & \ldots & 0 \\
1 & -1 & 0 & \ldots & 0 \\
1 & 0 & -1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & 0 & 0 & \ldots & -1
\end{array} \right)$$ Notice that this QMT corresponds to a different choice to the one considered in [@gl2]. When we perform the QMT of matrix (\[qmte4\]) on equations (\[ej42\]) the result is: $$\label{ej4f1}
\dot{z}_1 = z_1^2 \left( a_1 e^{\lambda _1 t} + \sum_{j=2}^n
\frac{a_j e^{\lambda _j t}}{z_j} \right)$$ and $$\label{ej4f2}
\dot{z}_i = 0 \;\: , \;\:\;\: i = 2 \ldots n$$ The outcome is that, in its final form (\[ej4f1\])–(\[ej4f2\]), the integrability of system (\[ej4\]) is made completely explicit —in fact, equations (\[ej4f1\])–(\[ej4f2\]) can be integrated trivially.
[**4. Final remarks**]{}
The QP formalism provides the natural operational framework for the changing variables method as given in [@gl2]: Not only allows its reformulation in simpler matrix terms, but also leads naturally to extensions, for example to the case of nonquadratic flows. The use of matrix algebra has also made possible the derivation of some simple criteria for the identification of reducible systems —criteria which are quite convenient for practical purposes.
It is worth insisting that this kind of approach should be especially appropriate in the context of 3D sets of ODEs. However, our treatment has been completely general in what concerns to the dimension of the system, since the possibility of finding higher-dimensional applications cannot be excluded, as our last example illustrates. In any case, the final goal has always been the reduction into a 2D flow: When this is possible, chaotic dynamics is precluded, and further analysis (on parameter space, for instance) can be carried out in much simpler terms.
[**Acknowledgements**]{}
This work has been supported by the DGICYT of Spain (grant PB94-0390) and by the E.U. (Esprit WG 24490). B. H. acknowledges a doctoral fellowship from Comunidad de Madrid.
[99]{} P. J. Olver, Applications of Lie Groups to Differential Equations, Second Ed. (Springer-Verlag, New York, 1993). K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization (World Scientific, Singapore, 1991). M. J. Prelle and M. F. Singer, Trans. Amer. Math. Soc. 279 (1983) 215. W.-H. Steeb and N. Euler, Nonlinear Evolution Equations and Painlevé Test (World Scientific, Singapore, 1988). M. Peschel and W. Mende, The Predator–Prey Model (Springer-Verlag, Vienna–New York, 1986). L. Brenig, Phys. Lett. A 133 (1988) 378. L. Brenig and A. Goriely, Phys. Rev. A, 40 (1989) 4119. J. L. Gouzé, Report INRIA 1308 (1990) 1. A. Goriely and L. Brenig, Phys. Lett. A 145 (1990) 245. A. Goriely, J. Math. Phys. 33 (1992) 2728. A. Figueiredo, T. M. Rocha Filho and L. Brenig, J. Math. Phys. 39 (1998) 2929. B. Hernández–Bermejo and V. Fairén, Phys. Lett. A 206 (1995) 31. B. Hernández–Bermejo, V. Fairén and L. Brenig, J. Phys. A: Math. Gen. 31 (1998) 2415. V. Fairén and B. Hernández–Bermejo, J. Phys. Chem. 100 (1996) 19023. B. Hernández–Bermejo and V. Fairén, Math. Biosci. 140 (1997) 1. S. Louies and L. Brenig, Phys. Lett. A 233 (1997) 184. B. Hernández–Bermejo and V. Fairén, Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems, J. Math. Phys. (1998), to be published. P. Gao and Z. Liu, Phys. Lett. A 244 (1998) 49. Z. Fu and J. Heidel, Nonlinearity 10 (1997) 1289. J. Hietarinta, B. Grammaticos, B. Dorizzi and A. Ramani, Phys. Rev. Lett. 53 (1984) 1707. V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Ed. (Springer-Verlag, New York, 1989). M. Halphen, C. R. Acad. Sci. Paris 92 (1881) 1101. M. Atiyah and N. Hitchin, The Geometry And Dynamics of Magnetic Monopoles (Princeton University Press, Princeton, New Jersey, 1988). H. Gümral and Y. Nutku, J. Math. Phys. 34 (1993) 5691. F. T. Arecchi, in: Order and Chaos in Nonlinear Physical Systems, eds. S. Lundqvist, N. H. March and M. P. Tosi (Plenum Press, New York, 1988) p. 193. A. V. Shapovalov and E. V. Evdokimov, Physica D 112 (1998) 441.
|
---
abstract: 'We introduce *neural particle smoothing*, a sequential Monte Carlo method for sampling annotations of an input string from a given probability model. In contrast to conventional particle filtering algorithms, we train a proposal distribution that [*looks ahead*]{} to the end of the input string by means of a right-to-left LSTM. We demonstrate that this innovation can improve the quality of the sample. To motivate our formal choices, we explain how our neural model and neural sampler can be viewed as low-dimensional but nonlinear approximations to working with HMMs over very large state spaces.=-1'
author:
- 'Chu-Cheng Lin'
- |
Jason Eisner\
Center for Language and Speech Processing\
Johns Hopkins University, Baltimore MD, 21218\
`{kitsing,jason}@cs.jhu.edu`
bibliography:
- 'psmooth.bib'
title: |
Neural Particle Smoothing\
for Sampling from Conditional Sequence Models
---
Introduction {#sec:intro}
============
Many structured prediction problems in NLP can be reduced to labeling a length-$T$ input string ${{{\boldsymbol{\mathbf{x}}}}}$ with a length-$T$ sequence ${{{\boldsymbol{\mathbf{y}}}}}$ of tags. In some cases, these tags are annotations such as syntactic parts of speech. In other cases, they represent actions that incrementally build an output structure: IOB tags build a chunking of the input [@ramshaw1999], shift-reduce actions build a tree [@yamada2003], and finite-state transducer arcs build an output string [@pereira1997]. One may wish to score the possible taggings using a recurrent neural network, which can learn to be sensitive to complex patterns in the training data. A globally normalized conditional probability model is particularly valuable because it quantifies uncertainty and does not suffer from label bias [@lafferty-mccallum-pereira-2001]; also, such models often arise as the predictive conditional distribution $p({{{\boldsymbol{\mathbf{y}}}}}\mid{{{\boldsymbol{\mathbf{x}}}}})$ corresponding to some well-designed generative model $p({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$ for the domain. In the neural case, however, inference in such models becomes intractable. It is hard to know what the model actually predicts and hard to compute gradients to improve its predictions.
In such intractable settings, one generally falls back on approximate inference or sampling. In the NLP community, beam search and importance sampling are common. Unfortunately, beam search considers only the approximate-top-$k$ taggings from an exponential set [@wiseman2016], and importance sampling requires the construction of a good proposal distribution [@dyer2016].
In this paper we exploit the sequential structure of the tagging problem to do [*sequential*]{} importance sampling, which resembles beam search in that it constructs its proposed samples incrementally—one tag at a time, taking the actual model into account at every step. This method is known as particle filtering [@doucet2009]. We extend it here to take advantage of the fact that the sampler has access to the entire input string as it constructs its tagging, which allows it to look ahead or—as we will show—to use a neural network to approximate the effect of lookahead. Our resulting method is called [*neural particle smoothing*]{}.
What this paper provides {#sec:setup}
------------------------
For ${{{\boldsymbol{\mathbf{x}}}}}= x_1 \cdots x_T$, let ${{{\boldsymbol{\mathbf{x}}}}}_{:t}$ and ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$ respectively denote the prefix $x_1 \cdots x_t$ and the suffix $x_{t+1} \cdots x_T$.
We develop *neural particle smoothing*—a sequential importance sampling method which, given a string ${{{\boldsymbol{\mathbf{x}}}}}$, draws a sample of taggings ${{{\boldsymbol{\mathbf{y}}}}}$ from $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. Our method works for any conditional probability model of the quite general form[^1] $$\begin{aligned}
\label{eq:condit}
p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) &{\mathrel{\stackrel{\mbox{\tiny def}}{\propto}}}\exp G_T\end{aligned}$$ where $G$ is an *incremental stateful global scoring model* that recursively defines scores $G_t$ of prefixes of $({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$ at all times $0 \leq t \leq T$: $$\begin{aligned}
\!\!\!\!G_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}G_{t-1} + g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) &\!\!\!\!\text{(with $G_0{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}0$)} \label{eq:G} \\
\!\!\!\!{{{\boldsymbol{\mathbf{s}}}}}_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}f_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) &\!\!\!\!\text{(with ${{{\boldsymbol{\mathbf{s}}}}}_0$ given)} \label{eq:s}\end{aligned}$$
These quantities implicitly depend on ${{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}$ and $\theta$. Here ${{{\boldsymbol{\mathbf{s}}}}}_t$ is the model’s *state* after observing the pair of length-$t$ prefixes $({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$. $G_t$ is the *score-so-far* of this prefix pair, while $G_T - G_t$ is the *score-to-go*. The state ${{{\boldsymbol{\mathbf{s}}}}}_t$ summarizes the prefix pair in the sense that the score-to-go depends only on ${{{\boldsymbol{\mathbf{s}}}}}_t$ and the length-$(T-t)$ suffixes $({{{\boldsymbol{\mathbf{x}}}}}_{t:}, {{{\boldsymbol{\mathbf{y}}}}}_{t:})$. The *local scoring function* $g_\theta$ and *state update function* $f_\theta$ may be any functions parameterized by $\theta$—perhaps neural networks. We assume $\theta$ is fixed and given.
This model family is expressive enough to capture any desired $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. Why? Take any distribution $p({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$ with this desired conditionalization (e.g., the true joint distribution) and factor it as $$\begin{aligned}
\!\!\!\! \log p({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})\!
&= \textstyle \sum_{t=1}^{T} \log p(x_t,y_t \mid {{{\boldsymbol{\mathbf{x}}}}}_{:t-1},{{{\boldsymbol{\mathbf{y}}}}}_{:t-1}) \nonumber \\
&= \textstyle \sum_{t=1}^{T} \underbrace{\log p(x_t,y_t \mid
{{{\boldsymbol{\mathbf{s}}}}}_{t-1})}_{\text{use as }g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t)} = G_T \label{eq:noindep}\end{aligned}$$ by making ${{{\boldsymbol{\mathbf{s}}}}}_t$ include as much information about $({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$ as needed for to hold (possibly ${{{\boldsymbol{\mathbf{s}}}}}_t=({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$).[^2] Then by defining $g_\theta$ as shown in , we get $p({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}) = \exp G_T$ and thus holds for each ${{{\boldsymbol{\mathbf{x}}}}}$.
Relationship to particle filtering {#sec:Hhat}
----------------------------------
Our method is spelled out in \[sec:sis\] (one may look now). It is a variant of the popular *particle filtering* method that tracks the state of a physical system in discrete time [@ristic2004]. Our particular *proposal distribution* for $y_t$ can be found in \[eq:Hhat,eq:sbar,eq:c,eq:q-local\]. It considers not only past observations ${{{\boldsymbol{\mathbf{x}}}}}_{:t}$ as reflected in ${{{\boldsymbol{\mathbf{s}}}}}_{t-1}$, but also future observations ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$, as summarized by the state ${{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t$ of a right-to-left recurrent neural network $\bar{f}$ that we will train: $$\begin{aligned}
\hat{H}_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}h_\phi({{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_{t+1},x_{t+1}) + \hat{H}_{t+1}\label{eq:Hhat} \\
{{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\bar{f}_\phi({{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_{t+1},x_{t+1}) &\!\!\!\!\!\!\!\!\text{(with ${{{\boldsymbol{\mathbf{s}}}}}_T$ given)}\label{eq:sbar}\end{aligned}$$ Conditioning the distribution of $y_t$ on future observations ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$ means that we are doing “smoothing” rather than “filtering” (in signal processing terminology). Doing so can reduce the bias and variance of our sampler. It is possible so long as ${{{\boldsymbol{\mathbf{x}}}}}$ is provided in its entirety before the sampler runs—which is often the case in NLP.
{width=".93\textwidth"}
Applications {#sec:apps}
------------
Why sample from $p_\theta$ at all? Many NLP systems instead simply search for the *Viterbi sequence* ${{{\boldsymbol{\mathbf{y}}}}}$ that maximizes $G_T$ and thus maximizes $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. If the space of states ${{{\boldsymbol{\mathbf{s}}}}}$ is small, this can be done efficiently by dynamic programming [@viterbi-1967]; if not, then $A^*$ may be an option (see \[sec:astar\]). More common is to use an approximate method: beam search, or perhaps a sequential prediction policy trained with reinforcement learning. Past work has already shown how to improve these approximate search algorithms by conditioning on the future [@bahdanau2016; @wiseman2016].
Sampling is essentially a generalization of maximization: sampling from $\exp \frac{G_T}{\mathrm{temperature}}$ approaches maximization as $\mathrm{temperature} \rightarrow 0$. It is a fundamental building block for other algorithms, as it can be used to take expectations over the whole space of possible ${{{\boldsymbol{\mathbf{y}}}}}$ values. For unfamiliar readers, \[app:apps\] reviews how sampling is crucially used in minimum-risk decoding, supervised training, unsupervised training, imputation of missing data, pipeline decoding, and inference in graphical models.=-1
Exact Sequential Sampling {#sec:exact}
=========================
\[sec:astar\]
To develop our method, it is useful to first consider exact samplers. Exact sampling is tractable for only some of the models allowed by \[sec:setup\]. However, the form and notation of the exact algorithms in \[sec:exact\] will guide our development of approximations in \[sec:approx\].
An *exact sequential sampler* draws $y_t$ from $p_\theta(y_t \mid {{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}_{:t-1})$ for each $t = 1, {} \ldots, T$ in sequence. Then ${{{\boldsymbol{\mathbf{y}}}}}$ is exactly distributed [as]{} $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$.
For each given ${{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}}_{:t-1}$, observe that $$\begin{aligned}
\makebox[1.5em][l]{$\displaystyle p_\theta(y_t \mid {{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}_{:t-1})$} \label{eq:q-exact} \\
& \propto p_\theta({{{\boldsymbol{\mathbf{y}}}}}_{:t} \mid {{{\boldsymbol{\mathbf{x}}}}})
= \textstyle\sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) \\
\quad
& \propto \textstyle \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \exp G_T \label{eq:GplusH} \\
&= \exp\;(G_t + \underbrace{\log \textstyle \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \!\exp\;(G_T - G_t)}_{\text{call this $H_t$}})\!\! \label{eq:H} \\
&= \exp\; (G_{t-1} + g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) + H_t) \\
&\propto \exp\;(g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) + H_t) \label{eq:q-exact-unnorm}\end{aligned}$$ Thus, we can easily construct the needed distribution by normalizing over all possible values of $y_t$. The challenging part of is to compute $H_t$: as defined in , $H_t$ involves a sum over exponentially many futures ${{{\boldsymbol{\mathbf{y}}}}}_{t:}$. (See \[fig:quantities\].)
We chose the symbols $G$ and $H$ in homage to the $A^*$ search algorithm [@ASTAR-1968]. In that algorithm (which could be used to find the Viterbi sequence), $g$ denotes the score-so-far of a partial solution ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$, and $h$ denotes the optimal score-to-go. Thus, $g+h$ would be the score of the *best* sequence with prefix ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$. Analogously, our $G_t+H_t$ is the log of the total exponentiated scores of *all* sequences with prefix ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$. $G_t$ and $H_t$ might be called the *logprob-so-far* and *logprob-to-go* of ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$.
Just as $A^*$ approximates $h$ with a “heuristic” $\hat{h}$, the next section will approximate $H_t$ using a neural estimate $\hat{H}_t$ (). However, the specific form of our approximation is inspired by cases where $H_t$ can be computed exactly. We consider those in the remainder of this section.
Exact sampling from HMMs {#sec:hmm}
------------------------
A *hidden Markov model* (HMM) specifies a normalized *joint* distribution $p_\theta({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}) = \exp G_T$ over state sequence ${{{\boldsymbol{\mathbf{y}}}}}$ and observation sequence ${{{\boldsymbol{\mathbf{x}}}}}$,[^3] Thus the posterior $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ is proportional to $\exp G_T$, as required by \[eq:condit\].
The HMM specifically defines $G_T$ by with ${{{\boldsymbol{\mathbf{s}}}}}_t = y_t$ and $g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) = \log p_\theta(y_t \mid y_{t-1}) + \log p_\theta(x_t \mid y_t)$.[^4]
In this setting, $H_t$ can be computed exactly by the *backward algorithm* [@rabiner-1989]. (Details are given in \[app:hmm-backward\] for completeness.)
Exact sampling from OOHMMs {#sec:oohmm}
--------------------------
For sequence tagging, a weakness of (first-order) HMMs is that the model state ${{{\boldsymbol{\mathbf{s}}}}}_t = y_t$ may contain little information: only the most recent tag $y_t$ is remembered, so the number of possible model states ${{{\boldsymbol{\mathbf{s}}}}}_t$ is limited by the vocabulary of output tags.
We may generalize so that the data generating process is in a latent state $u_t \in \{1,\ldots,k\}$ at each time $t$, and the observed $y_t$—along with $x_t$—is generated from $u_t$. Now $k$ may be arbitrarily large. The model has the form $$\begin{aligned}
p_\theta({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}) &= \exp G_T \label{eq:oohmm} \\
& = \sum_{{{{\boldsymbol{\mathbf{u}}}}}} \prod_{t=1}^T p_\theta(u_t \mid u_{t-1}) \cdot p_\theta(x_t,y_t \mid u_t) \nonumber\end{aligned}$$ This is essentially a pair HMM [@knudsen2003] without insertions or deletions, also known as an “$\epsilon$-free” or “same-length” probabilistic finite-state transducer. We refer to it here as an *output-output HMM* (OOHMM).[^5]
Is this still an example of the general model architecture from \[sec:setup\]? Yes. Since $u_t$ is latent and evolves stochastically, it cannot be used as the state ${{{\boldsymbol{\mathbf{s}}}}}_t$ in or . However, we [*can*]{} define ${{{\boldsymbol{\mathbf{s}}}}}_t$ to be the model’s *belief state* after observing $({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$. The belief state is the posterior probability distribution over the underlying state $u_t$ of the system. That is, ${{{\boldsymbol{\mathbf{s}}}}}_t$ deterministically keeps track of all possible states that the OOHMM might be in—just as the state of a determinized FSA keeps track of all possible states that the original nondeterministic FSA might be in.
We may compute the belief state in terms of a vector of *forward probabilities* that starts at ${{\boldsymbol{\mathbf{\alpha}}}}_0$, $$\begin{aligned}
({{\boldsymbol{\mathbf{\alpha}}}}_0)_u &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\begin{cases}
1 & \text{if }u={\textsc{bos}\xspace}\text{ (see \cref{fn:bos})} \\
0 & \text{if }u=\text{any other state}
\end{cases}
\raisetag{16pt}\end{aligned}$$ and is updated deterministically for each $0 < t \leq T$ by the *forward algorithm* [@rabiner-1989]: $$\begin{aligned}
({{\boldsymbol{\mathbf{\alpha}}}}_t)_u &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\sum_{u'=1}^k ({{\boldsymbol{\mathbf{\alpha}}}}_{t-1})_{u'} \cdot p_\theta(u \mid u') \cdot p_\theta(x_t,y_t \mid u) \raisetag{10pt} \label{eq:forward}\end{aligned}$$
$({{\boldsymbol{\mathbf{\alpha}}}}_t)_u$ can be interpreted as the logprob-so-far *if* the system is in state $u$ after observing $({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$. We may express the update rule by ${{\boldsymbol{\mathbf{\alpha}}}}_t^\top = {{\boldsymbol{\mathbf{\alpha}}}}_{t-1}^\top P$ where the matrix $P$ depends on $(x_t,y_t)$, namely $P_{u'u} {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}p_\theta(u \mid u') \cdot p_\theta(x_t,y_t \mid u)$.
The belief state ${{{\boldsymbol{\mathbf{s}}}}}_t {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}{\llbracket {{\boldsymbol{\mathbf{\alpha}}}}_t \rrbracket} \in {\mathbb{R}}^k$ simply normalizes ${{\boldsymbol{\mathbf{\alpha}}}}_t$ into a probability vector, where ${\llbracket {\boldsymbol{\mathbf{u}}} \rrbracket} {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}{\boldsymbol{\mathbf{u}}}/({\boldsymbol{\mathbf{u}}}^\top {\mathbf{1}})$ denotes the *normalization operator*. The state update now takes the form as desired, with $f_\theta$ a normalized vector-matrix product: $$\begin{aligned}
{{{\boldsymbol{\mathbf{s}}}}}_t^\top &= f_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}{\llbracket {{{\boldsymbol{\mathbf{s}}}}}_{t-1}^\top P \rrbracket} \label{eq:s-update}\end{aligned}$$
As in the HMM case, we define $G_t$ as the log of the generative prefix probability, $$\begin{aligned}
G_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\log p_\theta({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})
= \log \textstyle \sum_u ({{\boldsymbol{\mathbf{\alpha}}}}_t)_u \label{eq:sum-alpha}\end{aligned}$$ which has the form as desired if we put $$\begin{aligned}
g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}G_t - G_{t-1} \label{eq:g} \\
&= \log \frac{{{\boldsymbol{\mathbf{\alpha}}}}_{t-1}^\top P {\mathbf{1}}}{{{\boldsymbol{\mathbf{\alpha}}}}_{t-1}^\top{\mathbf{1}}}
= \log \; ({{{\boldsymbol{\mathbf{s}}}}}_{t-1}^\top P {\mathbf{1}}) \nonumber\end{aligned}$$
Again, exact sampling is possible. It suffices to compute . For the OOHMM, this is given by $$\textstyle \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \exp G_T = {{\boldsymbol{\mathbf{\alpha}}}}_t^\top {{\boldsymbol{\mathbf{\beta}}}}_t \label{eq:alpha-beta}$$ where ${{\boldsymbol{\mathbf{\beta}}}}_T{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}{\mathbf{1}}$ and the *backward algorithm* $$\begin{aligned}
({{\boldsymbol{\mathbf{\beta}}}}_t)_v \label{eq:beta-oohmm}
&{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}p_\theta({{{\boldsymbol{\mathbf{x}}}}}_{t:} \mid u_t=u) \\
&=\sum_{{{{\boldsymbol{\mathbf{u}}}}}_{t:},{{{\boldsymbol{\mathbf{y}}}}}_{t:}} p_\theta({{{\boldsymbol{\mathbf{u}}}}}_{t:},{{{\boldsymbol{\mathbf{x}}}}}_{t:},{{{\boldsymbol{\mathbf{y}}}}}_{t:} \mid u_t=u) \nonumber \\
&= \sum_{u'} \underbrace{p_\theta(u' \mid u) \cdot p(x_{t+1} \mid
u')}_{\text{call this }{{{\mkern 1.5mu\overline{\mkern-1.5muP\mkern-1.5mu}\mkern 1.5mu}}}_{uu'}} \cdot ({{\boldsymbol{\mathbf{\beta}}}}_{t+1})_{u'} \nonumber\end{aligned}$$ for $0 \leq t < T$ uses dynamic programming to find the total probability of all ways to generate the future observations ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$. Note that ${{\boldsymbol{\mathbf{\alpha}}}}_t$ is defined for a *specific prefix* ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$ (though it sums over all ${{{\boldsymbol{\mathbf{u}}}}}_{:t}$), whereas ${{\boldsymbol{\mathbf{\beta}}}}_t$ sums over *all suffixes* ${{{\boldsymbol{\mathbf{y}}}}}_{t:}$ (and over all ${{{\boldsymbol{\mathbf{u}}}}}_{t:}$), to achieve the asymmetric summation in .
Define ${{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}{\llbracket {{\boldsymbol{\mathbf{\beta}}}}_t \rrbracket} \in {\mathbb{R}}^k$ to be a normalized version of ${{\boldsymbol{\mathbf{\beta}}}}_t$. The ${{\boldsymbol{\mathbf{\beta}}}}_t$ recurrence can clearly be expressed in the form ${{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t =
{\llbracket {{{\mkern 1.5mu\overline{\mkern-1.5muP\mkern-1.5mu}\mkern 1.5mu}}}{{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_{t+1} \rrbracket}$, much like .=-1
The logprob-to-go for OOHMMs
----------------------------
Let us now work out the definition of $H_t$ for OOHMMs (cf. \[eq:H-as-diff\] in \[app:hmm-backward\] for HMMs). We will write it in terms of $\hat{H}_t$ from \[sec:Hhat\]. Let us define $\hat{H}_t$ symmetrically to $G_t$ (see ): $$\begin{aligned}
\hat{H}_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\log \sum_u ({{\boldsymbol{\mathbf{\beta}}}}_t)_u \;\;(= \log {\mathbf{1}}^\top {{\boldsymbol{\mathbf{\beta}}}}_t) \label{eq:sum-beta}\end{aligned}$$ which has the form as desired if we put $$\begin{aligned}
h_\phi({{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_{t+1},x_{t+1}) &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\hat{H}_t - \hat{H}_{t+1} = \log\;( {\mathbf{1}}^\top {{{\mkern 1.5mu\overline{\mkern-1.5muP\mkern-1.5mu}\mkern 1.5mu}}}{{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_{t+1}) \label{eq:h}\end{aligned}$$ From \[eq:H,eq:alpha-beta,eq:sum-alpha,eq:sum-beta\], we see $$\begin{aligned}
H_t &= \log \big( \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \exp G_T \big) - G_t \nonumber \\
&= \log \frac{{{\boldsymbol{\mathbf{\alpha}}}}_t^\top {{\boldsymbol{\mathbf{\beta}}}}_t}{({{\boldsymbol{\mathbf{\alpha}}}}_t^\top {\mathbf{1}})({\mathbf{1}}^\top {{\boldsymbol{\mathbf{\beta}}}}_t)} + \log\;({\mathbf{1}}^\top {{\boldsymbol{\mathbf{\beta}}}}_t) \nonumber \\
&= \underbrace{\log {{{\boldsymbol{\mathbf{s}}}}}_t^\top {{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t}_{\text{call this $C_t$}} + \hat{H}_t \label{eq:C}\end{aligned}$$ where $C_t \in {\mathbb{R}}$ can be regarded as evaluating the *compatibility* of the state distributions ${{{\boldsymbol{\mathbf{s}}}}}_t$ and ${{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t$.
In short, the generic strategy for exact sampling says that for an OOHMM, $y_t$ is distributed as=-1 $$\begin{aligned}
\lefteqn{p_\theta(y_t \mid {{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}}_{:t-1})
\propto \exp\; (g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) + H_t)} \nonumber \\
&\propto
\exp\; (\underbrace{g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t)}_{\ \ \text{depends on }{{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t}\ \ }
+ \underbrace{C_t}_{\text{on }{{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}_{:t}}
+ \underbrace{\hat{H}_t}_{\text{on }{{{\boldsymbol{\mathbf{x}}}}}_{t:}}
) \nonumber \\
&\propto \exp\; (g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) + C_t) \label{eq:q-exact-oohmm}\end{aligned}$$ This is equivalent to choosing $y_t$ in proportion to —but we now turn to settings where it is infeasible to compute exactly. There we will use the formulation but approximate $C_t$. For completeness, we will also consider how to approximate $\hat{H}_t$, which dropped out of the above distribution (because it was the same for all choices of $y_t$) but may be useful for other algorithms (see \[sec:sis\]).
Neural Modeling as Approximation {#sec:approx}
================================
Models with large state spaces
------------------------------
The expressivity of an OOHMM is limited by the number of states $k$. The state $u_t \in \{1,\ldots,k\}$ is a bottleneck between the past $({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$ and the future $({{{\boldsymbol{\mathbf{x}}}}}_{t:},{{{\boldsymbol{\mathbf{y}}}}}_{t:})$, in that past and future are *conditionally independent* given $u_t$. Thus, the mutual information between past and future is at most $\log_2 k$ bits.
In many NLP domains, however, the past seems to carry substantial information about the future. The first half of a sentence greatly reduces the uncertainly about the second half, by providing information about topics, referents, syntax, semantics, and discourse. This suggests that an accurate HMM language model $p({{{\boldsymbol{\mathbf{x}}}}})$ would require *very large $k$*—as would a generative OOHMM model $p({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$ of *annotated* language. The situation is perhaps better for discriminative models $p({{{\boldsymbol{\mathbf{y}}}}}\mid{{{\boldsymbol{\mathbf{x}}}}})$, since much of the information for predicting ${{{\boldsymbol{\mathbf{y}}}}}_{t:}$ might be available in ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$. Still, it is important to let $({{{\boldsymbol{\mathbf{x}}}}}_{:t},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$ contribute enough additional information about ${{{\boldsymbol{\mathbf{y}}}}}_{t:}$: even for short strings, making $k$ too small (giving $\leq \log_2 k$ bits) may harm prediction [@dreyer-smith-eisner-2008].
Of course, says that an OOHMM can express any joint distribution for which the mutual information is finite,[^6] by taking $k$ large enough for $v_{t-1}$ to capture the relevant info from $({{{\boldsymbol{\mathbf{x}}}}}_{:t-1},{{{\boldsymbol{\mathbf{y}}}}}_{:t-1})$.
So why not just take $k$ to be large—say, $k=2^{30}$ to allow 30 bits of information? Unfortunately, evaluating $G_T$ then becomes very expensive—both computationally and statistically. As we have seen, if we define ${{{\boldsymbol{\mathbf{s}}}}}_t$ to be the belief state ${\llbracket {{\boldsymbol{\mathbf{\alpha}}}}_t \rrbracket} \in {\mathbb{R}}^k$, updating it at each observation $(x_t,y_t)$ (\[eq:s\]) requires multiplication by a $k \times k$ matrix $P$. This takes time $O(k^2)$, and requires enough data to learn $O(k^2)$ transition probabilities.
Neural approximation of the model {#sec:neural-model}
---------------------------------
\[sec:globally\]
As a solution, we might hope that for the inputs ${{{\boldsymbol{\mathbf{x}}}}}$ observed in practice, the very high-dimensional belief states ${\llbracket {{\boldsymbol{\mathbf{\alpha}}}}_t \rrbracket} \in {\mathbb{R}}^k$ might tend to lie near a $d$-dimensional manifold where $d \ll k$. Then we could take ${{{\boldsymbol{\mathbf{s}}}}}_t$ to be a vector in ${\mathbb{R}}^d$ that compactly encodes the approximate coordinates of ${\llbracket {{\boldsymbol{\mathbf{\alpha}}}}_t \rrbracket}$ relative to the manifold: ${{{\boldsymbol{\mathbf{s}}}}}_t = \nu({\llbracket {{\boldsymbol{\mathbf{\alpha}}}}_t \rrbracket})$, where $\nu$ is the encoder.
In this new, nonlinearly warped coordinate system, the functions of ${{{\boldsymbol{\mathbf{s}}}}}_{t-1}$ in – are no longer the simple, essentially linear functions given by and . They become nonlinear functions operating on the manifold coordinates. ($f_\theta$ in should now ensure that ${{{\boldsymbol{\mathbf{s}}}}}_t^\top \approx \nu({\llbracket (\nu^{-1}({{{\boldsymbol{\mathbf{s}}}}}_{t-1}))^\top P \rrbracket})$, and $g_\theta$ in should now estimate $\log\; (\nu^{-1}({{{\boldsymbol{\mathbf{s}}}}}_{t-1}))^\top P {\mathbf{1}}$.) In a sense, this is the reverse of the “kernel trick” [@boser1992] that converts a low-dimensional nonlinear function to a high-dimensional linear one.
Our hope is that ${{{\boldsymbol{\mathbf{s}}}}}_t$ has enough dimensions $d \ll k$ to capture the useful information from the true ${\llbracket {{\boldsymbol{\mathbf{\alpha}}}}_t \rrbracket}$, **and** that $\theta$ has enough dimensions $\ll k^2$ to capture most of the dynamics of \[eq:s-update,eq:g\]. We thus proceed to fit the neural networks $f_\theta, g_\theta$ directly to the data, [*without ever knowing*]{} the true $k$ or the structure of the original operators $P \in {\mathbb{R}}^{k\times k}$.
We regard this as the implicit justification for various published probabilistic sequence models $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ that incorporate neural networks. These models usually have the form of \[sec:setup\]. Most simply, $(f_\theta,g_\theta)$ can be instantiated as one time step in an RNN [@aharoni2016], but it is common to use enriched versions such as deep LSTMs. It is also common to have the state ${{{\boldsymbol{\mathbf{s}}}}}_t$ contain not only a vector of manifold coordinates in ${\mathbb{R}}^d$ but also some unboundedly large representation of $({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}_{:t})$ (cf. \[eq:noindep\]), so the $f_\theta$ neural network can refer to this material with an attentional [@bahdanau2014] or stack mechanism [@dyer2015].
A few such papers have used *globally* normalized conditional models that can be viewed as approximating some OOHMM, e.g., the parsers of and . That is the case (\[sec:setup\]) that particle smoothing aims to support. Most papers are *locally* normalized conditional models [e.g., @kann2016; @aharoni2016]; these simplify supervised training and can be viewed as approximating IOHMMs (\[fn:iohmm\]). For locally normalized models, $H_t=0$ by construction, in which case particle filtering (which estimates $H_t=0$) is just as good as particle smoothing. Particle filtering is still useful for these models, but lookahead’s inability to help them is an expressive limitation (known as *label bias*) of locally normalized models. We hope the existence of particle smoothing (which learns an estimate $H_t$) will make it easier to adopt, train, and decode globally normalized models, as discussed in \[sec:apps\]. =-1
Neural approximation of logprob-to-go {#sec:q}
-------------------------------------
We can adopt the same neuralization trick to approximate the OOHMM’s logprob-to-go $H_t = C_t + \hat{H}_t$. We take ${{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t \in {\mathbb{R}}^d$ on the same theory that it is a low-dimensional reparameterization of ${\llbracket \beta_t \rrbracket}$, and define $(\bar{f}_\phi,h_\phi)$ in to be neural networks. Finally, we must replace the definition of $C_t$ in with another neural network $c_\phi$ that works on the low-dimensional approximations:[^7] $$\begin{aligned}
C_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}c_\phi({{{\boldsymbol{\mathbf{s}}}}}_t,{{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t) &\text{(except that $C_T{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}0$)}\label{eq:c}\end{aligned}$$ The resulting approximation to (which does not actually require $h_\phi$) will be denoted $q_{\theta,\phi}$: $$\begin{aligned}
\label{eq:q-local}
q_{\theta,\phi}(y_t \mid {{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}}_{:t-1}) {\mathrel{\stackrel{\mbox{\tiny def}}{\propto}}}\exp\; (g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t) + C_t)\end{aligned}$$
The neural networks in the present section are all parameterized by $\phi$, and are intended to produce an estimate of the logprob-*to-go* $H_t$—a function of ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$, which sums over all possible ${{{\boldsymbol{\mathbf{y}}}}}_{t:}$.
By contrast, the OOHMM-inspired neural networks suggested in \[sec:neural-model\] were used to specify an actual model of the logprob-*so-far* $G_t$—a function of ${{{\boldsymbol{\mathbf{x}}}}}_{:t}$ and ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$—using separate parameters $\theta$.=-1
Arguably $\phi$ has a harder modeling job than $\theta$ because it must implicitly sum over possible futures ${{{\boldsymbol{\mathbf{y}}}}}_{t:}$. We now consider how to get corrected samples from $q_{\theta,\phi}$ even if $\phi$ gives poor estimates of $H_t$, and then how to train $\phi$ to improve those estimates.
Particle smoothing {#sec:sis}
==================
In this paper, we assume nothing about the given model $G_T$ except that it is given in the form of (including the parameter vector $\theta$).=-1
Suppose we run the exact sampling strategy but approximate $p_\theta$ in with a *proposal distribution* $q_{\theta,\phi}$ of the form in –. Suppressing the subscripts on $p$ and $q$ for brevity, this means we are effectively drawing ${{{\boldsymbol{\mathbf{y}}}}}$ not from $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ but from $$\begin{aligned}
\label{eq:q}
q({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) &= \prod_{t=1}^T q(y_t \mid {{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}}_{:t-1})\end{aligned}$$ If $C_t \approx H_t + \text{const}$ within each $y_t$ draw, then $q \approx p$.
*Normalized importance sampling* corrects (mostly) for the approximation by drawing *many* sequences ${{{\boldsymbol{\mathbf{y}}}}}{^{(1)}}, \ldots {{{\boldsymbol{\mathbf{y}}}}}{^{(M)}}$ IID from and assigning ${{{\boldsymbol{\mathbf{y}}}}}{^{(m)}}$ a relative *weight* of $w{^{(m)}} {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\frac{p({{{\boldsymbol{\mathbf{y}}}}}{^{(m)}} \mid {{{\boldsymbol{\mathbf{x}}}}})}{q({{{\boldsymbol{\mathbf{y}}}}}{^{(m)}} \mid {{{\boldsymbol{\mathbf{x}}}}})}$. This *ensemble of weighted particles* yields a distribution $$\begin{aligned}
{{\hat{p}}}({{{\boldsymbol{\mathbf{y}}}}}) &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\tfrac{\sum_{m=1}^{M} w{^{(m)}} \mathbb{I}({{{\boldsymbol{\mathbf{y}}}}}= {{{\boldsymbol{\mathbf{y}}}}}{^{(m)}})}{\sum_{m=1}^{M} w{^{(m)}}} \approx p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})
\label{eq:phat}\end{aligned}$$ that can be used as discussed in \[sec:apps\]. To compute $w{^{(m)}}$ in practice, we replace the numerator $p({{{\boldsymbol{\mathbf{y}}}}}{^{(m)}} \mid {{{\boldsymbol{\mathbf{x}}}}})$ by the unnormalized version $\exp G_T$, which gives the same ${{\hat{p}}}$. Recall that each $G_T$ is a sum $\sum_{t=1}^T g_\theta(\cdots)$.
*Sequential importance sampling* is an equivalent implementation that makes $t$ the *outer* loop and $m$ the *inner* loop. It computes a *prefix ensemble* $$\begin{aligned}
Y_t &{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\{({{{\boldsymbol{\mathbf{y}}}}}_{:t}{^{(1)}},w_t{^{(1)}}),\ldots,({{{\boldsymbol{\mathbf{y}}}}}_{:t}{^{(M)}},w_t{^{(M)}})\}\end{aligned}$$ for each $0 \leq t \leq T$ in sequence. Initially, $({{{\boldsymbol{\mathbf{y}}}}}_{:0}{^{(m)}},w_0{^{(m)}}) = (\epsilon,\exp C_0)$ for all $m$. Then for $0 < t \leq T$, we extend these particles in parallel: $$\begin{aligned}
{{{\boldsymbol{\mathbf{y}}}}}_{:t}{^{(m)}} &= {{{\boldsymbol{\mathbf{y}}}}}_{:t-1}{^{(m)}} y_t{^{(m)}} \hspace{13mm}\text{(concatenation)}\\
w_t{^{(m)}} &= w_{t-1}{^{(m)}} \; \tfrac{\exp\; (g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t)\,+\,C_t\,-\,C_{t-1})}{q(y_t \mid {{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}}_{:t-1})} \label{eq:sis}\end{aligned}$$ where each $y_t{^{(m)}}$ is drawn from . Each $Y_t$ yields a distribution ${{\hat{p}}}_t$ over prefixes ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$, which estimates the distribution $p_t({{{\boldsymbol{\mathbf{y}}}}}_{:t}) {\mathrel{\stackrel{\mbox{\tiny def}}{\propto}}}\exp\;(G_t+C_t)$. We return ${{\hat{p}}}{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}{{\hat{p}}}_T \approx p_T = p$. This gives the same ${{\hat{p}}}$ as in : the final ${{{\boldsymbol{\mathbf{y}}}}}_T{^{(m)}}$ are the same, with the same final weights $w_T{^{(m)}} = \frac{\exp G_T}{q({{{\boldsymbol{\mathbf{y}}}}}{^{(m)}} \mid {{{\boldsymbol{\mathbf{x}}}}})}$, where $G_T$ was now summed up as $C_0 + \sum_{t=1}^T g_\theta(\cdots) + C_t - C_{t-1}$.
That is our basic *particle smoothing* strategy. If we use the naive approximation $C_t=0$ everywhere, it reduces to *particle filtering*. In either case, various well-studied improvements become available, such as various resampling schemes [@douc2005] and the particle cascade [@paige2014].[^8]
An easy improvement is *multinomial resampling*. After computing each ${{\hat{p}}}_t$, this replaces $Y_t$ with a set of $M$ new draws from ${{\hat{p}}}_t$ $(\approx p_t)$, each of weight 1—which tends to drop low-weight particles and duplicate high-weight ones.[^9] For this to usefully focus the ensemble on good prefixes ${{{\boldsymbol{\mathbf{y}}}}}_{:t}$, $p_t$ should be a good approximation to the true marginal $p({{{\boldsymbol{\mathbf{y}}}}}_{:t} \mid {{{\boldsymbol{\mathbf{x}}}}}) \propto \exp\; (G_t+H_t)$ from . That is why we arranged for $p_t({{{\boldsymbol{\mathbf{y}}}}}_{:t}) \propto \exp\;(G_t+C_t)$. Without $C_t$, we would have only $p_t({{{\boldsymbol{\mathbf{y}}}}}_{:t}) \propto \exp G_t$—which is fine for the traditional particle filtering setting, but in our setting it ignores future information in ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$ (which we have assumed is available) and also favors sequences ${{{\boldsymbol{\mathbf{y}}}}}$ that happen to accumulate most of their global score $G_T$ early rather than late (which is possible when the globally normalized model – is *not* factored in the generative form ). =-1
Training the Sampler Heuristic {#sec:optimization}
==============================
We now consider training the parameters $\phi$ of our sampler. These parameters determine the updates $\bar{f}_\phi$ in and the compatibility function $c_\phi$ in . As a result, they determine the proposal distribution $q$ used in \[eq:q,eq:sis\], and thus determine the stochastic choice of ${{\hat{p}}}$ that is returned by the sampler on a given input ${{{\boldsymbol{\mathbf{x}}}}}$.
In this paper, we simply try to tune $\phi$ to yield good proposals. Specifically, we try to ensure that $q_\phi({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ in \[eq:q\] is close to $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ from \[eq:condit\]. While this may not be necessary for the sampler to perform well downstream,[^10] it does guarantee it (assuming that the model $p$ is correct). Specifically, we seek to minimize $$(1 - \lambda) \text{KL}(p || q_\phi) + \lambda \text{KL}(q_\phi || p)\ \ \ \text{(with $\lambda \in [0,1]$)}$$ averaged over examples ${{{\boldsymbol{\mathbf{x}}}}}$ drawn from a training set.[^11] (The training set need not provide true ${{{\boldsymbol{\mathbf{y}}}}}$’s.)
The [*inclusive KL divergence*]{} $\text{KL} \left( p || q_{\phi} \right)$ is an expectation under $p$. We estimate it by replacing $p$ with a sample ${{\hat{p}}}$, which in practice we can obtain with our sampler under the current $\phi$. (The danger, then, is that ${{\hat{p}}}$ will be biased when $\phi$ is not yet well-trained; this can be mitigated by increasing the sample size $M$ when drawing ${{\hat{p}}}$ for training purposes.)
Intuitively, this term tries to encourage $q_\phi$ in future to re-propose those ${{{\boldsymbol{\mathbf{y}}}}}$ values that turned out to be “good” and survived into ${{\hat{p}}}$ with high weights.
The [*exclusive KL divergence*]{} $\text{KL}(q_\phi || p)$ is an expectation under $q_\phi$. Since we can sample from $q_\phi$ exactly, we can get an unbiased estimate of $\nabla_\phi \text{KL}(q_\phi || p)$ with the likelihood ratio trick [@glynn1990].[^12] (The danger is that such “REINFORCE” methods tend to suffer from very high variance.)
This term is a popular objective for variational approximation. Here, it tries to discourage $q_\phi$ from re-proposing “bad” ${{{\boldsymbol{\mathbf{y}}}}}$ values that turned out to have low $\exp G_T$ relative to their proposal probability.
Our experiments balance “recall” (inclusive) and “precision” (exclusive) by taking $\lambda = \frac{1}{2}$ (which compares to $\lambda \in \{0,1\})$. Alas, because of our approximation to the inclusive term, neither term’s gradient will “find” and directly encourage good ${{{\boldsymbol{\mathbf{y}}}}}$ values that have never been proposed. gives further discussion and formulas.
Models for the Experiments {#sec:model}
==========================
To evaluate our methods, we needed pre-trained models $p_\theta$. We experimented on several models. In each case, we trained a [*generative*]{} model $p_\theta({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$, so that we could try sampling from its posterior distribution $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. This is a very common setting where particle smoothing should be able to help. Details for replication are given in \[app:model-details\].
Tagging models {#sec:tagging}
--------------
We can regard a tagged sentence $({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$ as a string over the “pair alphabet” $\mathcal{X} \times \mathcal{Y}$. We train an RNN language model over this “pair alphabet”—this is a neuralized OOHMM as suggested in \[sec:neural-model\]: $$\begin{aligned}
\log p_\theta({{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}}) &= \sum_{t=1}^{T} \log p_\theta(x_t,y_t \mid {{{\boldsymbol{\mathbf{s}}}}}_{t-1})\end{aligned}$$
This model is locally normalized, so that $\log p_\theta({{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}})$ (as well as its gradient) is straightforward to compute for a given training pair $({{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}})$. Joint sampling from it would also be easy (\[sec:globally\]).
However, $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ is globally renormalized (by an unknown partition function that depends on ${{{\boldsymbol{\mathbf{x}}}}}$, namely $\exp H_0$). Conditional sampling of ${{{\boldsymbol{\mathbf{y}}}}}$ is therefore potentially hard. Choosing $y_t$ optimally requires knowledge of $H_t$, which depends on the future ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$.
As we noted in \[sec:intro\], many NLP tasks can be seen as tagging problems. In this paper we experiment with two such tasks: **English stressed syllable tagging**, where the stress of a syllable often depends on the number of remaining syllables,[^13] providing good reason to use the *lookahead* provided by particle smoothing; and **Chinese NER**, which is a familiar textbook application and reminds the reader that our formal setup (tagging) provides enough machinery to treat other tasks (chunking).
#### English stressed syllable tagging
This task tags a sequence of phonemes ${{{\boldsymbol{\mathbf{x}}}}}$, which form a word, with their stress markings ${{{\boldsymbol{\mathbf{y}}}}}$. Our training examples are the stressed words in the CMU pronunciation dictionary [@cmudict]. We test the sampler on held-out unstressed words.
#### Chinese social media NER
This task does named entity recognition in Chinese, by tagging the characters of a Chinese sentence in a way that marks the named entities. We use the dataset from @peng2015, whose tagging scheme is a variant of the BIO scheme mentioned in \[sec:intro\]. We test the sampler on held-out sentences.
String source separation {#sec:sourcesep}
------------------------
This is an artificial task that provides a discrete analogue of speech source separation [@zibulevsky2001]. The generative model is that $J$ strings (possibly of different lengths) are generated IID from an RNN language model, and are then combined into a single string ${{{\boldsymbol{\mathbf{x}}}}}$ according to a random [*interleaving*]{} string ${{{\boldsymbol{\mathbf{y}}}}}$.[^14] The posterior $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ predicts the interleaving string, which suffices to reconstruct the original strings. The interleaving string is selected from the uniform distribution over all possible interleavings (given the $J$ strings’ lengths). For example, with $J=2$, a possible generative story is that we first sample two strings and from an RNN language model. We then draw an interleaving string from the aforementioned uniform distribution, and interleave the $J$ strings deterministically to get .
$p({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$ is proportional to the product of the probabilities of the $J$ strings. The only parameters of $p_\theta$, then, are the parameters of the RNN language model, which we train on clean (non-interleaved) samples from a corpus. We test the sampler on random interleavings of held-out samples.
The state ${{{\boldsymbol{\mathbf{s}}}}}$ (which is provided as an input to $c_\theta$ in ) is the concatenation of the $J$ states of the language model as it independently generates the $J$ strings, and $g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y_t)$ is the log-probability of generating $x_t$ as the next character of the $y_t$^th^ string, given that string’s language model state within ${{{\boldsymbol{\mathbf{s}}}}}_{t-1}$. As a special case, ${{{\boldsymbol{\mathbf{x}}}}}_T={\textsc{eos}\xspace}$ (see \[fn:eos\]), and $g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{T-1},{\textsc{eos}\xspace},{\textsc{eos}\xspace})$ is the total log-probability of termination in all $J$ language model states.
String source separation has good reason for lookahead: appending character “o” to a reconstructed string “gh” is only advisable if “s” and “t” are coming up soon to make “ghost.” It also illustrates a powerful application setting—posterior inference under a generative model. This task conveniently allowed us to construct the generative model from a pre-trained language model. Our constructed generative model illustrates that the state ${{{\boldsymbol{\mathbf{s}}}}}$ and transition function $f$ can reflect interesting problem-specific structure.
#### CMU Pronunciation dictionary
The CMU pronunciation dictionary (already used above) provides sequences of phonemes. Here we use words no longer than $5$ phonemes. We interleave the (unstressed) phonemes of $J=5$ words.
#### Penn Treebank
The PTB corpus [@marcus1993] provides English sentences, from which we use only the sentences of length $\leq 8$. We interleave the words of $J=2$ sentences.
[.25]{} {width="97.00000%"}
[.25]{} {width="97.00000%"}
[.25]{} {width="97.00000%"}
[.25]{} {width="97.00000%"}
Experiments {#sec:exps}
===========
In our experiments, we are given a pre-trained scoring model $p_\theta$, and we train the parameters $\phi$ of a particle smoothing algorithm.[^15]
We now show that our proposed neural particle smoothing sampler does better than the particle filtering sampler. To define “better,” we evaluate samplers on the *offset KL divergence* from the true posterior.
Evaluation metrics {#sec:metrics}
------------------
Given , the “natural” goal of conditional sampling is for the sample distribution ${{\hat{p}}}({{{\boldsymbol{\mathbf{y}}}}})$ to approximate the true distribution $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) = \exp G_T / \exp H_0$ from . We will therefore report—averaged over all held-out test examples —the KL divergence $$\begin{aligned}
\text{KL}({{\hat{p}}}|| p) &= {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim {{\hat{p}}}}}\left[\log {{\hat{p}}}({{{\boldsymbol{\mathbf{y}}}}})\right]} \\ \nonumber
&\qquad - ({\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim {{\hat{p}}}}}\left[\log \tilde{p}({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})\right]} - \log Z({{{\boldsymbol{\mathbf{x}}}}})),\end{aligned}$$ where $\tilde{p}({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ denotes the [*unnormalized*]{} distribution given by $\exp G_T$ in , and $Z({{{\boldsymbol{\mathbf{x}}}}})$ denotes its normalizing constant, $\exp H_0 = \sum_{{{\boldsymbol{\mathbf{y}}}}}\tilde{p}({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$.
As we are unable to compute $\log Z({{{\boldsymbol{\mathbf{x}}}}})$ in practice, we replace it with an estimate $z({{{\boldsymbol{\mathbf{x}}}}})$ to obtain an *offset KL divergence*. This change of constant does not change the measured difference between two samplers, $\text{KL}({{\hat{p}}}_1 || p) - \text{KL}({{\hat{p}}}_2 || p)$. Nonetheless, we try to use a reasonable estimate so that the reported KL divergence is interpretable in an absolute sense. Specifically, we take $z({{{\boldsymbol{\mathbf{x}}}}}) = \log \sum_{{{{\boldsymbol{\mathbf{y}}}}}\in {\cal Y}} \tilde{p}({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) \leq \log Z$, where ${\cal Y}$ is the full set of distinct particles ${{{\boldsymbol{\mathbf{y}}}}}$ that we ever drew for input ${{{\boldsymbol{\mathbf{x}}}}}$, including samples from the beam search models, while constructing the experimental results graph.[^16] Thus, the offset KL divergence is a “best effort” lower bound on the true exclusive KL divergence $\text{KL}({{\hat{p}}}|| p)$.
Results
-------
In all experiments we compute the offset KL divergence for both the particle filtering samplers and the particle smoothing samplers, for varying ensemble sizes $M$. We also compare against a beam search baseline that keeps the highest-scoring $M$ particles at each step (scored by $\exp G_t$ with no lookahead). The results are in \[fig:div,fig:divcmu,fig:divweibo,fig:div-phone\].
Given a fixed ensemble size, we see the smoothing sampler consistently performs better than the filtering counterpart. It often achieves comparable performance at a fraction of the ensemble size.=-1
Beam search on the other hand falls behind on three tasks: stress prediction and the two source separation tasks. It does perform better than the stochastic methods on the Chinese NER task, but only at small beam sizes. Varying the beam size barely affects performance at all, across all tasks. This suggests that beam search is unable to explore the hypothesis space well.
We experiment with resampling for both the particle filtering sampler and our smoothing sampler. In source separation and stressed syllable prediction, where the right context contains critical information about how viable a particle is, resampling helps particle filtering [*almost*]{} catch up to particle smoothing. Particle smoothing itself is not further improved by resampling, presumably because its effective sample size is high. The goal of resampling is to kill off low-weight particles (which were overproposed) and reallocate their resources to higher-weight ones. But with particle smoothing, there are fewer low-weight particles, so the benefit of resampling may be outweighted by its cost (namely, increased variance).
Related Work {#sec:related}
============
Much previous work has employed sequential importance sampling for approximate inference of intractable distributions [e.g., @thrun99; @andrews2017]. Some of this work learns adaptive proposal distributions in this setting [e.g. @gu2015; @paige2016]. The key difference in our work is that we consider future inputs, which is impossible in online decision settings such as robotics. @klaas2006 did do particle smoothing, like us, but they did not learn adaptive proposal distributions.
Just as we use a right-to-left RNN to guide [*posterior sampling*]{} of a left-to-right generative model, @krishnan2016 employed a right-to-left RNN to guide [*posterior marginal inference*]{} in the same sort of model. @serdyuk2017 used a right-to-left RNN to regularize training of such a model.
Conclusion
==========
We have described neural particle smoothing, a sequential Monte Carlo method for approximate sampling from the posterior of incremental neural scoring models. Sequential importance sampling has arguably been underused in the natural language processing community. It is quite a plausible strategy for dealing with rich, globally normalized probability models such as neural models—particularly if a good sequential proposal distribution can be found. Our contribution is a neural proposal distribution, which goes beyond particle filtering in that it uses a right-to-left recurrent neural network to “look ahead” to future symbols of ${{{\boldsymbol{\mathbf{x}}}}}$ when proposing each symbol $y_t$. The form of our distribution is well-motivated.
There are many possible extensions to the work in this paper. For example, we can learn the generative model and proposal distribution jointly; we can also infuse them with hand-crafted structure, or use more deeply stacked architectures; and we can try training the proposal distribution end-to-end (\[fn:end2end\]). Another possible extension would be to allow each step of $q$ to propose a *sequence* of actions, effectively making the tagset size $\infty$. This extension relaxes our $|{{{\boldsymbol{\mathbf{y}}}}}|=|{{{\boldsymbol{\mathbf{x}}}}}|$ restriction from \[sec:intro\] and would allow us to do general sequence-to-sequence transduction.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been generously supported by a Google Faculty Research Award and by Grant No. 1718846 from the National Science Foundation.
The logprob-to-go for HMMs {#app:hmm-backward}
==========================
As noted in \[sec:hmm\], the logprob-to-go $H_t$ can be computed by the backward algorithm. By the definition of $H_t$ in \[eq:H\], $$\begin{aligned}
\exp H_t & = \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \exp\;(G_T - G_t) \label{eq:H-as-diff} \\
& = \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \exp \sum_{j=t+1}^T g_\theta({{{\boldsymbol{\mathbf{s}}}}}_{j-1},x_j,y_j) \\
& = \sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} \prod_{j=t+1}^T p_\theta(y_j \mid y_{j-1}) \cdot p_\theta(x_j \mid y_j) \nonumber \\
& = ({{\boldsymbol{\mathbf{\beta}}}}_t)_{y_t} \text{\ \ (backward prob of $y_t$ at time $t$)} \nonumber\end{aligned}$$ where the vector ${{\boldsymbol{\mathbf{\beta}}}}_t$ is defined by base case $({{\boldsymbol{\mathbf{\beta}}}}_T)_y = 1$ and for $0 \leq t < T$ by the recurrence $$\begin{aligned}
(\beta_t)_y
&{\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\sum_{{{{\boldsymbol{\mathbf{y}}}}}_{t:}} p_\theta({{{\boldsymbol{\mathbf{x}}}}}_{t:},{{{\boldsymbol{\mathbf{y}}}}}_{t:} \mid y_t=y) \label{eq:beta-hmm} \\
&= \sum_{y'} p_\theta(y' \mid y) \cdot p_\theta(x_{t+1} \mid y') \cdot ({{\boldsymbol{\mathbf{\beta}}}}_{t+1})_{y'} \nonumber\end{aligned}$$
The backward algorithm for OOHMMs in \[sec:oohmm\] is a variant of this.
Gradients for Training the Proposal Distribution {#app:gradient-derivation}
================================================
For a given ${{{\boldsymbol{\mathbf{x}}}}}$, both forms of KL divergence achieve their minimum of 0 when $(\forall {{{\boldsymbol{\mathbf{y}}}}})\; q_\phi({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) = p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. However, we are unlikely to be able to find such a $\phi$; the two metrics penalize $q_\phi$ differently for mismatches. We simplify the notation below by writing $q_\phi({{{\boldsymbol{\mathbf{y}}}}})$ and $p({{{\boldsymbol{\mathbf{y}}}}})$, suppressing the conditioning on ${{{\boldsymbol{\mathbf{x}}}}}$.
#### Inclusive KL Divergence
The inclusive KL divergence has that name because it is finite only when $\textrm{support}(q_\phi) \supseteq \textrm{support}(p)$, i.e., when $q_\phi$ is capable of proposing any string ${{{\boldsymbol{\mathbf{y}}}}}$ that has positive probability under $p$. This is required for $q_\phi$ to be a valid proposal distribution for importance sampling. $$\begin{aligned}
\MoveEqLeft[2] \text{KL} \left( p || q_{\phi} \right) \\
&= {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim p}}\left[ \log p\left({{{\boldsymbol{\mathbf{y}}}}}\right) - \log q_\phi({{{\boldsymbol{\mathbf{y}}}}}) \right]} \nonumber \\
&= {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim p}}\left[\log p \left( {{{\boldsymbol{\mathbf{y}}}}}\right)\right]} \nonumber \\
&\quad - {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim p}}\left[\log q_\phi \left( {{{\boldsymbol{\mathbf{y}}}}}\right)\right]} \nonumber\end{aligned}$$ The first term ${\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim p}}\left[\log p \left( {{{\boldsymbol{\mathbf{y}}}}}\right)\right]}$ is a constant with regard to $\phi$. As a result, the gradient of the above is just the gradient of the second term: $$\begin{aligned}
\nabla_{\phi} \text{KL}(p || q_\phi) &=
\nabla_{\phi} \underbrace{{\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim p}}\left[-\log q_\phi \left( {{{\boldsymbol{\mathbf{y}}}}}\right)\right]}}_{\text{the cross-entropy }H(p,q_{\phi})}\end{aligned}$$ We cannot directly sample from $p$. However, our weighted mixture ${{\hat{p}}}$ from \[eq:phat\] (obtained by sequential importance sampling) could be a good approximation: $$\begin{aligned}
\nabla_{\phi} \text{KL}(p || q_\phi) &\approx \nabla_\phi {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim {{\hat{p}}}}}\left[-\log q_\phi\left({{{\boldsymbol{\mathbf{y}}}}}\right)\right]}
\label{eq:cross-entropy} \\
&= \sum_{t=1}^{T} {\mathbb{E}_{{{{\hat{p}}}}}\left[ - \nabla_\phi \log q_\phi(y_t \mid y_{:t-1}, {{{\boldsymbol{\mathbf{x}}}}}) \right]} \nonumber\end{aligned}$$ Following this approximate gradient downhill has an intuitive interpretation: if a particular $y_t$ value ends up with high relative weight in the final ensemble ${{\hat{p}}}$, then we will try to adjust $q_\phi$ so that it would have had a high probability of proposing that $y_t$ value at step $t$ in the first place.
#### Exclusive KL Divergence
The exclusive divergence has that name because it is finite only when $\textrm{support}(q_\phi) \subseteq \textrm{support}(p)$. It is defined by $$\begin{aligned}
\MoveEqLeft[1]
\text{KL}(q_\phi || p) = {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi}}\left[ \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) - \log p({{{\boldsymbol{\mathbf{y}}}}}) \right]} \\
& = {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi}}\left[ \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) - \log \tilde{p}({{{\boldsymbol{\mathbf{y}}}}}) \right]} + \log Z \nonumber \\
&= \sum_{{{{\boldsymbol{\mathbf{y}}}}}} q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) \underbrace{\left[ \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) - \log \tilde{p}({{{\boldsymbol{\mathbf{y}}}}}) \right]}_{\textrm{call this }d_\phi({{{\boldsymbol{\mathbf{y}}}}})} + \log Z \nonumber\end{aligned}$$ where $p({{{\boldsymbol{\mathbf{y}}}}}) = \frac{1}{Z} \tilde{p}({{{\boldsymbol{\mathbf{y}}}}})$ for $\tilde{p}({{{\boldsymbol{\mathbf{y}}}}}) = \exp G_T$ and $Z = \sum_{{{{\boldsymbol{\mathbf{y}}}}}} \tilde{p}({{{\boldsymbol{\mathbf{y}}}}})$. With some rearrangement, we can write its gradient as an expectation that can be estimated by sampling from $q_\phi$.[^17] Observing that $Z$ is constant with respect to $\phi$, first write $$\begin{aligned}
\label{eq:exclusive-kl}
\MoveEqLeft[2] \nabla_\phi \text{KL}(q_\phi || p) \\
&= \sum_{{{{\boldsymbol{\mathbf{y}}}}}} \nabla_\phi \left( q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) \, d_\phi({{{\boldsymbol{\mathbf{y}}}}}) \right) \\
&= \sum_{{{{\boldsymbol{\mathbf{y}}}}}} \left( \nabla_\phi q_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right) \, d_\phi({{{\boldsymbol{\mathbf{y}}}}}) \nonumber
\\ &\qquad + \sum_{{{{\boldsymbol{\mathbf{y}}}}}} \underbrace{q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) \nabla_\phi \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}})}_{= \nabla_\phi q_\phi ({{{\boldsymbol{\mathbf{y}}}}})} \nonumber \\
&= \sum_{{{{\boldsymbol{\mathbf{y}}}}}} \left( \nabla_\phi q_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right) \, d_\phi({{{\boldsymbol{\mathbf{y}}}}}) \nonumber\end{aligned}$$ where the last step uses the fact that $\sum_{{{{\boldsymbol{\mathbf{y}}}}}}\nabla_\phi q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) = \nabla_\phi \sum_{{{{\boldsymbol{\mathbf{y}}}}}} q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) = \nabla_\phi 1 = 0$. We can turn this into an expectation with a second use of ’s observation that $\nabla_\phi q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) = q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) \nabla_\phi \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}})$ (the “likelihood ratio trick”): $$\begin{aligned}
\MoveEqLeft[2] \nabla_\phi \text{KL}(q_\phi || p) \nonumber \\
& = \sum_{{{{\boldsymbol{\mathbf{y}}}}}} q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) d_\phi({{{\boldsymbol{\mathbf{y}}}}}) \nabla_\phi \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}}) \nonumber \\
& = {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi}}\left[d_\phi({{{\boldsymbol{\mathbf{y}}}}}) \nabla_\phi \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right]} \label{eq:grad} \\
\intertext{which can, if desired, be further rewritten as}
& = {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi}}\left[d_\phi({{{\boldsymbol{\mathbf{y}}}}}) \nabla_\phi\, d_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right]} \nonumber \\
&= {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_{\phi}}}\left[\nabla_\phi \left( \tfrac{1}{2} d_\phi({{{\boldsymbol{\mathbf{y}}}}})^2 \right)\right]} \label{eq:squaredgrad}\end{aligned}$$ If we regard $d_\phi({{{\boldsymbol{\mathbf{y}}}}})$ as a signed error (in the log domain) in trying to fit $q_\phi$ to $\tilde{p}$, then the above gradient of KL can be interpreted as the gradient of the mean squared error (divided by 2).[^18]
We would get the same gradient for any rescaled version of the unnormalized distribution $\tilde{p}$, but the formula for obtaining that gradient would be different. In particular, if we rewrite the above derivation but add a constant $b$ to both $\log \tilde{p}({{{\boldsymbol{\mathbf{y}}}}})$ and $\log Z$ throughout (equivalent to adding $b$ to $G_T$), we will get the slightly generalized expectation formulas $$\begin{aligned}
& {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi}}\left[(d_\phi({{{\boldsymbol{\mathbf{y}}}}}) - b)\nabla_\phi \log q_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right]} \\
& {\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_{\phi}}}\left[\nabla_\phi \left( \tfrac{1}{2} \left( d_\phi ({{{\boldsymbol{\mathbf{y}}}}}) - b \right)^{2} \right)\right]}\end{aligned}$$ in place of \[eq:grad,eq:squaredgrad\]. By choosing an appropriate “baseline” $b$, we can reduce the variance of the sampling-based estimate of these expectations. This is similar to the use of a baseline in the REINFORCE algorithm [@williams92reinforce]. In this work we choose $b$ using an exponential moving average of past ${\mathbb{E}_{{}}\left[d_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right]}$ values: at the end of each training minibatch, we update $b \leftarrow 0.1\cdot b + 0.9\cdot \bar{d}$, where $\bar{d}$ is the mean of the estimated ${\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi(\cdot \mid {{{\boldsymbol{\mathbf{x}}}}})}}\left[d_\phi ({{{\boldsymbol{\mathbf{y}}}}})\right]}$ values for all examples ${{{\boldsymbol{\mathbf{x}}}}}$ in the minibatch.
![Offset KL divergence for the source separation task on phoneme sequences.[]{data-label="fig:cmu-beam"}](kl-sourcesep-cmu-beam-crop.pdf){width="\columnwidth"}
Implementation Details {#app:model-details}
======================
We implement all RNNs in this paper as GRU networks [@cho2014] with $d=32$ hidden units (state space ${\mathbb{R}}^{32}$). Each of our models (\[sec:model\]) always specifies the logprob-so-far in \[eq:G,eq:s\] using a 1-layer left-to-right GRU,[^19] while the corresponding proposal distribution (\[sec:q\]) always specifies the state $\overline{{{{\boldsymbol{\mathbf{s}}}}}}_t$ in using a 2-layer right-to-left GRU, and specifies the compatibility function $C_t$ in using a $4$-layer feedforward ReLU network.[^20] For the Chinese social media NER task (\[sec:tagging\]), we use the Chinese character embeddings provided by @peng2015, while for the source separation tasks (\[sec:sourcesep\]), we use the 50-dimensional GloVe word embeddings [@pennington2014]. In other cases, we train embeddings along with the rest of the network. We optimize with the Adam optimizer using the default parameters [@kingma2014] and $L_2$ regularization coefficient of $10^{-5}$.
Training Procedures {#app:training-procedures}
===================
In all our experiments, we train the incremental scoring models (the tagging and source separation models described in \[sec:tagging\] and \[sec:sourcesep\], respectively) on the training dataset $T$. We do early stopping, using perplexity on a held-out development set $D_1$ to choose the number of epochs to train (maximum of $3$).
Having obtained these model parameters $\theta$, we train our proposal distributions $q_{\theta,\phi}$ on $T$, keeping $\theta$ fixed and only tuning $\phi$. Again we use early stopping, using the KL divergence from \[sec:metrics\] on a separate development set $D_2$ to choose the number of epochs to train (maximum of $20$ for the two tagging tasks and source separation on the PTB dataset, and maximum of $50$ for source separation on the phoneme sequence dataset). We then evaluate $q_{\theta^{*},\phi^{*}}$ on the test dataset $E$.
**\[ appear in the supplementary material file.\]**
Applications of Sampling {#app:apps}
========================
In this paper, we evaluate our sampling algorithms “intrinsically” by how well a sample approximates the model distribution $p_\theta$—rather than “extrinsically” by using the samples in some larger method.
That said, \[sec:apps\] did list some larger methods that make use of sampling. We review them here for the interested reader.
*Minimum-risk decoding* seeks the output $$\label{eq:risk}
\operatorname*{argmin}_{{{\boldsymbol{\mathbf{z}}}}}\sum_{{{\boldsymbol{\mathbf{y}}}}}p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) \cdot \text{loss}({{{\boldsymbol{\mathbf{z}}}}}\mid {{{\boldsymbol{\mathbf{y}}}}})$$ In the special case where $\text{loss}({{{\boldsymbol{\mathbf{z}}}}}\mid {{{\boldsymbol{\mathbf{y}}}}})$ simply asks whether ${{{\boldsymbol{\mathbf{z}}}}}\neq {{{\boldsymbol{\mathbf{y}}}}}$, this simply returns the “Viterbi” sequence ${{{\boldsymbol{\mathbf{y}}}}}$ that maximises $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. However, it may give a different answer if the loss function gives partial credit (when ${{{\boldsymbol{\mathbf{z}}}}}\approx {{{\boldsymbol{\mathbf{y}}}}}$), or if the space of outputs ${{{\boldsymbol{\mathbf{z}}}}}$ is simply coarser than the space of taggings ${{{\boldsymbol{\mathbf{y}}}}}$—for example, if there are many action sequences ${{{\boldsymbol{\mathbf{y}}}}}$ that could build the same output structure ${{{\boldsymbol{\mathbf{z}}}}}$. In these cases, the optimal ${{{\boldsymbol{\mathbf{z}}}}}$ may win due to the combined support of many suboptimal ${{{\boldsymbol{\mathbf{y}}}}}$ values, and so finding the optimal ${{{\boldsymbol{\mathbf{y}}}}}$ (the Viterbi sequence) is not enough to determine the optimal ${{{\boldsymbol{\mathbf{z}}}}}$.
The risk objective is a expensive expectation under the distribution $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$. To approximate it, one can replace $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ with an approximation ${{\hat{p}}}({{{\boldsymbol{\mathbf{y}}}}})$ that has small support so that the summation is efficient. Particle smoothing returns such a ${{\hat{p}}}$—a non-uniform distribution over $M$ particles. Since those particles are randomly drawn, ${{\hat{p}}}$ is itself stochastic, but ${\mathbb{E}_{{}}\left[{{\hat{p}}}({{{\boldsymbol{\mathbf{y}}}}})\right]} \approx p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$, with the approximation improving with the quality of the proposal distribution (which is the focus of this paper) and with $M$.
In *supervised* training of the model by maximizing conditional log-likelihood, the gradient of $\log p({{{\boldsymbol{\mathbf{y}}}}}^*\mid{{{\boldsymbol{\mathbf{x}}}}})$ on a single training example $({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}}^*)$ is $\nabla_\theta \log p_\theta({{{\boldsymbol{\mathbf{y}}}}}^* \mid {{{\boldsymbol{\mathbf{x}}}}}) = \nabla_\theta G_T^* - \sum_{{{{\boldsymbol{\mathbf{y}}}}}} p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) \cdot \nabla_\theta G_T$. The sum is again an expectation that can be estimated by using ${{\hat{p}}}$. Since ${\mathbb{E}_{{}}\left[{{\hat{p}}}({{{\boldsymbol{\mathbf{y}}}}})\right]} \approx p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$, this yields a stochastic estimate of the gradient that can be used in the stochastic gradient ascent algorithm [@RobMon51Stochastic].[^21]
In *unsupervised or semi-supervised training* of a generative model $p_\theta({{{\boldsymbol{\mathbf{x}}}}},{{{\boldsymbol{\mathbf{y}}}}})$, one has some training examples where ${{{\boldsymbol{\mathbf{y}}}}}^*$ is unobserved or observed incompletely (e.g., perhaps only ${{{\boldsymbol{\mathbf{z}}}}}$ is observed). The Monte Carlo EM algorithm for estimating $\theta$ [@wei1990] replaces the missing ${{{\boldsymbol{\mathbf{y}}}}}^*$ with samples from $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}, \text{partial observation})$ (this is the Monte Carlo “E step”). This *multiple imputation* procedure has other uses as well in statistical analysis with missing data [@little-rubin-1987].
*Modular architectures* provide another use for sampling. If $p_\theta({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ is just one stage in an NLP annotation *pipeline*, recommend passing a diverse sample of ${{{\boldsymbol{\mathbf{y}}}}}$ values on to the next stage, where they can be further annotated and rescored or rejected. More generally, in a *graphical model* that relates multiple strings [@bouchard-EtAl:2007:EMNLP-CoNLL2007; @dreyer-eisner-2009; @E17-2120], inference could be performed by particle belief propagation [@ihler-mcallester-2009; @lienart-teh-doucet-2015], or with the help of stochastic-inverse proposal distributions [@stuhlmuller-et-al-2013]. These methods call conditional sampling as a subroutine.
Effect of different objective functions on lookahead optimization {#app:lastchar}
=================================================================
![Offset KL divergence on the *last char* task: a pathological case where a naive particle filtering sampler does really horribly, and an ill-trained smoothing sampler even worse. The logarithmic $x$-axis is the particle size used to train the sampler. At test time we evaluate with the same particle size ($M=32$).[]{data-label="fig:lastchar"}](kl-lastchar-crop.pdf){width="\columnwidth"}
discussed inclusive and exclusive KL divergences, and gave our rationale for optimizing an interpolation of the two. Here we study the effect of the interpolation weight. We train the lookahead sampler, and the joint language model, on a toy problem called “last char,” where ${{{\boldsymbol{\mathbf{y}}}}}$ is a deterministic function of ${{{\boldsymbol{\mathbf{x}}}}}$: either a lowercased version of ${{{\boldsymbol{\mathbf{x}}}}}$, or an identical copy of ${{{\boldsymbol{\mathbf{x}}}}}$, depending on whether the last character of ${{{\boldsymbol{\mathbf{x}}}}}$ is `0` or `1`. Note that this problem requires lookahead.
We obtain our ${{{\boldsymbol{\mathbf{x}}}}}$ sequences by taking the phoneme sequence data from the stressed syllable tagging task and flipping a fair coin to decide whether to append $0$ or $1$ to each sequence. Thus, the dataset may include $({{{\boldsymbol{\mathbf{x}}}}}, {{{\boldsymbol{\mathbf{y}}}}})$ pairs such as $(\texttt{K AU CH 0},\;\texttt{k au ch 1})$ or $(\texttt{K AU CH 1},\; \texttt{K AU CH 1})$, but not $(\texttt{K AU CH 1},\;\texttt{k au ch 1})$.
We treat this as a tagging problem, and treat it with our tagging model in \[sec:tagging\]. Results are in \[fig:lastchar\]. We see that optimizing for $\textrm{KL}({{\hat{p}}}||q)$ at a low particle size gives much worse performance than other methods. On the other hand, the objective function $\textrm{KL}(q||p)$ achieves constantly good performance. The middle ground $\frac{\textrm{KL}({{\hat{p}}}||q)+\textrm{KL}(q||p)}{2}$ improves when the particle size increases, and achieves better results than $\textrm{KL}(q||p)$ at larger particle sizes.
Generative process for source separation {#app:sourcesep-generation}
========================================
Given an alphabet $\Sigma$, $J$ strings ${{{\boldsymbol{\mathbf{x}}}}}{^{(1)}}, {{{\boldsymbol{\mathbf{x}}}}}{^{(2)}}, \ldots, {{{\boldsymbol{\mathbf{x}}}}}{^{(J)}} \in \Sigma^*$ are independently sampled from the respective distributions $p{^{(1)}}, \ldots p{^{(J)}}$ over $\Sigma^*$ (possibly all the same distribution $p{^{(1)}}=\cdots=p{^{(J)}}$). These source strings are then combined into a single observed string ${{{\boldsymbol{\mathbf{x}}}}}$, of length $K=\sum_j K_j$, according to an [**interleaving string**]{} ${{{\boldsymbol{\mathbf{y}}}}}$, also of length $K$. For example, ${{{\boldsymbol{\mathbf{y}}}}}=1132123$ means to take two characters from ${{{\boldsymbol{\mathbf{x}}}}}{^{(1)}}$, then a character from ${{{\boldsymbol{\mathbf{x}}}}}{^{(3)}}$, then a character from ${{{\boldsymbol{\mathbf{x}}}}}{^{(2)}}$, etc. Formally speaking, ${{{\boldsymbol{\mathbf{y}}}}}$ is an element of the mix language ${\cal Y}_{{{{\boldsymbol{\mathbf{x}}}}}} = \textsc{mix}(1^{k_1},2^{k_2},\ldots,j^{k_j})$, and we construct ${{{\boldsymbol{\mathbf{x}}}}}$ by specifying the character $x_k \in \Sigma$ to be $x{^{(y_k)}}_{|\{i \leq k: y_i=y_k\}|}$. We assume that ${{{\boldsymbol{\mathbf{y}}}}}$ is drawn from some distribution over ${\cal Y}_{{{{\boldsymbol{\mathbf{x}}}}}}$. The source separation problem is to recover the interleaving string ${{{\boldsymbol{\mathbf{y}}}}}$ from the interleaved string ${{{\boldsymbol{\mathbf{x}}}}}$.
We assume that each source model $p{^{(j)}}({{{\boldsymbol{\mathbf{x}}}}}{^{(j)}})$ is an RNN language model—that is, a locally normalized state machine that successively generates each character of ${{{\boldsymbol{\mathbf{x}}}}}{^{(j)}}$ given its left context. Thus, each source model is in some state ${{{\boldsymbol{\mathbf{s}}}}}_t{^{(j)}}$ after generating the prefix ${{{\boldsymbol{\mathbf{x}}}}}_{:t}{^{(j)}}$. In the remainder of this paragraph, we suppress the superscript ${^{(j)}}$ for simplicity. The model now stochastically generates character $x_{t+1}$ with probability $p(x_{t+1} \mid {{{\boldsymbol{\mathbf{s}}}}}_t)$, and from ${{{\boldsymbol{\mathbf{s}}}}}_t$ and this $x_{t+1}$ it deterministically computes its new state ${{{\boldsymbol{\mathbf{s}}}}}_{t+1}$. If $x_{t+1}$ is a special “end-of-sequence” character [<span style="font-variant:small-caps;">eos</span>]{}, we return ${{{\boldsymbol{\mathbf{x}}}}}={{{\boldsymbol{\mathbf{x}}}}}_{:t}$.
Given only ${{{\boldsymbol{\mathbf{x}}}}}$ of length $T$, we see that ${{{\boldsymbol{\mathbf{y}}}}}$ could be any element of $\{1,2,\ldots,J\}^T$. We can write the posterior probability of a given ${{{\boldsymbol{\mathbf{y}}}}}$ (by Bayes’ Theorem) as $$\begin{aligned}
p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}}) \;\propto\; p({{{\boldsymbol{\mathbf{y}}}}}) \prod_{j=1}^{J} p{^{(j)}}\left({{{\boldsymbol{\mathbf{x}}}}}{^{(j)}}\right)
\label{eq:perm}\end{aligned}$$ where (for this given ${{{\boldsymbol{\mathbf{y}}}}}$) ${{{\boldsymbol{\mathbf{x}}}}}{^{(j)}}$ denotes the subsequence of ${{{\boldsymbol{\mathbf{x}}}}}$ at indices $k$ such that $y_k=j$. In our experiments, we assume that $y$ was drawn uniformly from $\cal Y_{{{{\boldsymbol{\mathbf{x}}}}}}$, so $p({{{\boldsymbol{\mathbf{y}}}}})$ is constant and can be ignored. In general, the set of possible interleavings ${\cal Y}_{{{{\boldsymbol{\mathbf{x}}}}}}$ is so large that computing the constant of proportionality (partition function) for a given ${{{\boldsymbol{\mathbf{x}}}}}$ becomes prohibitive.
[^1]: \[fn:eos\]A model may require for convenience that each input end with a special end-of-sequence symbol: that is, $x_T = {\textsc{eos}\xspace}$.
[^2]: Furthermore, ${{{\boldsymbol{\mathbf{s}}}}}_t$ could even depend on all of ${{{\boldsymbol{\mathbf{x}}}}}$ (if ${{{\boldsymbol{\mathbf{s}}}}}_0$ does), allowing direct expression of models such as stacked BiRNNs.
[^3]: The HMM actually specifies a distribution over a pair of infinite sequences, but here we consider the marginal distribution over just the length-$T$ prefixes.
[^4]: \[fn:bos\]It takes ${{{\boldsymbol{\mathbf{s}}}}}_0 = {\textsc{bos}\xspace}$, a beginning-of-sequence symbol, so $p_\theta(y_1 \mid {\textsc{bos}\xspace})$ specifies the initial state distribution.
[^5]: \[fn:iohmm\]This is by analogy with the *input-output HMM* (IOHMM) of , which defines $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ directly and conditions the transition to $u_t$ on $x_t$. The OOHMM instead defines $p({{{\boldsymbol{\mathbf{y}}}}}\mid {{{\boldsymbol{\mathbf{x}}}}})$ by conditionalizing —which avoids the *label bias* problem [@lafferty-mccallum-pereira-2001] that in the IOHMM, $y_t$ is independent of future input ${{{\boldsymbol{\mathbf{x}}}}}_{t:}$ (given the past input ${{{\boldsymbol{\mathbf{x}}}}}_{:t}$).
[^6]: This is not true for the language of balanced parentheses.
[^7]: \[fn:CT\]$C_T=0$ is correct according to . Forcing this ensures $H_T=0$, so our approximation becomes exact as of $t=T$.
[^8]: The particle cascade would benefit from an estimate of $\hat{H}_t$, as it (like A$^*$ search) compares particles of different lengths.
[^9]: While resampling mitigates the degeneracy problem, it could also reduce the diversity of particles. In our experiments in this paper, we only do multinomial resampling when the effective sample size of $\hat{p}_t$ is lower than $\frac{M}{2}$. @doucet2009 give a more thorough discussion on when to resample.
[^10]: \[fn:end2end\]In principle, one could attempt to train $\phi$ “end-to-end” on some downstream objective by using reinforcement learning or the Gumbel-softmax trick [@jang2016; @maddison2016]. For example, we might try to ensure that ${{\hat{p}}}$ closely matches the model’s distribution $p$ (\[eq:phat\])—the “natural” goal of sampling. This objective can tolerate inaccurate local proposal distributions in cases where the algorithm could recover from them through resampling. Looking even farther downstream, we might merely want ${{\hat{p}}}$—which is typically used to compute expectations—to provide accurate guidance to some decision or training process (see \[app:apps\]). This might not require fully matching the model, and might even make it desirable to deviate from an inaccurate model.
[^11]: Training a single approximation $q_\phi$ for all ${{{\boldsymbol{\mathbf{x}}}}}$ is known as [*amortized inference*]{}.
[^12]: The normalizing constant of $p$ from can be ignored because the gradient of a constant is 0.
[^13]: English, like many other languages, assigns stress from right to left [@hayes1995].
[^14]: We formally describe the generative process in \[app:sourcesep-generation\].
[^15]: For the details of the training procedures and the specific neural architectures in our models, see \[app:training-procedures,app:model-details\].
[^16]: Thus, $\cal Y$ was collected across all samplings, iterations, and ensemble sizes $M$, in an attempt to make the summation over $\cal Y$ as complete as possible. For good measure, we added some extra particles: whenever we drew $M$ particles via particle smoothing, we drew an additional $2M$ particles by particle filtering and added them to $\cal Y$.
[^17]: This is an extension of the REINFORCE trick [@williams92reinforce], which estimates the gradient of ${\mathbb{E}_{{{{{\boldsymbol{\mathbf{y}}}}}\sim q_\phi}}\left[ \text{reward}({{{\boldsymbol{\mathbf{y}}}}}) \right]}$ when the reward is independent of $\phi$. In our case, the expectation is over a quantity that does depend on $\phi$.
[^18]: We thank Hongyuan Mei, Tim Vieira, and Sanjeev Khudanpur for insightful discussions on this derivation.
[^19]: For the tagging task described in \[sec:tagging\], $g_\theta ({{{\boldsymbol{\mathbf{s}}}}}_{t-1}, x_t, y_t) {\mathrel{\stackrel{\mbox{\tiny def}}{=}}}\log p_\theta (x_t, y_t \mid {{{\boldsymbol{\mathbf{s}}}}}_{t-1})$, where the GRU state ${{{\boldsymbol{\mathbf{s}}}}}_{t-1}$ is used to define a softmax distribution over possible $(x_t, y_t)$ pairs in the same manner as an RNN language model [@mikolov2010]. Likewise, for the source separation task (\[sec:sourcesep\]), the source language models described in \[app:sourcesep-generation\] are GRU-based RNN language models.
[^20]: As input to $C_t$, we actually provide not only ${{{\boldsymbol{\mathbf{s}}}}}_t,{{\bar{{{{\boldsymbol{\mathbf{s}}}}}}}}_t$ but also the states $f_\theta({{{\boldsymbol{\mathbf{s}}}}}_{t-1},x_t,y)$ (including ${{{\boldsymbol{\mathbf{s}}}}}_t$) that could have been reached for [*each*]{} possible value $y$ of $y_t$. We have to compute these anyway while constructing the proposal distribution, and we find that it helps performance to include them.
[^21]: Notice that the gradient takes this “difficult” form only because the model is globally normalized. If we were training a locally normalized conditional model [@mccallum-freitag-pereira-2000], or a locally normalized joint model like \[eq:noindep\], then sampling methods would not be needed, because the gradient of the (conditional or joint) log-likelihood would decompose into $T$ “easy” summands that each involve an expectation over the small set of $y_t$ values for some $t$, rather than over the exponentially larger set of strings ${{{\boldsymbol{\mathbf{y}}}}}$. However, this simplification goes away outside the fully supervised case, as the next paragraph discusses.
|
---
abstract: 'New nondiagonal $G_{2}$ inhomogeneous cosmological solutions are presented in a wide range of scalar-tensor theories with a stiff perfect fluid as a matter source. The solutions have no big-bang singularity or any other curvature singularities. The dilaton field and the fluid energy density are regular everywhere, too. The geodesic completeness of the solutions is investigated.'
address: |
Department of Theoretical Physics, Faculty of Physics\
Sofia University\
5 James Bourchier Boulevard\
1164 Sofia, Bulgaria
author:
- 'Stoytcho S. Yazadjiev[^1]'
title: |
Geodesically complete nondiagonal inhomogeneous cosmological solutions\
in dilatonic gravity with a stiff perfect fluid
---
All versions of string theory and higher dimensional gravity theories predict the existence of the dilaton field which determines the gravitational “constant” as a variable quantity. The existence of a scalar partner of the tensor graviton may have a serious influence on the space-time structure and important consequences for cosmology and astrophysics. A large amount of research has been done in order to unveil the possible cosmological significance of the dilaton [@GAS], [@LS]-[@M](and references therein). With a few exceptions most of the cosmological studies within the scalar-tensor theories were devoted to the homogeneous case. The homogeneous models are good approximations of the present universe. There is, however, no reason to assume that such a regular expansion is also suitable for a description of the early universe. Moreover, as is well known, the present universe is not exactly spacially homogeneous. That is why it is necessary to study inhomogeneous cosmological models. They allow us to investigate a number of long standing questions regarding the occurrence of singularities, the behaviour of the solutions in the vicinity of a singularity and the possibility of our universe arising from generic initial data.
In this work we shall address the question of the occurrence of singularities in inhomogeneous cosmologies within the framework of scalar-tensor theories.
As is well known, most of the homogeneous models (both in general relativity and in scalar-tensor theories) predict a universal space-like big-bang singularity in a finite past. It was, therefore, believed that this would be the usual singularity in general. The inclusion of inhomogeneities drastically changes this point of view. There are inhomogeneous cosmological solutions in general relativity which have no bing-bang or any other curvature singularity. The first such solution was discovered by Senovilla in 1990 Ref.[@SEN]. Senovilla’s solution represents a cylindrically symmetric universe filled with radiation. This solution has a diagonal metric and is also globally hyperbolic and geodesically complete [@CFJS]. Senovilla’s solution was generalized by Ruiz and Senovilla in Ref.[@RS] where a large family of singularity-free diagonal $G_{2}$ inhomogeneous perfect fluid solutions was found. Nonsingular diagonal inhomegeneous solutions in general relativity describing cylindrically symmetric universes filled with stiff perfect fluid were found by Patel and Dadhich in Ref.[@PATDAD]. Other examples of diagonal nonsingular solutions in general relativity can be found in Refs.[@PATDAD1]- [@LFJ1].
In Ref.[@MM], Mars found the first nondiagonal $G_{2}$ inhomogeneous cosmological solution of the Einstein equations with stiff perfect fluid as a source. This solution is globally hyperbolic and geodesically complete. Mars’s solution was generalized by Griffiths and Bicak in Ref.[@GB].
Within the framework of scalar-tensor theories there are also inhomogeneous cosmological solutions without big-bang or any other curvature singularity. In Ref.[@GIOV], Giovannini derived gravi-dilaton inhomogeneous cosmological solutions with everywhere regular curvature invariants and bounded dilaton in tree-level dilaton driven models. In a subsequent paper [@GIOV1], it was shown that these solutions describe singularity-free dilaton driven cosmologies. A nondiagonal inhomogeneous cosmological solution with regular curvature invariants and unbounded dilaton in the tree level effective string models was found by Pimentel [@PIM]. Very recently, inhomogeneous cosmological solutions without any curvature singularities were obtained by the author in a wide class of scalar-tensor theories with stiff perfect fluid as a source [@Y].
In this work we take a further step upwards and present new nondiagonal $G_{2}$ inhomogeneous cosmological stiff perfect fluid solutions with no curvature singularities in a wide range of scalar-tensor theories.
Scalar-tensor theories (without a cosmological potential) are described by the following action in Jordan (string) frame [@W1],[@W2]: $$\begin{aligned}
\label{JFA}
S = {1\over 16\pi G_{*}} \int d^4x \sqrt{-g}\left(F(\Phi)R -
Z(\Phi) g^{\mu\nu}\partial_{\mu}\Phi
\partial_{\nu}\Phi \right) \\+
S_{m}\left[\Psi_{m};g_{\mu\nu}\right] .\nonumber\end{aligned}$$
Here, $G_{*}$ is the bare gravitational constant and $R$ is the Ricci scalar curvature with respect to the space-time metric $g_{\mu\nu}$. The dynamics of the scalar field $\Phi$ depends on the functions $F(\Phi)$ and $Z(\Phi)$. In order for the gravitons to carry positive energy the function $F(\Phi)$ must be positive. The nonnegativity of the energy of the dilaton requires that $2F(\Phi)Z(\Phi)+ 3[dF(\Phi)/d\Phi ]^2\ge 0$. The action of matter depends on the material fields $\Psi_{m}$ and the space-time metric $g_{\mu\nu}$ but does not involve the scalar field $\Phi$ in order for the weak equivalence principle to be satisfied.
As a matter source we consider a stiff perfect fluid with equation of state $p=\rho$.
The general form of the solutions is given by
$$\begin{aligned}
d{s}^2= F^{-1}(\Phi(t))\left[ e^{\gamma a^2r^2}\cosh(2at)(-dt^2 +
dr^2) \right. \nonumber \\ \left. + \,\, r^2\cosh(2at)d\phi^2 +
{1\over \cosh(2at)}(dz + ar^2d\phi)^2 \right] \nonumber ,\end{aligned}$$
$$\begin{aligned}
\label{STS}
8\pi G_{*}\rho = f(\lambda) {a^2 (\gamma - 1)F^3(\Phi(t))
e^{-\gamma a^2 r^2} \over \cosh(2at)} ,\end{aligned}$$
$$\begin{aligned}
u_{\mu} =F^{-1/2}(\Phi(t)) e^{(1/2)\gamma a^2 r^2}
\cosh^{1/2}(2at)\,\delta^{0}_{\mu} \nonumber .\end{aligned}$$
The solution depends on three parameters - $a$, $\gamma$ ($\gamma
>1 $) and $\lambda$. The range of the coordinates is
$$-\infty <t, z < \infty ,\,\,\, 0\leq r < \infty , \,\,\, 0\leq
\phi \leq 2\pi .$$
The explicit form of the functions $\Phi(t)$ and $f(\lambda)$, and the range of the parameter $\lambda$ depend on the particular scalar tensor theory. These solutions can be generated[^2] from the general relativistic Mars’s solution [@MM] using the solution generating methods developed in Ref.[@Y].
Below we consider the explicit form of the general solution for some particular scalar-tensor theories.
Barker’s theory
---------------
Barker’s theory is described by the functions $F(\Phi)=\Phi$ and $Z(\Phi)= (4-3\Phi)/2\Phi(\Phi - 1)$.
In the case of Barker’s theory the explicit forms of the functions $\Phi(t)$ and $f(\lambda)$ are:
$$\begin{aligned}
\Phi^{-1}(t)& =& 1 -\lambda \cos^2\left(a\sqrt{\gamma
-1}t\right),
\\
f(\lambda) &=& 1-\lambda\end{aligned}$$
where the range of $\lambda$ is $0< \lambda < 1$. This range can be extended to $0\le \lambda \le 1$. For $\lambda=0$ and $\lambda=1$ we obtain the Mars’s solution and gravi-dilaton vacuum solution, respectively. That is why we consider only $0<\lambda
<1$. It should be noted that the range of the parameter $\lambda$ is crucial for the curvature invariants. It is easy to see that the gravi-dilaton vacuum solution corresponding to $\lambda=1$ has divergent curvature invarinats because of the conformal factor $\Phi^{-1}(t)= \sin^2\left(a\sqrt{\gamma -1}t\right)$.
Brans-Dicke theory
------------------
Brans-Dicke theory is described by the functions $F(\Phi)=\Phi$ and $Z(\Phi) = \omega /\Phi$ where $\omega$ is a constant parameter. Here we consider the case $\omega >-3/2$. The explicit form of the functions $\Phi(t)$ and $f(\lambda)$ in the Brans-Dicke case is the following:
$$\begin{aligned}
\Phi^{-1/2}(t) &=& \lambda \exp\left(a\sqrt{\gamma -1\over 3+
2\omega}\,t\right) \nonumber \\ &+& (1-\lambda)\exp\left( -
\,a\sqrt{\gamma
-1\over 3+2\omega}\,t\right), \\
f(\lambda) &=& 4\lambda(1-\lambda).\end{aligned}$$
Here the range of the parameter $\lambda$ is $0< \lambda <1$. The solution exists for $\lambda=0$ and $\lambda=1$, too. In these cases, however, we obtain a gravi-dilaton vacuum solution which is just the Pimentel’s solution [@PIM]. That is the reason we do not consider these limiting values of $\lambda$. The solution is invariant under the trasformations $\lambda \longleftrightarrow
1-\lambda$ and $t\longleftrightarrow -t$. In this generalized sense, we can say that the solution is even in time.
Theory with “conformal” coupling
---------------------------------
The theory with “conformal” coupling is described by the functions $F(\Phi) = 1 - {1\over 6}\Phi^2 $ and $Z(\Phi)= 1$. In this case we have:
$$\begin{aligned}
F^{-1}(\Phi(t)) &=& 1 +
4\lambda(1-\lambda)\sinh^2\left(a\sqrt{\gamma
-1\over 3}\, t\right), \\
f(\lambda) &=& (1-2\lambda)^2 .\end{aligned}$$
The range of the parameter $\lambda$ is $0<\lambda \le 1/2$. For $\lambda=1/2$ we obtain a gravi-dilaton vacuum solution which is well-behaved and can be included as a limiting case.
$Z(\Phi)= {(\Omega^2 - 3\Phi)/ 2\Phi^2}$ theory
-----------------------------------------------
Here we consider the scalar-tensor theory described by the functions $F(\Phi)= \Phi$ and $Z(\Phi)= (\Omega^2 - 3\Phi)/
2\Phi^2$ where $\Omega>0$. The explicit forms of $\Phi(t)$ and $f(\lambda)$ are:
$$\begin{aligned}
\Phi^{-1}(t) &=& \left(1 + {1 \over \Omega}a\sqrt{\gamma -1}t
\right)^2 + \lambda , \\
f(\lambda) &=& \lambda .\end{aligned}$$
Here, the range of the parameter is $0< \lambda < \infty$.
$Z(\Phi)= {1\over 2}(\Phi^2 - 3\Phi + 3)/\Phi(\Phi -1)$ theory
---------------------------------------------------------------
The theory with $F(\Phi)=\Phi$ and $Z(\Phi)= {1\over 2}(\Phi^2 -
3\Phi + 3)/\Phi(\Phi -1)$ possesses the following solution:
$$\begin{aligned}
\Phi^{-1}(t) &=& {\lambda^2 \over \lambda^2 +
(1-\lambda^2)\sin^2(\lambda a\sqrt{\gamma -1} t) } ,\\
f(\lambda) &=& \lambda^2.\end{aligned}$$
In order for the dilaton field in this solution to have positive energy we should restrict the range of the parameter $\lambda$ to $0<\lambda<1$.
Using the solution generating methods developed in Ref.[@Y] we can generate nondiagonal $G_{2}$ inhomogeneous cosmological solutions in many other scalar-tensor theories different from those considered above. However, the solutions we have presented here are expressed in a closed analytic form and they are also representative and cover a wide range of the possible behaviors of the scalar-tensor solutions which can be generated from Mars’s solution.
Let us consider the main properties of the found solutions. The metric functions, the gravitational scalar (the dilaton) and the fluid energy density are everywhere regular. The space-times described by our solutions have no big-bang nor any other curvature singularity - the curvature invariants $I_{1}=
C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$, $I_{2}=
R_{\mu\nu}R^{\mu\nu}$, and $I_{3} = R^2$ are regular everywhere. The solution possesses a two dimensional abelian group of isometries inherited from the seed Mars’s solution and generated by the Killing vectors $\partial/\partial z$ and $\partial/\partial \phi$. In addition, the metrics have a well defined axis of symmetry and the elementary flatness condition [@KSHMC] is satisfied. Since the presented solutions are conformally related to the Mars’s solution, the spacetimes descibed by them are globally hyperbolic. In fact, the global hyperbolicity can be proved independently as a consequence of the poof of the geodesic completeness presented below.
The existence of two Killing vectors gives rise to two constants of motion along the geodesics:
$$\begin{aligned}
\label{FIN}
K= F^{-1}(\Phi(t)) \nonumber \\\times
\left[\cosh(2at)r^2{d\phi\over ds} + {ar^2\over
\cosh(2at)}\big({dz\over ds} + ar^2 {d\phi \over ds}\big) \right], \\
L= {F^{-1}(\Phi(t))\over \cosh(2at)} ({dz\over ds} + ar^2 {d\phi
\over ds}\nonumber ).\end{aligned}$$
The affinely parameterized causal geodesics satisfy
$$\begin{aligned}
\label{AFP}
F^{-1}(\Phi(t))\{e^{\gamma a^2r^2}\cosh(2at) [({dt\over ds })^2 -
({dr\over ds })^2 ] \nonumber \\ - {L^2 \cosh(2at)\over
F^{-2}(\Phi(t))} - {(K -Lar^2)^2\over r^2 F^{-2}(\Phi(t))
\cosh(2at)}\} = \epsilon\end{aligned}$$
where $\epsilon =0$ and $1$ for null and timelike geodesics, respectively. Taking into account (\[FIN\]) and (\[AFP\]) the geodesic equations for $t$ and $r$ can be written in the following form:
$$\begin{aligned}
{d\over ds }\left( F^{-1}(\Phi(t))e^{\gamma a^2
r^2}\cosh(2at){dt\over ds } \right) \\ = F(\Phi(t))e^{-\gamma a^2
r^2} \cosh^{-1}(2at)M(t,r)\partial_{t}M(t,r) ,
\nonumber \\
{d\over ds }\left( F^{-1}(\Phi(t))e^{\gamma a^2
r^2}\cosh(2at){dr\over ds } \right) \\ = - F(\Phi(t))e^{-\gamma
a^2 r^2} \cosh^{-1}(2at)M(t,r)\partial_{r}M(t,r) ,\nonumber\end{aligned}$$
where
$$\begin{aligned}
M(t,r) = F^{-1/2}(\Phi(t))e^{(1/2)\gamma a^2 r^2}\cosh^{1/2}(2at)
\nonumber \\ \times \left[\epsilon + {L^2 \cosh(2at)\over
F^{-1}(\Phi(t))} + {(K - Lar^2)^2 \over r^2 F^{-1}(\Phi(t))
\cosh(2at)} \right]^{1/2}.\end{aligned}$$
To demonstrate the geodesic completeness of our metric, we have to show that all non-spacelike (i.e. causal) geodesics can be extended to arbitrary values of the affine parameter. We shall consider only future directed geodesics. The past directed geodesics can be treated analogously.
First we consider null geodesics with $K=L=0$. For them we have ${dt\over ds}=|{dr\over ds}|$ and
$${d\over ds }\left( F^{-1}(\Phi(t))e^{\gamma a^2
r^2}\cosh(2at){dt\over ds } \right) \\ =0.$$
After integrating we obtain
$${dt\over ds} = C \left(F^{-1}(\Phi(t))e^{\gamma a^2
r^2}\cosh(2at)\right)^{-1}$$
where $C>0$ is a constant. Taking into account that for each of our solutions there exists a constant $B$ such that
$$0 <B\le F^{-1}(\Phi(t))$$
for arbitrary values of $t$ and fixed parameter $\lambda$, we obtain
$${dt\over ds}=|{dr\over ds}|\le {C\over B}.$$
Therefore the geodesics under consideration are complete.
Now let us turn to the general case when at least one of the constants $\epsilon$, $K$ or $L$ is different from zero. Here we shall use a method similar to that for diagonal metrics described in Ref.[@CFJS]. Let us parameterize $dt\over ds$ and $dr\over ds$ by writing :
$$\begin{aligned}
\label{PGE1}
{dt\over ds} = {F(\Phi(t))e^{-\gamma a^2r^2}\over \cosh(2at)}
M(t,r)\cosh(\upsilon), \\
{dr\over ds} = {F(\Phi(t))e^{-\gamma a^2r^2}\over \cosh(2at)}
M(t,r)\sinh(\upsilon).\end{aligned}$$
Substituting these expressions in the equations for $t$ and $r$ we obtain
$$\begin{aligned}
\label{PGE2}
{d\upsilon \over ds } = -{F(\Phi(t))e^{-\gamma a^2r^2}\over
\cosh(2at)}\left[\partial_{t}M(t,r)\sinh(\upsilon) \right.
\nonumber \\ \left. +
\partial_{r}M(t,r)\cosh(\upsilon) \right]\end{aligned}$$
or equivalently
$$\begin{aligned}
\label{EV}
{d\upsilon\over ds } = - {1\over 2 M(t,r)} \{
\Gamma_{+}(t,r)e^{\upsilon} + \Gamma_{-}(t,r)e^{-\upsilon}\}\end{aligned}$$
where
$$\begin{aligned}
\label{GPM}
\Gamma_{+}(t,r) = \epsilon \left[ a\tanh(2at) + {1\over
2}{d\ln[F^{-1}(\Phi(t))] \over dt} + \gamma a^2 r \right] \nonumber \\
+ { (K - Lar^2)^2 \over r^2 F^{-1}(\Phi(t))\cosh(2at)}
\left[\gamma a^2 r - {1\over r} - {2Lar \over K - Lar^2} \right]
\nonumber \\ + {L^2 \cosh(2at)\over F^{-1}(\Phi(t))}
\left[2a\tanh(2at) + \gamma a^2 r \right], \\
\Gamma_{-}(t,r) = \epsilon \left[- a\tanh(2at) - {1\over
2}{d\ln[F^{-1}(\Phi(t))] \over dt} + \gamma a^2 r \right] \nonumber \\
+ { (K - Lar^2)^2 \over r^2 F^{-1}(\Phi(t))\cosh(2at)}
\left[\gamma a^2 r - {1\over r} - {2Lar \over K - Lar^2} \right]
\nonumber \\ + {L^2 \cosh(2at)\over F^{-1}(\Phi(t))}
\left[-2a\tanh(2at) + \gamma a^2 r \right].\nonumber\end{aligned}$$
In order for the geodesics to be complete ${dt \over ds }$ and ${dr\over ds}$ have to remain finite for finite values of the affine parameter. In fact, it is sufficient to consider only ${dt\over ds}$, since ${dr\over ds}$ and ${dt\over ds}$ are related via (\[AFP\]). The derivatives ${d\phi \over ds}$ and ${dz \over ds}$ are regular functions of $t$ and $r$, and the only problem we could have appear when $r$ approaches $r=0$ for $K\ne 0$. We shall show, however, that $r$ cannot become zero for $K\ne 0$.
First we consider geodesics with increasing $r$ (i.e. $\upsilon
> 0$). In this case it is not difficult to see that the term
$${F(\Phi(t))e^{-\gamma a^2r^2}\over \cosh(2at)} M(t,r)$$
in eqn. (\[PGE1\]) can not become singular (for increasing $r$). Therefore, $dt \over ds $ could become singular only for $\upsilon$. We shall show, however, that $\upsilon$ can not become singular for finite values of the affine parameter. For increasing $r$, $\upsilon$ cannot diverge since for large $t$ (large $r$ ) the derivative ${d\upsilon\over ds}$ becomes negative. Indeed, for all exact solutions presented here, there exists a constant $B_{1}>0$ such that
$$\mid{d\ln[F^{-1}(\Phi(t))]\over dt }\mid< B_{1}$$
for arbitrary $t$ and fixed $\lambda$.
Therefore, as can be seen from eqns. (\[GPM\]), the terms associated with the constant $\varepsilon$, $K$ and $L$ are all positive for large values of $t$.As a consequence we obtain that the functions $\Gamma_{+}(t,r)$ and $\Gamma_{-}(t,r)$ are positive, i.e. ${d\upsilon\over ds}<0$ for large $t$(large $r$).[^3]
In the second case,when $r$ decreases ($\upsilon<0$), the problem comes from $r=0$ when $K \ne 0$. The geodesics with $K=0$ can reach the axis $r=0$ without problems and then continue with ${dr\over ds}>0$ ($\upsilon>0$). When $K\ne 0$, $\upsilon$ cannot diverge for finite values of the affine parameter. This follows from the fact that the derivative ${d\upsilon\over ds}$ becomes positive for small $r$ (large $t$) as can be seen from Eqns. (\[GPM\]) and (\[EV\]), taking into account that $\Gamma_{+}(t,r)$ is exponentially suppressed compared with $\Gamma_{-}(t,r)$. The positiveness of the derivative ${d\upsilon\over ds}$ when the geodesics are close to the axis $r=0$ prevents the radial coordinate from collapsing too quickly and reaching the axis. The fact that $r$ can not become zero for $K\ne 0$ may be seen more explicitly as follows. When $r$ approaches zero the dominant term is that associated with $K$ and the other terms can be ignored. So, for small $r$ the geodesics behaves as null geodesics with $L=0$:
$$\begin{aligned}
{dt\over ds} = {F(\Phi(t))e^{-\gamma a^2r^2}\over \cosh(2at)}
M(r)\cosh(\upsilon), \\
{dr\over ds} = {F(\Phi(t))e^{-\gamma a^2r^2}\over \cosh(2at)}
M(r)\sinh(\upsilon), \\
{d\upsilon \over ds } = -{F(\Phi(t))e^{-\gamma a^2r^2}\over
\cosh(2at)}
\partial_{r}M(r)\cosh(\upsilon)\end{aligned}$$
where $M(r)= {|K|\over r}e^{(1/2)\gamma a^2r^2}$. Hence, we obtain the orbit equation
$${dr\over d\upsilon} = -{M(r)\over \partial_{r}M(r)}
\tanh(\upsilon).$$
Integrating, we have
$$e^{-(1/2)\gamma a^2r^2}r = C_{1}\cosh(\upsilon)$$
where $C_{1}>0$ is a constant. Since $\cosh(\upsilon)\ge 1$, $r$ can not become zero.
From the proof of the geodesic completeness it follows that every maximally extended null geodesic intersects any of the hypersurfaces $t=const$. According to [@Geroch], this a sufficient condition that the hypersurfaces $t=const$ are global Cauchy surfaces. Therefore, the solutions are globally hyperbolic.
We have explicitly proven the geodesic completeness of the solutions using their particular properties. The geodesic completeness can be proved independently by considering the solutions from a more general point of view. In Ref.[@LFJ2](see also Ref.[@LFJ3]), Fernandez-Jambrina presented a general theorem providing wide sufficient conditions for an orthogonally transitive cylindrical space-time to be geodesically complete. It can be verified that the solutions presented here satisfy all conditions in the Fernandez-Jambrina’s theorem and therefore they are geodesically complete.
New diagonal solutions can be obtained from (\[STS\]) as a limiting case. Taking $a\rightarrow 0$ and keeping $a^2\gamma=\beta$ fixed, we obtain the following diagonal inhomogeneous cosmological scalar-tensor solutions:
$$\begin{aligned}
ds^2 &=& F^{-1}(\Phi(t)) \left[e^{\beta r^2}(-dt^2 + dr^2) +
r^2d\phi^2 + dz^2 \right], \nonumber\\
8\pi G_{*}\rho &=& \beta f(\lambda)e^{-\beta r^2}F^{3}(\Phi(t)) ,\\
u_{\mu}&=& F^{-1/2}(\Phi(t)) e^{(1/2)\beta
r^2}\delta^{0}_{\mu}\nonumber\end{aligned}$$
where $a\sqrt{\gamma -1}$ should be replaced by $\sqrt{\beta}$ in the explicit formulas for $F^{-1}(\Phi(t))$.
We have proven that the solutions presented in the present paper are geodesically complete. This result is not in contradiction with the well-known singularity theorems because in our case the strong energy condition is violated in the Jordan frame. This can be explicitly seen by calculating the components of the Ricci tensor. All components are bounded except for $R_{tr}= -r \gamma
a^2 \partial_{t} \ln\{F[\Phi(t)]\}$. Therefore, for large enough $r$, one can always find timelike and null vectors $\upsilon^{\mu}$ such that $R_{\mu\nu}\upsilon^{\mu}\upsilon^{\nu}<0$ i.e. the strong energy condition is violated. However, the situation is different in the Einstein frame. The Einstein frame metric $g^{E}_{\mu\nu}$ is just the Mars’s metric and it is geodesically complete as we have already mentioned. Since the energy conditions are satisfied in the Einstein frame it remains to see which other conditions of the singularity theorems are violated. The space-time described by the metric $g^{E}_{\mu\nu}$ does not contain closed trapped surfaces. In order to prove this we will employ the techniques of differential geometry described in Refs.[@SEN2] and [@SEN3]. Let us consider a closed spacelike surface $\cal{S}$ and suppose that it is trapped. Since the surface is compact it must have a point $q$ where $r$ reaches its maximum. Let us denote $r_{max}=R$ on a constant time hypersurface $t=T$. For the traces of both null second fundamental forms at $q$, it can be shown that (see Refs.[@SEN2] and[@SEN3])
$$\begin{aligned}
K^{+}_{\cal{S}}|_{q}\ge {e^{-(1/2)\gamma a^2R^2}\over
\sqrt{2}R \cosh^{1/2}(2aT)}>0 \nonumber , \\
K^{-}_{\cal{S}}|_{q}\le - {e^{-(1/2)\gamma a^2R^2}\over \sqrt{2}R
\cosh^{1/2}(2aT)}<0 .\end{aligned}$$
The traces have opposite signs so that there are no trapped surfaces.
Our solutions are stiff perfect fluid cosmologies and, therefore, the natural question which arises is what happens if the fluid is not stiff. In this case,however, the situation is much more complicated. In contrary to the stiff fluid case, the dilaton-matter sector does not posses nontrivial symmetries which allow us to generate new solutions from known ones.The only way to find exact solutions is to attack directly the corresponding system of coupled partial differential equations. This question is currently under investigation.
Summarizing, in this work we have presented new nondiagonal $G_{2}$ inhomogeneous stiff perfect fluid cosmological solutions in a wide range of scalar-tensor theories. The found solutions have no big-bang nor any other curvature singularity. The gravitational scalar (dilaton) and fluid energy density (pressure) are regular everywhere, too. Moreover, the solutions are globally hyperbolic and geodesically complete. To the best of our knowledge, these solutions are the first examples of nonsingular $G_{2}$ inhomogeneous perfect fluid scalar-tensor cosmologies with a nondiagonal metric.
I would like to thank V. Rizov for discussions and especially L. Fernandez-Jambrina for his valuable comments on the geodesic completeness of the orthogonally transitive cylindrical spacetimes. My thanks also go to J. Senovilla for sending me some valuable papers. This work was supported in part by Sofia University Grant No 459/2001.
M. Gasperini’s web page, http://www.to.infn.it${\tilde{}}$gasperin
, [P. Steinhardt]{}, Phys. Rev. Lett. [**62**]{}, 376 (1989)
, [J. Stein-Schabes]{}, Phys. Lett. [**B 216**]{}, 25 (1989)
, [K. Maeda]{}, Nucl. Phys. [**B 341**]{}, 294 (1990)
, Phys. Rev. [**D 47**]{}, 5329 (1993)
, [J. Mimoso]{}, Phys. Rev. [**D 50**]{}, 3746 (1994)
, [P. Steinhardt]{}, Phys.Rev. [**D 52**]{}, 628 (1995)
, [D. Wands]{}, Phys. Rev. [**D 51**]{}, 477 (1995)
, [D. Wands]{}, Phys. Rev. [**D 52**]{}, 5612 (1995)
, [P. Parsons ]{}, Phys. Rev. [**D 55**]{}, 1906 (1997)
, [P. Martins]{}, Phys. Rev. [**D 61**]{}, 064007-1 (2000)
, [M. Pietroni]{}, Phys. Rev. [**D 61**]{}, 023518 (2000)
, [D. Polarski]{}, Phys. Rev. [**D 63**]{}, 063504 (2001)
, Class.Quant.Grav. [**18**]{} 2977 (2001)
J. Senovilla, Phys. Rev. Lett. [**64**]{}, 2219 (1990)
F. Chinea, L. Fernandez-Jambrina, J. Senovilla, Phys. Rev [**D45**]{}, 481 (1992)
E.Ruiz, J. Senovilla, Phys.Rev.[**D45**]{}, 1995 (1992)
L. Patel, N. Dadhich, Report No. IUCAA-1/93; gr-qc/ 9302001
L. Patel, N. Dadhich, Class. Quant. Grav. [**10**]{}, L85 (1993)
N. Dadhich, L. Patel, R Tikekar, Current Science [**65**]{}, 694 (1993)
J. Senovilla, Phys. Rev. [**D53**]{}, 1799 (1996)
N. Dadhich, J. Astrophys. Astr. [**18**]{}, 343 (1997)
N. Dadhich, A. Raychaudhuri, Mod. Phys. Lett [**A14**]{}, 2135 (1999).
M. Mars, Class. Quant. Grav.[**12**]{}, 2831 (1995)
L. Fernandez-Jambrina, Class. Quant. Grav.[**14**]{}, 3407 (1997)
M. Mars, Phys. Rev. [**D51**]{}, R3989 (1995)
J. Griffiths, J. Bicak, Class. Quant. Grav.[**12**]{}, L81 (1995)
M. Giovannini, Phys. Rev. [**D57**]{}, 7223 (1998)
M. Giovannini, Phys. Rev. [**D59**]{}, 083511 (1999)
L. Pimentel, Mod. Phys. Lett [**A**]{}, [**Vol 14**]{}, 43 (1999)
S. Yazadjiev, Phys. Rev. [**D65**]{}, 084023 (2002)
C. Will, The Confrontation between General Relativity and Experiment,Living Rev.Rel. 4 (2001) 4; E-print gr-qc/0103036
C. Will [*Theory and experiment in graviattional physics*]{} (Cambridge University Press, Cambridge, 1993)
S. Yazadjiev, Talk given at the Meeting of the researchers in physics, Sofia University, Bulgaria, 2002
D. Kramer, H. Stephani, E. Herlt, M. MacCallum, [*Exact Solutions of Einstein’s Field Equations*]{}(Cambridge University Press, Cambridge, 1980)
R. Geroch, J. Math. Phys. [**11**]{}, 437 (1970)
L. Fernandez-Jambrina, J. Math. Phys. [**40**]{}, 4028 (1999)
L. Fernandez-Jambrina, L. M. Gonzalez-Romero, Class.Quant.Grav. [**16**]{}, 953, (1999)
J. Senovilla, Gen. Relativ. Gravit. [**30**]{}, 701 (1998)
J. Senovilla, “Trapped surfaces, horizons, and exact solutions in higher dimensions”, hep-th/0204005
[^1]: E-mail: yazad@phys.uni-sofia.bg
[^2]: Some of the solutions were first obtained by solving the corresponding system of partial differential equations for nondiagonal $G_{2}$ cosmologies [@Y1].
[^3]: In fact, the function $\Gamma_{-}(t,r)$ is exponentially small compared with $\Gamma_{+}(t,r)$ and may not be considered.
|
---
abstract: 'We review some astrophysical and cosmological properties and implications of neutrino masses and mixing angles. These include: constraints based on the relic density of neutrinos, limits on their masses and lifetimes, BBN limits on mass parameters, neutrinos and supernovae, and neutrinos and high energy cosmic rays.'
---
2.0cm
[**Astrophysical and Cosmological Constraints on Neutrino masses [^1]**]{}
1.5 cm
[Kimmo Kainulainen]{} [^2]\
.1cm Theory Division, CERN, CH-1211, Geneva, Switzerland and\
NORDITA, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
.5cm [Keith A. Olive]{} [^3]\
.1cm Theoretical Physics Institute, School of Physics and Astronomy,\
University of Minnesota, Minneapolis, MN 55455, USA
1.5cm
2truecm
Introduction
============
The role of neutrinos in cosmology and astrophysics can not be understated [@dolgovreview]. They play a critical role in the physics of the early Universe, at temperatures scales of order 1 MeV, and strongly determine the abundances of the light elements produced in big bang nucleosynthesis. They almost certainly play a key role in supernova explosions, and if they have mass, could easily contribute to the overall mass density of the Universe. At the present time, the only indicators of neutrino masses are from astrophysical sources, the inferred neutrino oscillations of neutrinos produced in the Sun, and those produced in cosmic-ray collisions in the atmosphere. Indeed, their elusive character has proven that a great deal of information on neutrino properties can be gained by studying their behavior in astrophsyical and cosmological environments. Here, we will try to elucidate some of these constraints on neutrino masses.
In our discussion below, we will assume that the early Universe is well described by a standard Friedmann-Lemaitre-Robertson-Walker metric ds\^2 = dt\^2 - R\^2(t) We further assume that thermal equilibrium was established at some early epoch and that we can describe the radiation by a black body equation of state, $p = \rho/3$ at a temperature $T$. Solutions to Einstein’s equations allow one to determine the expansion rate of the Universe defined to be the Hubble parameter in terms of the energy density in radiation, the curvature and the cosmological constant. In the early Universe the latter two quantities can be neglected and we write H\^2 ( )\^2 = \[H\] where the energy density is = ( \_B g\_[B]{} + [7 8]{} \_F g\_[F]{} ) [\^[ 2]{} 30]{} T\^[4]{} N(T) T\^[4]{} \[rho\] The present neutrino contribution to the total energy density, relative to the critical density (for a spatially flat Universe) is \_= \[omegaratio\] where $\rho_c = 1.06 \times 10^{-5} {h}^2 {\rm GeV/cm}^3$ and $h = H/100$km/Mpc/s is the scaled Hubble parameter. For a recent review of standard big bang cosmology, see [@OP].
The Cosmological Relic Density of Stable Neutrinos {#sec:1.1}
==================================================
The simplicity of the standard big bang model allows one to compute in a straightforward manner the relic density of any stable particle if that particle was once in thermal equilibrium with the thermal radiation bath. At early times, neutrinos were kept in thermal equilibrium by their weak interactions with electrons and positrons. Equilibrium is achieved whenever some rate $\Gamma$ is larger than the expansion rate of the Universe, or $\Gamma_{ i} > H$. Recalling that the age of the Universe is determined by $H^{-1}$, this condition is equivalent to requiring that on average, at least one interaction has occurred over the life-time of the Universe. On dimensional grounds, one can estimate the thermally averaged low-energy weak interaction scattering cross section v g\^4 T\^[ 2]{} /m\_[W]{}\^4 for $T \ll m_W$. Recalling that the number density scales as $n \propto T^3$, we can compare the weak interaction rate $\Gamma \sim n \langle \sigma v \rangle$, with the expansion rate given by eqs. (\[H\]) and (\[rho\]). Neutrinos will be in equilibrium when $\Gamma_{\rm wk} > H$ or T\^3 > m\_[W]{}\^4/M\_[P]{} where $M_{P} = G_{N}^{-1/2} = 1.22 \times 10^{19}$ GeV is the Planck mass. For $N = 43/4$ (accounting for photons, electrons, positrons and three neutrino flavors) we see that equilibrium is maintained at temperatures greater than ${\cal O}(1)$ MeV (for a more accurate calculation see [@Enqvist:gx]).
The decoupling scale of ${\cal O}(1)$ MeV has an important consequence on the final relic density of massive neutrinos. Neutrinos more massive than 1 MeV will begin to annihilate prior to decoupling, and while in equilibrium, their number density will become exponentially suppressed. Lighter neutrinos decouple as radiation on the other hand, and hence do not experience the suppression due to annihilation. Therefore, the calculations of the number density of light ($m_\nu \la 1$ MeV) and heavy ($m_\nu \ga 1$ MeV) neutrinos differ substantially.
The number of density of light neutrinos with $m_\nu \la 1$ MeV can be expressed at late times as \_= m\_Y\_n\_ \[rhonus\] where $Y_\nu = n_\nu/n_\gamma$ is the density of $\nu$’s relative to the density of photons, which today is 411 photons per cm$^3$. It is easy to show that in an adiabatically expanding universe $Y_\nu =
3/11$. This suppression is a result of the $e^+ e^-$ annihilation which occurs after neutrino decoupling and heats the photon bath relative to the neutrinos. In order to obtain an age of the Universe, $t > 12$ Gyr, one requires that the matter component is constrained by h\^2 0.3. \[omegabound\] From this one finds the strong constraint (upper bound) on Majorana neutrino masses: [@cows] m\_[tot]{} = \_m\_28 [eV]{}. \[ml1\] where the sum runs over neutrino mass eigenstates. The limit for Dirac neutrinos depends on the interactions of the right-handed states (see discussion below). As one can see, even very small neutrino masses of order 1 eV, may contribute substantially to the overall relic density. The limit (\[ml1\]) and the corresponding initial rise in $\Omega_\nu h^2$ as a function of $m_\nu$ is displayed in the Figure \[fig:1\] (the low mass end with $m_\nu \la 1$ MeV).
![Summary plot of the relic density of Dirac neutrinos (solid) including a possible neutrino asymmetry of $\eta_\nu =
5\times 10^{-11}$ (dotted).[]{data-label="fig:1"}](fig1a.eps){height="6.0cm"}
The calculation of the relic density for neutrinos more massive than $\sim 1$ MeV, is substantially more involved. The relic density is now determined by the freeze-out of neutrino annihilations which occur at $T \la m_\nu$, after annihilations have begun to seriously reduce their number density [@lw]. The annihilation rate is given by \_[ann]{} = v \_[ann]{} n\_\~ (m\_T)\^[3/2]{} e\^[-m\_/T]{} \[annrate\] where we have assumed, for example, that the annihilation cross section is dominated by $\nu {\bar \nu} \rightarrow f {\bar f}$ via $Z$-boson exchange[^4] and $\langle \sigma v \rangle_{ann} \sim m_\nu^2/m_Z^4$. When the annihilation rate becomes slower than the expansion rate of the Universe the annihilations freeze out and the relative abundance of neutrinos becomes fixed. Roughly, $Y_\nu \sim (m \langle \sigma v \rangle_{ann} )^{-1}$ and hence $\Omega_\nu h^2 \sim {\langle \sigma v \rangle_{ann}}^{-1}$, so that parametrically $\Omega_\nu h^2 \sim 1/{m_\nu^2}$. As a result, the constraint (\[omegabound\]) now leads to a [*lower*]{} bound [@lw; @ko; @wso] on the neutrino mass, of about $m_\nu \ga 3-7$ GeV, depending on whether it is a Dirac or Majorana neutrino. This bound and the corresponding downward trend $\Omega_\nu h^2 \sim
1/m^2_\nu$ can again be seen in Figure \[fig:1\]. The result of a more detailed calculation is shown in Figure \[fig:2\] [@wso] for the case of a Dirac neutrino. The two curves show the slight sensitivity on the temperature scale associated with the quark-hadron transition. The result for a Majorana mass neutrino is qualitatively similar. Indeed, any particle with roughly weak scale cross-sections will tend to give an interesting value of $\Omega h^2 \sim 1$.
![The relic density of heavy Dirac neutrinos due to annihilations [@wso]. The curves are labeled by the assumed quark-hadron phase transition temperature in MeV.[]{data-label="fig:2"}](swo7.eps){height="5cm"}
The deep drop in $\Omega_\nu h^2$, visible in Figure \[fig:1\] at around $m_\nu = M_Z/2$, is due to a very strong annihilation cross section at $Z$-boson pole. For yet higher neutrino masses the $Z$-annihilation channel cross section drops as $\sim 1/m_\nu^2$, leading to a brief period of an increasing trend in $\Omega_\nu h^2$. However, for $m_\nu \ga m_W$ the cross section regains its parametric form $\langle \sigma v \rangle_{ann} \sim m_\nu^2$ due to the opening up of a new annihilation channel to $W$-boson pairs [@Enqvist:1988we], and the density drops again as $\Omega_\nu h^2 \sim 1/m^2_\nu$. The tree level $W$-channel cross section breaks the unitarity at around ${\cal O}({\rm few})$ TeV [@Enqvist:yz] however, and the full cross section must be bound by the unitarity limit [@Griest:1989wd]. This behaves again as $1/m_\nu^2$, whereby $\Omega_\nu h^2$ has to start increasing again, until it becomes too large again at 200-400 TeV [@Griest:1989wd; @Enqvist:yz] (or perhpas somewhat earlier as the weak interactions become strong at the unitarity breaking scale).
Neutrinos as Dark Matter {#sec:1.2}
========================
Based on the leptonic and invisible width of the $Z$ boson, experiments at LEP have determined that the number of neutrinos is $N_\nu = 2.9841 \pm 0.0083$ [@RPP]. Conversely, any new physics must fit within these brackets, and thus LEP excludes additional neutrinos (with standard weak interactions) with masses $m_\nu
\la 45$ GeV. Combined with the limits displayed in Figures \[fig:1\] and \[fig:2\], we see that the mass density of ordinary heavy neutrinos is bound to be very small, $\Omega_\nu {h}^2 < 0.001$ for masses $m_\nu > 45$ GeV up to $m_\nu \sim {\cal O}(100)$ TeV.
A bound on neutrino masses even stonger than Eqn. (\[ml1\]) can be obtained from the recent observations of active-active mixing in both solar- and atmospheric neutrino experiments. The inferred evidence for $\nu_\mu-\nu_\tau$ and $\nu_e-\nu_{\mu,\tau}$ mixings are on the scales $m_\nu^2 \sim 1-10 \times 10^{-5}$ and $m_\nu^2 \sim 2-5 \times 10^{-3}$. When combined with the upper bound on the electon-like neutrino mass $m_{\nu} < 2.8$ eV [@Mainz], and the LEP-limit on the number of neutrino species, one finds the constraint on the sum of neutrino masses: 0.05 [eV]{} m\_[tot]{} 8.4 eV. Conversely, the experimental and observational data then implies that the cosmological energy density of all light, weakly interacting neutrinos can be restricted to the range 0.0005 \_h\^2 0.09. \[range\] Interestingly there is now also a lower bound due to the fact that at least one of the neutrino masses has to be larger than the scale $m^2
\sim 10^{-3}$ eV$^2$ set by the atmospheric neutrino data. Combined with the results on relic mass density of neutrinos and the LEP limits, the bound (\[range\]) implies that the ordinary weakly interacting neutrinos, once the standard dark matter candidate [@ss], can be ruled out completely as a dominant component of the dark matter.
This conclusion can be evaded if neutrinos are Dirac particles, and have a nonzero asymmetry however, since then the relic density could be governed by the asymmetry rather than by the annihilation cross section. Indeed, it is easy to see that the neutrino mass density corresponding to the asymmetry $\eta_\nu \equiv (n_\nu - n_{\bar \nu})/n_\gamma$ is given by [@ho] = m\_\_n\_, which implies \_h\^2 0.004 \_[10]{} (m\_/[GeV]{}). where $\eta_{\nu 10}\equiv 10^{10}\eta_\nu$. We have shown the behaviour of the energy density of neutrinos with an asymmetry by the dotted line in the Figure \[fig:1\]. At low $m_\nu$, the mass density is dominated by the symmetric, relic abundance of both neutrinos and antineutrinos which have already frozen out. At higher values of $m_\nu$, the annihilations suppress the symmetric part of the relic density until $\Omega_\nu h^2$ eventually becomes dominated by the linearly increasing asymmetric contribution. In the figure, we have assumed an asymmetry of $\eta_\nu \sim 5 \times 10^{-11}$ for neutrinos with standard weak interaction strength. In this case, $\Omega_\nu h^2$ begins to rise when $m_\nu \ga 20$ GeV. Obviously, the bound (\[omegabound\]) is saturated for $m_\nu = 75 \, {\rm GeV}/\eta_{\nu 10}$.
There are also other cosmolgical settings that give rise to interesting mass constraints on the eV scale. Indeed, light neutrinos were problematic in cosmology long before the imporoved mass limits leading to (\[range\]) were established, due to their effect on structure formation. Light particles which are still relativistic at the time of matter domination erase primordial perturbations due to free streaming out to very large scales [@free]. Given a neutrino with mass $m_\nu$, the smallest surviving non-linear structures are determined by the Jean’s mass M\_J = 3 10\^[18]{} [M\_(eV)]{}. \[mj\] Thus, for eV mass neutrinos the large scale structures, including filaments and voids [@nu1; @nu2], must form first and galaxies whose typical mass scale is $\simeq 10^{12} M_\odot$ are expected to fragment out later. Particles with this property are termed hot dark matter (HDM). It seemed that neutrinos were ruled out because they tend to produce too much large scale structure [@nu3], and galaxies formed too late [@nu2; @nu4], at $z \le 1$, whereas quasars and galaxies are seen out to redshifts $z \ga 6$.
Subsequent to the demise of the HDM scenario, there was a brief revival for neutrino dark matter as part of a mixed dark matter model, using now more conventional cold dark matter along with a small component of hot (neutrino) dark matter. The motivation for doing this was to recover some of the lost power on large scales that is absent in CDM models [@chdm]. However, galaxies still form late in these models, and more importantly, almost all evidence now points away from models with $\Omega_m = 1$, and strongly favor models with a cosmological constant ($\Lambda$CDM).
Combining the rapidly improving data on key cosmological parameters with the better statistics from large redshift surveys has made it possible to go a step forward along this path. It is now possible to set stringent limits on the light neutrino mass density $\Omega_\nu h^2$, and hence on neutrino mass based on the power spectrum of the Ly $\alpha$ forest [@strong], $m_{\rm tot} < 5.5$ eV, and the limit is even stronger if the total matter density, $\Omega_m$ is less than 0.5. This limit has recently been improved by the 2dF Galaxy redshift [@2dF] survey by comparing the derived power spectrum of fluctuations with structure formation models. Focussing on the the presently favoured $\Lambda$CDM model, the neutrino mass bound becomes $m_{\rm tot} < 1.8 $ eV for $\Omega_m < 0.5$.
Finally, right handed or sterile neutrinos may also contribute to the dark matter. The mass limits for neutrinos with less than full weak strength interactions are relaxed [@OT]. For Dirac neutrinos, the upper limit varies between 100 – 200 eV depending on the strength of their interactions. For Majorana neutrinos, the limit is further relaxed to 200 – 2000 eV. This relaxation is primarily due to the dilution of the number density of super-weakly interacting neutrinos due to entropy production by decay and annihilation of massive states after their decoupling from equilibrium [@oss]. Such neutrinos make excellent warm dark matter candidates, albeit the viable mass range for galaxy formation is quite restricted [@warm].
Neutrinos and Big Bang Nucleosynthesis {#sec:1.3}
======================================
Big bang nucleosynthesis is the cosmological theory of the origin of the light element isotopes D, $^3$He, $^4$He, and $^7$Li [@bbn]. The success of the theory when compared to the observational determinations of the light elements allows one to place strong constraints on the physics of the early Universe at a time scale of 1-100 seconds after the big bang. $^4$He is the most sensitive probe of deviations from the standard model and its abundance is determined primarily by the neutron to proton ratio when nucleosynthesis begins at a temperature of $\sim 100$ keV (to a good approximation all neutrons are then bound to form $^4$He). The ratio $n/p$ is determined by the competition between the weak interaction rates which interconvert neutrons and protons, $$p + e^- \leftrightarrow n + \nu_e \, , \qquad
n + e^+ \leftrightarrow p + \bar \nu_e \, , \qquad
n \leftrightarrow p + e^- + \bar \nu_e
\label{weakrates}$$ and the expansion rate, and is largely given by the Boltzmann factor n/p \~e\^[-(m\_n - m\_p)/T\_f]{} \[nperp\] where $m_n-m_p$ is the neutron to proton mass difference. As in the case of neutinos discussed above, these weak interactions also freeze out at a temperature of roughly 1 MeV when \^2 [T\_f]{}\^5 \~\_[wk]{}(T\_f) = H(T\_f) \~\^2 \[comp\] \[freeze\] The freeze-out condition implies the scaling $T_f^3 \sim \sqrt{N}$. From Eqs. (\[nperp\]) and (\[freeze\]), it is then clear that changes in $N$, caused for example by a change in the number of light neutrinos $N_\nu$, would directly influence $n/p$, and hence the $^4$He abdundance. The dependence of the light element abundances on $N_\nu$ is shown in Figure \[fig:3\] [@cfo2], where plotted is the mass fraction of $^4$He, $Y$, and the abundances by number of the D, $^3$He, and $^7$Li as a function of the baryon-to-photon ratio, $\eta$, for values of $N_\nu = 2 - 7$. As one can see, an upper limit to $Y$, combined with a lower limit to $\eta$ will yield an upper limit to $N_\nu$ [@ssg].
![The light element abundances as a function of the baryon-to-photon ratio for different values of $N_\nu$ [@cfo2].[]{data-label="fig:3"}](bbn1.eps "fig:"){height="9cm"} -1.0truecm
Assuming no new physics at low energies, the value of $\eta$ is the sole input parameter to BBN calculations. It is fixed by the comparison between BBN predictions and the observational determinations of the isotopic abundances [@cfo1]. From $^4$He and $^7$Li, one finds a relatively low value [@Fields:1996yw; @cfo1] for $\eta \sim 2.4 \times 10^{-10}$ corresponding to a low baryon density $\Omega_B h^2 = 0.009$ with a 95% CL range of 0.006 – 0.017. Deuterium, on the other hand, implies a large value of $\eta$ and hence a large baryon density: $\eta \sim 5.8 \times 10^{-10}$ and $\Omega_B h^2 \sim 0.021$ with a 95% CL range of 0.018 – 0.027. The value of the baryon density has also been determined recently from measurements of microwave background anisotropies. The recent result from DASI [@dasi] indicates that $\Omega_B h^2 =
0.022^{+0.004}_{-0.003}$, while that of BOOMERanG-98 [@newboom], $\Omega_B h^2 = 0.021^{+0.004}_{-0.003}$ (using 1$\sigma$ errors).
With the value of $\eta$ fixed, one can use He abundance measurements to set limits on new physics. In particular one can set upper limits the number of neutrino flavors. Taking $Y_p = 0.238 \pm 0.002 \pm
0.005$ (see e.g. [@yp]), we show in Figure \[fig:4\] the likelihood functions for $N_\nu$ based on both the low and high values of $\eta$ [@cfo2]. The curves show the impact of an increasingly accurate determination of $\eta$ from 30% to 3%. If one assumes a 20% uncertainty in $\eta$ (the current uncertainty level), these calculations provide upper limits of $$\begin{aligned}
N_\nu < 3.9 & \qquad \eta = 2.4 \times 10^{-10} \nonumber \\
N_\nu < 3.6 & \qquad \eta = 5.8 \times 10^{-10}
\label{BBNnubounds}\end{aligned}$$ at the 95% CL. Although, as noted above, LEP has already placed very stringent limit to $N_\nu$, the limit (\[BBNnubounds\]) is useful, because it actually applies to the total number of new particle degrees of freedom and is not tied specifically to neutrinos. In fact, more generally, the neutrino limit can be translated into a limit on the expansion rate of the Universe at the time of BBN, which can be applied to a host of other constraints on particle properties.
![(a) The distribution in $N_\nu$ assuming a value of $\eta = 2.4 \times 10^{-10}$ from $^4$He and $^7$Li and the CBI measurement of the microwave background anisotropy [@cfo2]. The curves show the effect of the expected increased accuracy in the CMB determination of $\eta$. (b) (b) As in (a), but assuming a value of $\eta = 5.8 \times 10^{-10}$ from D and the DASI and BOOMERanG measurements of the microwave background anisotropy.[]{data-label="fig:4"}](cfo24.eps "fig:"){height="6.0cm"} -0.5truecm
BBN limits on Neutrino masses and lifetimes {#sec:1.3.1}
-------------------------------------------
As discussed above, the nucleosynthesis prediction for light element abundances is sensitive to the changes in the expansion rate of the universe, which depends on the energy density of the universe during the BBN era (\[H\]). This extra energy density could be in the form of new massless degrees of freedom, in which case their number is directly constrained by equation (\[BBNnubounds\]). Equally well the extra energy density could reside in the form of massive long lived but unstable neutrinos, in which case nucleosynthesis provides interesting constraints on their masses and life-times.
We already pointed out that the relic density of neutrinos strongly depends on whether they decouple while relativistic or nonrelativistic. Here the calculations also depend on how the neutrino life-times relate to the BBN time-scale of about 100 seconds. Also, in order to get reliable results for the light element abundances, one must keep track of the induced perturbations (electron neutrino heating) in the weak reaction rates (\[weakrates\]) in addition to computing changes in the expansion rate. Nevertheless, even in this case it is customary to measure the change in helium abundance in units of equivalent effective neutrino degrees of freedom $N_{\rm eff}$, such that the limit (\[BBNnubounds\]) can be applied on $N_{\rm eff}(\Delta Y(m_\nu,\tau_\nu))$.
When the neutrino life-time is much larger than 100 seconds, neutrinos are effectively stable on a nucleosynthesis scale [@Kolb:1991sn; @FKOandKK; @HaMaDoHa]. While accounting for changes in the rates in Eq. (\[weakrates\]) is important for the detailed bounds, the bulk behaviour of $N_{\rm eff}(m_\nu)$ is dictated by the neutrino mass contribution to the energy density. Obviously, when $m_\nu
\ll 0.1$ MeV, neutrinos are effectively massless during BBN, and $N_{\rm
eff}(m_\nu)
\rightarrow 3$ when $m_\nu \rightarrow 0$. For masses in excess of $0.1$ MeV, but below the neutrino decoupling temperature of ${\cal O}({\rm few})$ MeV, their number density is unsuppressed and their mass density can be large during BBN, causing $N_{\rm eff}$ to increase. For yet larger masses however, the Boltzmann factor shown in (\[annrate\]) begins to suppress the mass density and eventually turns $N_{\rm eff}$ down again. This behaviour is shown in Figure \[fig:5\] for a massive Dirac and Majorana type tau-neutrino [@FKOandKK].
![Plot of the effective number of neutrino degrees of freedom during BBN for a Dirac (dashed) and a Majorana neutrino (solid) [@FKOandKK].[]{data-label="fig:5"}](figure5a.eps){height="5.5cm"}
The bound (\[BBNnubounds\]) yields an excluded region for stable neutrino masses centered around ${\cal O}({\rm few})$ MeV. For $N_\nu < 3.6$ the lower bound is $m_\nu > 42$ MeV (Majorana) and $m_\nu > 30$ MeV (Dirac) [@FKOandKK]. This is only relevant for $\nu_\tau$, and is complementary to the present laboratory limit on the $\tau$-like neutrino, $m_{\nu} < 18$ MeV [@taulab]. Due to contributions from pion decays and inverse decays to neutrinos, the upper bound from BBN depends on the QCD-phase transition temperature, $T_{\rm QCD}$, and is also different for $\tau$- and $\mu$-neutrinos because of their different scattering rates off muons. Imposing again the constraint $N_\nu < 3.6$, and taking $T_{QCD}= 200~{\rm MeV}$ gives [@DKR]: $$\begin{aligned}
m_{\nu} & \la & 230~{\rm KeV} \qquad {\rm \mu-like} \nonumber \\
m_{\nu} & \la & 290~{\rm KeV} \qquad {\rm \tau-like}.\end{aligned}$$ The laboratory limit on the muon-like neutrino, for comparision, is $m_{\nu} \la 170$ keV [@mubound]. To improve this, the BBN limit should be improved to $N_\nu \la 3.4$ [@DKR].
When the neutrino life-time is small or comparable to the nucleosynthesis time scale, one has to account for neutrino decay processes as well. This involves solving for the distributions of the final state decay products, which might include new particles like majorons, and their possible direct effect on BBN. (For example energetic photons would cause the dissociation of the newly generated light nuclei.) Such calculations have been done by many groups [@groupref; @DolHan], and the results are given in exclusion plots in the mass-vs-life-time plane. Constraints are possible for masses of order ${\cal O}(1)$ MeV and life-times of order ${\cal O}(1)$ second. In Figure \[fig:6\], we show a constraint on tau-neutrino masses and life-times as an example (data taken from the reference [@DolHan]).
![Plot of BBN-constraint on masses and life-times of an unstable tau neutrino. Contours are labeled by one (dashed) and two sigma (solid) deviations from the observed value of $Y_p$. The upper right corner is excluded due to too much and the lower part of the graph by too little $^4$He being produced.[]{data-label="fig:6"}](massltime.eps){height="7cm"}
BBN limits on Neutrino mixing parameters {#1.3.2}
----------------------------------------
Despite losing the competitive edge [*w.r.t. *]{} masses and life-times, BBN continues to put interesting limits on other neutrino mass parameters, relevant for neutrino oscillations. Indeed, the LEP-limit of course only applies to neutrinos with weak interactions, while neutrinos without weak interactions, or [*sterile*]{} neutrinos, have been proposed in many different contexts over the years. At present, a prime motivation to introduce sterile neutrinos is to explain the LSND neutrino anomaly [@LSND] in conjunction with the solar and atmospheric neutrino deficits.
BBN on the other hand is sensitive to any type of energy density changing the expansion rate in the ${\cal O}$(0.1-1) MeV range, irrespective of their interactions. It is therefore very interesting to observe that even if no sterile neutrinos were created at very early times, they could be excited by mixing effects in the early universe prior to nucleosynthesis. The basic mechanism is very simple. Suppose that an active state $\nu_{\alpha}$ ($\alpha = e,\mu,\tau$) [*mixes*]{} with a sterile state $\nu_s$. That is, the neutrino mass matrix and hence the Hamiltonian, is not diagonal in the interaction bases. The mixing is further affected by the forward scattering interactions with the background plasma, felt by the active state. As a result, even though a neutrino state was initially produced in purely active projection, after some time $t$ it has become some coherent linear combination of both active and sterile states: $$\nu(t) = c_{\rm e}(t)\nu_{\rm e} + c_{\rm s}(t)\nu_{\rm s}.
\label{mixture}$$ The coherent evolution of this state is interrupted by collisions, which effect a sequence of quantum mechanical measurements of the flavour content of the propagating state. Since the sterile state has no interactions, each measurement is complete, and collapses the wave-function to the sterile state with a probability $P_{\nu_{\rm e} \rightarrow \nu_{\rm s}}(t) = {| c_{\rm s}(t) |}^2$. As a result, the sterile states are populated roughly with an average rate $$\Gamma_{\nu_{\rm s}} = \Gamma_{\nu}{\langle
{| c_{\rm s}(t) |}^2\rangle}_{\rm coll} =
\frac{1}{2}\sin^22\theta_m \Gamma_{\nu_\alpha}.$$ where $\Gamma_{\nu_\alpha}$ is the weak interaction rate of the active state $\nu_\alpha$, and we have assumed that the oscillation time is short in comparision with the collision time scale. The matter mixing angle $\theta_m$ is given by [@EKTBig] $$\sin^22\theta_{\rm m} = \frac{\sin^22\theta_0}
{{1-2 \chi\cos2\theta_0 + \chi^2}}$$ where $\sin2\theta_0$ is the mixing angle in vacuum and $\chi\equiv 2p|V|/\delta m^2$ where $\delta m^2$ is the mass squared difference of the vacuum mass eigenstates, $p\sim T$ is the momentum and $|V|$ is the matter induced effective potential to the Hamiltonian [@Raffelt; @EKTBig]. The weak rate scales as $\Gamma_{\nu_\alpha} \sim T^5$. Moreover $\chi \sim T^6$ at very high temperatures, which causes a strong matter suppression for mixing and hence $\Gamma_{\nu_{\rm s}} \sim T^{-7}$. At very small temperatures $\theta_m \rightarrow \theta_0$ on the other hand, and hence $\Gamma_{\nu_s} \sim T^5$. The rate is thus suppressed both at very large and at very small temperatures [@Kainulainen:bn]. In the intermediate region of ${\cal O}({\rm few})$ MeV however, $\Gamma_{\nu_{\rm s}}$ can exceed the expansion rate bringing a significant amount of sterile neutrinos into equilibrium. An accurate treatment of the problem requires a numerical solution of the appropriate quantum kinetic equations, and the results depend on whether the mostly active state is heavier ($\delta
m^2 <0$) or lighter ($\delta m^2 >0$) of the mixing states. We show the results of such a calculation in Figure \[fig:7\] below [@EKTBig]. The lines are labeled by constant effective number of degrees of freedom during BBN: $\delta N_\nu \equiv
N_\nu - 3$. The most recent limits corresponding to (\[BBNnubounds\]) can be interpolated from the curves shown. It is the area above the curves which is excluded by BBN-limit.
0.5truecm ![Plotted are the BBN constraints on the active sterile neutrino mixing parameters [@EKTBig] for $\nu_\alpha = \nu_e$ (left) and $\nu_\alpha = \nu_{\mu,\tau}$ (right). Regions to the right from the contours labeled by the bound on effective neutrino degrees of freedom $\delta N_{\rm eff} = N_{\rm eff}-3$ are excluded. Shown are also the current regions corresponding to active-sterile mixing parameters for atmospheric (ATM) and the large mixing angle (LMA) solar neutrino solutions.[]{data-label="fig:7"}](EKTfig.eps "fig:"){height="6.5cm"}
The BBN-limit can be converted to an upper bound on the sterile neutrino flux [@Kainulainen:2001cb] in the atmospheric and the solar neutrino observations. Using $N_\nu < 3.6$ one finds $$\begin{aligned}
\sin^2\theta_{\mu \rm s} &\la 0.03& \qquad \rm (Atmospheric) \nonumber\\
\sin^2\theta_{\rm es} &\la 0.06& \qquad \rm (Solar \;\; LMA),
\label{fluxbounds}\end{aligned}$$ whereas the bounds from the atmospheric and solar neutrino experiments are about an order of mangitude weaker: $\sin^2\theta_{\mu \rm s} \la
0.48$ and $\sin^2\theta_{\rm es} \la 0.72$.
The constraints shown in the Figure \[fig:7\] and in Equation (\[fluxbounds\]) depend on the assumption that the primordial lepton asymmetry is not anomalously large [@Enqvist:1990dq]. In ref. [@australia] it was suggested that a large effective asymmetry violating this assumption could actually be generated by oscillations given a particular neutrino mass and mixing hierarchy. For a while these ideas generated a lot of interest, as they would have allowed reconciling all observed anomalies (including LSND) with the nucleosynthesis constraints. However, in these scenarios at least one of the active states would have to be much heavier than the two others, which is not allowed by the atmospheric and solar neutrino flux observations. As a result, the bounds (\[fluxbounds\]) hold, and in particular nucleosynthesis is very much at odds with the possible existence of the LSND-type sterile state. To see this, observe that creating a large enough effective mixing between $\bar \nu_\mu$ and $\bar \nu_{\rm e}$ to explain the anomaly [@LSND] would require a sterile intermediare with $m_{\nu_s} \simeq 1$ eV, and $\sin^22\theta_{\mu\rm e} \simeq
\frac{1}{2}\sin^22\theta_{\mu\rm s} \sin^22\theta_{\rm se} \ga
10^{-2}(\delta m^2/{\rm eV}^2)^{-2}$. In other words at least one of the active-sterile mixings should satisfy $\sin^22\theta_{(\mu, {\rm e})\rm s}
\ga 0.15(\delta m^2/{\rm eV}^2)^{-1}$, which is well within the BBN excluded regions shown in Figure \[fig:7\]. (It would be excluded even by $N_{\rm eff} \la 3.9$, although we do not show that contour in Figure \[fig:7\]).
It should finally be noted that active-active type oscillations have hardly any effect on the expansion rate or the weak interaction rates [@Langacker:1986jv], and hence are not constrained in the above sense by BBN. However, large mixing angle active-active oscillations could equilibrate the lepton [*asymmetries*]{} prior to BBN. This has been used to put strong bounds on muon- and tau-lepton asymmetries [@Lbounds], which exclude the possibility of the degenerate nucleosynthesis.
Neutrinos and Supernovae {#sec:1.4}
========================
Neutrinos have for long been known to play important role in the physics of supernovae. To be sure, it is clear that by far the largest part, roughly 99$\%$, of the gravitational binding energy of about $3\times 10^{53}$ erg involved in the explosion of a type II supernova is carried out by neutrinos, while just 1$\%$ powers the shock wave responsible for blowing out the mantle of the star, and only a tiny fraction of about 0.1$\%$ escapes in the form of light responsible for the spectacular sights observed in the telescopes watching the sky.
The formation of the neutrino burst in the collapse of a type II supernova is rather well understood. The temporal structure of the burst and the energy spectrum of the emitted neutrinos can be computed fairly well [@Giorg]. Existing or planned large scale neutrino detectors [@futurenuexps] can be used to observe deviations from these signatures and to obtain interesting information on neutrino masses and mixing parameters [@SmirnovandX], given a future observation of a Galactic supernova. We show an example of a compilation of neutrino fluxes and spectra in Figure \[fig:8\].
A number of constraints on new physics and in particular on neutrino parameters have already been deduced from the famous SN1987A event in the Small Magellanic Cloud. Of these, perhaps the most direct is the upper bound on the neutrino mass derivable from the maximum duration of the observed duration of the neutrino pulse of about 10 seconds. Given the initial energy spectrum of neutrinos and the distance to the supernova, one can compute the expected spread in the arrival times of the neutrinos to earth as a function of neutrino mass. Comparing the predictions with the observations has been shown to yield the bound [@SNnumassbnd] $$m_{\nu_e} \la 6 - 20 \; {\rm eV}.$$
The observed pulse length also lends to the classic [*cooling argument*]{}: any new physics that would enhance the neutrino diffusion such that the cooling time drops below the observed duration, must be excluded. Cooling arguments have been used to set limits on various neutrino properties [@giorgbook] such as active-sterile neutrino mixing [@itejamuut] and neutrino magnetic moments. The magnetic dipole moment bound in particular was recently revised by Ayala etal. [@Ayala] to $$\mu_{\nu_e} \la 1-4 \times 10^{-12} \mu_B
\label{mubound}$$ which is a couple of orders of magnitude more stringent than the best laboratory bounds, and comparable to the bound coming from the globular cluster red giant cooling arguments [@Raffelt:gv], which give $\mu_{\nu_e} \la 3 \times 10^{-12} \mu_B$.
![Time evolution of neutrino luminosities and average energies: $\nu_x$ represents the spectrum of $\nu_\mu$, $\nu_\tau$, $\bar \nu_\mu$ and $\bar \nu_\tau$. Figure taken from [@futurenuexps]. []{data-label="fig:8"}](nuflux.eps){height="6cm"}
At present, most of the activity concerning neutrinos in supernovae has focussed on the effects of neutrino transport in supernova explosion dynamics rather than with finding constraints on neutrino mixing parameters or masses. Indeed, the details of the physics responsible for the actual visibly observed supernova explosion, including blowing out the stellar mantle, are not very well understood. In particular, the shock wave, which forms deep within the iron core as the infall of matter is reversed due to the stiffening of the nuclear matter equation of state, is typically found to be too weak to explode the star. This is believed to be due to energy loss from the shock and dissociating iron nuclei on its way out from the core to the mantle. In the popular “delayed explosion scenario", the stalling shock wave is rejuvenated by energy transfer to the shock from the huge energy-flux of neutrinos free streaming away from the core. Recent numerical simulations including diffusive neutrino transport do not verify this expectation however [@bosnbang]; while neutrinos definitely help, they do not appear to solve the problem. These results are not conclusive because the diffusive transport equations used so far [@bosnbang] did not include all relevant neutrino interactions, most notably the nuclear brehmsstralung processes [@nubrehm; @Giorg]. Furthermore, processes other than diffusive processes, such as convective flows in the core and behind the shock, appear to play an important role as well [@giorgreview].
It is of course possible that a succesfull SN explosion requires help of some new physics to channel energy more efficiently to the shock, and neutrino-oscillations have already been considered for this role [@fulleretal]. The idea is that $\nu_\mu$ and $\nu_\tau$, interact more weakly and hence escape more energetic from deeper in the core than do electron neutrinos. Arranging mixing parameters so that $\nu_{\mu,\tau}$ turn resonantly to $\nu_{\rm e}$ in the mantle behind the shock could increase the energy deposited to the shock significantly. Unfortunately the mass difference needed for the resonant transition would be very big: $$\delta m^2 \sim 2 E V_{\rm eff} \simeq 1.5 \times 10^5
\rho_e E_\nu^{100}\; {\rm eV}^2$$ where $\rho_e$ is the electron density in units $10^{10}\,\rm g/cm^3$, which at the shock front is about $10^{-3}$, and $ E_\nu^{100}$ is the neutrino energy in units of 100 MeV. So one gets $m_\nu \ga 10$ eV, which is excluded by the present data. A similar idea was behind the suggestion [@Voloshin88], that neutrino magnetic moments induce resonant transitions from $\nu_R$ (which escape energetic from deep in the core) to $\nu_L$ behind the mantle. This mechanism could actually be used to evade the bound (\[mubound\]), but as it demands a magnetic moment of order $\mu_\nu
\sim 10^{-11}\mu_B$ it has problems in coping with the red giant cooling bound [@Raffelt:gv].
Sterile neutrinos could also be relevant for supernovae by alleviating the problems with the r-process nucleosynthesis, which is thought to be responsible for creation of the most heavy elements. In the standard SN calculations the r-process nucleosynthesis is not effective due to too efficient de-neutronization by the processes $\nu_e + n \rightarrow e^- + p$. If electron neutrinos mixed with a sterile state however, these processes could be made less effective, increasing the neutron density in the mantle, and hence improving the r-process efficiency [@rprocSN].
Finally, there is also the old problem of the “kick"-velocities of pulsars (neutron star remnants of supernova explosions). It has proven difficult to arrange for these velocities, which average around 450 km/sec, based on the normal fluid dynamics in asymmetrical SN-explosions. The momentum carried out by neutrinos $p_\nu \simeq E_\nu$, on the other hand, is about 100 times larger than the pulsar kinetic energy, so that a mere one per cent asymmetry in the neutrino emission would be enough to power the pulsar velocities. Interesting attempts have been made to explain such asymmetric emission by an asymmetric distribution of inhomogeneities in the SN magnetic fields, combined with a large neutrino magnetic moment [@Voloshin88], or just the magnetic field induced deformation of the neutrino spheres [@kusenkosegre]. However, the former would again need probably too large magnetic moment to work, and a detailed analysis of the latter suggests that the a symmetric flux is very suppressed, requiring perhpas unrealistically large magnetic fields; according to [@jankaraffelt] the field needs to be in excess of $10^{17} G$, while [@KusenkoLater] argue that a field of $10^{14-15} G$ would suffice. The true nature of the physics explaining the kick velocities may remain ambiguous for some time to come, but a neutrino solution looks definitely appealing from the pure energetics point of view.
Before we conclude this section, we would like to mention, several other astrophysical limits on neutrino properties. A sure limit to the mean life/mass ratio is obtained from solar x- and $\gamma$- ray fluxes [@raffmean]. The limit is $\tau/m_{\nu_1} > 7
\times 10^9$ s/eV. This is far superior to the laboratory bound of 300 s/eV [@reines]. Other much stronger limits ($> O(10^{15}$) s/eV) are available from the lack of observation of $\gamma$-rays in coincidence with neutrinos from SN 1987A [@other]. This latter limit applies to the heavier neutrino mass eigenstates, $\nu_2$ and $\nu_3$, as well.
Neutrinos and Cosmic Rays {#sec:1.5}
=========================
One of the most interesting puzzles in astrophysics today concerns the observations of ultra high energy cosmic rays (UHECR), beyond the so called Greisen-Zatsepin-Kuzmin (GZK) cutoff $$E_{\rm GZK} \simeq 5 \times 10^{19} \, \rm eV.$$ The problem is that cosmic rays at these energies necessarily need to be of extragalactic origin, since their gyromagnetic radius within the galactic magnetic field far exceeds galactic dimensions. Yet, the attenuation lengths of both protons and photons are rather small in comparision with intergalactic distances, and neither can have originated by further than about $50$ Mpc away from us, due to their scattering off the intergalactic cosmic photon background. As a result, one would expect that the cosmic ray spectrum would abruptly end around $E \sim E_{\rm GZK}$ due to scattering off of the microwave background. This cutoff is represented by the dotted line in Figure \[fig:9\]. In contrast, several groups, most notably the HiRes [@hires] and AGASA [@Agasa] collaborations, have reported events with energies well above the GZK-cutoff: the latest compliation of AGASA, for example, contains 10 events above the scale $E > 10^{20}$ eV, observed since 1993.
![Sceled by $E^3$ spectrum of highest energy cosmic rays near the GZK cutoff. The dotted line corresponds to the expectation from uniformly distributed extragalactic sources, and the solid line shows the prediction of a $Z$-burst model of ref. [@gelmini]. (Figure modified from the orginal found at the AGASA web page [@Agasa].)[]{data-label="fig:9"}](spectrum.eps "fig:"){height="6.5cm"} -0.4truecm
The origin of UHECR’s has been the subject of lively discussions over the last few years. Apart from having astrophysical orgins, being acclerated in extreme environments at extragalactic distances (in AGN’s, GRB’s or Blazars), they have been attributed to the decay products of very heavy particles or of topological defects. All these explanations have problems, however. Astrophysical explanations face the difficult task of accelerating particles to the extreme energies required, with little or no associated sub-TeV-scale photonic component (as none have been ever observed). This is in addition to the above mentioned problem of the propagation of UHECR’s over extragalactic distances. Decay explanations are somewhat disfavoured by the growing evidence of doublets and triplets in AGASA data, and by the correlation between the UHECR arrival directions with far compact Blazars [@tinyakov], which appear rather to be pointing towards astrophysical origin.
The attenuation problem for extragalactical UHECR’s can be avoided however, if they cross the universe in the form of a neutrino beam, since neutrinos travel practically free over super-Hubble distances. Indeed, should this be the case, the initial $\nu_{\rm UHE}$’s could occasionally interact with the cosmological relic neutrino background close to us, giving rise to “$Z$-bursts" of hadrons and photons [@weiler], which then would give rise to the observed UHECR events. Indeed, given a neutrino mass $m_\nu$, such a collision has sufficient CM-energy for resonant $Z$ production if $$E_\nu = \frac{M^2_Z}{2m_\nu} \simeq 4.2 \times 10^{21} {\rm eV}
\left( \frac{{\rm eV}}{m_\nu} \right).$$ The requirement of super-GZK-energies for the initial $\nu_{\rm UHE}$ beam leads immediately to the interesting mass scale for neutrinos: $m_\nu \sim {\cal O}(1)$ eV.
$Z$-burst models have been extensively studied lately [@fodor]. For example, in ref. [@gelmini] it was shown that a $Z$-burst model with $m_\nu = 0.07$ eV, corresponding to a degenerate neutrino spectrum, could reproduce the AGASA-data including the spectral features, such as the “ankle" and the “bump" observed at $E \la E_{\rm GZK}$. The best fit model of ref. [@gelmini] is shown by the solid line in Figure \[fig:9\]. While the agreement with AGASA data is good, it should be noted that this model predicts that cosmic ray primaries are exclusively photons above $E \ga 10^{20}$ eV, whereas the $E \simeq 3 \times 10^{20}$ eV event observed by Fly’s Eye is almost certainly not caused by a photon [@Haltzen]. The model also predicts a large increase of the cosmic ray flux above few$\times 10^{20}$ eV (as a direct result of the huge initial energy needed: $E_{\nu_{\rm UHE}}
\simeq 6 \times 10^{22}$ eV), which excacerbates the already outstanding problem of the origin of UHECR’s. These problems could be ameliorated by assuming somewhat larger neutrino masses, and it has been argued by Fodor [*etal. *]{}[@fodor], that the $Z$-burst scenario can already be used to [*constrain*]{} the mass, plausibly to within $m_\nu
\sim 0.08 - 1.3$ eV, in very good agreement with other mass determinations. The analysis of ref. [@fodor], was restricted to $Z$-resonance interactions however, while for higher CM-energies the cross section [@Enqvist:1988we] for pair production of gauge bosons $\nu\nu\rightarrow ZZ,WW$ becomes important. These reactions have been shown to give accptable solutions with larger neutrino masses $m_\nu \ga 3$ eV [@fargion].
In summary, although the origin of UHECR’s remains a mystery today, it is almost certain that if it has to do with extragalactic sources, neutrino physics plays a very important role in the interpretation of these events.
Acknowledgments {#acknowledgments .unnumbered}
===============
K.K. would like to thank Steen Hannestad and Petteri Keranen for useful advice. The work of K.A.O. was supported partly by DOE grant DE–FG02–94ER–40823.
[99]{} A. D. Dolgov, arXiv:hep-ph/0202122. K. Olive and J.A. Peacock, to appear in the Review of Particle Properties, 2002.
K. Enqvist, K. Kainulainen and V. Semikoz, Nucl. Phys. B [**374**]{}, 392 (1992).
S. S. Gerstein and Ya. B. Zeldovich, JETP Letters [**4**]{} 647 (1966); R. Cowsik and J. McClelland, Phys. Rev. Lett. [**29**]{}, 669 (1972); A.S. Szalay and G. Marx, Astron. Astrophys. [**49**]{}, 437 (1976).
P. Hut, Phys. Lett. B[**69**]{}, 85 (1977); B.W. Lee and S. Weinberg, Phys. Rev. Lett. [**39**]{}, 165 (1977); M. I. Vysotsky, A. D. Dolgov and Y. B. Zeldovich, Pisma Zh. Eksp. Teor. Fiz. [**26**]{} 200 (1977). E.W. Kolb and K.A. Olive, Phys. Rev. [**D33**]{}, 1202 (1986); E: [**34**]{}, 2531 (1986)
R. Watkins, M. Srednicki and K.A. Olive, Nucl. Phys. [**B310**]{}, 693 (1988).
K. Enqvist, K. Kainulainen and J. Maalampi, Nucl. Phys. B [**317**]{}, 647 (1989). K. Enqvist and K. Kainulainen, Phys. Lett. B [**264**]{}, 367 (1991). K. Griest and M. Kamionkowski, Phys. Rev. Lett. [**64**]{}, 615 (1990). D. Abbaneo [*et al.*]{} \[ALEPH Collaboration\], arXiv:hep-ex/0112021.
Ch. Weinheimer, et al, Phys. Lett. B[**460**]{} (1999) 219.
D.N. Schramm and G. Steigman, Ap. J. [**243**]{} (1981) 1.
P. Hut, K.A. Olive, Phys. Lett. B[**87**]{} (1979) 144.
J.R. Bond, G. Efstathiou and J. Silk, [ Phys. Lett. ]{} [**45**]{} (1980) 1980; Ya.B. Zeldovich and R.A. Sunyaev, [ Sov. Ast. Lett. ]{} [**6**]{} (1980) 457.
P.J.E. Peebles, [ Ap. J.]{} [**258**]{} (1982) 415; A. Melott, [ M.N.R.A.S. ]{} [**202**]{} (1983) 595; A.A. Klypin, S.F. Shandarin, [ M.N.R.A.S. ]{} [**204**]{} (1983) 891.
C.S. Frenk, S.D.M. White and M. Davis, [ Ap. J. ]{} [**271**]{} (1983) 417.
S.D.M. White, C.S. Frenk and M. Davis, [ Ap. J. ]{} [**274**]{} (1983) 61.
J.R. Bond, J. Centrella, A.S. Szalay and J. Wilson, in [*Formation and Evolution of Galaxies and Large Structures in the Universe*]{}, ed. J. Andouze and J. Tran Thanh Van, (Dordrecht-Reidel 1983) p. 87.
M. Davis, F. J. Summers and D. Schlegel, Nature [**359**]{} (1992) 393; A. Klypin, J. Holtzman, J. Primack and E. Regos, Astrophys. J. [**416**]{}, 1 (1993). R. A. Croft, W. Hu and R. Dave, Phys. Rev. Lett. [**83**]{}, 1092 (1999) \[arXiv:astro-ph/9903335\]. Ø. Elgaroy et. al. , \[arXiv:astro-ph/0204152\].
K. A. Olive and M. S. Turner, Phys. Rev. D [**25**]{}, 213 (1982). G. Steigman, K. A. Olive and D. N. Schramm, Phys. Rev. Lett. [**43**]{} (1979) 239; K. A. Olive, D. N. Schramm and G. Steigman, Nucl. Phys. B [**180**]{}, 497 (1981). S. H. Hansen, J. Lesgourgues, S. Pastor and J. Silk, \[arXiv:astro-ph/0106108\]. K. Olive, G. Steigman, and T.P. Walker, Phys. Rep. [**333**]{}, 389 (2000); B.D. Fields and S. Sarkar, to appear in the Review of Particle Properties, 2002.
R. H. Cyburt, B. D. Fields and K. A. Olive, Astropart. Phys. [**17**]{}, 87 (2002) \[arXiv:astro-ph/0105397\]. G. Steigman, D.N. Schramm, and J. Gunn, Phys. Lett. [**B66**]{}, 202 (1977)
R.H. Cyburt, B.D. Fields, and K.A. Olive, New Astron. [**6**]{}, 215 (2001)
B. D. Fields, K. Kainulainen, K. A. Olive and D. Thomas, New Astron. [**1**]{}, 77 (1996). C. Pryke et al., Ap. J. [**568**]{}, 46 (2002), \[arXiv:astro-ph/0104490\]
C.B. Netterfield et al., (2001), \[arXiv:astro-ph/0104460\]
B.D. Fields and K.A. Olive, Astrophys. J. [**506**]{}, 177 (1998); A. Peimbert, M. Peimbert, and V. Luridiana, Ap. J. [**565**]{}, 668 (2002), \[arXiv:astro-ph/0107189\]
E. W. Kolb, M. S. Turner, A. Chakravorty and D. N. Schramm, Phys. Rev. Lett. [**67**]{}, 533 (1991). B. D. Fields, K. Kainulainen and K. A. Olive, Astropart. Phys. [**6**]{}, 169 (1997); K. Kainulainen, in Neutrino physics and astrophysics 1996, Helsinki, Finland \[arXiv:hep-ph/9608215\]. S. Hannestad and J. Madsen, Phys. Rev. Lett. [**76**]{}, 2848 (1996) \[Erratum-ibid. [**77**]{}, 5148 (1996)\]; A. D. Dolgov, S. H. Hansen and D. V. Semikoz, Nucl. Phys. B [**524**]{} (1998) 621. L. Passalacqua, Nucl. Phys. Proc. Suppl. [**55C**]{} (1997) 435.
A. D. Dolgov, K. Kainulainen and I. Z. Rothstein, Phys. Rev. D [**51**]{}, 4129 (1995). K. Assamagan, et al, Phys. Rev. D [**53**]{} (1996) 6065.
M. Kawasaki, P. Kernan, H. S. Kang, R. J. Scherrer, G. Steigman and T. P. Walker, Nucl. Phys. B [**419**]{}, 105 (1994); S. Dodelson, G. Gyuk and M. S. Turner, Phys. Rev. D [**49**]{}, 5068 (1994); S. Hannestad, Phys. Rev. D [**57**]{} (1998) 2213. A. D. Dolgov, S. H. Hansen, S. Pastor and D. V. Semikoz, Nucl. Phys. B [**548**]{} (1999) 385. A. Aguilar et al, Phys. Rev. D[**64**]{} (2001) 112007.
K. Enqvist, K. Kainulainen and M. J. Thomson, Nucl. Phys. B [**373**]{}, 498 (1992). D. Notzold and G. Raffelt, Nucl. Phys. B [**307**]{}, 924 (1988).
K. Kainulainen, Phys. Lett. B [**244**]{}, 191 (1990).
K. Kainulainen and A. Sorri, JHEP [**0202**]{}, 020 (2002). K. Enqvist, K. Kainulainen and J. Maalampi, Phys. Lett. B [**244**]{}, 186 (1990). R. Foot and R. R. Volkas, Phys. Rev. Lett. [**75**]{}, 4350 (1995); R. Foot, M. J. Thomson and R. R. Volkas, Phys. Rev. D [**53**]{}, 5349 (1996). P. Langacker, S. T. Petcov, G. Steigman and S. Toshev, Nucl. Phys. B [**282**]{} (1987) 589. A. D. Dolgov, S. H. Hansen, S. Pastor, S. T. Petcov, G. G. Raffelt and D. V. Semikoz, Nucl. Phys. B [**632**]{} (2002) 363 \[arXiv:hep-ph/0201287\]; K. N. Abazajian, J. F. Beacom and N. F. Bell, \[arXiv:astro-ph/0203442\]; Y. Y. Wong, arXiv:hep-ph/0203180. G. G. Raffelt, \[arXiv:astro-ph/0105250\]. T. Totani, K. Sato, H. E. Dalhed and J. R. Wilson, Astrophys. J. [**496**]{}, 216 (1998). A. S. Dighe and A. Y. Smirnov, Phys. Rev. D [**62**]{}, 033007 (2000); J. F. Beacom, R. N. Boyd and A. Mezzacappa, Phys. Rev. Lett. [**85**]{}, 3568 (2000).
T. J. Loredo and D. Q. Lamb, Ann. N.Y. Acad. Sci. [**571**]{} (1989) 601; Phys. Rev. D [**65**]{}, 063002 (2002). G. Raffelt, Ann. Rev. Nucl. Part. Sci. [**49**]{} (1999) 163.
D. Notzold, Phys. Lett. B [**196**]{}, 315 (1987). K. Kainulainen, J. Maalampi and J. T. Peltoniemi, Nucl. Phys. B [**358**]{} (1991) 435. A. Ayala, J. C. D’Olivo and M. Torres, Phys. Rev. D [**59**]{} (1999) 111901. G. G. Raffelt, Phys. Rept. [**320**]{} (1999) 319. M. Rampp and H. T. Janka, Astrophys. J. [**539**]{} (2000) L33; \[arXiv:astro-ph/0005438\]. M. Liebendorfer et al, Phys. Rev. D [**63**]{}, 103004 (2001). S. Hannestad and G. Raffelt, Astrophys. J. [**507**]{} (1998) 339. G. G. Raffelt, arXiv:hep-ph/0201099. G. M. Fuller et al, Astrophys. J. [**389**]{} (1992) 517.
M. B. Voloshin, Phys. Lett. B [**209**]{} (1988) 360. C. J. Horowitz, \[arXiv:astro-ph/0108113\] (2002); Y.-Z. Qian, \[arXiv:astro-ph/0203194\] (2002).
A. Kusenko and G. Segre, Phys. Rev. Lett. [**77**]{}, 4872 (1996). H. T. Janka and G. G. Raffelt, Phys. Rev. D [**59**]{}, 023005 (1999) \[arXiv:astro-ph/9808099\]. A. Kusenko and G. Segre, Phys. Rev. D [**59**]{} (1999) 061302 \[arXiv:astro-ph/9811144\]; M. Barkovich, J. C. D’Olivo, R. Montemayor and J. F. Zanella, \[arXiv:astro-ph/0206471\].
G. G. Raffelt, Phys. Rev. D [**31**]{} (1985) 3002. F. Reines, H. W. Sobel and H. S. Gurr, Phys. Rev. Lett. [**32**]{} (1974) 180. F. Von Feilitzsch and L. Oberauer, Phys. Lett. B [**200**]{} (1988) 580. E. L. Chupp, W. T. Vestrand and C. Reppin, Phys. Rev. Lett. [**62**]{} (1989) 505. E. W. Kolb and M. S. Turner, Phys. Rev. Lett. [**62**]{}, 509 (1989). S. A. Bludman, Phys. Rev. D [**45**]{} (1992) 4720.
D. J. Bird [ et al.]{} \[HIRES Collaboration\], Astrophys. J. [**424**]{} (1994) 491. M. Takeda [ et al.]{}, Phys. Rev. Lett. [**81**]{} (1998) 1163. The AGASA home page: www-akeno.icrr.u-tokyo.ac.jp/AGASA/
G. Gelmini and G. Varieschi, arXiv:hep-ph/0201273. P. G. Tinyakov and I. I. Tkachev, JETP Lett. [**74**]{} (2001) 445. T. J. Weiler, Astropart. Phys. [**11**]{} (1999) 303. Z. Fodor, S. D. Katz and A. Ringwald, arXiv:hep-ph/0203198. See for example, F. Halzen,\[arXiv:astro-ph/0111059\].
D. Fargion, P. G. De Sanctis Lucentini, M. Grossi, M. De Santis and B. Mele, arXiv:hep-ph/0112014; D. Fargion, B. Mele and A. Salis, Astrophys. J. [**517**]{} (1999) 725.
[^1]: to appear in “Neutrino Mass”, Springer Tracts in Modern Physics, ed. by G. Altarelli and K. Winter.
[^2]: e-mail address: kimmo.kainulainen@cern.ch; Permanent address: University of Jyväskylä, Finland.
[^3]: e-mail address: olive@umn.edu
[^4]: While this is approximately true for Dirac neutrinos, the annihilation cross section of Majorana neutrinos is $p$-wave suppressed and is proportional of the final state fermion masses rather than $m_\nu$.
|
---
abstract: |
In this paper, we give new sparse interpolation algorithms for black box polynomial $f$ whose coefficients are from a finite set. In the univariate case, we recover $f$ from one evaluation $f(\beta)$ for a sufficiently large number $\beta$. In the multivariate case, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial to that of the univariate case. Both algorithms have polynomial bit-size complexity.
. Sparse polynomial interpolation, modified Kronecker substitution, polynomial time algorithms.
author:
- |
Qiaolong Huang and Xiao-Shan Gao\
KLMM, UCAS, Academy of Mathematics and Systems Science\
Chinese Academy of Sciences, Beijing 100190, China
title: |
Sparse Polynomial Interpolation with\
Finitely Many Values for the Coefficients[^1]
---
Introduction
============
The interpolation for a sparse multivariate polynomial $f(x_1,\ldots,x_n)$ given as a black box is a basic computational problem. Interpolation algorithms were given when we know an upper bound for the terms of $f$ [@1] and upper bounds for the terms and the degrees of $f$ [@71]. These algorithms were significantly improved and these works can be found in the references of [@72]. In this paper, we consider the sparse interpolation for $f$ whose coefficients are taken from a known finite set. For example, $f$ could be in $\Z[x_1,\ldots,x_n]$ with an upper bound on the absolute values of coefficients of $f$, or $f$ is in $\Q[x_1,\ldots,x_n]$ with upper bounds both on the absolute values of coefficients and their denominators. This kind of interpolation is motivated by the following applications. The interpolation of sparse rational functions leads to interpolation of sparse polynomials whose coefficients have bounded denominators [@812 p.6]. In [@8123], a new method is introduced to reduce the interpolation of a multivariate polynomial $f$ into the interpolation of univariate polynomials, where we need to obtain the terms of $f$ from a larger set of terms and the method given in this paper is needed to solve this problem.
In the univariate case, we show that if $\beta$ is larger than a given bound depending on the coefficients of $f$, then $f$ can be recovered from $f(\beta)$. Based on this idea, we give a sparse interpolation algorithm for univariate polynomials with rational numbers as coefficients, whose bit complexity is $\mathcal{O}((td\log H(\log C+\log H))$ or $\widetilde{\mathcal{O}}(td)$, where $t$ is the number of terms of $f$, $d$ is the degree of $f$, $C$ and $H$ are upper bounds for the coefficients and the denominators of the coefficients of $f$. It seems that the algorithm has the optimal bit complexity $\widetilde{\mathcal{O}}(td)$ in all known deterministic and exact interpolation algorithms for black box univariate polynomials as discussed in Remark \[rem-1\].
In the multivariate case, we show that by choosing a good prime, the interpolation of a multivariate polynomial can be reduced to that of the univariate case in polynomial-time. As a consequence, a new sparse interpolation algorithm for multivariate polynomials is given, which has polynomial bit-size complexity. We also give its probabilistic version. There exist many methods for reducing the interpolation of a multivariate polynomial into that of univariate polynomials, like the classical Kronecker substitution, randomize Kronecker substitutions[@7], Zipple’s algorithm[@71], Klivans-Spielman’s algorithm[@5], Garg-Schost’s algorithm [@21], and Giesbrecht-Roche’s algorithm[@6]. Using the original Kronecker substitution [@2], interpolation for multivariate polynomials can be easily reduced to the univariate case. The main problem with this approach is that the highest degree of the univariate polynomial and the height of the data in the algorithm are exponential. In this paper, we give the following modified Kronecker substitution $$x_i=x^{\mathbf{mod}((D+1)^{i-1},p)},i=1,2,\dots,n$$ to reduce multivariate interpolations to univariate interpolations. Our approach simplifies and builds on previous work by Garg-Schost[@21], Giesbrecht-Roche[@6], and Klivans-Spielman[@5]. The first two are for straight-line programs. Our interpolation algorithm works for the more general setting of black box sampling.
The rest of this paper is organized as follows. In Section 2, we give interpolation algorithms about univariate polynomials. In Section 3, we give interpolation algorithms about multivariate polynomials. In Section 4, experimental results are presented.
Univariate polynomial interpolation
===================================
Sparse interpolation with finitely many coefficients
----------------------------------------------------
In this section, we always assume $$\label{eq-f}
f(x)=c_1x^{d_1}+c_2x^{d_2}+\dots+c_tx^{d_t}$$ where $d_1,d_2,\dots,d_t\in \N,d_1<d_2<\cdots<d_t$, and $c_1,c_2,\cdots,c_t\in A$, where $A\subset \mathbb{C}$ is a finite set. Introduce the following notations $$\label{eq-e}
C:=\max_{a\in A}(|a|),\quad \varepsilon:=\min (\varepsilon_1,\varepsilon_2)$$ where $\varepsilon_1 :=\min_{a,b\in A,a\neq b}|a-b|$ and $\varepsilon_2 :=\min_{a\in A,a\neq 0} |a|$.
\[the-5\] If $\beta\geq \frac{2C}{\varepsilon}+1$, then $f(x)$ can be uniquely determined by $f(\beta)$.
Firstly, for $\forall k=1,2,\cdots$, we have $$\begin{aligned}
\beta\geq \frac{2C}{\varepsilon}+1\Longrightarrow{} & \beta-1\geq \frac{2C}{\varepsilon}\notag \\
\Longrightarrow{} & \beta-1>\frac{2C}{\varepsilon}\frac{\beta^k-1}{\beta^k}\notag\\
\Longrightarrow{} & \varepsilon \beta^k>2C\frac{\beta^k-1}{\beta-1}\notag\\
\Longrightarrow{} & \varepsilon \beta^k>2C(\beta^{k-1}+\beta^{k-2}+\cdots+\beta+1)\notag\end{aligned}$$ From , we have $f(\beta)=c_1\beta^{d_1}+c_2\beta^{d_2}+\cdots+c_t\beta^{d_t}$. Assume that there is another form $f(\beta)=a_1\beta^{k_1}+a_2\beta^{k_2}+\dots+a_s\beta^{k_s}$, where $a_1,a_1,\dots,a_s \in A$ and $k_1<k_2<\cdots<k_s$. It suffices to show that $c_t\beta^{d_t}=a_s\beta^{k_s}$. The rest can be proved by induction. First assume that $d_t\neq k_s$. Without loss of generality, let $d_t>k_s$. Then we have $$\begin{aligned}
0=&|(c_1\beta^{d_1}+c_2\beta^{d_2}+\cdots+c_t\beta^{d_t})-(a_1\beta^{k_1}+a_2\beta^{k_2}+\cdots+a_s\beta^{k_s})|\notag\\
\geq{} & |c_t| \beta^{d_t}-C(\beta^{d_t-1}+\cdots+\beta+1)-C(\beta^{k_s}+\cdots+\beta+1)\notag\\
\geq{} &|c_t| \beta^{d_t}-2C(\beta^{d_t-1}+\cdots+\beta+1)\notag\\
>{} &|c_t| \beta^{d_t}-\varepsilon \beta^{d_t}\geq0\notag
\end{aligned}$$ It is a contradiction, so $d_t=k_s$. Assume $c_t\neq a_s$, then $$\begin{aligned}
0=&|(c_1\beta^{d_1}+c_2\beta^{d_2}+\cdots+c_t\beta^{d_t})-(a_1\beta^{k_1}+a_2\beta^{k_2}+\cdots+a_s\beta^{k_s})\notag\\
\geq{} &|c_t-a_s| \beta^{d_t}-2C(\beta^{d_t-1}+\cdots+\beta+1)\notag\\
>{} &|c_t-a_s| \beta^{d_t}-\varepsilon \beta^{d_t}\geq0\notag\end{aligned}$$ It is a contradiction, so $c_t=a_s$. The theorem has been proved.
The sparse interpolation algorithm
----------------------------------
The idea of the algorithm is first to obtain the maximum term $m$ of $f$, then subtract $m(\beta)$ from $f(\beta)$ and repeat the procedure until $f(\beta)$ becomes $0$. We first show how to compute the leading degree $d_t$.
\[the-1\] If $\beta\geq \frac{2C}{\varepsilon}+1$, then $|\frac{f(\beta)}{\beta^k}|=\begin{cases}
>\frac{\varepsilon}{2},&\text{if } k\leq d_t\\
<\frac{\varepsilon}{2},&\text{if } k> d_t
\end{cases}
$
From $|f(\beta)|=|c_1\beta^{d_1}+c_2\beta^{d_2}+\cdots+c_t\beta^{d_t}|\leq{} C(\beta^{d_t}+\cdots+\beta+1)={} C(\frac{\beta^{d_t+1}-1}{\beta-1})$ and $|f(\beta)|={} |c_1\beta^{d_1}+c_2\beta^{d_2}+\cdots+c_t\beta^{d_t}|
\geq{} |c_t|\beta^{d_t}-C(\beta^{d_t-1}+\cdots+\beta+1)
={} |c_t|\beta^{d_t}-C\frac{\beta^{d_t}-1}{\beta-1},$ we have $$|c_t|\beta^{d_t}-C\frac{\beta^{d_t}-1}{\beta-1}\leq |f(\beta)| \leq C (\frac{\beta^{d_t+1}-1}{\beta-1}).$$ When $k\leq d_t$, $|\frac{f(\beta)}{\beta^k}|\geq{} |c_t|\beta^{d_t-k}-\frac{C}{\beta-1}(\beta^{d_t-k}-\frac{1}{\beta^k})
\geq{} \varepsilon \beta^{d_t-k}-\frac{\varepsilon}{2}(\beta^{d_t-k}-\frac{1}{\beta^k})
\geq{} \frac{\varepsilon}{2}\beta^{d_t-k}+\frac{\varepsilon}{2}\frac{1}{\beta^k}>\frac{\varepsilon}{2}$. When $k>d_t$, $|\frac{f(\beta)}{\beta^k}|\leq{} \frac{C}{\beta-1}(\beta^{d_t+1-k}-\frac{1}{\beta^k})
\leq{} \frac{\varepsilon}{2}(\beta^{d_t+1-k}-\frac{1}{\beta^k})
\leq{} \frac{\varepsilon}{2}\beta^{d_t+1-k}-\frac{\varepsilon}{2}\frac{1}{\beta^k}<\frac{\varepsilon}{2}$.
If we can use logarithm operation, we can change the above lemma into the following form.
\[lm-1\] If $\beta\geq \frac{2C}{\varepsilon}+1$, then $d_t=\lfloor\log_\beta \frac{2|f(\beta)|}{\varepsilon}\rfloor$.
By lemma \[the-1\], we know $\frac{|f(\beta)|}{\beta^{d_t}}>\frac{\varepsilon}{2}$ and $\frac{|f(\beta)|}{\beta^{d_t+1}}<\frac{\varepsilon}{2}$. Then we have $\log_\beta \frac{|f(\beta)|}{\beta^{d_t}}>\log_\beta \frac{\varepsilon}{2}$ and $\log_\beta\frac{|f(\beta)|}{\beta^{d_t+1}}<\log_\beta\frac{\varepsilon}{2}$, this can be reduced into $\log_\beta \frac{2|f(\beta)|}{\varepsilon}-1<d_t<\log_\beta \frac{2|f(\beta)|}{\varepsilon}$. As $d_t$ is an integer, then we have $d_t=\lfloor\log_\beta \frac{2|f(\beta)|}{\varepsilon}\rfloor$.
Based on Lemma \[lm-1\], we have the following algorithm which will be used in several places.
\[alg-2\]
[**Input:**]{} $f(\beta),\varepsilon,$ where $\beta\geq \frac{2C}{\varepsilon}+1$.
[**Output:**]{} the degree of $f(x)$.
Step 1:
: return $\lfloor \log_\beta (\frac{2|f(\beta)|}{\varepsilon})\rfloor$.
\[rem-1\] If we cannot use logarithm operation, then it is easy to show that we need $\mathcal{O}(\log^2 D)$ arithmetic operations to obtain the degree based on Lemma \[the-1\]. In the following section, we will regard logarithm as a basic step.
Now we will show how to compute the leading coefficient $c_t$.
\[the-2\] If $\beta\geq \frac{2C}{\varepsilon}+1$, then $c_t$ is the only element in $A$ that satisfies $|\frac{f(\beta)}{\beta^{d_t}}-c_t|<\frac{\varepsilon}{2}$.
First we show that $c_t$ satisfies $|\frac{f(\beta)}{\beta^{d_t}}-c_t|<\frac{\varepsilon}{2}$. We rewrite $f(\beta)$ as $f(\beta)=c_t\beta^{d_t}+g(\beta)$, where $g(x):=c_{t-1}x^{d_{t-1}}+c_{t-2}x^{d_{t-2}}+\cdots+c_1x^{d_1}$. So $\frac{f(\beta)}{\beta^{d_t}}=c_t+\frac{g(\beta)}{\beta^{d_t}}$. As $\deg(g)<d_t$, by Lemma \[the-1\], we have $|\frac{g(\beta)}{\beta^{d_t}}|<\frac{\varepsilon}{2}$. So $|\frac{f(\beta)}{\beta^{d_t}}-c_t|<\frac{\varepsilon}{2}$.
Assume there is another $c\in A$ also have $|\frac{f(\beta)}{\beta^{d_t}}-c|<\frac{\varepsilon}{2}$, then $|c_t-c|\leq |\frac{f(\beta)}{\beta^{d_t}}-c|+|\frac{f(\beta)}{\beta^{d_t}}-c_t|<\varepsilon$. This is only happen when $c_t=c$, so we prove the uniqueness.
Based on Lemma \[the-2\], we give the algorithm to obtain the leading coefficient.
\[alg-uvlc\]
[**Input:**]{} $f(\beta),\beta,\varepsilon, d_t$
[**Output:**]{} the leading coefficient of $f(x)$
Step 1:
: Find the element $c$ in $A$ such that $|\frac{f(\beta)}{\beta^{d_t}}-c|<\frac{\varepsilon}{2}
$.
Step 2:
: Return $c$.
Now we can give the complete algorithm.
\[alg-1\]
[**Input:**]{} A black box univariate polynomial $f(x)$, whose coefficients are in $A$.
[**Output:**]{} The exact form of $f(x)$.
Step 1:
: Find the bounds $C$ and $\varepsilon$ of $A$, as defined in .
Step 2:
: Let $\beta:=\frac{2C}{\varepsilon}+1$.
Step 3:
: Let $g:=0,u:=f(\beta)$.
Step 4:
: $\mathbf{while}$ $u\neq 0$ $\mathbf{do}$
$d:=$[**UDeg**]{}$(u,\varepsilon,\beta)$;
$c:=$[**ULCoef**]{}$(u,\beta,\varepsilon,d)$;
$u:=u-c\beta^d$;
$g:=g+cx^d$;
$\mathbf{end\ do}$.
Step 5:
: Return $g$.
Note that, the complexity of Algorithm \[alg-uvlc\] depends on $A$, which is denoted by $O_A$. Note that $O_A\le |A|$. We have the following theorem.
The arithmetic complexity of the Algorithm \[alg-1\] is $\mathcal{O}(tO_A)\le \mathcal{O}(t|A|)$, where $t$ is the number of terms in $f$.
Since finding the maximum degree needs one operation and finding the coefficient of the maximum term needs $O_A$ operations, and finding the maximum term needs $\mathcal{O}(O_A)$ operations. We prove the theorem.
The rational number coefficients case
-------------------------------------
In this section, we assume that the coefficients of $f(x)$ are rational numbers in $$\label{eq-rat}
A=\{\frac b a \,|\, 0<a\leq H,|\frac{b}{a}|\leq C,a,b\in \Z\}$$ and we have $\varepsilon=\frac{1}{H(H-1)}$. Notice that in Algorithm \[alg-1\], only Algorithm \[alg-uvlc\] (${\bf ULCoef}$) needs refinement. We first consider the following general problem about rational numbers.
\[lm-uvrc2\] Let $0<r_1<r_2$ be rational numbers. Then we can find the smallest $d>0$ such that a rational number with denominator $d$ is in $(r_1,r_2)$ with computational complexity $\mathcal{O}(\log(r_2-r_1))$.
We consider three cases.
$1$. If one of the $r_1$ and $r_2$ is an integer and the other one is not, then the smallest positive integer $d$ such that $(r_2-r_1)d>1$ is the smallest denominator, and $d=\lceil\frac{1}{r_2-r_1}\rceil$.
$2$. Both of $r_1,r_2$ are integers. If $r_2-r_1>1$, then $1$ is the smallest denominator. If $r_2-r_1=1$, then $2$ is the smallest denominator.
$3$. Both of $r_1,r_2$ are not integers. This is the most complicated case. First, we check if there exists an integer in $(r_1,r_2)$. If $\lceil r_1\rceil<r_2$, then $\lceil r_1\rceil$ is in the interval which has the smallest denominator $1$.
Now we consider the case that $(r_1,r_2)$ does not contain an integer. Assume $r_1<\frac{d_1}{d}<r_2$, where $d>1$ is the smallest denominator. Denote $w:=$trunc$(r_1)$, $\epsilon_1:=r_1-w$,$\epsilon_2:=r_2-w$. Then $\epsilon_1<\epsilon_2<1$ and $d$ is the smallest positive integer such that $(dr_1,dr_2)$ contains an integer. Since $dr_1=d(w+\epsilon_1),dr_2=d(w+\epsilon_2)$, $d$ is the smallest positive integer such that interval $(d\epsilon_1,d\epsilon_2)$ contains an integer. Since $dr_1=d(w+\epsilon_1),dr_2=d(w+\epsilon_2)$, $d$ is the smallest positive integer such that interval $(d\epsilon_1,d\epsilon_2)$ contains an integer. We still denote it $d_1$. Then $d\epsilon_1<d_1< d\epsilon_2$, so $\frac{d_1}{\epsilon_2}< d<\frac{d_1}{\epsilon_1}$, and we can see that $d_1$ is the the smallest integer such that $(\frac{d_1}{\epsilon_2},\frac{d_1}{\epsilon_1})$ contains an integer. Suppose we know how to compute the number $d_1$. Then $d=\lceil\frac{d_1}{\epsilon_2}\rceil$ when $\frac{d_1}{\epsilon_2}$ is not an integer, and $d=\frac{d_1}{\epsilon_2}+1$ when $\frac{d_1}{\epsilon_2}$ is an integer. Note that $d_1$ is the smallest denominator such that some rational number $\frac{d}{d_1}$ is in $(\frac{1}{\epsilon_2},\frac{1}{\epsilon_1})$. To find $d_1$, we need to repeat the above procedure to $(\frac{1}{\epsilon_2},\frac{1}{\epsilon_1})$ and obtain a sequence of intervals $(r_1,r_2)\rightarrow (\frac{1}{\epsilon_2},\frac{1}{\epsilon_1})\rightarrow \cdots$. The denominators of end points of the intervals becomes smaller after each repetition. So the algorithm will terminates.
Now we prove that the number of operations of the procedure is $\mathcal{O}(\log(r_2-r_1))$. First, we know the length of the interval $(r_1,r_2)$ is $r_2-r_1$. Now we prove that every time we run one or two recursive steps, the length of the new interval will be $2$ times bigger. Let $(\frac{b_1}{a_1},\frac{b_2}{a_2})$ be the first interval. If it contains an integer, then we finish the algorithm. We assume that case does not happen, so we can assume $|\frac{b_1}{a_1}|\leq1,|\frac{b_2}{a_2}|\leq1$. Then the second interval is $(\frac{a_2}{b_2},\frac{a_1}{b_1})$. Now the new interval length is $\frac{a_1}{b_1}-\frac{a_2}{b_2}$. If $\frac{b_1}{a_1}\leq\frac{1}{2}$, then we have $\frac{\frac{a_1}{b_1}-\frac{a_2}{b_2}}{\frac{b_2}{a_2}-\frac{b_1}{a_1}}=\frac{\frac{a_1b_2-a_2b_1}{b_1b_2}}{\frac{a_1b_2-a_2b_1}{a_1a_2}}=\frac{a_1a_2}{b_1b_2}\geq2$.
If $\frac{b_1}{a_1}>\frac{1}{2}$, then we let $a_1=b_1+c_1,a_2=b_2+c_2$ and the third interval is $(\frac{b_1}{c_1},\frac{b_2}{c_2})$.
Then we have $\frac{\frac{b_2}{c_2}-\frac{b_1}{c_1}}{\frac{b_2}{a_2}-\frac{b_1}{a_1}}=\frac{\frac{c_1b_2-c_2b_1}{c_1c_2}}{\frac{a_1b_2-a_2b_1}{a_1a_2}}=\frac{a_1a_2}{c_1c_2}>2$. In this case, if we have an interval whose length is bigger than $1$, then the recursion will terminate. So if $(r_2-r_1)2^k\geq 1$, then $2k$ is the upper bound of the number of recursions. So the complexity is $\mathcal{O}(\log (r_2-r_1))$. We proved the lemma.
Based on Lemma \[lm-uvrc2\], we present a recursive algorithm to compute the rational number in an interval $(r_1,r_2)$ with the smallest denominator.
\[alg-mindn\]
[**Input:**]{} $r_1,r_2$ are positive rational numbers.
[**Output:**]{} the minimum denominator of rational numbers in $(r_1,r_2)$
Step 1:
: $\mathbf{if}$ one of $r_1,r_2$ is an integer and the other one is not an integer $\mathbf{then}$ return $\lceil \frac{1}{r_2-r_1}\rceil$.
Step 2:
: $\mathbf{if}$ both of $r_1$ and $r_2$ are integers and $r_2-r_1>1$ $\mathbf{then}$ return $1$.
$\mathbf{if}$ both of $r_1$ and $r_2$ are integers and $r_2-r_1=1$ $\mathbf{then}$ return $2$.
Step 3:
: $\mathbf{if}$ $\lceil r_1\rceil<r_2$, $\mathbf{then}$ return 1.
Step 4:
: let $w:=$trunc$(r_1)$, $\epsilon_1:=r_1-w$,$\epsilon_2:=r_2-w$; $d_1:=\mathbf{MiniDenom}(\frac{1}{\epsilon_2},\frac{1}{\epsilon_1})$;
$\mathbf{if}$ $\frac{d_1}{\epsilon_2}$ is a integer $\mathbf{then}$ return $\frac{d_1}{\epsilon_2}+1$ $\mathbf{else}$ return $\lceil\frac{d_1}{{\epsilon_2}}\rceil$.
We now show how to compute the leading coefficient of $f(x)$.
\[the-3\] Suppose $c_t=\frac{b}{a}$, where $\gcd(a,b)=1,a>0$, and $I_i=(\frac{f(\beta)}{\beta^{d_t}} i-\frac{\varepsilon}{2} i,\frac{f(\beta)}{\beta^{d_t}} i+\frac{\varepsilon}{2} i),i=1,2,\dots,H$. Then $I_a\cap\Z=\{b\}$ and if $I_{a_0}\cap\Z=\{b_0\}$ then $\frac{b}{a}=\frac{b_0}{a_0}$.
By lemma \[the-2\], we have $\frac{f(\beta)}{\beta^{d_t}}-\frac{\varepsilon}{2}<\frac{b}{a}<\frac{f(\beta)}{\beta^{d_t}}+\frac{\varepsilon}{2}$, so $\frac{f(\beta)}{\beta^{d_t}} a-\frac{\varepsilon}{2} a<b<\frac{f(\beta)}{\beta^{d_t}} a+\frac{\varepsilon}{2} a$, and the existence is proved. As the length of $(\frac{f(\beta)}{\beta^{d_t}} a-\frac{\varepsilon}{2} a,\frac{f(\beta)}{\beta^{d_t}} a+\frac{\varepsilon}{2} a)$ is $<2\frac{\varepsilon}{2} a\leq \varepsilon H\leq \frac{1}{H-1}\leq1$, so $b$ is the unique integer in the interval.
Assume that there is another $a_0\in\{1,2,\dots,H\}$, such that $(\frac{f(\beta)}{\beta^{d_t}} a_0-\frac{\varepsilon}{2} a_0,\frac{f(\beta)}{\beta^{d_t}} a_0+\frac{\varepsilon}{2} a_0)$ contains the integer $b_0$. Then $\frac{f(\beta)}{\beta^{d_t}} a_0-\frac{\varepsilon}{2} a_0<b_0<\frac{f(\beta)}{\beta^{d_t}} a_0+\frac{\varepsilon}{2} a_0$, so $\frac{f(\beta)}{\beta^{d_t}}-\frac{\varepsilon}{2}<\frac{b_0}{a_0}<\frac{f(\beta)}{\beta^{d_t}}+\frac{\varepsilon}{2}$. If $\frac{a}{b}\neq\frac{a_0}{b_0}$, then $|\frac{a}{b}-\frac{a_0}{b_0}|=|\frac{ab_0-a_0b}{bb_0}|\geq \frac{1}{H(H-1)}=\varepsilon$, which contradicts to that the length of the interval is less than $\varepsilon$.
Let $r_1:=\frac{f(\beta)}{\beta^{d_t}} -\frac{\varepsilon}{2},r_2:=\frac{f(\beta)}{\beta^{d_t}} +\frac{\varepsilon}{2}$. By Lemma \[the-3\], if $a_0$ is the smallest positive integer such that $(a_0r_1,a_0r_2)$ contains the unique integer $b_0$, then we have $c_t=\frac{b_0}{a_0}$. Note that $a_0$ is the smallest integer such that $(a_0r_1,a_0r_2)$ contains the unique integer $b_0$ if and only if $a_0$ is the smallest integer such that $b_0/a_0$ is in $(r_1,r_2)$, and such an $a_0$ can be found with Algorithm \[alg-mindn\]. This observation leads to the following algorithm to find the leading coefficient of $f(x)$.
\[alg-uvlcr\]
[**Input:**]{} $f(\beta),\beta,\varepsilon,d_t$
[**Output:**]{} the leading coefficient of $f(x)$.
Step 1
: $\mathbf{if}$ $\frac{f(\beta)}{\beta^{d_t}}>0$, $\mathbf{then}$ $r_1:=\frac{f(\beta)}{\beta^{d_t}}-\frac{\varepsilon}{2}$, $r_2:=\frac{f(\beta)}{\beta^{d_t}}+\frac{\varepsilon}{2}$; $\mathbf{else}$ $r_1:=-\frac{f(\beta)}{\beta^{d_t}}-\frac{\varepsilon}{2}$, $r_2:=-\frac{f(\beta)}{\beta^{d_t}}+\frac{\varepsilon}{2}$;
Step 2:
: Let $a:=$ [**MiniDenom**]{} $(r_1,r_2)$;
Step 3:
: Return $\frac{\lceil a(\frac{f(\beta)}{\beta^{d_t}}-\frac{\varepsilon}{2})\rceil}{a}$
Replacing Algorithm [**[ULCoef]{}**]{} with Algrothm [**[ULCoefRat]{}**]{} in Algorithm [**[UPolySI]{}**]{}, we obtain the following interpolation algorithm for sparse polynomials with rational coefficients.
\[alg-uprc\]
[**Input:**]{} A black box polynomial $f(x)\in \Q[x]$ whose coefficients are in $A$ given in .
[**Output:**]{} The exact form of $f(x)$.
\[the-11\] The arithmetic operations of Algorithm \[alg-uprc\] are $\mathcal{O}(t\log H)$ and the bit complexity is $\mathcal{O}(td\log H(\log C+\log H))$, where $d$ is the degree of $f(x)$.
In order to obtain the degree, we need one log arithmetic operation in field $\Q$, while in order to obtain the coefficient, we need $\mathcal{O}(\log H)$ arithmetic operations, so the total complexity is $\mathcal{O}(t\log H)$.
Assume $f(\beta)=\frac{a_1}{h_1}\beta^{d_1}+\frac{a_2}{h_2}\beta^{d_2}+\cdots+\frac{a_t}{h_t}\beta^{d_t}$ and let $H_i:=h_1\cdots h_{i-1}h_{i+1}\cdots h_t$. Then we have
$$f(\beta)=\frac{a_1H_1\beta^{d_1}+a_2H_2\beta^{d_2}+\cdots+a_tH_t\beta^{d_t}}{h_1h_2\cdots h_t}$$ Then $|a_1H_1\beta^{d_1}+a_2H_2\beta^{d_2}+\cdots+a_tH_t\beta^{d_t}|\leq H^{t-1}C(\beta^{d_t}+\cdots+\beta+1)=H^{t-1}\frac{C}{\beta-1}(\beta^{d_t+1}-1)$, so its bit length is $\mathcal{O}(t\log H+d\log C+d\log H)$. It is easy to see that the bit length of $h_1h_2\cdots h_t$ is $\mathcal{O}(t\log H)$. So the total bit complexity is $\mathcal{O}((t\log H)(t\log H+D\log C+D\log H))$. As $t\leq d$, the bit complexity is $\mathcal{O}(td\log H(\log C+\log H))$.
\[cor-11\] If the coefficients of $f(x)$ are integers in $[-C,C]$, then Algorithm \[alg-uprc\] computes $f(x)$ with arithmetic complexity $\mathcal{O}(t)$ and with bit complexity $\mathcal{O}(td\log C)$.
\[rem-1\]
The bit complexity of Algorithm \[alg-uprc\] is $\widetilde{\mathcal{O}}(td)$, which seems to be the optimal bit complexity for deterministic and exact interpolation algorithms for a black box polynomial $f(x)\in Q[x]$. For a $t$-sparse polynomial, $t$ terms are needed and the arithmetic complexity is at least $\mathcal{O}(t)$. For $\beta\in \C$, we have $|f(\beta)|\leq C \frac{\beta^{d+1}-1}{\beta-1}$, where $C$ is defined in (\[eq-e\]). If $|\beta|\ne1$, then the height of $f(\beta)$ is $d|\log\beta|+\log C$ or $\widetilde{\mathcal{O}}(d)$. For a deterministic and exact algorithm, $\beta$ satisfying $|\beta|=1$ seems not usable. So the bit complexity is at least $\widetilde{\mathcal{O}}(td)$. For instance, the height of the data in Ben-or and Tiwari’s algorithm is already $\widetilde{\mathcal{O}}(td)$ [@1; @813].
Multivariate polynomial sparse interpolation with modified Kronecker substitution
=================================================================================
In this section, we give a deterministic and a probabilistic polynomial-time reduction of multivariate polynomial interpolation to univariate polynomial interpolation.
Find a good prime
-----------------
We will show how to find a prime number which can be used in the reduction.
We assume $f(x_1,x_2,\dots,x_n)$ is a multivariate polynomial in $\Q[x_1,x_2,\dots,x_n]$ with a degree bound $D$, a term bound $T$, and $p$ is a prime. We use the substitution $$\label{eq-sub1}
x_i=x^{\mathbf{mod}((D+1)^{i-1},p)},i=1,2,\dots,n.$$ For convenience of description, we denote $$\label{eq-sub11}
f_{x,p}:=f(x,x^{\mathbf{mod}((D+1),p)},\dots,x^{\mathbf{mod}((D+1)^{n-1},p)}).$$ Then the degree of $f_{x,p}$ is no more than $D(p-1)$ and the number of terms of $f_{x,p}$ is no more than $T$.
If the number of terms of $f_{x,p}$ is the same as that of $f(x_1,x_2,\dots,x_n)$, there is no collision in different monomials and we call such prime as a [*good prime*]{} for $f(x_1,x_2,\dots,x_n)$. If $p$ is a good prime, then we can consider a new substitution: $$\label{eq-sub2}
x_i=q_ix^{\mathbf{mod}((D+1)^{i-1},p)},i=1,2,\dots,n,$$ where $q_i,i=1,2,\dots,n$ is the $i$-th prime. In this case, each coefficient will change according to monomials of $f$. Note that in [@21], the substitution is $f(x,x^{(D+1)},\ldots,x^{(D+1)^{n-1}}) \mathbf{mod}(x^p-1)$.
We show how to find a good prime $p$. We first give a lemma.
\[lm-4\] Suppose $p$ is a prime. If $\mathbf{mod}(a_1+a_2(D+1)+\cdots+a_n(D+1)^{n-1},p)\neq 0$, then $a_1+a_2\mathbf{mod}(D+1,p)+\cdots+a_n\mathbf{mod}((D+1)^{n-1},p)\neq 0$.
If $a_1+a_2\mathbf{mod}(D+1,p)+\cdots+a_n\mathbf{mod}((D+1)^{n-1},p)=0$, then $\mathbf{mod}(a_1+a_2(D+1)+\cdots+a_n(D+1)^{n-1},p)=0$, which contradicts to the assumption.
Now, we have the following theorem to find the good prime.
\[th-gp1\] Let $f(x_1,x_2,\dots,x_n)$ be polynomial with degree at most $D$ and $t\leq T$ terms. If $$N>\frac{T(T-1)}{2}\log_2[(D+1)^n-1]-\frac14 T^2+\frac12 T$$ then there at least one of $N$ distinct odd primes $p_1,p_2,\dots,p_N$ is a good prime for $f$.
Assume $m_1,m_2,\dots,m_t$ are all the monomials in $f$, and $m_i=x_1^{e_{i,1}}x_2^{e_{i,2}}\cdots x_n^{e_{i,n}}$. In order for $p$ to be a good prime, we need $e_{i,1}+e_{i,2}(\mathbf{mod}(D+1,p))+\cdots+e_{i,n}(\mathbf{mod}((D+1)^{n-1},p))\neq e_{j,1}+e_{j,2}(\mathbf{mod}(D+1,p))+\cdots+e_{j,n}(\mathbf{mod}((D+1)^{n-1},p))$, for all $i\neq j$. This can be change into $(e_{i,1}-e_{j,1})+(e_{i,2}-e_{j,2})(\mathbf{mod}(D+1, p))+\cdots+(e_{i,n}-e_{j,n})(\mathbf{mod}((D+1)^{n-1}, p))\neq 0$. By Lemma \[lm-4\], it is enough to show $$\mathbf{mod}((e_{i,1}-e_{j,1})+(e_{i,2}-e_{j,2})(D+1)+\cdots+(e_{i,n}-e_{j,n})(D+1)^{n-1}, p)\neq 0, i\neq j$$ Firstly, $|(e_{i,1}-e_{j,1})+(e_{i,2}-e_{j,2})(D+1)+\cdots+(e_{i,n}-e_{j,n})(D+1)^{n-1}|\leq D(1+(D+1)+\cdots+(D+1)^{n-1})=(D+1)^n-1$.
We assume that $\overline{f}(x)=a_1x^{k_1}+a_2x^{k_2}+\cdots+a_tx^{k_t}$ is the polynomial after the Kronecker substitution, where $k_i=e_{i,1}+e_{i,2}(D+1)+\cdots+e_{i,n}(D+1)^{n-1}$. If $t=2$, it is trivial. So now we assume $t>2$ and we analyse how many kinds of primes the number $\prod_{i>j}(k_i-k_j)$ has. Without lose of generality, assume $k_1,k_2\dots,k_w$ are even, $k_{w+1},k_{w+2}\dots,k_t$ are odd, denote $v:=t-w$. It is easy to see that $k_i-k_j$ has factor $2$ if $1\leq i\neq j\leq w$ or $w+1\leq i \neq j\leq t$.
If one of the $w$ and $v$ is zero, then $\prod_{i>j}(k_i-k_j)$ has a factor $2^{\frac{t(t-1)}{2}}$.
If both $w,v$ are not zero, then $\prod_{i>j}(k_i-k_j)$ has a factor $2^{\frac{w(w-1)}{2}+\frac{v(v-1)}{2}}$.
We give a lower bound of $\frac{w(w-1)}{2}+\frac{v(v-1)}{2}$.
As $\frac{w(w-1)}{2}+\frac{v(v-1)}{2}=\frac{w^2+v^2-t}{2}\geq\frac{1/2 (w+v)^2-t}{2}=\frac{1}{4}t^2-\frac{1}{2}t$, $\prod_{i>j}(k_i-k_j)$ at least has a factor $2^{\frac{1}{4}t^2-\frac{1}{2}t}$.
Since $|k_i-k_j|\leq (D+1)^n-1$, we have $\prod_{i>j}(k_i-k_j)\leq [(D+1)^n-1]^{\frac{t(t-1)}{2}}$.
If $p_1,p_2,\dots,p_N$ are distinct primes satisfying $$p_1p_2\dots p_N>\frac{[(D+1)^n-1]^{\frac{t(t-1)}{2}}}{2^{\frac{1}{4}t^2-\frac{1}{2}t}}$$ Then at least one of the primes is a good prime. Since $p_i\geq 2$, $N>\frac{t(t-1)}{2}\log_2[(D+1)^n-1]-\frac14 t^2+\frac12 t$.
As we just know the upper bound $T$ of $t$, we can choose $T-t$ different positive integer $k_{t+1},k_{t+2},\dots,k_{T}$ which are different from $k_1,k_2,\dots,k_t$. So we still can use $T$ as the number of the terms. We have proved the lemma.
A deterministic algorithm
-------------------------
\[lm-203\] Assume $f=\frac{c_1}{H_1}x^{d_1}+\frac{c_2}{H_2}x^{d_2}+\dots+\frac{c_t}{H_t}x^{d_t}$, where $c_1,c_2,\dots,c_t\in\Z,H_1,H_2,\dots,H_t\in \Z_{+},d_1,d_2,\dots,d_t\in \N,d_1<d_2<\cdots<d_t,|\frac{c_i}{H_i}|\leq C$, $H_1,H_2,\dots,H_t,d_1,d_2,\dots,d_t$ are known. Let $H_{\max}:=\max\{H_1,H_2,\dots,H_t\}$. If $\beta\geq 2CH_{\max}+1$, then we can recover $c_1,c_2,\dots,c_t$ from $f(\beta)$.
It suffices to show that $c_t$ can be recovered from $f(\beta)$. As $\beta-1\geq 2CH_{\max}\geq 2CH_{t}$, then $\frac12\geq \frac{CH_t}{\beta-1}$. So $|f(\beta)H_t-c_t\beta^{d_t}|=|\frac{c_1H_t}{H_1}\beta^{d_1}+\frac{c_2H_t}{H_2}\beta^{d_2}+\dots+\frac{c_{t-1}H_t}{H_{t-1}}p^{d_{t-1}}|\leq CH_t (\frac{\beta^{d_t}-1}{\beta-1})\leq\frac12 (\beta^{d_t}-1)$. So $|\frac{f(\beta)H_t}{\beta^{d_t}}-c_t|<\frac12$. That is $\frac{f(\beta)H_t}{\beta^{d_t}}-\frac12<c_t<\frac{f(\beta)H_t}{\beta^{d_t}}+\frac12$. Since $c_t$ is an integer, $c_t=\lceil\frac{f(\beta)H_t}{\beta^{d_t}}-\frac12 \rceil$. The rest can be proved by induction.
\[alg-mp1\]
[**Input:**]{} A black box polynomial $f(x_1,x_2,\dots,x_n)\in A[x_1,x_2,\dots,x_n]$, whose coefficients are in $A$ given in , an upper bound $D$ for the degree, an upper bound $T$ of the number of terms, a list of $n$ different primes $q_1,q_2,\dots,q_n(q_1<\cdots<q_n)$.
[**Output:**]{} The exact form of $f(x_1,x_2,\dots,x_n)$.
Step 1:
: Randomly choose $N$ different odd primes $p_1,p_2,\dots,p_N$, where
$N=\lfloor\frac{T(T-1)}{2}\log_2[(D+1)^n-1]-\frac14 T^2+\frac12 T\rfloor+1$.
Step 2:
: $\mathbf{for}$ $i=1,2,\dots,N$ $\mathbf{do}$
Let $f_i:=\mathbf{UPolySIRat}(f_{x,p_i},A,T)$ via Algorithm \[alg-uprc\], where $f_{x,p_i}$ is defined in .
Step 3:
: Let $S:=\{\}$;
$\mathbf{for}$ $i=1,2,\dots,N$ $\mathbf{do}$
if $f_i\neq failure$, then $S:=S\bigcup\{f_i\}$.
$\mathbf{end\ do}$;
Step 4:
: $\mathbf{Repeat}$:
Choose one integer $i$ such that $f_{i}$ has the most number of the terms in $S$.
$\mathbf{if}$ $f_i(j)=f_{x,p_i}(j)$ for $j=1,2,\dots,D(p_i-1)+1$ $\mathbf{then}$ break $\mathbf{Repeat}$;
$S:=S\backslash \{f_i\}$
$\mathbf{end\ Repeat}$
Let $i_0$ be the integer found and $f_{i_0}=\frac{c_1}{H_1}x^{d_1}+\frac{c_2}{H_2}x^{d_2}+\cdots+\frac{c_t}{H_t}x^{d_t},d_1<d_2<\cdots<d_t$
Step 5:
: Let $\beta:=2Cq_n^D\max\{H_1,H_2,\dots,H_t\}+1$.\[Lemma \[lm-203\]\]
Denote $g= f(q_1x,q_2x^{\mathbf{mod}(D+1,p_{i_0})},\dots,q_nx^{\mathbf{mod}((D+1)^{n-1},p_{i_0})})$.
Let $u:=g(\beta)$.
Step 6:
: Let $h:=0$.
$\mathbf{for}$ $i=t,t-1,\dots,1$ $\mathbf{do}$
Let $b:=\lceil \frac{u}{\beta^{d_i}}H_i-\frac12\rceil$
Factor $\frac{b}{c_i}$ into $q_1^{e_1}q_2^{e_2}\cdots q_n^{e_n}$.
$h:=h+\frac{c_i}{H_i}x_1^{e_1}x_2^{e_2}\cdots x_n^{e_n}$.
$u:=u-\frac{b}{H_i}\beta^{d_i}$
$\mathbf{end\ do}$;
Step 7:
: return $h$.
\[remar-1\] If $p_i$ is not a good prime for $f$, then the substitution $f_{x,p_i}$ of $f$ has collisions. $f_{x,p_i}$ may have some coefficients not in $A$. So we need to modify Step 4 of Algorithm \[alg-uprc\] as follows, with $T$ as an extra input. For $c=\frac ab$, if $|c|>C$, $|b|>H$, or the number of the terms of $f_i$ are more than $T$, then we let $f_i=failure$.
\[th-algmp1\] Algorithm \[alg-mp1\] is correct and its bit complexity is $\widetilde{\mathcal{O}}(n^2T^5D\log H\log C+n^2T^5D\log^2 H+n^3T^6D^2)$.
First, we show the correctness. If $p_i$ is a good prime for $f $, then all the coefficients of $f_{x,p_i}$ are in $A$. So in step 2, Algorithm \[alg-uprc\] can be used to find $f_i=f_{x,p_i}$. It is sufficient to show that the prime $p_{i_0}$ that corresponding to $i_0$ obtained in step $4$ is a good prime. In step 4, if there exists a $j_0$ such that $f_i(j_0)\neq f_{x,p_i}(j_0)$, then $f_i\neq f_{x,p_i}$. This only happens when some of the coefficients of $f_{x,p_i}$ are not in $A$. That is, $p_i$ is not a good prime for $f $. So we throw it away. If $f_{i_0}(j)=f_{x,p_{i_0}}(j)$ for $j=1,2,\dots,D(p_{i_0}-1)+1$ for some $i_0$. Since $\deg f_{i_0}\leq D(p_{i_0}-1)$, we have $f_{i_0}=f_{x,p_{i_0}}$. Assume by contradiction that $p_{i_0}$ is not a good prime for $f$, then the number of terms of $f_{i_0}$ is less than that of $f$. Since $S$ includes at least one $f_{i_1}$ such that $p_{i_1}$ is good prime for $f$, the number of terms in $f_{i_1}$ is more than $f_{i_0}$. It contradicts to that $f_{i_0}$ has the most number of the terms in $S$. So $p_{i_0}$ is a good prime for $f$. As $f_{i_0}=\frac{c_1}{H_1}x^{d_1}+\frac{c_2}{H_2}x^{d_2}+\cdots+
\frac{c_t}{H_t}x^{d_t},d_1<d_2<\cdots<d_t$, we can assume $f=\frac{c_1}{H_1}m_1+\frac{c_2}{H_2}m_2+\cdots+
\frac{c_t}{H_t}m_t$, where $m_i=x_1^{e_{i,1}}x_2^{e_{i,2}}\cdots x_n^{e_{i,n}}$. We can write $g$ as $g=f(q_1x,q_2x^{\mathbf{mod}(D+1,p_{i_0})},\\\dots,q_nx^{\mathbf{mod}((D+1)^{n-1},p_{i_0})})=\frac{c_1 q_1^{e_{1,1}}q_2^{e_{1,2}}\cdots q_n^{e_{1,n}}}{H_1}x^{d_1}+\frac{c_2 q_1^{e_{2,1}}q_2^{e_{2,2}}\cdots q_n^{e_{2,n}}}{H_2}x^{d_2}+\cdots+
\frac{c_t q_1^{e_{t,1}}q_2^{e_{t,2}}\cdots q_n^{e_{t,n}}}{H_t}x^{d_t}$. Since $|\frac{c_i q_1^{e_{i,1}}q_2^{e_{i,2}}\cdots q_n^{e_{i,n}}}{H_i}|\leq Cq_n^D$, by Lemma \[lm-203\], in step 6, $b=c_i q_1^{e_{i,1}}q_2^{e_{i,2}}\cdots q_n^{e_{i,n}}$. By factoring $\frac{b}{c_i}=q_1^{e_{i,1}}q_2^{e_{i,2}}\cdots q_n^{e_{i,n}}$, we obtain the degrees of $m_i$. We have proved the correctness.
We now analyse the complexity. In step 2, we call Algorithm $\mathbf{UPolySIRat}$ $\mathcal{O}(nT^2\log D)$ times. The degree of $f_{x,p_i}$ is bounded by $D(p_i-1)$. Since the $i$-th prime is $\mathcal{O}(i\log i)$ and we use at most $\mathcal{O}(nT^2\log D)$ primes, the degree bound is $\widetilde{\mathcal{O}}(nT^2D)$. So by Theorem \[the-11\], the bit complexity of getting all $f_i$ is $\widetilde{\mathcal{O}}((nT^3D\log H)(\log C+\log H)(nT^2\log D))$, this is $\widetilde{\mathcal{O}}(n^2T^5D\log H\log C+n^2T^5D\log^2 H)$.
In step 4, since $\deg f_i$ is $\widetilde{\mathcal{O}}(nT^2D)$, by fast multipoint evaluation [@8 p.299], it needs $\widetilde{\mathcal{O}}(nT^2D)$ operations. The number of the $f_i$ that we need to check is at most $\widetilde{\mathcal{O}}(nT^2\log D)$, so the total arithmetic operation for evaluations is $\widetilde{\mathcal{O}}(n^2T^4 D)$. As the coefficients of $f_i$ are in $A$ and the number of terms is less than $T$, the data is $\widetilde{\mathcal{O}}(TC(nT^2D)^{nT^2D}H^T)$. So the height of the data is $\widetilde{\mathcal{O}}(nT^2D+\log C+T\log H)$. The total bit complexity of step 4 is $\widetilde{\mathcal{O}}(n^3T^6D^2+n^2T^4D\log C+n^2T^5 D\log H)$.
In step 6, we need to obtain $t$ terms of $g$. We analyse the bit complexity of one step of the cycle. To obtain $b$, we need $\mathcal{O}(1)$ arithmetic operations. The height of the data is $\widetilde{\mathcal{O}}(nT^2D(\log C+D\log n+\log H))$, so the bit complexity is $\widetilde{\mathcal{O}}(nT^2D\log C+nT^2D^2+nT^2D\log H)$. To factor $\frac{b}{c_i}$, we need $n\log^2 D$ operations. The data of $b$ and $c_i$ is $\widetilde{\mathcal{O}}(Cq_n^DH)$, so the bit complexity is $\widetilde{\mathcal{O}}(n\log^2 D\log C+nD+n\log^2D\log H)$. So the total bit complexity of step 6 is $\widetilde{\mathcal{O}}(nT^3D\log C+nT^3D^2+nT^3D\log H)$.
Therefore, the bit complexity is $\widetilde{\mathcal{O}}(n^2T^5D\log H\log C+n^2T^5D\log^2 H+n^3T^6D^2)$.
\[remar-10\] If $A=\{a|C\geq |a|,a\in \Z\}$, we can modified the Algorithm \[alg-mp1\]. Assume $A_{T}=\{a|TC\geq |a|,a\in \Z\}$. In step 2, we let $f_i:=\mathbf{UPolySIRat}(f_{x,p_i},A_{T})$. As $f_{x,p_i}$ is an integer polynomial with coefficients bounded by $TC$, $f_i=f_{x,p_i}$. So in step 4, we just find the smallest integer $i_0$ that $f_{i_0}$ has the most number of the terms in $S$. In this case, $p_{i_0}$ is a good prime for $f$. The bit complexity of the algorithm will be $\widetilde{\mathcal{O}}(n^2T^5D\log C+nT^3D^2)$.
Probabilistic Algorithm
-----------------------
Giesbrecht and Roche [@6 Lemma 2.1] proved that if $\lambda=\max\{21,\frac 5 3 nT(T-1)\ln D\}$, then a prime $p$ chosen at random in $[\lambda,2\lambda]$ is a good prime for $f(x_1, \dots,x_n)$ with probability at least $\frac1 2$. Based on this result, we give a probabilistic algorithm.
\[alg-pmp1\]
[**Input:**]{} A black box polynomial $f(x_1, \dots,x_n)\in A[x_1, \dots,x_n]$, whose coefficients are in $A$ given in , an upper bound $D$ for the degree, an upper bound $T$ of the number of terms, a list of $n$ different primes $q_1,q_2,\dots,q_n(q_1<\dots<q_n)$.
[**Output:**]{} The exact form of $f(x_1, \dots,x_n)$ with probability $\ge\frac12$.
Step 1:
: Let $\lambda:=\max\{21,\frac 5 3 nT(T-1)\ln D\}$, randomly choose a prime $p$ in $[\lambda,2\lambda]$.
Step 2:
: Let $f_p:=\mathbf{UPolySIRat}(f_{x,p},A,T)$ via Algorithm \[alg-uprc\].
$\mathbf{if}$ $f_p=failure$ $\mathbf{then}$ return failure;
Assume $f_p=\frac{c_1}{H_1}x^{d_1}+\frac{c_2}{H_2}x^{d_2}+\cdots+\frac{c_t}{H_t}x^{d_t},d_1<d_2<\cdots<d_t$
Step 3:
: Let $\beta:=2Cq_n^D\max\{H_1,H_2,\dots,H_t\}+1$.\[Lemma \[lm-203\]\]
Denote $g(x)= f(q_1x,q_2x^{\mathbf{mod}(D+1,p)},\dots,q_nx^{\mathbf{mod}((D+1)^{n-1},p)})$.
Let $u:=g(\beta)$;
Step 4:
: Let $s:=0$;
$\mathbf{for}$ $i=t,t-1,\dots,1$ $\mathbf{do}$
Let $b:=\lceil \frac{u}{\beta^{d_i}}H_i-\frac12\rceil$
Factor $\frac{b}{c_i}=kq_1^{e_1}q_2^{e_2}\cdots q_n^{e_n}$, where $q_i\nmid k,i=1,2,\dots,n$
$\mathbf{if}$ $k\neq 1$ or $e_1+e_2+\cdots+e_n>D$ $\mathbf{then}$ return failure;
$s:=s+\frac{c_i}{H_i}x_1^{e_1}x_2^{e_2}\cdots x_n^{e_n}$.
$u:=u-\frac{b}{H_i}x^{d_i}$
$\mathbf{end\ do}$;
$\mathbf{if}$ $u=0$ $\mathbf{then}$ return $s$
$\mathbf{else}$ return failure;
\[th-qalgmp1\] The bit complexity of Algorithm \[alg-pmp1\] is $\widetilde{\mathcal{O}}(nT^3D\log H\log C+nT^3D\log^2 H+nT^3D^2)$.
In step 2, the degree of $f_{x,p}$ is bounded by $D(p-1)$. Since the $p$ is $\mathcal{O}(nT^2\log D)$, the degree bound is $\widetilde{\mathcal{O}}(nT^2D)$. By Theorem \[the-11\], the complexity is $\widetilde{\mathcal{O}}(( nT^3D\log H)(\log C+\log H))$, or $\widetilde{\mathcal{O}}(nT^3D\log H\log C+nT^3D\log^2 H)$.
In step 4, we need to obtain $t$ terms of $g$. We analyse the bit complexity of one step of the cycle. To obtain $b$, we need $\mathcal{O}(1)$ arithmetic operations. The height of the data is $\widetilde{\mathcal{O}}(nT^2D(\log C+D\log n+\log H))$, so the bit complexity is $\widetilde{\mathcal{O}}(nT^2D\log C+nT^2D^2+nT^2D\log H)$. To factor $\frac{b}{c_i}$, we need $n\log^2 D$ operations. The height of $b$ and $c_i$ is $\widetilde{\mathcal{O}}(Cq_n^DH)$, so the bit complexity is $\widetilde{\mathcal{O}}(n\log^2 D\log C+nD+n\log^2D\log H)$. So the total bit complexity of step 4 is $\widetilde{\mathcal{O}}(nT^3D\log C+nT^3D^2+nT^3D\log H)$. Therefore, the total bit complexity of the algorithm is $\widetilde{\mathcal{O}}(nT^3D\log H\log C+nT^3D^2+nT^3D\log^2 H)$.
In Algorithm \[alg-pmp1\], we also modify step 4 of Algorithm \[alg-uprc\] as remark \[remar-1\].
Experimental results
====================
In this section, practical performances of the algorithms will be presented. The data are collected on a desktop with Windows system, 3.60GHz Core $i7-4790$ CPU, and 8GB RAM memory. The implementations in Maple can be found in
http://www.mmrc.iss.ac.cn/~xgao/software/sicoeff.zip
We randomly construct five polynomials, then regard them as black box polynomials and reconstruct them with the algorithms. The average times are collected.
The results for univariate interpolation are shown in Figures \[fig1\], \[fig2\], \[fig3\], \[fig4\]. In each figure, three of the parameters $C,H,D,T$ are fixed and one of them is variant. From these figures, we can see that Algorithm $\mathbf{UPolySIRat}$ is linear in $T$, approximately linear in $D$, logarithmic in $C$ and $H$.
The results in the multivariate case are shown in Figures \[fig5\], \[fig6\]. We just test the probabilistic algorithm. From these figures, we can see that Algorithm $\mathbf{ProMPolySIMK}$ are polynomial in $T$ and $D$.
![$\mathbf{UPolySIRat}$: average running times with varying $D$ []{data-label="fig2"}](ur1.jpg)
![$\mathbf{UPolySIRat}$: average running times with varying $D$ []{data-label="fig2"}](ur2.jpg)
![$\mathbf{UPolySIRat}$: average running times with varying $H$ []{data-label="fig4"}](ur3.jpg)
![$\mathbf{UPolySIRat}$: average running times with varying $H$ []{data-label="fig4"}](ur4.jpg)
![$\mathbf{ProMPolySIMK}$: average running times with varying $D$ []{data-label="fig6"}](pm1.jpg)
![$\mathbf{ProMPolySIMK}$: average running times with varying $D$ []{data-label="fig6"}](pm2.jpg)
Conclusion
==========
In this paper, a new type of sparse interpolation is considered, that is, the coefficients of the black box polynomial $f$ are from a finite set. Specifically, we assume that the coefficients are rational numbers such that the upper bounds of the absolute values of these numbers and their denominators are given, respectively. We first give an interpolation algorithm for a univariate polynomial $f$, where $f$ is obtained from one evaluation $f(\beta)$ for a sufficiently large number $\beta$. Then, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial into the univariate case. Both algorithms have polynomial bit-size complexity and the algorithms can be used to recover quite large polynomials.
[99]{}
A. Arnold. Sparse Polynomial Interpolation and Testing. PhD Thesis, Waterloo Unversity, Canada,2 016.
A. Arnold and D.S. Roche. Multivariate sparse interpolation using randomized Kronecker substitutions. ISSAC’14, July 23-25, 2014, Kobe, Japan.
M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. 20th Annual ACM Symp. Theory Comp., 301-309, 1988.
S. Garg and E. Schost. Interpolation of polynomials given by straight-line programs. Theoretical Computer Science, 410(27-29):2659-2662, 2009.
M. Giesbrecht and D.S. Roche. Diversification improves interpolation. Proc. ISSAC’11, 123-130, ACM Press, 2011.
Q.L. Huang and X.S Gao. Sparse sational function interpolation with finitely many values for the coefficients. arXiv:1706.00914, 2017.
Q.L. Huang and X.S Gao. New algorithms for sparse interpolation and identity testing of multivariate polynomials. Preprint, 2017.
E. Kaltofen and L. Yagati. Improved sparse multivariate polynomial interpolation algorithms. Proc. ISSAC’88, 467-474, 1988.
A.R. Klivans and D. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proc. STOC ’01, 216-223, ACM Press, 2001.
L. Kronecker. Grundz$\ddot{u}$ge einer arithmetischen theorie der algebraischen gr$\ddot{o}$ssen. Journal f$\ddot{u}$r die reine und angewandte Mathematik, 92:1-122, 1882.
Y.N. Lakshman and B.D. Saunders. Sparse polynomial interpolation in nonstandard bases. SIAM J. Comput., 24(2), 387-397, 1995.
J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999.
R. Zippel. Interpolating polynomials from their values. Journal of Symbolic Computation, 9(3), 375-403, 1990.
[^1]: Partially supported by a grant from NSFC No.11688101.
|
---
abstract: 'We have analyzed the clustering of and absorption–line systems on comoving scales from 1 to 16 , using an extensive catalog of heavy–element QSO absorbers with mean redshift $\langle z\rangle_{\rm \CIV} = 2.2$ and $\langle z\rangle_{\rm \MgII} = 0.9$. For the sample as a whole, the absorber line–of–sight correlation function is well fit by a power law of the form $\xi_{\rm aa}(r)={\left(r_0/r\right)}^\gamma$, with maximum–likelihood values of $\gamma = 1.75\,^{+0.50}_{-0.70}$ and comoving $r_0 = 3.4\,^{+0.7}_{-1.0}$ ($q_0=0.5$). This clustering is of the [*same form*]{} as that for galaxies and clusters at low redshift, and of amplitude such that absorbers are correlated on scales of clusters of galaxies. We also trace the [*evolution*]{} of the mean amplitude $\xi_0(z)$ of the correlation function from $z=3$ to $z=0.9$. We find that, when parametrized in the conventional manner as $\xi_0(z)\propto (1+z)^{-(3+\epsilon)+\gamma}$, the amplitude grows [*rapidly*]{} with decreasing redshift, with maximum–likelihood value for the evolutionary parameter of $\epsilon = 2.05 \pm 1.0 $ ($q_0=0.5$). The rapid growth seen in the clustering of absorbers is consistent with gravitationally induced growth of perturbations.'
address:
- 'Dept. of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637'
- 'Dept. of Astronomy, University of Texas at Austin, Austin, TX 78712'
author:
- 'Jean M. Quashnock$^{1}$, Daniel E. Vanden Berk$^{1,2}$'
---
Introduction
============
In a previous paper [@Quash96], Quashnock, Vanden Berk, & York analyzed line–of–sight correlations of and absorption–line systems on large scales, using an extensive catalog [@Y97] of 2200 heavy–element absorption–line systems in over 500 QSO spectra. Here, we extend that analysis to smaller comoving scales — from 1 to 16 — and relate the small–scale clustering of absorbers to galaxy clustering in general.
The and data sample is drawn from the catalog of Vanden Berk [@Y97], using the same selection criteria as those in [@Quash96]. It consists of 260 absorbers, drawn from 202 lines of sight, with redshifts ranging from $1.2 < z < 3.6$ and mean redshift $\langle z\rangle_{\rm \CIV} = 2.2$, and 64 absorbers, drawn from 278 lines of sight, with redshifts ranging from $0.3 < z < 1.6$ and mean redshift $\langle z\rangle_{\rm \MgII} = 0.9$.
Unless otherwise noted, we take $q_0=0.5$ and $\Lambda=0$. We follow the usual convention and take the Hubble constant to be 100$h$ km s$^{-1}$ Mpc$^{-1}$. A more detailed version of this work, including more results and outlining our maximum–likelihood method, has appeared elsewhere [@Quash97].
Form and Evolution of the Correlation Function
==============================================
Figure 1 shows the line–of–sight correlation function $\xi_{\rm aa}(r)$, for the entire sample of absorbers (with mean redshift $\langle z\rangle_{\rm \CIV} = 2.2$), as a function of absorber comoving separation $r$ from 1 to 16 , in 4 octaves. The vertical error bars through the data points are 1-$\sigma$ errors in the estimator for $\xi_{\rm aa}$. The correlation function and error bars are computed in the same fashion and using the same selection criteria as those in [@Quash96], except that we have combined all absorbers lying within 1.0 (instead of 3.5) comoving of each other into a single system.
Using the maximum–likelihood method of [@Quash97], we find that, for the sample as a whole, the line–of–sight correlation function is well described by a power law of the form $\xi_{\rm aa}(r)={\left(r_0/r\right)}^\gamma$, with maximum–likelihood values of $\gamma = 1.75\,^{+0.50}_{-0.70}$ and comoving correlation length $r_0 = 3.4\,^{+0.7}_{-1.0}$ ($q_0=0.5$). The clustering of absorbers at high redshift is thus of the [*same form*]{} as that found for galaxies and clusters at low redshift ($\gamma = 1.77 \pm 0.04$ for galaxies [@DP83], $\gamma = 2.1 \pm 0.3$ for clusters [@Nichol92]), and of amplitude such that absorbers are correlated on scales of clusters of galaxies. It appears that the absorbers are tracing the large–scale structure seen in the distribution of galaxies and clusters, and are doing so at high redshift. The finding strengthens the case for using absorbers in probing large–scale structure.
We have investigated the evolution of the clustering of absorbers by dividing the absorber sample into three approximately equal redshift sub–samples, and comparing these to the sample. Figure 2 shows the mean of the correlation function, $\xi_0(z)$, averaged over comoving scales $r$ from 1 to 16 , for the low ($1.2<z<2.0$), medium ($2.0<z<2.8$), and high ($2.8<z<3.6$) redshift sub–samples, as well as for the sample ($0.3<z<1.6$). The amplitude of the correlation function is clearly growing [*rapidly*]{} with decreasing redshift.
We have used the maximum–likelihood formalism of [@Quash97]to describe the evolution of the correlation function. We have fixed $\gamma$ at its maximum–likelihood value of 1.75 in our analysis, and parametrized the amplitude of the correlation function in the usual manner as $\xi_0(z) \propto (1+z)^{-(3+\epsilon)+\gamma}$, where $\epsilon$ is the evolutionary parameter [@Efstat91; @GP77]. Using all the data sets, we find that the rapid growth is reflected in a large value for the evolutionary parameter, namely $\epsilon = 2.05 \pm 1.0$. This value is 3.3-$\sigma$ from the no–evolution value ($\epsilon = -1.25$); thus, at the 99.95 % confidence level, growth of the correlation function has been detected. (These results are with $q_0=0.5$. With $q_0=0.1$, we expect our estimate of $\epsilon$ to decrease by about 1.3 .)
The rapid growth in the correlation function, and the correspondingly large value of the evolutionary parameter ($\epsilon = 2.05 \pm 1.0$) that is implied, is what is expected in a critical universe ($\Omega_0 = 1$), both from linear theory of gravitational instability [@Peeb80; @Peeb93], with $\xi \propto {\left(1+z\right)}^{-2}$ (or $\epsilon = 0.75$, if $\gamma=1.75$), and from numerical simulations [@Carlberg97; @Colin97]: For $\Omega_0 = 1$, $\epsilon = 1.0 \pm 0.1$, whereas for $\Omega_0 = 0.2$, $\epsilon = 0.2 \pm 0.1$.
Evidence for a trend of increasing clustering of Ly$\alpha$ absorbers ($N({\rm\HI}) > 6.3 \times 10^{13}~{\rm cm}^{-2}$) with decreasing redshift has been found by Cristiani [@Cr97]. These authors also find a clear trend of increasing Ly$\alpha$ absorber clustering with increasing column density, and find that an extrapolation to column densities typical of heavy–element systems ($N({\rm\HI}) > 10^{16}~{\rm cm}^{-2}$) is consistent with the clustering observed for absorbers [@PB94; @SC96]. Our finding of growth in the clustering of heavy–element systems with decreasing redshift supports both a continuity scenario between Ly$\alpha$ and heavy–element systems [@Cr97], and the common action of gravitational instability.
The strong clustering that we find in the heavy–element absorption–line systems is thus not surprising, given that most of the sample consists of the strongest systems with relatively large equivalent widths (order 0.4 Å and greater), and the recent claims [@Cr97; @Dod97] of a strong dependence of clustering strength on the column density of the systems. We do confirm that the weaker systems (equivalent widths 0.2 Å and less) are less clustered than the stronger ones, by a factor of two or so; unfortunately, most of the spectra used to assemble the Vanden Berk catalog [@Y97] are not of sufficient quality to yield a large number of weak systems.
We wish to acknowledge the long–term direction of Don York, in compiling the extensive catalog of heavy–element absorbers used in this study, and in providing intellectual leadership for the project.
[99]{}
Carlberg, R. G., Cowie, L. L., Songaila, A., & Hu, E. M. 1997, , 484, 538
Colin, P., Carlberg, R. G., & Couchman, H. M. P. 1997, , in press
Cristiani, S., D’Odorico, S., D’Odorico, V., Fontana, A., Giallongo, E., & Savaglio, S. 1997, , 285, 209
Davis, M., & Peebles, P. J. E. 1983, , 267, 465
D’Odorico, V., Cristiani, S., D’Odorico, S., Fontana, A., & Giallongo, E. 1997, , in press
Efstathiou, G., Bernstein, G., Katz, N., Tyson, J. A., & Guhathakurta, P. 1991, , 380, L47
Groth, E. J., & Peebles, P. J. E. 1977, , 217, 385
Nichol, R. C., Collins, C. A., Guzzo, L., & Lumsden, S. L. 1992, , 255, 21p
Peebles, P. J. E. 1980, The Large–Scale Structure of the Universe (Princeton: Princeton Univ. Press)
Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton: Princeton Univ. Press)
Petitjean, P., & Bergeron, J. 1994, , 283, 759
Quashnock, J. M., Vanden Berk, D. E., & York, D. G. 1996, , 472, L69
Quashnock, J. M., & Vanden Berk, D. E. 1997, , submitted
Songaila, A., Cowie, L. L. 1996, , 112, 839
Vanden Berk, D. E., 1997, Suppl., submitted
|
---
abstract: 'Ca II triplet spectroscopy has been used to derive stellar metallicities for individual stars in four LMC fields situated at galactocentric distances of 3, 5, 6 and 8 to the north of the Bar. Observed metallicity distributions show a well defined peak, with a tail toward low metallicities. The mean metallicity remains constant until 6 (\[Fe/H\]$\sim$-0.5 dex), while for the outermost field, at 8, the mean metallicity is substantially lower than in the rest of the disk (\[Fe/H\]$\sim$-0.8 dex). The combination of spectroscopy with deep CCD photometry has allowed us to break the RGB age–metallicity degeneracy and compute the ages for the objects observed spectroscopically. The obtained age–metallicity relationships for our four fields are statistically indistinguishable. We conclude that the lower mean metallicity in the outermost field is a consequence of it having a lower fraction of intermediate-age stars, which are more metal-rich than the older stars. The disk age–metallicity relationship is similar to that for clusters. However, the lack of objects with ages between 3 and 10 Gyr is not observed in the field population. Finally, we used data from the literature to derive consistently the age–metallicity relationship of the bar. Simple chemical evolution models have been used to reproduce the observed age–metallicity relationships with the purpose of investigating which mechanism has participated in the evolution of the disk and bar. We find that while the disk age–metallicity relationship is well reproduced by close-box models or models with a small degree of outflow, that of the bar is only reproduced by models with combination of infall and outflow.'
author:
- 'R. Carrera and C. Gallart'
- Eduardo Hardy
- 'A. Aparicio'
- 'R. Zinn'
title: The Chemical Enrichment History of the Large Magellanic Cloud
---
Introduction
============
Despite decades of work, there are still significant gaps in our knowledge of the LMC’s star formation and chemical enrichment histories [@ol96]. This is motivated in part by the vastness of its stellar populations and to our limitations in observing sizable samples of stars. The age distribution of star clusters is relatively well known [e.g. @geisler97]: there is an age interval between 10 and 3 Gyr with almost no clusters. This so-called age-gap may also correspond to an abundance gap [@ol91], since the old clusters are metal-poor while the young ones are relatively metal-rich. The star formation history (SFH) of field stars is much less precisely known. Studies using [*HST*]{} data in small fields, suggest that the SFH of the LMC disk has been more or less continuous, with some increase in the star formation rate (SFR) in the last few Gyr [e.g. @Smecker-Hane02; @castro01; @holtzman99]. This contrasts with previous results (based on much shallower data), which found a relatively young age (a few Gyr) for the dominant LMC population [@hardy84; @bertelli92; @westerlund95; @vallenari96]. The SFH of the LMC can now be studied using sufficiently deep ground-based data (e.g. reaching the oldest main-sequence turn-off with good photometric precision) in large areas and in different positions of the galaxy [e.g. @gshpz04; @gshpz05 hereafter Paper I].
Less is known about the LMC chemical enrichment history. The age–metallicity relation (AMR) normally used for the LMC is defined by star clusters [e.g. @ol91; @geisler97; @bica98; @dirsch00]. All works based on clusters obtain similar results. The mean metallicity jumps from \[Fe/H\] $\sim -1.5$ for the oldest clusters to \[Fe/H\] $\sim -0.5$ for the youngest ones. There were no studies on the field population AMR until the last decade. @dopita97, from a study of $\alpha$-elements in planetary nebulae, obtained a result qualitatively similar to that found in the clusters although only ten objects were used. Later, @bica98, using Washington photometry, and @c00 and @dirsch00, using Ströngrem photometry, obtained the metallicity of RGB stars in different positions of the LMC. @c00 also observed 20 stars with the infrared CaII triplet (CaT). All of them found that the age-gap observed in the clusters is not found in the field population. More recently, @c05 have obtained stellar metallicities for almost 400 stars in the bar of the LMC, also using CaT lines. The metallicity for each star has been combined with its position in the color–magnitude diagram (CMD) to estimate its age. The obtained AMR shows a similar behavior to that of the clusters for the oldest population. However, while for the clusters the metallicity has increased over the last 2 Gyr, this has not happened in the bar. Finally @gratton04 measured the metallicity of about 100 bar RR Lyrae, obtaining an average metallicity of \[Fe/H\] $=-1.48$.
Another point that it is necessary to investigate is the presence, or not, of an abundance gradient in the LMC. From observations of clusters, @ol91 and @santos99 found no evidence for the presence of a radial metallicity gradient in the LMC. The only evidence for a radial metallicity gradient in the LMC cluster system was reported by @kon93 based on six outer LMC clusters ($\geq$ 8 kpc). @hill95 were the first to report evidences of a gradient in the field population from high-resolution spectroscopy, using a sample of nine stars in the bar and in the disk. They found that the bar is on average 0.3 dex more metal-rich than the disk population, but the disk stars studied are located within a radius of 2. Subsequent work by @cioni03, detected that the C/M ratio between number of asymptotic giant branch stars of spectral types C and M increased when moving away from the bar and within a radius of 6$\fdg$7. As the C/M ratio is anticorrelated with metallicity, this increment implies a decrease in metallicity. Finally, an outward radial gradient of decreasing metallicity was also found by @alves04 from infrared CMD using the 2MASS survey.
We have obtained deep photometry with the Mosaic II CCD Imager on the CTIO 4m telescope in four disk fields at different distances from the center of the LMC (RA=5$^h$23$^m$34$\fs$5, $\delta$=-69$\arcdeg$45’22“) with a quality similar to that obtained by [*HST*]{} in more crowded areas (see Paper I). The position of these fields, together with a description of their CMDs, are presented in Paper I, and detailed SFHs will be published in forthcoming papers. In the present investigation we focus on obtaining stellar metallicities for a significant number of individual RGB stars in these four fields using spectra obtained with the HYDRA spectrograph at the CTIO 4m telescope. In Section \[targetselection\] we present our target selection. The observations and data reduction are presented in Section \[datareduction\]. The radial velocities of the stars in our sample are obtained in Section \[radialvelocities\]. In Section \[cat\] we discuss the calculation of the CaT equivalent widths and the determination of metallicities. Section \[agedetermination\] presents the method used to derive ages for each star by combining the information on their metallicity and position on the CMD. The analysis of the data is presented in Section \[analysis\], where the possible presence of a kinematically hot halo is discussed and the AMR and the SFH in each of our fields are described and compared. In the final section, the AMRs are compared to theoretical models and conclusions are drawn about the chemical evolution of the LMC.
Target Selection\[targetselection\]
===================================
Our starting point are deep CMDs of four $36'\times 36'$ fields located $\sim$3$\arcdeg$, 5$\arcdeg$, 6$\arcdeg$ and 8$\arcdeg$ from the LMC center, which are presented and discussed in Paper I. All fields are located north of the LMC bar, and their CMDs reach the oldest main-sequence turn-offs. A clear gradient exists in the amount of intermediate-age and young populations from the inner to the outermost fields, in the sense that the young population is more prominent in the inner parts (see Paper I for a more detailed discussion).
In each field we have selected the stars to be observed with HYDRA in two windows in the CMD, which are plotted in Figure \[dcmbox\]. The bluest color limit has been selected to avoid upper red-clump and red supergiant stars and the lower magnitude limit to avoid including red-clump stars in the spectroscopic sample. The blue edge of the window is sufficiently blue to ensure that no metal poor stars would be excluded (the position of the blue ridge corresponds approximately with the position of $\sim$2 Gyr old stars with \[Fe/H\]=-3). Similarly, the opposite edge is sufficiently red so that even old stars with solar metalicities, or even higher, fit within the box. These selected stars have been ordered from the brightest to the faintest ones, with no color restriction. This list has been used as input for the configuration task of the instrument, which tries to optimize the number of allocated fibers. The first stars in the list, the brightest ones, have priority over the others.
Observations and Data Reduction\[datareduction\]
================================================
The observations were carried out in two different runs, in December 2002 and in January 2005, at the CTIO 4 m telescope with the HYDRA multifiber spectrograph. We used the KPGLD grating, which provides a central wavelength of 8500 Å, and a OG590 order-blocking filter. The physical pixels were binned 2 $\times$ 1 in the spectral direction yielding a dispersion of 0.9 Å/pix. HYDRA has a field of view of about 40 arcmin, which on account of the high density of stars in our fields, allowed us to observe more than 100 stars in each configuration, with the 140 available fibers. Because of technical problems and bad weather, on our first run we were able to observe only two configurations located at 5$\arcdeg$ and 8$\arcdeg$ respectively. In the more successful second run, one fiber configuration in each field was observed except for the closest one, where we observed two configurations. Stars in two globular clusters, NGC 4590 and NGC 3201, were also observed. Those in the first were used as radial velocity standards, but also to compare the equivalent widths determined through the present set-up with those setups used for the cluster observations reported in @carrera06 [hereafter Paper II]. Since the differences were negligible (see Figure 1 of paper II), the second cluster was included in the calibration of the CaT as metallicity indicator. We concluded that the LMC equivalent widths obtained with the present observational setup could be directly transformed into \[Fe/H\] using the calibration in paper II. Equivalent widths, magnitudes and radial velocities for the 702 stars observed in the LMC are listed in Table \[starobs\].
The data reduction was done following the procedure described in [*Knut Hydra notes*]{}[^1]. First, cosmic rays were removed from the images using the IRAF[^2] Laplacian edge-detection routine [@dokkun01]. All images were then bias- and overscan-subtracted, and trimmed. Flat exposures were taken during the day with a diffusing screen installed in the spectrograph to obtain the so-called “milk-flat”, which is used to correct the effect of the different sensitivities among pixels. It was applied to all images using *ccdproc*. The spectra were extracted, flat-field corrected (see bellow), and calibrated in wavelength using *dohydra*, a program developed specifically to reduce the data obtained with this spectrograph. The lamp flats acquired at the beginning of each configuration were used to define and trace each aperture. The sky flats were used to eliminate the residuals after the flat-field correction with the milk-flat. Arcs, obtained before and after each configuration, were used for wavelength calibration.
*Dohydra* can also perform sky subtraction. However, we noticed that after sky subtraction, there remained significant sky line residues. We therefore developed our own procedure to remove the contribution of the sky lines from the object spectra. For a given configuration, we obtain an average high signal–to–noise ratio (S/N) sky spectrum from all fibers placed on the sky. Before subtracting this averaged sky from each star spectrum, we need to know the relation between the intensity of the sky in each fiber (which varies from fiber to fiber owing to the different fiber responses) and the average sky. This relation is used as a weight (which may depend on wavelength) that multiplies the average sky spectrum before subtracting it from each star. Our task minimizes the sky line residuals over the whole spectral region considered and allows very accurate removal of the sky emission lines. An example of raw, sky and final object is shown in Figure \[sky\].
We obtained between three and four exposures of 2700 s in each configuration. To minimize the contribution of residuals arising from cosmic rays and bad pixels, all spectra of the same object were combined. Finally, each object spectrum was normalized with the *continuum* task by fitting a polynomial, making sure to exclude all absorption lines in our wavelength range, such as the CaT lines themselves.
Radial Velocities\[radialvelocities\]
=====================================
Our main purpose in obtaining radial velocities for our program stars is to reject possible foreground objects. However, radial velocities can provide more useful information, as discussed in depth in Section \[kinematics\]. The radial velocities were calculated with the *fxcor* task in IRAF, which performs the cross-correlation described by @td79 between the spectra of the target and selected templates of known radial velocities. As templates, we used three stars in the globular cluster NGC 4590 (M68) with radial velocities obtained by @geisler95. The final radial velocities for all the stars in our sample were obtained as the average of the velocities obtained with the three templates, weighted by the width of the corresponding correlation peaks. Radial velocity histograms for each field are plotted in Figure \[vel\_dist\]. A Gaussian function was fitted in order to obtain the central peak and the $\sigma$ of each distribution (values listed in Table \[radialvelocity\]). The velocity distribution of the field located 3 from the center shows two peaks, which are marked in Figure \[vel\_dist\]. Stars with radial velocity in the range 170 $\leq V_r\leq$ 380 km s$^{-1}$ [@zhao03] are considered LMC members. Only about 20 stars in each configuration of the instrument have been excluded based on their radial velocity. These results are discussed in depth in Section \[kinematics\].
CaT Equivalent Widths and Metallicity Determination\[cat\]
==========================================================
The metallicity of the RGB stars is obtained following the procedure described in Paper II. In short, the equivalent width is the line area normalized to the continuum. The continuum is calculated from the linear fit to the mean value of each bandpass, defined to obtain the continuum position. We have used the continuum and line bandpasses defined by @cen01, which are listed in Table \[bandastable\]. The equivalent widths are calculated from profile fitting using a Gaussian plus a Lorentzian. This combination provides the best fit to line core and wings as discussed in Paper II. The CaT index is defined as the sum of the equivalent widths of the three CaT lines, denoted as $\Sigma Ca$. The $\Sigma Ca$ calculated for each observed star and its uncertainty are given in Table \[starobs\], together with the star magnitude and radial velocity. The reduced equivalent width, $W'_I$, for each star has been calculated as the value of $\Sigma Ca$ at M$_I$=0, using the slope obtained in Paper II for the calibration clusters in the M$_I$–$\Sigma Ca$ plane ($\beta_I$ = $-0.611$ Åmag$^{-1}$).
In Paper II we obtained relationships between the reduced equivalent widths, $W'_V$ and $W'_I$, and metallicity, on the @zw84, @cg97 [hereafter CG97] and @ki03 scales. In this case, we will only use the relationships obtained on the CG97 metallicity scale, because it is the only one in which the metallicities of our open and globular calibration clusters have been obtained in a homogeneous way from high-resolution spectroscopy.
In order to verify that the $V$ and $I$ relationships give similar results, we have applied both to obtain the metallicities of the stars observed in the field located at 5. The differences are on average $-$0.01 dex, and never larger than 0.15 dex for stars with $(V-I)\leq$ 2.5. For stars with colors larger than this, the differences are on average 0.2 dex and never get above 0.35 dex. The calibration stars used in Paper II have $(V-I)<2.5$, so the metallicities calculated for the reddest stars are necessarily extrapolations of the relationships. On average, the relationship based on M$_I$ yields metallicities slightly lower than those obtained from M$_V$, but always within the uncertainties. In what follows we will use the relationship based on M$_I$, because: i) the photometric accuracy in $I$ is slightly better than in $V$, especially for the reddest stars on the RGB; ii) the RGB is better resolved in the $I$ band; and iii) from a theoretical point of view [@pont04], it is likely that the relationship based on M$_I$ is less sensitive to age. In short, then, the metallicity for each star is given by:
$$\label{metaleq}
[Fe/H]_{CG97}=-2.95+0.38\Sigma Ca+0.23M_I$$
Binarity could affect the derived metallicity only in the unlikely case of the two stars in the system being RGB stars of a very similar mass. In this case the system would be $\sim$0.7 mag brighter than a single star. As the binarity might not affect the equivalent width of the CaT lines, the derived metallicity would be lower by $\sim$0.4 dex than the actual metallicity.
Figure \[misigmaca\] shows the position of LMC radial velocity member stars in the M$_I$–$\Sigma$Ca plane for our four fields, from the innermost one (top left) to the outermost field (bottom right). The star with $\Sigma Ca$ = 11 and M$_I$ = $-$2 in the field situated at 5$\arcdeg$ has a well measured $\Sigma Ca$ and radial velocity V$_r$ = 264 km s$^{-1}$ (the mean V$_r$ of the sample is 278 $\pm$ 20 km s$^{-1}$), so it seems to be an LMC RGB star with unusually large metallicity. However, a reliable metallicity cannot be obtained with the present calibration, which is only valid for \[Fe/H\] $\leq$ +0.47. The extrapolation of our calibration gives \[Fe/H\] = +0.72 $\pm$ 0.4 for this star.
The metallicity distribution of each field is shown in Figure \[histo\_fields\] and will be discussed *in extenso* in Section \[analisismetallicitydistribution\]. In Figure \[dcm\_feh\] the observed stars have been plotted in the color–magnitude diagram employing different symbols as a function of their metallicity. It is important to highlight that there is no correlation between the metallicity of the stars and their position in the color–magnitude diagram, as was also found in other galaxies [e.g. @pont04; @c05; @koch05]. This shows once more that the metallicities derived from the position of the stars in the RGB are unreliable in systems with complex SFHs.
Determination of stellar ages\[agedetermination\]
=================================================
For a given age, the position of the RGB on the color–magnitude diagram depends mainly on metallicity in the sense that more metal-rich stars are redder than more metal-poor ones. At the same time, for a fixed metallicity, older stars are also redder than younger ones. If we take into account the logical assumption that younger stars are also more metal-rich, age may partly counteract the effect of metallicity. The combination of both effects is the well known RGB age–metallicity degeneracy, which sets a limit on the amount of information that can be retrieved from the RGB using only photometric data. However, if stellar metallicities are obtained independently from another source, such as spectroscopy, we should in principle be able to break the age–metallicity degeneracy. @pont04 and @c05 have derived stellar ages from isochrones for stars whose metallicity had been previously obtained from low-resolution spectroscopy. In our case, instead of directly comparing the position of the star in the CMDs with isochrones, we have obtained a relation for the age as a function of color, magnitude and metallicity from a synthetic CMD. This allows to easily obtain ages for the large number of stars in our sample.
The reference synthetic CMD has been computed using IAC-STAR[^3] [@aparicio04]. Its main input parameters are the SFR and the chemical enrichment law, as a function of time. Since we are interested in a general relation, we have chosen a constant SFR for the whole range of ages (13 $\geq$ (Age/Gyr) $\geq$ 0) and a chemical enrichment law such that a star of any age can have any metallicity in the range $-2.5\leq$ \[Fe/H\] $\leq$ +0.5. Both the constant SFR and the chemical enrichment law ensure that there is no age or metallicity bias. We have computed two synthetic CMDs, one using the BaSTI stellar evolution library [@pie04] and another using the Padova library [@girardi02] respectively. The ranges of ages and metallicities covered, as well as other important features of each library are listed in Table 1 of @gallartza05.
Since the uncertainty of the age assigned by stellar evolution models to bright AGB stars is large, we will only compute ages for RGB and AGB stars fainter than the RGB tip (M$_I\sim$-4 in Figure \[dcmbox\]) . We have chosen the synthetic stars in the box below the tip of the RGB shown in Figure \[dcmbox\]. We have computed a polynomial relationship which gives the age as a function of \[Fe/H\] $\equiv$ log(Z/Z$_\odot$), (V-I) and M$_V$. In order to minimize the $\sigma$ and to improve the correlation coefficient of the relation, different linear, quadratic and cubic terms of each observed magnitude have been added. When the addition of a term did not improve the relation we rejected it. The best combination of these parameters is in the form of Equation \[rela\] and the values obtained for each stellar evolution library are shown in Table \[tableage\]. Note that each stellar evolution library has its own age scale, and some differences might exist among them.
$$\label{rela}
log(age)=a+b(V-I)+cM_V+d[Fe/H]+f(V-I)^2+g[Fe/H]^2+h(V-I)^3$$
The uncertainties of each term on Equation \[rela\] are given in Table \[tableage\]. However, the way in which they propagate into the error of the age is complex. For this reason a Monte Carlo test has been performed to estimate age errors. The synthetic stellar population computed from BaSTI stellar evolution models has been used to this purpose. That based on the Padua library would have produced similar results. The test consists in computing, for each synthetic star, several age values for stochastically varying $[Fe/H]$, $(V-I)$ and $M_{V}$ according to a gaussian probability distribution of the corresponding $\sigma$ ($\sigma_{[Fe/H]}\sim$0.15 dex; $\sigma_{(V-I)}\sim$0.001 and $\sigma_{M_V}\sim$ 0.001). The $\sigma$ value of the obtained ages provide an estimation of the age error when Equation \[rela\] is used. The values obtained for different age intervals are shown in Figure \[erroredad\]. The error increases for older ages.
We have also investigated the accuracy with which the age of a real stellar population is measured. The stars in open and globular clusters in Paper II, which ages are independently known, are used for this purpose. We adopt the ages estimated by @sw02 and @swp04, which are in a common scale, for a globular and open clusters, respectively. They are listed in Table 1 of Paper II.
The age of each cluster star has been estimated from Equation \[rela\] assuming the distance modulus and reddening listed in Paper II. The metallicity of each star has been calculated following the same procedure as for the LMC stars (see Section \[cat\]). The age of each cluster has been obtained as the mean age derived from all the stars in each cluster and the quoted uncertainty is the standard deviation of this mean. Ages derived for each cluster from the BaSTI and Padova relationships in Table \[tableage\], versus the reference values for each cluster, have been plotted in Figure \[cluster\]. The relationships saturate for ages older than 10 Gyr because differences of 1 or 2 Gyr produce only negligible variations of position in the CMD. In both cases, the age of clusters younger than 10 Gyr are well reproduced. Differences between the age derived with each model and the reference value are within the error bars, with the exception of cluster NGC 6819 for the BaSTI relationship. The large errorbar of the youngest cluster, NGC 6705, is due to the fact that the stars observed in this cluster are fainter that the synthetic stars used to derive the relationships. Therefore, we are extrapolating the relations to obtain the age of this cluster. As both models produce similar values, we adopted for simplicity the relationships derived from the BaSTI stellar library [@pie04].
Analysis\[analysis\]
====================
Stellar kinematics of the LMC\[kinematics\]
-------------------------------------------
In Figure \[kunkel\] we have plotted the mean of the velocity distribution for each field [including the @c05 bar field] as a function of its Galactic longitude. The mean velocity changes from field to field due to the disk rotation of the LMC. The radial velocities of carbon stars derived by @kunkel97 have also been plotted. Our data are in good agreement with their result, which is consistent with the presence of a rotational disk.
If a classical Milky Way-like halo existed in the LMC (i.e. old, metal-poor and with high velocity dispersion), a dependency between the velocity dispersion and metallicity/age of the stars in our sample might be observed. The first evidence of the presence of a kinetically hot spheroidal population was reported by @hughes91. They found that the LMC long period variables, related to an old stellar population, have a high velocity dispersion (33 km s$^{-1}$), with a low rotational component. @miniti03 measured the kinematics of 43 RR Lyrae stars in the inner regions of the LMC and found that the velocity dispersion of these stars is 53 $\pm$ 10 km s$^{-1}$, which they associated with the presence of a kinematically hot halo populated by old metal-poor stars. Some studies of the intermediate-age and old populations have found that the velocity dispersion increases with age [e.g. @hughes91; @schommer92; @graff00]. In fact @graff00, using C stars, have found the stars of the disk belong to two populations: a young disk population containing 20% of stars with a velocity dispersion of 8 km s$^{-1}$, and an old disk population containing the remaining stars with a velocity dispersion of 22 km s$^{-1}$.
We can check the presence of a hot halo with our sample stars. Assuming that stars with similar metallicities are of similar ages (see below), a possible dependency of velocity dispersion with metallicity could indicate whether different stellar populations have different kinematics.
The procedure has been the follow up. The mean velocity in each field has been subtracted from the radial velocity of each star to eliminate the rotational component of the disk. The total metallicity range has been divided into three bins. The velocity dispersion of stars in each metallicity bin is listed in Table \[vrmetaltable\] and plotted in Figure \[vrmetal\]. The velocity dispersion for old and metal-poor stars, 26.4 km s$^{-1}$, is slightly smaller than the value found by @hughes91 for the old long period variables ($\sigma$=33 km s$^{-1}$) and significantly smaller than the dispersion found in RR Lyrae stars [$\sigma\sim$53 km s$^{-1}$, @miniti03]. @c05 found a velocity dispersion of $\sigma$ = 40.8 km s$^{-1}$ for the most metal-poor stars in the bar, which is also higher than the value found here. The metal-poor stars in our sample seem to be members of the thick disk instead of the halo. In short, from our data we have found no clear evidence for the presence of a kinematically hot halo populated by old stars.
Metallicity distribution\[analisismetallicitydistribution\]
-----------------------------------------------------------
The metallicity distribution for each field is shown in Figure \[histo\_fields\]. We have fitted a Gaussian to each one in order to obtain its mean value and dispersion (Table \[metallicitybin\]). The mean value is constant at \[Fe/H\]$\sim$-0.5 dex up to 8$\arcdeg$, where the mean metallicity decreases by about a factor of two (\[Fe/H\]=-0.8 dex). All metallicity distributions are similar. They show a clear peak with a tail toward low metallicities. We complement our sample with the results by @c05, who derived stellar metallicities in the bar in a similar way as here. As demonstrated in Paper II, their CaT index is equivalent to ours. Using their measurements of the $\Sigma Ca$ we can calculate the metallicities from Equation \[metaleq\]. We have also fitted a Gaussian to the resulting bar metallicity distribution to obtain its mean value. On average, the bar is slightly more metal-rich than the inne disk \[Fe/H\]$\sim$-0.4 dex).
Age–metallicity relationships\[agemetallicity\]
-----------------------------------------------
To understand the nature of the observed metallicity distributions, and to join insight on SFR and the chemical enrichment history of the LMC fields, we have estimated the age of each star in our sample following the procedure described in Section \[agedetermination\]. These individual age determinations have a much larger uncertainty than the metallicity calculations. However, they are still useful because we are interested in the general trend rather than in obtaining precise values for individual stars.
We assumed that the oldest stars have the same age as the oldest cluster in our galaxy, for which we adopt 12.9 Gyr [NGC 6426, @sw02]. This agrees with a 13.7$\pm$0.2 Gyr old Universe [@spergel03] where the first stars formed about 1 Gyr after the Big Bang. Another important point is the youngest age that we can find in our sample. According to stellar evolution models [@pie04], we do not expect to find stars younger than 0.8 Gyr in the region of the RGB where we selected our objects. However, application of Equation \[rela\] may result in ages younger than this value. To avoid this contradiction, and taking into account that the age determination uncertainty for the younger stars is about 1 Gyr, we assigned an age of 0.8 Gyr to those stars for which Equation \[rela\] gives younger values. Finally, as the relationships used to estimate the age have been calculated for stars below the tip of the RGB, only stars fainter than this point have been used.
The age of each star versus its metallicity (i.e. the AMR) has been plotted in Figure \[amrfields\] for each field. The many stars with the same age that appear both at the old (12.9 Gyr) and young (0.8 Gyr) limits are the consequence of the boundary conditions imposed on the ages, and the exact values should not be taken at face value. The age error in each age interval, computed in Section \[agedetermination\], is indicated in the top panel. The age distribution for each field has been plotted in inset panels. The histogram is the age distribution without taking into account the age uncertainty. The solid line is the same age distribution computed by taking into account the age error. To do this, a Gaussian probability distribution is used to represent the age of each star. The mean and $\sigma$ of the Gaussian are the age obtained for the star and its error, respectively. The area of each Gaussian is unity. In stars near the edges, the wings of the distributions may extend further than the age physical limits. In such cases, we truncated the wing and rescaled the rest of the distribution such that the area remains unity.
Stars brighter than M$_I$=-3.5 were not used in Paper II to obtain the metallicity calibration of $\Sigma$Ca (Equation \[metaleq\]). However, we observed stars brighter than M$_I$=-3.5 in the LMC. Therefore, extrapolation of Equation \[metaleq\] has been used to obtain the metallicity of these stars, but only for those below the tip of the RGB. It is necessary to check whether this has introduced any bias in the obtained AMR. Paper II demonstrated that the sequences described by cluster stars used for the calibration in the M$_I$–$\Sigma Ca$ plane are not exactly linear and have a quadratic component. In the interval M$_I$=0 – -3.5 the difference between the quadratic and linear behaviors are negligible. However, for brighter stars, the deviation from the linear behavior might be important, resulting in a possible underestimation of the metallicity of these stars. To check this point, in Figure \[testamr\] we have plotted as filled squares stars with magnitudes within the range of the calibration. Open circles are stars whose metallicities have been obtained by extrapolating the relationship (-3.5$\geq M_I\geq$-4). The mean metallicity and its dispersion for several age intervals have been also plotted for both cases. The differences between both groups are smaller than the observed dispersion and therefore we conclude that no strong bias is produced as a consequence of extrapolating Equation \[metaleq\] for bright stars.
Figure \[amrfields\] shows that the AMR is, within the uncertainties, very similar for all fields. As expected, the most metal-poor stars in each field are also the oldest ones. A rapid chemical enrichment at a very early epoch is followed by a period of very slow metallicity evolution until around 3 Gyr ago, when the galaxy started another period of chemical enrichment that is still ongoing. Furthermore, the age histograms for the three innermost fields are similar, although the total number of stars decreases when we move away from the centre. The outermost field has a lower fraction of ”young” (1–4 Gyr) intermediate-age stars. This indicates that its lower mean metallicity is related to the lower fraction of intermediate-age, more metal-rich stars rather than to a different chemical enrichment history (for example, a slower metal enrichment).
With the aim of obtaining a global AMR for the disk, in the following we will quantitatively address the question of whether the AMRs of all our disk fields are statistically the same. We have calculated the mean metallicity of the stars and its dispersion in six age intervals, for each field and for the combination of the four. The values obtained are listed in Table \[testchi2\]. To compare the values obtained in each field with those for the combined sample, we performed a $\chi^2$ test such that $\chi^2=\sum_{i=1}^6\frac{(Z_i^{field}-Z_i^{comb})^2}{\sigma_i^2}$, where $\sigma^2_i$ is the sum of the uncertainties squared in the age bin $i$ of the field and the combined AMR. The last column in Table \[testchi2\] shows the values $\chi^2_\nu=\chi^2/\sqrt{5}$. From these values we may conclude that the AMR of each field is equivalent to that of the combined sample to a probability of more than 99 per cent.
Now we will compare the global AMR obtained for the disk with that observed in the bar. The bar stellar ages have been obtained using the relation calculated in Section \[agedetermination\]. The age versus metallicity for each star in the bar (*left*) and disk (*right*) have been plotted in Figure \[amrbardisk\]. Notice that with a different method and different stellar evolution models, the AMR obtained here for the bar is quite similar to that obtained by @c05. We performed a $\chi^2$ test as before between the galaxy disk and bar AMRs. They are the same to within a probability of 90 per cent. However, a visual comparison would indicate that, while in the disk the metallicity has risen monotonically over the last few Gyr, this is not the case for the bar. However, from the comparison of the mean metallicities and the errorbars for stars in this youngest bin (1–3 Gyr) it appears that this difference is statistically significant.
The global disk AMR is qualitatively similar to that of the LMC cluster system [e.g. @ol91; @dirsch00], except for the lack of intermediate-age clusters (Figure \[cumulos\]). Clusters show a rapid enrichment phase around 10 Gyr ago which is also observed in the disk and the bar. The second period of chemical enrichment in the last 3 Gyr is also observed both in the field and in the clusters. The field intermediate-age stars may have contributed to the chemical enrichment at recent times. Note that the @grocholski06 sample has a maximum metallicity \[Fe/H\]=-0.6, i.e. it does not seem to participate on the high metallicity tail at young ages that we and the other authors analysing clusters samples find. In the three works, there are a number of young clusters with low metallicity for their age, which @bekki07 associate with metal-poor gas stripped from the SMC due to tidal interaction between the SMC, LMC and the Galaxy over the last 2 Gyr. A few stars also have low metallicity for their age in our sample, but they represent a minor contribution to the total.
The Star Formation History {#sfhcal}
--------------------------
More information can be retrieved from the AMRs. In particular we can also derive an approximate SFH of the LMC if we can establish that the observed stars are representative of the total population. The number of stars in each age bin can be transformed into stellar total mass, after accounting for the number of stars with a given age that have already died. To evaluate this correction we have computed a synthetic CMD using IAC-star [@aparicio04]. We have assumed a constant SFR and used the relations derived for the chemical enrichment. As we have only observed a fraction of the total stars in the region of the RGB, we have rescaled the result to the total number of stars in this region.
In order to check whether the procedure used to select the stars observed spectroscopically introduces any bias in the computed SFH, we have calculated a synthetic CMD with a known chemical enrichment history and SFR. It was computed such that the number of stars in the selection region was the same as in the field at 3. The input SFR as a function of age, $\psi(t)$, is plotted as a solid line in Figure \[pruebaconf\]. To check that the selection criteria did not introduce biases, we have selected objects from this synthetic CMD in the same way as from the LMC fields (see Figure \[dcmbox\]). We assigned random positions to each synthetic star and, using the HYDRA configuration software, we computed 20 test configurations. The number of objects selected in each configuration, about 115, is similar to the observed stars in the LMC fields. For each test configuration, we calculated $\psi(t)$ following the procedure described above. The mean of the 20 total tests is the dashed line in Figure \[pruebaconf\]. The error bars are the standard deviation of the 20 tests. We repeated the same test, but obtaining the ages of each synthetic star using Equation \[rela\]. The result is the dotted line in Figure \[pruebaconf\]. As we can see in this Figure, the recovered $\psi(t)$ are consistent with the real one. In half of the bins the differences are within the error bars and in the rest they are within 3$\sigma$.
The $\psi(t)$ derived from our four fields are shown in Figure \[hfecampos\] (*solid line*), which has been computed from the age distribution, taking into account the age error (see Figure \[erroredad\]). The uncertainty in the computed $\psi(t)$ has been estimated as the square root of the SFR value for each age (*dotted lines*). However, values of the SFR for different ages are not independent from each other because the integral for the full age interval is a boundary condition of the solution. In other words, fluctuations above the best solution should be compensated by others below it, and a solution close to, for example, the upper (or the lower) dashed line is very unlike. As a comparison, the $\psi(t)$ computed without taking into account the age error has also been plotted in Figure \[hfecampos\]. All fields have a first episode of star formation more than 10 Gyrs ago. Then, $\psi(t)$ decreases until $\sim$5 Gyr ago, when it rises again in the inner regions of the LMC (r$\leq$6). This enhancement of $\psi(t)$ is not observed in the outermost field.
We followed the same procedure to calculate $\psi(t)$ for the bar stars. The bar $\psi(t)$ is shown in Figure \[hfe\]. In this case we do not know the total number of objects in the region where the observed stars were selected. For this reason, we have plotted the percentage, over the total, that the SFR represents in each age bin. In the right panel of the same Figure, the $\psi(t)$ of the disk, obtained as a combination of the four fields in our sample, is shown. The bar $\psi(t)$ is similar to that derived in small fields from [*HST*]{} deep photometry [@holtzman99; @Smecker-Hane02], as indeed is expected since the [*HST*]{} fields are in the same region in which the RGB stars were observed. Both in the bar and in the disk, the galaxy started forming stars more than 10 Gyr ago, although in the bar the initial enhancement of $\psi(t)$ is not observed. For intermediate ages, the SFR was low until about 3 Gyr ago, when another increase in the SFR, which is particularly intense in the bar, is observed. This is consistent with previous investigations [e.g. @hardy84; @Smecker-Hane02] and with the SFR derived from the clusters. The cluster age gap coincides with the age interval in which star formation has been less efficient.
A Chemical Evolution Model for the LMC\[modelsec\]
--------------------------------------------------
In what follows we use the AMR to obtain information on the physical parameters governing the chemical evolution of the LMC. As a first approximation, we will try to reproduce the derived relations with simple models.
Under the assumption of instantaneous recycling approximation, following @tinsley80 and @peimbert94, the heavy-element mass fraction in the ISM, $Z$, evolves via:
$$\label{variaz}
\mu\frac{dZ}{d\mu}=\frac{y(1-R)\psi+(Z_f-Z)f_I}{-(1-R)\psi+(f_I-f_O)(1-\mu)}$$
where $f_I$ and $f_O$ are the inflow and outflow rates respectively, $Z_f$ is the metallicity of the infall gas, $y$ denotes the yield, $R$ is the mass fraction returned to the interstellar medium by a generation of stars, relative to the mass locked in stars and stellar remnants in that generation, $\psi$ is the SFR and $\mu$ is defined as $\mu=M_g/M_b$, where M$_g$ is the gas mass and M$_b$ is the total baryonic mass, i.e., the mass participating in the chemical evolution process. In general, $Z, \mu, \psi, f_I$ and $f_O$ are explicit functions of time, while $R$ and $y$ may change according to characteristics of the stellar population like metallicity and initial mass function, but are usually assumed to be constant for a given population.
When there are no gas flows in and out of the system ($f_I=f_O=0$), the model is called a closed-box model and the solution to Equation \[variaz\] provides the variation of $Z(t)$ as a function of $\mu(t)$ via
$$Z(t)=Z_i+y\ln\mu(t)^{-1}$$
In an infall scenario, in which gas flows into the system at a rate proportional to the amount of star formation, $f_I=\alpha(1-R)\psi$, $\alpha$ being a free parameter, we can integrate Equation \[variaz\] for $\alpha\neq1$, assuming that $f_O=0$, and obtaining:
$$Z(t)=Z_i+\frac{y+Z_f\alpha}{\alpha}\left\lbrack 1-
\left(\alpha-\frac{1-\alpha}{\mu(t)}\right)^{-\alpha/(1-\alpha)}\right\rbrack$$
For $\alpha$=1, the solution of Equation \[variaz\] is
$$Z=Z_i+y\left\lbrack 1-e^{\left(1-\mu^{-1}\right)}\right\rbrack$$
In an outflow scenario, in which gas scapes from the system at a rate $f_O=\lambda(1-R)\psi$, $\lambda$ being a free parameter, we obtain from Equation \[variaz\] assuming $f_I=0$, and for ($\lambda\neq1$):
$$Z(t)=Z_i+\frac{y}{\lambda-1}\ln\left\lbrack
\frac{\lambda-1}{\mu(t)}-\lambda\right\rbrack$$
Finally, we will consider a model combination of inflow and outflow, and assume that the system gas flow is proportional to the mass that has taken part in the star formation: $f=(\alpha-\lambda)(1-R)\psi$, where we can define $\beta=\alpha-\lambda$. In this case, we can integrate Equation \[variaz\] for $\beta\neq$1, to obtain
$$Z(t)=Z_i+\frac{y+Z_f\alpha}{\alpha}\left\lbrack
1-\left(\beta-\frac{1-\beta}{\mu(t)}\right)^{-\beta/(1-\beta)}\right\rbrack$$
In summary, the chemical evolution of the galaxy is given by the yield $y$, the initial metallicity of the gas Z$_i$, the fraction of gas mass $\mu(t)$, which is a function of $\psi(t)$, and $R$, the mass fraction returned to the interstellar medium. In the case of inflow and/or outflow, also by the additional parameters $\alpha$ and $\lambda$.
We can now apply the former relations to the LMC. In the previous section we have derived $\psi(t)$ for the LMC bar and disk, the latter being obtained from the combination of the four fields in our sample. $\mu$ is computed explicitly as a function of time from $\psi(t)$, the balance of gas flowing to and from the system, and using as a boundary condition the current LMC gas fraction. In the case of the LMC disk, we have assumed the value derived by @kim98 from observations of H [I]{}. These authors derived $M_g=5.2\times10^8 M_{\odot}$ and estimated that the total mass of the disk is $M_b=2.5\times10^9 M_{\odot}$. With these values, we find the current gas fraction in the disk, $\mu_f=0.21$. For the bar we have assumed the value of $\mu=0.08$ given by @westerlund90. For the yield, we have assumed the value obtained by @pagel95 for the solar neighborhood, $y=0.014$. Assuming the @scalo86 initial mass function, @maeder93 calculated a returned fraction associated to this yield of $R=0.44$. Putting all these ingredients together, we will now explore several scenarios in order to reproduce the AMR of the disk and the bar.
In Figure \[modeldisk\] we have plotted the age and metallicity of each star in the disk, together with the mean metallicity and its dispersion, for stars in each age bin. In this figure, chemical evolution models have been superposed for different scenarios and parameter assumptions (see the figure caption). In the case of the infall models (*green dashed lines*), we have assumed a zero-metallicity infalling gas, in order to obtain the current metallicity. They do not reproduce the general behavior of the observed AMR. The disk AMR is well reproduced by outflow models (*blue dot-dashed lines*) with a relatively large range of $\lambda$. Finally, the combination of inflow and outflow (*pink and cyan long–short dashed lines*) models also reproduce the metallicity tendency (in this case the model that best matches the observed data has $\alpha$ = 0.2 and $\lambda$ = 0.05). The previous models have been computed assuming the yield observed in the solar neighborhood which is probably too large for the LMC. With a smaller yield, $y=0.008$ (*red thick solid line*), the observed AMR can be reproduced by a closed-box model. This yield would correspond to stars with $Z=0.001$ [see @maeder93 for details], which is slightly low compared with the mean LMC metallicity. Bursting and smooth models computed by @pagel98 have been plotted as comparison (*brown lines*). The bursting model was computed assuming two burst of star formation, one about 12 Gyr ago and a strong one about 3 Gyr ago, and some fraction of infall and outflow. Note that their predicted values agree with our mean metallicity for each age bin, within the errorbars, specially at old and intermediate–age. It also qualitatively agrees with the episode of faster chemical evolution with started around $\sim$2 Gyr ago. The smooth model was computed assuming a constant star formation rate and does not match with the observations.
In the case of the bar (Figure \[modelbar\]), the very slowly rising metallicity, almost constant over the last few Gyr, is best reproduced by the combination of models with inflow and outflow with infall parameter $\alpha$ = 1.2 and $\lambda$ = 0.4-0.6 (*pink long–short dashed lines*). Infall models (*green dashed lines*) with $\alpha$ = 0.4-0.6 marginally reproduce the observed trend, but they predict higher metallicities than observed in the last few Gyr, unless a smaller yield would be assumed. Finally, outflow and closed-box models predict a rise of metallicity in the last few Gyr more steep than observed. The @pagel98 bursting model agrees worse with the bar AMR than with that of the disk, specially in the youngest age bins.
Conclusions
===========
Using infrared spectra in the CaT region, we have obtained metallicities and radial velocities for a sample of stars in four LMC fields. Metallicities have been calculated using the relationships between the equivalent width of the CaT lines, $\Sigma Ca$, and metallicity derived in Paper II. In addition, we have estimated the age of each star using a relationship derived from a synthetic CMD which, from the color, magnitude and metallicity of a star, allow us to estimate its age. The main results of this paper are:
- The velocity distribution observed in each field agrees with the rotational thick disk kinematics of the LMC. The velocity dispersion is slightly larger for the most metal-poor stars. However, the values obtained do not indicate the presence of a kinematically hot halo.
- The metallicity distribution of each field has a well defined peak with a tail toward low metallicities. The mean metallicity is constant until the field at 6 (\[Fe/H\]$\sim$-0.5 dex), and is a factor two more metal-poor for the outermost field (\[Fe/H\]$\sim$-0.8 dex).
- The AMR observed in each disk field is compatible with a single global relationship for the disk. We conclude that the outermost field is more metal-poor on average because it contains a lower fraction of relatively young stars (age$\leq$5 Gyr), which are also more metal-rich.
- The disk AMR shows a prompt initial chemical enrichment. Subsequently, the metallicity increased very slowly until about 3 Gyr ago, when the rate of metal enrichment increased again. This AMR is similar to that of the cluster system, except for the lack of clusters with ages between 3 and 10 Gyr. The recent fast enrichment observed in the disk and in the cluster system is not observed in the bar.
- The $\psi(t)$ of the three innermost fields show a first episode of star formation until about 10 Gyr ago, followed by a period with a low SFR until $\sim$5 Gyr ago, when the SFR increases, and reaches its highest values $\sim$2-3 Gyr ago. The outermost field does not show the recent increase of SFR. The second main episode is also observed, and is more prominent in the bar, where an increased SFR at old ages is not observed. The lower SFR between 5 and 10 Gyr ago is probably related to the age-gap observed in the clusters.
- Under the assumption of a solar yield, the disk AMR is well reproduced either by a chemical evolution model with outflow with $\lambda$ between 1 and 2, so the disk losses the same amount of gas that has taken part in the star formation, or by models combining infall and outflow with $\alpha$=0.2 and $\lambda$=0.05, which means that the galaxy is almost a closed-box system. With a smaller yield, the AMR could also be reproduced with a closed-box model. The bar AMR is well reproduced by models with a combination of inflow and outflow with $\alpha$=1.2 and $\lambda$ between 0.4 and 0.5. This suggests that the amount of infalling gas was larger than the amount that participated in the star formation in the bar, and also that the amount of gas that escaped the bar was 50% of the total that participated in star formation.
Discussion
==========
The main result in this paper is that all our fields, covering galactocentric radius from 3 to 8 from the center (2.7 to 7.2 kpc) share a very similar AMR. The mean metallicity is very similar in the three innermost fields (\[Fe/H\] $\simeq$-0.5 dex), and it is a factor of $\simeq 2$ smaller in the outermost field (\[Fe/H\] $\simeq$-0.8 dex). Because the AMRs are the same, this has to be related with the lower fraction of relatively young stars (age $\le$ 5 Gyr), which are also more metal-rich, in the outer part. We find, therefore, a change in the age composition of the disk population beyond a certain radius ($\simeq$ 6 kpc; note however that in this study we are only sensitive to populations older than $\simeq$ 0.8 Gyr, and that differences among fields for younger ages could also exist), while the chemical enrichment history seems to be shared by all fields.
The comparison of the AMR observed in the disk with simple chemical evolution models suggests that the LMC is most likely losing some of its gas. This is in agreement with the work by @nidever07 which suggests that the main contribution to the Magellanic Stream comes from the LMC instead of from the SMC as it was believed until now [e.g. @putman03]. Alternatively, the LMC could be an almost closed-box system with small gas exchanges ($\alpha$=0.2, $\lambda$=0.05). This could be in agreement with the models by @bekki07 which suggest that the LMC has received metal-poor gas from the SMC in the last 2 Gyr. In our field we observed some stars with a low metallicity to which we assign an age younger than 2 Gyr. However, the uncertainty in the age determination is large.
We have also obtained the AMR of the LMC bar from the data by @c05. The bar AMR differs from the one of the disk in the last 5 Gyr: while in the disk the metallicity has increased in this time, in the bar it has remained approximately constant. This feature is best reproduced by models combination of outflow and a relatively large infall of pristine gas. Models with a smaller yield, but with an infall of previously enriched gas also reproduce the observed bar AMR. This would be in agreement with the prediction that a typical bar instability pushes the gas of the disk towards the center [@sellwood93], so the infall gas is expected to be previously pre-enrichment by the disk chemical evolution. This would be also the case in the scenario suggested by @bekki05, in which the bar would have formed from disk material as a consequence of tidal interactions between the LMC, the SMC and the Milky Way about 5 Gyr ago, the moment when the bar AMR differs from that of the disk. Also, a bar is expected to destroy any metallicity gradient within a certain radius, as it is observed.
@gshpz04 derived the LMC surface brightness profile using deep resolved star photometry of the four fields in the current spectroscopic study, and found that it remains exponential to a radius of 8 ($\simeq$ 7 kpc), with no evidence of disk truncation. Combining this information with that on the deep CMD of the outermost field, which contains a large fraction of intermediate-age stars, they concluded that the LMC disk extends (and dominates over a possible halo) at a distance of at least 7 kpc from its center. In the present study, the kinematics of the stars in these four fields confirm this conclusion: the velocity dispersion of all four fields is similar, around 20 km/sec. If the stars are binned by metallicity, the velocity dispersion of the most metal-poor bin is slightly larger than that of the metal-richer bins ($\sigma_V
\simeq 25$ km/sec as opposed to $\simeq 20$ km/sec) but still not large enough to indicate the presence of a halo, even one formed as a consequence of the interaction with the Galaxy. In this case, [@bekki04] predicted a velocity dispersion $\simeq$ 40 km/s at a distance of 7.5 kpc from the LMC center (similar to the distance of our outermost field). Other authors have also failed to find a kinematically hot halo [@freeman83; @schommer92; @graff00; @zhao03]. The first evidence of the presence of a kinetically hot, old spheroidal population was reported by @hughes91 using a sample of long period variables, which are related to an old stellar population. Recently, @miniti03 observed spectroscopically a sample of 43 bar RR Lyrae stars, and obtained a large velocity dispersion (53 $\pm$ 10 km s$^{-1}$) for them. It is possible that our failure (and that of other authors) to find evidence of a hot stellar halo is related with a low contrast of the halo population with respect to the disk one, even at large galactocentric radius as our outermost field.
AA, CG, and RC acknowledge the support from the Spanish Ministry of Science and Technology (Plan Nacional de Investigación Científica, Desarrollo, e Investigación Tecnológica, AYA2004-06343) and from the Instituto de Astrofísica de Canarias (grants P3/94 and 3I1902). RZ acknowledges the support of National Science Foundation grant AST05-07362. This work has made use of the IAC-STAR Synthetic CMD computation code. IAC-STAR is supported and maintained by the computer division of the Instituto de Astrofísica de Canarias.
Facilities: .
Alves, D. R. 2004, , 49, astro-ph0408336 Aparicio, A., & Gallart, C. 2004, , 128, 1465 Bekki, K., Couch, W. J., Beasley, M. A., Forbes, M. A., Chiba, M., & Da Costa, G. S. 2004, , 610, L93 Bekki, K., & Chiba, M., 2005, , 356, 680 Bekki, K., & Chiba, M., 2007, , in press Bertelli, G., Mateo, M., Chiosi, C. & Bressan, A. 1992, , 388, 400 Bica, E., Geisler, D., Dottori, H., Clari[á]{}, J. J., Piatti, A. E., & Santos, J. F. C., Jr. 1998, , 116, 723 Carrera, R., Gallart, C., Pancino, E. & Zinn, R. 2007, , 134, 1298 (Paper II) Carretta, E., & Gratton, R. G. 1997, , 121, 95 (CG97) Castro, R., Santiago, B. X., Gilmore, G. F., Beaulieu, S. & Johnson, R. A. 2001, , 326, 333 Cenarro, A. J., Cardiel, N., Gorgas, J., Peletier, R. F., Vazdekis, A., & Prada, F. 2001, , 326, 959 Cioni, M.-R. L., & Habing, H. J. 2003, , 402, 133 Cole, A. A., Smecker-Hane, T. A., & Gallagher III, J. S. 2000, , 120, 1808 Cole, A. A., Tolstoy, E., Gallagher III, J. S. & Smecker-Hane, T. A. 2005, , 129, 1465 Dirsch, B., Richtler, T., Gieren, W.P. & Hilker, M. 2000, , 360, 133 Dopita M. A., Vassiliadis, E., Wood, P.R., Meatheringhan, J. S., Harrington, J. P., Bohlin, R. C., Ford, H.C., Stecher, T. P., & Maran, S. P. 1997, , 474, 188 Freeman, K. C., Illingworth, G., & Oemler Jr, A. 1983, , 272, 488 Gallart, C., Stetson, P. B., Hardy, E., Pont, F., & Zinn, R. 2004, , 614, L109 Gallart, C., Zoccali, M., & Aparicio, A. 2005, , 43, 387 Gallart, C., Stetson, P. B., Pont, F., Hardy, E., & Zinn, R. 2007, in preparation, (Paper I) Geisler, D., Piatti, A. E., Clariá, J. J., & Minniti, D. 1995, , 109, 605 Geisler, D., Bica, E., Dottori, H., Clariá, J. J., Piatti, A. E., & Santos Jr., J. F. C. 1997, , 114, 1920
Girardi, L., Bertelli, G., Bressan. A., Chiosi, C., Groenewegen, M: A. T., Marigo, P., Salasnich, B., & Weiss, A. 2002, , 391, 195 Gratton, R. G., Bragaglia, A., Clementini, G., Carretta, E., Di Fabrizio, L., Maio, M., & Taribello, E. 2004, , 421, 937 Graff, D. S., Gould, A. P., Suntzeff, N. B., Schommer, R. A., & Hardy, E. 2000, , 540, 211 Grocholski, A. J., Cole, A. A., Sarajedini, A., Geisler, D., & Smith, V. V. 2006, , 132, 1630 Hardy, E., Buonanno, R., Corsi, C. E., Janes, K. A. & Schommer, R. A. 1984, , 278, 592 Hill, V., Andrievsky, S., & Spite, M. 1995, , 293, 347 Holtzman, J. A., Gallagher, J. S., III, Cole, A. A., Mould, J. R., Grillmair, C. J., Ballester, G. E., Burrows, C. J., Clarke, J. T., Crisp, D., Evans, R. W., Griffiths, R. E., Hester, J. J., Hoessel, J. G., Scowen, P. A., Stapelfeldt, K. R., Trauger, J. T. & Watson, A. M., 1999, , 118, 2262 Hughes, S. M. G., Wood, P. R., & Reid, N. 1991, , 101, 1304 Kim, S., Staveley-Smith, L., Dopita, M. A., Freeman, K. C., Sault, R. J., Kesteven, M. J. & McConnell, D. 1998, , 503, 674 Koch, A., Grebel, E. K., Wyse, R. F. G., Kleyna, J. T., Wilkinson, M. I., Harbeck, D. R., Gilmore, G. F., & Evans, N. W.. 2006, , 131, 895 Kontizas, M., Kontizas, E., & Michalitsiano, A. G. 1993, , 269, 107 Kraft, R. P., & Ivans, I. I. 2003, , 115, 143 Kunkel, W. E., Demers, S., Irwin, M. J., & Albert, L. 1997, , 488, L129 Maeder, A. 1993, , 268, 833 Minniti, D., Borissova, J., Rejkuba, M., Alves, D. R., Cook, K. H. & Freeman, K. C., 2003, Science, 301, 5639 Nidever, D. L., Majewski, S. R. & Butler Burton, W. 2007, , submitted Olszewski, E. W., Schommer, R. A., Suntzeff, N. B., & Harris, H., C. 1991, , 101, 515 Olszewski, E. W., Suntzeff, N. B., & Mateo, M. 1996, , 34, 511 Pagel, B. E. J., & Tautvaisiene, G. 1995, , 276, 505 Pagel, B. E. J., & Tautvaisiene, G. 1998, , 299, 535 Peimbert, M., Colin, P. & Sarmiento, A. 1994, in [*“Violent Star Formation From 20 Doradus to QSOs”*]{} ed. G. Tenorio-Tagle. Cambridge University Press. Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, , 612, 168 Pont, F., Zinn, R., Gallart, C., Hardy, E., & Winnick, R. 2004, , 127, 840 Putman, M. E., Staveley-Smith, L., Freeman K. C., Gibson, B. K. & Barnes, D. G. 2003, , 586, 170 Salaris, M., & Weiss, A. 2002, , 388, 492 Salaris, M., Weiss, A., & Percival, S. M. 2004, , 414, 163 Santos Jr., F. F. C., Piatti, A. E., Cariá, J. J., Geisler, D., & Dottori, H. 1999, , 117, 2841 Scalo, J. M. 1986, , 11, 1 Schommer, R. A., Suntzeff, N. B., Olszewski, E. W. & Harris, H. C. 1992, , 103, 447 Sellwood, J. A. & Wilkinson, A. 1993, Reports on Progress in Physics, 56, 173 Smecker-Hane, T. A., Cole, A. A., Gallagher, J. S., III & Stetson, P. B. 2002, , 566, 239 Spergel, D. N., Verde, L., Peiris, H. V., Komatsu, E., Nolta, M. R., Bennett, C. L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S. S., Page, L., Tucker, G. S., Weiland, J. L., Wollack, E., Wright, E. L. 2003, , 148, 175 Tinsley, B. M., 1980, , 5, 287 Tonry, J., & Davis, M. 1979, , 84, 1511 van Dokkun, P. G. 2001, , 113, 1420 Vallenari, A., Chiosi, C., Bertelli, G. & Ortolani, S. 1996, , 309, 358 Westerlund, B. E. 1990, , 2, 29 Westerlund, B. E., Linde, P. & Lynga, G. 1995, , 298, 39 Zinn, R., & West, M. J. 1984 , 55, 45 Zhao, H. S., Ibata, R., Lewis, G. F., & Irwin, M. J. 2003 , 339, 701
[ccccccccc]{} 05:08:53.92 & -66:49:55.1 & 7.8 & 0.8 & 17.67 & 16.36 & 287.3 & 0.7 &\
05:08:55.21 & -66:47:34.7 & 8.7 & 0.2 & 16.66 & 14.70 & 240.9 & 0.6 &\
05:08:53.89 & -67:02:50.0 & 2.4 & 0.2 & 17.19 & 14.25 & 23.0 & 0.6 & No member\
05:08:55.40 & -66:53:45.4 & 8.9 & 0.4 & 16.79 & 15.09 & 292.3 & 0.5 &\
05:08:58.58 & -66:45:49.2 & 9.4 & 0.2 & 16.00 & 13.99 & 291.9 & 0.5 &\
[ccc]{} Bar & 260 & 24\
3& 269 & 27\
5& 278 & 20\
6& 293 & 15\
8& 282 & 14\
[cc]{} 8484-8513 & 8474-8484\
8522-8562 & 8563-8577\
8642-8682 & 8619-8642\
& 8799-8725\
& 8776-8792\
[lcccccccc]{} BaSTI & 2.57$\pm$0.10 & 9.72$\pm$0.15 & 0.70$\pm$0.003 & -1.51$\pm$0.007 & -3.86$\pm$0.08 &-0.19$\pm$0.007 & 0.49$\pm$0.01 & 0.38\
Padova & 1.69$\pm$0.07 & 10.6$\pm$0.1 & 0.75$\pm$0.002 & -1.87$\pm$0.007 & -4.14$\pm$0.06 &-0.24$\pm$0.006 & 0.51$\pm$0.009 &0.37\
[ccccc]{} $\geq$-0.5 & $<$3 & 226 & 0.1 & 20.5\
-0.5 to -1 & 2-10 & 178 & 1.9 & 20.5\
$\leq$-1 & $>$10 & 100 & 1.8 & 26.4\
[ccc]{} Bar & -0.39 & 0.19\
3& -0.47 & 0.31\
5& -0.50 & 0.37\
6& -0.45 & 0.31\
8& -0.79 & 0.44\
[cccccccc]{} Bar & -0.27$\pm$0.13 & -0.32$\pm$0.16 & -0.53$\pm$0.26 & -0.65$\pm$0.30 & -1.02$\pm$0.41 & -1.25$\pm$0.56 & ...\
Disk & -0.17$\pm$0.21 & -0.39$\pm$0.15 & -0.60$\pm$0.16 & -0.71$\pm$0.18 & -0.80$\pm$0.27 & -1.25$\pm$0.41 & ...\
3& -0.16$\pm$0.17 & -0.40$\pm$0.14 & -0.58$\pm$0.15 & -0.67$\pm$0.20 & -0.69$\pm$0.07 & -1.13$\pm$0.43 & 0.05\
5& -0.16$\pm$0.28 & -0.39$\pm$0.15 & -0.63$\pm$0.18 & -0.72$\pm$0.16 & -0.83$\pm$0.35 & -1.39$\pm$0.42 & 0.02\
6& -0.15$\pm$0.16 & -0.38$\pm$0.16 & -0.60$\pm$0.16 & -0.75$\pm$0.20 & -0.83$\pm$0.14 & -1.34$\pm$0.24 & 0.02\
8& -0.16$\pm$0.21 & -0.45$\pm$0.13 & -0.62$\pm$0.20 & -0.89$\pm$0.16 & -0.92$\pm$0.18 & -1.25$\pm$0.32 & 0.16\
[^1]: http://www.ctio.noao.edu/spectrographs/hydra/hydra-knutnotes.html
[^2]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^3]: available on the Web at http://iac-star.iac.es
|
---
abstract: 'In this talk it is reported on an analysis of hard exclusive $\pi^+$ electroproduction within the handbag approach. Particular emphasis is laid on single-spin asymmetries. It is argued that a recent HERMES measurement of asymmetries measured with a transversely polarized target clearly indicate the occurrence of strong contributions from transversely polarized photons. Within the handbag approach such $\gamma^{\,*}_T\to \pi$ transitions are described by the transversity GPDs accompanied by a twist-3 pion wave function. It is shown that this approach leads to results on cross sections and single-spin asymmetries in fair agreement with experiment.'
address: |
Fachbereich Physik, Universität Wuppertal,\
Wuppertal, D-42097, Germany\
$^*$E-mail: kroll@physik.uni-wuppertal.de
author:
- 'P. Kroll'
title: Exclusive pion electroproduction and transversity
---
Introduction
============
In this article it will be reported upon an analysis of hard exclusive electroproduction of positively charged pions [@GK5] within the frame work of the so-called handbag approach which offers a partonic description of meson electroproduction provided the virtuality of the exchanged photon, $Q^2$, is sufficiently large. The theoretical basis of the handbag approach is the factorization of the process amplitudes in hard partonic subprocesses and soft hadronic matrix elements, parameterized as generalized parton distributions (GPDs), as well as wave functions for the produced mesons, see Fig. \[fig:1\]. In collinear approximation factorization has been shown [@rad96; @col96] to hold rigorously for exclusive meson electroproduction in the limit $Q^2\to\infty$. It has also been shown that the transitions from a longitudinally polarized photon to the pion, $\gamma^{\,*}_L\to \pi$, dominates at large $Q^2$. Transitions from transversely polarized photons to pions, $\gamma^{\,*}_T\to \pi$, are suppressed by inverse powers of the hard scale.
However, as has been argued in Ref. , $\gamma^{\,*}_T\to \pi$ transitions are quite large at experimentally accessible values of $Q^2$ which are typically of the order of a few ${\rm GeV}^2$. This follows from data of asymmetries measured with a transversely polarized target [@Hristova] and is also seen in the transverse cross section measured by the $F_\pi-2$ collaboration [@horn06]. It is demonstrated in Ref. that within the handbag approach, these $\gamma^{\,*}_T\to \pi$ transitions can be calculated as a twist-3 effect consisting of the leading-twist helicity-flip GPDs [@diehl01; @hoodbhoy] combined with the twist-3 pion distribution amplitude [@braun90]. In the following the main ideas of the approach advocated for in Ref. will be briefly described and some of the results will be discussed and compared to experiment.
The handbag approach
====================
Within the handbag approach the amplitudes for pion electroproduction through longitudinally polarized photons read $$\begin{aligned}
{\cal M}^{\pi^+}_{0+,0+} &=& \sqrt{1-\xi^2}\, \frac{e_0}{Q}
\,\Big[\langle \widetilde{H}^{(3)}\rangle
-\frac{\xi^2}{1-\xi^2}\langle \widetilde{E}^{(3)}\rangle
- \frac{2\xi mQ^2}{1-\xi^2}\frac{\rho_\pi}{t-m_\pi^2}\Big]\,,\nonumber\\
{\cal M}^{\pi^+}_{0-,0+} &=& \frac{e_0}{Q}\,\frac{\sqrt{-t^\prime}}{2m}\,\Big[ \xi
\langle \widetilde{E}^{(3)}\rangle + 2mQ^2\frac{\rho_\pi}{t-m_\pi^2}\Big]\,.
\label{eq:L-amplitudes}\end{aligned}$$ Here, the usual abbreviation $t^\prime=t-t_0$ is employed where $t_0=-4m^2\xi^2/(1-\xi^2)$ is the minimal value of $t$ corresponding to forward scattering. The mass of the nucleon is denoted by $m$ and the skewness parameter, $\xi$, is related to Bjorken-$x$ by $$\xi =\frac{x_{\rm Bj}}{2-x_{Bj}}\,.$$ Helicity flips at the baryon vertex are taken into account since they are only suppressed by $\sqrt{-t^\prime}/m$. In contrast to this, effects of order $\sqrt{-t^\prime}/Q$ are neglected. The last term in each of the above amplitudes is the contribution from the pion pole (see Fig. \[fig:1\]). Its residue reads $$\rho_\pi = \sqrt{2}g_{\pi NN} F_\pi(Q^2) F_{\pi NN}(t^\prime)\,,
\label{residue}$$ where $g_{\pi NN}$ is the familiar pion-nucleon coupling constant. The structure of the pion and the nucleon is taken into account by form factors, the electromagnetic one for the pion, $F_\pi(Q^2)$, whereby the small virtuality of the exchanged pion is as usual ignored, and $F_{\pi NN}(t)$ for the $\pi$-nucleon vertex. The pion-pole term has been used in essentially this form in the measurement of the pion form factor [@horn06]. In contrast to other work on hard exclusive pion electroproduction ( an exception is Ref.) the full pion form factor is taken into account this way and not only its so-called perturbative contribution which only amounts to about a third of its experimental value. It is to be stressed that the pion pole also contributes to the amplitudes for transversely polarized photons. However, these contributions are very small [@GK5].
The convolutions $\langle F\rangle$ in (\[eq:L-amplitudes\]) have been worked out in Ref. with subprocess amplitudes calculated within the modified perturbative approach [@sterman]. In this approach the quark transverse momenta are retained in the subprocess and Sudakov suppressions are taken into account. The partons are still emitted and re-absorbed by the nucleon collinearly, i.e. we still have collinear factorization in GPDs and hard subprocess amplitudes. It has been shown [@GK1] that within this variant of the handbag approach the data on cross sections and spin density matrix elements for vector-meson production are well fitted for small values of skewness ($\;\xi\simeq x_{Bj}/2\,{\raisebox{-4pt}{$\,\stackrel{\textstyle
<}{\sim}\,$}}\, 0.1\;$).
$\gamma^{\,*}_T\to \pi$ transitions
===================================
The electroproduction cross sections measured with a transversely or longitudinally polarized target consist of many terms, each can be projected out by $\sin{\varphi}$ or $\cos{\varphi}$ moments where $\varphi$ is a specific linear combination of $\phi$, the azimuthal angle between the lepton and the hadron plane and $\phi_s$, the orientation of the target spin vector. A number of these moments have been measured recently [@Hristova; @hermes02]. A particularly striking result is the $\sin{\phi_S}$ moment. The data on it, displayed in Fig. \[fig:2\], exhibit a mild $t$-dependence and do not show any indication for a turnover towards zero for $t^\prime\to 0$. This behavior of $A_{UT}^{\sin{\phi_s}}$ at small $-t^\prime$ can only be produced by an interference term between the two helicity non-flip amplitudes ${\cal M}_{0+,0+}$ and ${\cal M}_{0-,++}$ which are not forced to vanish in the forward direction by angular momentum conservation. The amplitude ${\cal M}_{0-,++}$ has to be sizeable because of the large size of the $\sin{\phi_s}$ moment. We therefore have to conclude that there are strong contributions from $\gamma^*_T \to\pi$ transitions at moderately large values of $Q^2$.
\[tab:1\]
How can this amplitude be modeled in the frame work of the handbag approach? From Fig. \[fig:1\] where the helicity configuration of the amplitude ${\cal M}_{0-,++}$ is shown, it is clear that contributions from the usual helicity non-flip GPDs, $\widetilde{H}$ and $\widetilde{E}$, to this amplitude do not have the properties required by the data on the $\sin{\phi_s}$ moment. For these GPDs the emitted and re-absorbed partons from the nucleon have the same helicity. Consequently, there are net helicity flips of one unit at both the parton-nucleon vertex and the subprocess. Angular momentum conservation therefore forces both parts to vanish as $\sqrt{-t^\prime}$. Thus, a contribution from the ordinary GPDs to ${\cal M}_{0-,++}$ vanishes $\propto t^\prime$. There is a second set of leading-twist GPDs, the helicity-flip or transversity ones $H_T, E_T, \ldots$ [@diehl01; @hoodbhoy] for which the emitted and re-absorbed partons have opposite helicities. As an inspection of Fig. \[fig:1\] reveals the parton-nucleon vertex as well as the subprocess amplitude ${\cal H}_{0-,++}$ are now of helicity non-flip nature and are therefore not forced to vanish in the forward direction. The prize to pay is that quark and anti-quark forming the pion have the same helicity. Therefore, the twist-3 pion wave function is needed instead of the familiar twist-2 one. The dynamical mechanism building up the amplitude ${\cal M}_{0-,++}$ is so of twist-3 accuracy. It has been first proposed in Ref. for wide-angle photo- and electroproduction of mesons where $-t$ is considered to be the large scale [@huang].
Allowing only for $H_T$ as the only transversity GPD in an admittedly rough approximation the twist-3 mechanism only contributes to the amplitude $${\cal M}_{0-,++}^{\rm twist-3} = e_0 \int_{-1}^1 dx H^{(3)}_T(x,\xi,t) {\cal H}_{0-,++}
\label{twist-3}$$ For the calculation of the subprocess amplitude ${\cal H}_{0-,++}$ the twist-3 pion wave function is taken from Ref. with the three-particle Fock component neglected [@GK5]. This wave function contains a pseudo-scalar and a tensor component. The latter one provides a contribution to ${\cal M}_{0-,++}$ which is proportional to $t^\prime/Q^2$ and, hence, neglected. The contribution from the pseudo-scalar component to ${\cal M}_{0-,++}$ has the required properties. It is proportional to the parameter $\mu_\pi=m^2_\pi/(m_u+m_d)$ which appears as a consequence of the divergency of the axial-vector current. Since $m_u$ and $m_d$ are current quark masses $\mu_\pi$ is large, actually $\simeq 2\,{\rm GeV}$ at the scale of $2\,{\rm GeV}$. Thus, although parametrically suppressed by $\mu_\pi/Q$ as compared to the longitudinal amplitudes, the twist-3 effect is sizeable for $Q$ of the order of a few GeV.
The GPDs at small skewness
==========================
For $\pi^+$ electroproduction the GPDs, namely $\widetilde{H}$, the non-pole part of $\widetilde{E}$ and the most important one of the transversity GPDs, $H_T$, contribute in the isovector combination $$F_i^{(3)} = F_i^u - F_i^d\,.$$ The GPDs are constructed with the help of double distributions ansatz [@rad98] consisting of the product of the zero-skewness GPDs and an appropriate weight function, actually parameterized as a power of the valence Fock state meson distribution amplitude. This weight function generates the skewness dependence of the GPD. It is important to note that other methods to generate the skewness dependence, namely the Shuvaev transform [@martin] or the dual parameterization [@semenov] lead to very similar results for the GPDs at small skewness. The zero-skewness GPDs in the double distribution ansatz are assumed to be given by products of their respective forward limits and Regge-like $t$ dependences, $\exp{[f_i(x,t) t]}$, with profile functions that read $$f_i(x,t) = b_i-\alpha_i^\prime \ln{x}
\label{profile1}$$ where $\alpha_i^\prime$ is the slope of an appropriate Regge trajectory (pole or cut). These profile functions can be regarded as small-$x$ approximations of more complicated versions used for the determination of the zero-skewness GPDs from the nucleon form factors [@DFJK4] $$f_i(x,t) = b_i(1-x)^3 - \alpha_i^\prime(1-x)^3 \ln{x} +A_i x(1-x)^2
\label{profile2}$$ which hold at all $x$. There is a strong correlation between $t$ and $x$ in this ansatz: the behavior of moments or convolutions of a GPD at small (large) $-t$ is determined by the small (large) $x$ behavior of this GPD. It is to be stressed that the analysis performed in Ref. as well as recent results from lattice QCD [@haegler] clearly rule out a factorization of the zero-skewness GPDs in $x$ and $t$.
The forward limit of $\widetilde{H}$ is given by the polarized parton distributions $\Delta q(x)$, that of $H_T$ by the transversity distribution $\delta(x)$ for which the results of an analysis of the asymmetries in semi-inclusive electroproduction have been taken [@anselmino]. Finally, the forward limit of the non-pole part of $\widetilde{E}$ is parameterized as $$\tilde{e}^{(3)}(x)= \widetilde{E}^{(3)}_{\rm n.p.}(x,\xi=t=0) =
\widetilde{N}_{\tilde{e}}^{(3)} x^{-0.48} (1-x)^5\,,
\label{Enp}$$ in analogy to the PDFs. The normalization $\widetilde{N}_{\tilde{e}}^{(3)}$ is fitted to experiment. The full set of parameters used in the analysis of $\pi^+$ electroproduction can be found in Ref. .
The full GPD $\widetilde{E}^{(3)}$ is the sum of the pole and non-pole contribution where the first one reads $$\widetilde{E}^u_{\rm pole} =- \widetilde{E}^d_{\rm pole} = \Theta(|x|\leq \xi)
\frac{F_P^{\rm pole}(t)}{4\xi} \Phi_\pi((x+\xi)/(2\xi))\,.
\label{e-tilde-pole}$$ Here, $\Phi_\pi$ is the pion’s distribution amplitude and $F_P$ is the pseudo-scalar form factor of the nucleon being related to $\widetilde{E}^{(3)}$ by the sum rule $$\int^1_{-1} dx \widetilde{E}^{(3)}(x,\xi,t) = F_P(t)\,.$$ The evaluation of the pion electroproduction amplitude from the graph shown on the left hand side of Fig. \[fig:1\] just using $\widetilde{E}_{\rm pole}$ leads to the pion-pole contribution as given in (\[eq:L-amplitudes\]) but with only the perturbative contribution to the pion’s electromagnetic form factor occurring in the residue (\[residue\]). Other graphs have to be considered in addition for the pion pole, e.g. the Feyman mechanism. In order to avoid this complication the pion-pole contribution is simply worked out from the graph shown on the right hand side of Fig. \[fig:1\].
In summary: the GPDs used in Ref. are valid at small skewness ($\xi \,{\raisebox{-4pt}{$\,\stackrel{\textstyle <}{\sim}\,$}}\, 0.1\;$) and are probed by experiment for $x \,{\raisebox{-4pt}{$\,\stackrel{\textstyle <}{\sim}\,$}}\, 0.6\;$. Due to the double distribution ansatz they satisfy polynomiality and the reduction formulas. It has also been checked numerically that the lowest moments of the GPDs $\widetilde{H}$ and $\widetilde{E}$ are in agreement with the data on the axial-vector [@kitagaki] and pseudo-scalar [@choi] form factors of the nucleon (see Fig. \[fig:3\]) and respect various positivity bounds [@poby; @diehl-haegler]. Comparison with recent lattice QCD studies [@haegler; @goeckeler] reveals that there is good agreement with the relative strength of moments and their relative $t$ dependences. At small $t$ even the absolute values of the moments agree quite well but the $t$ dependence of the moments obtained from lattice QCD are usually flatter than those from the GPDs and the form factor data. An exception is the lowest moment of $H_T$ for $u$ quarks for which we have a value that is about $25\%$ smaller than the lattice result. Similar observation can be made for the GPDs $H$ and $E$ which have been constructed analogously and probed in vector meson electroproduction [@GK1]. As an example the axial-vector form factor obtained from $\widetilde{H}$ as used in Ref. is compared with experiment and with the lattice QCD results in Fig. \[fig:3\].
Results
=======
It is shown in Ref. that with the described GPDs, the $\pi^+$ cross sections as measured by HERMES [@HERMES07] are nicely fitted as well as the transverse target asymmetries [@Hristova]. This can be seen for instance from Fig. \[fig:1\] where $A_{UT}^{\sin{\phi_s}}$ is displayed. Also the $\sin(\phi-\phi_s)$ moment which is dominantly fed by an interference term of the two amplitudes for longitudinally polarized photons, is fairly well described as is obvious from Fig. \[fig:5\]. Very interesting is also the asymmetry for a longitudinally polarized target. It is dominated by an interference term between ${\cal M}_{0-,++}$ which comprises the twist-3 effect, and the nucleon helicity-flip amplitude for $\gamma^{\,*}_L\to \pi$ transition, ${\cal M}_{0-,0+}$. Results for $A_{UL}^{\sin \phi}$ are displayed and compared to the data in Fig. \[fig:6\]. In both the cases, $A_{UT}^{\sin{\phi_s}}$ and $A_{UL}^{\sin \phi}$, the prominent role of the twist-3 mechanism is clearly visible. Switching it off one obtains the dashed lines which are significantly at variance with experiment. In this case the transverse amplitudes are only fed by the pion-pole contribution.
Although the main purpose of the work presented in Ref. is focused on the analysis of the HERMES data one may also be interested in comparing this approach to the Jefferson Lab data on the cross sections [@horn06]. With the GPDs $\widetilde{H}, \widetilde{E}$ and $H_T$ in their present form the agreement with these data is poor. I remind the reader that the approach advocated for in Refs. and is optimized for small skewness. At larger values of it the parameterizations of the GPDs are perhaps to simple and may require improvements as for instance the replacement of the profile function (\[profile1\]) by (\[profile2\]). As mentioned above the GPDs are probed by the HERMES data only for $x$ less than about 0.6. One may therefore change the GPDs at large $x$ to some extent without changing much the results for cross sections and asymmetries in the kinematical region of small skewness. For Jefferson Lab kinematics, on the other hand, such changes of the GPDs may matter. Finally one should be aware that at larger values of skewness the other transversity GPDs may not be negligible. In a recent lattice study [@haegler06] the moments of the combination $2\widetilde{H}_T+E_T$ have been found to be rather large in comparison to those of $H_T$. Including this combination of GPDs into the analysis of pion electroproduction one would have $${\cal M}_{0+,\mu +}^{\rm twist-3} = -e_0 \frac{\sqrt{-t^\prime}}{4m}
\int_{-1}^1 dx \big[2\widetilde{H}^{(3)}_T+E_T^{(3)}\big] {\cal H}_{0-,++}
\label{twist-3-2}$$ in addition to (\[twist-3\]). Here, $\mu$ ($\pm 1$) labels the photon helicity. The amplitude (\[twist-3-2\]) holds up to corrections of order $\xi$.
Summary and outlook
===================
In summary, there is strong evidence for transversity in hard exclusive electroproduction of pions. A most striking effect is seen in the target asymmetry $A_{UT}^{\sin \phi_s}$. The interpretation of this effect requires a large helicity non-flip amplitude ${\cal M}_{0-,++}$. Within the handbag approach this amplitude is generated by the helicity-flip or transversity GPDs in combination with a twist-3 pion wave function. This explanation establishes an interesting connection to transversity parton distributions measured in inclusive processes. Further studies of transversity in exclusive reactions are certainly demanded. Good data on $\pi^0$ electroproduction would also be welcome. They would not only allow for further tests of the twist-3 mechanism but also give the opportunity to verify the model GPDs $\widetilde{H}$ and $\widetilde{E}$ as used in Ref. . An intriguing issue is whether or not the handbag approach in its present form for pion electroproduction works for the kinematics presently accessible at Jlab. It is known that it cannot accommodate the CLAS data on $\rho^0$, $\rho^+$ and $\omega$ production [@clas]. Other applications and tests of the handbag approach including the twist-3 mechanism are pion electroproduction measured at the upgraded Jlab accelerator or by the COMPASS collaboration and the measurement of the time-like process $\pi^- p \to \mu^+\mu^-n$ [@pire]. The extension of this approach to electroproduction of other pseudoscalar mesons, in particular the $\eta$ and $\eta^\prime$, is also of interest. In principle this would give access to the GPDs for strange quarks. As has been shown in Ref. there is no complication in the analysis of the electroproduction data due to the two-gluon Fock components of the $\eta$ and $\eta^\prime$ since they are suppressed by $t^\prime/Q^2$.
Acknowledgments
===============
The author wishes to thank Anatoly Radyushkin and Paul Stoler for inviting him to present this talk at the workshop on Exclusive Reactions at High Momentum Transfer. This work is supported in part by the BMBF under contract 06RY258.
[99]{} S.V. Goloskokov and P. Kroll, *Eur. Phys. J.* [**C**65]{} (2010) 137. A. V. Radyushkin, *Phys. Lett.* [**B**385]{} (1996) 333. J.C. Collins, L. Frankfurt and M. Strikman, *Phys. Rev.* [**D**56]{} (1997) 2982. A. Airapetian [*et al.*]{} \[HERMES Collaboration\], *Phys. Lett.* [**B682**]{}, 345 (2010). H. P. Blok [*et al.*]{} \[Jefferson Lab Collaboration\], *Phys. Rev.* [**C**78]{} (2008) 045202. M. Diehl, *Eur. Phys. J.* [**C**19]{} (2001) 485. P. Hoodbhoy and X. Ji, *Phys. Rev.* [**D**58]{} (1998) 054006. V. M. Braun and I. E. Halperin, *Z. Phys.* [**C**48]{} (1990) 239. \[*Sov. J. Nucl. Phys.* [**5**2]{} (1990 YAFIA,52,199-213.1990) 126\]. C. Bechler and D. Mueller, arXiv:0906.2571.
J. Botts and G. Sterman, *Nucl. Phys.* [**B**325]{} (1989) 62.
S. V. Goloskokov and P. Kroll, *Eur. Phys. J.* [**C**42]{} (2005) 281; [*ibid.*]{} [**C**53]{} (2008) 367.
A. Airapetian [*et al.*]{} \[HERMES Collaboration\], *Phys. Lett.* [**B**535]{} (2002) 85.
H. W. Huang [*et al.*]{}, *Eur. Phys. J.* [**C**33]{} (2004) 91. H. W. Huang and P. Kroll, *Eur. Phys. J.* [**C**17]{} (2000) 423. T. Kitagaki [*et al.*]{}, *Phys. Rev.* [**D**28]{} (1983) 436. Ph. Hagler [*et al.*]{} \[LHPC Collaborations\], *Phys. Rev.* [**D**77]{} (2008) 094502. A. V. Radyushkin, *Phys. Lett.* [**B**449]{} (1999) 81. A. D. Martin, C. Nockles, M. G. Ryskin, A. G. Shuvaev and T. Teubner, *Eur. Phys. J.* [**C**63]{} (2009) 57. M. V. Polyakov and K. M. Semenov-Tian-Shansky, *Eur. Phys. J.* [**A**40]{} (2009) 181.
M. Diehl, T. Feldmann, R. Jakob and P. Kroll, *Eur. Phys. J.* [**C**39]{} (2005) 1. M. Anselmino [*et al.*]{}, *Nucl. Phys. Proc. Suppl* [**1**91]{} (2009) 98. S. Choi [*et al.*]{}, *Phys. Rev. Lett.* [**7**1]{} (1993) 3927.
P. V. Pobylitsa, *Phys. Rev.* [**D**65]{} (2002) 114015.
M. Diehl and Ph. Hagler, *Eur. Phys. J.* [**C**44]{} (2005) 87. M. Gockeler [*et al.*]{} \[QCDSF Collaboration and UKQCD Collaboration\], *Phys. Lett.* [**B**627]{} (2005) 113.
A. Airapetian [*et al.*]{} \[HERMES Collaboration\], *Phys. Lett.* [**B**659]{} (2008) 486. M. Gockeler [*et al.*]{} \[QCDSF Collaboration and UKQCD Collaboration\], *Phys. Rev. Lett.* [**9**8]{} (2007) 222001. S. A. Morrow [*et al.*]{} \[CLAS Collaboration\], *Eur. Phys. J.* [**A**39]{} (2009) 5. E. R. Berger, M. Diehl and B. Pire, *Phys. Lett.* [**B**523]{} (2001) 265. P. Kroll and K. Passek-Kumericki, *Phys. Rev.* [**D**67]{} (2003) 054017.
|
---
abstract: 'We give a presentation of the Kauffman (BMW) skein algebra of the torus, which is the “type $BCD$” analogue of the Homflypt skein algebra of torus which was computed by the first and third authors. In the appendix we show this presentation is compatible with the Frohman-Gelca description of the Kauffman bracket (Temperley-Lieb) skein algebra of the torus [@FG00].'
address:
- 'Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK'
- 'Department of Mathematics, University of California, Riverside'
- 'Department of Mathematics, University of California, Riverside'
author:
- Hugh Morton
- Alex Pokorny
- Peter Samuelson
bibliography:
- 'somerefs.bib'
title: The Kauffman skein algebra of the torus
---
Introduction
============
The *skein* ${\mathrm{Sk}}(M)$ of the 3-manifold $M$ is a vector space formally spanned by embedded links in $M$, modulo local relations which hold in a ball. These relations are referred to as *skein relations*, and there are several versions, each of which encodes a linear relation between $R$-matrices for some quantum group. *A priori*, ${\mathrm{Sk}}(M)$ is just a vector space, but if $M = F \times [0,1]$, then ${\mathrm{Sk}}(M)$ is an algebra, where the product is given by stacking in the $[0,1]$ direction.
In recent years there have been several results relating skein algebras of surfaces with algebras constructed using representation theoretic methods. Let us somewhat imprecisely state two of these results which are most relevant to the present paper.
\[thm:past\] We have the following isomorphisms.
1. The Kauffman bracket skein algebra of the torus is isomorphic to the $t=q$ specialization of the $\mathfrak{sl}_2$ spherical double affine Hecke algebra. [@FG00]
2. The Homflypt skein algebra of the torus is isomorphic to the $t=q$ specialization of the elliptic Hall algebra of Burban and Schiffmann. [@MS17]
The double affine Hecke algebra was defined by Cherednik (for general ${\mathfrak{g}}$, see, e.g. [@Che05]) as part of his proof of the Macdonald conjectures. The elliptic Hall algebra was defined by Burban and Schiffmann, who showed it is the “generic” Hall algebra of coherent sheaves over an elliptic curve over a finite field. This algebra can also be viewed as a DAHA (the spherical $\mathfrak{gl}_\infty$ DAHA) by work of Schiffmann and Vasserot in [@SV11]. These algebras have found many applications over the years, including the following:
- Cherednik’s proof of the Macdonald conjectures [@Che95],
- algebraic combinatorics (proofs and generalizations of the shuffle conjecture [@CM18; @BGLX16]),
- knot theory (computations of Homflypt homology of positive torus knots [@Mel17]),
- mathematical physics (a mathematical version of the AGT conjectures [@SV13]),
- enumerative geometry (counting stable Higgs bundles [@Sch16]).
These results suggest that it is worthwhile to understand skein algebras of surfaces, and in particular of the torus, for various versions of the skein relations. In this paper we give a presentation of the algebra $\sd$ associated to the torus by the Kauffman[^1] skein relations given in equations and .
\[thm:main\] The Kauffman skein algebra $\sd$ of the torus has a presentation with generators $D_{{{\mathbf{x}}}}$ for ${{\mathbf{x}}}\in \Z^2$, with relations $$\begin{aligned}
[D_{{\mathbf{x}}}, D_{{\mathbf{y}}}] &= (s^d - s^{-d})\left( D_{{{\mathbf{x}}}+{{\mathbf{y}}}} - D_{{{\mathbf{x}}}-{{\mathbf{y}}}}\right)\\
D_{{\mathbf{x}}}&= D_{-{{\mathbf{x}}}}
\end{aligned}$$
If ${{\mathbf{x}}}\in \Z^2$ is primitive (i.e. its entries are relatively prime), then $D_{{\mathbf{x}}}$ is the (unoriented) simple closed curve on the torus of slope ${{\mathbf{x}}}$. However, the definition for nonprimitive ${{\mathbf{x}}}$ is more complicated, and in some sense is the key difficulty in the paper. The strategy for the proof follows the ideas of [@MS17], which in turn was motivated by some proofs in [@BS12]. Using skein theoretic calculations and some identities involving idempotents in the BMW algebra, we prove two special cases of these relations (see Proposition \[prop:specialcases\]). We then use the natural $SL_2(\Z)$ action and an induction argument to show that all the claimed relations follow from these special ones.
We also note that the $D_{{\mathbf{x}}}$ come from elements $B_k \in BMW_k$ that are analogues of the power sum elements $P_k \in H_k$ in the Hecke algebra that were found and studied by Morton and coauthors. In Theorem \[thm:aid\], which may be of independent interest, we show that the $B_k$ satisfy a “fundamental identity” which is an analogue of an identity for the $P_k$ found by Morton and coauthors. We aren’t aware of a previous appearance of this identity in the literature.
In both of the statements in Theorem \[thm:past\], the skein algebra was computed explicitly and then shown to be isomorphic to a DAHA-type algebra that had already appeared in representation theory. However, at present we are not aware of a previous appearance of the algebra described in Theorem \[thm:main\]. In particular, the discussion above suggests that there should be an interesting $t$-deformation of the algebra presented in Theorem \[thm:main\].
We remark that these skein relations are the relations used in defining the BMW algebra [@BW89; @Mur87], which is the centralizer algebra of tensor powers of the defining representation of quantum groups in type $BCD$. The algebra $\sd$ could therefore optimistically be viewed as a “type $BCD$ elliptic Hall algebra.” We believe making this statement more precise is an interesting question, but it will have to wait for future work. We do note[^2], however, that this algebra is unlikely to literally be the Hall algebra of a category because it does not have a $\Z^2$ grading[^3].
A summary of the contents of the paper is as follows. In Section 2, we recall some background and prove some identities in the $BMW$ algebra. In Section 3, we state the main theorem and reduce its proof to 2 families of identities, which are then proved in Sections 4 and 5. The Appendix regards a classical theorem of Przytycki and the relation of our theorem to a theorem of Frohman and Gelca in [@FG00] describing the Kauffman *bracket* skein algebra of the torus.
**Acknowledgements:** This paper was initiated during a Research in Pairs stay by the first and third authors at Oberwolfach in 2015, and we appreciate their support and excellent working conditions. Further work was done at conferences at Banff and the Isaac Newton Institute, and the third author’s travel was partially supported by a Simons travel grant. The authors would like to thank A. Beliakova, C. Blanchet, C. Frohman, F. Goodman, A. Negut, A. Oblomkov, and A. Ram for helpful discussions.
The BMW algebra and symmetrizers
================================
In this section we define the Kauffman skein algebra and the BMW algebra $BMW_k$, define elements $B_k \in BMW_k$ which are fundamental in the paper. We relate the closure of $B_k$ in the annulus to elements $D_k$ defined using power series, and we prove some identities involving the $D_k$ that will be needed in later sections.
Notation and definitions
------------------------
Here we will record some notation and definitions. Everywhere our base ring will be $$R := {\mathbb{Q}}(s,v)$$ We will use standard notation for quantum integers: $$\{n\} := s^n - s^{-n},\quad \quad \{n\}^+ := s^n + s^{-n},\quad \quad
[n] := \{ n \} / \{ 1 \}$$
The Kauffman skein $\sd(M)$ of a 3-manifold $M$ is the vector space spanned by framed unoriented links in $M$, modulo the Kauffman skein relations: $$\begin{aligned}
\label{eq:sk1}
\vcenter{\hbox{\includegraphics[height=2cm]{poscross.eps}}} \,\, - \,\, \vcenter{\hbox{\includegraphics[height=2cm]{negcross.eps}}} \quad = \quad (s-s^{-1}) \left( \vcenter{\hbox{\includegraphics[height=2cm]{idresolution.eps}}} \right. \,\, - \,\, \left. \vcenter{\hbox{\includegraphics[height=2cm]{capcupresolution.eps}}} \right),
\end{aligned}$$ $$\begin{aligned}
\label{eq:sk2}
\vcenter{\hbox{\includegraphics[height=2cm, keepaspectratio]{invvh.eps}}} \quad = \quad v^{-1} \,\, \vcenter{\hbox{\includegraphics[height=2cm, keepaspectratio]{frameresolution.eps}}} \,\, , \qquad \qquad \vcenter{\hbox{\includegraphics[height=2cm, keepaspectratio]{vh.eps}}} \quad = \quad v \,\, \vcenter{\hbox{\includegraphics[height=2cm, keepaspectratio]{frameresolution.eps}}}
\end{aligned}$$
In particular, we will abbreviate the notation for the thickened torus and solid torus.
- $\sd := \sd(T^2\times [0,1])$ is the skein algebra of the thickened torus,
- $\cc := \sd(S^1 \times [0,1]^2)$ is the skein of the solid torus.
The Hecke and BMW Algebras {#sec:bmw}
--------------------------
The Birman-Murakami-Wenzl algebra $BMW_n$ is the $R$-algebra generated by $\sigma_1, \cdots, \sigma_{n-1}, h_1, \cdots, h_{n-1}$ subject to certain relations, which can be found in [@BW89; @Mur87]. They showed that the BMW algebra is isomorphic to the Kauffman algebra (see also [@Mor10; @GH06]), which consists of tangles in the square with $n$ points on both the top and bottom edges, with orientation removed, modulo the Kauffman skein relations, where the multiplication is stacking diagrams vertically. By convention, the diagram of the product $XY$ is obtained by joining the top edge of $X$ with the bottom edge of $Y$. This is sometimes referred to the relative skein module on the square with $2n$ boundary points. A general theory of relative skein modules exist, and we will discuss another example of such a skein in Section \[sec:perpendicular\].
The Hecke algebra $H_n$ is similarly defined as the relative Homflypt skein module of the square with $2n$ boundary points. As an algebra, it is generated by $\sigma_1,\cdots,\sigma_{n-1}$, modulo the quadratic relations $\sigma_i - \sigma^{-1}_i = s - s^{-1}$. There is an algebra projection $\pi_n: BMW_n \to H_n$ sending $\sigma_i$ to $\sigma_i$ and sending $h_i$ to 0.
In this subsection we first recall a very useful result of Beliakova and Blanchet in [@BB01] relating the Hecke algebra $H_n$ and the BMW algebra, which we use to prove an identity involving the $D_k$. Let $\pi_n: BMW_n \to H_n$ be the projection from the BMW algebra to the Hecke algebra, and let $I_n$ be the kernel of $\pi_n$.
\[thm:bb01\] There exists a unique (non-unital[^4]) algebra map $s_n: H_n \to BMW_n$ which is a section of $\pi_n$ such that $$s_n(x)y = 0 = y s_n(x)\quad \mathrm{for} \,\, x \in H_n,\,\, y \in I_n$$
The Hecke algebra has minimal central idempotents $z_\lambda \in H_n$ indexed by partitions of $n$: $$\{\lambda = (\lambda_1,\cdots,\lambda_k)\mid \lambda_j \in \mathbb{N}_{\geq 0},\, \lambda_i \geq \lambda_{i+1},\,\, \sum \lambda_i = n\}$$ The section $s_n$ transfers these to minimal central idempotents in the BMW algebra. In particular, if $z_{(n)} \in H_n$ is the symmetrizer, then $s_n(z_{(n)}) = f_n$ is the symmetrizer in the BMW algebra. We use these symmetrizers to make the key definition of the paper.
\[def:dk\]Given an element $x \in BMW_n$, let $\hat x \in \cc = \sd(S^1\times D^2)$ be the annular closure of a diagram representing $x$. Define elements $D_k \in \cc$ using the following equality of power series: $$\label{eq:dk}
\sum_{k=1}^\infty \frac{D_k}{k} t^k:=\ln\left(1 + \sum_{j \geq 0} \hat f_j t^j\right)$$ Note the right hand side makes sense because $\cc$ is a commutative algebra.
Relating $D_k$ to hook idempotents
----------------------------------
Write $\widehat{f}_n \in \cc$ for the standard closure of the symmetriser $f_n \in BMW_n$ and similarly $\widehat{e}_n \in \cc$ for that of the antisymmetriser. We will also use the notation $\widehat{Q}_{(i|j)} \in \cc$ for the image in $\cc$ of the class $[s_n(z_{(i|j)})]$ corresponding to the hook $\lambda =(i|j)$, which has an arm of length $i$ and a leg of length $j$, and whose Young diagram contains $i+j+1$ cells. Then we have as a special case $\widehat{f}_n=\widehat{Q}_{(n-1|0)}$ and $\widehat{e}_n=\widehat{Q}_{(0|n-1)}$.
We can use the decomposition of the product of $\widehat{f}_i$ and $\widehat{e}_j$ as a sum of hooks to analyse the element $D_k$ in terms of hooks. Write $F(t), E(t)$ for the formal generating functions $$F(t):=1+\sum_{i=1}^\infty \widehat{f}_i t^i, \ E(t)=1+\sum_{i=1}^\infty \widehat{e}_if t^i.$$ Recall that we have defined $D_k$ by $$\sum_{k=1}^\infty \frac{D_k}{k} t^k:=\ln(F(t)).$$ Taking the formal derivative of these series gives the alternative description $$\sum_{k=1}^\infty D_k t^{k-1}=\frac{F'(t)}{F(t)}.$$
Let us also introduce the formal 2-variable hook series $S(t,u)$ by setting $$S(t,u):=\sum_{k,l\ge 0} \widehat{Q}_{(k|l)} t^k u^l.$$
\[hooks\] The product of the two series $F(t)$ and $E(u)$ satisfies $$F(t) E(u) = (1+tu)\left(1+(t+u)S(t,u)\right).\label{hookeqn}$$
From the paper of Koike and Terada [@KT87] we have an expansion for the product $\widehat{f}_i \widehat{e}_j$ in terms of hooks: $$\widehat{f}_i \widehat{e}_j = \widehat{Q}_{(i-1|j)} + \widehat{Q}_{(i|j-1)} + \widehat{Q}_{(i-2|j-1)} + \widehat{Q}_{(i-1|j-2)}$$ This relation is true for $i,j>1$, and also holds when one of $i$ and $j$ is equal to $1$, with the convention that $\widehat{Q}_{(k|l)}=0$ when $k<0$ or $l<0$. The one exception is that when $i=j=1$ there is an extra term $$\widehat{f}_1 \widehat{e}_1 = \widehat{Q}_{(0|1)}+ \widehat{Q}_{(1|0)} +1.$$
This accounts for the term $tu$ on the right hand side of equation . Otherwise the coefficient of $t^i u^j$ in $(t+u+t^2u+tu^2)S(t,u)$ is $$\widehat{Q}_{(i-1|j)} + \widehat{Q}_{(i|j-1)} + \widehat{Q}_{(i-2|j-1)} + \widehat{Q}_{(i-1|j-2)}$$ with the same convention, and so agrees with the coefficient of $t^i u^j$ in $F(t)E(u)$.
In Proposition \[hooks\] we can take $u=-t$ to get the relation $F(t)E(-t)=1-t^2$.
\[thm:hooksum\] We have $$D_k=\sum_{i+j+1=k} (-1)^j \widehat{Q}_{(i|j)} + c_k,$$ where $$\label{eq:ck}
c_k :=
\begin{cases}
0 & k \textrm{ odd} \\
-1 & k \textrm{ even}
\end{cases}$$
Differentiate equation with respect to $t$. Then $$F'(t)E(u)=u\left(1+(t+u)S(t,u)\right) +(1+tu)\left( S(t,u)+(t+u)\frac{\partial S}{\partial t}\right).$$
Divide this by $F(t)E(u)$ and set $u=-t$. Then $$\frac{F'(t)}{F(t)}=\frac{-t +(1-t^2)S(t,-t)}{1-t^2} = S(t,-t)-\frac{t}{1-t^2}.$$ Now $D_k$ is the coefficient of $t^{k-1}$ in this series, which is as claimed, with $c_k=-1$ when $k$ is even, and $c_k=0$ when $k$ is odd.
Lifting from the Hecke algebra
------------------------------
We now recall elements in the Hecke algebra which were studied by Morton and coauthors, and were useful in [@MS17]. We will show that the section of [@BB01] maps these elements $B_k \in BMW_k$ whose closure is (essentially) the elements $D_k$. We use this to “lift” some identites involving these elements from the Hecke algebra to the BMW algebra.
\[def:pk\] Let $P_n \in H_n$ be given by the formula $$P_n = \frac{s-s^{-1}}{s^n-s^{-n}} \sum_{i=0}^{n-1} \sigma_1 \cdots \sigma_i \sigma_{i+1}^{-1}\cdots \sigma_{n-1}^{-1}$$ where, by convention, the first term is $\sigma_1^{-1}\cdots \sigma_{n-1}^{-1}$ and the last term is $\sigma_1 \cdots \sigma_{n-1}$.
Recall that the Hochschild homology $HH_0(A)$ of an algebra $A$ is the vector space defined by $$HH_0(A) := \frac{A} {\mathrm{span}_{R}\, \{ab - ba \mid a,b \in A\} }$$ This is useful for us because of the following lemma.
\[lemma:close\] There is an injective linear map $cl: HH_0(H_n) \to Homflypt(S^1\times D^2)$ from the Hochschild homology of the Hecke algebra into the Homflypt skein algebra of the annulus. There is a (noninjective) linear map $cl: HH_0(BMW_n) \to BMW(S^1\times D^2)$ from the Hochschild homology of the BMW algebra to the Kauffman skein algebra of the annulus.
The existence of both linear maps follows from the fact that the Hecke algebra (BMW algebra) is isomorphic to the algebra of tangles in the disk modulo the Homflypt (Kauffman) skein relations. The injectivity in the Homflypt case is well known (see, e.g. [@CLLSS18 Thm. 3.25] for a stronger statement).
In the BMW case this closure map does not induce an injection from $HH_0(BMW_n)$. For example, the class of the “cup-cap” element $h_1 \in BMW_2$ is not a multiple of the class of the identity since $BMW_2$ is commutative, but $1-\delta^{-1}h_1$ is sent to zero in the skein algebra.
We write $[a] \in HH_0(A)$ for the class of an element $a \in A$. If we expand the class $[P_k]$ in terms of the classes of the central idempotents $z_\lambda$, we obtain the following identity involving the so-called “hook” idempotents: $$\label{eq:pkidem}
[P_k]=\sum_{i+j+1=k} (-1)^j [z_{(i|j)}]$$ (This follows e.g. from [@MS17 Eq. (4.2)] and the injectivity statement in Lemma \[lemma:close\].) This equation explains the notation for $P_k$, since this identity is satisfied by the power sums in the ring of symmetric functions.
\[def:bk\] The image in the BMW algebra of the power sum element defined above is denoted $$\label{eq:bk}
B_k := s_k(P_k) \in BMW_k$$
To separate notation between the Hecke and BMW algebras, we write $Q_\lambda := s_n(z_\lambda)$ for the image in the BMW algebra of idempotents in the Hecke algebra. The following identity will be used to relate the closure of $B_k$ to the element $D_k$ defined using the power series .
\[cor:skpkhook\] The $B_k$ satisfy the following identities: $$\begin{aligned}
[B_k]&=\sum_{i+j+1=k} (-1)^j [Q_{(i|j)}] \label{eq:hooksum}\\
D_k &= \hat{B}_k + c_k \label{eq:dvsb}
\end{aligned}$$
The identity follows immediately from Theorem \[thm:bb01\] and from equation , and the second follows from this and Theorem \[thm:hooksum\].
We will also need a more general identity which will be used to describe “part” of the action of the skein algebra of the torus on the skein module of the annulus. In the Hecke/Homflypt case this was described completely in [@MS17], and here we slightly rephrase some of these results in terms of Hochschild homology.
Let $k \in {\mathbb{N}}_{\geq 1}$ and $n\in {\mathbb{N}}_{\geq 2}$, and let $m \in \Z$ be relatively prime to $n$. Here we write an element $T_{km,kn} \in H_{kn}$ in the Hecke algebra whose closure is the image of the $(km,kn)$ torus link in the annulus. $$\begin{aligned}
\alpha_{i,j} &:= \sigma_{i}\sigma_{i+1}\cdots \sigma_{j-1}\\
T_{km,kn} &:= \left( \alpha_{k,n}\alpha_{k-1,n-1}\cdots \alpha_{1,n-k}\right)^m\end{aligned}$$ Let $H_k \hookrightarrow H_{k+1}$ be the standard inclusion of Hecke algebras. We abuse notation and write $P_k \in H_n$ for the image of $P_k$ under compositions of these inclusions. We next define elements in the Hecke algebra whose closures are positive torus links colored by $P_k$ (the factor of $v$ is to compensate for framing). $$\tilde T_{km,kn} := v^{-km} P_k T_{km,kn}$$ The following identity is a special case of [@MS17 Thm. 4.6, (4.3)] when the partitions $\lambda$ and $\mu$ are empty.
In $HH_0(H_{kn})$, we have the following identity: $$[\tilde T_{km,kn}] = \sum_{a+b+1=kn} (-1)^b v^{-km} s^{km(a-b)} [z_{(a\mid b)}]$$
Since the closure map $HH_0(H_n) \to Homflypt(S^1\times D^2)$ is injective, we can transfer the skein-theoretic identity [@MS17 Thm. 4.6, (4.3)] into the claimed identity in $HH_0(H_n)$.
We then have the following corollary which describes part of the action of the BMW skein algebra of the torus on the BMW skein module of the annulus.\
\[cor:projection\] In $HH_0(BMW_{kn})$, we have the identity $$[s_{kn} (\tilde T_{km,kn})] = \sum_{a+b+1=kn} (-1)^b v^{-km} s^{km(a-b)} [Q_{(a|b)}]$$
Some of the identities involving the $B_k$ and $D_k$ are reminiscent of identities in [@DRV13; @DRV14], and we hope to clarify this in future work.
Explicit formulas for $n=2$ {#sec:D_2}
---------------------------
In this section we write explicitly the statements above in the case $n=2$, both as an example and as a sanity check. This explicit formula will also be useful in equation . By [@BB01], when $n=2$, the section $s_2: H_2 \to BMW_2$ is given by $s_n(x) = p_1^+ x p_1^+$, where $$p_1^+ := 1 - \delta^{-1}h_1$$ We have $P_2 = (\sigma + \sigma^{-1})/(s+s^{-1})$, and a short computation gives $$B_2 = s_2(P_2) = \frac{\sigma + \sigma^{-1}}{s+s^{-1}} - \frac{v+v^{-1}}{s+s^{-1}}$$ The symmetrizer $z_2 \in H_2$ is $z_2 = (1+s\sigma)/(s^2+1)$, and another computation gives $$s_2(z_2) = (1+s\sigma + \beta_1 h)/(s^2+1) = f_2$$ which agrees with the $BMW_2$ symmetrizer (see, e.g. [@She16]). (Here $\beta_1 = -\delta^{-1}(sv^{-1}+1)$.) The power series definition of $D_2\in \cc$ shows $D_2 = 2\widehat f_2 - 1$, where as above have written $\widehat f_2$ for the closure of the symmetrizer in the annulus. Using relations in $BMW_2$, one can manipulate the above equations to obtain $$D_2 = \widehat{B}_2 - 1$$ which agrees with equation .
Recursion between symmetrizers {#sec:recursion}
------------------------------
In this section, we will recall some facts about the symmetrizers $f_n$ in the BMW algebras.
\[def:symmetrizer\] The *symmetrizer* $f_n \in BMW_n$ is the unique idempotent such that $$\begin{aligned}
f_n \sigma_i &= \sigma_i f_n = s \sigma_i \\
f_n h_i &= h_i f_n = 0
\end{aligned}$$ for all $i$, where $\sigma_i$ is the standard positive simple braid and $h_i$ is the cap-cup in the $i$ and $i+1$ positions.
Note that this implies that the $f_n$ are central in $BMW_n$. In the analogous case of the Hecke algebra, the symmetrizers $z_n$ are well known. In fact, each $f_n$ is the image of $z_n$ under the section of the projection map $\pi:BMW_n \to H_n$ defined in [@BB01].
\[prop:recursion\] The symmetrizers $f_n \in BMW_n$ satisfy the following recurrence relation: $$[n+1]f_{n+1} = [n]s^{-1}\left( f_n \otimes 1 \right) \left( 1 \otimes f_n \right) + \sigma_n \cdots \sigma_1 \left( 1 \otimes f_n \right) + [n]s^{-1}\beta_n\left( f_n \otimes 1 \right) h_n \cdots h_1 \left( 1 \otimes f_n \right)$$
In terms of diagrams, $$[n+1] \vcenter{\hbox{\includegraphics[width=2.9cm]{f_nplus1.eps}}} = [n]s^{-1} \vcenter{\hbox{\includegraphics[width=2.9cm]{f_notimes11otimesf_n.eps}}} + \vcenter{\hbox{\includegraphics[width=2.9cm]{sigma_ndotssigma_11otimesf_n.eps}}} + [n]s^{-1} \beta_n \vcenter{\hbox{\includegraphics[width=2.9cm]{f_notimes1H1otimesf_n.eps}}}$$ This recurrence relation will prove to be useful in Section \[sec:perpendicular\].
All relations
=============
In this section we state the main theorem of the paper, and we reduce its proof to several independent propositions that we prove in later sections. The structure of this proof is similar to that of Proposition 3.7 in [@MS17]. Recall that the algebra $\sd$ is the Kauffman skein algebra $\sd(T^2 \times [0,1])$ of the torus, and that $D_k \in \cc = \sd(S^1\times [0,1] \times [0,1])$ is the element in the skein of the annulus defined by the power series identity .
\[def:D\_x\] Given ${{\mathbf{x}}}= (a,b) \in \Z^2$ with $k = gcd(a,b)$, we define $D_{{\mathbf{x}}}\in \sd$ to be the image of $D_k$ under the map $\cc \to \sd$ that embeds the annulus along the simple closed curve ${{\mathbf{x}}}/ d$ on the torus $T^2$.
\[rmk:simplecurve\] Note that if ${{\mathbf{x}}}$ is primitive (i.e. its entries have gcd 1), then $D_{{{\mathbf{x}}}}$ is just the simple closed curve on the torus of slope ${{\mathbf{x}}}$. The reason that the $D_{{\mathbf{x}}}$ are the “correct” choice of generators for non-primitive ${{\mathbf{x}}}$ is that they satisfy surprisingly simple relations.
We first state special cases of the relations, which we prove in later sections. We then state and prove the general case of these relations.
\[prop:specialcases\] The elements $D_{{{\mathbf{x}}}}$ satisfy the following relations: $$\begin{aligned}
[D_{1,0}, D_{0,n}] &= \{n\}(D_{1,n} - D_{1,-n})\label{eq:rel1}\\
[D_{1,0},D_{1,n}] &= \{n\}(D_{2,n} - D_{0,n})\label{eq:rel2}
\end{aligned}$$
These are proved in Sections \[sec:perpendicular\] and \[sec:angled\].
Suppose we picked some other generators $D'_{{\mathbf{x}}}$, and we required the $D'_{{\mathbf{x}}}$ to be equivariant with respect to the $SL_2(\Z)$ action, and we choose $D'_{1,0}$ to be a simple closed curve. Then equation determines the $D'_{n,0}$ uniquely, and it can be shown that equation determines $D'_{n,0}$ up to addition of a scalar. This means the choice of generators is essentially uniquely determined by either special case of our desired relations. The key point is that the $D'_{n,0}$ are simultaneous solutions to these two equations.
In what proceeds, we will show that the relations of Proposition \[prop:specialcases\] imply the relations desired in the main Theorem. We will write $d({{\mathbf{x}}}, {{\mathbf{y}}}) = \det\left[{{\mathbf{x}}}\,\, {{\mathbf{y}}}\right]$ for ${{\mathbf{x}}}, {{\mathbf{y}}}\in \Z^2$ and $d({{\mathbf{x}}}) = gcd(m,n)$ when ${{\mathbf{x}}}= (m,n)$ and we will also use the following terminology: $$({{\mathbf{x}}},{{\mathbf{y}}}) \in \Z \times \Z \textrm{ is \emph{good} if } [D_{{\mathbf{x}}},D_{{\mathbf{y}}}] = \{d({{\mathbf{x}}},{{\mathbf{y}}})\} \left( D_{{{\mathbf{x}}}+{{\mathbf{y}}}}-D_{{{\mathbf{x}}}-{{\mathbf{y}}}} \right)$$
\[remark\_goodsymmetry\] Note that because $D_{{{\mathbf{x}}}}=D_{-{{\mathbf{x}}}}$, if $({{\mathbf{x}}}, {{\mathbf{y}}})$ is good, then the pairs $(\pm{{\mathbf{x}}}, \pm{{\mathbf{y}}})$ are good as well.
The idea of the proof is to induct on the absolute value of the determinant of the matrix with columns ${{\mathbf{x}}}$ and ${{\mathbf{y}}}$. To induct, we write ${{\mathbf{x}}}= {{\mathbf{a}}}+ {{\mathbf{b}}}$ for carefully chosen vectors ${{\mathbf{a}}}, {{\mathbf{b}}}$ and then use the following lemma.\
It is easy to see that Lemma \[lemma\_trueforab\] applies to the vectors in this example.
\[lemma\_trueforab\] Assume ${{\mathbf{a}}}+ {{\mathbf{b}}}= {{\mathbf{x}}}$ and that $({{\mathbf{a}}},{{\mathbf{b}}})$ is good. Further assume that the five pairs of vectors $({{\mathbf{y}}}, {{\mathbf{a}}})$, $({{\mathbf{y}}}, {{\mathbf{b}}})$, $({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}})$, $({{\mathbf{y}}}+{{\mathbf{b}}},{{\mathbf{a}}})$, and $({{\mathbf{a}}}-{{\mathbf{b}}}, {{\mathbf{y}}})$, are good. Then the pair $({{\mathbf{x}}},{{\mathbf{y}}})$ is good.
By the first assumption, we have $[D_{{{\mathbf{a}}}}, D_{{{\mathbf{b}}}}] = \{d({{\mathbf{a}}}, {{\mathbf{b}}})\} \big( D_{{{\mathbf{a}}}+{{\mathbf{b}}}} - D_{{{\mathbf{a}}}-{{\mathbf{b}}}} \big)$. We then use the Jacobi identity and the remaining assumptions to compute $$\begin{aligned}
&-&\{d({{\mathbf{a}}}, {{\mathbf{b}}})\} [D_{{{\mathbf{a}}}+{{\mathbf{b}}}}, D_y] + \{d({{\mathbf{a}}}, {{\mathbf{b}}})\} [D_{{{\mathbf{a}}}-{{\mathbf{b}}}}, D_y] \\
=&-&[[D_{{{\mathbf{a}}}}, D_{{{\mathbf{b}}}}], D_{{{\mathbf{y}}}}] \\
=&&[[D_{{{\mathbf{y}}}}, D_{{{\mathbf{a}}}}], D_{{{\mathbf{b}}}}] + [[D_{{{\mathbf{b}}}}, D_y], D_{{{\mathbf{a}}}}] \\
=&&\{d({{\mathbf{y}}}, {{\mathbf{a}}}\} [D_{{{\mathbf{y}}}+{{\mathbf{a}}}} - D_{{{\mathbf{y}}}-{{\mathbf{a}}}}, D_{{{\mathbf{b}}}}] + \{d(b,y)\} [D_{{{\mathbf{b}}}+{{\mathbf{y}}}} - D_{{{\mathbf{b}}}-{{\mathbf{y}}}}], D_{{{\mathbf{a}}}}] \\
=&&\{d({{\mathbf{y}}},{{\mathbf{a}}})\} \left( \{d({{\mathbf{y}}}+{{\mathbf{a}}}, {{\mathbf{b}}})\} \left( D_{{{\mathbf{y}}}+{{\mathbf{a}}}+{{\mathbf{b}}}} - D_{{{\mathbf{y}}}+{{\mathbf{a}}}-{{\mathbf{b}}}} \right) \right. \\
&& \left. \qquad\qquad -\{d({{\mathbf{y}}}-{{\mathbf{a}}},{{\mathbf{b}}})\} \left( D_{{{\mathbf{y}}}-{{\mathbf{a}}}+{{\mathbf{b}}}} - D_{{{\mathbf{y}}}-{{\mathbf{a}}}-{{\mathbf{b}}}} \right) \right) \\
&+&\{d({{\mathbf{b}}},{{\mathbf{y}}})\} \left( \{d({{\mathbf{b}}}+{{\mathbf{y}}}, {{\mathbf{a}}})\} \left( D_{{{\mathbf{b}}}+{{\mathbf{y}}}+{{\mathbf{a}}}} - D_{{{\mathbf{b}}}+{{\mathbf{y}}}-{{\mathbf{a}}}} \right) \right. \\
&& \left. \qquad\qquad -\{d({{\mathbf{b}}}-{{\mathbf{y}}}, {{\mathbf{a}}})\} \left( D_{{{\mathbf{b}}}-{{\mathbf{y}}}+{{\mathbf{a}}}} - D_{{{\mathbf{b}}}-{{\mathbf{y}}}-{{\mathbf{a}}}} \right) \right) \\
=&& \left( \{d({{\mathbf{y}}},{{\mathbf{a}}})\} \{d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}})\} + \{d({{\mathbf{b}}},{{\mathbf{y}}})\} \{d({{\mathbf{b}}}+{{\mathbf{y}}},{{\mathbf{a}}})\} \right) D_{{{\mathbf{a}}}+{{\mathbf{b}}}+{{\mathbf{y}}}} \\
&+& \left( \{d({{\mathbf{y}}},{{\mathbf{a}}})\} \{d({{\mathbf{y}}}-{{\mathbf{a}}},{{\mathbf{b}}})\} - \{d({{\mathbf{b}}},{{\mathbf{y}}})\} \{d({{\mathbf{b}}}-{{\mathbf{y}}},{{\mathbf{a}}})\} \right) D_{{{\mathbf{a}}}+{{\mathbf{b}}}-{{\mathbf{y}}}} \\
&-& \left( \{d({{\mathbf{y}}},{{\mathbf{a}}})\} \{d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}})\} - \{d({{\mathbf{b}}},{{\mathbf{y}}})\} \{d({{\mathbf{b}}}-{{\mathbf{y}}},{{\mathbf{a}}})\} \right) D_{{{\mathbf{a}}}-{{\mathbf{b}}}+{{\mathbf{y}}}} \\
&-& \left( \{d({{\mathbf{y}}},{{\mathbf{a}}})\} \{d({{\mathbf{y}}}-{{\mathbf{a}}},{{\mathbf{b}}})\} + \{d({{\mathbf{b}}},{{\mathbf{y}}})\} \{d({{\mathbf{b}}}+{{\mathbf{y}}},{{\mathbf{a}}})\} \right) D_{{{\mathbf{a}}}-{{\mathbf{b}}}-{{\mathbf{y}}}} \\
=&& c_1 D_{{{\mathbf{a}}}+{{\mathbf{b}}}+{{\mathbf{y}}}} + c_2 D_{{{\mathbf{a}}}+{{\mathbf{b}}}-{{\mathbf{y}}}} - c_3 D_{{{\mathbf{a}}}-{{\mathbf{b}}}+{{\mathbf{y}}}} - c_4 D_{{{\mathbf{a}}}-{{\mathbf{b}}}-{{\mathbf{y}}}}
\end{aligned}$$ Using some simple algebra, we can show $$\begin{aligned}
c_1 \,\,\, =& & \left( \{d({{\mathbf{y}}},{{\mathbf{a}}})\} \{d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}})\} + \{d({{\mathbf{b}}},{{\mathbf{y}}})\} \{d({{\mathbf{b}}}+{{\mathbf{y}}},{{\mathbf{a}}})\} \right) \\
=&&\left( \{d({{\mathbf{y}}},{{\mathbf{a}}})+d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}})\}^+ - \{d({{\mathbf{y}}},{{\mathbf{a}}})-d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}})\}^+ \right. \\
&+& \left. \{d({{\mathbf{b}}},{{\mathbf{y}}})+d({{\mathbf{b}}}+{{\mathbf{y}}},{{\mathbf{a}}})\}^+ - \{d({{\mathbf{b}}},{{\mathbf{y}}})-d({{\mathbf{b}}}+{{\mathbf{y}}},{{\mathbf{a}}})\}^+ \right) \\
=&&\left( \{d({{\mathbf{y}}},{{\mathbf{a}}}+{{\mathbf{b}}})+d({{\mathbf{a}}},{{\mathbf{b}}})\}^+ - \{d({{\mathbf{y}}},{{\mathbf{a}}}-{{\mathbf{b}}})-d({{\mathbf{a}}},{{\mathbf{b}}})\}^+ \right. \\
&+& \left. \{d({{\mathbf{y}}},{{\mathbf{a}}}-{{\mathbf{b}}})-d({{\mathbf{a}}},{{\mathbf{b}}})\}^+ - \{d({{\mathbf{a}}},{{\mathbf{b}}})-d({{\mathbf{y}}},{{\mathbf{a}}}+{{\mathbf{b}}})\}^+ \right) \\
=&& \left( \{d({{\mathbf{a}}},{{\mathbf{b}}})+d({{\mathbf{y}}},{{\mathbf{a}}}+{{\mathbf{b}}})\}^+ - \{d({{\mathbf{a}}},{{\mathbf{b}}})-d({{\mathbf{y}}},{{\mathbf{a}}}+{{\mathbf{b}}})\}^+ \right) \\
=&& \{d({{\mathbf{a}}},{{\mathbf{b}}})\} \{d({{\mathbf{y}}},{{\mathbf{a}}}+{{\mathbf{b}}})\}
\end{aligned}$$ Similar computations for the other $c_i$ show that $$\begin{aligned}
c_1 D_{{{\mathbf{a}}}+{{\mathbf{b}}}+{{\mathbf{y}}}} + c_2 D_{{{\mathbf{a}}}+{{\mathbf{b}}}-{{\mathbf{y}}}} - c_3 D_{{{\mathbf{a}}}-{{\mathbf{b}}}+{{\mathbf{y}}}} - c_4 D_{{{\mathbf{a}}}-{{\mathbf{b}}}-{{\mathbf{y}}}} =& -&\{d({{\mathbf{a}}},{{\mathbf{b}}})\} \{d({{\mathbf{a}}}+{{\mathbf{b}}},{{\mathbf{y}}})\} \left( D_{{{\mathbf{a}}}+{{\mathbf{b}}}+{{\mathbf{y}}}} - D_{{{\mathbf{a}}}+{{\mathbf{b}}}-{{\mathbf{y}}}} \right) \\
&+&\{d({{\mathbf{a}}},{{\mathbf{b}}})\} \{d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}})\} \left( D_{{{\mathbf{a}}}-{{\mathbf{b}}}+{{\mathbf{y}}}} - D_{{{\mathbf{a}}}-{{\mathbf{b}}}-{{\mathbf{y}}}} \right)
\end{aligned}$$
Since $({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}})$ is assumed to be good, we see that $({{\mathbf{a}}}+{{\mathbf{b}}},{{\mathbf{y}}})=({{\mathbf{x}}},{{\mathbf{y}}})$ is good. This completes the proof of the lemma.
We next prove the following elementary lemma (which is a slight modification of [@FG00 Lemma 1]). This lemma is used to make a careful choice of vectors ${{\mathbf{a}}}, {{\mathbf{b}}}$ so that the previous lemma can be applied.
\[lemma\_diophantine\] Suppose $p,q \in \Z$ are relatively prime with $0 < q < p$ and $p > 1$. Then there exist $u,v,w,z \in \Z$ such that the following conditions hold: $$\begin{aligned}
u + w &=& p,\quad v + z = q\notag\\
0 < u, w &<& p\label{equation_conditionsonuvwz}\\
uz - wv &=& 1\notag
\end{aligned}$$
Since $p$ and $q$ are relatively prime, there exist $a,b \in \Z$ with $ bq - ap = 1$. This solution can be modified to give another solution $a' = a + q$ and $b' = b + p$, so we may assume $0 \leq b < p$. We then define $$u=b,\quad v=a,\quad w=p-b,\quad z=q-a$$ By definition, $u,v,w,z$ satisfy the first condition of , and the inequalities $0 \leq b < p$ and $p > 1$ imply the second condition. To finish the proof, we compute $$uz - wv = b(q-a) - a(p-b) = bq - ap = 1$$
It might be helpful to point out that the numbers in the previous lemma satisfy $$\left[ \begin{array}{cc} u&w\\v&z\end{array}\right]\left[\begin{array}{c}1\\1\end{array}\right] = \left[ \begin{array}{c}p\\q\end{array}\right]$$
\[remark\_gl2z\] There is a natural $R$-linear anti-automorphism $\tau:\sd \to \sd$ which “flips $T^2 \times [0,1]$ across the $y$-axis and inverts $[0,1]$.” In terms of the elements $D_{a,b}$, we have $\tau(D_{a,b}) = D_{a,-b}$. We therefore have an *a priori* action of $\mathrm{GL}_2(\Z)$ on $\sd$, where elements of determinant $1$ act by algebra automorphisms, and elements of determinant $-1$ act by algebra anti-automorphisms. It will be important that the orbits of this action on the $D_{{{\mathbf{x}}}}$ are the fibers of the assignment $D_{{{\mathbf{x}}}} \mapsto d({{\mathbf{x}}})$ (which is essentially the statement of Lemma \[lemma\_diophantine\]).
\[lemma\_allfromsome\] Suppose $A$ is an algebra with elements $D_{{\mathbf{x}}}$ for ${{\mathbf{x}}}\in \Z^2/\langle -{{\mathbf{x}}}= {{\mathbf{x}}}\rangle$ that satisfy equations and . Furthermore, suppose that there is a ${\mathrm{GL}}_2(\Z)$ action by (anti-)automorphisms on $A$ as in Remark \[remark\_gl2z\], and that the action of ${\mathrm{GL}}_2(\Z)$ is given by $\gamma(D_{{\mathbf{x}}}) = D_{\gamma({{\mathbf{x}}})}$ for $\gamma \in {\mathrm{GL}}_2(\Z)$. Then the $D_{{\mathbf{x}}}$ satisfy the equations $$[D_{{\mathbf{x}}}, D_{{\mathbf{y}}}] = (s^d - s^{-d})\left( D_{{{\mathbf{x}}}+{{\mathbf{y}}}} - D_{{{\mathbf{x}}}-{{\mathbf{y}}}}\right)$$
The proof proceeds by induction on $\lvert d({{\mathbf{x}}},{{\mathbf{y}}})\rvert $, and the base case $\lvert d({{\mathbf{x}}},{{\mathbf{y}}})\rvert = 1$ is immediate from Remark \[remark\_gl2z\] and the assumption for ${{\mathbf{x}}}= (1,0)$ and ${{\mathbf{y}}}= (0,1)$. We now make the following inductive assumption: $$\label{assumption1}
\textrm{For all } {{\mathbf{x}}}',{{\mathbf{y}}}' \in \Z^2\textrm{ with } \lvert d({{\mathbf{x}}}',{{\mathbf{y}}}')\rvert < \lvert d({{\mathbf{x}}},{{\mathbf{y}}}) \rvert, \textrm{ we have that } ({{\mathbf{x}}}',{{\mathbf{y}}}') \textrm{ is good.}$$ We would like to show that $[D_{{{\mathbf{x}}}},D_{{{\mathbf{y}}}}] = \{d({{\mathbf{x}}},{{\mathbf{y}}})\} \big( D_{{{\mathbf{x}}}+ {{\mathbf{y}}}} - D_{{{\mathbf{x}}}-{{\mathbf{y}}}} \big)$. By Remark \[remark\_gl2z\], we may assume $${{\mathbf{y}}}= \begin{pmatrix}0\\r\end{pmatrix},\quad {{\mathbf{x}}}= \begin{pmatrix}p\\q\end{pmatrix},\quad d({{\mathbf{x}}}) \leq d({{\mathbf{y}}}),\quad 0 \leq q < p$$
If $p=1$, then this equation follows from , so we may also assume $p > 1$. Furthermore, we may assume that $r>0$ by Remark \[remark\_goodsymmetry\].
We will now show that if either $d({{\mathbf{x}}})=1$ or $d({{\mathbf{y}}})=1$, then $({{\mathbf{x}}},{{\mathbf{y}}})$ is good as follows. By symmetry of the above construction of ${{\mathbf{x}}}$ and ${{\mathbf{y}}}$, we may assume $d({{\mathbf{x}}})=1$, which immediately implies $q>0$. Furthermore, we may now assume that $r>1$ by the relation . We apply Lemma \[lemma\_diophantine\] to $p, q$ to obtain $u,v,w,z \in \Z$ satisfying $$\label{assumption1.49}
uz - vw = 1,\quad uq - vp = 1,\quad u + w = p,\quad v + z = q,\quad 0 < u,w < p$$ We then define vectors ${{\mathbf{a}}}$ and ${{\mathbf{b}}}$ as follows: $$\label{assumption1.9}
{{\mathbf{a}}}:= \begin{pmatrix} u\\ v\end{pmatrix},\quad
{{\mathbf{b}}}:= \begin{pmatrix} w\\ z\end{pmatrix},
\quad {{\mathbf{a}}}+ {{\mathbf{b}}}= {{\mathbf{x}}},\quad d({{\mathbf{a}}}, {{\mathbf{b}}}) = 1 $$
Using Lemma \[lemma\_trueforab\] and Assumption (\[assumption1\]), it is sufficient to show that each of $\lvert d({{\mathbf{a}}}, {{\mathbf{b}}}) \rvert$, $\lvert d({{\mathbf{y}}},{{\mathbf{b}}}) \rvert$, $ \lvert d({{\mathbf{y}}},{{\mathbf{a}}}) \rvert$, $\lvert d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}}) \rvert$, $\lvert d({{\mathbf{y}}}+{{\mathbf{b}}},{{\mathbf{a}}}) \rvert$, and $\lvert d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}}) \rvert$ are strictly less than $pr = \lvert d({{\mathbf{x}}},{{\mathbf{y}}}) \rvert$. First, $ \lvert d({{\mathbf{a}}},{{\mathbf{b}}}) \rvert = 1$ is strictly less than $pr$ since $p>1$ and $r>0$. Second, $ \lvert d({{\mathbf{y}}},{{\mathbf{a}}}) \rvert = ur $ and $ \lvert d({{\mathbf{y}}},{{\mathbf{b}}}) \rvert = wr $ are strictly less than $pr$ by the inequalities in (\[assumption1.49\]). Third, we compute $$\begin{aligned}
\vert d({{\mathbf{y}}}+{{\mathbf{a}}}, {{\mathbf{b}}}) \rvert &=&\vert - d({{\mathbf{y}}}+{{\mathbf{a}}}, {{\mathbf{b}}}) \rvert \\
&=& \lvert - d({{\mathbf{y}}},{{\mathbf{b}}}) - d({{\mathbf{a}}}, {{\mathbf{b}}}) \rvert \\
&=& \lvert wr - 1 \rvert \\
&=& wr-1 \\
&<& wr \\
&<& pr
\end{aligned}$$
Fourth, we compute $$\begin{aligned}
\lvert -d({{\mathbf{y}}}+{{\mathbf{b}}}, {{\mathbf{a}}}) \rvert &=& \lvert -d({{\mathbf{y}}}+{{\mathbf{b}}}, {{\mathbf{a}}}) \rvert \\
&=& \lvert -d({{\mathbf{y}}},{{\mathbf{a}}}) - d({{\mathbf{b}}}, {{\mathbf{a}}}) \rvert \\
&=& \lvert ur + 1 \rvert \\
&=& ur+1 \\
&\leq& (p-1)r+1 \\
&=& pr-r+1 \\
&<& pr
\end{aligned}$$
Finally, we compute $$\begin{aligned}
\lvert d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}}) \rvert &=& \lvert d({{\mathbf{a}}},{{\mathbf{y}}}) - d({{\mathbf{b}}},{{\mathbf{y}}}) \rvert \\
&=& \lvert d({{\mathbf{y}}},{{\mathbf{a}}}) - d({{\mathbf{y}}},{{\mathbf{b}}}) \rvert \\
&=& \lvert ur - wr \rvert \\
&=& \lvert u-w \rvert r \\
&<& \lvert u+w \rvert r \\
&=& pr
\end{aligned}$$
So we have shown that $({{\mathbf{x}}},{{\mathbf{y}}})$ is good if $d({{\mathbf{x}}})=1$ or $d({{\mathbf{y}}})=1$. Let us now turn our attention to the more general case. We will immediately split this into cases depending on $q$.
*Case 1:* Assume $0 < q$.
Let $p' = p / d({{\mathbf{x}}})$ and $q' = q / d({{\mathbf{x}}})$. By the assumption $0 < q$, we see that $d({{\mathbf{x}}}) < p$, so $p' > 1$. We can therefore apply Lemma \[lemma\_diophantine\] to $p',q'$ to obtain $u,v,w,z \in \Z$ satisfying $$\label{assumption1.5}
uz - vw = 1,\quad uq' - vp' = 1,\quad u + w = p',\quad v + z = q',\quad 0 < u,w < p'$$ In a way similar to the above, we may pick vectors ${{\mathbf{a}}}$ and ${{\mathbf{b}}}$ as follows: $$\label{assumption2}
{{\mathbf{a}}}:= \begin{pmatrix} d({{\mathbf{x}}})u\\ d({{\mathbf{x}}})v\end{pmatrix},\quad
{{\mathbf{b}}}:= \begin{pmatrix} d({{\mathbf{x}}})w\\ d({{\mathbf{x}}})z\end{pmatrix},
\quad {{\mathbf{a}}}+ {{\mathbf{b}}}= {{\mathbf{x}}},\quad d({{\mathbf{a}}}, {{\mathbf{b}}}) = d({{\mathbf{x}}})^2 $$
As before, it is sufficient to show that each of $\lvert d({{\mathbf{a}}}, {{\mathbf{b}}}) \rvert$, $\lvert d({{\mathbf{y}}},{{\mathbf{b}}}) \rvert$, $\lvert d({{\mathbf{y}}},{{\mathbf{a}}}) \rvert$, $\lvert d({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}}) \rvert$, $\lvert d({{\mathbf{y}}}+{{\mathbf{b}}},{{\mathbf{a}}}) \rvert$, and $\lvert d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}}) \rvert$ are strictly less than $pr = \lvert d({{\mathbf{x}}},{{\mathbf{y}}}) \rvert$. First, $$\lvert d({{\mathbf{a}}},{{\mathbf{b}}}) \rvert = d({{\mathbf{x}}})^2 \leq d({{\mathbf{x}}})d({{\mathbf{y}}}) = d({{\mathbf{x}}})r < pr$$ where the last inequality follows from the assumption $0 < q < p$. Second, we can compute $\lvert d({{\mathbf{y}}},{{\mathbf{b}}}) \rvert = d({{\mathbf{x}}})r w $ and $ \lvert d({{\mathbf{y}}},{{\mathbf{a}}}) \rvert = d({{\mathbf{x}}})r u $ are strictly less than $pr$ by the inequalities in (\[assumption1.5\]). Third, we compute $$\begin{aligned}
\lvert d({{\mathbf{y}}}+{{\mathbf{a}}}, {{\mathbf{b}}}) \rvert &=& \lvert -d({{\mathbf{y}}}+{{\mathbf{a}}}, {{\mathbf{b}}}) \rvert \\
&=& \lvert -d({{\mathbf{y}}},{{\mathbf{b}}}) - d({{\mathbf{a}}}, {{\mathbf{b}}}) \rvert\\
&=& \lvert d({{\mathbf{x}}})wr - d({{\mathbf{x}}})^2 \rvert \\
&=& d({{\mathbf{x}}})wr - d({{\mathbf{x}}})^2 \\ &<& d({{\mathbf{x}}})wr\\
&\leq& pr
\end{aligned}$$ Finally, we compute $$\begin{aligned}
\lvert d({{\mathbf{y}}}+{{\mathbf{b}}}, {{\mathbf{a}}}) \rvert &=& \lvert -d({{\mathbf{y}}}+{{\mathbf{b}}}, {{\mathbf{a}}}) \rvert \\
&=& \lvert -d({{\mathbf{y}}},{{\mathbf{a}}}) - d({{\mathbf{b}}}, {{\mathbf{a}}}) \rvert \\
&=& d({{\mathbf{x}}})ur + d({{\mathbf{x}}})^2 \\
&\leq& \left(d({{\mathbf{x}}})u + d({{\mathbf{x}}})\right)d({{\mathbf{y}}})\\
&=& (u+1)d({{\mathbf{x}}})r
\end{aligned}$$
Therefore, we will be finished once we show that $(u+1)d({{\mathbf{x}}})$ is strictly less than $p$. We now split into subcases:\
*Subcase 1a:* If $u + 1 < p'$, then $(u+1)d({{\mathbf{x}}})r < p'd({{\mathbf{x}}})r = pr$, and we are done.
*Subcase 1b:* Assume $u + 1 = p'$. By equation (\[assumption1.5\]), we have $$1 = uq' - vp' = (p'-1)q' - vp' \implies p'(q'-v) = 1 + q' < 1 + p'$$ Since $p' > 1$, the last inequality implies $q' - v = 1$, which implies $v = q'-1$ and $z = 1$. Since $uz-vw = 1$, this implies $(p'-1)-(q'-1) = 1$, which implies $q' = p'-1$. If we write $g = d({{\mathbf{x}}})$, we then have $$\lvert d({{\mathbf{y}}}+ {{\mathbf{b}}}, {{\mathbf{a}}}) \rvert = \lvert -d({{\mathbf{y}}}+ {{\mathbf{b}}}, {{\mathbf{a}}}) \rvert = \Bigg| -\det\left[ \begin{array}{cc}g&p -g\\g + r&p-2g\end{array}\right] \Bigg| = \lvert rp + g(g-r) \rvert = rp + d({{\mathbf{x}}})(d({{\mathbf{x}}})-r) \leq rp$$ Where the last inequality comes from the assumption $d({{\mathbf{x}}}) \leq r$. If this inequality is strict, then we are done. Otherwise, we move onto the next Subcase.
*Subcase 1c:* In this subcase, we are reduced to showing the following vectors are good: $${{\mathbf{y}}}= (0,r),\quad \quad {{\mathbf{x}}}= (rp', rp'-r)$$ If $r=1$, then $d({{\mathbf{y}}})=1$, which makes ${{\mathbf{y}}}$ good. Thus, we may assume that $r>1$.
We must replace our previous choice of ${{\mathbf{a}}}$ and ${{\mathbf{b}}}$ with a choice which is better adapted to this particular subcase. We define $${{\mathbf{a}}}:= \begin{pmatrix} 1\\-1 \end{pmatrix},\quad {{\mathbf{b}}}:= \begin{pmatrix} rp'-1\\rp' - r + 1 \end{pmatrix}$$
We know that the pairs $({{\mathbf{a}}}, {{\mathbf{b}}}), ({{\mathbf{y}}}, {{\mathbf{a}}}), ({{\mathbf{y}}}+{{\mathbf{b}}},{{\mathbf{a}}})$ are good since $d({{\mathbf{a}}})=1$. Since $r > 1$ and $p'>1$, we can compute that $({{\mathbf{y}}},{{\mathbf{b}}}), ({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}}), ({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}})$ are strictly less than $r^2p' = \lvert d({{\mathbf{x}}},{{\mathbf{y}}}) \rvert$ as follows: $$\begin{aligned}
\lvert d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}})\rvert &=& \lvert r^2p' - 2r \rvert \\
&=& r^2p' - 2r \\
&<& r^2p'
\end{aligned}$$
$$\begin{aligned}
\lvert d({{\mathbf{y}}},{{\mathbf{b}}})\rvert &=& \lvert r^2p' - r \rvert \\
&=& r^2p' - r \\
&<& r^2p'
\end{aligned}$$
$$\begin{aligned}
\lvert d({{\mathbf{y}}}+{{\mathbf{a}}}, {{\mathbf{b}}})\rvert &=& \lvert r^2p' - (rp'+r-1) \rvert \\
&=& r^2p' - (rp'+r-1) \\
&<& r^2p'
\end{aligned}$$
This together with Assumption (\[assumption1\]) and Lemma \[lemma\_trueforab\] shows that $({{\mathbf{x}}},{{\mathbf{y}}})$ finishes the proof of this subcase, which finishes the proof of Case 1.\
*Case 2:* In this case we assume $q=0$. We define ${{\mathbf{a}}}, {{\mathbf{b}}}$ similarly to Subcase 1c, so we have $${{\mathbf{y}}}= \begin{pmatrix} 0\\r\end{pmatrix},\quad {{\mathbf{x}}}= \begin{pmatrix} p\\0\end{pmatrix},\quad {{\mathbf{a}}}:= \begin{pmatrix} 1\\-1 \end{pmatrix},\quad {{\mathbf{b}}}:= \begin{pmatrix} p-1\\ 1 \end{pmatrix}$$
Since $d({{\mathbf{a}}}) = d({{\mathbf{b}}}) = 1$, the pairs $({{\mathbf{a}}},{{\mathbf{b}}}), ({{\mathbf{y}}},{{\mathbf{a}}}), ({{\mathbf{y}}},{{\mathbf{b}}}), ({{\mathbf{y}}}+{{\mathbf{a}}},{{\mathbf{b}}}), ({{\mathbf{y}}}+{{\mathbf{b}}},{{\mathbf{a}}})$ are all good. We must check that $\lvert d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}}) \rvert < pr = \lvert d({{\mathbf{x}}},{{\mathbf{y}}}) \rvert$. If $r=1$, then the relation implies that the pair $({{\mathbf{x}}},{{\mathbf{y}}})$ is good. Thus, we may asume that $r>1$. We may also assume that $p>1$. Finally, we check $\lvert d({{\mathbf{a}}}-{{\mathbf{b}}},{{\mathbf{y}}}) \rvert = \lvert rp-2r \rvert = rp-2r < rp$. By using Assumption (\[assumption1\]) and Lemma \[lemma\_trueforab\], this completes Case 2 which completes the proof.
Using the technical results above, we give a presentation of the algebra $\sd$.
\[thm:presentation\] The algebra $\sd$ is generated by the elements $D_{{\mathbf{x}}}$, and these satisfy the following relations: $$\begin{aligned}
\label{eq:allrelations}
[D_{{\mathbf{x}}},D_{{\mathbf{y}}}] &= (s^d-s^{-d})(D_{{{\mathbf{x}}}+{{\mathbf{y}}}}-D_{{{\mathbf{x}}}-{{\mathbf{y}}}})\\
D_{{\mathbf{x}}}&= D_{-{{\mathbf{x}}}} \notag\end{aligned}$$ where $d = \det({{\mathbf{x}}}\, {{\mathbf{y}}})$. This gives a presentation of $\sd$ as an algebra.
In the simple case ${{\mathbf{x}}}=(1,0)$ and ${{\mathbf{y}}}= (0,1)$, the claimed relation is exactly the Kauffman skein relation. Also, note that if we take the first equation and replace ${{\mathbf{x}}}$ with $-{{\mathbf{x}}}$, the left hand side is invariant because of the second relation, and the right hand side is invariant because both factors switch sign.
First we will show that $\sd$ is generated by the $D_{{{\mathbf{x}}}}$. Using the skein relation, we may write an arbitrary element of $\sd$ as a sum of products of knots. Using further skein relations, any knot can be written as a sum of products of annular knots (i.e. knots which are contained in an annulus inside the torus). Therefore, it is sufficient to prove that the $D_k$ generate the skein algebra of the annulus. By [@MS17 Lemma 3.1], the $P_k$ generate the Homflypt skein algebra of the annulus. Combining this with the inclusion $s_n: H_n \to BMW_n$ from [@BB01] shows that $D_k$ generate the Kauffman skein algebra of the annulus (since all idempotents in $BMW_n$ come from idempotents in $H_{n-2k}$ for some $k$). It is clear that $D_{{{\mathbf{x}}}}=D_{-{{\mathbf{x}}}}$ due to the lack of orientation on the links. The other relations have already been shown to hold in Proposition \[lemma\_allfromsome\].
To show that these relations give a presentation of $\sd$, we first note that Theorem \[thm:basis\] shows that as a vector space, $\sd$ has a basis given by unordered words in the elements $D_{{\mathbf{x}}}$. The relations allow any $D_{{\mathbf{x}}}$ and $D_{{\mathbf{y}}}$ to be reordered, which shows that they give a presentation of $\sd$.
Perpendicular relations {#sec:perpendicular}
=======================
In this section we prove the perpendicular relations $$\label{eq:perp}
[D_{1,0}, D_{0,n}] = \{n\}(D_{1,n} - D_{1,-n})$$
The main tool we use will be the recursion relation between the symmetrizers in the BMW algebra from Section \[sec:recursion\]. We will actually show a stronger identity, which is the analogue of an identity involving the Hecke algebra elements $P_k$ found by Morton and coauthors.
Let $\sa$ be the skein algebra of the annulus with one marked point on each boundary component. For $x \in \cc$, the element $l(x) \in \sa$ is defined to be a horizontal arc joining the boundary points with the element $x$ placed below the arc. Similarly, we may define $r(x)$, but the element $x$ is placed below the arc. $$l(x) = \vcenter{\hbox{\includegraphics[width=5cm]{lx.eps}}} \qquad\qquad\qquad r(x) = \vcenter{\hbox{\includegraphics[width=5cm]{rx.eps}}}$$
Let $a$ be the arc that goes once around the annulus in the counter-clockwise direction: $$a=\vcenter{\hbox{\includegraphics[width=5cm]{ainA.eps}}}$$ The vector space $sa$ is an algebra, where the multiplication is given by “nesting” annuli. The identity in this algebra is $l(\varnothing)$, where $\varnothing \in \cc$ is the empty link. In fact $\sa$ is commutative, and is isomorphic to $\cc \otimes_R R[a^{\pm 1}]$ (see [@She16]), but we won’t need this fact. We will, however, use the fact that $l(-)$ and $r(-)$ are algebra maps from $\cc$ to $\sa$.
\[thm:aid\] We have the following identity in $\sa$: $$\label{eq:aid}
l(D_n) - r(D_n) = \{n\}(a^n - a^{-n})$$
Note that this implies . In particular, by Remark \[rmk:simplecurve\], after applying the wiring $cl_1:\sa\to\sd$ which closes the marked points around the torus as shown: $$\vcenter{\hbox{\includegraphics[width=5cm]{cl_1AtoD.eps}}}$$ equation becomes $$D_{1,0}D_{0,n} - D_{0,n}D_{1,0} = \{n\}(D_{1,n} - D_{1,-n})$$
The rest of the section is dedicated to the proof of Theorem \[thm:aid\], and we begin with some lemmas. In an effort to reduce notation, define the following elements of $\sa$: $$\begin{aligned}
c&:=l(D_n) - r(D_n) - \{n\}(a^n - a^{-n}) \\
l_n &:= l \left( \widehat{f}_n \right)\\
r_n &:= r \left( \widehat{f}_n \right)\end{aligned}$$ As a convention, let $l_0=r_0=l_{-1}=r_{-1}=1_{\sa}$.
\[lemma:perprel\] The relation $c=0$ in $\sa$ is equivalent to the following: $$\begin{aligned}
l_{n+2} - r_{n+2} &= (s^{-1}a + sa^{-1}) l_{n+1} - (sa + s^{-1}a^{-1}) r_{n+1} - (l_{n} - r_{n}); \quad n \geq -1 \label{eq:perprel3}\end{aligned}$$
We may rephrase the relation $c=0$ as one of power series in $\sa[[t]]$. $$\sum_{n=1}^{\infty} \frac{l(D_n) - r(D_ n)}{n}t^n = \sum_{n=1}^{\infty} \frac{\{n\}(a^n-a^{-n})}{n}t^n$$ Using linearity of $l$ and $r$ on the left hand side, while expanding out the right hand side we get the equation: $$\begin{aligned}
l\left( \sum_{n=1}^{\infty} \frac{D_n}{n}t^n \right) - r\left( \sum_{n=1}^{\infty} \frac{D_n}{n}t^n \right) &= \sum_{n=1}^{\infty} \frac{s^{n}a^{n}t^n}{n} + \sum_{n=1}^{\infty} \frac{s^{-n}a^{-n}t^n}{n} - \sum_{n=1}^{\infty} \frac{s^{n}a^{-n}t^n}{n} - \sum_{n=1}^{\infty} \frac{s^{-n}a^{n}t^n}{n}\end{aligned}$$
Define the power series $$F(t) := 1+\sum_{n=1}^{\infty} \widehat{f}_n t^n \in \cc[[t]]$$ Then on the left hand side we may use equation and on the right side we may apply Newton’s power series identity to obtain: $$l\left( \mathrm{ln}\left( F(t) \right) \right) - r\left( \mathrm{ln}\left( F(t) \right) \right) = -\mathrm{ln}(1-sat) - \mathrm{ln}(1-s^{-1}a^{-1}t) + \mathrm{ln}(1-sa^{-1}t) + \mathrm{ln}(1-s^{-1}at)$$ Moving terms around and using properties of natural log, we arrive at the equation: $$\mathrm{ln}( l(F(t)) (1-sat-s^{-1}a^{-1}t +t^2)) = \mathrm{ln}( r( F(t)) (1-sa^{-1}t-s^{-1}at+t^2))$$ Exponentiating both sides, we get: $$l(F(t)) (1-sat-s^{-1}a^{-1}t +t^2) = r( F(t)) (1-sa^{-1}t-s^{-1}at+t^2)$$ Equate the coefficients of to obtain a system of equations in $\sa$: $$\begin{aligned}
l_1 - (sa + s^{-1}a^{-1}) &= r_1 - (sa^{-1} + s^{-1}a)\\
l_2 - (sa + s^{-1}a^{-1})l_1 +1 &= r_2 - (sa^{-1}+s^{-1}a)r_1 +1\\
l_{n} - sal_{n-1} - s^{-1}a^{-1}l_{n-1} + l_{n-2} &= r_{n} - sa^{-1}r_{n-1} - s^{-1}ar_{n-1} + r_{n-2}; \quad n \geq 2\end{aligned}$$ These terms can be rearranged to those in the statement, thus completing the proof.
Before showing the relation , we will first set up some machinery in $\sa$. First, consider the following wirings $W_n, \widetilde{W}_n:BMW_{n+1}\to\sa$ defined by: $$W_n = \vcenter{\hbox{\includegraphics[width=5cm]{W_n.eps}}} \qquad\qquad\qquad \widetilde{W}_n = \vcenter{\hbox{\includegraphics[width=5cm]{tildeW_n.eps}}}$$
At the risk of abusing notation, we will set $W_n := W_n \left( \widehat{f}_{n+1}\right)$ and $\widetilde{W}_n := \widetilde{W}_n \left( \widehat{f}_{n+1}\right)$. We can see diagramatically the properties: $$\begin{aligned}
W_n \left( x \otimes 1 \right) = W_n \left( 1 \otimes x \right) = a W_n \left( x \right) \\
\widetilde{W}_n \left( x \otimes 1 \right) = \widetilde{W}_n \left( x \otimes 1 \right) = a^{-1} \widetilde{W}_n \left( x \right)\end{aligned}$$
We will also need the constants\
\
$$\beta_n:=\frac{1-s^2}{s^{2n-1}v^{-1}-1}$$
\[rmk:involutions\] There exist maps $$\tau: BMW_n \to BMW_n \qquad \overline{\,\cdot\,}: BMW_n \to BMW_n$$ induced by the diffeomorphisms of the thickened square $(x, y, t) \mapsto (x, 1-y, 1-t)$ and $(x, y, t) \mapsto (x, y, 1-t)$, respectively. The map $\overline{\,\cdot\,}$ is often called the *mirror map* and is an $R$-anti-linear involution, while $\tau$ will be called the *flip map* and is an $R$-linear anti-involution which extends the dihedral symmetry of the square to the thickened square. As noted in [@She16], the symmetrizers $f_n$ are fixed under these maps. Using the quotient map defined by the equivalence relation $(x, 0, t) \sim (x, 1, t)$, we may analagously define maps $$\tau: \sa \to \sa, \qquad \overline{\,\cdot\,}: \sa \to \sa$$ which are linear and anti-linear involutions, respectively. We will also call these the flip map and the mirror map; it will be clear from the context which is being applied. These maps satisfy: $$\begin{aligned}
\tau \left( W_n \left( x \right) \right) &=& \widetilde{W}_n \left( \tau \left( x \right) \right) \label{tauW} \\
\overline{W_n \left( x \right)} &=& W_n \left( \overline{x} \right)\end{aligned}$$
\[lemma:recursionina\] The following relations hold in $\sa$. $$\begin{aligned}
l_n &= [n+1]W_n - [n]s^{-1}aW_{n-1} - [n]s^{-1}\beta_na^{-1}\widetilde{W}_{n-1} \label{eq:recursionina1} \\
l_n &= [n+1]\widetilde{W}_n - [n]sa^{-1}\widetilde{W}_{n-1} - [n]s\bar{\beta}_naW_{n-1} \label{eq:recursionina2} \\
r_n &= [n+1]W_n - [n]saW_{n-1} - [n]s\bar{\beta}_na^{-1}\widetilde{W}_{n-1} \label{eq:recursionina3} \\
r_n &= [n+1]\widetilde{W}_n - [n]s^{-1}a^{-1}\widetilde{W}_{n-1} - [n]s^{-1}\beta_naW_{n-1} \label{eq:recursionina4}
\end{aligned}$$
This was more or less observed in \[She16\] using different notation. An outline of the proof is as follows. First, notice that the equations and are the result of applying the mirror map to equations and respectively. By , one can quickly verify that equation is the image of equation under the flip map. Thus, it suffices to prove equation . To do this, we will use Proposition 3 from [@She16] which provides a recurrence relation of the symmetrizers of $BMW_{n+1}$. Taking the image of this relation under the maps $\widetilde{W_n}$ produces relation as follows.
From Proposition \[prop:recursion\], the recurrence relation for the symmetrizers in $BMW_{n+1}$ is $$[n+1]f_{n+1} = [n]s^{-1}\left( f_n \otimes 1 \right) \left( 1 \otimes f_n \right) + \sigma_n \cdots \sigma_1 \left( 1 \otimes f_n \right) + [n]s^{-1}\beta_n\left( f_n \otimes 1 \right) h_n \cdots h_1 \left( 1 \otimes f_n \right)$$ where $\sigma_i$ is the standard simple braiding of the $i$ and $i+1$ strands and $h_i$ is the horizontal edge between the $i$ and $i+1$ strands. One may verify diagramatically that applying $\widetilde{W_n}$ to the above gives us relation : $$\begin{aligned}
[n+1]\widetilde{W}_{n} =& [n]s^{-1}\widetilde{W}_n\left( \left( f_n \otimes 1 \right) \left( 1 \otimes f_n \right) \right) + \widetilde{W}_n\left( \sigma_n \cdots \sigma_1 \left( 1 \otimes f_n \right) \right) + \\
&[n]s^{-1}\beta_n \widetilde{W}_n \left( \left( f_n \otimes 1 \right) h_n \cdots h_1 \left( 1 \otimes f_n \right) \right) \\
=&[n]s^{-1}\widetilde{W}_n\left( f_n \otimes 1 \right) + r_n + [n]s^{-1}\beta_n W_n \left( f_n \otimes 1 \right) \\
=&[n]s^{-1}a\widetilde{W}_{n-1} + r_n + [n]s^{-1}\beta_n a^{-1} W_{n-1}\end{aligned}$$ This completes the proof.
We will also need the following identities for the $\beta_n$.
\[lemma:ring\] The following relations hold in the $\C(s^{\pm1}, v^{\pm1})$: $$\begin{aligned}
s - s^{-1}\beta_n &= s^{-1} - s\bar{\beta}_n\\
\left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( s-s^{-1}\beta_n \right) &= - \left( s - s^{-1} \right)\end{aligned}$$
\[lemma:annfund\] For all $n\geq 1$, $$l_n-r_n = \{n\}\left( aW_{n-1} - a^{-1}\widetilde{W}_{n-1} \right)$$
This is proven in Proposition 7 of \[She16\]. We can prove it here by taking the difference of equations and and using that $s^{-1}\beta_n - s\bar{\beta}_n = s-s^{-1}$, as provided by Lemma \[lemma:ring\]. $$\begin{aligned}
l_n - r_n =& [n]\left( s-s^{-1} \right) aW_{n-1} - \left( s^{-1}\beta_n - s\bar{\beta}_n \right) a^{-1}\widetilde{W}_{n-1} \\
=& [n]\left( s-s^{-1} \right) \left( aW_{n-1} - a^{-1}\widetilde{W}_{n-1} \right) \\
=& \{n\}\left( aW_{n-1} - a^{-1}\widetilde{W}_{n-1} \right)\end{aligned}$$
The relations of Lemma \[lemma:perprel\] hold.
In the case $n=-1$, the relation we want to show becomes $$l_1-r_1 = \{1\} \left( a - a^{-1} \right)$$ This can easily be verified by applying the skein relation once to the left-hand side of the equation. By Lemma \[lemma:annfund\], the relation is equivalent to: $$\label{eq_perprel4}
\{n+2\} \left( aW_{n+1} - a^{-1}\widetilde{W}_{n+1} \right) = \left( sa+s^{-1}a^{-1} \right) l_{n+1} - \left( sa^{-1}+s^{-1}a \right) r_{n+1} - \{n\}\left( aW_{n-1}-a^{-1}\widetilde{W}_{n-1} \right)$$
Now we will prove (\[eq:perprel3\]). By Lemma \[lemma:annfund\], the relation (\[eq:perprel3\]) is equivalent to: $$\label{eq_perprel4}
\{n+2\} \left( aW_{n+1} - a^{-1}\widetilde{W}_{n+1} \right) = \left( sa+s^{-1}a^{-1} \right) l_{n+1} - \left( sa^{-1}+s^{-1}a \right) r_{n+1} - \{n\}\left( aW_{n-1}-a^{-1}\widetilde{W}_{n-1} \right)$$
We will show that the left hand side of the equation above may be reduced to the right hand side by a series of applications of Lemma \[lemma:recursionina\]. $$\begin{aligned}
& \{n+2\} \left( aW_{n+1} - a^{-1}\widetilde{W}_{n+1} \right) \\
\overset{\mathrm{(1) and (4)}}{=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=}& \{n+2\} \left( \frac{a}{[n+2]} \left( l_{n+1} + [n+1]s^{-1}aW_n + [n+1]s^{-1}\beta_{n+1}a^{-1}\widetilde{W}_n \right) \right.- \\
&\left.\qquad\qquad\frac{a^{-1}}{[n+2]} \left( r_{n+1} + [n+1]s^{-1}a^{-1}\widetilde{W}_n + [n+1]s^{-1}\beta_{n+1}aW_n \right) \right) \\
=& \left( s - s^{-1} \right) \left( \left( al_{n+1} + [n+1]s^{-1}a^2W_n + [n+1]s^{-1}\beta_{n+1}\widetilde{W}_n \right)- \right. \\
&\qquad\qquad\,\,\,\,\left.\left( a^{-1}r_{n+1} - [n+1]s^{-1}a^{-2}\widetilde{W}_n - [n+1]s^{-1}\beta_{n+1}W_n \right)\right) \\
\overset{\mathrm{(3) and (2)}}{=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=\mathrel{\mkern-3mu}=}&\left( s-s^{-1} \right) \left( \left( al_{n+1} + \left( [n+2]s^{-2}aW_{n+1} - s^{-2}ar_{n+1} - [n+1]s^{-1}\bar{\beta}_{n+1}\widetilde{W}_n \right) + [n+1]s^{-1}\beta_{n+1}\widetilde{W}_n \right)- \right. \\
&\qquad\qquad\,\,\,\, \left. \left( a^{-1}r_{n+1} + \left( [n+2]s^{-2}a^{-1}\widetilde{W}_{n+1} - s^{-2}a^{-1}l_{n+1} - [n+1]s^{-1}\bar{\beta}_{n+1}W_n \right) + [n+1]s^{-1}\beta_{n+1}W_n \right) \right) \\
=&\left( \left( sa + s^{-1}a^{-1} \right) l_{n+1} - \left( sa^{-1} + s^{-1}a \right) r_{n+1} \right) + \left( s^{-1}a^{-1} + s^{-3}a \right) r_{n+1} - \left( s^{-1}a + s^{-3}a^{-1} \right) l_{n+1} + \\
&\qquad\qquad\,\,\,\, \{n+2\}s^{-2}\left( aW_{n+1} - a^{-1} \widetilde{W}_{n+1} \right) + \{n+1\}s^{-1}\left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( W_n - \widetilde{W}_n \right)\end{aligned}$$
We break the computation here to note that the first two terms in the last line also appear on the right hand side of (\[eq\_perprel4\]). Thus, we would like to prove the following equality: $$\begin{aligned}
\begin{split}
& \left( s^{-1}a^{-1} + s^{-3}a \right) r_{n+1} - \left( s^{-1}a + s^{-3}a^{-1} \right) l_{n+1} + \{n+2\}s^{-2}\left( aW_{n+1} - a^{-1} \widetilde{W}_{n+1} \right) - \\
&\qquad\qquad \{n+1\}s^{-1}\left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( W_n - \widetilde{W}_n \right) = -\{n\} \left( aW_{n-1} - a^{-1}\widetilde{W}_{n-1} \right)
\end{split}\end{aligned}$$
We will show this by continuing to expand the right hand side using the identities from Lemma \[lemma:recursionina\]. $$\begin{aligned}
&\left( s^{-1}a + s^{-3}a^{-1} \right) l_{n+1} - \left( s^{-1}a^{-1} + s^{-3}a \right) r_{n+1} - \{n+2\}s^{-2}\left( aW_{n+1} - a^{-1} \widetilde{W}_{n+1} \right) - \\
&\qquad\qquad \{n+1\}s^{-1}\left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( W_n - \widetilde{W}_n \right) \\
=& \left( s^{-1}a^{-1} + s^{-3}a \right) r_{n+1} - \left( s^{-1}a+s^{-3}a^{-1} \right) l_{n+1} + [n+2]\left( s^{-1}-s^{-3} \right) \left(aW_{n+1} - a^{-1}\widetilde{W}_{n+1} \right) + \\
&\qquad\qquad [n+1]\left( 1-s^{-2} \right) \left( \bar{\beta}_{n+1}-\beta_{n+1} \right) \left(W_n - \widetilde{W}_n \right) \\
=& s^{-1}a\left( [n+2]W_{n+1} - l_{n+1} \right) + s^{-3}a^{-1}\left( [n+2]\widetilde{W}_{n+1} - l_{n+1} \right) - s^{-3}a\left( [n+2]W_{n+1} - r_{n+1} \right) - \\
&\qquad\qquad s^{-1}a^{-1}\left( [n+2]\widetilde{W}_{n+1} - r_{n+1} \right) + [n+1]\left( 1-s^{-2} \right) \left( \bar{\beta}_{n+1} -\beta_{n+1} \right) \left( W_n - \widetilde{W}_n \right) \\
=& s^{-1}a\left( [n+1]s^{-1}aW_n + [n+1]s^{-1}\beta_{n+1}a^{-1}\widetilde{W}_n \right) + s^{-3}a^{-1}\left( [n+1]sa^{-1}\widetilde{W}_n + [n+1]s\bar{\beta}_{n+1}aW_n \right) - \\
&s^{-3}a\left( [n+1]saW_n + [n+1]s\bar{\beta}_{n+1}a^{-1}\widetilde{W}_n \right) - s^{-1}a^{-1}\left( [n+1]s^{-1}a^{-1}\widetilde{W}_n + [n+1]s^{-1}\beta_{n+1}aW_n \right) + \\
& [n+1]\left(1-s^{-2} \right) \left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( W_n - \widetilde{W}_n \right)\\
=& [n+1]\left( s^{-2}a^2W_n + s^{-2}\beta_{n+1}\widetilde{W}_n + s^{-2}a^{-2}\widetilde{W}_n + s^{-2}\bar{\beta}_{n+1}W_n - s^{-2}a^{2}W_n - s^{-2}\bar{\beta}_{n+1}\widetilde{W}_n - \right. \\
&\qquad\quad \,\, \left. s^{-2}a^{-2}\widetilde{W}_n - s^{-2}\beta_{n+1}W_n + \bar{\beta}_{n+1}W_n - \bar{\beta}_{n+1}\widetilde{W}_n - \beta_{n+1}W_n + \beta_{n+1}\widetilde{W}_n - \right. \\
&\qquad\quad\,\, \left. s^{-2}\bar{\beta}_{n+1}W_n + s^{-2}\bar{\beta}_{n+1}\widetilde{W}_n + s^{-2}\beta_{n+1}W_n - s^{-2}\beta_{n+1}\widetilde{W}_n \right) \\
=& [n+1]\left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( W_n - \widetilde{W}_n \right) \\
=& \left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( \left( [n+1]W_n \right) - \left( [n+1]\widetilde{W}_n \right) \right) \\
=& \left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( \left( l_n + [n]s^{-1}aW_{n-1} + [n]s^{-1}\beta_{n}a^{-1}\widetilde{W}_{n-1} \right) - \left( l_n + [n]sa^{-1}\widetilde{W}_{n-1} + [n]s\bar{\beta}_{n}aW_{n-1} \right) \right) \\
=& \left( \bar{\beta}_{n+1} - \beta_{n+1} \right) \left( [n]\left( s^{-1} - s\bar{\beta}_{n} \right) aW_{n-1} - [n]\left( s-s^{-1}\beta_{n} \right) a^{-1}\widetilde{W}_{n-1} \right) \\
=& [n]\left(\bar{\beta}_{n+1} - \beta_{n+1} \right) \left( s - s^{-1}\beta_{n} \right) \left( aW_{n-1} - a^{-1}\widetilde{W}_{n-1} \right) \\
=& - \{n\}\left( aW_{n-1} - a^{-1}\widetilde{W}_{n-1} \right)\end{aligned}$$
Angled relations {#sec:angled}
================
In this section we prove the angled relations $$\label{eq:angleinsec}
[D_{1,0},D_{1,n}] = \{n\}(D_{2,n} - D_{0,n})$$ The argument here is more intricate than in the previous section, so we first give the main steps, and then prove the main two steps in independent subsections.
Let $a = [D_{1,0},D_{1,n}] - \{n\}(D_{2,n} - D_{0,n})$, so that our goal is to show $a=0$. Recall that $\cc$ is the skein of the annulus, and let $\cc_{0,1}$ be the image of $\cc$ when the annulus is embedded in $T^2$ along the $(0,1)$ curve. Also, recall that $\sd$ acts on $\cc$, where in this action $(0,1) \in T^2$ corresponds to the core of the annulus, and $(1,0)$ is the meridian of the annulus. After making these choices, if we compose the algebra maps $\cc \to \cc_{0,1} \subset \sd \to \mathrm{End}(\cc)$, then the (commutative) algebra $\cc$ is acting on itself by multiplication.
The claimed relations then follow from the following statements:
1. First, in Subsection \[sec:acting\] we show that $a \cdot \varnothing = 0$, where $\varnothing $ is the empty link in $\cc$.
2. Then, using skein-theoretic arguments, in Section \[sec:skein\] we show that $a$ is in the subalgebra $\cc_{0,1}$.
3. Finally, the following composition is an isomorphism $$\cc \to \cc_{0,1} \to \cc$$ where the second map is given by $b \mapsto b\cdot \varnothing$. This follows from the fact that there is a deformation retraction comprised of smooth embeddings which takes $S^1 \times D^2$ to the image of a neighborhood of the $(0,1)$ curve under the gluing map $$\left( T^2\times [0,1] \right) \sqcup \left( S^1 \times D^2 \right) \to S^1 \times D^2$$
Combining these three facts shows $a=0$ in the algebra $\sd$.
Acting on the empty link {#sec:acting}
------------------------
Here we partially describe the action of the skein algebra of the torus on the skein algebra of the annulus. The formula we give here is similar in spirit to the formula for torus link invariants in [@RJ93], but it is simpler because of our choice to color the torus link by $D_k$. (For uncolored torus links there are idempotents other than hooks on the right hand side of the formula in [@RJ93].)
\[lemma:projection\] Suppose that $k \in {\mathbb{N}}_{\geq 1}$, that $m,n \in \Z$ are relatively prime, and that $n \geq 1$. Then $$D_{km,kn}\cdot\varnothing = c_k + \sum_{a+b+1=kn} v^{-km} s^{km(a-b)} (-1)^b \widehat{Q}_{(a|b)}$$
The left hand side is the $(km,kn)$ torus link colored by the element $D_k$, and projected into the skein of the annulus. Since the left hand side of Corollary \[cor:projection\] is an explicit expression in the BMW algebra for an element which closes to the $B_k$-colored $(km,kn)$ torus link in the annulus, the formula claimed in this lemma follows from Corollary \[cor:projection\] and equation .
Since the constants $c_k$ from equation only depend on the parity of $k$, and the parity of $d(2,n)$ and $d(0,n)$ is the same, Lemma \[lemma:projection\] implies the following identity: $$\label{eq:projection1}
\{n\}\left( D_{2,n} - D_{0,n} \right) \cdot \varnothing = \{n\} \sum_{a+b+1=n} \left( v^{-2}s^{2(a-b)} - 1 \right) (-1)^b \widehat{Q}_{(a|b)}.$$
By [@LZ02 Proposition 2.1], $\widehat{Q}_{\lambda}$ is an eigenvector of the action by $D_{1,0}$ on $\cc$ with eigenvalue $$c_{\lambda} = \delta + \left( s - s^{-1} \right) \left( v^{-1} \sum_{x \in \lambda} s^{2cn(x)} - v \sum_{x \in \lambda} s^{-2cn(x)} \right)$$ where if $c$ is a cell of index $(i,j)$ in $\lambda$, it has *content* $cn(c)= j - i$. When $\lambda = (a|b)$, one may compute $$c_{(a|b)} = \delta + v^{-1} \left( s^n - s^{-n} \right) s^{a-b} - v \left( s^n - s^{-n} \right) s^{-(a-b)}$$ This together with gives us $$\begin{aligned}
[D_{1,0},D_{1,n}] \cdot \varnothing &=& D_{1,0} \cdot \left( D_{1,n} \cdot \varnothing \right) - D_{1,n} \cdot \left( D_{1,0} \cdot \varnothing \right) \\
&=& D_{1,0} \cdot \left( \sum_{a+b+1=n} v^{-1} s^{a-b} (-1)^b \widehat{Q}_{(a|b)} \right) - D_{1,n} \cdot \left( \delta \varnothing \right) \\
&=& \sum_{a+b+1=n} \left( c_{(a|b)} - \delta \right) v^{-1} s^{a-b} (-1)^b \widehat{Q}_{(a|b)} \\
&=& \sum_{a+b+1=n} \left( v^{-1} \left( s^n - s^{-n} \right) s^{a-b} - v \left( s^n - s^{-n} \right) s^{-(a-b)} \right) v^{-1} s^{a-b} (-1)^b \widehat{Q}_{(a|b)} \\
&=& \{n\} \sum_{a+b+1=n} \left( v^{-2}s^{2(a-b)}- 1 \right) (-1)^b \widehat{Q}_{(a|b)}\\
&=& \{n\} (D_{2,n}\cdot \varnothing -D_{0,n} \cdot \varnothing).\end{aligned}$$
This shows that $a \cdot \varnothing = 0$.
The affine BMW algebra {#sec:skein}
----------------------
In this section, we will show that the element $a \in \sd$ lies in $\cc_{0,1}$. We will do this in the following skein-based calculations, starting with the use of the affine BMW algebra on 2 strands, $\dot{BMW}_2$. The algebra $\dot{BMW}_n$ is isomorphic to the algebra of $n$-tangles in the thickened annulus, modulo the Kauffman skein relations, where the product is stacking digrams [@GH06]. We will think of the diagrams as living in the square with the top and bottom horizontal edged identified, where a product of diagrams $XY$ is the diagram obtained from connecting the right vertical edge of $X$ with the left vertical edge of $Y$.
Making a further identification of the vertical edges allows us to pass to the skein of the torus $T^2$, and induces a linear map $cl_2:\dot{BMW}_2\to\sd$. This closure map $cl_2$ has the standard ‘trace’ property, namely that $cl_2(XY)=cl_2(YX)$ for any $X,Y\in\dot{BMW}_2$. Similar to $cl_1$, this map may also be defined using a careful choice of wiring diagram: $$\vcenter{\hbox{\includegraphics[width=6cm]{cl_2BMW_2toD.eps}}}.$$
From the diagrams $A$ and $\bar A$ as shown: $$A = \vcenter{\hbox{\includegraphics[width=3cm]{AinBMW_2.eps}}} \quad \qquad \qquad \bar{A} = \vcenter{\hbox{\includegraphics[width=3cm]{AbarinBMW_2.eps}}}$$ we can picture $A^n$ and $\bar{A}^n$. The closures of these satisfy $cl_2(A^n)=D_{1,0}D_{1,n}$ and $cl_2(\bar{A}^n)=D_{1,n}D_{1,0}$. Hence $$[D_{1,0},D_{1,n}]=cl_2(A^n-\bar{A}^n).$$
By the skein relation we can write $A-\bar{A}=(s-s^{-1})(C-D)$ in $\dot{BMW}_2$, where $C$ and $D$ are the diagrams: $$C = \vcenter{\hbox{\includegraphics[width=3cm]{CinBMW_2.eps}}} \quad \qquad \qquad D = \vcenter{\hbox{\includegraphics[width=3cm]{DinBMW_2.eps}}}$$
Using Definition \[def:D\_x\] and the explicit computation of $D_2$ in Section \[sec:D\_2\], one can verify that $$\label{eq:deven}
D_{2,n}=
\begin{cases}
cl_2 (C^n) & n \textrm{ odd} \\
cl_2 \left( \frac{1}{s+s^{-1}} \left( A C^{n-1}+\bar A C^{n-1} \right) \right) - c & n \textrm{ even} \\
\end{cases}$$ where $$c=\frac{v+v^{-1}}{s+s^{-1}} - 1$$
The product of $D$ with any element of $\dot{BMW}_2$ closes in $T^2$ to an element in the skein of $T^2$ lying entirely in an annulus around the vertical curve, $(0,1)$: $$\vcenter{\hbox{\includegraphics[width=6cm]{cl_2XDinC_01.eps}}}$$\
In algebraic terms, $cl_2(XD)\in\cc_{0,1}$ for any $X\in\dot{BMW}_2$. Write ${\mathcal{I}}$ for the two-sided ideal of $\dot{BMW}_2$ generated by $D$. Then $cl_2({\mathcal{I}})\subset \cc_{0,1}$.
Note that $D_{0,n}$ is in $\cc_{0,1}$ by definition. Our argument then, for $n$ odd, is to show that $cl_2(A^n-\bar{A}^n-(s^n-s^{-n})C^n) \in \cc_{0,1}$, with the appropriate modification for $n$ even. In the case where $n$ is even, we may forget about the constant term in $D_{2,n}$ since $-c\in\cc_{0,1}$.
A few observations in the algebra $\dot{BMW}_2$ will get us much of the way towards our goal.
Firstly we can see that $C^2$ commutes with $A$ and $\bar{A}$ at the diagrammatic level. Indeed it is central in the algebra, and so commutes with $D$, although more strongly we have that $C^2 D=DC^2=D$.
We see also that $AC\bar{A}=C^3 =\bar ACA$, leading to the more general cancellation of crossings $AC^{2k-1}\bar{A}=C^{2k+1}$ for $k>0$, again true at the diagrammatic level.
In $\dot{BMW}_2$ we have $$A^n-\bar{A}^n=(s-s^{-1})\left(A^{n-1}C +A^{n-3}C^3 + \dots + C^3\bar{A}^{n-3}+C\bar{A}^{n-1} -\sum_{i=1}^n A^{n-i}D\bar{A}^{i-1}\right)$$
$$\begin{aligned}
A^n-\bar{A}^n &=& \sum_{i=1}^n A^{n-i}(A-\bar{A})\bar{A}^{i-1}\\
&=& (s-s^{-1})\sum_{i=1}^n A^{n-i}(C-D)\bar{A}^{i-1}\\
&=& (s-s^{-1})\left(A^{n-1}C +A^{n-3}C^3 + \dots + C^3\bar{A}^{n-3}+C\bar{A}^{n-1} -\sum_{i=1}^n A^{n-i}D\bar{A}^{i-1}\right)\end{aligned}$$
The terms in a product can be reordered cyclically before closure so that $$cl_2 \left(\sum_{i=1}^n A^{n-i}D\bar{A}^{i-1}\right) = {\mathrm{cl}}_2 \left(\sum_{i=1}^n \bar{A}^{i-1}A^{n-i}D\right)= X_n$$ which represents the element $X_n\in\cc_{0,1}=\cc$ in the skein of the annulus which has the same diagrammatic representation as the $P_k$ in the Hecke algebra in Definition \[def:pk\].
\[expansion\] We have the expansion $$\begin{aligned}
[D_{1,0},D_{1,n}] &= cl_2 (A^n-\bar A^n)\\
&= cl_2 \left((s-s^{-1})(A^{n-1}C+C\bar A^{n-1})+ (A^{n-2}-\bar A^{n-2})C^2\right) -(s-s^{-1})(X_n-X_{n-2})
\end{aligned}$$
\[commutator\] In $\sd$ we have $[D_{1,0},D_{1,n}] =(s^n-s^{-n})D_{2,n}+Z_n$ for some $Z_n\in \cc_{0,1}$.
We prove this by induction on $n$ in two separate cases, either when $n$ is odd, or when $n$ is even.
The case $n=1$ is immediate since $A-\bar A-(s-s^{-1})C=-(s-s^{-1})D\in{\mathcal{I}}$ and so $cl_2 (A-\bar A-(s-s^{-1})C)= [D_{1,0},D_{1,1}] - (s^1-s^{-1})D_{2,1}$ lies in $\cc_{0,1}$.
When $n=2$ we have from lemma (\[expansion\]) that $A^2-\bar A^2 -(s-s^{-1})(AC +C\bar A) \in {\mathcal{I}}$. Then $cl_2 (A^2-\bar A^2 -(s-s^{-1})(AC +C\bar A))=[D_{1,0},D_{1,2}] - (s^1-s^{-1})(s+s^{-1})D_{2,2}$ lies in $\cc_{0,1}$.\
Assume now that $n>2$. By our induction hypothesis, theorem \[commutator\], for $n-2$ we know that $$\begin{aligned}
&=& (s^{n-2}-s^{-(n-2)})D_{2,n-2} +Z_{n-2},\end{aligned}$$ where $Z_{n-2}\in\cc_{0,1}$. Apply a single Dehn twist about the $(0,1)$ curve. Then $$[D_{1,1},D_{1,n-1}]= (s^{n-2}-s^{-(n-2)})D_{2,n} +Z_{n-2},$$ since $\cc_{0,1}$ is unaffected by this Dehn twist. We can see diagrammatically that $$cl_2 (A^{n-2}C^2-\bar A^{n-2}C^2)=[D_{1,1},D_{1,n-1}].$$ Our induction hypothesis then shows that $$\begin{aligned}
\label{commutatorcentre}
cl_2 (A^{n-2}C^2-\bar A^{n-2}C^2)&=&(s^{n-2}-s^{-(n-2)})D_{2,n}\ \rm{modulo}\ \cc_{0,1}.\end{aligned}$$ This provides us with one part of the expression for $[D_{1,0},D_{1,n}] $ in corollary \[expansion\]. We calculate the other part by means of the following lemma.
\[endpart\] We have $cl_2 \left(A^{n-1}C+C\bar A^{n-1}\right)=(s^{n-1}+s^{-(n-1)})D_{2,n} + Y_n$ for some $Y_n\in\cc_{0,1}$.
Lemma \[endpart\] is immediate for $n=1,2$.
For $n>2$ use induction on $n$, and equation (\[commutatorcentre\]).
Now $ cl_2 (A^{n-1}C+C\bar A^{n-1}) = cl_2 (A^{n-2}CA+\bar AC\bar A^{n-2}).$ Expand $A^{n-2}CA$ and $\bar AC\bar A^{n-2}$ as $$A^{n-2}CA=A^{n-2}C\left(\bar A+(s-s^{-1})(C-D)\right)=A^{n-3}C^3 +(s-s^{-1})A^{n-2}C^2 -(s-s^{-1})A^{n-2}CD$$ and $$\bar A C\bar A^{n-2}= \left( A-(s-s^{-1})(C-D)\right)C\bar A^{n-2}=C^3\bar A^{n-3} -(s-s^{-1})\bar A^{n-2}C^2 +(s-s^{-1})DC\bar A^{n-2}.$$
Then $$\begin{aligned}
cl_2 (A^{n-2}CA+\bar A C\bar A^{n-2})
&=& cl_2 \left(A^{n-3}C^3+C^3\bar A^{n-3}+(s-s^{-1})(A^{n-2}C^2-\bar A^{n-2}C^2) \right)\ \rm{modulo}\ \cc_{0,1}\\
&=& (s^{n-3}+s^{-(n-3)})D_{2,n}+(s-s^{-1})(s^{n-2}-s^{-(n-2)})D_{2,n}\ \rm{modulo}\ \cc_{0,1}\\
&=& (s^{n-1}+s^{-(n-1)})D_{2,n} \ \rm{modulo}\ \cc_{0,1}, \end{aligned}$$ using Lemma \[endpart\] for $n-2$ and equation (\[commutatorcentre\]).\
This establishes Lemma \[endpart\].
We complete the induction step for Theorem \[commutator\] using Corollary \[expansion\] and equation (\[commutatorcentre\]). This shows that $$\begin{aligned}
[D_{1,0},D_{1,n}]&=& cl_2 \left((s-s^{-1})(A^{n-1}C +C\bar{A}^{n-1}) + (A^{n-2}-\bar A^{n-2})C^2\right) \ \rm{modulo}\ \cc_{0,1}\\
&=& (s-s^{-1})(s^{n-1}-s^{-(n-1)})D_{2,n} +(s^{n-2}-s^{-(n-2)})D_{2,n}\ \rm{modulo}\ \cc_{0,1}\\
&=&(s^n-s^{-n})D_{2,n} \ \rm{modulo}\ \cc_{0,1}.\end{aligned}$$
Appendix
========
In the first subsection we state the BMW analogue of a classical result of Przytycki. In the second we show that our results on the Kauffman skein algebra are compatible with the analogous result about the Kauffman *bracket* skein algebra obtained by Frohman and Gelca [@FG00].
Przytycki
---------
In [@Prz91], Przytycki described a linear basis of the Homflypt skein algebra of a surface. This basis is the set of unordered words on the set $\{\pi_1(F) \setminus 1\}/\sim$, the set of non-identity conjugacy classes in the fundamental group of $F$. For the torus, this implies that the Homflypt skein algebra has a basis given by unordered words in $P_{{\mathbf{x}}}$ for ${{\mathbf{x}}}\in \Z^2$. A rough statement of the idea of the proof is that the diamond-lemma-type equalities that are used to show that the Homflypt polynomial is well-defined in $S^3$ generalize to 3-manifolds of the form $F \times [0,1]$. However, the details are quite technical, so we state the following analogue of his theorem for the Kauffman skein algebra without proof.
\[thm:basis\] The algebra $\sd$ has a basis given by unordered words in the elements $D_{{\mathbf{x}}}$, subject to the relation $D_{{\mathbf{x}}}= D_{-{{\mathbf{x}}}}$.
Frohman and Gelca
-----------------
Here we show that our presentation of the Kauffman skein algebra of the torus is compatible with Frohman and Gelca’s presentation of the Kauffman *bracket* skein algebra of the torus, which we recall below. This algebra is defined in the same way as the Kauffman skein algebra, but using the Kauffman bracket skein relations: $$\begin{aligned}
\label{eq:kb}
\vcenter{\hbox{\includegraphics[height=2cm]{poscross.eps}}} \quad &= \quad s \,\,\vcenter{\hbox{\includegraphics[height=2cm]{idresolution.eps}}} + s^{-1} \,\, \vcenter{\hbox{\includegraphics[height=2cm]{capcupresolution.eps}}}\\
\vcenter{\hbox{\includegraphics[height=2cm, keepaspectratio]{invvh.eps}}} \quad &= \quad -s^{-3}\,\, \vcenter{\hbox{\includegraphics[height=2cm, keepaspectratio]{frameresolution.eps}}} \notag\end{aligned}$$
Given any 3-manifold, there is a natural map ${\mathrm{Sk}}_s(M) \to K_s(M)$, which follows from the fact that the Kauffman bracket skein relations imply the Kauffman skein relations. If $M = F \times [0,1]$ for some surface $F$, then this is an algebra map.
The algebra map ${\mathrm{Sk}}_s(T^2\times [0,1]) \to K_s(T^2\times [0,1])$ takes the generator $D_{{\mathbf{x}}}$ to $e_{{\mathbf{x}}}$.
The closure of symmetrizer $f_n$ in the annulus gets sent to the (closure of the) Jones-Wenzl idempotent in the Kauffman bracket skein algebra of the annulus, because both symmetrizers are uniquely determined by the way they absorb crossings and caps. Under the identification $K_s(S^1\times D^2) = \C[x]$, the closure of the Jones-Wenzl idempotent is sent to the Chebyshev polynomial $S_n(x)$, which is defined by $$S_n(X+X^{-1}) = \frac{X^{n+1}-X^{-n-1}}{X-X^{-1}}$$ The elements $e_n$ are other version $T_n(x)$ of Chebyshev polynomials, which are defined by $$T_n(X+X^{-1}) = X^n+X^{-n}$$ Then all that is left to show is that the $S_n(x)$ and $T_n(x)$ satisfy the power series identity in Definition \[def:dk\].
Now let us recall the description of the Kauffman bracket skein algebra of the torus given by Frohman and Gelca.
The Kauffman bracket skein algebra $K_s(T^2)$ has a presentation with generators $e_{{\mathbf{x}}}$ and relations $$\begin{aligned}
e_{{\mathbf{x}}}e_{{\mathbf{y}}}&= s^d e_{{{\mathbf{x}}}+ {{\mathbf{y}}}} + s^{-d} e_{{{\mathbf{x}}}-{{\mathbf{y}}}}\\
e_{{{\mathbf{x}}}} &= e_{-{{\mathbf{x}}}}
\end{aligned}$$ where $d = \det[{{\mathbf{x}}}\,{{\mathbf{y}}}]$.
A short computation using this presentation shows the commutator identity $$[e_{{\mathbf{x}}}, e_{{\mathbf{y}}}] = (s^d-s^{-d}) (e_{{{\mathbf{x}}}+{{\mathbf{y}}}} - e_{{{\mathbf{x}}}-{{\mathbf{y}}}})$$ In other words, the relations we show for $D_{{\mathbf{x}}}\in {\mathrm{Sk}}(T^2)$ are compatible with the relations Frohman and Gelca show for the $e_{{\mathbf{x}}}\in K_s(T^2)$.
\[rmk:sobig\] We emphasize that the algebra we describe is much bigger – ${\mathrm{Sk}}(T^2)$ has a linear basis given by unordered words in the $D_{{\mathbf{x}}}$, while the Kauffman bracket skein algebra $K_s(T^2)$ has a linear basis given by the $e_{{\mathbf{x}}}$ themselves. This happens because the Kauffman bracket skein relations allow all crossings to be removed from a diagram, so the algebra is spanned (over $R$) by curves without crossings. However, the more general Kauffman relations only allow crossings to be flipped, and not removed, which means the algebra is spanned (over $R$) by products of knots (which may have self-crossings).
[^1]: To clarify a possible point of confusion, the Kauffman *bracket* skein relations are defined in the Appendix. Computations with the Kauffman bracket skein relations are much simpler because “all crossings can be removed.” See also Remark \[rmk:sobig\].
[^2]: We thank Negut for pointing this out.
[^3]: The Hall algebra is graded by $K$-theory, and in many cases of interest this is a free abelian group of rank 2 or greater.
[^4]: To be clear, when we say “non-unital algebra map,” we mean $s_n(x+y)=s_n(x) + s_n(y)$ and $s_n(xy) = s_n(x)s_n(y)$, but that $s_n(1)$ is not the identity in $BMW_n$, but instead is another idempotent.
|
---
abstract: |
The web of collaborations between individuals and group of researchers continuously grows thanks to online platforms, where people can share their codes, calculations, data and results. These virtual research platforms are innovative, mostly browser-based, community-oriented and flexible. They provide a secure working environment required by modern scientific approaches. There is a wide range of open source and commercial solutions in this field and each of them emphasizes the relevant aspects of such a platform differently.
In this paper we present our open source and modular platform, [`KOOPLEX`]{}[^1], that combines such key concepts as dynamic collaboration, customizable research environment, data sharing, access to datahubs, reproducible research and reporting. It is easily deployable and scalable to serve more users or access large computational resources.
address:
- 'Department of Physics of Complex Systems, E[ö]{}tv[ö]{}s Lor[á]{}nd University, H-1117, Pázmány Péter sétány 1/a. Budapest, Hungary'
- 'Department of Information Systems, E[ö]{}tv[ö]{}s Lor[á]{}nd University, H-1117, Pázmány Péter sétány 1/c. Budapest, Hungary'
- 'Department of Computational Sciences, Wigner Research Centre for Physics of the HAS, Konkoly-Thege Miklós út 29-33., Budapest 1121, Hungary'
author:
- 'D. Visontai'
- 'J. Stéger'
- 'J. M. Szalai$-$Gindl'
- 'L. Dobos'
- 'L. Oroszl[á]{}ny'
- 'I. Csabai'
bibliography:
- 'ref.bib'
title: 'Kooplex: collaborative data analytics portal for advancing sciences'
---
kooplex ,collaboration ,platform ,jupyter ,notebook ,scalable ,open source ,data science ,reporting ,rstudio ,business intelligence ,gitea ,seafile ,kubernetes ,docker
Summary
=======
[`KOOPLEX`]{} is a platform for easy access to datahubs, for collaborative work, for developing new workflows and for creating and publishing static or interactive reports. It is clear from the user feedback regarding the platform instances mentioned in **Section \[sec:kooplex-usecase\].** that the combination of such integrated services is attractive. Accessing the various modules in the same user space speeds up analysis, code development work and sharing, not having to move data and files around between disconnected components. The platform has been designed in a way that the integration of new tools taken up by the research community is straight forward. This feature helps to keep up with the ever evolving user requirements.
Acknowledgements {#acknowledgements .unnumbered}
================
This study has received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 643476 (COMPARE) and from National Research, Development and Innovation Fund of Hungary Project (FIEK\_16-1-2016-0005 to I.C.). The authors are grateful to G. Vattay, S. Laki and students at Eötvös University for help with thorough testing of the system. This work was completed in the ELTE Excellence Program (783-3/2018/FEKUTSRAT) supported by the Hungarian Ministry of Human Capacities. The work was supported by the Hungarian National Research, Development and Innovation Office (NKFIH) through Grants No. K120660, K109577, K124351, K124152, KH129601, and the Hungarian Quantum Technology National Excellence Program (Project No. 2017-1.2.1-NKP-2017- 00001).
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors declare that there is no conflict of interest regarding the publication of this manuscript.
[^1]: https://kooplex.github.io
|
---
abstract: 'We investigate the abundance of large-scale hot and cold spots in the temperature maps and find considerable discrepancies compared to Gaussian simulations based on the $\Lambda$CDM best-fit model. Too few spots are present in the reliably observed cosmic microwave background (CMB) region, i.e., outside the foreground-contaminated parts excluded by the KQ75 mask. Even simulated maps created from the original estimated multipoles contain more spots than visible in the measured CMB maps. A strong suppression of the lowest multipoles would lead to better agreement. The lack of spots is reflected in a low mean temperature fluctuation on scales of several degrees ($4^\circ$–$8^\circ$), which is only shared by less than $1\%$ ($0.16\%$–$0.62\%$) of Gaussian $\Lambda$CDM simulations. After removing the quadrupole, the probabilities change to $2.5\%$–$8.0\%$. This shows the importance of the anomalously low quadrupole for the statistical significance of the missing spots. We also analyze a possible violation of Gaussianity or statistical isotropy (spots are distributed differently outside and inside the masked region).'
author:
- 'Youness Ayaita, Maik Weber, Christof Wetterich'
bibliography:
- 'cmbspots.bib'
title: Too few spots in the cosmic microwave background
---
Introduction
============
The precise measurement of anisotropies in the cosmic microwave background (CMB) has played a key role in amplifying our knowledge about the structure and evolution of the Universe. The best data available today is provided by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite mission from five years of observation. Its results are powerful enough to put various cosmological models to stringent tests. They helped establishing today’s standard model of a spatially flat universe with Gaussian initial perturbations, possibly generated during an early inflationary epoch. According to the standard $\Lambda$CDM model, the present Universe is essentially made up from dark energy in the form of a cosmological constant $\Lambda$ and cold dark matter (CDM). Under the assumptions of Gaussianity and statistical isotropy, all the information about the temperature fluctuations in the CMB are encoded in the angular power spectrum $C_\ell$ from a harmonic decomposition of the temperature field. A crucial result of the WMAP analysis therefore is an estimate of the multipoles $C_\ell$ which is in good agreement with the $\Lambda$CDM best fit [@Nolta09] except for the well-known discrepancies of the low multipoles, especially the quadrupole $C_2$ [@Hinshaw07]. Nonetheless, many issues are still under intense discussion. Repeatedly, authors have claimed to detect non-Gaussian signals [@McEwen08; @Yadav07] or statistical anisotropy [@Eriksen03; @Hansen08; @deOliveiraCosta03; @Hoftuft09; @Land05; @Bernui06]. Since the power spectrum is insensitive to these anomalies, it is necessary to perform additional investigations of the temperature sky map. These are done in harmonic, wavelet, and pixel space [@Cabella04]. Even if Gaussianity holds, it may still give new insights to switch to another representation of the statistical properties of the temperature maps since a phenomenon can be more easily detected in one representation than in another.
The goal of this work is to provide a clear and intuitive analysis in pixel space regarding abundances of large-scale hot and cold spots identified as regions whose mean temperature contrasts exceed some (variable) threshold. We analyze both observed CMB maps and Gaussian simulations based on $\Lambda$CDM. The comparison reveals severe deviations. Other authors who worked with statistics of local extrema in the temperature field also observed significant anomalies [@Larson04; @Larson05; @Hou09].
We start by recalling some basic results that connect pixel-space analyses with the angular power spectrum in Sec. \[sec:preliminary\]. A comprehensive description of our method follows in Sec. \[sec:method\] including the preparation of adequate Gaussian simulations, the working principle of our spot searching algorithm, and an error estimation. Our results are presented in Sec. \[sec:results\]. We consider both cut-sky maps (with unreliable pixels excluded by the KQ75 temperature analysis mask) and the Internal Linear Combination (ILC) full-sky map, and quantify deviations from Gaussian simulations. We sum up and conclude in Sec. \[sec:discussion\].
Preliminary considerations {#sec:preliminary}
==========================
The most robust comparison between predicted and observed spot abundances of CMB sky maps relies on simulated maps since analytic methods can hardly care for complications due to masking and beam properties. Creating a number of simulated maps and treating them in exactly the same way as the original map therefore is the clearest method. Nonetheless, it is instructive to recall some well-known analytic results that connect the pixel-space analysis to familiar harmonic space.
The spot abundances in a CMB sky map are dictated by the angular correlations of temperature fluctuations. The most popular theories stick to Gaussianity and statistical isotropy. Then, the ensemble average of the angular correlation between two directions $(\theta,\varphi)$ and $(\theta',\varphi')$ only depends on the angle $\Theta$ between them. This leads to the definition of the angular correlation function $$C(\Theta) = \left< \frac{\Delta T}{\bar
T}(\theta,\varphi)\times \frac{\Delta T}{\bar T}(\theta',\varphi')
\right>.
\label{eq:correlation_fct}$$ We can switch to harmonic space by decomposing the temperature field into spherical harmonics: $$\frac{\Delta
T}{\bar T}(\theta,\varphi)=\sum_{\ell,m} a_{\ell m}
Y_{\ell m}(\theta,\varphi),
\label{eq:T_decomposed}$$ where the crucial assumption of statistical isotropy implies $$\left< a_{\ell m} a_{\ell'm'}^* \right> = \delta_{\ell \ell'}
\delta_{mm'} C_\ell.
\label{eq:stat_isotropy}$$ So, in this case, all the statistical information is in the coefficients $C_\ell$, the angular power spectrum. More generally, we may define $$C_\ell = \frac{1}{2 \ell+1} \sum_{m}^{} \left< |a_{\ell m}|^2 \right>.
\label{eq:Cl_def}$$
When searching for spots of a given size, we will average the temperature fluctuations in regions of that size. These regions are defined by window functions $W(\theta,\varphi)$. The mean temperature contrast in such a region is $$\Delta T = \int_{}^{} {\mathrm{d}}\Omega\, \Delta
T(\theta,\varphi)\, W(\theta,\varphi).
\label{DTrmsWindow}$$ In our sense, a [*spot*]{} is characterized as follows. When a threshold $\Delta \mathfrak T$ is fixed, a [*hot spot*]{} is found if $\Delta T \ge \Delta \mathfrak T$, whereas a [*cold spot*]{} is found if $\Delta T \le - \Delta \mathfrak T$. The characteristic scale for $\Delta T$ is the mean temperature contrast for these regions $\Delta T_{rms} = \sqrt{\left< \Delta T^2
\right>}$. Clearly, if $\Delta \mathfrak T \ll \Delta T_{rms}$, most regions will be spots, if $\Delta \mathfrak T \gg \Delta T_{rms}$, only a few or none.
The transformation to harmonic space can be done by decomposing the window function $W(\theta,\varphi)$ into spherical harmonics with coefficients $W_{\ell m}$ and defining $$W_\ell = \sum_{m}^{} |W_{\ell m}|^2.
\label{eq:Wl}$$ Together with Eqs. (\[eq:T\_decomposed\]) and (\[eq:stat\_isotropy\]), it is straightforward to calculate $$\Delta T_{rms}^2 = \sum_{\ell}^{}
\frac{2 \ell+1}{4\pi}\,C_\ell\,W_\ell\,\bar T^2.
\label{eq:DTrms}$$ This result shows that the mean temperature fluctuation $\Delta
T_{rms}$ is given by the multipoles $C_\ell$ weighted by $W_\ell$. The $W_\ell$ strongly depend on the angular scale of the regions. Their magnitude will suppress large $\ell$ values corresponding to scales smaller than the window. By virtue of the addition theorem for spherical harmonics, we can write $$W_\ell = \int {\mathrm{d}}\Omega\ \int
{\mathrm{d}}\Omega'\
W(\theta,\varphi)\, W(\theta',\varphi')\, P_\ell(\cos\Theta).
\label{Wl_integration}$$ This allows us to calculate the $W_\ell$ for a chosen window. An example is shown in Fig. \[fig:Wls\].
\[B\]\[c\]\[.8\]\[0\][$\ell$]{} \[B\]\[c\]\[.8\]\[0\][$W_\ell$]{} ![Coefficients $W_\ell$ for the top hat circle window function at scales $a=1^\circ$ (right plot), $6^\circ$ (left plot). The plots show which multipoles predominantly determine $\Delta T_{rms}$. For smaller angular scale $a$, higher $\ell$ values enter the analysis.[]{data-label="fig:Wls"}](Wls.eps "fig:"){width=".4\textwidth"}
In our case, it is adequate to approximate the sphere by the tangent plane at a region, replacing the direction $(\theta,\varphi)$ by points ${\ensuremath{\mathbf{x}}}$ on the plane. For our purposes, it is most convenient to work with top hat windows because they have clear boundaries. This is the easiest way to avoid ambiguities arising from overlapping spots. Exemplary choices may be the top hat circle with window function $$W({\ensuremath{\mathbf{x}}}) = \frac{1}{\pi R^2}\Theta(R-|{\ensuremath{\mathbf{x}}}|)
\label{eq:W_circle}$$ or a square with window function $$W({\ensuremath{\mathbf{x}}}) = \frac{1}{a^2}\, \Theta(a-x_1)\,
\Theta(a-x_2).
\label{eq:W_square}$$ Following @Durrer08 [p. 218], we can approximate the $W_\ell$ by an angular average over the Fourier transform of $W({\ensuremath{\mathbf{x}}})$ which considerably reduces the computational effort: $$W_\ell \approx \frac{1}{2\pi} \int_{0}^{2\pi} {\mathrm{d}}\alpha \, |\tilde
W({\ensuremath{\mathbf{l}}})|^2.
\label{eq:durrer}$$ For the aforementioned window functions, we can use this equation to easily calculate $\Delta T_{rms}$ by Eq. (\[eq:DTrms\]). The results are plotted for the $\Lambda$CDM best-fit power spectrum in Fig. \[fig:DTrms\]. For the sake of comparability, we use the parameter $a$ which equals the square root of the windows’ area; in the case of squares, it simply is the side length. We also show the relative deviation due to the different window functions. We conclude that the result is not sensitive to the exact geometry if the covered surface area is the same.
\[B\]\[c\]\[.8\]\[0\][$a$ \[$^\circ$\]]{} \[B\]\[c\]\[.8\]\[0\][$\Delta T_{rms}$ $[\upmu \text K ]$]{} ![Mean temperature fluctuation for various spot sizes and the $\Lambda$CDM power spectrum. The plots for circles and squares are visually indistinguishable. The difference between the result for circles and the result for squares is shown in the second figure.[]{data-label="fig:DTrms"}](sigmacalc.eps "fig:"){width=".4\textwidth"} \[B\]\[c\]\[.8\]\[0\][$a$ \[$^\circ$\]]{} \[B\]\[c\]\[.8\]\[0\][Relative deviation in %]{} ![Mean temperature fluctuation for various spot sizes and the $\Lambda$CDM power spectrum. The plots for circles and squares are visually indistinguishable. The difference between the result for circles and the result for squares is shown in the second figure.[]{data-label="fig:DTrms"}](sigmacalcdifference.eps "fig:"){width=".4\textwidth"}
Method {#sec:method}
======
Our strategy consists of performing an identical analysis of spot abundances both for observational maps and maps generated from simulations of Gaussian fluctuations. For the simulated maps, we use the best-fit $\Lambda$CDM model and a Gaussian fluctuation model based on the $C_\ell$ quoted by the WMAP collaboration. The comparison with maps from observation tests Gaussianity.
Because of the excellent data products of the WMAP team available at the legacy archive[^1] and the comprehensive HEALPix package[^2] [@HEALPIX], it is possible to obtain reliable CMB sky maps and to create maps from Gaussian simulations. We summarize the steps in Sec. \[sec:data\]. We developed an algorithm searching for hot and cold spots (in the sense of Sec. \[sec:preliminary\]) within these temperature sky maps. Its working principle and properties are presented in Sec. \[sec:algorithm\]. The treatment of statistical errors is described in Sec. \[sec:errors\].
Maps and data preparation {#sec:data}
-------------------------
Whenever the original signal is to be extracted from CMB data, it is crucial to minimize the influence of foreground contamination. The frequency dependence of the foreground components (e.g., synchrotron emission, free-free emission, and thermal dust) allows to reduce the contamination with the help of various foreground models [@Gold09]. The WMAP team provides foreground-reduced maps for the $Q$ (35–46 GHz), $V$ (53–69 GHz), and $W$ (82–106 GHz) bands. Since the $V$ band has a better signal-to-noise ratio than the $W$ band and is less foreground contaminated than the $Q$ band [@Hinshaw07], it is the natural choice to use the foreground-reduced $V$ map. Further noise minimization by constructing linear combinations of the maps is possible but does not affect our analysis which focuses on large scales. But still, large parts of the temperature map are unreliable and must be excluded from the analysis. We therefore apply the KQ75 mask, cutting out the contaminated galaxy region and point sources [@Gold09]. Finally, the residual monopole and dipole are removed with the HEALPix routine . Figure \[fig:maps\] shows the foreground-reduced $V$ map and the KQ75 mask.
Gaussian simulations based on some input $C_\ell$ spectrum and a beam window function are achieved with the help of the [synfast]{} HEALPix facility. These input data can be obtained from the legacy archive. The power spectra we used are the $\Lambda$CDM best fit and the original estimate both shown in Fig. 1 of @Nolta09. Subsequently, we will refer to them by “$\Lambda$CDM” and “” power spectrum for short. We take care of treating simulated and original maps as equally as possible. This necessitates the additional simulation of the instruments’ noise, masking, and removal of monopole and dipole. Since the WMAP design minimizes noise correlation between neighboring pixels in a map [@Page03], it is legitimate to add white noise with the properties described by the WMAP team at the legacy archive.
When studying possible anisotropy of the CMB, we need a full-sky (unmasked) map. Since the foreground contaminations usually force us to mask parts of the sky, it is not a trivial task to reconstruct the full-sky CMB signal. However, the WMAP team tries to tackle this job by combining the measurements of all bands and merge them into a single (ILC) map of the full sky [@Gold09]. The applied procedure is independent of foreground models but has the disadvantage of being doubtful on scales below approximately $10^\circ$ according to the WMAP product description at the legacy archive. But since we are lacking any better alternative, we employ the 5-year WMAP ILC map for full-sky analyses.
Spot searching algorithm {#sec:algorithm}
------------------------
The primary goal of the algorithm is to count hot and cold spots in CMB sky maps on various scales and temperature contrasts. A typical application will be to plot spot abundances against the threshold on the temperature contrast $\Delta \mathfrak T$ for a specific angular scale. This application directly imposes several features the algorithm should have:
It must define [*sectors on the sphere of equal surface area*]{} (for some desired scale). Their mean temperature contrasts will decide whether they are counted as spots.
The areas must be chosen such that one can [ *smoothly scan*]{} through the map. Between two distinct areas, there must exist many others allowing for a smooth transition.
Double counting of spots has to be excluded. The easiest way to achieve this is working with [*top hat windows*]{} which have clear boundaries. Overlapping spots will be counted as a single.
For a statistically satisfactory comparison between observed and simulated CMB maps, the algorithm will have to analyze many sky maps. Given the huge amount of data, one has to implement the algorithm carefully in order to make this [*numerically tractable*]{}.
The algorithm is designed such that it allows for an approximate pixelization of the sphere into distinct areas of a given scale. Calculating their temperature contrasts determines the mean temperature fluctuation $\Delta T_{rms}$ on that scale. By virtue of the ergodic theorem, this is a good estimate for the ensemble average introduced in Sec. \[sec:preliminary\].
### Working principle {#sec:principle}
The first task is to define the sectors $S$ of equal surface area on the sphere satisfying the requirements explained above. We choose them to be intersections of latitude and longitude rings. A latitude ring $\mathcal R^{lat}$ consists of all points between two latitude angles $\theta_0$ and $\theta_1$, a longitude ring $\mathcal R^{lon}$ of all points between two longitude angles $\varphi_0$ and $\varphi_1$. The rings have two nice properties. First, as needed for spot searching, one can smoothly go from one ring to any other ring by smoothly changing its boundary angles; second, as needed for calculating $\Delta T_{rms}$, it is an easy task to discretize a sphere into distinct rings. Since sectors are intersections $S = \mathcal R^{lat}
\cap \mathcal R^{lon}$, they share these properties. We thereby satisfy the requirement of smooth scanning to all directions.
We impose \[meeting the requirement (i) above\] equal area $A$ for all sectors: $$A = \int_{\mathcal S}^{} {\mathrm{d}}\Omega =
\int_{\varphi_0}^{\varphi_1}{\mathrm{d}}\varphi\
\int_{\theta_0}^{\theta_1} {\mathrm{d}}\theta\ \sin\theta.
\label{eq:equal_area}$$ Once we have decided to define sectors like this, we still have some freedom to choose the boundaries $\theta_0$, $\theta_1$, $\varphi_0$, $\varphi_1$. In order to avoid the influence of small scales, we must reasonably choose the sectors such that they are by no means degenerated. We therefore fix this freedom by adding another constraint. For any sector $\mathcal S$, the boundary lines in the north-south direction and the longer boundary in the east-west direction are chosen to be of equal length: $$(\varphi_1-\varphi_0)\, \sin\theta_* =
(\theta_1-\theta_0).
\label{eq:equal_length}$$ On the northern hemisphere $\theta_*=\theta_1$, on the southern hemisphere $\theta_*=\theta_0$. Note that these sectors behave well. In the limiting case near the equator, they correspond to squares in flat space. At the poles ($\theta_0=0$ or $\theta_1=\pi$), they become equilateral triangles.
In practice though, the temperature field is not given as a smooth function of $\theta$ and $\varphi$. The WMAP temperature sky maps are lists assigning a temperature contrast $\Delta T_i$ to each HEALPix pixel $p_i$. The mapping $p_i \mapsto (\theta,\varphi)$ is given in the form of a table. But since our approach defines sectors by means of angles, we need the reverse. Given the list $p_i \mapsto
(\theta,\varphi)$, finding the appropriate pixel $p_i$ for given angles $(\theta,\varphi)$ corresponds to searching through the list. Whereas searching in an unsorted list is very expensive, an adequate sorting may considerably reduce the effort. The algorithm performs the following steps starting at the north pole :
For given $\theta_0$ and area $A$, calculate $\theta_1$ and $\Delta \varphi$ by solving Eqs. (\[eq:equal\_area\]) and (\[eq:equal\_length\]).
Collect the pixels $\{p_i\}$ belonging to the latitude ring $\mathcal R^{lat}$ between $\theta_0$ and $\theta_1$. This can be done efficiently if the map was prepared by transforming to sorted latitude angles (HEALPix [ring]{} ordering).
Using a fast routine, sort the list $\{p_i\}$ with respect to longitude angles. This new sorting allows one to directly identify the pixels out of $\{p_i\}$ belonging to a longitude ring $\mathcal R^{lon}$ with boundaries $\varphi_0$ and $\varphi_1$—these pixels form the sector $\mathcal S =
\mathcal R^{lat} \cap \mathcal R^{lon}$. Start at $\varphi_0=0$ and $\varphi_1=\Delta\varphi$ and smoothly scan (by increasing $\varphi_0$, $\varphi_1$ by a small step size $h$) through all longitude rings. For every sector, calculate the sector’s mean temperature contrast $\Delta T$ by averaging over the pixel values $\Delta T_i$ and compare it with the threshold $\Delta \mathfrak T$. If it exceeds the threshold, count a spot if the sector does not overlap with a previously found spot.
Choose the next ring by slightly increasing $\theta_0
\mapsto \theta_0 + h$. It is profitable to exploit the fact that the sorting for longitude angles (point 3) need not be repeated completely. The algorithm saves the previous sorting and uses it for a presort such that as much information is transferred as possible.
Having increased the threshold $\Delta\mathfrak T$, again searching for spot abundances in a map can be optimized by noticing that spots at a higher threshold cannot be found where there were not spots at a lower threshold. Our algorithm can focus on areas around previously found spots once this becomes advantageous.
If we slightly adapt the algorithm, we can use it to measure $\Delta
T_{rms}$. Now, the algorithm jumps between distinct sectors instead of smoothly transforming them. The distinct sectors are visualized in Fig. \[fig:sectors\].
![Exemplary decomposition of the sky into $N_{sec}$ distinct sectors $\mathcal S_j$ for measuring $\Delta T_{rms}$. For searching spots, the algorithm analyzes many more sectors $\mathcal S$ (those in between, sharing pixels with the illustrated sectors $\mathcal S_j$). Nonetheless, $N_{sec}$ limits the maximum number of spots since overlapping spots are not multiply counted.[]{data-label="fig:sectors"}](sectors.eps){width=".4\textwidth"}
In every distinct sector, the mean temperature fluctuation is calculated. The squares are averaged to give $\Delta T_{rms}$. Although the shapes of the sectors vary, the results of Sec. \[sec:preliminary\] ensure that $\Delta T_{rms}$ is only marginally affected.
### Treatment of masked maps {#sec:mask_treatment}
The sectors defined by our algorithm may include none, some, or many masked pixels. We must define selection rules determining which sectors are to be included in the statistics. We used the following two rules. The most restrictive choice is to only consider sectors with no mask overlap ([*strict selection*]{} for short). These sectors will only contain reliable pixels. But since especially on large scales, only a minority of sectors will belong to this group, bad statistics are the price to pay. The alternative choice is to also consider sectors with a slight mask overlap ([*tolerant selection*]{}). This is a compromise between good statistics on the one hand and reliable results on the other. We typically allow for $5\%$ masked area within a sector which guarantees that usually the majority of sectors fall into this group. In any case, we emphasize that masked pixels, even if included in the statistics, are assigned zero temperature fluctuation. This will avoid misinterpreting foregrounds as a CMB signal. Note however, that the pixels of zero temperature fluctuation reduce $\Delta T_{rms}$. For comparisons between observed maps and Gaussian simulations, we employ the tolerant selection for the sake of better statistics; the comparison is still trustworthy.
### Alternative shapes {#sec:shapes}
The algorithm works with the shapes defined in Sec. \[sec:principle\] and illustrated in Fig. \[fig:sectors\]. But we can easily treat other shapes by embedding them into the previous sectors. This corresponds to a multiplication of the previous window function $W_0$ with the window function $W_1$ of the desired shape where $W_0$ must be large enough to ensure $W_0\equiv 1$ where $W_1$ is non-zero. The condition (\[eq:equal\_area\]) of equal area now concerns the new shape and reads $$\int_{\varphi_0}^{\varphi_1}{\mathrm{d}}\varphi\
\int_{\theta_0}^{\theta_1} {\mathrm{d}}\theta\ \sin\theta \ W_1(\theta,\varphi)
= A.
\label{eq:equal_area_shapes}$$ As an example, we compare the standard shape with top hat circles of equal area \[cf. Eq. (\[eq:W\_circle\])\] and plot the result in Fig. \[fig:circles\].
\[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of spots]{} ![Mean spot abundances in 100 simulated $\Lambda$CDM full-sky maps showing the results for different window functions of scale $a=\sqrt{A}=6^\circ$.[]{data-label="fig:circles"}](circles.eps "fig:"){width=".45\textwidth"}
For low thresholds, the abundances are systematically higher for the standard window function. This is due to the fact that circles do not exhaust the area without space in between. The effect becomes important where many spots are found and overlapping is frequent but disappears for large thresholds where the results agree.
### Step size dependence {#sec:stepsize}
In the ideal case, the boundary angles of the sectors would vary in a perfectly smooth manner when searching for spots in a map. But numerically, we have to choose a finite step size $h$ (introduced in Sec. \[sec:principle\]). A good choice of $h$ balances sensitivity and numerical effort. Figure \[fig:stepsize\] shows detected spot abundances against $h$ in simulated maps. We chose $h=0.3^\circ$ for which we conclude that our sensitivity to detect spots is satisfactory.
\[B\]\[c\]\[.8\]\[0\][Step size $h$ \[$^\circ$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of spots]{} ![Mean spot abundances for a fixed threshold (80 $\upmu
\text{K}$) against a varying step size. 100 masked $\Lambda$CDM simulated maps were scanned for $a=6^\circ$, the error bars quantify the statistical error.[]{data-label="fig:stepsize"}](stepsize.eps "fig:"){width=".4\textwidth"}
Errors and cosmic variance {#sec:errors}
--------------------------
There are statistical uncertainties simply due to the finite number of Gaussian simulations. Moreover, the CMB signal itself can be regarded as the outcome of a statistical process. It is therefore subject to statistical variation, quantified by the concept of cosmic variance.
Let us assume that $N$ Gaussian maps are analyzed for spots (area and threshold fixed). If $n^{(k)}$ spots are detected in map $k$, the mean spot abundance is $$\bar n = \sum_{k=1}^{N}\frac{n^{(k)}}{N}.
\label{eq:mean_abundance}$$ The statistical uncertainty of the mean value $\bar n$ and the statistical deviation of the single values $\bar n^{(k)}$ are $$\sigma_{\bar n}^2 = \frac{\sum_{k=1}^{N}(n^{(k)}-\bar n)^2}{N
(N-1)}, \ \sigma_{n^{(k)}}^2 = N \, \sigma_{\bar n}^2.
\label{eq:errors_abundance}$$ The same procedure applies if we measure the mean temperature fluctuations $\Delta T_{rms}^{(k)}$ in the maps and calculate a mean value $\Delta \bar T_{rms}^{(k)}$.
We now consider cosmic variance. When we observe a spot abundance $n$, we must expect a certain deviation from the theoretically predicted ensemble average $\left< n \right>$. The expectation value of this deviation, $\sigma_n^2 = \left< (n - \left< n \right>)^2 \right>$, quantifies cosmic variance. For a very large number $N$ of simulated maps, we may replace the ensemble average by an averaging over the set of simulations. We then obtain $\sigma_n \approx \sigma_{n^{(k)}}$ with the latter calculated according to Eq. (\[eq:errors\_abundance\]). This can be done equally for the mean temperature contrast $\Delta T_{rms}$. Whenever we specify cosmic variance (e.g., in the form of error bars), we estimated it by this method.
Results {#sec:results}
=======
The application of the spot-searching algorithm described in Sec. \[sec:method\] shows that the standard model $\Lambda$CDM together with Gaussianity predicts more large-scale hot and cold spots than are actually present in cut-sky data (see Sec. \[sec:cut-sky\]). Removing the quadrupole or using the original $C_\ell$ (instead of the $\Lambda$CDM fit) considerably reduces the discrepancies. While only $0.16\%$–$0.62\%$ of Gaussian $\Lambda$CDM simulations fall below the observed mean temperature fluctuations on angular scales of $4^\circ$–$8^\circ$, this increases to $2.5\%$–$8\%$ if the quadrupole is removed. We also investigate full-sky maps in Sec. \[sec:ilc\] and modifications of the first multipoles in Sec. \[sec:modified\].
Cut-sky maps {#sec:cut-sky}
------------
The spots’ size is characterized by their area $A$. We use the parameter $a=\sqrt{A}$ to specify the angular scale of this size. Since on the one hand, we aim at large scales, and on the other hand, we want reasonable statistics, we are forced to find a compromise. We chose an angular scale $a=6^\circ$. The spot abundances of the $V$ map and 500 Gaussian $\Lambda$CDM simulations (created as described in Sec. \[sec:data\]) are found for varying threshold $\Delta \mathfrak T$. The HEALPix resolution parameter of the maps is 8, corresponding to $N_{pix}=12\times256^2=786,432$ pixels. Statistical uncertainties and cosmic variance are displayed as error bars even though the spot abundances for different thresholds are of course correlated. The results for hot and cold spots are plotted in Fig. \[fig:spots\_V\_LCDM\].
\[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of hot spots]{} ![Spot abundances in the CMB sky (with cosmic variance) as compared to 500 $\Lambda$CDM simulations (with statistical errors) on an angular scale of $a=6^\circ$. The fractions of Gaussian simulations with smaller values of $s$ \[Eq. (\[eq:s\])\] are $p_s^\text{hot} =
0.2\%$ and $p_s^\text{cold} = 1.8\%$.[]{data-label="fig:spots_V_LCDM"}](spots_V_LCDM_hot.eps "fig:"){width=".45\textwidth"} \[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of cold spots]{} ![Spot abundances in the CMB sky (with cosmic variance) as compared to 500 $\Lambda$CDM simulations (with statistical errors) on an angular scale of $a=6^\circ$. The fractions of Gaussian simulations with smaller values of $s$ \[Eq. (\[eq:s\])\] are $p_s^\text{hot} =
0.2\%$ and $p_s^\text{cold} = 1.8\%$.[]{data-label="fig:spots_V_LCDM"}](spots_V_LCDM_cold.eps "fig:"){width=".45\textwidth"}
The striking feature of the plots is the discrepancy between theory and observation. They only agree in the limit of very small thresholds $\Delta\mathfrak T$ where it is obvious that almost every area is counted as a spot anyway. The discrepancy is seemingly more drastic for hot spots. In the plot for cold spots, it is seen that there is one considerable cold spot nearly reaching $150\,\upmu \text{K}$. But even this spot does not surpass the $\Lambda$CDM prediction. We note that this spot is localized in the region of the famous Vielva cold spot [@Vielva03]. It is insightful to look at the spot distributions of single Gaussian simulations in order to get an impression of their typical behavior. Five examples are plotted in Fig. \[fig:gaussians\].
\[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of hot spots]{} ![Spot abundances in five randomly chosen Gaussian simulations based on the $\Lambda$CDM best-fit power spectrum and the mean curve from Fig. \[fig:spots\_V\_LCDM\] (hot spots). []{data-label="fig:gaussians"}](gaussians.eps "fig:"){width=".45\textwidth"}
Because of the strong correlation between the spot abundances $n_i$ for different thresholds $\Delta\mathfrak T_i$, it is difficult to judge the significance of the discrepancies by eye. A possible quantity that can be used for a comparison of the observed CMB map with Gaussian simulations is obtained by summing up the spot abundances at different thresholds, $$s = \sum_{i}^{} n_i,
\label{eq:s}$$ where the lowest threshold included is chosen to be the characteristic scale $\Delta\bar T_{rms}$. We denote the fraction of Gaussian simulations $k$ with $s^{(k)}$ smaller than found in the $V$ map by $p_s$. For the spot abundances shown in Fig. \[fig:spots\_V\_LCDM\], we find $p_s^\text{hot} = 0.2\%$ for hot spots and $p_s^\text{cold} =
1.8\%$ for cold spots.
The discrepancies are reflected in the mean temperature fluctuation $\Delta T_{rms}$ which on large scales is higher in $\Lambda$CDM simulations than in the observed CMB sky. We have simulated 5000 $\Lambda$CDM maps and compared their mean temperature fluctuations to the value of the $V$ map. We employed the tolerant selection of our algorithm (see Sec. \[sec:mask\_treatment\]). For $a=6^\circ$, we find the mean value $\Delta T_{rms} = 39.4\,\upmu \text{K}$ for the $V$ map, as compared to the mean value $\Delta \bar T_{rms} = 47.9 \pm 0.1 \,
\upmu \text{K}$ for $\Lambda$CDM, where the error is only statistical while cosmic variance amounts to $4.2\, \upmu \text{K}$. Only a fraction $p=0.6\%$ of the simulations had a smaller $\Delta T_{rms}$ than the $V$ map. This does not improve at other large angular scales which can be seen in Table \[tab:significance\].
Scale $a$ Fraction $p$
----------- -- -- -- --------------
$4^\circ$ $0.50\%$
$5^\circ$ $0.62\%$
$6^\circ$ $0.60\%$
$7^\circ$ $0.16\%$
$8^\circ$ $0.36\%$
: The fraction $p$ of Gaussian $\Lambda$CDM simulations with a $\Delta T_{rms}$ smaller than found in the $V$ map on the angular scale $a$.
\[tab:significance\]
It is interesting to see how this behavior changes when going to smaller scales. However, the results on smaller scales (approaching $1^\circ$) become sensitive to noise and beam properties. Since the WMAP team offers the latter for the differencing assemblies $V$1 and $V$2 [@Hill09] instead of the combined $V$ map, it is the easiest to switch to the $V$1-band map and simulations thereof. The impact on large scales is negligible. Figure \[fig:sigmas\_original\] shows $\Delta
T_{rms}$ against the scale $a$ for the $V$1 map and $\Lambda$CDM simulations (again with tolerant selection).
\[B\]\[c\]\[.8\]\[0\][Angular scale $a$ \[$^\circ$\]]{} \[B\]\[c\]\[.8\]\[0\][$\Delta T_{rms}$ \[$\upmu
\text{K}$\]]{} ![The mean temperature fluctuation for different angular scales $a$ in 50 Gaussian $\Lambda$CDM simulations and the $V$1 map.[]{data-label="fig:sigmas_original"}](sigmas_original.eps "fig:"){width=".45\textwidth"}
We see that the deviations decrease when going to smaller scales. This is also suggested by the $C_\ell$ spectrum which is in good agreement with the $\Lambda$CDM fit for large $\ell$ which dominate on small scales. But still, @Monteserin07 find a too low [*CMB variance*]{} which approximately corresponds to $\Delta T_{rms}$ on scales even smaller than $1^\circ$.
For the results in Fig. \[fig:sigmas\_original\], we used the highest available HEALPix resolution 9 corresponding to $N_{pix}=12\times 512^2 = 3,145,728$ pixels in a map. As stated above, the plots are highly influenced by the beam function and noise. The beam function acts as an extra window function which suppresses the growth of $\Delta T_{rms}$ for small scales. The white noise instead leads to a diverging $1/a$ behavior on the smallest scales (with an effective pixel noise amplitude $\sigma_{pix}$ and the number of pixels $N_a = N_{pix}\times a^2/4\pi$ within a sector of scale $a$, the noise contribution will be $\Delta T_{rms}^\text{noise}=\sigma_{pix}/\sqrt{N_a}\propto 1/a$).
On large scales, the first multipoles of the $C_\ell$ spectrum play an important role (see, e.g., Fig. \[fig:Wls\]). It is therefore a natural idea to suspect the well-known quadrupole anomaly [@Hinshaw07] to be responsible for the observed discrepancies. We check this by repeating the analysis after removing the quadrupole from the $\Lambda$CDM simulations as well as the observed CMB map. The results, summarized in Table \[tab:remquadrupole\], confirm the influence of the quadrupole anomaly. Now, the fractions $p$ of Gaussian $\Lambda$CDM simulations reach $p=7.3\%$ for $a=6^\circ$. These numbers still do not show good agreement, but they are not statistically significant anymore.
Scale $a$ Fraction $p$
----------- -- -- -- --------------
$4^\circ$ $6.5\%$
$5^\circ$ $8.0\%$
$6^\circ$ $7.3\%$
$7^\circ$ $2.5\%$
$8^\circ$ $6.4\%$
: The fraction $p$ of $1000$ Gaussian $\Lambda$CDM simulations with a $\Delta T_{rms}$ smaller than found in the $V$ map on the angular scale $a$, after removing the quadrupole from the maps.
\[tab:remquadrupole\]
We now investigate whether there are still discrepancies if we compare the observed $V$ map with Gaussian simulations based on the original $C_\ell$ spectrum rather than the $\Lambda$CDM best fit. This tests whether the observed map is a typical Gaussian realization of the power spectrum. Figure \[fig:spots\_V\_measured\] shows the spot abundances.
\[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of hot spots]{} ![Spot abundances in the CMB sky (with cosmic variance) as compared to 500 simulations (with statistical errors) based on the $C_\ell$ spectrum on an angular scale of $a=6^\circ$. The fractions of Gaussian simulations with smaller values of $s$ \[Eq. (\[eq:s\])\] are $p_s^\text{hot} =
3.4\%$ and $p_s^\text{cold} = 13.2\%$.[]{data-label="fig:spots_V_measured"}](spots_V_measured_hot.eps "fig:"){width=".45\textwidth"} \[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of cold spots]{} ![Spot abundances in the CMB sky (with cosmic variance) as compared to 500 simulations (with statistical errors) based on the $C_\ell$ spectrum on an angular scale of $a=6^\circ$. The fractions of Gaussian simulations with smaller values of $s$ \[Eq. (\[eq:s\])\] are $p_s^\text{hot} =
3.4\%$ and $p_s^\text{cold} = 13.2\%$.[]{data-label="fig:spots_V_measured"}](spots_V_measured_cold.eps "fig:"){width=".45\textwidth"}
The effect arising from changing the power spectrum is clearly visible and reduces the discrepancies to some extent. But although closer to the spot abundances in the observed cut-sky CMB map, the numbers of hot and cold spots are still too high. Again, this is reflected in the fact that most simulated maps have a larger $\Delta
T_{rms}$ than the $V$ map. Even though the values, listed in Table \[tab:significance\_measured\], are less drastic, we emphasize that the estimation of the $C_\ell$ relies on similar data, i.e., cut-sky CMB maps. If the observed CMB map was a typical Gaussian realization of the extracted $C_\ell$ spectrum, we would expect agreement.
Scale $a$ Fraction $p$
----------- -- -- -- --------------
$4^\circ$ $4.2\%$
$5^\circ$ $5.4\%$
$6^\circ$ $5.8\%$
$7^\circ$ $2.4\%$
$8^\circ$ $4.1\%$
: The fraction $p$ of 1000 Gaussian simulations ( $C_\ell$) with a $\Delta T_{rms}$ smaller than found in the $V$ map on the angular scale $a$.
\[tab:significance\_measured\]
Bearing in mind, however, that power spectra refer to the full sky whereas we only look at regions outside the mask, an explanation could be that the missing spots were located in the hidden part of the sky. In the next section, we investigate whether the ILC map indicates this violation of isotropy.
ILC full-sky map {#sec:ilc}
----------------
The five-year ILC map is the best approximate full-sky CMB map available. We therefore analyze it even though the quality of the reconstruction is not high enough to guarantee robustness of the results (see, also, Sec. \[sec:data\]). We analyze the ILC full-sky map and 100 Gaussian full-sky simulations based on the power spectrum and separately consider the results in three sky regions. First, we analyze the full sky. Second, we collect the spots of those regions that have also been studied in the $V$ map, i.e., regions with no or little overlap with the KQ75 mask (tolerant selection). Finally, we consider the remaining spots that consequently lie in sectors completely inside the mask or with considerable mask overlap (rejected by tolerant selection). We loosely refer to the three regions as [*full sky*]{}, [*outside*]{}, and [*inside mask*]{}. The results are plotted in Fig. \[fig:spots\_ILC\].
In the previously analyzed region (outside the mask), we see too few spots, as before. But there are by far too many spots in the complementary region. The variances providing the error bars are, as always, obtained from Eq. (\[eq:errors\_abundance\]). Although there are less statistics inside the mask than in the full sky, the error bars in the corresponding figure are smaller. This can be intuitively understood as follows. If, for simplicity, we assumed that the spot abundances outside and inside the mask were statistically independent, the variances $\sigma_{\text{in}}^2$, $\sigma_{\text{out}}^2$ would add to $\sigma_\text{full}^2$ in the full sky, whence $\sigma_{\text{in}} < \sigma_\text{full}$. The loss of statistics when counting spots inside the masked region only causes the [*relative*]{} fluctuations between two Gaussian simulations to increase. The error bars in the central figure (outside the mask) visually appear larger due to the logarithmic plotting but are in fact smaller than for the full sky.
The values of $p_s$ confirm the uneven distribution of spots in the ILC map. For the full sky, we have $p_s^\text{full} = 58\%$ in good agreement with the simulations. Outside the mask, there are too few spots, $p_s^\text{out} = 7\%$, whereas inside the mask, we find $p_s^\text{in} = 96\%$. The ILC map is clearly anisotropic. Other authors draw the same conclusion [@Hajian07; @Copi08; @Bernui08].
Anisotropy of the CMB is a possible explanation of the discrepancies revealed in Sec. \[sec:cut-sky\] and quantified in Table \[tab:significance\], and indeed, the ILC map contains this anisotropy. But since there is not enough reliable information about the CMB signal in the galactic plane, we cannot finally judge whether this is the true solution to the problem. We have also studied if, additional to the galactic plane, the orientation of the galactic halo defines a preferred direction. Therefore, we divided the ILC map into two halves, one around the galactic center and one covering the opposite direction. We have seen no signal of anisotropy in this direction.
Modified power spectra {#sec:modified}
----------------------
We have pointed out that anisotropy is a potential explanation. It is however unsatisfying to assume that so many additional spots lie in the contaminated regions hidden by the KQ75 mask. This would be a surprising coincidence of CMB signals and the orientation of the galactic plane. Alternatively, we may stick to statistical isotropy; then, our results may be due to some non-Gaussian signal.
In this section, we investigate whether our results imply non-Gaussianity or statistical anisotropy by themselves. We do this by analyzing the effect of modifications to the $C_\ell$ spectrum.
So far, $\Delta T_{rms}$ has proved to be a good parameter to quantify the visible effects. We can perform a quick check whether our data supports the hypothesis that $\Delta T_{rms}$ is the decisive parameter. Out of the 500 simulations with the power spectrum used in Sec. \[sec:cut-sky\], we collect those with a $\Delta T_{rms}$ smaller or equal than those found in the $V$ map. Figure \[fig:conform\] shows their spot abundances which agree well with the $V$ map.
If there is a $C_\ell$ spectrum that produces $\Delta T_{rms}$ values similar to the ones found in the $V$ map, our results alone do not imply non-Gaussianity or statistical anisotropy. In order to keep the analysis as generic as possible, we do not use any specific cosmological model but only modify the $C_\ell$ of the original spectrum. Figure \[fig:sigmas\_original\] suggests that only large scales are affected which is why we concentrate on a few low multipoles $\ell$. @Copi08 found that the correlation function is essentially zero on angular scales above $\approx 60^\circ$. Since this scale is roughly linked to the multipole range $\ell \le 3$, our first modification simply consists in setting $C_\ell\equiv 0$ for $\ell \le 3$ (although of course the correlation function does not translate this easily). Another example may be to halve the $C_\ell$ for $\ell \le 5$ (modification II). Figure \[fig:modified\] shows the resulting values of $\Delta
T_{rms}$.
The plots show the discrepancies between the $\Lambda$CDM prediction, the spectrum, and observation. We also show the results for a combined power spectrum, replacing the first $32$ multipoles by the values quoted by [@Hinshaw03]. For this range of multipoles the WMAP analysis changed after the 1-year release, following the suggestion of @Efstathiou03. The difference between and may serve as an illustration that the extraction of reliable $C_\ell$ for low $\ell$ is a nontrivial task. Modifications I and II of the power spectrum succeeded in reconciling Gaussian simulations and observed CMB sky. This is confirmed by measuring the spot abundances in simulated maps based on the modified spectra, seen in Fig. \[fig:spots\_V\_modified\].
\[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of hot spots]{} {width=".45\textwidth"} \[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of hot spots]{} {width=".45\textwidth"} \[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of cold spots]{} {width=".45\textwidth"} \[B\]\[c\]\[.8\]\[0\][Threshold $\Delta\mathfrak T$ \[$\upmu \text{K}$\]]{} \[B\]\[c\]\[.8\]\[0\][Abundances of cold spots]{} {width=".45\textwidth"}
We conclude that our results are not incompatible with Gaussianity. However, if we stick to Gaussianity, they favor (although statistically not significant, cf. Table \[tab:significance\_measured\]) even lower values of the first multipoles than currently estimated.
Conclusions {#sec:discussion}
===========
The study of spot abundances has revealed discrepancies between the cut-sky CMB temperature maps and the standard best-fit $\Lambda$CDM model or, but less significant, a Gaussian spectrum for the $C_\ell$ estimated by . We have shown in Sec. \[sec:modified\] that a good parameter to quantify them is the mean temperature fluctuation $\Delta T_{rms}$ which we investigated on large scales. On scales $a$ between $4^\circ$ and $8^\circ$, only $0.16\%$ to $0.62\%$ of Gaussian simulations based on the $\Lambda$CDM best-fit power spectrum fall below the $\Delta T_{rms}$ value of the observed $V$ map. If this merely was an imprint of the anomalously low quadrupole, we would expect the discrepancies to disappear when removing the quadrupole from the Gaussian simulations and the $V$ map. The difference in fact reduces, the aforementioned fractions change to $2.5\%$ to $8.0\%$. These numbers are not significant and do not allow for a clear interpretation whether our results go beyond the quadrupole anomaly. Similar fractions are obtained when exchanging the $\Lambda$CDM best-fit spectrum by the originally published $C_\ell$, yielding $2.4\%$ to $5.8\%$. This is difficult to understand if we bear in mind that the $C_\ell$ themselves are estimated from the cut-sky CMB maps [@Nolta09].
Non-Gaussianity and also statistical anisotropy are possible explanations. In our case, anisotropy means that many spots have to be hidden behind the masked region. Unfortunately, this hypothesis can hardly be tested as there is currently no method to reliably extract the CMB signal in the highly foreground-contaminated regions. Nonetheless, we have employed the ILC full-sky CMB map and found evidence for anisotropy in this map. This agrees with results obtained by @Hajian07 and @Copi08 who found that most of the power on the largest scales comes from the (masked) galaxy region. Though possible, this unnatural alignment of the CMB signal with the galactic plane would be intriguing and lacks so far any explanation.
Our analysis of Sec. \[sec:modified\] shows that our results for cut-sky maps do not suggest non-Gaussianity or statistical anisotropy by themselves. They agree well with Gaussian fluctuations if one performs a modification of the lowest multipoles. In doing so, no fine-tuning of the $C_\ell$ is necessary in order to reconcile the spot abundances from Gaussian simulations and the observed CMB. It is sufficient to lower the first multipoles by a substantial amount. When studying local extrema in the temperature field, @Hou09 similarly found discrepancies that disappeared when excluding the first multipoles. We recall, however, that the $C_\ell$ and the assumption of Gaussianity completely fix the expected spot abundances. If both the extraction of the $C_\ell$ by and our analysis of spot abundances are correct, our results may indicate non-Gaussianity or statistical anisotropy.
If the discrepancies are not caused by mere statistical coincidence or unknown secondary effects, we have to leave open the question whether we see the consequence of non-Gaussianity or anisotropy, or whether our results strengthen the evidence for a severe lack of large-scale power. The first case would challenge fundamental assumptions, the second would make it difficult to understand the CMB maps on large scales within standard $\Lambda$CDM cosmology. If the discrepancies between the $C_\ell$, as determined by , and the spot abundances persist, this can be interpreted as a signal for non-Gaussian fluctuations.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Christian T. Byrnes for useful discussions. We also thank the WMAP team for producing great data products and publishing them on LAMBDA (the Legacy Archive for Microwave Background Data Analysis). Support for LAMBDA is provided by the NASA Office of Space Science. We acknowledge the use of the HEALPix package [@HEALPIX] that we employed for many tasks, most notably the creation and preparation of Gaussian simulations.
[^1]: http://cmbdata.gsfc.nasa.gov
[^2]: http://healpix.jpl.nasa.gov
|
---
abstract: 'The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.'
author:
- Amalaswintha Wolfsdorf
date: March 2006
title: |
A.M.D.G.\
Factorising Polynomials over Finite Fields
---
Preface
=======
The study of prime numbers has been puzzling Number Theorists for several centuries. Certainly since the remarkable results found by Pierre de Fermat in the $17^{th}$ century, a new wave of motivation has triggered some of the most talented mathematicians to research this field in more depth.
The field is vast, and possibly one of the most challenging ones: whilst the statement of a problem in this field may at first sound like a lunchtime brainteaser for a hobby - number-cruncher, its solution will in general be extremely complex and in many cases has taken centuries to find, if this has been achieved at all yet!\
But this field is not only of high importance to theoretical research. The most recent developments in this field have been concentrated on computational number theory. The fact that still so little is known about prime numbers, and that it is such a difficult field to make much progress in, has been exploited by the computer industry during the last century.
Secure transmission of data is made possible by prime numbers, and hence research in Number Theory is nowadays mainly revolved around finding ways to ensure that this level of security is maintained.\
At the heart of this lies the problem of finding roots of polynomials modulo prime numbers.
Whilst it is in theory *possible* to do this, the procedures that we know about so far are not very efficient and would in general take far too long to be of any practical use.\
This dissertation (unfortunately) does not provide us with a magic key to cracking such codes. I will show that there exists a deterministic algorithm that can, under certain circumstances, find the factors of polynomials modulo a prime number. However the running time of this algorithm is still much higher than some probabilistic (and fairly reliable!) algorithms that are in use already.\
The result that we will obtain here is hence rather of interest to mathematicians working in Algorithmic Number Theory than of practical use. Perhaps, however, similar techniques will eventually be developed that might be put to more use in practice. Perhaps the purely theoretical side of mathematics will find its applications in practice, and Albert Einstein will be proved wrong for his remark “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
Introduction {#ch:intro}
============
The idea for this project originates from a claim made by Dr Neeraj Kayal in 2005, together with some further refinements added by Prof Bjorn Poonen (University of California, Berkeley).\
A well-known open problem in Algorithmic Number Theory is the efficient calculation of roots of polynomials modulo a prime number $q$ in deterministic polynomial-time.\
A very basic example of this is the following: let $q$ be prime and $a$ and number between $0$ and $q-1$. The study of Elementary Number Theory provides us with easy tools to check whether there exists a number $b$ between $0$ and $q-1$ such that $b^2 = a$ (mod $q$) - in that case, $a$ is called a *quadratic residue*.\
For example, we could apply what is called “Euler’s Criterion”. It says that if $q$ is an odd prime, then for all $a \in \mathbb{N}$, we have
$$\bigg(\frac{a}{q}\bigg) \equiv a^{(q-1)/2}
\textrm{ (mod } q) \, ,$$
where the fraction on the left hand side denotes the Legendre Symbol, defined by
$$\bigg(\frac{a}{q}\bigg): = \left\{
\begin{array}{ll}
1 &
\textrm{if such a number $b$ exists} \\
-1 & \textrm{if no such $b$ exists} \\
0 & \textrm{if $q$ divides $a$}
\end{array} \right.$$
But how can we calculate this number $b$, if it exists? We would need to solve the equation $h(z):= z^2 - a \equiv 0$ modulo $q$. This is a much harder problem, if it is to be solved efficiently. Of course we could try substituting every value in $\{0,1,\dots,q-1\}$ for $z$ to check whether the equation is satisfied; however as we are in practical applications more concerned with large primes, this could take quite a while.\
The Claim
---------
Kayal claimed that we can factorise such a polynomial $h(z)$ defined over a finite field $\mathbb{F}_q$ using a deterministic algorithm with running time bounded by a universal polynomial in $(log \, q)$, given certain circumstances (Poonen’s input to this claim will be discussed later). Roughly speaking, the underlying condition is that we can construct an algebraic family of bivariate polynomials $C_z(X, Y)$, each member of which has a different number of solutions modulo $q$.\
I will restrict the detailed proof to the case $deg \, (h) = 2$. The case for higher degree polynomials will be discussed briefly afterwards.\
I will then also give a brief discussion about the running time of the algorithm.\
The idea of the proof
---------------------
The idea of the proof can be outlined as follows:\
We have a deterministic algorithm, known as “Schoof’s Algorithm”, which is used to compute the number of rational points on an elliptic curve given in Weierstrass Form and defined over a finite field.\
Let $h(z)$ denote the polynomial that we wish to factorise, and $\mathbb{F}_q$ the finite field over which $h$ is defined. We consider the ring $R:= \mathbb{F}_q[z]/(h(z))$.\
If we are given an elliptic curve $C$ over $R$ that satisfies the underlying condition, and we attempt to apply Schoof’s Algorithm to count the number of rational points on $C$, the algorithm will at some point break down, and thereby reveal the factors of $h(z)$.\
Now why does this happen?
Consider the difference between a ring and a field: a field contains all its inverses, which is not necessarily true for a ring. This is precisely why the algorithm will not work when it is working with a ring: whilst trying to compute the inverse of an element (which the algorithm can easily do when in a field), it will at some point not be able to find that inverse and will therefore stop running.\
At this point we know that it must have found an element of the ring that has no inverse. But by inspecting this ring $R$ more closely, we can see which elements in the ring do not have an inverse: it is precisely the set of elements in $\mathbb{F}_q[z]$ spanned by the factors of $h(z)$.\
So all we need to do is compute the greatest common divisor of this element that made the algorithm stop, and $h(z)$ (since this element may be a multiple of a factor of a factor), to obtain a non-trivial factor of $h(z)$!\
To present a detailed proof however requires a lot more careful explanation; this is what will follow now.\
Structure of this Paper
-----------------------
In Chapter (\[ch:ecs\]) of this dissertation, I will define elliptic curves and explain some of their elementary properties that we will need to be aware of in order to understand Schoof’s Algorithm.
Chapter (\[ch:counting\]) contains a brief discussion about counting rational points on elliptic curves, which is followed by a rather technical section explaining the essential tools that underlie Schoof’s Algorithm.\
I will give a full description of Schoof’s Algorithm for elliptic curves over a finite field in Chapter (\[ch:schoof\]).
For the purpose of a clear and thorough understanding of the theorem and its proof, I will then include a short chapter on elementary Ring Theory; it will be a collection of standard results that should only serve as a reference to the following chapter.\
A slightly simplified version of the actual assertion will finally be explained in Chapter (\[ch:theorem\]), together with a detailed proof. The next chapter will then explain how this simplified version differs from the original claim made by Kayal & Poonen, and what changes might be made to the proof in the previous chapter in order to adapt it to the “full version”.\
Finally I will, in chapter (\[ch:running\]), provide the reader with some background about algorithms and computations, and give a brief discussion about the running time of the algorithm.
Elliptic Curves {#ch:ecs}
===============
I will first of all state a few definitions and standard results from the study of elliptic curves. As some of the proofs require a few technical lemmas that are not directly relevant to this dissertation I will omit most of them; they are standard bookwork and can be found e.g. in [@sil3] and [@flynn] (N.B. those sources also provide the interested reader with a thorough insight into Elliptic Curves).
Preliminary Definitions
-----------------------
Throughout these definitions, we shall denote by $K$ some field.
\[section\]
$A_n(K) = \{(x_1,\dots,x_n): x_1,\dots,x_n \in K\}$, is called *affine n-space*.
When $P \in
A_n(K)$, we say that $P$ is *K-rational* or *defined over K*.
Let $P_n(K):=
\{(x_0,\dots,x_n): x_0,\dots,x_n \in K, \textrm{not all 0}\}$, subject to the relation that $(x_0,\dots,x_n) = (y_0,\dots,y_n) \in
P_n(K)$ if there exists $r \in K$, $r \ne 0$, such that $(y_0,\dots,y_n) = (rx_0,\dots,rx_n)$. $P_n(K)$ is called *projective n-space over K*.
A polynomial in $n$ projective variables is an *(n + 1)-variable homogeneous polynomial*.
A *projective curve in $P_2$* is defined by a homogeneous polynomial in 3 variables $F(X, Y, Z) = 0$.
Let $C: f(x,y) = 0$ be an affine curve and let $P =
(x_0,y_0)$ be a point on C. We say that P is a *singular point on C* if $$\frac{d}{dx} (f) |_P = \frac{d}{dy} (f) |_P = 0.$$
A curve is called *non-singular* if it does not contain any singular points.
Finally, we are in a position to unambiguously define elliptic curves:\
An *elliptic curve over a field K* is a non-singular, projective cubic curve, defined over $K$, with a $K$-rational point.
Let $C: f(x, y) = 0$ and $C': f(x, y) = 0$ be curves over $K$. A *rational map $\phi$* over $K$ from $C$ to $C'$ is a map given by a pair $\phi_1, \phi_2$ of rational functions in $(x, y)$, defined over $K$, with the property that given any point $P = (x_0,
y_0)$ on $C$, then $(\phi_1(x_0, y_0), \phi_2(x_0, y_0))$ lies on $C'$.\
If there also exists a rational map $\psi$ from $C'$ to $C$ such that $\psi\cdot\phi$ is the identity on $C$ and $\phi\cdot\psi$ is the identity on $C'$ then we say that $\phi$ is a *birational transformation* over $K$ from $C$ to $C'$, and that $C$ and $C'$ are *birationally equivalent* over $K$.
##### Remark about the terminology
An elliptic curve is not to be confused with an ellipse, which is a plane algebraic curve usually given in the form $$\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$$ for some non-zero constants $a, b$ in some field. There is however an explanation for the terminology.\
Consider the relationship between the trigonometric functions sine, cosine and tangent, and the arc lengths of a circle. The further study of elliptic curves shows that there is a similar relationship between elliptic curves and arc lengths on ellipses. These give rise to so-called elliptic integrals of the form
$$\label{eq:I} \int \frac{dx}{4x^3 + Ax + B}$$
\
Integrals like (\[eq:I\]) are multi-valued and only well-defined modulo a period lattice $L$. The “inverse” function of those integrals is a doubly periodic function called an elliptic function.\
In fact every such function $P$ with periods independent over $L$ satisfies an equation of the form $$P'^2 = 4 P^3 + AP
+ B$$
If we consider $(P, P')$ as a point in space then we can define a mapping from the solutions of this equation to the curve
$$Y^2 = X^3 + AX + B$$
This is the standard form for an elliptic curve that we shall be concerned with throughout this dissertation.
Arithmetic on Elliptic Curves
-----------------------------
Let $C$ be an elliptic curve over a field $K$. Let $o$ be its K-rational point. For any two points $a$, $b$ on $C$, denote by $l_{a,b}$ the line through $a$ and $b$; if $a=b$ then $l_{a,b}$ is defined to be the tangent to $C$ at $a = b$. Let $d$ be the third point of intersection of $l_{a,b}$ with $C$. Define $c$ to be the third point of intersection between $C$ and $l_{o,d}$, the line through $o$ and $d$. We then define $a + b := c$.\
Let $k$ be the third point of intersection between $C$ and $l_{o,o}$, the tangent to $C$ at $o$. Let $a'$ be the third point of intersection between $C$ and $l_{a,k}$. Define $-a:= a'$.
#### Comment
It is often convenient to write elliptic curves in affine form, although it should be understood that we always mean a projective curve. For example, $C: y^2 = x^3 + 1$ will be used as the shorthand notation for the projective curve $C: ZY^2 = X^3 + Z^3$.\
It can be shown that any elliptic curve over K can be birationally transformed over K to the Weierstrass form
$$\label{eq:complicated} C: y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$$
To further simplify the equation we can use the following theorem.
\[predef\][Theorem]{}
Let $K$ be a field with $char(K) \ne 2$. Then any elliptic curve over K is birationally equivalent over $K$ to a curve of the form $Y^2 =$ cubic in $x$. If $char(K) \ne 2 \textrm{ or } 3$ then we can further reduce (\[eq:complicated\]) to the form $$\label{eq:standard} Y^2 = X^3 + AX + B.$$
For the purpose of this dissertation, we shall only be concerned with elliptic curves over fields of characteristic $q > 3$, hence (\[eq:standard\]) will be treated as our standard equation for an elliptic curve. We will also adapt the convention to choose $o =
(0,
1, 0)$, the point at infinity.\
Note that $Z = 0$ meets $C$ at $o$ three times. Given $a = (x, y,
z)$, the third point of intersection between the curve and the line through $a$ and $o$ is $(x, -y, z)$, which must then be $-a$. This leads to the following result.
\[predef\][Lemma]{}
\[def:gp\] For an elliptic curve $C$ written in our standard form (\[eq:standard\]), we can simplify the formulae for addition and inverses of points on $C$ as follows:
- $-(x, y) := (x, -y)$\
- If $d:= (x_3, y_3)$, the third point of intersection of $C$ and $l_{a,b}$, then $a + b = (x_3, -y_3)$.
The Group Structure
-------------------
Now let us have a closer look at the rational points on an elliptic curve $C$. With the above definitions of addition and $o$ we can show the following:\
After a few computations it is easy to see that for all $a$ and $b$ on $C$ we have
- $a + b = b + a$ ,\
- $a + o = o + a = a$ , and\
- $a + (-a) = (-a) + a = o$ .
Moreover, further computations that involve a few technical lemmas, will reveal that for $c$ on $C$ we also have
- $(a + b) + c = a + (b + c) \, .$
From this we can deduce the following theorem:
Let $C$ be an elliptic curve over $K$. The points on $C$, together with the operation $a + b$ as defined in Lemma \[def:gp\], form a group. The point $o$ acts as the identity in this group, and inverses are given by $-a$ as in the definition above.
For a natural number $m \in \mathbb{N}$ we will from now on adapt the notation $[m]P:= P + P + \dots + P$ ($m$ times). This map is also known as the “multiplication-by-$m$-map” from the curve to itself. We can extend the definition of this to $m \in \mathbb{Z}$ by defining $[o]P:= 0$ and $[-m]P:= -[m]P$.\
So for example, if we have $P = (0, 1)$ on $C: Y^2 = X^3 +1$, then $-[2]P = -(P+P)$. Computing the tangent at $P$, we obtain the line $L: Y = 1$, and the “third point of intersection” of $C$ and $L$ being again $P$. So $[2]P = -P = (0, -1)$, and hence $-[2]P =P$.\
This map plays a central part in elliptic curve cryptography; its applications will later on be extremely useful in this dissertation.
Elliptic Curves over Finite Fields
----------------------------------
Now let us consider an elliptic curve $C$ over a finite field $\mathbb{F}_q$. Recall that we are only considering fields of characteristic $q>3$ here. The cases for $q = 2$ or $3$ are similar, and some of our computations and notations could be adapted to include those cases, too. However, for the entire purpose of this dissertation, those two cases will be irrelevant and we will therefore exclude them in all our computations.\
Consider the group of rational points on $C$ over $\mathbb{F}_q$. The following result should be immediately obvious:
Over a finite field $\mathbb{F}_q$, the number of rational points on an elliptic curve $C$ is finite.
We shall denote this number by $\#C(\mathbb{F}_q)$. It may be asked whether we can find out anything about this quantity. The answer to this is that we can indeed, and in fact the computation of this number lies right at the heart of the proof of the theorem.\
Before I give an in-depth discussion of how to compute the actual value of $\#C(\mathbb{F}_q)$, I will give an upper and lower bound on it, and define a few tools that we will later on need in our computations.
#### Discussion
Consider a straight line $L$ over $\mathbb{F}_q$, given by $L: Y =
aX + b$. What do we know about the number of points on $L$?\
For every possible value $x \in \mathbb{F}_q$, i.e. $x \in
{0,1,\dots,q-1}$, there exists exactly one solution in $\mathbb{F}_q$ for $y$, so we obtain $q$ rational points. Also, the point at infinity is always a rational point; in total we therefore have exactly $q + 1$ rational points on $L$.\
Now we can consider the number of points on a curve $C$ of the form $Y^2 = f(X)$ in a similar way: for each of the $q$ possible values for $X$ we have one of the three cases:\
- If $f(x)$ is a quadratic residue modulo $q$, we obtain *two* solutions for $Y$, namely $y = \pm \sqrt{f(x)}$;
- If $f(x)$ is a quadratic non-residue modulo $q$, we will have *no* solutions for Y;
- If $f(x) = 0$ we have precisely *one* solution for Y, namely $y = 0$.\
From elementary Number Theory we know that in $\mathbb{F}_q$, exactly half the values in $\{1,\dots,q-1\}$ are quadratic residues, so we would expect the number of rational points on $C$ to be roughly $q$ to represent the “fifty-fifty chance of $f(x)$ being a quadratic residue”, hence yielding $2$ solutions. We then add the point at infinity, and obtain as a rough estimate $q + 1$ rational points.
For a given curve $C$ over $\mathbb{F}_q$, the *trace of Frobenius $t$* is the quantity defined by the relation $\#C(\mathbb{F}_q) = q + 1 - t \, .$
$t$ can therefore be regarded as the“error term” in our estimate of $\#C(\mathbb{F}_q)$. The following theorem gives a bound on this error term:
\[hasse\]
$$| t | \leq 2 \sqrt{q}$$
A detailed proof of this can be found in [@sil1]. In Subsection \[subs:zeta\] of Chapter \[ch:orig\] in this paper we will see an alternative argument to deduce this.\
A map that should be well-known to anyone who has studied basic algebra and number theory is the Frobenius map. It has a very interesting property that has important applications in the study of elliptic curves, as we shall soon see.
The $q^{th}$-power Frobenius map $\phi$ defined on an elliptic curve $C$ over $\mathbb{F}_q$ maps points on $C$ to points on $C$ as follows:
$$\phi = \left\{ \begin{array}{ll} C(\mathbb{F}_q) \to &
C(\mathbb{F}_q) \\
(x, y) \, \mapsto & (x^q, y^q)\\
o \qquad \mapsto & o
\end{array} \right.$$
It is easily verified that $\phi$ is a group endomorphism for the group of rational points on $C$ over $\mathbb{F}_q$ and is therefore most commonly referred to as the Frobenius endomorphism.\
As mentioned above, a deeper study of $\phi$ will reveal several interesting results; the following property of $\phi$ is crucial to this dissertation, and deserves particular attention.
The Frobenius endomorphism has characteristic polynomial\
$$\chi (X): X^2 - tX + q \, .$$
#### Outline Proof
A full proof of this involves a lot of technical Lemmas; I will therefore only state the main idea of the proof.\
It relies on the fact that $$\#Ker(\phi - 1) =
deg(\phi - 1) = q + 1 - t \, .$$
From this we can deduce that if we take an integer $m \ge 1$ with $gcd(m, q) = 1$, then we have $$det(\phi_m) \equiv q \,(\textrm{mod }m), \qquad
tr(\phi_m) \equiv t\, (\textrm{mod }m) \, .$$
The details of this proof can be found in [@sil1].\
Clearly this is the same as writing
$$\label{eq:frob} \phi ^2
- [t]\phi + [q] = [o].$$
\[predef\][Corollary]{}\[cor:cru\]
Hence we have that for a point $P:= (x, y)$ on $C$: $$(x^{q^2}, y^{q^2}) - [t] (x^q, y^q) + [q] (x, y) = o.$$
Counting Rational Points on Elliptic Curves {#ch:counting}
===========================================
As mentioned in Chapter (\[ch:intro\]), the key to the proof of our assertion is part of an algorithm that reveals the factors of $h(z)$.\
Although this dissertation is not about the efficient computation of the number of rational points on elliptic curves, the algorithm that we will later on adapt is in its original form a point-counting algorithm for elliptic curves over finite fields. I will therefore give a brief introduction to such algorithms in general.\
As mentioned earlier, over a finite field the number of rational points on an elliptic curve is clearly finite. In Chapter (\[ch:ecs\]) we have seen an upper and lower bound for the number of such points.\
Now it may be asked if we can actually compute the precise number of rational points on a given curve. The answer is that we can indeed, and there are several methods that can be applied to do this.\
An explicit formula for the number of rational points on an elliptic curve $C: Y^2 = X^3 + AX + B$ over $\mathbb{F}_q$ is given by the following sum: $$\#C(\mathbb{F}_q) = 1 + \sum_{x \textrm{ mod } q}
\Bigg( \bigg(\frac{X^3 + AX + B}{p} \bigg) + 1 \Bigg).$$
where $(\frac{a}{p})$ denotes the Legendre Symbol.\
Computing $\#C(\mathbb{F}_q)$ this way takes $O(q^{1+\epsilon})$ bit operations [^1]; this is clearly not very practical when $q$ is a large prime number (which, in practical applications of our theorem, it usually will be!).\
Due to Rene Schoof however, we have a more efficient way of computing $\#C(\mathbb{F}_q)$. In his paper [@schoof], Schoof gives an explicit deterministic algorithm to compute the exact number of points on any given curve over a finite field.\
In the next chapter I will go into detail about this particular algorithm, but first we will need yet more technical tools in order to understand the algorithm better.\
As Corollary (\[cor:cru\]) suggests, the algorithm involves calculating coordinates of rational points of the form $[m]P$ where $P$ is a rational point and $m$ a positive integer. We will therefore need techniques to efficiently compute those coordinates.\
It should be clear that the coordinates of $P_1 + P_2$ are rational functions of the coordinates $(x_1, y_1)$ of $P_1$ and $(x_2, y_2)$ of $P_2$. By repetition of this calculation we can see that multiplication by $[m]$ given by $$(x,y) \mapsto [m](x, y)$$
can also be expressed in terms of rational functions in $x$ and $y$. Explicitly, we have the formulae given in the following section.
The Division Polynomials
------------------------
Let $C$ be an elliptic curve defined over a field $K$ and let $m$ be a positive integer. There exist polynomials $\psi_m, \theta_m, \omega_m \in K[x, y]$ such that for $P = (x, y) \in C(K)$ with $[m]P \ne o$ we have $$\label{eq:mP} [m]P = \bigg( \frac{\theta_m(x, y)}{\psi_m(x, y)^2},
\frac{\omega_m(x, y)}{\psi_m(x, y)^3} \bigg)$$
The polynomial $\psi_m$ is generally referred to as the *$m^{th}$ Division Polynomial* of $C$. $\theta_m$ and $\omega_m$ can both be expressed in terms of $\psi_m$ as shown in the explicit recursive expressions for $\psi_m$ below.
#### Remark
The expressions for $\psi_m$ that I will give are simplified for the case where we can write our curve in the form $C: Y^2 = X^3 + AX +
B$. They are given in a more general form in [@blake] for the general curve
$$C: Y^2 + a_1 XY + a_3 Y = X^3 + a_2X^2 + a_4 X + a_6 \,
,$$
which could also be defined over fields of characteristic 2 or 3.
### Explicit Expressions for $\psi$ {#fdef}
Let $C: Y^2 = X^3 + AX + B$ be defined over $K$.\
Then $\psi_m$ can be computed as follows:
$$\begin{aligned}
\lefteqn{\psi_0 = 0, \psi_1 = 1, \psi_2 = 2y, {}} \nonumber \\
& & {} \psi_3 = 3x^4 + 6Ax^2 + 12Bx - A^2, {} \nonumber \\
& & {}\psi_4 =
4y(x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - 8B^2 - A^3), {} \\
& & {}\psi_{2m+1} =
\psi_{m+2} \psi_m^3 - \psi_{m-1} \psi_{m+1}^3, \, m \ge 2, {} \nonumber \\
& & {}\psi_{2m} = \frac{(\psi_{m+2} \psi_{m-1}^2 - \psi_{m-2}
\psi_{m+1}) \psi_m}{2y}, \, m > 2 {} \nonumber
\end{aligned}$$
We can now in turn define $\theta_m$ and $\omega_m$ in terms of the division polynomials: $$\begin{aligned}
\lefteqn{\theta_m = x \psi_m^2 - \psi_{m-1} \psi_{m+1} {}}\\
& & {}\omega_m = \frac{\psi_{2m}}{2 \psi_m}
\end{aligned}$$
Finally, we define $$\label{eq:f} f_m := \left\{ \begin{array}{ll}
\psi_m & m\textrm{ odd} \\
\psi_m / (2y) & m\textrm{ even} \end{array} \right.$$
The proof of these formulae involves straightforward but lengthy calculations and will therefore be omitted; some more detail is included in [@lang1]. It is however important to note the following two facts:
Let $f_m$ be defined as in (\[eq:f\]). Then
1. $f_m$ is a polynomial in $x$ only.
2. \[degf\] The degree of $f_m$ is at most $(m^2 -1)/2$ if $m$ is odd,\
and at most $(m^2 - 4)/2$ if $m$ is even.
The latter fact will be relevant in Chapter \[ch:running\] when calculating the running time of the algorithm.
The $m$-Torsion Subgroup
------------------------
Clearly, when $K$ is a finite field, $\mathbb{F}_q$ for some prime $q$, then $C(K)$ is a torsion group, i.e. every point on the curve has finite order (since $C(K)$ itself is finite). For a non-negative integer $m$, the set of *m-Torsion points* on $C(K)$ is defined by $$C[m]:= \{P \in C(K) | [m]P
= o\}.$$
We can now also express Corollary \[cor:cru\] in terms of elements of this subgroup:
For points of order $m$ on $C$, i.e. $P = (x,y) \in C[m]$, we have
$$\phi_m^2(P) - [\tau]\phi_m(P) + [k](P) = [o](P) = o \, ,$$
where we define $\tau \equiv t$ (mod $m$) and $k \equiv q$ (mod $m$).
It is easily verified that this is a subgroup of $C(K)$. By definition, $o \in C[m]$. The $m^{th}$ division polynomial $\psi_m$ characterises the $m$-Torsion subgroup as stated in the following theorem.
Let $P \in C(K) \backslash {o}$, and let $m \ge 1$. Then $$P \in C[m] \Leftrightarrow \psi_m(P)
= 0.$$
Clearly, this condition is equivalent to the following corollary, which is more useful for our computations later:
Let $P = (x, y) \in C(K) \backslash \{o\}$ be such that $[2]P \ne o$ and let $m \ge 2$. Then $$\label{le:tor} P \in C[m] \Leftrightarrow f_m(x) =
0.$$
The 2-torsion points are excluded in this, since they satisfy $\psi_2{P} = 0$, which we need to divide $\psi_m$ by in order to obtain $f_m$ if $m$ is even. However, we can immediately recognise points of order 2 due to the fact that their $y$-coordinate is always equal to zero (the reader may check this as an easy exercise to become familiar with the arithmetic on elliptic curves).\
To finish this rather technical section off, I will give an explicit expression for $[m]P$, which is again just a straightforward transformation of (\[eq:mP\]):
$$\label{eq:mP2}
[m]P = \bigg(x - \frac{\psi_{m-1} \psi_{m+1}}{\psi_m^2},
\frac{\psi_{m+2} \psi_{m-1}^2 - \psi_{m-2} \psi_{m+1}^2}{4y
\psi_m^3}\bigg)$$
In the actual application of this result, $\psi_m$ will be replaced by $f_m$ so that $[m]P$ is a rational function of $x$ only. We will show this explicitly later.\
In the following chapter I will give a detailed explanation of the deterministic algorithm that provides us with an efficient method to count the number of rational points on a given elliptic curve over a finite field.
Schoof’s Algorithm {#ch:schoof}
==================
Schoof’s Algorithm (published in April 1985) provides us with a tool to compute $\#C(K)$, where $C$ is an elliptic curve given in Weierstrass form, and $K = \mathbb{F}_q$ is a finite field. The algorithm takes $O((\log q)^8)$ elementary operations and is deterministic [^2]; it does not depend on any unproved hypotheses. As usual, we will restrict ourselves to the case where $char(K) \ne 2$ or $3$; those cases again need separate treatment, however as mentioned before, they are irrelevant for our purposes.\
I will first of all list the main steps of the algorithm, so that the reader can refer to them when working through the following section. Note that the purpose of some of those steps may not seem immediately obvious, and some notation may be unfamiliar, but the details will of course be filled in afterwards.
Outline
-------
`INPUT: An elliptic curve C: Y^2 = X^3 + AX + B defined over \mathbb{F}_q where q is a prime \ne 2 or 3.`
Explanation {#expl}
-----------
Now the above looks very abstract and clearly requires explanation.\
Note that only in the last step we are concerned with $\#C(\mathbb{F}_q)$. The algorithm actually computes the trace of Frobenius; this is clearly equivalent to computing $\#C(\mathbb{F}_q)$ due to the one-to-one correspondence between the two quantities, $\#C(\mathbb{F}_q) = q + 1 - t$.\
Since Hasse’s Theorem \[hasse\] provides us with a bound on $t$, it will be sufficient to compute $t$ modulo a sufficiently large number of primes and then recover the value of $t$ by an application of the Chinese Remainder Theorem.\
I will explain the algorithm for $\tau \equiv 0$ in thorough detail first; the case for $\tau \in \{1,\dots,(l-1)/2\}$ will then only be outlined. It follows the same idea, but involves computing slightly more complicated polynomials.\
Since the algorithm is in its abstract form very technical and hence somewhat difficult to follow, I will, as an example of its application, demonstrate each step by performing it on the elliptic curve $C: Y^2 = X^3 +1$ over the finite field $\mathbb{F}_5$.\
Note that this example is almost trivial, since over $\mathbb{F}_5$ we can find the number of points by inspection rather easily.
The example will however also show that the computations over such small fields already involve very complicated looking polynomials. In practice, we would apply Schoof’s Algorithm to fields of characteristic a large prime. [^3]\
We begin by defining $l_{max}$ to be the smallest prime such that $$\label{eq:defl} \prod_{\substack{l \textrm{ prime} \\ 2\le l \le l_{max}}} l
> 4 \cdot \sqrt{q} \, .$$
This bound is sufficient for us to obtain enough values of $\tau$ (mod $l$) to recover $t$ using the Chinese Remainder Theorem.\
We know from (\[eq:frob\]) that the trace of Frobenius $t$ satisfies $$\phi ^2 + q = t \phi,$$ so if we reduce this equation modulo $l$ we have $$\label{eq:tau} \phi _l ^2 + k = \tau \phi_l \textrm{
, where } \tau \equiv t \textrm{ mod }l, \, k \equiv q \textrm{ mod
} l$$
for all points on $C$ of order $l$, i.e. for $P = (x, y) \in C[l]$.
In order to find $\tau$ we need to check for which $\tau \in
\{0,1,\dots,l\}$ the relation (\[eq:tau\]) holds. To do this, we will test for $\tau \in \{0,1,\dots,(l-1)/2\}$ whether a point $P = (x, y)$ exists in $C[l]$ such that $$\label{eq:t} \phi_l^2 P + [k]P = \pm [\tau] \phi P \, .$$
By applying (\[eq:mP2\]), we can see that this is equivalent to testing for which $\tau$ we have $$\begin{aligned}
\label{eq:expl} (x^{q^2}, y^{q^2}) + \Big(x - \frac{\psi_{q-1}
\psi_{q+1}}{\psi_q^2}, \frac{\psi_{q+2} \psi_{q-1}^2 - \psi_{q-2}
\psi_{q+1}^2}{4y \psi_q^3}\Big) \nonumber \\
\nonumber \\
= \left\{
\begin{array}{ll} 0 &
\textrm{if } \tau \equiv 0 \textrm{ (mod l)}\\
\bigg(x^q - \Big(\frac{\psi_{\tau - 1} \psi_{\tau + 1}}{\psi_\tau
^2}\Big)^q, \Big(\frac{\psi_{\tau + 2} \psi_{\tau - 1} ^2 -
\psi_{\tau - 2} \psi_{\tau + 1}^2}{4y \psi_\tau ^3} \Big)^q \bigg) &
otherwise. \end{array} \right. \end{aligned}$$
So let us now run through the algorithm to see what happens at each step. First of all we compute $l_{max}$.
### Step (\[step2\])
We set $l = 3, \tau = 0$. We then test whether there exists a point in $C[l]$ such that $\phi_l^2P = \pm [k]P$.\
Comparing the x-coordinates of both sides in (\[eq:expl\]), we can see that this holds if and only if $$x^{q^2} = x - \frac{\psi_{k-1}\psi_{k+1}}{\psi_k^2}$$.
In order to obtain a univariate polynomial in $x$ only, we replace the $\psi_n$ by $f_n$ and multiply through by the denominator. Let us now define $H_{k,0}(x)$ as follows: $$\label{eq:H0} H_{k,0}(x) : = \left\{
\begin{array}{ll}
(x^{q^2} - x) f_k^2(x)(x^3 + Ax + B) + f_{k-1}(x)f_{k+1}(x) &
(k \textrm{ even)} \\
(x^{q^2} - x)f_k^2(x) + f_{k-1}(x)f_{k+1}(x)(x^3 + Ax + B) &
(k \textrm{ odd).}
\end{array} \right.$$
### Step (\[gcd\])
We have now reduced the problem of testing whether relation (\[eq:tau\]) holds for $\tau \equiv 0$ (mod $l$) to checking whether there exists a $P = (x, y) \in C[l]$ such that $H_{k,0}(x) = 0$.\
Although it may not seem immediately obvious, why this is any easier than the original problem, it is indeed a simplification: We recall from (\[le:tor\]) that $$P = (x, y) \in C[l]
\Leftrightarrow [l]P = 0 \Leftrightarrow f_l(x) = 0 \,
.$$ On the other hand we know that if our chosen $\tau$ is indeed the trace of Frobenius, then for all such $x$, we have $H_{k, \tau}(x) = 0$.\
From this we deduce that all roots of $f_l$ are also roots of $H_{k, \tau}$, and the two polynomials therefore have a non-trivial greatest common divisor.\
So rather than attempting to solve the equation $H_{k,0}(x) = 0$, we only need to compute the greatest common divisor of $H_{k,\tau}$ and $f_l$; this explains step (\[gcd\]) of the algorithm. Note here that in order to compute the greatest common divisor we use the Euclidean Algorithm, which the reader should be familiar with; it is briefly outlined in the next chapter.\
Now consider the case where $gcd(H_{k,0}, f_l) = 1$. This happens if and only if $H_{k,0}$ and $f_l$ have no roots in common. In that case we have $H_{k,0}(x) \ne 0 \textrm{ for any }P = (x,y) \in C[l]$ and so we conclude that there exists no $P \in C[l]$ such that (\[eq:tau\]) holds. Clearly this means that $t \ne \tau$ (mod $l$), and we go to step (\[nextt\]) to set $\tau = \tau + 1$ and try again for this new value of $\tau$.[^4] I will return to this case later.\
If, on the other hand, the greatest common divisor is non-trivial, then we know that if we have a point $P$ in $C[l]$, it will necessarily satisfy the desired property (\[eq:tau\]).\
\
#### gcd $\ne 1, \, \tau =0$
Now there are two “subcases” to be considered: Namely when $\phi_l^2 P = [-q]P$ and when $\phi_l^2 P = [+q]P$. We now run through a “sub-algorithm” for the case $\tau = 0$.
- Test whether $\phi_l^2P = - [k]P$ or $\phi_l^2P = + [k]P$ by checking the $y$-coordinate in a similar way.\
- If $\phi_l^2P = - [k]P$, go to step (\[output\]) of the main algorithm.\
- If $\phi_l^2P = + [k]P$, test whether $q$ is a square modulo $l$\
- If $\big(\frac{q}{l}\big) = - 1$, go to step (\[output\]) of the main algorithm.\
- If $\big(\frac{q}{l}\big) = 1$, let $\omega^2 = q$ (mod $l$) and test whether $\phi_l P = [\omega]P$ or $\phi_l P = -[\omega]P$, . Set $\omega_0 = \pm \omega$ accordingly. Set $\tau = 2 \omega_0$, go to step (\[output\]) of the main algorithm.\
Now why are we doing all this?\
##### Case 1
First assume that $\phi_l^2 P = [-k]P$. So we know that $\tau \phi_l P =0$, and since $\phi_l P \ne 0$, we can conclude that $t \equiv 0$ (mod $l$). So we proceed to Step (\[output\]) in the main algorithm and then run the algorithm for the next prime $l$.\
##### Case 2
On the other hand, if $\phi_l^2 P = [+k]P$, then $$(2q - \tau \phi_l)P = 0 \qquad \textrm{and so} \qquad \phi_lP
= \frac{2q}{t}P.$$ Let us apply $\phi_l$ to both sides and use the equality satisfied by $P$; so we get $$qP = \phi_l^2P = \phi_l\big(\frac{2q}{t}P\big) = \big(\frac{2q}{t}\big)^2P \,
,$$
and hence that $t^2 \equiv 4q$ (mod $l$). Again, we must split this into two subcases: When $q$ is a quadratic residue modulo $l$ and when it is not.
- $\big(\frac{q}{l}\big) = - 1$: In this case we can conclude that $\tau \equiv 0$ (mod $l$) and go to Step (\[output\]).
- $\big(\frac{q}{l}\big) = 1$: Let $0 < \omega < q-1$ denote a square root of $q$ modulo $l$. Since we have $(2q - \tau \phi_l)P =
0$, we can see that $2q/t$ is an eigenvalue of $\phi_l$; but $t/2 =
\pm \sqrt{q}$, so either $\sqrt{q}$ or $-\sqrt{q}$ is an eigenvalue of $\phi_l$. To test this, we proceed exactly as before in checking whether $\phi_lP = [\pm \sqrt{q}]P$. If we denote by $\omega$ the correct eigenvalue, we can finally set $\tau = 2 \omega$ and proceed to Step (\[output\]).
\
\
#### gcd $= 1$
Here we have that for no point $P \in C[l]$, relation (\[eq:t\]) is satisfied. From this we conclude that $t \ne \tau$ (mod $l$) and so we need to check whether the next value of $\tau$ is the trace of Frobenius mod $l$, i.e. whether $\phi_l^2 P + [k] P = \pm
[\tau]\phi_l P$ for $P\in C[l]$. So we go back to step (\[test\]) and compute $H_{k, \tau}$ for this new value of $\tau$. Referring to (\[eq:expl\]) again, we know that $$\label{eq:tx} \big(\phi_l^2P + kP\big)_X = x^{q^2} + x +
\frac{f_{k-1}f_{k+1}}{f_k^2} + \lambda^2 + \lambda ,$$
where $$\lambda =
\frac{(y^{q^2}+y+x)xf_{k}^3+f_{k-2}f_{k+1}^2
+(x^2+x+y)(f_{k-1}f_kf_{k+1})}{xf_k^3 (x+x^{q^2})+
xf_{k-1}f_kf_{k+1}}.$$
On the other side we have that
$$\big(\pm\tau\phi_lP\big)_X = x^q +
\bigg(\frac{f_{\tau+1}f_{\tau-1}}{f_\tau^2} \bigg)^q.$$
Now we can, in a similar way as above, transform the equation by reducing modulo the curve equation so that we have polynomials of degree at most one in $y$, since we can substitute $(x^3 + Ax + B)^m$ for any $y^{2m}$.\
We then obtain an equation of the form $a(x) - yb(x) =0$, hence $y =
a(x)/b(x)$ for some $a(x), b(x) \in \mathbb{F}_q(x)$. Again substituting for $y$ in the curve equation we therefore finally get $$y^2 = (x^3 + Ax + B) = \Big(\frac{a(x)}{b(x)}\Big)^2 \,
.$$
So we define $$\label{eq:Htau}
H_{k,\tau}:=a(x)^2 - (x^3 + Ax + B)b(x)^2,$$
a polynomial in $x$ only.\
Now we can proceed precisely as before: We want to check whether for $x$ such that $f_l(x) = 0$ we also have $H_{k,\tau} = 0$, i.e. whether the roots of $f_l$ are also roots of $H_{k,\tau}$.\
So in Step (\[gcd\]) we compute the greatest common divisor of $H_{k,\tau}$ and $f_l$.\
If the points on $C[l]$ do not satisfy the Frobenius relation and hence the greatest common divisor is 1, we conclude that this value of $\tau$ is also not the correct value.\
We are therefore sent to Step (\[nextt\]) to proceed to the next possible value of $\tau$ and then return to to Step (\[test\]), where we run the same test for that new value.\
Otherwise we have that $t \equiv \pm \tau (l)$ for our chosen $\tau$. In this case we need to check which sign is correct: We refer to (\[eq:expl\]) again, this time comparing the $y$-coordinates of both sides, and check in a similar manner which is the correct sign.\
### The final steps
This way we eventually obtain enough values for $t$ (mod $l$) so that we can finally proceed to Step (\[crt\]) and apply the Chinese Remainder Theorem to the pairs $(\tau, l)$. Finally we can calculate $\#C(\mathbb{F}_q)$, which completes the algorithm.\
\
\
\
The topic of point-counting algorithms, improvements of Schoof’s Algorithm and its applications is an extremely interesting and wide-ranging one; I refer the interested reader to [@blake] and [@kobl1] for the further study of this subject. Any deeper discussion about this subject is however irrelevant to this dissertation.
Some Ring Theory {#ch:ring}
================
For the purpose of a clearer understanding of the proof that will follow in the next chapter, we will need to recall some elementary theory about rings and fields.\
The following results should be known to the reader. I will therefore omit proofs to the assertions made; they should be regarded as a list of results that the reader may refer to in some steps of the proof of the theorem.\
Details of proofs can be found in e.g. [@cohn1], [@cohn2] or [@hers].
Elliptic Curves defined over a Ring
-----------------------------------
Let $h(z)$ in $\mathbb{F}_q[z]$ be a nonzero polynomial of degree 2 with distinct roots in $\mathbb{F}_q$, and consider a curve $C$ over the ring $R: = \mathbb{F}_q[z]/(h(z))$. We may view $C$ as a pair of curves over $\mathbb{F}_q$ as follows:\
$C$ has coefficients of the form $(r z + s)$, where $r, s \in
\mathbb{F}_q$.\
Since $R \cong \mathbb{F}_q \times \mathbb{F}_q$, we can apply the isomorphic map $(r z + s) \mapsto (r a + s, - r a + s)$, where $ \pm
a$ are the roots of $h(z)$ in $\mathbb{F}_q$, to the curve to obtain a pair of curves, both defined over $\mathbb{F}_q$. I.e.: $$\label{eq:C} C: Y^2 = X^3 + (\alpha z + \beta)X +
(\gamma z + \delta) \mapsto \left\{ \begin{array}{ll} C_{+}: & Y^2 =
X^3 + (\alpha
a + \beta)X + (\gamma a + \delta) \\
C_{-}: & Y^2 = X^3 + (-\alpha a + \beta)X + (-\gamma a + \delta)
\end{array} \right.$$\
Rings and Fields
----------------
The fundamental difference between a ring and a field is that in a ring we may have non-units. That is, we may have elements $r \in
R$, such that there exists no $s \in R$ with $r \cdot s = 1_R$.\
A *zero divisor* is an element $r \in R, r\ne 0$ such that there exists $s \in R \backslash\{0\}$ with $r\cdot s = 0_R$.
If $r \in R$ is a zero divisor, then it is a non-unit.
##### Example
For instance, in the ring $A = \mathbb{Z} / 4\mathbb{Z}$, we have that $2 \ne 0$, but $2 \cdot 2 = 4 = 0_A$. 2 is therefore a zero divisor. It is also a non-unit: there is no element $s \in A$ such that $s \cdot 2 = 1_A$.\
Although this should be obvious, the following result is worth some particular attention:
\[nonunits\] In $R \backslash \{0\} =
\big(\mathbb{F}_q[z] / (z^2 - a^2)\big) \, \backslash \{0\}$, the non-units are $(z - a)$ and $(z + a)$.
Euclid’s Polynomial Division Algorithm
--------------------------------------
Let us consider two univariate polynomials $f(x), g(x)$ defined over some field $F$. Euclid’s Polynomial Division Algorithm provides us with an efficient tool to compute the greatest common divisor of $f$ and $g$.
### Long Division of Polynomials
Recall from school how we divide polynomials: First we divide the leading term of the higher degree polynomial by the leading term of the lower degree polynomial. Now think about what “dividing” means: we try to find an element $a$ such that $a \cdot b = c$, where $b$ is the leading coefficient of the lower degree polynomial and $c$ that of the higher degree polynomial. All this should of course be clear, but as it will be crucial later on, it is again worth noting down the following result:\
\[division\] We have $a = c\cdot b^{-1}$.\
Hence in order to find $a$, we compute the inverse of $b$ and premultiply it by $c$.
### Euclid’s Algorithm
This is just a brief outline of the algorithm. Details can be found in any undergraduate book on linear algebra, e.g. [@cohn1], [@cohn2] or [@hers].
#### Proposition
For $f(x), g(x) \in F[X]$ with $0<deg(g)<deg(f)$, there exist $r_1(x), q_0(x) \in F[X]$ such that we can write\
$f(x) = g(x)q_0(x) + r_1(x)$, with $deg(r_1) < deg(f) \textrm{ or } r_1 \equiv 0$.\
The polynomials $r_1(x)$ and $q_0(x)$ are computed by long division of polynomials. As the next step in the Algorithm, we define a sequence $r_i(x)$ as follows:\
$r_0(x) = g(x)$\
$r_i(x) = r_{i+1}(x)q_{i+1}(x) + r_{i+2}(x) \textrm{ with } deg(r_{i+2})<deg(r_{i+1}) \textrm{ or } r_{i+2} = 0$ .\
We eventually obtain\
$r_{n-1}(x) = r_{n}(x)q_{n}(x) + r_{n+1}(x), \qquad r_{n+1}(x) = 0, r_{n}(x) \ne 0$ .\
At this point the algorithm ends, and returns $r_{n}(x)$ as the greatest common divisor.\
Now we are finally ready to tackle the actual problem we are aiming to solve.
The Theorem {#ch:theorem}
===========
In this chapter I will discuss the theorem to be proved. First of all I will give the already simplified version of the theorem and prove it. The original statement of it is somewhat more complicated and requires a few more definitions; this will be discussed in Chapter \[ch:orig\].
Statement of the Theorem
------------------------
### The problem
The problem to be solved here is:\
Find a deterministic algorithm with
- .
### The hypothesis
We are given a polynomial $h(z)$ of degree 2 with roots in the finite field $\mathbb{F}_q$, where $q$ is a prime.\
Assume that there exists an elliptic curve $C: Y^2 = X^3 + AX + B$ over the ring $R: = \mathbb{F}_q[z]/(h(z))$ for which there exists a prime $l$, such that we have $\#C_+(\mathbb{F}_q) \ne \#C_-(\mathbb{F}_q)$ (mod $l$). $$\label{qu:hyp}$$
Given (\[qu:hyp\]), there exists an algorithm as in (\[eq:alg\]).
The Proof
---------
Let us define $h(z):= (z^2 - a^2) \in \mathbb{F}_q[z]$ (where we do not know $a$ but only $a^2$) and let $$C: Y^2 = X^3 + (\alpha z + \beta)X + (\gamma z + \delta)$$
over the ring $R$ as above. Let $l$ be the prime number that satisfies the hypothesis of the theorem, i.e. such that $\#C_{+a}(\mathbb{F}_q) \ne \#C_{-a}(\mathbb{F}_q)$ (mod $l$).
Let $t_+, t_-$ be the respective traces of Frobenius of $C_+$ and $C_-$. Then we have that $t_+ \ne t_-$ (mod $l$).\
The idea of the proof is that Schoof’s point counting algorithm *is* an algorithm that solves our problem of factorising $h(z)$. I claimed earlier that when we apply it to $C$, it will at some point reveal the factors of $h$.\
Schoof’s Algorithm is defined for elliptic curves over finite fields, whereas $C$ is defined over a ring. Note that if the underlying hypothesis for the theorem were not fulfilled, we could in general run the algorithm over curves defined over a ring without any problems.\
Running Schoof’s Algorithm over $C$ is equivalent to running it over $C_+$ and $C_-$ simultaneously. Every operation that we are performing on $C$ can be thought of as performing the same operations on $C_+$ and $C_-$ if we map $z$ to $\pm a$ accordingly.\
Assume that we are in Step (\[begin\]) of the algorithm with $l$, the prime number with the desired property. Checking every value of $\tau \in \{0,\dots,(l-1)/2\}$ to see whether it satisfies $\phi^2 -
\tau \phi + q = 0$ is hence the same as checking whether there exists a point $P^+ = (x^+, y^+)$ in $C_+[l]$ such that the relation $\phi^2 - \tau \phi + q = 0$ is satisfied, and whether for a point $P^- = (x^-, y^-)$ in $C_-[l]$, this equation holds.\
Let $f_l^+$ and $f_l^-$ denote the $l^{th}$ division polynomial on $C_+$ and $C_-$ respectively, and let $H^+_{k,\tau}$ and $H^-_{k,\tau}$ be as defined in (\[eq:Htau\]) for the two curves accordingly.\
Now, without loss of generality, we assume that $t_+ < t_-$. Consider Step (\[gcd\]) in the algorithm with $\tau = \tau_+
\equiv t_+$ (mod $l$). We compute $H^+_{k,\tau}$ and $H^-_{k,\tau}$. Since $\tau = \tau_+$, we will find that for all points $P^+ = (x^+, y^+)$ in $C_+[l]$, we have $H^+_{k,\tau}(x^+) = 0$.\
On the other hand however, since $\tau \ne \tau_- \equiv t_-$ (mod $l$), we know that *no* point in $C_-[l]$ satisfies $H^-_{k,\tau} = 0$.\
Now consider this step of the algorithm over $C$ itself. So we attempt to compute $gcd(H_{k,\tau}, f_l)$, as usual, using the Euclidean Algorithm.\
Suppose we are trying to divide some polynomial $r_i(x)$ by $r_{i+1}(x)$ where $deg(r_i) > deg(r_{i+1})$. I will now make the following claim:
\[predef\][Proposition]{}
\[finalprop\]
The leading coefficient $c(z)$ of $r_{i+1}(x)$ is a non-unit in $R$, for some $i$.
\[finalcor\] Proposition \[finalprop\] completes the proof.
##### Proof of Corollary \[finalcor\]
If Proposition \[finalprop\] is true, then from Lemma \[division\], we know that we are trying to compute the inverse of $c(z)$. Since this is a non-unit in $R$, it has no inverse, and hence the algorithm “crashes”.\
Now, the non-units in $R \backslash \{0\}$ are $(z \pm a)$ as noted in Corollary \[nonunits\] (possibly multiplied by a constant in $\mathbb{F}_q)$.\
So if we compute $gcd(c(z), h(z))$, we obtain a non-trivial factor of $h(z)$, as required.\
##### Proof of Proposition \[finalprop\]
Imagine that the lower degree polynomial $r_{i+1}(x)$ *never* has leading coefficient a non-unit in $R$. The Euclidean Algorithm will just run smoothly over the ring as if it were a field.\
Now recall that everything we are doing with the curve $C$ is equivalent to performing the same operations on the pair of curves $C_+$ and $C_-$ simultaneously.\
Let us once more consider in detail the relationship between the polynomials $r_i$ for $C$ and the $r_i^+$ and $r_i^-$ for $C_+$ and $C_-$. The latter two are just evaluations of the coefficients of $r_i$ at $z = a$ and $z = -a$ respectively.\
So if we assume that the leading coefficient of $r_i$ is a unit for every $i = 0,\dots,n$ (where we have that $r_{n+1} = 0)$, then the leading coefficient of $r_i$ *never vanishes* on $C_+$ and $C_-$.\
This implies that the degree of the polynomials $r_i$ is the same as the degree of the $r_i^+$ and $r_i^-$. In particular, we note that for all $i = 0,\dots,n$, the degrees of $r_i^+$ and $r_i^-$ are the same.\
So finally, we conclude that the degree of $r_n^+$ is the same as the degree of $r_n^-$. But recall that $r_n^+$ and $r_n^-$ are the greatest common divisors of $H_{k,\tau}^+$ and $f_l^+$, and $H_{k,\tau}^-$ and $f_l^-$ respectively.\
By assumption however, the greatest common divisor of $H_{k,\tau}^-$ and $f_l^-$ is 1, since $\tau \ne \tau_-$ (mod $l$), whereas that of $H_{k,\tau}^+$ and $f_l^+$ is strictly non-trivial!\
This is clearly a contradiction.\
We can now see that at some point we must encounter a non-unit as the leading coefficient of some $r_i$.\
By Corollary \[finalcor\], this completes the proof.\
The original statement of the Theorem {#ch:orig}
=====================================
Polynomials of higher degree
----------------------------
As mentioned earlier, the full assertion made by Dr Kayal is slightly more advanced. Instead of restricting himself to polynomials of degree 2, he claimed that the assertion would hold for any polynomial with distinct roots in a finite field.\
On closer inspection, one can see that this is plausible, and that in fact the proof will be very similar to the one given above. One needs to think of an elliptic curve over the ring $$R:= \mathbb{F}_q[z] / (h(z)) = \mathbb{F}_q[z] /
\big((z-\alpha_1)(z-\alpha_2)(\dots)(z-\alpha_n)\big)$$
as a family of curves over $\mathbb{F}_q$ in the same way as above, i.e. with $z$ evaluated at $\alpha_i$ for each “subcurve” $C_i$.\
If we are then given a curve $C$ over this ring, such that for some prime $l$ the number of rational points on $C_i$ is not congruent to the number of rational points on $C_j$ modulo $l$, for some $i < j$, the same problem as discussed above will arise in Schoof’s Algorithm.\
Let $\tau_i$ be the trace of Frobenius of $C_i$ modulo $l$, which is hence not equivalent to the trace of Frobenius of $C_j$. Adapting a similar notation as before, and using the same arguments, we can deduce that $gcd(H_{k,\tau_i}^{(i)}, f_l^{(i)})$ is strictly non-trivial, whereas $gcd(H_{k,\tau_i}^{(j)}, f_l^{(j)}) = 1$ .\
When computing $gcd(H_{k,\tau_i}, f_l)$, on $C$, the Euclidean Algorithm will again break down in an attempt to compute the inverse of a non-unit in $R$, which we will inevitably encounter as the leading coefficient of some $r_i$. The reason for this is precisely the same as in the case for $deg(h) = 2$: if this never happened, then we would be able to conclude from this fact that the greatest common divisors $gcd(H_{k,\tau_i}^{(i)}, f_l^{(i)})$ and $gcd(H_{k,\tau_i}^{(j)}, f_l^{(j)})$ have the same degree.\
The above very brief outline of the proof already shows that a detailed proof of this version of the assertion would have involved a lot of careful, possibly confusing, notation (just imagine a detailed account of Schoof’s Algorithm with this notation!). It should be clear however, that the proof follows the same string of arguments.
Advanced topics
---------------
There are some further simplifications of the assertion that I have made. Some of the topics underlying the full claim made by Dr Kayal, and Prof Poonen’s addition to this, are somewhat too advanced to give a “brief” explanation of them before being able to prove the theorem. Details of such topics however can be found e.g. in [@poonen], [@sil1] and [@kobl1].\
In its original form, the assertion has the following underlying hypothesis:
> Let $h(t)$ in $\mathbb{Z}[t]$ be a nonzero polynomial, and let $C$ be a smooth projective curve of genus g over $\mathbb{Z}[t] /
> (h(t))$.
>
> The hypothesis is: There exists a $C$ as above such that for each sufficiently large primes $p$, the zeta functions of the curves $C_{p,\alpha_i}$ are distinct.
### Curves of Genus g
In this dissertation I have restricted myself to the case of very “simple” curves, i.e. elliptic curves, which are also often defined as non-singular curves of genus 1.\
Giving a detailed discussion about curves of higher genus would take us too far afield in this dissertation. As a brief description of “genus” however, I will just say that any curve $F(x, y) = 0$ has a non-negative integer $g$ associated with it; $g$ is referred to as the genus. In general (for example if the curve is non-singular), the genus increases as the degree of $F$ increases.\
For curves of the form $Y^2 = F(X)$, we have that $deg(F) = 2g +1$. Hence in our case the genus is 1, and a curve of the form $Y^2 = X^5
+ a_4X^4 + \dots + a_0$ has genus 2.\
The addition that Prof Poonen made to Dr Kayal’s initial claim is in fact to do with such curves of genus greater or equal to 2.\
In trying to find “applicable” curves for this problem, he remarked that if one used a curve of higher genus, the probability of the hypothesis being fulfilled would be much higher than for elliptic curves.\
In fact, he claimed that the probability of finding an elliptic curve over a ring $\mathbb{F}_q[z]/(h(z))$, for which the fibres at each root $\alpha_i$ (that is, the “subcurves” $C_i$ for all $i$) have the same number of rational points modulo some prime number $l$, is of order $1/q$.\
For curves of higher genus, that probability is, according to Prof Poonen, much smaller - in fact it is of order $1/q^2$. This is of course an important result if one tries to find curves to apply this theorem to.\
We can however find suitable elliptic curves, too, that fulfil our hypothesis. For example, consider the curve $$C: Y^2 = X^3 + zX \textrm{ over } R =
\mathbb{F}_5[z]/(z^2-1) \, .$$
The subcurves, i.e. $C$ evaluated at the roots of $(z^2 - 1) =
(z+1)(z-1)$ are given by $$C_+: Y^2 = X^3 + X \textrm{ and } C_-: Y^2 = X^3
- X \, ,$$ both defined over $\mathbb{F}_5$.\
By inspection, we can see that the points in $C_+(\mathbb{F}_5)$ are $$\{o, (0,0),(2,0),(3,0)\},$$ so there are 4 of them.\
On the other hand, $C_-(\mathbb{F}_5)$ has the points $$\{o, (0,0), (1,0), (2, 1), (2, -1), (3, 2), (3, -2), (4,
0)\}$$ - a set of 8 rational points!\
Clearly, $4 \ne 8$ (mod $3$). So this curve satisfies our hypothesis and could be used in applications of the theorem (although of course it would be a fairly trivial and pointless example).\
Now in order to show that what we have proved above is (almost) the same as the original assertion by Dr Kayal, we will just need to understand what the “Zeta function of $C$” is.
### The Zeta function {#subs:zeta}
Let $C$ be a curve defined over $\mathbb{F}_q$. Clearly if $C$ is defined over $\mathbb{F}_q$ then it is also defined over $\mathbb{F}_{q^n}$ for all $n \ge 1$. It may therefore be interesting to consider $$N_n=\#C(\mathbb{F}_{q^n})$$ for $n \ge 1$, i.e. the number of rational points on $C$ over $\mathbb{F}_{q^n}$.
Define the series $$\label{eq:zeta} Z(E;T) = \exp\bigg(\sum_{n \ge 1} \frac{N_n}{n} \cdot
T^n\bigg)$$ for an indeterminate T. This is called the *Zeta function* of C over $\mathbb{F}_q$.
Due to work by Hasse - and for a more general case extending to curves of genus higher than 1, by Weil - we can show that the Zeta function has a simpler form:
Let $C$ be a curve defined over $\mathbb{F}_q$. Denote by $c_n$ the trace of Frobenius of $C$ over $\mathbb{F}_{q^n}$, i.e. $c_n =
\#C(\mathbb{F}_{q^n}) - q^n - 1$. The Zeta function is a rational function of T and takes the form $$\label{eq:ZetaT}
Z(C;T) = \frac{P(T)}{(1-T)(1-qT)}$$ where $P(T) = 1 -
c_1T + qT^2 = (1 - \alpha)(1 - \bar{\alpha})$. Furthermore, the discriminant of $P(T)$ is non-positive and the magnitude of $\alpha$ is $\sqrt{q}$.
A proof of this theorem can be found e.g. in [@kobl2], for the case $g = 1$, or [@sil2], for curves of higher genus. Note in particular that the last line of the theorem implies that $c_1^2 <
4q$, which is Hasse’s Theorem.\
More importantly however, if we take the derivative of the logarithm of both sides in (\[eq:ZetaT\]), substituting in (\[eq:zeta\]) for the left hand side, we can show after some straightforward series manipulations and partial fraction expansions that this implies $$\label{eq:Nn} N_n = \#C(\mathbb{F}_{q^n}) = q^n + 1
- \alpha^n - \bar{\alpha}^n = |1 - \alpha^n|^2 \, .$$
Since both $c_1$ and $\alpha$ can be immediately derived from knowledge of $N_1$, we can uniquely determine $N_n$ for all $n \ge
1$ once we know the number of rational points of $C$ over the base field $\mathbb{F}_q$.\
It should be clear that this result is an extremely important one which has many useful applications not only in attempting to prove the above theorem. However, if we return to Kayal/Poonen’s claim, we can now simplify the proof of the theorem as follows:\
The underlying hypothesis for our factorisation is a different Zeta function for the fibres at $\alpha_i$ (mod $l$) and $\alpha_j$ (mod $l$) for some prime $l$. Now that we know that we can determine $N_n$ unambiguously from computing $N_1$, and since $Z(C;T)$ only depends on the $N_n$, it will be sufficient to use a curve for which the fibres at $\alpha_i$ and $\alpha_j$ have a different number of rational points over $\mathbb{F}_q$, modulo $l$. This is clearly what we have done above.
Running Time {#ch:running}
============
To finish this dissertation off, I will now give an account of the running time of the algorithm.\
Since algorithms and computations thereof is a rather broad mathematical subject on its own, and one which I assume the reader to be unfamiliar with, I will restrict this discussion to the key points. I recommend in particular [@bach] to the interested reader; it contains a comprehensive introduction to this subject, and will also fill in some details about the running time of the individual steps in our algorithm that I will omit.\
Introduction to running times
-----------------------------
In order to calculate the running time of an algorithm, we count the number of basic operations performed by the algorithm on the “worst-case input”. The *worst-case input* is the input for which the most basic operations are required. We count the basic operations as follows:
### Definition
Let $n \in \mathbb{Z}$. Define $$lg \, n:= \left\{
\begin{array}{ll}
1, & \textrm{if } n = 0; \\
1 + \lfloor log_2 |n| \rfloor , & \textrm{if } n \ne 0. \end{array}
\right.$$
Then $lg \, n$ counts the number of bits in the binary representation of $n$.\
A *step* is the fundamental unit of computation. Now, different situations require different units. For example, analysing a sorting algorithm would require counting the number of comparison steps, whereas in the case of an algorithm that computes the evaluation of a polynomial at a certain point we may want to count each addition, subtraction and multiplication as a single step.\
In general we therefore adapt the convention to equate “step” with “bit operation”: We write all integers in binary code, so we are only working with variables that take the values 0 or 1. We then perform logical operations on these variables: conjunction ($\land$), disjunction ($\lor$) and negation ($\sim$). Each of those operations takes 1 bit.\
The *running time*, or *cost of computation*, is then the total number of such logical operations performed in an algorithm. It depends on the size of the input.\
For example, the operation $a + b$ takes $lg \, a + lg \, b$ bit operations. However we usually just aim to find an upper bound of the running time, rather than an exact number; we therefore only note that the running time is $O(lg \, a + lg \, b)$ (where the $O$ is the “Big-Oh-notation”, which should be well-known to the reader).
\[runningtime\]
If we have a sequence of operations in an algorithm, say $P$ and $Q$, then we have that the running time of the algorithm “operation $P$ followed by operation $Q$” is\
$Time(P \, ; Q) = Time(P) + Time(Q)$.
Running time of our algorithm
-----------------------------
Lemma \[runningtime\] tells us that in order to compute the precise running time of the algorithm, we need to add up the running times required for each individual part of the algorithm.\
However, to find an upper bound of the running times, it will be sufficient to find an upper bound for the part of the algorithm that has the largest running time, as stated in the following Lemma:
\[bigolemma\] If $f(x) = O(g(x))$ then $f(x) + g(x) = O(g(x))$.
Let us now go through the steps of Schoof’s Algorithm, and analyse the amount of computations involved in each step. I will use several standard results for running times without proof; details can be found e.g. in [@bach].\
It is a well-known result that there exists a universal constant $C$ such that $$\prod_{\substack{l \textrm{ prime} \\ 2\le l \le l_{max}}} l > C \cdot
e^L.$$
for every $L>0$. For a proof of this, see e.g. [@ros].\
So we can take $l_{max}:= O(lg \, q)$. The number of primes occurring in the product is $O(lg \, q)$ and the primes $l$ themselves are clearly also $O(lg \, q)$.\
Now consider the running time involved in Step (2) of the algorithm. This step clearly requires the largest amount of computation, so that its running time will in the end dominate over the others.\
We now need to state some more results from complexity theory.
Let $f(x) \in R[X]$ for some ring $R$ with $|R| = p^m$. Define $$lg \, f:= \left\{
\begin{array}{ll}
1, & \textrm{if } f = 0; \\
(1 + deg (f)) lg \, |R|, & \textrm{if } f \ne 0.
\end{array} \right.$$
Let $f$, $g$ be polynomials in $R[X]$. Then
1. $f \pm g$ can be computed with $O(lg \,f + lg \, g)$ bit operations.
2. $f \cdot g$ can be computed using $O((lg \, f)(lg \, g))$ bit operations.
3. Computing the greatest common divisor of $f$ and $g$ also requires $O((lg \, f)(lg \, g))$ bit operations.
From Corollary \[fdef\], Part \[degf\], we have that $deg(f_l) =
O(l^2)$, and from above we know that $l_{max} = O(lg \, q)$.\
Computing the $H_{k, \tau}$ will involve computing $x^q, y^q,
x^{q^2}$ and $y^{q^2}$ (reduced modulo the curve equation) modulo $f_l$.\
For $x^q$ and $x^{q^2}$, this will require $O(lg \, q)$ multiplications in the ring each - hence $O((lg \, q)^2)$ together. Reducing modulo $f_l$ takes $O(l^2) = O((lg \, q)^2)$ bit operations, so the computation of $x^q$ and $x^{q^2}$ will require $O((lg \, q)^4)$ multiplications in the ring.\
Since the order of the ring in our case is $|\mathbb{F}_q| = O(lg \,
q)$, multiplication of any two elements in $R$ takes $O((lg \, q)^2)$ bit operations.\
We therefore need $O((lg \, q)^6)$ bit operations in total to compute $x^q$ and $x^{q^2}$. For $y^q$ and $y^{q^2}$, the computations are similar and hence their complexity will not affect the asymptotic upper bound.\
Now the $x^q, y^q, x^{q^2}$ and $y^{q^2}$ are computed once for each prime $l$, so $O(lg \, q)$ times, and then stay the same for each $\tau$. Now $\tau$ is also $O(lg \,q)$, so we have $O((lg \, q)^7)$ bit operations for each prime $l$.\
Finally, we have $O(lg \, q)$ primes $l$, so the complexity in the entire Step (2) of Schoof’s Algorithm amounts to $O((lg \, q)^8)$ bit operations.\
This does indeed dominate the computations of both $l_{max}$ and the Chinese Remainder Theorem in the last step, so we will not have to compute the complexities involved in those (we are not concerned with the latter anyway though, as in *our* algorithm, we will never get as far as computing the group order!).\
In fact, one can make improvements to find a slightly lower “upper bound” for the complexity, but let us finish this dissertation with the conclusion that our algorithm computes the factors of $h(z)$ over $\mathbb{F}_q$ using at most $O((lg \, q)^8)$ bit operations.\
If we convert our “Big-Oh-notation” to a polynomial, we can say that, indeed, the running time of the algorithm is bounded by a polynomial in $log \, q$, as asserted at the beginning of this dissertation.\
[99]{}
Bach & Shallit, *Algorithmic Number Theory*, Volume 1: Efficient Algorithms. The MIT Press, 1996.\
Blake, Seroussi & Smart, *Elliptic Curves in Cryptography*. CUP, 1999.\
Cohn, *Basic Algebra*. Springer Verlag, 2003.\
Cohn, *Classic Algebra*. Springer Verlag, 2000.\
Flynn, *Elliptic Curves HT 2005/06, Preliminary Reading*. Lecture Notes, www.maths.ox.ac.uk, 2005.\
Guy, *Unsolved Problems in Number Theory, $3^{rd}$ Edition*. Springer Verlag, 2004.\
Herstein, *Topics in Algebra, $2^{nd}$ Edition*. John Wiley & Sons, 1975.\
Knapp, *Elliptic Curves*. Princeton University Press, 1993.\
Koblitz, *A Course in Number Theory and Cryptography*. Springer Verlag, 1987.\
Koblitz, *Algebraic Aspects of Cryptography. 3, Algorithms and Computation in Mathematics*. Springer Verlag, 1998.\
Koblitz, *Introduction to Elliptic Curves and Modular Forms*. Springer Verlag, 1984.\
Lang, *Elliptic Curves: Diophantine Analysis*. Springer Verlag, 1978.\
Lang, *Introduction to Modular Forms*. Springer Verlag, 1976.\
Poonen, ‘Computational Aspects of Curves of Genus at least 2’, *Expository Articles*, Section V. http://math.berkeley.edu/ poonen/papers/ants2.pdf, 1996.\
Rosser & Schoenfeld, ‘Approximate Formulas for some Functions of Prime Numbers’, *Illinois Journal of Mathematics* 6, 1962.\
du Sautoy, *Music of the Primes*. Harper Perennial, 2004.\
Schoof, ‘Elliptic Curves over Finite Fields and the Computation of square roots mod $p$’, *Mathematics of Computation,* Vol. 44, No. 170, April 1985. (http://www.jstor.org/view/00255718/di970594/97p00836/0)\
Silverman, *Advanced Topics in the Arithmetic of Elliptic Curves*. Springer Verlag, 1994.\
Silverman, *The Arithmetic of Elliptic Curves*. Springer Verlag, 1986.\
Silverman & Tate, *Rational Points on Elliptic Curves*. Springer Verlag, 1992.
Acknowledgements
================
I would like to thank Dr Lauder, both for drawing my attention to the idea for this project, and for all his support throughout the last two terms.
I would also like to express my gratitude to Dr Flynn, who in his lecture course on Elliptic Curves has provided me with a strong background in the study of this field and whose ideas for my dissertation have been extremely helpful.\
The insightful email conversations with Prof Poonen have been invaluable, and I am very grateful for all his inspirations.\
The online guide to LaTeXprovided by the Mathematical Institute (written by Tobias Oetiker) has been a fantastic way for me to teach myself in a very limited amount of time how to use this language. Also the resources provided by the University Library Services and the Computing Services ought to be mentioned here - in terms of availability of books, software and general help, they have been extremely efficient.\
Last but certainly not least, I am extremely grateful to my friends for their “moral support”.\
In particular, I thank my parents for all their advice and care. Without it, the final few weeks would have been unthinkable.
*Thank you.*\
L.D.S.
[^1]: For a definition of “bit operations”, see Chapter (\[ch:running\]).
[^2]: Note that in his paper, Schoof shows that the running time of his algorithm is $O((\log q)^9)$; it can however be shown that one can make improvements on this bound. This will be discussed in more depth in Chapter \[ch:running\].
[^3]: In order to avoid confusion I will print the example in blue so that the reader can easily distinguish more easily between “theory” and “practice”, since I will often skip between the two.
[^4]: Note here that we never hit $\tau + 1 > (l-1)/2$ as we know that *exactly* one $\tau \in \{0,\dots,(l-1)/2\}$ will satisfy (\[eq:tau\]) and so $(l-1)/2$ is the largest value that $|\tau|$ can take. Once we have hit this value we know it is the correct solution and we find ourselves in Step (\[gcd\]).ii, from which we proceed to (\[nextt\]) straight away.
|
---
abstract: 'The effects of chemical disorder on the electronic and optical properties of semiconductor alloy multilayers are studied based on the tight-binding theory and single-site coherent potential approximation. Due to the quantum confinement of the system, the electronic spectrum breaks into a set of subbands and the electronic density of states and hence the optical absorption spectrum become layer-dependent. We find that, the values of absorption depend on the alloy concentration, the strength of disorder, and the layer number. The absorption spectrum in all layers is broadened because of the influence of disorder and in the case of strong disorder regime, two optical absorption bands appear. In the process of absorption, most of the photon energy is absorbed by the interior layers of the system. The results may be useful for the development of optoelectronic nanodevices.'
author:
- Alireza Saffarzadeh
- Leili Gharaee
title: Optical absorption spectrum in disordered semiconductor multilayers
---
Introduction
============
Low-dimensional quantum structures, such as quantum wells and superlattices, wires and dots have attracted great interest in the last few years, due to their possible application in nanodevices [@Harrison]. Among these structures, multilayers and superlattices, with thickness of few nanometers, are of considerable experimental and theoretical interest because of their specific electronic and optical properties and many promising areas of applications. For example, multilayer semiconductors are very useful for laser diodes, leading to low threshold current, high power, and weak temperature dependence devices [@Razeghi].
It has been demonstrated that, by using semiconductor nanostructures instead of bulk structures, the threshold current may be reduced by more than ten times due to the abrupt energy dependence of the density of states in low-dimensional structures which can enhance the light amplification mechanisms and thus allows lasing to occur at lower currents [@Asada]. Furthermore, the optical absorption in semiconductors which is governed by the electronic density of states, is strongly affected by the impurities [@Fuchs; @Shinozuka1; @Shinozuka2]. For instance, using a semiconducting alloy A$_{1-x}$B$_x$, composed of two semiconductors A and B with different band gap energies, one can tune the operation of an optical device at desired wavelength.
There are many different theoretical approaches that can be used for studying the electronic properties of disordered semiconductors (see for instance [@Gonis]). Among them, the coherent potential approximation (CPA) [@Soven; @Velick; @Gonis] is one of the most widely used methods to study chemically disordered systems. In this approximation the multiple scattering on a single site is taken into consideration and this approach is fairly good for any values of bandwidth and scattering potential.
According to the work of Onodera and Toyozawa [@Onodera; @Toyozawa], the optical absorption spectra of substitutional binary solid solutions can be classified into persistence and amalgamation types in a unified theory. In the persistence type, two exciton peaks corresponding to the two constituent substances are observed, as seen in alkali halides with mixed halides such as KCl$_{1-x}$Br$_x$. In the amalgamation type, however, only one exciton state appears, which can be seen in alkali halides with mixed alkalis such as K$_x$Rb$_{1-x}$Cl. These two typical cases can be distinguished by a parameter which is the ratio of the difference of the atomic energies in the two constituent substances to the energy-band width. In the persistence type, this difference is large compared to the bandwidth and the electronic spectrum is split into two bands, while, when this ratio is small, the impurity band is united with the host band and in this case, which corresponds to the amalgamation type, a single band appears. We should note that the transition from the amalgamation type to the persistence type also depends on the value of impurity concentration [@Onodera].
Based on the Onodera-Toyozawa theory and the CPA, optical properties of various disorder systems have been investigated [@Yoshikawa; @Shinozuka1; @Shinozuka2; @Bakalis; @Takahashi]. Recently, Shinozuka *et al.* [@Shinozuka1; @Shinozuka2] studied the effect of chemical disorder on the electronic and optical properties of bulk, quantum well and wire systems within the CPA. Because of the absence of periodicity in the confinement directions in the quantum wells and wires, the electronic and optical properties of these systems become site-dependent. In their calculations, however, such a dependence has not been included. This site-dependence may be important in operation of nanoscale devices and will be included in our calculations.
The aim of this work is to extend the CPA for the doped semiconductor multilayers to investigate the influence of substitutional disorder on the local density of states (LDOS) and optical absorbtion spectrum in each layer of the system and in both the persistence and amalgamation types. The paper is organized as follows: in Sec. II, by applying the single-site approximation to the layered structures in the presence of chemical disorder, we present our theory and derive equations for the LDOS and optical absorption. The numerical results and discussion for the behavior of desired quantities in terms of the alloy concentration and the scattering-strength parameter are given in Sec. III. The concluding remarks are given in section IV.
Model and formalism
===================
We consider a multilayer structure where the layers are stacked along the $z$-direction and labeled by integer number $n$. The number of layers in this direction is $N_z$, hence $1\leq n\leq
N_z$. The lattice structure of the system is assumed to be a simple cubic with lattice spacing $a$ and the (001) orientation of the layers is taken to be normal to these layers. The multilayer is a semiconducting alloy of the form $A_{1-x}B_x$, where $A$ atom (such as Si) and $B$ atom (such as Ge) occupy randomly the lattice sites with concentration $x$. We use the single-band tight-binding approximation with nearest-neighbor hopping and the on-site delta function-like potential. The Hamiltonian for a single electron (or a Frenkel exciton) in this system is given by $$\label{1}
H=\sum_{\mathbf{r},n}\sum_{\mathbf{r}',n'}[u_{\mathbf{r},n}\delta_{n,n'}\delta_{\mathbf{r},\mathbf{r}'}
-t_{\mathbf{r}n,\mathbf{r}'n'}]\mid \mathbf{r},n\rangle\langle
\mathbf{r}',n'\mid \ ,$$ where $\mathbf{r}$ denotes the positional vector in the $x-y$ plane and the on-site potential $u_{\mathbf{r},n}$ is assumed to be $E_A$ and $E_B$ with probabilities $1-x$ and $x$ when the lattice site ($\mathbf{r},n$) is occupied by the $A$ and $B$ atoms, respectively. The arrangement of these atoms is completely random. The hopping integral $t_{\mathbf{r}n,\mathbf{r}'n'}$ is equal to $t$ for the nearest neighbor sites and zero otherwise. We assume that the hopping integral only depends on the relative position of the lattice sites, that is, only the diagonal disorder is considered. The introduced randomness is treated by the CPA [@Soven; @Gonis]. In this approximation, one seeks to replace a disordered alloy with an effective periodic medium. This CPA medium is characterized by a local (i.e., momentum-independent), energy-dependent self-energy, $\Sigma(\omega)$, called the coherent potential. In some disordered low dimensional systems such as the quantum wires [@Nikolic; @Saffar] and the multilayers [@Okiji; @Hasegawa], the self-energy depends on the position of atomic sites, due to the absence of translational symmetry in some directions. Therefore, in this study the self-energy depends on the layer number, i.e., $\Sigma(\omega)\equiv\Sigma_n(\omega)$ because of the quantum confinement along the $z$-direction. Accordingly, the multilayer Hamiltonian $H$ is replaced by an effective medium Hamiltonian defined by $$\label{2}
\mathcal{H}_{eff}=\sum_{\mathbf{r},n}\sum_{\mathbf{r}',n'}
[\Sigma_{n}(\omega)\delta_{n,n'}\delta_{\mathbf{r},\mathbf{r}'}-t_{\mathbf{r}n,\mathbf{r}'n'}]\mid\mathbf{r},n\rangle\langle
\mathbf{r}',n'\mid\ .$$
{width="0.85\linewidth"}
The physical properties of the real system can be obtained from the configurationally averaged Green’s function $\langle
G\rangle_{\mathrm{av}}=\langle(\omega-H)^{-1}\rangle_{\mathrm{av}}$ which is replaced in the CPA with an effective medium Green’s function $\bar G=(\omega-\mathcal{H}_{eff})^{-1}$ defined by the following Dyson equation [@Saffar] $$\begin{aligned}
\label{3}
\bar{G}_{n,n'}(\mathbf{r},\mathbf{r}';\omega)&=&G^0_{n,n'}(\mathbf{r},\mathbf{r}';\omega)
+\sum_{n''=1}^{N_z}\sum_{\mathbf{r}''}G^0_{n,n''}(\mathbf{r},\mathbf{r}'';\omega)
\nonumber\\
&&\times\Sigma_{n''}(\omega)
\bar{G}_{n'',n'}(\mathbf{r}'',\mathbf{r}';\omega)\ ,\end{aligned}$$ where $G^0$ is the clean system Green’s function and its matrix element is given by $$\label{4}
G^0_{n,n'}(\mathbf{r},\mathbf{r}';\omega)=\frac{1}{N_\parallel}\sum_{\ell=1}^{N_z}
\sum_{\mathbf{k}_\parallel}G^0_{n,n'}(\ell,\mathbf{k}_\parallel;\omega)
\,e^{i\mathbf{k}_\parallel\cdot(\mathbf{r}-\mathbf{r}')} \ ,$$ here, $G^0_{n,n'}(\ell,\mathbf{k}_\parallel;\omega)$ is the mixed Bloch-Wannier representation of $G^0$ and is expressed as $$\label{GLK}
G^0_{n,n'}(\ell,\mathbf{k}_\parallel;\omega)=\frac{\frac{2}{(N_z+1)}\sin(\frac{\ell\pi}{N_z+1}n)
\sin(\frac{\ell\pi}{N_z+1}n')}{\omega+i\eta-\varepsilon_{\ell}(\mathbf{k}_\parallel)}\
,$$ and $$\label{42}
\varepsilon_{\ell}(\mathbf{k}_\parallel)=-2t[\cos(k_xa)+\cos(k_ya)+\cos(\frac{\ell\pi}{N_z+1})]\
,$$ is the clean system band structure. In Eq. (\[4\]), $\mathbf{k}_\parallel\equiv(k_x,k_y)$ is a wave vector parallel to the layer and the summation is over all the wave vectors in the first Brillouin zone (1BZ) of the two-dimensional lattice [@Gonis], $\ell$ is the mode of the subband, $N_\parallel$ is the number of lattice sites in each layer, and $\eta$ is a positive infinitesimal.
Now, to determine the coherent potential in each layer (say $n$), we consider an arbitrary chosen site in the effective medium (layer) describe by $\Sigma_n$. Then, we apply the condition that the average scattering of a carrier by the chosen site in the medium is zero. This condition for any site $\mathbf{r}$ in the $n$th layer is given by $$\label{5}
\langle
\tilde{t}_{n,\mathbf{r}}\rangle_\mathrm{av}=\frac{(1-x)[E_A-\Sigma_n]}{1-[E_A-\Sigma_n]F_n}
+\frac{x[E_B-\Sigma_n]}{1-[E_B-\Sigma_n]F_n}=0\ ,$$ where, $\tilde{t}_{n,\mathbf{r}}$ is the single-site $t$-matrix which represents the multiple scattering of carriers by $A$ and $B$ atoms in the effective medium, $\langle\cdots\rangle_\mathrm{av}$ denotes average over the disorder in the system, and $F_n(\omega)=\bar{G}_{n,n}(\mathbf{r},\mathbf{r};\omega)$ is the diagonal matrix element of the Green’s function of the $n$th effective layer and for $n=1,\cdots,N_z$ can be written as $$\label{6}
F_n(\omega)=\frac{a^2}{{4\pi^2}}\sum^{N_z}_{\ell=1}\int_{\mathrm{1BZ}}d\mathbf{k}_\parallel\,\bar{G}_{n,n}
(\ell,\mathbf{k}_\parallel;\omega)\ .$$ We should note that, the effective Green’s function, $\bar G$, depends on $\ell$ and the two dimensional wave vector $\mathbf{k}_\parallel$ via $G^0$.
From Eq. (\[5\]) one can derive an equation for the self-energy of $n$th layer. Then, using such an equation and also Eq. (\[3\]), which gives a system of linear equations, one can obtain self-consistently the self-energy $\Sigma_n(\omega)$ and the Green’s function $F_n(\omega)$ in each layer. Then, the LDOS in the $n$th layer is calculated by $$\label{7}
g_n(\omega)=-\frac{1}{\pi}\,\mathrm{Im}\,F_n(\omega)\ .$$
In order to calculate the optical absorption spectrum, we assume that both the $A$ and $B$ atoms have equal transition dipole moments [@Onodera; @Toyozawa]. Accordingly, when the $\ell$th subband is optically excited the layer-dependent optical absorption is given by the $\mathbf{k}_\parallel=0$ component of the LDOS. The reason is that $\mathbf{k}_\parallel=0$ (i.e., $\Gamma$ point in the inset of Fig. 1) is the bottom of the conduction band at each layer. We should note that, in the case of bulk materials, $\mathbf{k}=0$ (in three dimensions) is the minimum point in the exciton band. In this study $\varepsilon_\ell(\mathbf{k}_\parallel=0)=-2t[2+\cos(\ell\pi/(N_z+1)]$ gives the minimum energy for each subband. Therefore, the optical absorption of the $n$th layer, due to the creation of an exciton in the system, can be expressed as $$\label{optic}
A_n(\omega)=-\frac{1}{\pi}\,\mathrm{Im}\sum_{\ell=1}^{N_z}\bar{G}_{n,n}(\ell,\mathbf{k}_\parallel=0;\omega)\
.$$
If it is assumed that, the self-energy does not depend on the layer number (i.e., the assumption of translational invariant along the $z$-direction), then a simple expression for Eq. (\[optic\]) is obtained, as it was assumed in Refs. [@Shinozuka1; @Shinozuka2]. In the present work, the behavior of this function in each layer of the system which depends on the number of layers, alloy concentration and the strength of scattering processes via the self-energy will be studied in Sec. III.
{width="1.1\linewidth"}
{width="1.1\linewidth"}
Results and discussion
======================
To study the electronic and optical properties of the system, we perform our numerical results for $N=5$ and characterize the scattering-strength parameter by $\Delta=\frac{|E_A-E_B|}{W}$, where $W\simeq 6t$ is the half-bandwidth of either constituent. We also emphasize that, the calculation of $A_n(\omega)$ is not restricted to the lowest $\ell=1$ subband, because the results showed that $\mathbf{k}_\parallel=0$ component in other $\ell$ subbands is finite and hence, all the subband contributions must be included.
In order to see the effect of quantum confinement on the clean system, $\varepsilon_{\ell}(\mathbf{k}_\parallel)$ is shown in Fig. 1 along the high symmetry directions in the 1BZ. We observe five energy subbands due to the five atomic layers. Each of the subbands crosses the energy axis at the symmetry points with vanishing slope. In other words, there are states with zero group velocity which are responsible for the singularities in the LDOS of each layer and will be shown below in the case of $x=0$ and $x=1$. Since the behavior of electronic states and the modification of energy band due to the influence of chemical disorder play a significant role in determining the optical properties of the disordered system, the LDOS should be analyzed in detail.
In Figs. 2 and 3 we plot the layer-dependence of LDOS and the optical absorption spectrum for $\Delta=0.5$ which corresponds to $E_B=-3\,t$. Because of the symmetry of the system in the $z$-direction, the electronic states for the layers $n$=1 and $n=5$, and also for $n=2$ and $n=4$ are similar to each other. For this reason, we have investigated our desired quantities in the layers $n$=1, 2, and 3. The figures show how the value of alloy concentration influences the $g_n(\omega)$ and $A_n(\omega)$. In the case of $x=0$ and $x=1$, the LDOS of all layers shows a steplike behavior for the states around the bottom and the top of the band, while, sharp features are observed for energies around the center of band. Both the steplike and sharp features (van-Hove singularities) in the LDOS, which correspond to the symmetry points of the 1BZ, comes from the two-dimensional nature of the multilayer. The position of sharp peaks in the electronic density of states is different in various layers due to the quantum confinement along the confined direction. With increasing $x$ the electronic spectra are shifted to the low-energy side and the steplike behavior and sharp peaks gradually disappeared. In the case of maximum disorder, i.e., at $x=0.5$ the LDOS is completely symmetric with respect to the center of band. The steplike and sharp features in the localized states are reproduced for $x>0.5$. Also, it is clear that, a cusp may appear in the LDOS, depending on the value of alloy concentration. This effect, which is related to the value of $\Delta$, confirms that such electronic spectra belong to the amalgamation type.
{width="1.1\linewidth"}
{width="1.1\linewidth"}
In a clean system, i.e., in the case of $x=0$ or $x=1$, $A_n(\omega)$ shows sharp peaks at the edges of the subbands. However, in a system with substitutional disorder, the absorption peaks are broadened and their heights decrease. Based on the behavior of electronic states, with increasing $x$, as shown in Fig. 3, the optical absorption spectrum shifts to the low-energy side, due to its dependence on the bottom energy of the subbands. The maximum value of $A_n(\omega)$ corresponds to the cases of $x=0$ and $x=1$, because the van Hove singularities for these values of $x$ are significantly large. Thus, we can conclude that, in doped semiconductor multilayers, impurities with lower (higher) site energy in comparison with the energy of host atoms and with finite concentration $x$, decrease the strength of optical absorption, i.e., the height of peaks in $A_n(\omega)$, and incident photons with low (high) energy are needed.
{width="0.99\linewidth"}
Another interesting feature is that optical absorption of the surface layers, i.e., layers $n=1$ and $n=N$, is smaller than that of the interior layers, due to the influence of surface states \[compare the maximum value of $A_1(\omega)$ with that of $A_2(\omega)$ and $A_3(\omega)$ at the lowest energies in Fig. 3\]. This indicates that in multilayer systems, the value of optical absorption in each layer depends on the layer number, and the interior layers play a major role in such a process. This feature, which has not been predicted in the previous studies [@Shinozuka1; @Shinozuka2], is important in operation of semiconductor optoelectronic nanodevices.
To investigate the another aspect of the electronic and optical properties of the system, we have shown in Figs. 4 and 5, $g_n(\omega)$ and $A_n(\omega)$ for $\Delta=1.33$ which corresponds to $E_B=-8\,t$. We can clearly see that a gap appears in the LDOS for each value of $x\,(\neq 0,1)$ and the electronic spectra split into two bands corresponding to the two constituent crystals; the lower- (higher-) energy band in this figure corresponds to the $B$ ($A$) atoms. Accordingly, these electronic spectra belong to the persistence type. This gap opening is a consequence of scattering process of carriers by random distribution of the elements in the alloy, and its magnitude depends on the strength of disorder. Our analysis of the behavior of LDOS is similar to that of Fig. 2.
Fig. 5 clearly indicates that in the case of strong disorder regime, $A_n(\omega)$ consists of two bands and the spectrum stretches toward the higher-energy side. With increasing $x$, the optical absorption spectrum of the $A$-type band decreases and shifts to the high-energy side, while the spectrum of the $B$-type band increases and shifts to the low-energy side. Here, we should mention that, under certain conditions, the width of optical absorption spectrum in disordered quantum wires can be considerably stronger than that of the present system [@Fuchs]. An interesting point in Fig. 5 is that, in spite of the symmetry of $g_n(\omega)$ for $x=0.5$ with respect to the center of electronic spectrum, the optical spectrum is not symmetric and the lower band shows a higher value of absorption which is due to the fact that only the $\mathbf{k}_\parallel=0$ component contributes to this process.
It is important to mention that the optical absorption spectrum from each layer is not a measurable quantity in experiment and only the bulk (total) optical absorption spectrum, $A(\omega)$, is possible. In order to obtain this quantity we should sum over the optical absorption of all layers, i.e., $A(\omega)=\sum_{n=1}^{N_z}A_n(\omega)$. The total optical absorption are shown in Fig. 6(a) and 6(b) for $E_B=-3\,t$ and $E_B=-8\,t$, respectively. From the figures we see that, in the case of $E_B=-3\,t$, the absorption spectrum has sharp peaks at the bottom of the optical band for all values of $x$, and shows that the value of total optical absorption at the band minimum of the amalgamation type (Fig. 6(a)) is higher than that of the persistence type (Fig. 6(b)). The numerical results also confirm that, as in the bulk materials, the transition from the amalgamation type to the persistence type in multilayers depends on two parameters $\Delta$ and $x$ [@Onodera; @Toyozawa].
Conclusion
==========
Using the single-band tight-binding theory and the single-site CPA, we have studied the influence of substitutional disorder on the electronic and optical properties of disordered semiconductor multilayers. The results of this theory which is able to predict either the persistent or amalgamated bands, indicate that the optical absorption is governed by the electronic density of states, which is stronger near the bottom of the band. The values of absorption depend on the layer number, the concentration of chemical disorder, and the scattering-strength parameter. In addition, we found that the interior layers in comparison with the surface layer, have significant contribution in the optical absorption of the system.
The obtained results clearly indicate that, studying the dependence of quantum size effects on the optical properties of doped semiconductors is important for better understanding the process of optical absorption in layered structures, and will be helpful for designing optoelectronic nanodevices.
[99]{}
|
---
abstract: 'An isolated quantum gas with a localized loss features a non-monotonic behavior of the particle loss rate as an incarnation of the quantum Zeno effect, as recently shown in experiments with cold atomic gases. While this effect can be understood in terms of local, microscopic physics, we show that novel many-body effects emerge when non-linear gapless quantum fluctuations become important. To this end, we investigate the effect of a local dissipative impurity on a one-dimensional gas of interacting fermions. We show that the escape probability for modes close to the Fermi energy vanishes for an arbitrary strength of the dissipation. In addition, transport properties across the impurity are qualitatively modified, similarly to the Kane-Fisher barrier problem. We substantiate these findings using both a microscopic model of spinless fermions and a Luttinger liquid description.'
author:
- Heinrich Fröml
- Alessio Chiocchetta
- Corinna Kollath
- Sebastian Diehl
bibliography:
- 'biblio.bib'
title: 'Fluctuation-Induced Quantum Zeno Effect'
---
*Introduction —* The quantum Zeno effect (QZE) entails that, perhaps surprisingly, a frequent measurement of a microscopic quantum system suppresses transitions between quantum states [@Misra1977]. Recently, experiments with ultracold atoms have revealed the QZE in many-body systems. Here, different loss processes play the role of a continuous measurement. For example, it has been demonstrated that strong two-body losses give rise to an effective two-body hardcore constraint, in this way turning losses into a tool to create strong correlations [@Syassen2008; @GarciaRipoll2009]. For strong three-body losses, this effect has also been predicted theoretically to give rise to intriguing many-body phenomena such as dimer superfluids and -solids [@Daley2009], or the fractional quantum Hall effect [@Roncaglia2010]. Another paradigmatic setup was introduced in Refs. [@Barontini2013; @Labouvie2016; @Muellers2018], where a local loss process is induced by shining a focused electron beam onto an atomic Bose-Einstein condensate. In particular, the QZE manifests itself in a non-monotonic behavior of the number of atoms lost from the condensate [[@Barontini2013; @Labouvie2016]]{}: while it scales $\sim \gamma$ for a small dissipation strength $\gamma$, in the Zeno regime the inverse scaling $\sim 1/\gamma$ is obtained – the fast scale $\gamma$ locally prevents the loss site to be entered by nearby particles (cf. Fig. \[fig:fig1\]).
Although the QZE occurs here in many-body systems, the effect is understood in terms of the local, microscopic loss physics. In this work, we reveal a new incarnation of the QZE with a genuine many-body origin, induced by the interplay of strong quantum correlations, gapless modes, and a localized loss. To this end, we study a one dimensional wire of interacting fermions prepared in their ground state: this constitutes the dissipative nonequilibrium counterpart of the paradigmatic Kane-Fisher problem [@Kane1992; @Kane1992Long].
We show that fluctuations strongly renormalize the loss barrier in the vicinity of the [initial]{} Fermi momentum $k_F$, even if it is weak on the microscopic scale. For repulsive interactions, the loss barrier is indefinitely enhanced at $k_F$ (cf. Fig. \[fig:fig1\]). A *fluctuation-induced* QZE then manifests itself by the loss barrier becoming fully opaque for momenta $\sim k_F$. The opposite behavior with a renormalization group (RG) flow towards a transparent fixed point is observed for attractive interactions. This leads to a fluctuation-induced transparency.
Although the defining feature of the localized dissipation is the absence of a unitarity constraint for scattering off it, unitarity is thus *emerging* exactly at the Fermi level. In fact, the fixed points are analogous to the ones of Kane and Fisher [@Kane1992]. However, the approach to the fixed points, i.e. the physics in the vicinity of the Fermi surface, strongly deviates from the Kane-Fisher scenario. This is highlighted for attractive interactions: here, observables scale *logarithmically* with, e.g., temperature, instead of the more common algebraic behavior.
The open-system nature of the setup provides an opportunity to probe the system via its output and thus calls for suitable new observables without closed system counterpart. We show that the momentum or energy resolved escape probability turns out to be a realistic measure to detect the fluctuation-induced QZE in future experiments with ultracold atoms.
In the following, we substantiate these findings in two complementary approaches: first, within a minimalistic bosonization approach, providing a simple qualitative picture. Second, via a microscopic calculation, taking into account the dynamical nature of the open system problem and elucidating the physical mechanism behind our results.
![(Color online). Non-monotonic behavior of the escape probability as a function of the dissipation strength. For momenta close to $k_F$, gapless fluctuations renormalize the escape probability, reaching the Zeno (rightmost black dot) or transparent (leftmost black dot) fixed points for repulsive or attractive interactions, respectively.[]{data-label="fig:fig1"}](figure_1.pdf){width="8cm"}
*Microscopic model —* We consider a wire of spinless fermions with mass $m$, interacting through a short-range potential $V(x)$, thus obeying the Hamiltonian $$\label{eq:Hamiltonian-microscopic}
H = -\int_x \psi^\dagger(x)\frac{\nabla^2}{2m}\psi(x) + \int_{x,y} V(x-y) n(x)n(y),$$ with $\psi^\dagger,\psi$ fermionic operators and $n(x) =\psi^\dagger(x)\psi(x)$. We assume the wire to be infinitely long, so that $\int_x = \int_{-\infty}^{+\infty}{\mathrm{d}}x$, and to be prepared at $T=0$. At time $t=0$, a localized loss is switched on at $x=0$: this will generate particle currents, thus driving the system out of equilibrium. We model the loss as a localized coupling to an empty Markovian bath, thus describing the irreversible loss of atoms from the wire. Its dynamics will be then conveniently described by the quantum master equation [@Zoller_book] $$\label{eq:master-equation}
\partial_t \rho = - i [H,\rho ] + \int_x \Gamma(x) \left[L \rho L^\dag - \tfrac{1}{2} \{L^\dag L, \rho\}\right],$$ with $L(x) = \psi(x)$, $\Gamma(x) = \gamma \delta(x)$, $\gamma$ being the dissipation strength. *Luttinger liquid description —* To obtain a first simple picture, we consider a long-wavelength description of Hamiltonian in terms of a Luttinger liquid [@Giamarchi_book] $$\label{eq:H-Luttinger}
H = \frac{v}{2\pi} \int_x \left[ g\, (\partial_x \phi)^2 +g^{-1}\, (\partial_x \theta)^2 \right]$$ with $v$ the speed of sound, $g$ a parameter encoding the effect of interactions, and $\theta$ and $\phi$ bosonic fields related to density and phase fluctuations, respectively. Given the nonequilibrium nature of the system, the usual equilibrium techniques are inadequate to treat the problem, and therefore it is convenient to resort to a Keldysh description [@Kamenev_book; @Sieberer_review]. In order to include the local loss in the bosonization language, we map the master equation onto a Keldysh action, and then bosonize the fields [@suppmat; @Mitra2011; @Schiro2015; @Buchhold2015]. Analogously to the case of a potential barrier [@Kane1992; @Kane1992Long], this yields a local backscattering term $$\label{eq:backscattering}
S_\text{back} = - 2i \gamma \int_{x,t} \delta(x) \left( {\mathrm{e}}^{i \phi_q} - \cos \theta_q \right)\cos \theta_c ,$$ where the labels $c,q$ denote the classical and quantum fields, respectively [@Kamenev_book]. Notice that both $\theta$ and $\phi$ appear, differently from the case of a potential barrier, where only $\theta$ is involved: the field $\phi$ accounts for the currents flowing towards the impurity. Following Refs. [@Kane1992; @Kane1992Long], we study the renormalization of the barrier in the limiting case of weak dissipation $\gamma \to 0$. The renormalization of $\gamma$ at long wavelengths is then determined [within a momentum-shell RG scheme [@suppmat], and]{} produces the flow equation $$\label{eq:RG-luttinger}
\frac{{\mathrm{d}}\gamma}{{\mathrm{d}}\ell} = (1-g) \gamma.$$ This entails that the particle loss is expected to be suppressed for slow modes. For attractive interactions, the perturbation is irrelevant in the RG sense as $\gamma\to 0$ and, thus, the flow suppresses the dissipation strength. In contrast, for repulsive interactions ($g < 1$) the strength of the localized loss [is relevant in the RG sense and flows to infinity, so that losses become suppressed by the QZE.]{} Eq. is remarkably similar to the one obtained for the renormalization of a potential barrier [@Kane1992; @Kane1992Long], despite the fact that the present system is subject to dissipation and is out of equilibrium. In order to certify the domain of validity of Eq. during the time evolution of the system, and its effect on the observables, we will analyze directly the microscopic model in Eq. . Moreover, while the previous analysis is perturbative in $\gamma$, the following, complementary approach is exact in $\gamma$ and perturbative in the microscopic interaction.
*Dynamical regimes —* As the dynamics following the quench of the dissipative impurity is remarkably complex [@Barmettler2011; @Kepesidis2012; @Vidanovic2014; @Kiefer2017], we clarify its different stages by first solving numerically the non-interacting model on a lattice with periodic boundary conditions, described by the Hamiltonian $H = -\sum_{j=1}^{L}(\psi_{j+1}^\dag \psi_j + \text{h.c.})$, with $L$ the size of the system and $\psi_j,\psi^\dagger_j$ the fermionic operators on site $j$. The characterizing parameters are the system length $L$, the initial density $n_0$, and the dissipation strength $\gamma$.
![(Color online). Upper panel: Particle loss rate from the lattice model as a function of time elapsed from the quench, for different system sizes $L$. The dashed line indicates the regime of constant loss. Lower panel, main plot: Density profile from the lattice model [with $L=501$]{}, for different times elapsed from the quench. Inset: Friedel oscillations around the loss site. For all curves $\gamma = 3$ and $N(0)/L = 0.25 $, [with $N(t)$ the number of particles]{}. []{data-label="fig:fig2"}](figure_2.pdf){width="8.3cm"}
Fig. \[fig:fig2\] (upper panel) shows the particle loss rate as a function of time, from which one can identify three regimes: i) For $t < t_\text{I} \sim \gamma^{-1}$ an initial, fast depletion of particles occurs close to the loss impurity. ii) For $t_\text{I} < t < t_\text{II}$, a steady particle current is established, flowing from the yet unperturbed regions of the wire at $x > vt$ (with $v$ the speed of sound) towards the loss site. This regime, which we will focus on, lasts up to a *macroscopic* time $t_\text{II} \sim L$. iii) For $t>t_\text{II}$, the entire system experiences the effect of the dissipation and the particle loss rate slows down, until the system is eventually depleted. In Fig. \[fig:fig2\] (lower panel) we show the density profile during the second regime $t_\text{I}<t<t_\text{II}$. At the impurity site the density is strongly depleted, while the density in the surrounding region exhibits a less pronounced depletion, which heals back to the initial value of the density on a distance $\sim v t$. Crucially, the density around the impurity displays Friedel oscillations, originating from the Fermi step in the momentum distribution of the initial state. *Friedel oscillations —* As the second dynamical regime is extensively long, we will focus on it in the remaining part of this Letter. We substantiate the existence of Friedel oscillations by an analytical solution of the non-interacting continuum problem with a localized loss. By using the Green’s functions method [@suppmat], we derive the time-dependent density profile $n(x,t) = \int_k \left|G_R(x,-k,t)\right|^2\, n_0(k)$, with $G_R(x,k,t)$ the single-particle retarded Green’s function, and $n_0(k) = \theta(k_F^2-k^2)$ the momentum distribution in the initial state, [with $k_F = \pi n_0$ the Fermi momentum and $n_0$ the initial density.]{} The single-particle retarded Green’s function $G_R$ is then evaluated exactly by solving the corresponding Dyson equation [@Kamenev_book], and its explicit form is reported in [@suppmat]. By taking the limit $t\to \infty$, we obtain a stationary value for $n(x)$ corresponding to the second regime discussed above: in fact, by having already taken the thermodynamic limit $L\to\infty$ we implicitly assumed $t_\text{II} \to +\infty$, so that the system is “frozen” in the second regime. We then find [@suppmat] $ n(x) - n_\text{ness} \propto \sin(2k_Fx)/x $, which holds for $x \gg k_F^{-1}$, with $n_\text{ness}$ the uniform background of the stationary state. [Remarkably, the discontinuity in the momentum distribution remains at the initial value of $k_F$ [@suppmat]]{}. These density modulations will generate, in an interacting system, an additional barrier renormalizing the original one. In fact, for momenta close to $k_F$, the virtual scattering processes between the two barriers add up to an effective impenetrable one (for repulsive interactions) or a vanishing one (for attractive interactions) [@Matveev1993; @Yue1994]. In the following, we show that this mechanism also applies to the present nonequilibrium, dissipative case.\
*Transport properties —* To gain further insight on Eq. , we consider the transport properties. The transmission and reflection probabilities $\mathcal{T}(k)$ and $\mathcal{R}(k)$, respectively, for a particle with momentum $k > 0$ impinging upon the dissipative impurity can be read off the retarded Green’s function [@suppmat]. The loss of unitarity related to the scattering off the loss barrier is then quantified by $$\eta(k) = 1 - \mathcal{T}(k) - \mathcal{R}(k).$$ The escape probability $\eta(k)$ describes the probability that a particle with momentum $k$ is absorbed into the bath. $\eta(k)$, which is related to the Fourier transform of $\langle \psi^\dagger(t,x=0)\psi(t',x=0)\rangle$ [@suppmat], is the key quantity of the present analysis, for three reasons: i) it bears signatures of the QZE, ii) as a momentum-resolved quantity, it is sensitive to the renormalization of long-wavelength modes, and iii) it can be directly related to experimentally measurable quantities.\
For the interactionless case $\eta_0(k) = 2\gamma v_k/(\gamma+v_k)^2$, with $v_k = |k|/m$ the group velocity, showing that losses are suppressed for both $v_k/\gamma \to 0$ and $v_k/\gamma \to \infty$. The perturbative corrections to $\mathcal{T}, \mathcal{R}$ due to the interaction potential $V(x)$ can be computed in analogy to equilibrium [@Matveev1993; @Yue1994; @Aristov2010] and they yield [@suppmat]
\[eq:perturbative-corrections\] $$\begin{aligned}
\delta\mathcal{T} & = 2\alpha\, \mathcal{T}_0 \mathcal{R}_0 \, \log|d(k-k_F)|, \\
\delta\mathcal{R} & = \alpha\, \mathcal{R}_0\left(\mathcal{R}_0 + \mathcal{T}_0 - 1\right) \, \log|d(k-k_F)|,\end{aligned}$$
with $\mathcal{T}_0,\mathcal{R}_0$ the bare values, $d$ a length scale to be chosen as the largest between the spatial extent of the interaction $V(x)$ and the Fermi wavelength, and $\alpha = [\widetilde{V}(0)-\widetilde{V}(2k_F)]/(2\pi v_F)$, $\widetilde{V}(k)$ being the Fourier transform of $V(x)$ and $v_F\equiv|k_F|/m$ the Fermi velocity. The two contributions $\widetilde{V}(0)$ and $\widetilde{V}(2k_F)$ derive from the exchange and Hartree part of the interaction, respectively; $\alpha>0$ corresponds to repulsive interactions, and $\alpha<0$ to attractive ones. While the perturbative corrections are in principle controlled by an expansion in $\alpha\ll 1$, they actually diverge logarithmically for $k\to k_F$. These divergences can be resummed by an RG treatment [@Matveev1993; @Yue1994; @suppmat], leading to the RG flow equations $$\label{eq:RG-equations}
\frac{{\mathrm{d}}\mathcal{T}}{{\mathrm{d}}\ell} = - 2\alpha \, \mathcal{T} \mathcal{R} , \qquad \frac{{\mathrm{d}}\mathcal{R}}{{\mathrm{d}}\ell} = - \alpha\, \mathcal{R}\left(\mathcal{R} + \mathcal{T} - 1\right),$$ with the flow to be stopped at $\ell = -\log|d(k-k_F)|$. Eqs. have one stable fixed point: $\mathcal{T}^*=0$, $\mathcal{R}^*=1$ for $\alpha>0$ and $\mathcal{T}^*=1$, $\mathcal{R}^*=0$ for $\alpha<0$. Physically, this entails that tunneling through the dissipative impurity is suppressed at $k=k_F$ for repulsive interactions, while it is maximally enhanced for attractive interactions, similarly to the case of a potential barrier [@Matveev1993; @Yue1994]. However, two novel remarkable features, emerge from the solutions of Eqs. . First, for both attractive and repulsive interactions, $\eta^* = 0$, implying that particles with $k=k_F$ are not emitted into the bath, but are actually “trapped” inside the wire, signaling emergent unitarity at the Fermi level. Second, $\eta(k)$ approaches its fixed point value in qualitatively different ways, depending on the sign of $\alpha$: $$\label{eq:fluctuation-QZ}
\eta(k) \sim
\begin{cases}
|k-k_F|^{\alpha} & \text{for} \quad \alpha > 0, \\
-1/\log|d(k-k_F)| & \text{for}\quad \alpha < 0.
\end{cases}$$ This asymmetry, also visible in the behaviour of $\mathcal{T}(k)$ and $\mathcal{R}(k)$ for $k \to k_F $, does not occur for the case of a potential barrier, where the fixed-point values are approached algebraically in both cases [@Matveev1993; @Yue1994]. Eq. is the key result of this work: the escape probability at the Fermi momentum is strongly renormalized by fluctuations, which suppress it. For $\alpha>0$, this happens as if $\gamma \to +\infty$, thus producing a QZE; for $\alpha<0$, instead, this happens as if $\gamma \to 0$, the impurity thus becoming transparent. Fig. \[fig:fig4\] (upper panels) shows the RG flow of $\mathcal{T}$, $\mathcal{R}$, and $\eta$, for both repulsive and attractive interactions. The flow of $\eta$ may be non-monotonic depending on the sign of $\alpha$ and on the bare value $\eta_0$. To make contact with the Luttinger formulation and Fig. \[fig:fig1\], it is possible to reparametrize $\mathcal{T}(k)$, $\mathcal{R}(k)$ and $\eta(k)$ in terms of a single function $\widetilde{\gamma}(k)$ [@suppmat]. For the escape probability one finds $\eta(k)= 2\widetilde{\gamma}(k)/[1+\widetilde{\gamma}(k)]^2$. The RG flow of $\widetilde{\gamma}$ can be determined from Eqs. as $$\label{eq:RG-gamma}
\frac{{\mathrm{d}}\widetilde{\gamma}}{{\mathrm{d}}\ell} = \alpha \frac{\widetilde{\gamma}^2}{1+\widetilde{\gamma}}.$$ The fixed points of Eq. translate then to $\widetilde{\gamma}^*= \infty$ for $\alpha>0$ and $\widetilde{\gamma}^*= 0$ for $\alpha<0$. $\widetilde{\gamma}$ can therefore be interpreted as the effective strength of the localized dissipation, thus bridging the Luttinger result with the one obtained from the microscopic model. In fact, for a Luttinger parameter $g \simeq 1 $, one has $ 1 - g \simeq g^{-1}-1\simeq \alpha$ [@Fisher_review], and therefore Eqs. and coincide for $\widetilde{\gamma} \gg 1$, upon the identifications $\widetilde{\gamma}\equiv\gamma$. The discrepancy for $\widetilde{\gamma} \ll 1$ can be understood as the limits $\gamma\to 0$ and $\alpha \to 0$ do not commute for the nonequilibrium stationary state.
The behavior of $\eta(k)$ under RG can be therefore rationalized in terms of the flow of $\gamma(\ell)$ (see Fig. \[fig:fig1\]): $\eta$ reaches its fixed point $\eta^*=0$ either for $\gamma\to \infty$ or for $\gamma \to 0$, in the former case thus resulting in a fluctuation-induced QZE, and in the latter one a fluctuation-induced transparency. In Fig. \[fig:fig4\] (lower panels) we show the value of $\eta(k)$ as a function of $k$ reconstructed from the RG flow: its value drops to zero at $k_F$ for both attractive and repulsive interactions. As a consequence of the non-monotonicity of the RG flow, $\eta(k)$ may increase for momenta in the vicinity of $k_F$ (right panel).
![(Color online). Upper panel: RG flow of $\mathcal{T}$, $\mathcal{R}$, and $\eta$. For $\alpha>0$ (left, $\gamma = 0.2$) a fully reflective fixed point is approached, while for $\alpha<0$ (right, $\gamma = 20$) the system is perfectly transmissive at the fixed point. Lower panel: Renormalized $\eta$ as a function of momentum in comparison to the non-interacting value $\eta_0$, for $\alpha > 0$ (left panel) and $\alpha < 0$ (right panel). For all curves $\gamma = 4$.[]{data-label="fig:fig4"}](figure_3.pdf){width="8.6cm"}
*Observables —* The fluctuation-induced QZE can be naturally detected by harnessing the energy or momentum resolved flow of particles leaving the wire, without destructive measurements. For illustrative purposes, we consider the following model, inspired by the input-output formalism of quantum optics [@Zoller_book]. The local dissipation originates from the wire being coupled to a continuum of fermionic modes outside the wire. For definiteness, we assume particles to exit the wire by expanding isotropically in the surrounding vacuum, which could, e.g., be realized in the setup of Ref. [@Lebrat2018] by a local transfer to an untrapped internal state. The particles could be described by operators $c_{\mathbf{q}}, c^\dagger_{\mathbf{q}}$, with ${\mathbf{q}}$ a three-dimensional momentum. These modes are coupled to the wire at $x = 0$ through the Hamiltonian $H_\text{int} = \sum_{{\mathbf{q}}}(g_{\mathbf{q}}c^\dagger_{\mathbf{q}}\psi_0+ \text{h.c.})$, with $g_{\mathbf{q}}$ the coupling of the ${\mathbf{q}}$-mode to the wire and $\psi_0 \equiv \psi(x=0)$. In the second regime, the constant rate of particles with momentum ${\mathbf{q}}$ [and energy $\epsilon_{\mathbf{q}}$]{} leaving the wire is then given by [@suppmat] $$\label{eq:particle-production}
\frac{{\mathrm{d}}\langle c^\dagger_{\mathbf{q}}c_{\mathbf{q}}\rangle}{{\mathrm{d}}t} = \theta(E_F-\epsilon_{\mathbf{q}}) \frac{|g_{\mathbf{q}}|^2}{\gamma} \eta(\epsilon_{\mathbf{q}}),$$ with $E_F = k_F^2/2m$, thus providing a connection between $\eta$ and an experimentally accessible quantity. The bare Fermi distribution enters Eq. as the approach is perturbative in the interaction: it is expected to be smeared out by stronger interactions [@Giamarchi_book].\
*Finite temperature and size —* The unavoidable presence of a finite temperature $T$ and system size $L$ in realistic systems can be accounted for by our RG analysis, and their variation actually leveraged to disclose the novel collective behaviors described above. In fact, a finite $T$ (resp. $L$) cuts off the RG flow at a scale $\ell_T =-\log(d\, T)$ (resp. $\ell_L = \log(L/d)$). As a consequence $\eta(k_F)$ can exhibit a non-monotonic behaviour as a function of the considered length-scale (cf. Fig. \[fig:fig4\], upper panels). For instance, the value of $\eta(k_F)$ *increases* up to $\sim$ 100% by *reducing* the temperature of a gas from $T_\text{Fermi}$ to $0.1T_\text{Fermi}$, thus suggesting that the effects above discussed are observable within the current experimental setups [@Brantut_review; @Lebrat2018]. Although in nonequilibrium systems the (effective) temperature may change during the RG flow [@Mitra2011; @DellaTorre2012; @Schiro2014], we argue that this is not the case in our setup. In fact, we expect the temperature to be enforced by the extensive “reservoir” constituted by the far ends of the wire, rather than by the impurity, which is a local perturbation. A quantitative answer requires to extend our RG analysis to two loops, which we leave to future work.
*Conclusions —* We have shown that a one dimensional ultracold gas of fermions displays novel many-body effects in presence of a localized loss. Loss is suppressed close to the Fermi energy, effectively restoring unitarity. This consequence of the renormalization of the dissipation strength can be interpreted as an incarnation of the QZE. Moreover, transport properties are modified similarly to the case of a potential barrier. These effects would be experimentally accessible by analyzing the ejected flow of particles energy or momentum resolved, without further destructive measurements. An analogous situation to the considered stationary regime could be obtained in experiments with systems coupled to reservoirs at both ends [@suppmat] (cf. Ref. [@Lebrat2018]) . *Acknowledgements —* We thank J.-P. Brantut, L. Corman, J. Marino, A. Rosch and F. Tonielli for useful discussions. We acknowledge support by the Institutional Strategy of the University of Cologne within the German Excellence Initiative (ZUK 81), by the funding from the European Research Council (ERC) under the Horizon 2020 research and innovation program, Grant Agreement No. 647434 (DOQS) and No. 648166 (Phonton), and by the DFG [(TR 185 projects B3 and B4, Collaborative Research Center (CRC) 1238 Project No. 277146847 - projects C04 and C05)]{}. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.
Supplemental Material
=====================
I. Exact results for the non-interacting continuum model with localized loss
============================================================================
Retarded Green’s function {#sec:retarded}
-------------------------
The quantum master equation (see Eq.(2) in the main text) can be mapped onto a Keldysh action $S = S_0 + S_\text{loss}$ , with $$S_0 = \int_{x,t}
\left[
i\psi_+^*\dot{\psi}_+ - H(\psi_+^*,\psi_+) - i\psi_-^*\dot{\psi}_- + H(\psi_-^*,\psi_-)
\right],$$ and $$\label{eq:Lindblad-to-Keldysh}
S_\text{loss} = - i \int_{x,t} \Gamma(x) \left[L_-^* L_+ - \frac{1}{2} \left( L_+^* L_+ + L_-^* L_- \right) \right],$$ where $\Gamma (x) = \gamma \delta(x)$ and $L_\pm(x) = \psi_\pm(x)$. The retarded Green’s function $G(x,x',\omega)$ can be then computed conveniently by using the Dyson’s equation : $$\label{eq:Dyson}
G(x,x',\omega) = G_0(x,x',\omega) + \int_y G_0(x,y,\omega)\Sigma(y)G(y,x',\omega) ,$$ where $$G_0(x,x',\omega) = \frac{1}{2i\sqrt{\omega}} {\mathrm{e}}^{i\sqrt{\omega}|x-x'|}
\label{eq:G0xx}$$ is the (translation invariant) retarded Green’s function of the system in absence of localized loss, while $\Sigma(y)$ is the self-energy associated with the localized loss. Since the loss appear as a quadratic operator in the Keldysh action, the self-energy is a field-independent function which reads $\Sigma(y) = -i\gamma\delta(y)$, so that Eq. can be rewritten as $$G(x,x',\omega) = G_0(x,x',\omega) -i\gamma G_0(x,0,\omega)G(0,x',\omega),$$ which can readily be solved, yielding $$\label{eq:Gxx}
G(x,x',\omega) = G_0(x,x',\omega) -i\gamma\frac{G_0(x,0,\omega)G_0(0,x',\omega)}{1+i\gamma G_0(0,0,\omega)} = \frac{1}{2i\sqrt{\omega}} \left[ {\mathrm{e}}^{i\sqrt{\omega}|x-x'|} + r(\omega)\, {\mathrm{e}}^{i\sqrt{\omega}(|x|+|x'|)} \right],$$ where $$r(\omega) = \frac{-\gamma}{2\sqrt{\omega} + \gamma}
\label{eq:romega}$$ is a function to be discussed below. We report, for later convenience, also the expression for the Green’s function in a mixed momentum-real space representation $G(x,q,\omega)$: $$\label{eq:Gxq}
G(x,q,\omega) = \frac{1}{\omega+i\epsilon - q^2}\left[ {\mathrm{e}}^{iqx} + r(\omega)\, {\mathrm{e}}^{i\sqrt{\omega}|x|} \right],$$ with $\epsilon$ an infinitesimal dissipation needed in order to ensure causality.
Transmission and reflection coefficients
----------------------------------------
The transmission and reflection coefficients for the scattering of a particle off the lossy barrier can be inferred from Eqs. and . For instance, Eq. can be interpreted as the response to an external field $h(x,t) = h\exp(iqx-i\omega t)$ corresponding to a plane wave incoming from left. From the on-shell condition $\omega \simeq q^2$ one finds $$G(x,q,\omega\simeq q^2) \propto
\begin{cases}
{\mathrm{e}}^{iqx} + r(q^2){\mathrm{e}}^{-iqx} & x<0 \\
t(q^2){\mathrm{e}}^{iqx} & x > 0 ,
\end{cases}
\label{eq:GScattering}$$ where $r(q^2)$ is defined in Eq. and corresponds to the reflection coefficient, while $t(q^2) = 1+r(q^2)$ corresponds to the transmission coefficient. Using Eq. one finds $$t(q^2) = \frac{2|q|}{2|q| + \gamma}, \quad r(q^2) = -\frac{\gamma}{2|q| + \gamma}.$$
One-particle correlation function
---------------------------------
We study the evolution of the one-particle correlation function $C(x,y,t,t') = \langle \psi^\dagger(x,t)\psi(y,t')\rangle$ after the quench of the impurity at time $t_0$. Since the wire is coupled to an empty bath which absorbs irreversibly particles from it, $C(x,y,t,t')$ can be written as a propagation of the correlation at time $t_0$, by using the retarded Green’s function previously computed: $$\label{eq:Cxy-definition}
C(x,y,t,t') = \int_{x',y'} G^*(x,x',t-t_0)G(y,y',t'-t_0) C(x',y',t_0,t_0).$$ With the initial condition $C(x,y,t_0,t_0) = \int_q{\mathrm{e}}^{iq(y-x)}n_F(q)$, where $n_F(q)$ is the Fermi distribution at $T=0$, Eq. becomes $$C(x,y,t,t') = \int_{-k_F}^{k_F}\frac{{\mathrm{d}}q}{2\pi}\, G^*(x,-q,t)G(y,-q,t').$$ We now compute the stationary value of $C(x,y,t,t')$ by taking the limit $t_0\to -\infty$. For the sake of clarity, we consider the case $x=y$ and $t=t'$, but the reasoning can be straightforwardly generalized to the case $x\neq y$ and $t\neq t'$. $$\begin{aligned}
|G(x,-q,t-t_0)|^2
& = \left| \int\frac{{\mathrm{d}}\omega}{2\pi}\, \frac{{\mathrm{e}}^{-i\omega(t-t_0)}}{\omega + i\epsilon - q^2}\left[{\mathrm{e}}^{-iqx} + r(\omega){\mathrm{e}}^{i\sqrt{\omega}|x|} \right] \right|^2 \nonumber \\
& = \left| \int\frac{{\mathrm{d}}\omega}{2\pi} {\mathrm{e}}^{-i\omega(t-t_0)} \left[{\mathrm{e}}^{-iqx} + r(\omega){\mathrm{e}}^{i\sqrt{\omega}|x|} \right] \int{\mathrm{d}}t'\, (-i)\theta(t') {\mathrm{e}}^{i(\omega -q^2)t'}\right|^2 \nonumber \\
& = \left|\int\frac{{\mathrm{d}}\omega}{2\pi} {\mathrm{e}}^{-i\omega t} \left[{\mathrm{e}}^{-iqx} + r(\omega){\mathrm{e}}^{i\sqrt{\omega}|x|} \right]\,2\pi\, \delta(\omega-q^2)\right|^2 \nonumber \\
& = \left| {\mathrm{e}}^{-iqx} + r(q^2){\mathrm{e}}^{i|q||x|} \right|^2 ,
\label{eq:CalcG} \end{aligned}$$ were in the third step we took the limit $t_0\to-\infty$. The local density of particles at time $t$ is given by $n(x,t) = C(x,x,t,t)$. By making use of the previous equation we obtain $$n(x) = \frac{1}{\pi}\int_0^{k_F}{\mathrm{d}}q\bigg[ 1+r(q^2)^2 + r(q^2) + r(q^2) \cos(2q|x|) \bigg] \equiv n_\text{ness} + \delta n(x), \label{eq:densityProfile}$$ where $n_\text{ness}$ is the homogeneous background and $\delta n(x)$ the density modulations, which exhibits for $|x|\gg k_F^{-1}$ Friedel oscillations $$\delta n(x) = r(k_F) \frac{\sin(2k_F|x|)}{2\pi|x|}.$$
Preservation of the Fermi momentum and experimental observability
-----------------------------------------------------------------
We investigate the time-evolution of the momentum occupation number in the non-interacting system after a quench of the localized loss both in the lattice system and in the continuum model. In the lattice system we evaluate $n_k(t) = \langle \psi_k^\dagger \psi_k \rangle$ numerically. The system is initialized in its zero-temperature ground state characterized by a Fermi momentum $k_F=\pi N(0)/L$, with $N(0)$ the initial particle number and $L$ the system size. Crucially, we observe in the dynamics after the quench that the step at the initial Fermi momentum $k_F$ remains well-defined (see Fig. \[fig:fig1SM\]), although the overall number of particles depletes. At all times the distribution exhibits a discontinuity at the initial Fermi momentum.
![ (Color online). Numerical study of the momentum occupation in the second time regime for different times after the quench ($L=401, \gamma = 1, N(0)/L=0.25$ ). []{data-label="fig:fig1SM"}](figure_4.pdf){width="9cm"}
In order to obtain an analytical expression for the momentum distribution $n(k,t)$, we evaluate $ \langle \psi^\dagger(k',t) \psi(k,t) \rangle$ for the continuum model. By taking the Fourier transform of Eq. we obtain: $$\langle \psi^\dagger(k,t) \psi(k,t) \rangle = \int \frac{{\mathrm{d}}q}{2\pi} G^*(k,-q,t-t_0) G(k,-q,t-t_0) n_0(q) ,$$ with $ n_0(q) = \theta(k_F^2-q^2)$ the initial zero-temperature Fermi distribution. We are left to evaluate $$\begin{gathered}
\label{eq:momentum-distro}
\langle \psi^\dagger(k,t) \psi(k,t) \rangle \\
= \int \frac{{\mathrm{d}}q}{2\pi} n_0(q) \left[ |G_0(k,t-t_0)|^2 \delta^2(k+q) + \delta(k+q) (G_0^*(k,t-t_0) \widetilde G(k,-q,t-t_0) +c.c.) + |\widetilde G(k,-q,t-t_0)|^2 \right],\end{gathered}$$ with $$G(k,k',\omega) = \delta(k-k') G_0 (k,\omega) + \widetilde G(k,k',\omega) ,\qquad \widetilde G(k,k',\omega) = i2 \sqrt{\omega} r(\omega) G_0(k ,\omega) G_0(k' ,\omega) ,$$ where $G_0(k,\omega)$ is the Fourier transform of Eq. , and $r(\omega)$ is given in Eq. . By noticing that $n(k,t)$ and $\langle \psi^\dagger(k,t) \psi(k,t) \rangle$ are proportional by a factor diverging as the system’s volume (hence the square delta function in Eq. ), and by taking the limit $t_0 \rightarrow -\infty$, we obtain the stationary distribution $$n(k) = n_0(k) \left( 1 - \frac{\gamma / k}{ 1 + \gamma / (2k)} + \frac{1}{2} \frac{\gamma^2 / k^2}{(1+ \gamma/(2k) )^2} \right) = n_0(k) (1- \eta_0(k)),
\label{eq:MomentumDistEta}$$ where we recognized the escape probability $\eta_0(k)$. Hence, we find that the discontinuity at the initial Fermi momentum $k_F$ persists in the stationary state, as $\eta_0(k)$ is a smooth function of $k$. Eq. can be generalized to account for interactions.
In experiments where the stationary regime could approximately be reached one could obtain the value of $\eta(k)$ by measuring the momentum distribution of the system in the presence of the impurity, and by comparing it to the one without impurity. An analogous situation to the discussed stationary regime could be achieved in systems coupled to reservoirs at the ends (cf. Ref. ), both at $T=0$ and $\mu = k_F$, thus sustaining the particle currents.
II. Bosonization of a dissipative impurity
==========================================
The low-energy properties of a system interacting one-dimensional fermions is captured by the Luttinger Hamiltonian described in the main text (see Eq.(3)). The mapping between fermions and bosons is done via the transformation (considering the two leading harmonics) : $$\label{eq:bosonization-mapping}
\psi \sim \sqrt{\rho_0} \, \left[ e^{i k_F x} e^{i (\phi + \theta)} + e^{-i k_F x} e^{i (\phi - \theta)} \right],$$ where $\phi$ and $\theta$ are bosonic fields describing phase fluctuations and density fluctuations, respectively. The associated Keldysh action $S_0$ can be written as $$S_0 = \int_{k,\omega}\ \chi^\dagger (k, \omega) G^{-1}(k, \omega) \chi(k, \omega)$$ where $\int_{k,\omega} \equiv \int {\mathrm{d}}k {\mathrm{d}}\omega/(2\pi)^2 $, $\chi \equiv (\phi_c, \theta_c, \phi_q, \theta_q )^T$ , $$G^{-1}(k, \omega) =
\begin{pmatrix}
0 & P_A (k, \omega)\\
P_R(k, \omega) & P_K(k, \omega)
\end{pmatrix},$$ and $$\label{eq:PR-PK}
P_R = P_A^\dagger = \frac{1}{2\pi}
\begin{pmatrix}
v g k^2 + i \epsilon \omega & - k \omega \\
- k \omega & v g^{-1} k^2 + i \epsilon \omega
\end{pmatrix}
,\qquad
P_K = \frac{1}{2\pi}
\begin{pmatrix}
2 i \epsilon \omega \coth \frac{\omega}{2 T} & 0 \\
0 & 2 i \epsilon \omega \coth \frac{\omega}{2 T}
\end{pmatrix}.$$ The retarded and Keldysh Green’s functions, $G_R$ and $G_K$, respectively, can be obtained as $G_R = P_R^{-1}$ and $G_K = P_K^{-1}$ . By using the mapping and the bosonization formula , the backward scattering due to the impurity reads: $$\label{eq:SD}
S_\text{back} = - i 2 \gamma \int_{x,t} \delta(x) \left[ \cos \theta_c \left( {\mathrm{e}}^{i\phi_q} - \cos \theta_q \right) \right],$$ where $\theta_{c,q} = \theta_+ \pm \theta_-$ and $\phi_{c,q} = \phi_+ \pm \phi_-$.
A renormalization group analysis is then performed by integrating out fast modes lying in the momentum-shells $k \in \left[ \Lambda {\mathrm{e}}^{-\ell}, \Lambda \right]$ and subsequently rescaling as $(x,t) \rightarrow ({\mathrm{e}}^\ell x, {\mathrm{e}}^\ell t)$. In the weak coupling limit ($\gamma \to 0 $) we consider the leading term in the expansion in $\gamma$: $$\langle {\mathrm{e}}^{ i S_\text{back} } \rangle_\text{fast} \simeq \langle 1 + i S_\text{back} \rangle_\text{fast} \simeq {\mathrm{e}}^{ \langle i S_\text{back}\rangle_\text{fast} } .$$ By making use of the Gaussian identity $\langle {\mathrm{e}}^{i x} \rangle = \exp[ - \langle x^2\rangle/2 ]$ and the correlation functions from Eq. , we obtain from Eq. $$\left\langle S_\text{loss}[\phi_\text{fast} + \phi_\text{slow} , \theta_\text{fast} + \theta_\text{slow}] \right\rangle_\text{fast} =
S_\text{loss}[\phi_\text{slow}, \theta_\text{slow} ] {\mathrm{e}}^{- \left\langle \theta_{c}^2(x,t) \right\rangle_\text{fast} } .$$ This first order corrections can thus be calculated from , and for $T=0$ it reads $$\langle \theta_c^2(x,t)\rangle_\text{fast} = g \int_{\Lambda {\mathrm{e}}^{-\ell}}^\Lambda \frac{{\mathrm{d}}k}{k} = g \ell .$$ Finally, by noticing that the canonical scaling dimension of the dissipation strength $\gamma$ is $[\gamma] = 1$, we obtain the flow equation (5) in the main text.
III. Renormalization of scattering amplitudes
=============================================
We proceed by computing the corrections of the scattering probabilities $\mathcal{T}$ and $\mathcal{R}$ due to interactions $V$ (see Eq. (1) in the main text) to first order in perturbation theory in the interactions. To this end, its convenient to focus on the transmission and reflection amplitudes, which relate to the corresponding probabilities via $\mathcal{T} = |t|^2$ and $\mathcal{R} = |r|^2$. $t$ and $r$ are defined via the retarded Green’s function as discussed in Sec. I. The corrections can be obtained from the perturbed retarded Green’s function $G = G_{0} + \delta G$, where $G_0$ denotes the unperturbed Green’s function ($V=0$): $$\delta G(x,y,\omega) = \int_{x',y'} G_0(x,x',\omega)\left[ V_H(x',y') + V_{ex}(x',y') \right] G_0(y',y,\omega) ,
\label{eq:GPert}$$ where the Hartree and exchange potentials $V_H$ and $V_{ex}$ are given by $$V_H(x,y) = \delta(x-y)\int_{x'}V(x-x')C(x',x',t,t), \qquad V_{ex}(x,y) = -V(x-y)\,C(y,x,t,t) ,$$ with $C(x,y,t,t)$ evaluated in the stationary limit. From Eq. one obtains the following corrections (see Ref. ),
$$\begin{aligned}
t & = t_0 + \alpha\, t_0r_0^2 \, \log|d(k-k_F)| ,\\
r & = r_0 + \frac{\alpha}{2} r_0\bigg(r_0^2 + t_0^2 - 1 \bigg) \, \log|d(k-k_F)|,\end{aligned}$$
with $t,r \equiv t(k), r(k)$ for $k\sim k_F$, $d$ is the typical length scale of $V(x)$, and $\alpha = [V(2k_F)-V(0)]/(2\pi v_F)$. The logarithmic divergences of the perturbation theory remain at higher orders and are cured by a proper resummation, achieved by a real space or a frequency RG leading to
\[eq:RG-equationsSM\] $$\begin{aligned}
\frac{{\mathrm{d}}t}{{\mathrm{d}}\ell} & = - \alpha \, t r^2 , \\
\frac{{\mathrm{d}}r}{{\mathrm{d}}\ell} & = - \frac{\alpha}{2} \, r \bigg( t^2+r^2 - 1 \bigg) .\end{aligned}$$
From here it is straightforward to derive the flow equations of the scattering probabilities (Eqs. (8) in the main text).
Since both, the continuity relation $t=1+r$ is preserved and $t,r$ remain real-valued along the flow, it is possible to parametrize the flow of $r$ and $t$ by a single function $\widetilde{\gamma}$, such that $$r = -\frac{\widetilde{\gamma}}{1+\widetilde{\gamma}}, \qquad t = \frac{1}{1+\widetilde{\gamma}}.$$ The flow equation for $\widetilde{\gamma}$ is then easily derived from Eq. and reads $$\frac{{\mathrm{d}}\widetilde{\gamma}}{{\mathrm{d}}\ell} = \alpha \frac{\widetilde{\gamma}^2}{1+\widetilde{\gamma}}.$$
IV. Input-output formalism
==========================
We investigate the loss of particles from the system at $x=0$. To this end, we study $G(x=0, p, \omega)$, describing the response at $x=0$ to a plane wave perturbation, as discussed in Sec. I (cf. Eq. ): $$\label{eq:t_eta}
G(x=0, p, \omega) = t_{\eta}(\omega) G_0(p, \omega) .$$ Here, we introduced the amplitude $ t_{\eta}$, related to $\eta$ through $$\frac{\gamma}{k} t_\eta^2 = 1- \mathcal{T}-\mathcal{R} \equiv \eta.
\label{eq:EtaTmSqIdentity}$$ Eqs. and can then be regarded as an operative definition to evaluate $\eta$ directly from the response function.\
We then introduce $g(t_2 - t_1) = \langle \psi^\dagger(x=0, t_1) \psi(x=0, t_2) \rangle$ and obtain as a generalization of the calculation in Sec. I $$\begin{aligned}
g(t_2 - t_1) &= \int_{-k_F}^{k_F} \frac{dp}{2 \pi} G(x=0, -p, t_1- t_0) G(x=0, -p, t_2 - t_0) = \int_{-k_F}^{k_F} \frac{dp}{2 \pi} {\mathrm{e}}^{- p^2 (t_2 - t_1)} | t_{\eta}(p^2)|^2 .\end{aligned}$$ As discussed in the main text, we consider the wire being coupled to a continuum of free fermionic modes, described by $H_\text{int} = \sum_{{\mathbf{q}}}(g_{\mathbf{q}}c^\dagger_{\mathbf{q}}\psi_0+ \text{h.c.})$. The momentum resolved loss rate is then related to $t_{\eta}$ by $$\frac{{\mathrm{d}}\langle c^\dagger_{\mathbf{q}}c_{\mathbf{q}}\rangle}{{\mathrm{d}}t} = |g_{\mathbf{q}}|^2 g(\omega=\epsilon_{\mathbf{q}}) = |g_{\mathbf{q}}|^2 \frac{\theta(E_F - \epsilon_{\mathbf{q}}) }{\sqrt{\epsilon_{\mathbf{q}}}} | t_{\eta}(\omega=\epsilon_{\mathbf{q}})|^2 ,
\label{eq:momResolvedLossRate}$$ with $E_F = k_F^2/2m$.\
The renormalization procedure from Sec. III can be adapted to obtain the renormalization of $t_{\eta}$. As the starting point, the perturbative correction of $G(x=0,p,\omega)$ is given by $$\delta G(x=0,p,\omega) = \int_{x',y'} G(x=0,x',\omega)\left[ V_H(x',y') + V_{ex}(x',y') \right] G(y',p,\omega).$$ Crucially, we observe that $ t_{\eta}$ enters the expression via $G(x=0,x',\omega)$, while the logarithmic divergencies are generated by the asymptotic forms of $G(y',y,\omega)$, $V_H(x',y')$ and $ V_{ex}(x',y')$ in which accordingly $t$ and $r$ appear.\
The flow equation of $t_\eta \equiv t_\eta(k)$ for $ k\sim k_F$ is found based on these considerations as $$\frac{{\mathrm{d}}t_\eta}{{\mathrm{d}}\ell} = - \frac{\alpha}{2} t_\eta \left( r^2 + t r \right) .$$ One can readily check that the flow equations of $\eta = 1 - t^2 - r^2$ and $t_\eta^2$ coincide, in agreement with Eq. . According to Eqs. and , we can relate the momentum resolved loss rate to the escape probability $\eta$ as the key loss indicator of the system and arrive at the central result given in Eq. (11) of the main text.
|
---
abstract: 'The divergent integral $\int_a^b f(x)(x-x_0)^{-n-1}\mathrm{d}x$, for $-\infty<a<x_0<b<\infty$ and $n=0, 1, 2, \dots$, is assigned, under certain conditions, the value equal to the simple average of the contour integrals $\int_{C^{\pm}} f(z)(z-x_0)^{-n-1}\mathrm{d}z$, where $C^+$ ($C^-$) is a path that starts from $a$ and ends at $b$, and which passes above (below) the pole at $x_0$. It is shown that this value, which we refer to as the Analytic Principal Value, is equal to the Cauchy principal value for $n=0$ and to the Hadamard finite-part of the divergent integral for positive integer $n$. This implies that, where the conditions apply, the Cauchy principal value and the Hadamard finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox Theorem with integrals along some arbitrary paths. The utility of the Analytic Principal Value in the numerical, analytical and asymptotic evaluation of the principal value and the finite-part integral is discussed and demonstrated.'
address: 'Theoretical Physics Group, National Institute of Physics, University of the Philippines, Diliman Quezon City, 1101 Philippines'
author:
- 'Eric A. Galapon'
title: The Cauchy Principal Value and the Hadamard finite part integral as values of absolutely convergent integrals
---
Introduction
============
Divergent integrals arise naturally in many areas of physics and engineering [@shiekh; @lee; @ioa2; @lifanov]. In this paper we consider the class of non-converging integrals given by $$\label{divergent}
\int_a^b \frac{f(x)}{(x-x_0)^{n+1}} \mbox{d}x,\;\;\; -\infty < a<x_0<b<\infty,\;\; n=0,1,2,\dots$$ for some function $f(x)$ not vanishing at $x=x_0$. These have been assigned meaningful values by a symmetric removal of the singular point, $x=x_0$, of the integrand. In particular the integral, for a fixed $n$, is replaced with the limit $$\label{limit}
\lim_{\epsilon\rightarrow 0^+}\left[\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x+ \int_{x_0+\epsilon}^{b} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x\right],$$ and a finite value is extracted which is assigned as the value of the divergent integral. Under some continuity conditions on $f(x)$, only the case $n=0$ leads to a well-defined limit, which is the well-known Cauchy Principal Value (CPV) [@pipkin]. For positive integer $n=1,2, \dots$ the limit does not exist. However, expression \[limit\] can be cast into a form with a group of terms possessing a finite value in the limit $\epsilon\rightarrow 0$ and into another group of terms that diverge in the same limit. The divergent integral is assigned a value by hand by dropping the diverging term, leaving the group of terms with a finite value in the limit, the limit of which is assigned as the value of the divergent integral [@ang; @kanwal; @cohen; @chan; @monegato]. This manner of assigning value to a divergent integral is due to Hadamard [@hadamard], and the value is now known as the Hadamard finite part or the Finite-Part Integral (FPI).
Historically it was Fox [@fox] who made the first investigation of the integral \[divergent\] following earlier Hadamard’s introduction of the finite-part of a divergent integral [@hadamard]. He obtained the explicit form of the Finite-Part Integral of equation \[divergent\], with the Cauchy Principal Value as a special case. It is given by $$\begin{aligned}
\label{fox}
&&\#\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}} \mbox{d}x \nonumber \\
&&\hspace{12mm}=\lim_{\epsilon\rightarrow 0^+}\left[\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x+ \int_{x_0+\epsilon}^{b} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x - H_n(x_0,\epsilon)\right],\end{aligned}$$ where $$\begin{aligned}
H_n(x_0,\epsilon)&=&0, \;\;\; n=0\label{fox1} \\
H_n(x_0,\epsilon)&=& \sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \frac{(1-(-1)^{n-k})}{\epsilon^{n-k}},\;\;\; n=1, 2, \dots , \label{fox2}\end{aligned}$$ in which $\#$ is the Cauchy Principal Value for $n=0$ and $\#$ is the Finite-Part Integral for positive integer $n$. Fox referred to equation \[fox\] as the principal value, appropriately so as the CPV is a special case; however, we will continue to call equation \[fox\] as the finite part integral, in accordance with current literature. The limit in equation \[fox\] exists if $f(x)$ possesses derivatives up to order $n$ and satisfies a Holder condition.
While the Cauchy principal value and the finite-part of the divergent integrals given by expression \[divergent\] have been studied extensively and have reached the status of an orthodoxy in applied mathematics [@lifanov; @ang; @pipkin; @kanwal], in this paper we point to another perspective on the integral which, we believe, should have had been much earlier appreciated but somehow had escaped the researchers of the field. The basic idea promoted here has already been intimated by Hadamard in considering a family of divergent integrals similar to equation \[divergent\] [@hadamard], but it was not developed further. Under certain conditions on the function $f(x)$, here we assign a value to the integral \[divergent\] by the average of the values of the same integral when the contour of integration is displaced above and below the pole at $z=x_0$ while keeping the limits of the integration fixed. We will refer to this value as the Analytic Principal Value (APV), for a reason to made clear below. Our introduction of the APV is motivated by the Sokhotski-Plemelj theorem (SPT) for the CPV [@sokho; @plemelj] and its generalization due to Fox [@fox]. According to SPT the CPV is the average of the boundary values of the analytic function, $\Phi(z)$, obtained from the integral \[divergent\] (with $n=0$) by replacing $x_0$ with the complex variable $z$; the boundary values are the values of $\Phi(z)$ as $z$ approaches the singular point $x_0$ from above and below (the real-axis).
In our definition of the Analytic Principal Value, we do the opposite in Sokhostski-Plemelj theorem. In the SPT the singular point $x_0$ is lifted out of the path of integration with the replacement $x_0\rightarrow z$, the path being fixed. Here the point $x_0$ is fixed but the contour of integration is deformed above and below the singularity, defining two families of paths separated by the pole, with each family having a unique value. The simple average of these values is our APV. We will show that the Analytic Principal Value is equal to the Cauchy Principal Value for $n=0$ and to the Finite-Part Integrals for positive integer $n$. The APV then obviates the need to introduce boundary values. In fact, the SPT and its generalization can be seen as a specific realization of the Analytic Principal Value; close scrutiny, for example, of a particular implementation of the limiting procedure in the SPT shows that it is no more than a special computation of the APV [@pipkin].
A feature of the APV is that its computation does not require the need for a limiting process. It assumes the form of an absolutely convergent integral, so that the CPV and the FPI are themselves values of absolutely convergent integrals. That is the divergent integrals \[divergent\], interpreted as APVs, are convergent integrals in disguise. Beside its conceptual appeal, such integral representation of the CPV and the FPI make them amenable to computation using standard numerical quadratures, obviating the need for specialized algorithms specifically tailored for them [@cris; @bia]. Furthermore, an (absolutely convergent) integral representation allows an asymptotic analysis of the CPV and the FPI [@guer] using the already established methods of asymptotic analysis for regular integrals [@wong].
This paper is organized as follows. In Section-\[principalvalue\] we formalize the definition of the Analytic Principal Value of the given family of divergent integrals. In Section-\[cpvfpi\] we show that the Principal Value equals the Cauchy Principal Value and the Finite-Part Integral. In Section-\[spft\] we discuss the relationship between our results and the Sokhotski-Plemelj theorem for the CPV and its generalization due to Fox for the FPI. In Section-\[example\] we apply the definition of the Analytic Principal Value to a specific case, and demonstrate the utility of the absolutely convergent integral representation of the CPV and the PFI. In Section-\[conclusion\] we conclude and point to further applications of the Principal Value and raise an open problem.
The Analytic Principal Value Integral {#principalvalue}
=====================================
Given the function $f(x)$ in the divergent integral \[divergent\], let us introduce the complex valued function $f(z)$ obtained by replacing the real variable $x$ with the complex variable $z$ in $f(x)$. We will refer to $f(z)$ as the complex extension of $f(x)$. We require that there exists a neighborhood $R$ that encloses the strip $[a,b]$ and in this region $f(z)$ is analytic. $f(z)$ can have an infinite number of poles as long as none of them are in $R$. Denote the punctured domain by $R_{x_0}=R\ \backslash\{x_0\}$. Also denote $\Gamma^+$ the set of all continuous, non-self-intersecting paths contained in $R_{x_0}$ that start at $a$ and end at $b$, and that pass the pole at $z=x_0$ above the real axis; and $\Gamma^-$ the set of all similar paths in $R_{x_0}$ that pass the pole below the real axis (see Figure-1).
Due to the analyticity of $f(z)/(z-x_0)^{n+1}$ in the region $R_{x_0}$, the value of the integral $$\label{contour}
\int_{\gamma}\frac{f(z)}{(z-x_0)^{n+1}} \mathrm{d}z$$ does not depend on $\gamma$ when $\gamma$ is restricted on $\Gamma^+$ or $\Gamma^-$ only. That is there is a single value to the integral \[contour\] for all paths $\gamma^+$ in $\Gamma^+$, which we denote by $\mathrm{Int}^+(x_0)$; similarly for all paths $\gamma^-$ in $\Gamma^-$, which we denote by $\mathrm{Int}^-(x_0)$. However, due to the pole at $z=x_0$, the value of the integrals $\mathrm{Int}^+(x_0)$ and $\mathrm{Int}^-(x_0)$ are generally not equal. We now define the Analytic Principal Value of the divergent integral \[divergent\] as follows.
Let $f(x)$ admit a complex extension $f(z)$ that is analytic in a neighborhood containing the strip $[a,b]$. Then the Analytic Principal Value of the divergent integral, to be denoted by ${\scriptstyle \backslash\!\!\!\!\backslash\!\!\!\!} { \int}$, is given by the simple average of $\mathrm{Int}^+(x_0)$ and $\mathrm{Int}^-(x_0)$, $$\backslash\!\!\!\!\backslash\!\!\!\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}}\,\mathrm{d}x =\frac{1}{2}\left[ \mathrm{Int}^+(x_0) + \mathrm{Int}^-(x_0)\right]\label{principal} .$$
For $n=0, 1, 2, \dots$ $$\label{relation}
\mathrm{Int}^-(x_0) - \mathrm{Int}^+(x_0)= 2 \pi i \frac{f^{(n)}(x_0)}{n!},$$ $$\label{form1}
\backslash\!\!\!\!\backslash\!\!\!\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}}\,\mathrm{d}x = \mathrm{Int}^+(x_0) + i \pi \frac{f^{(n)}(x_0)}{n!}.$$ $$\label{form2}
\backslash\!\!\!\!\backslash\!\!\!\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}}\,\mathrm{d}x = \mathrm{Int}^-(x_0) - i \pi \frac{f^{(n)}(x_0)}{n!}.$$
Equation \[relation\] follows from the residue theorem and the fact that the closed path $\gamma^-+(-\gamma^+)$, for every $\gamma^{\pm}\in\Gamma^{\pm}$, encloses the pole $z=x_0$. Equations \[form1\] and \[form2\] are consequences of the definition of the Principal Value given by equation \[principal\] and the relationship between $\mathrm{Int}^+(x_0)$ and $\mathrm{Int}^-(x_0)$ given by equation \[relation\].
Equations \[principal\], \[form1\] and \[form2\] are equivalent and any one of them can be used to compute the Analytic Principal Value. Equation \[principal\] requires two paths, while equations \[form1\] and \[form2\] require one path each. The integrals involved in these expressions are absolutely convergent, so that the family of divergent integrals given by equation \[divergent\] are in fact absolutely convergent integrals when interpreted as Analytic Principal Values. But since the choice of path or paths in $\Gamma^{\pm}$ is arbitrary, the divergent integrals assume various but equivalent (convergent) integral representations. Any one of these representation can be chosen to conveniently compute the principal value, either analytically or numerically.
The above definition of the Analytic Principal Value extends readily when the path lies in the complex plane.
![The mapping of the region of integration from the real line to a region in the complex plane. If $f(z)$ has poles, the region $R$ is chosen such that the poles, indicated by the hollow circles, are outside $R$.[]{data-label="fig:boat1"}](principalvalue.jpg)
The Cauchy Principal Value and the Finite Part Integrals as Values of the Analytic Principal Value {#cpvfpi}
==================================================================================================
For $n=0$ the Analytic Principal Value is equal to the Cauchy principal value, $$\label{cpv}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}}\frac{f(x)}{x-x_0} \mathrm{d}x = \lim_{\epsilon\rightarrow 0}\left[\int_a^{x_0-\epsilon} \frac{f(x)}{x-x_0}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{x-x_0}\mathrm{d}x\right] .$$
To compute the analytic principal value, it is sufficient to chose a convenient path in $\Gamma^+$ and use equation \[form1\]. We use the contour $\bar{\gamma}^+$ shown in Figure-2. Then $$\label{precpv}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}} \frac{f(x)}{x-x_0}\mathrm{d}x=\int_a^{x_0-\epsilon} \frac{f(x)}{x-x_0}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{x-x_0}\mathrm{d}x + \int_{C^+}\frac{f(z)}{z-x_0}\mathrm{d}z + i\pi f(x_0)$$ where $C^+$ is the semi-circle centered at the origin with radius $\epsilon$. We only need to evaluate the integral along $C^+$. We parametrize the semi-circle by $z=x_0+\epsilon \mathrm{e}^{i\theta}$, where $\pi>\theta>0$. Because $f(z)$ is analytic in $R$, we can use the Taylor expansion theorem to expand $f(x_0+\epsilon \mathrm{e}^{i\theta})$ about $z=x_0$ up to the order $O(\epsilon)$, $$f(x_0+\epsilon\mathrm{e}^{i\theta})=f(x_0) + \epsilon \mathrm{e}^{i\theta} f_1(x_0+\epsilon\mathrm{e}^{i\theta}),$$ where $f_1(z)$ is an analytic function [@ahlfors]. From the Taylor expansion theorem, we have the bound $$\label{bound}
\left| f_1(x_0+\epsilon\mathrm{e}^{i\theta})\right|\leq \frac{M}{\rho(\rho-\epsilon)},$$ where $M$ is the maximum of $|f(z)|$ in $R$, and $\rho$ is any positive constant with $\rho>\epsilon$ and such that the disk $|z-x_0| \leq \rho$ is contained in $R$. We can choose $\epsilon$ sufficiently small to satisfy the last requirement. Then we have $$\int_{C^+}\frac{f(z)}{z-x_0}\mathrm{d}z= -i\pi f(x_0) +\epsilon i \int_{\pi}^{0} \mathrm{e}^{i\theta} f_1(x_0+\epsilon\mathrm{e}^{i\theta}) \mathrm{d}\theta .$$
We substitute this back into equation \[precpv\] and the residue term cancels out. We obtain $$\label{precpv2}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}} \frac{f(x)}{x-x_0}\mathrm{d}x=\int_a^{x_0-\epsilon} \frac{f(x)}{x-x_0}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{x-x_0}\mathrm{d}x +\epsilon i \int_{\pi}^{0}\mathrm{e}^{i\theta} f_1(x_0+\epsilon\mathrm{e}^{i\theta}) \mathrm{d}\theta .$$ Using the bound given by \[bound\] we obtain the inequality $$\begin{aligned}
\label{ineq1}
&&\left|{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}} \frac{f(x)}{x-x_0}\mathrm{d}x-\left[\int_a^{x_0-\epsilon} \frac{f(x)}{x-x_0}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{x-x_0}\mathrm{d}x \right] \right|\nonumber \\
&&\hspace{44mm}\leq \epsilon \int^{\pi}_{0} |f_1(x_0+\epsilon\mathrm{e}^{i\theta})| \mathrm{d}\theta \leq \frac{\pi M \epsilon}{\rho(\rho-\epsilon)}
\end{aligned}$$ Since the right hand side of the inequality \[ineq1\] becomes arbitrarily small for arbitrarily small $\epsilon$, we obtain the equality \[cpv\] in the limit as $\epsilon$ approaches zero.
For positive integer $n$, the Analytic Principal Value is equal to the finite-part of the divergent integral, $$\begin{aligned}
\label{general}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}} \frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x&=&\lim_{\epsilon\rightarrow 0}\left[\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x \right. \nonumber \\
&& \left.\hspace{18mm} -\sum_{k=0}^{n-1}\frac{f^{(k)}(x_0)}{k! (n-k)} \frac{(1-(-1)^{n-k})}{\epsilon^{n-k}} \right] .
\end{aligned}$$
We use the same contour of integration to calculate the analytic principal value and obtain $$\begin{aligned}
\label{ddd}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}} \frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x&=&\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x \nonumber \\
&&\hspace{12mm}+ \int_{C^+}\frac{f(z)}{(z-x_0)^{n+1}}\mathrm{d}z + i\pi \frac{f^{(n)}(x_0)}{n!} .
\end{aligned}$$ To evaluate the integral around the semi-circle, we again parametrize the semi-circle in the same way we did above and expand $f(x_0+\epsilon \mathrm{e}^{i\theta})$ at least up to the order $O(\epsilon^{n+1})$, $$\label{expand2}
f(x_0+\epsilon \mathrm{e}^{i\theta}) = \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!} \epsilon^k \mathrm{e}^{i k \theta} + \epsilon^{n+1} \mathrm{e}^{i\theta (n+1) } f_{n+1}(x_0+\epsilon\mathrm{e}^{i\theta}),$$ where $f_{n+1}(z)$ is an analytic function. Again from the Taylor expansion theorem, we have the bound $$| f_{n+1}(x_0+\epsilon\mathrm{e}^{i\theta})|\leq \frac{M}{\rho^n (\rho-\epsilon)},$$ where $M$ and $\rho$ are as above. Substituting the expansion \[expand2\] back into equation \[ddd\] and performing the integrations, we obtain $$\begin{aligned}
\label{ddd2}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{a}^{b}}} \frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x&=&\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x + \int_{x_0+\epsilon}^b\frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x \nonumber \\
&&\hspace{-8mm} -\sum_{k=0}^{n-1}\frac{f^{(k)}(x_0)}{k! (n-k)} \frac{(1-(-1)^{n-k})}{\epsilon^{n-k}} + i\epsilon \int_{\pi}^{0}\mathrm{e}^{i\theta} f_{n+1}(x_0+\epsilon\mathrm{e}^{i\theta}) \mathrm{d}\theta.
\end{aligned}$$ By the same arguments we used in the previous Proposition, we obtain equation \[general\] from equation \[ddd2\] in the limit as $\epsilon$ approaches zero.
![The contour of integration $\gamma^+$ is deformed into the contour $\bar{\gamma}^+$. The value of the contour integral along these paths are equal, in particular, the integral does not depend on $\epsilon$.[]{data-label="fig:boat2"}](contour.jpg)
As pointed out earlier, the APV assumes an absolutely convergent integral representation. This, together with Propositions-1 and-2, leads us to the following statement of our main result.
If the function $f(x)$ in the divergent integral \[divergent\] has an analytic complex extension, $f(z)$, in a region in the complex plane containing the integration interval $[a,b]$, then there exists a family of absolutely convergent integral with a value equal to the Cauchy principal value for $n=0$ or to the finite part integral for $n=1, 2, \dots$. The family of integrals is precisely the different integral representations of the Analytic Principal Value of the divergent integral. Each integral representation corresponds to a path in $\Gamma^{\pm}$ or to a pair of paths, one from $\Gamma^+$ and another from $\Gamma^-$.
The Sokhotski-Plemelj-Fox Theorem and the Analytic Principal Value {#spft}
==================================================================
Lemma-1 looks exactly as the Sokhotski-Plemelj theorem (for $n=0$) and its generalized version by Fox. However, they are not the same, but their similarity in form indicates an existing relationship between our Lemma and them.
Let us consider the version of the Sokhotski-Plemelj and Fox Theorem in the real line, which we will refer to as the Sokhotski-Plemelj-Fox Theorem (SPFT). Central to the SPFT is the function $$\Phi(z)=\int_a^b \frac{f(x)}{(x-z)^{n+1}}\mathrm{d}x ,$$ where $z$ does not lie along the contour of integration which is the straight line from $a$ to $b$. The value of $\Phi(z)$ as it approaches any value $x_0\in(a,b)$ depends on the approach, depending on whether the limit is from above or from below the real axis. The limiting values are known as the boundary values of $\Phi(z)$ and they are given by the limiting values $$\Phi^{\pm}(x_0)=\lim_{y\rightarrow 0} \int_a^b \frac{f(x)}{(x-(x_0\pm iy))^{n+1}}\mbox{d}x .$$
The Sokhotski-Plemelj-Fox Theorem is a statement on the relationship between these values and the Cauchy Principal Value or the Finite-Part Integrals, in particular $$\label{boundary1}
\Phi^{+}(x_0)= \#\!\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}} \mbox{d}x + i \pi \frac{f^{(n)}(x_0)}{n!} .$$ $$\label{boundary2}
\Phi^{-}(x_0)= \#\!\!\!\int_a^b \frac{f(x)}{( x-x_0)^{n+1}} \mbox{d}x - i \pi \frac{f^{(n)}(x_0)}{n!} ,$$ where $\#$ takes on either the CPV ($n=0$) or the FPI ($n=1,2,\dots$). Adding this two gives $$\label{plem}
\#\!\!\!\int_a^b \frac{f(x)}{(x-x_0)^{n+1}} \mbox{d}x = \frac{1}{2}\left[\Phi^+(x_0)+\Phi^-(x_0)\right] .$$ That is the CPV and the FPI are the averages of the boundary values of $\Phi(z)$.
We observe that there is an exact correspondence between equations \[principal\] and \[plem\], equations \[form1\] and \[boundary1\], equations \[form2\] and \[boundary2\]. Since the Analytic Principal Value and the Cauchy Principal Value/Finite-Part Integrals are equal, we must have the equality $$\begin{aligned}
\Phi^{\pm}(x_0)=\mathrm{Int}^{\mp}(x_0) .\end{aligned}$$ That is the value of the function $\Phi(z)$ at the boundary is equal to the integral on any path connecting $a$ and $b$ that is deformable to the cut, with the value depending on which side of the cut the path passes through. Then the boundary value in the SPT can be replaced by a contour integral. Since the boundary values $\Phi^{\pm}(x_0)$ can be interpreted as values for particular paths, the SPFT can be seen as a special case of Lemma-1.
Of course we can still maintain the interpretation of the Sokohotski-Plemelj-Fox Theorem as a statement on the relationship between the boundary values of $\Phi(z)$ and the Cauchy Principal Value or the Finite-Part Integral. With this interpretation, the SPFT stands independent from our Lemma. However, Lemma-1, together with Propositions-1 and 2, now provides a way of computing for the boundary values without explicit evaluation of the function $\Phi(z)$ and then taking the required limit. Conversely we can take the SPFT to be a means of computing the integral values $\mathrm{Int}^{\pm}(x_0)$ in terms of the boundary values of $\Phi(z)$. Either way we have now more ways of obtaining the Cauchy Principal Value, and the Finite-Part Integral.
The proof of Proposition-1 is exactly the implementation of the limiting operation in Sokhotski-Plemelj Theorem in obtaining the boundary values as performed in [@pipkin]. It can now be seen that such implementation is more properly interpreted as a specific computation of the Analytic Principal Value rather than the computation of the boundary value coming from the function $\Phi(z)$ itself; that is because the contour is supposed to be fixed in the SPT, and it is the complex variable $z$ that is supposed to approach the singular point $x_0$.
The Cauchy principal value and the finite-part integral involving functions with entire complex extensions {#example}
==========================================================================================================
Let us consider the case when the complex extension, $f(z)$, is entire, so that it posses a Taylor series expansion at any point with an infinite radius of convergence. In particular the following expansion holds $$\label{ps}
f(z)=\sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!} (z-x_0)^k$$ for all $z$ in the complex plane. This gives us the opportunity to evaluate explicitly the CPV and the FPI and compare them with the APV. Moreover, it will be instructive to demonstrate the independence of the APV on the parameter $\epsilon$.
The Cauchy Principal Value and the Finite-Part Integral
-------------------------------------------------------
The CPV and the FPI are known, but we give their derivations here to aid in the derivation of the Analytic Principal Value. Term by term integration, which we can do because the expansion \[ps\] has an infinite radius of convergence, yields the following integrals, $$\begin{aligned}
\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{(-\epsilon)^{n-k}} - \frac{1}{(a-x_0)^{n-k}}\right) \nonumber \\
&& + \frac{f^{(n)}(x_0)}{n!} \left(\ln \epsilon - \ln(x_0-a)\right)\nonumber \\
&& + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k!(k-n)} \left((-\epsilon)^{k-n}-(a-x_0)^{k-n}\right)\end{aligned}$$ $$\begin{aligned}
\int_{x_0+\epsilon}^b \frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{(b-x_0)^{n-k}} - \frac{1}{\epsilon^{n-k}}\right) \nonumber \\
&& + \frac{f^{(n)}(x_0)}{n!} \left(\ln (b-x_0) - \ln\epsilon\right)\nonumber \\
&& + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k!(k-n)} \left((b-x_0)^{k-n}-\epsilon^{k-n}\right)\end{aligned}$$
Adding these two gives the desired integral in the extraction of the finite-part of the divergent integral, $$\begin{aligned}
\label{desired}
&&\int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x+ \int_{x_0+\epsilon}^{b} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x \nonumber \\
&&\hspace{10mm}= \frac{f^{(n)}(x_0)}{n!}\left[\ln(b-x_0)-\ln(x_0-a)\right]+\left[F_n(b-x_0)-F_n(a-x_0)\right]\nonumber\\
&&\hspace{40mm} - \left[F_n(\epsilon) - F_n(-\epsilon)\right]\end{aligned}$$ where $$\label{fn}
F_n(s)=-\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k! (n-k)} \frac{1}{s^{n-k}} + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k! (k-n)} s^{k-n} .$$
The first two terms in equation \[desired\] are independent of $\epsilon$, and the third term contains a group of terms that diverge as $\epsilon\rightarrow 0$. For $n=0$ the diverging group of term in \[desired\] are not present. Then in the limit we obtain the Cauchy principal value $$\begin{aligned}
\mathrm{CPV}\!\!\int_a^b \frac{f(x)}{x-x_0} \mathrm{d}x\!\!\!&=&\!\!\! \sum_{k=1}^{\infty} \frac{f^{(k)}(x_0)}{k! \times k} \left[(b-x_0)^k-(a-x_0)^k\right] \nonumber \\
&&\hspace{12mm} + f(x_0) \left[\ln(b-x_0)-\ln(x_0-a)\right]\label{CPV}\end{aligned}$$ For $n=1, 2, \dots$ the diverging terms are present to make the integral diverge in the limit. The finite part is obtained by dropping the diverging term and is given by $$\begin{aligned}
\mbox{FPI}\!\!\int_a^b\frac{f(x)}{(x-x_0)^{n+1}}\mbox{d}x\!\!\!& =&\!\!\! F_n(b-x_0)-F_n(a-x_0)\nonumber\\
&&\hspace{12mm} + \frac{f^{(n)}(x_0)}{n!} \left[\ln(b-x_0)-\ln(x_0-a)\right].\label{FPI}\end{aligned}$$ The group of terms that we dropped is precisely the sum given by equation \[fox2\].
The Analytic Principal Value
----------------------------
To obtain the APV, let us first consider the integral in $\Gamma^+$. We deform the path $\gamma^+$ into the path consisting of the straight path from $a$ to $x_0-\epsilon$, and the semi-circle with radius $\epsilon$ centered at the origin in the positive direction, and then the straight path from $x_0+\epsilon$ to $b$ for some $\epsilon>0$, as depicted in Figure-2. Then the integral in the upper half-plane becomes $$\begin{aligned}
\int_{\gamma^+} \frac{f(z)}{(z-x_0)^{n+1}} \mathrm{d}z &=& \int_a^{x_0-\epsilon} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x +\int_{x_0+\epsilon}^{b} \frac{f(x)}{(x-x_0)^{n+1}} \mathrm{d}x \nonumber \\
&& \hspace{20mm} + \int_{C^+} \frac{f(z)}{(z-x_0)^{n+1}} \mathrm{d}z\end{aligned}$$ Notice that the first and second terms are just the integrals appearing in the definition of the finite part integral.
The integral around the semi-circle is performed with the parametrization $z=x_0+ \epsilon \mathrm{e}^{i\theta}$, with $\pi>\theta>0$. Expanding $f(z)$ along the contour of integration $$f\left(x_0+\epsilon \mathrm{e}^{i\theta}\right)=\sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!} \epsilon^{k} \mathrm{e}^{i k \theta} ,$$ and substituting its expansion in the integral yield $$\begin{aligned}
\int_{C^+} \frac{f(z)}{(z-x_0)^{n+1}} \mbox{d}z &=& -\sum_{k=0}^{n-1} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\frac{1}{\epsilon^{n-k}} - \frac{1}{(-\epsilon)^{n-k}}\right) - i \pi \frac{f^{(n)}(x_0)}{n!}\nonumber \\
&& + \sum_{k=n+1}^{\infty} \frac{f^{(k)}(x_0)}{k!(n-k)} \left(\epsilon^{n-k}-(-\epsilon)^{n-k}\right) .\end{aligned}$$ Adding all the terms, we find terms involving $\epsilon$ cancel out, leaving the integral independent of $\epsilon$, as it should be. We obtain $$\begin{aligned}
\label{upper}
\int_{\gamma^+} \frac{f(z)}{(z-x_0)^{n+1}}\mbox{d}x\!\!\!& =&\!\!\! F_n(b-x_0)-F_n(a-x_0) \nonumber \\
&&\hspace{1mm}+ \frac{f^{(n)}(x_0)}{n!} \left[\ln(b-x_0)-\ln(x_0-a)\right] - i \pi \frac{f^{(n)}(x_0)}{n!} .\end{aligned}$$
To obtain the integral in $\Gamma^-$, we use the contour which is the mirror image of the contour in the evaluation of the integral in $\Gamma^+$. Following similar steps, we obtain the integral in the lower half-plane $$\begin{aligned}
\label{lower}
\int_{\gamma^-} \frac{f(z)}{(z-x_0)^{n+1}}\mbox{d}x\!\!\!& =&\!\!\! F_n(b-x_0)-F_n(a-x_0) \nonumber \\
&&\hspace{1mm}+ \frac{f^{(n)}(x_0)}{n!} \left[\ln(b-x_0)-\ln(x_0-a)\right] + i \pi \frac{f^{(n)}(x_0)}{n!} .\end{aligned}$$ Observe that equations \[upper\] and \[lower\] differ only in sign in the second term. which is due to the fact that the two paths are oppositely oriented.
While equations \[upper\] and \[lower\] are computed using particular paths, their values, however, do not depend on the chosen path. Their right-hand sides give the values $\mathrm{Int}^+(x_0)$ and $\mathrm{Int}^-(x_0)$, respectively. Any of the equations \[principal\], \[form1\] and \[form2\] can now be used to reproduce the Cauchy principal value and the finite-part integrals obtained above.
Example
-------
Let us compute the Cauchy principal value and the finite-part integral of the following divergent integral, $$\int_{-1}^{1} \frac{\cos x}{x^{n+1}} \mathrm{d}x, \;\;\; n=0,1,\dots ,$$ using equations \[CPV\] and \[FPI\], respectively. It is evident from equation \[CPV\] that the CPV vanishes. Also the FPI vanishes for even $n$. Only when $n$ is odd that the FPI does not vanish. Using equation \[FPI\], the first two non-zero FPI are determined to be $$\begin{aligned}
\mathrm{FPI}\!\!\int_{_-1}^{1} \frac{\cos x}{x^2}\mathrm{d}x \!\!\!&=&\!\!\! -2 + 2\sum_{j=1}^{\infty} \frac{(-1)^j}{(2j)! (2j-1)}\nonumber \\
\!\!\!&=&\!\!\! - 2 \left[\cos(1)+\mathrm{Si}(1)\right],\label{fpi1}\end{aligned}$$ $$\begin{aligned}
\mathrm{FPI}\!\!\int_{_-1}^{1} \frac{\cos x}{x^4}\mathrm{d}x \!\!\!&=&\!\!\! \frac{1}{3} +2 \sum_{j=2}^{\infty} \frac{(-1)^j}{(2j)! (2j-3)}\nonumber \\
\!\!\!&=&\!\!\! \frac{1}{3} \left[\mathrm{Si}(1)+\sin(1)-\cos(1)\right] \label{fpi2},\end{aligned}$$ where $\mathrm{Si}(z)$ is the sine-integral function.
Now we obtain the Analytic Principal Value of the same divergent integrals. The complex extension of $f(x)=\cos x$ is $f(z)=\cos z$, which is an entire function. Let us use the definition of the Principal Value given by equation \[principal\]. We chose for $\gamma^+$ the semi-circle in the upper-half plane centered at the origin with a unit radius; and for $\gamma^-$ the semi-circle in the lower half plane centered at the origin as well. Parameterizing the paths as $z=a \mathrm{e}^{i\theta}$, where $\pi<\theta<0$ for $\gamma^+$ and $-\pi<\theta<0$ for $\gamma^-$, and averaging the integrals for these paths yield the APV $$\begin{aligned}
\label{pvex}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{-1}^{1}}} \frac{\cos x}{x^{n+1}}\mathrm{d}x &=& -\int_0^{\pi}\sin(\cos\theta) \sinh(\sin\theta) \cos(n\theta) \mathrm{d}\theta \nonumber \\
&& -\int_0^{\pi}\cos(\cos\theta) \cosh(\sin\theta) \sin(n\theta) \mathrm{d}\theta \end{aligned}$$ The right hand side of the equation now involves an integration that is absolutely convergent which is amenable to standard methods of integration, either analytically or numerically. For even $n$ the principal value vanishes. Moreover, equations \[fpi1\] and \[fpi2\] are reproduced from equation \[pvex\] by explicit evaluation of the integral (which we did using Mathematica 10.3), in accordance with the equality of the Analytic Principal Value and the Finite-Part Integral.
We may be interested in obtaining the behavior of the FPI for arbitrarily large $n$. It is not immediately clear from the definition of the FPI or from the power series how the asymptotic expansion can be obtained. This is where the integral representation comes in handy in obtaining the asymptotic expansion. The integral representation given by equation \[pvex\] suggests that the expansion can be obtained by the standard method of integration by parts for Fourier integrals [@wong]. Successive integration by parts yields the expansion $$\begin{aligned}
{\ensuremath{\backslash\!\!\!\!\backslash\!\!\!\!\!\int_{-1}^{1}}} \frac{\cos x}{x^{n+1}}\mathrm{d}x\!\!\! &\sim& \!\!\!((-1)^n-1) \left[\frac{\cos(1)}{n}-\frac{\cos(1)+\sin(1)}{n^3}+ \frac{5\sin(1)-6\cos(1)}{n^5} - \cdots \right] \nonumber \\
&& \hspace{-14mm}+ ((-1)^n-1) \left[-\frac{\sin(1)}{n^2}+\frac{3\cos(1)}{n^4} - \frac{5\cos(1)-23\sin(1)}{n^6} + \dots \right],\;\;\; n\rightarrow \infty .\end{aligned}$$ The expansion identically vanishes for even $n$, which is consistent with the fact that the principal value vanishes for such values of $n$. This example demonstrates how the entire repertoire of asymptotic analysis for regular integrals can be used in the asymptotic analysis of hypersingular integrals using their (absolutely convergent) integral representations.
Conclusion
==========
We have shown that, under certain conditions, the Cauchy Principal Value and the Finite-Part Integral are values of absolutely convergent integrals. We have seen that such integral representation of them provides another way of computing their values, analytically, numerically or asymptotically, using standard methods applicable to regular integrals. Furthermore, it offers another way to look at problems at a different perspective. A convergent integral representation may allow us to cast, for example, integral equations involving singular kernels into integral equations involving regular kernels. But of course this cannot be done without complication: the domain of the unknown function will now have to be extended beyond the original domain. It is possible though that new insight can be gained from such rewriting of the original problem. Conversely our results here may allow us to rewrite integral equations involving regular kernels into hypersingular integral equations which, too, may offer new insights not available in the original formulation of the problem.
Clearly there is an enormous potential in absolutely convergent integral representations of divergent integrals such as the family of integrals considered here. However, it is not apparent at the moment if such representation exists for every divergent integral. For our present case, such representation is possible under analyticity condition on the complex extension of the relevant function. However, such condition is a stringent one. It is only necessary for the function to satisfy a Holder condition for the CPV to exist; and for it to further posses derivatives up to order $n$ for the FPI to exist as well. Our results here then do not cover all possible cases of the CPV and the FPI; it is an open question whether an absolutely convergent integral representation exist for the rest of the cases. We leave it to future developments in the exploration of the possibility of obtaining an absolutely convergent integral representation of any given divergent integral.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was funded by the UP System Enhanced Creative Work and Research Grant (ECWRG 2015-2-016).
[00]{} Shiekh, A.Y. [*Zeta-function regularization of quantum field theory*]{} Can. J. Phys. [**68**]{} 620 (1990).
Lee, H. and Milgram, M. [*On the Regularization of a Class of Divergent Feynman Integrals in Covariant and Axial Gauges*]{} Ann. Phys. [**147**]{} 408 (1984).
Ioakimidis, N.J. [*Generalized Mangler-type principal value integrals with an application to fracture mechanics*]{} J. Comp. and Appl. Math. [**30**]{} 227 (1990).
Davies, K.T.R, and Davies, R.W. [*Evaluation of a class of integrals occurring in mathematical physics via a higher order generalization of the principal value*]{} Can. J. Phys. [**67**]{} 759 (1989).
L.K. Lifanov, L.N. Poltavskii, and G.M. Vianiko [*Hypersingular Integral Equations and their applications*]{} Chapmann & Hall/CRC (2004).
A.C. Pipkin [*A course of integral equations*]{} Text in Applied Mathematics [**9**]{} Springer (1991).
W. Ang [*Hypersingular integral equations in fracture analysis*]{} Woodhead Publishing (2013).
R. Estrada and R.P. Kanwal [*Singular Integral Equations* ]{} Springer (2000).
Scott M. Cohen, S.M., Davies, Scott M. Cohen, K.T.R. Davies, R.W. Davies, and M. Howard Lee, Davies, R.W. and Lee, H.M. [*Principal-value integrals — Revisited*]{} Can. J. Phys. [**83**]{} 565 (2005).
Y. Chan, A.C. Fannjiang, G.H. Paulino, and B. Feng [*Finite Part Integrals and Hypersingular Kernels*]{} DCDIS A Supplement, Advances in Dynamical Systems, Vol. 14(S2)264 –269 (2007).
G. Monegato [*Definitions, properties and applications of finite-part integrals*]{} Journal of Computational and Applied Mathematics [**229**]{}, 425–439 (2009).
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1952).
C. Fox [*A generalization of the Cauchy principal value*]{} Canad. J. Math. [**9**]{} 110-117 (1957).
Yu.V. Sokhotski [*On Definite Integrals and Functions Employed in Expansions into Series*]{} St. Petersburg (1873).
J. Plemelj, Monat. Math. Phys. [**19**]{} 205-210 (1908).
E.G. Ladopoulos [*New aspects for the generalization of the Sokhotski-Plemelj formulae for the solution of finite-part singular integrals used in fracture mechanics*]{} International Journal of Fracture [**54**]{} 317-328 (1992).
Ioakimidis, N.J. [*A new interpretation of Cauchy type singular integrals with an application to singular integral equations*]{} J. Comp. and Appl. Math. [**14**]{} 271 (1986).
DeKlerk J. [*Cauchy principal value and hypersingular integrals*]{} (2011). Last accessed on 13 November 2015 at http://www.nwu.ac.za/sites/www.nwu.ac.za/files/files/p-ubmi/documents/hipersinintegrale.pdf
Criscuolo G. [*A new algorithm for Cauchy principal value and Hadamard finite-part integrals*]{} J. Comp. and Appl. Math [**78**]{} 255 (1997).
Bialecki, B. [*A Sine Quadrature Rule for Hadamard Finite-part Integrals*]{} Numer. Math. [**57**]{} 263 (1990).
J.L. Guermond [*A systematic formula for the asymptotic expansion of singular integrals*]{} J. Appl. Math. and Phys (ZAMP) [**38**]{} 717-728 (1987).
R. Wong [*Asymptotic Approximations of Integrals*]{} SIAM (2001).
L.V. Ahlfors [*Complex Analysis*]{} 3rd Edition McGraw-Hill Inc. (1979).
|
---
author:
- Ilaria Castellano
- Anna Giordano Bruno
date: '*Dedicated to the 70eth birthday of Luigi Salce*'
title: Algebraic entropy in locally linearly compact vector spaces
---
Introduction {#intro}
============
In [@AKM] Adler, Konheim and McAndrew introduced the notion of topological entropy $h_{top}$ for continuous self-maps of compact spaces, and they concluded the paper by sketching a definition of the algebraic entropy $h_{alg}$ for endomorphisms of abelian groups. This notion of algebraic entropy, which is appropriate for torsion abelian groups and vanishes on torsion-free abelian groups, was later reconsidered by Weiss in [@Weiss], who proved all the basic properties of $h_{alg}$. Recently, $h_{alg}$ was deeply investigated by Dikranjan, Goldsmith, Salce and Zanardo for torsion abelian groups in [@Aeag], where they proved in particular the Addition Theorem and the Uniqueness Theorem.
Later on, Peters suggested another definition of algebraic entropy for automorphisms of abelian groups in [@Peters1]; here we denote Peters’ entropy still by $h_{alg}$, since it coincides with Weiss’ notion on torsion abelian groups; on the other hand, Peters’ entropy is not vanishing on torsion-free abelian groups. In [@DGBpet] $h_{alg}$ was extended to all endomorphisms and deeply investigated, in particular the Addition Theorem and the Uniqueness Theorem were proved in full generality. In [@Peters2] Peters gave a further generalization of his notion of entropy for continuous automorphisms of locally compact abelian groups, which was recently extended by Virili in [@Virili] to continuous endomorphisms.
Weiss in [@Weiss] connected the algebraic entropy $h_{alg}$ for endomorphisms of torsion abelian groups with the topological entropy $h_{top}$ for continuous endomorphisms of totally disconnected compact abelian groups by means of Pontryagin duality. Moreover, the same connection was shown by Peters in [@Peters1] between $h_{alg}$ for topological automorphisms of countable abelian groups and $h_{top}$ for topological automorphisms of metrizable compact abelian groups. These results, known as Bridge Theorems, were recently extended to endomorphisms of abelian groups in [@DGB-BT], to continuous endomorphisms of locally compact abelian groups with totally disconnected Pontryagin dual in [@GBD], and to topological automorphisms of locally compact abelian groups in [@Virili-BT] (in the latter two cases on the Potryagin dual one considers an extension of $h_{top}$ to locally compact groups based on a notion of entropy introduced by Hood in [@H] as a generalization of Bowen’s entropy from [@B] – see also [@GBVirili]).
A generalization of Weiss’ entropy in another direction was given in [@SZ], where Salce and Zanardo introduced the $i$-entropy $\ent_i$ for endomorphisms of modules over a ring $R$ and an invariant $i$ of $\mathrm{Mod}(R)$. For abelian groups (i.e., ${\mathbb Z}$-modules) and $i=\log|-|$, $\ent_i$ coincides with Weiss’ entropy. Moreover, the theory of the entropies $\ent_L$ where $L$ is a length function was pushed further in [@SVV; @SV].
In [@GBSalce] the easiest case of $\ent_i$ was studied, namely, the case of vector spaces with the dimension as invariant, as an introduction to algebraic entropy in the most convenient and familiar setting. The *dimension entropy* $\ent_{\dim}$ is defined for an endomorphism $\phi:V\to V$ of a vector space $V$ as $$\ent_{\dim }(\f)=\sup\{H_{\dim}(\f,F): F\leq V,\ \dim F<\infty\},$$ where $$H_{\dim }(\f,F)=\lim_{n\to\infty}\frac{1}{n}\dim(F+\phi F+\ldots+\phi^{n-1}F).$$ All the basic properties of $\ent_{\dim}$ were proved in [@GBSalce], namely, Invariance under conjugation, Monotonicity for linear subspaces and quotient vector spaces, Logarithmic Law, Continuity on direct limits, weak Addition Theorem (see Section \[ss:Properties\] for the precise meaning of these properties).
Moreover, compared to the Addition Theorem for $h_{alg}$ and other entropies, a simpler proof was given in [@GBSalce Theorem 5.1] of the Addition Theorem for $\ent_{\dim}$, which states that if $V$ is a vector space, $\phi:V\to V$ an endomorphism and $W$ a $\phi$-invariant (i.e., $\phi W\leq W$) linear subspace of $V$, then $$\ent_{\dim}(\phi)=\ent_{\dim}(\phi\restriction_W)+\ent_{\dim}(\overline \phi),$$ where $\overline\phi:V/W\to V/W$ is the endomorphism induced by $\phi$.
Also the Uniqueness Theorem is proved for the dimension entropy (see [@GBSalce Theorem 5.3]), namely $\ent_{\dim}$ is the unique collection of functions $\ent_{\dim}^V:\End(V)\to{\mathbb N}\cup\{\infty\}$, $\phi\mapsto\ent_{\dim}(\phi)$, satisfying for every vector space $V$: Invariance under conjugation, Continuity on direct limits, Addition Theorem and $\ent_{\dim}(\beta_F)=\dim F$ for any finite-dimensional vector space $F$, where $\beta_F:\bigoplus_{\mathbb N}F\to \bigoplus_{\mathbb N}F$, $(x_0,x_1,x_2,\ldots)\mapsto (0,x_0,x_1,\ldots)$ is the right Bernoulli shift.
Inspired by the extension of $h_{alg}$ from the discrete case to the locally compact one, and by the approach used in [@intrinsic] to define the intrinsic algebraic entropy, we extend the dimension entropy to continuous endomorphisms of locally linearly compact vector spaces. Recall that a linearly topologized vector space $V$ over a discrete field ${\mathbb K}$ is *locally linearly compact* (briefly, l.l.c.) if it admits a local basis at $0$ consisting of linearly compact open linear subspaces; we denote by ${{\mathcal{B}}(V)}$ the family of all linearly compact open linear subspaces of $V$ (see [@Lef]). Clearly, linearly compact and discrete vector spaces are l.l.c.. (See Section \[s:llc\] for some background on linearly compact and locally linearly compact vector spaces.)
Let $V$ be an l.l.c. vector space and $\f\colon V\to V$ a continuous endomorphism. The *algebraic entropy of $\phi$ with respect to $U\in{{\mathcal{B}}(V)}$* is $$\label{hdef}
H(\f,U)=\lim_{n\to\infty}\frac{1}{n}\dim\frac{U+\phi U+\ldots+\phi^{n-1}U}{U},$$ and the *algebraic entropy* of $\f$ is $$\ent(\f)=\sup\{ H(\f,U)\mid U\in\mathcal B(V)\}.$$ In Section \[s:ent\] we show that the limit in exists. Moreover, we see in Corollary \[lc0\] that $\ent$ is always zero on linearly compact vector spaces. On the other hand, if $V$ is a discrete vector space, then $\ent(\f)$ turns out to coincide with $\ent_{\dim}(\phi)$ (see Lemma \[entdim\]). Moreover, if $V$ is an l.l.c. vector space over a finite field $\mathbb F$, then $V$ is a totally disconnected locally compact abelian group and $h_{alg}(\phi)=\ent(\phi)\cdot\log|\mathbb F|$ (see Lemma \[halg\]).
In Section \[ss:Properties\] we prove all of the general properties that the algebraic entropy is expected to satisfy, namely, Invariance under conjugation, Monotonicity for linear subspaces and quotient vector spaces, Logarithmic Law, Continuity on direct limits, weak Addition Theorem. As a consequence of the computation of the algebraic entropy for the Bernoulli shifts (see Example \[bernoulli\]), we find in particular that the algebraic entropy for continuous endomorphisms of l.l.c. vector spaces takes all values in ${\mathbb N}\cup\{\infty\}$.
In Section \[lff-sec\] we prove the so-called Limit-free Formula for the computation of the algebraic entropy, that permits to avoid the limit in the definition in (see Proposition \[lf-ent\]). Indeed, taken $V$ an l.l.c. vector space and $\phi:V\to V$ a continuous endomorphism, for every $U\in{{\mathcal{B}}(V)}$ we construct an open linear subspace $U^-$ of $V$ (see Definition \[def:uminus\]) such that $\phi^{-1}U^-$ is an open linear subspace of $U^-$ of finite codimension and $$H(\phi,U)=\dim \frac{U^-}{\phi^{-1}U^-}.$$
A first Limit-free Formula for $h_{alg}$ in the case of injective endomorphisms of torsion abelian groups was sketched by Yuzvinski in [@Y] and was later proved in a slightly more general setting in [@DGB-lff]; this result was extended in [@yuzapp Lemma 5.4] to a Limit-free Formula for the intrinsic algebraic entropy of automorphisms of abelian groups. In [@DGB-lff] one can find also a Limit-free Formula for the topological entropy of surjective continuous endomorphisms of totally disconnected compact groups, which was extended to continuous endomorphisms of totally disconnected locally compact groups in [@GBVirili Proposition 3.9], using ideas by Willis in [@Willis]. Our Limit-free Formula is inspired by all these results, mainly by ideas from the latter one.
The Limit-free Formula is one of the main tools that we use in Section \[AT-sec\] to extend the Addition Theorem from the discrete case (i.e., the Addition Theorem for $\ent_{\dim}$ [@GBSalce Theorem 5.1]) to the general case of l.l.c vector spaces (see Theorem \[thm:ATllc\]). If $V$ is an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $W$ a closed $\phi$-invariant linear subspace of $V$, consider the following commutative diagram $$\xymatrix{
0\ar[r]&W\ar[r]\ar[d]^{\f\restriction_W}&V\ar[r]\ar[d]^{\f}&V/W\ar[r]\ar[d]^{\overline\f}&0\\
0\ar[r]&W\ar[r]&V\ar[r]&V/W\ar[r]&0}$$ of continuous endomorphisms of l.l.c. vector spaces, where $\f\restriction_W$ is the restriction of $\f$ to $W$ and $\overline\f$ is induced by $\phi$; we say that the Addition Theorem holds if $$\ent(\f)=\ent(\f\restriction_W)+\ent(\overline\f).$$
While it is known that $h_{alg}$ satisfies the Addition Theorem for endomorphisms of discrete abelian groups (see [@DGBpet]), it is still an open problem to establish whether $h_{alg}$ satisfies the Addition Theorem in the general case of continuous endomorphisms of locally compact abelian groups; from the Addition Theorem for the topological entropy in [@GBVirili] and the Bridge Theorem in [@GBD] one can only deduce that the Addition Theorem holds for $h_{alg}$ in the case of topological automorphisms of locally compact abelian groups which are compactly covered (i.e., they have totally disconnected Pontryagin dual). Here, Theorem \[thm:ATllc\] shows in particular that the Addition Theorem holds for $h_{alg}$ on the small subclass of compactly covered locally compact abelian groups consisting of all locally linearly compact spaces over finite fields.
With respect to the Uniqueness Theorem for $\ent_{\dim}$ mentioned above, we leave open the following question.
Does a Uniqueness Theorem hold also for the algebraic entropy $\ent$ on locally linearly compact vector spaces?
In other words, we ask whether $\ent$ is the unique collection of functions $\ent^V:\End(V)\to{\mathbb N}\cup\{\infty\}$, $\phi\mapsto\ent(\phi)$, satisfying for every l.l.c. vector space $V$: Invariance under conjugation, Continuity on direct limits, Addition Theorem and $\ent(\beta_F)=\dim F$ for any finite-dimensional vector space $F$, where $V=\bigoplus_{n=-\infty}^0 F\oplus \prod_{n=1}^\infty F$ is endowed with the topology inherited from the product topology of $\prod_{n\in{\mathbb Z}}F$, and $\beta_F:V \to V$, $(x_n)_{n\in{\mathbb Z}}\mapsto (x_{n-1})_{n\in{\mathbb Z}}$ is the right Bernoulli shift (see Example \[bernoulli\]).
We end by remarking that in [@CGB] we introduce a topological entropy for l.l.c. vector spaces and connect it to the algebraic entropy studied in this paper by means of Lefschetz Duality, by proving a Bridge Theorem in analogy to the ones recalled above for $h_{alg}$ and $h_{top}$ in the case of locally compact abelian groups and their continuous endomorphisms.
Background on locally linearly compact vector spaces {#s:llc}
====================================================
Fix an arbitrary field ${\mathbb K}$ endowed always with the discrete topology. A topological vector space $V$ over ${\mathbb K}$ is said to be *linearly topologized* if it is Hausdorff and it admits a neighborhood basis at $0$ consisting of linear subspaces of $V$. Clearly, a discrete vector space $V$ is linearly topologized, and if $V$ has finite dimension then the vice-versa holds as well (see [@Lef p.76, (25.6)]).
If $W$ is a linear subspace of a linearly topologized vector space $V$, then $W$ with the induced topology is a linearly topologized vector space; if $W$ is also closed in $V$, then $V/W$ with the quotient topology is a linearly topologized vector space as well. Given a linearly topologized vector space $V$, a *linear variety* $M$ of $V$ is a subset $v+W$, where $v\in V$ and $W$ is a linear subspace of $V$. A linear variety $M=v+W$ is said to be *open* (respectively, *closed*) in $V$ if $W$ is open (respectively, closed) in $V$.
A linearly topologized vector space $V$ is *linearly compact* if any collection of closed linear varieties of $V$ with the finite intersection property has non-empty intersection (equivalently, any collection of open linear varieties of $V$ with the finite intersection property has non-empty intersection) (see [@Lef]).
For reader’s convenience, we collect in the following proposition all those properties concerning linearly compact vector spaces that we use further on.
\[prop:lc properties\] Let $V$ be a linearly topologized vector space.
- If $W$ is a linearly compact subspace of $V$, then $W$ is closed.
- If $V$ is linearly compact and $W$ is a closed linear subspace of $V$, then $W$ is linearly compact.
- If $W$ is a linearly topologized vector space and $\phi:V\to W$ is a surjective continuous homomorphism, then $W$ is linearly compact.
- If $V$ is discrete, then $V$ is linearly compact if and only if it has finite dimension (hence, if $V$ has finite dimension then $V$ is linearly compact).
- If $W$ is a closed linear subspace of $V$, then $V$ is linearly compact if and only if $W$ and $V/W$ are linearly compact.
- The direct product of linearly compact vector spaces is linearly compact.
- An inverse limit of linearly compact vector spaces is linearly compact.
- A linearly compact vector space is complete.
A proof for (a), (b), (c) and (d) can be found in [@Lef page 78]. Properties (e) and (f) are proved in [@Mat Propositions 2 and 9]. Finally, (g) follows from (b) and (f). Let $\iota\colon V\to\tilde V$ be the topological dense embedding of $V$ into its completion $\tilde{V}$, thus (a) implies (h).
A linearly topologized vector space $V$ is *locally linearly compact* (briefly, l.l.c.) if there exists an open linear subspace of $V$ that is linearly compact (see [@Lef]). Thus $V$ is l.l.c. if and only if it admits a neighborhood basis at $0$ consisting of linearly compact linear subspaces of $V$. Linearly compact and discrete vector spaces are l.l.c. vector spaces, of course. The structure of an l.l.c. vector space can be characterized as follows.
\[thm:dec\] If $V$ is an l.l.c. vector space, then $V$ is topologically isomorphic to $V_c\oplus V_d$, where $V_c$ is a linearly compact linear subspace of $V$ and $V_d$ is a discrete linear subspace of $V$.
By Proposition \[prop:lc properties\] and Theorem \[thm:dec\], one may prove that an l.l.c. vector space verifies the following properties.
\[rem:complete\] Let $V$ be a linearly topologized vector space.
- If $V$ is l.l.c., then $V$ is complete.
- If $W$ is an l.l.c. linear subspace of $V$, then $W$ is closed.
- If $W$ is a closed linear subspace of $V$, then $V$ is l.l.c. if and only if $W$ and $V/W$ are l.l.c..
Given an l.l.c. vector space $V$, for the computation of the algebraic entropy we are interested in the neighborhood basis ${{\mathcal{B}}(V)}$ at $0$ of $V$ consisting of all linearly compact open linear subspaces of $V$. We see now how the local bases ${\mathcal{B}}(W)$ and ${\mathcal{B}}(V/W)$ of a closed linear subspace $W$ of $V$ and the quotient $V/W$ depend on ${{\mathcal{B}}(V)}$.
\[prop:basis\] Let $V$ be an l.l.c. vector space and $W$ a closed linear subspace of $V$. Then:
(a) ${\mathcal{B}}(W)=\{U\cap W\mid U\in{{\mathcal{B}}(V)}\}$;
(b) ${\mathcal{B}}(V/W)=\{(U+W)/W\mid U\in{{\mathcal{B}}(V)}\}$.
\(a) Clearly, $\{U\cap W\mid U\in{{\mathcal{B}}(V)}\}\subseteq{\mathcal{B}}(W)$. Conversely, let $U_W\in\mathcal B(W)$. Since $U_W$ is open in $W$, there exists an open subset $A\subseteq V$ such that $U_W=A\cap W$. As $A$ is a neighborhood of $0$, there exists $U'\in{{\mathcal{B}}(V)}$ such that $U'\subseteq A$. In particular, $U'\cap W\subseteq U_W$ is an open subspace of the linearly compact space $U_W$, and so $U_W/(U'\cap W)$ has finite dimension by Proposition \[prop:lc properties\](d,e). Therefore, there exists a finite-dimensional subspace $F\leq U_W$ such that $U_W=F+(U'\cap W)$. Finally, let $U:=F+U'\in{{\mathcal{B}}(V)}$. Hence, for $F\leq W$ we have $U_W=F+(U'\cap W)=(F+U')\cap W=U\cap W$.
\(b) Since the canonical projection $\pi:V\to V/W$ is continuous and open, the set $\{\pi(U)\mid U\in {{\mathcal{B}}(V)}\}$ is contained in ${\mathcal{B}}(V/W)$. To prove that ${\mathcal{B}}(V/W)\subseteq\{(U+W)/W\mid U\in{{\mathcal{B}}(V)}\}$, consider $\overline U\in\mathcal B(V/W)$ and let $\pi:V\to V/W$ be the canonical projection. Then $\pi^{-1} \overline U$ is an open linear subspace of $V$, hence it contains some $U\in{{\mathcal{B}}(V)}$. Then $\pi U\leq\overline U$ and $\pi U$ has finite codimension in $\overline U$ by Proposition \[prop:lc properties\](d,e). Therefore, there exists a finite-dimensional linear subspace $\overline F$ of $V/W$ such that $\overline F\leq \overline U$ and $\overline U=\pi U+\overline F$. Let $F$ be a finite-dimensional linear subspace of $V$ such that $F\leq \pi^{-1} \overline U$ and $\pi F=\overline F$. Now $\pi(U+F)=\overline U$ and $U+F\in{{\mathcal{B}}(V)}$ by Proposition \[prop:lc properties\](c).
As consequence of Lefschetz Duality Theorem, every linearly compact vector space is topologically isomorphic to a direct product of one-dimensional vector spaces (see [@Lef Theorem 32.1]). From this result, we derive the known properties that if a linearly topologized vector space $V$ over a finite discrete field is linearly compact then it is compact, and if $V$ is l.l.c. then it is locally compact.
\[prop:compact\] Let $V$ be a linearly compact vector space over a discrete field ${\mathbb K}$. Then $V$ is compact if and only if ${\mathbb K}$ is finite.
Write $V=\prod_{i\in I} {\mathbb K}_i$ with ${\mathbb K}_i={\mathbb K}$ for all $i\in I$. If ${\mathbb K}$ is finite, then ${\mathbb K}_i$ is compact for all $i\in I$, and so $V$ is compact. Conversely, if $V$ is compact, then each ${\mathbb K}_i$ is compact as well, hence ${\mathbb K}$ is a compact discrete field, so ${\mathbb K}$ is finite.
\[cor:tdlc\] An l.l.c. vector space $V$ over a finite discrete field ${\mathbb F}$ is a totally disconnected locally compact abelian group.
By Proposition \[prop:compact\], ${{\mathcal{B}}(V)}$ is a local basis at 0 of $V$ consisting of compact open subgroups, thus van Dantzig Theorem yields the claim.
Existence of the limit and basic properties {#s:ent}
===========================================
Let $V$ be an l.l.c. vector space, $\f:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. For $n\in{\mathbb N}_+$ and a linear subspace $F$ of $V$, the *$n^{th}$ partial $\phi$-trajectory* of $F$ is $$T_n(\phi,F)= F+\phi F+\phi^2 F+\ldots +\phi^{n-1} F.$$ If $U\in{{\mathcal{B}}(V)}$, notice that for every $n\in{\mathbb N}_+$, $T_n(\phi,U)\in{{\mathcal{B}}(V)}$ as well, as it is open being the union of cosets of $U$, and linearly compact by Proposition \[prop:lc properties\](c,f). Moreover, $T_n(\phi,U)\leq T_{n+1}(\phi,U)$ for all $n\in{\mathbb N}_+$, thus we obtain an increasing chain of linearly compact open linear subspaces of $V$, namely $$U=T_1(\f,U)\leq T_2(\phi,U)\leq\ldots\leq T_{n}(\phi,U)\leq T_{n+1}(\phi,U)\leq\ldots.$$ Moreover, the *$\phi$-trajectory* of $U$ is $T(\phi,U)=\bigcup_{n\in{\mathbb N}_+} T_n (\phi,U),$ which is open and it is the smallest $\f$-invariant linear subspace of $V$ containing $U$. Hence, the algebraic entropy of $\phi$ with respect to $U$ introduced in can be written as $$\label{eq:rel ent}
H(\phi,U)=\lim_{n \to \infty} \frac{1}{n}\dim\frac{T_n(\phi,U)}{U}.$$ Notice that since $T_n(\f,U)$ is linearly compact and $U$ is open, $U$ has finite codimension in $T_n(\phi,U)$, that is, $\frac{T_n(\phi,U)}{U}$ has finite dimension by Proposition \[prop:lc properties\](d,e). Moreover, the following result shows that the limit in exists.
\[entvalue\] Let $V$ be an l.l.c. vector space and $\phi:V\to V$ a continuous endomorphism. For every $n\in{\mathbb N}_+$ let $$\alpha_n=\dim\frac{T_{n+1}(\phi,U)}{T_n(\phi,U)}.$$ Then the sequence of non-negative integers $\{\alpha_n\}_n$ is stationary and $H(\phi,U)=\alpha$ where $\alpha$ is the value of the stationary sequence $\{\alpha_n\}_n$ for $n$ large enough.
For every $n>1$, $T_{n+1}(\phi,U)=T_n(\phi,U)+\phi^nU$ and $\phi T_{n-1}(\phi,U)\leq T_{n}(\phi,U)$. Thus $$\frac{T_{n+1}(\phi,U)}{T_n(\phi,U)}\cong\frac{\phi^n U}{T_n(\phi,U)\cap\phi^n U}$$ is a quotient of $$B_n=\frac{\phi^n U}{\phi T_{n-1}(\phi,U)\cap \phi^n U}.$$ Therefore $\alpha_{n}\leq\dim B_n$. Moreover, since $\phi T_n(\phi,U)=\phi T_{n-1}(\phi,U)+\phi^n U$, $$B_n\cong\frac{\phi T_{n-1}(\phi,U)+\phi^n U}{\phi T_{n-1}(\phi,U)}=\frac{\phi T_n(\phi,U)}{\phi T_{n-1}(\phi,U)}\cong \frac{T_n(\phi,U)}{T_{n-1}(\phi,U)+(T_n(\phi,U)\cap \ker\phi)};$$ the latter vector space is a quotient of ${T_n(\phi,U)}/{T_{n-1}(\phi,U)}$, so $\dim B_n\leq \alpha_{n-1}$. Hence $\alpha_{n}\leq \alpha_{n-1}$. Thus $\{\alpha_n\}_n$ is a decreasing sequence of non-negative integers, therefore stationary. Since $U\leq T_n(\f,U)\leq T_{n+1}(\f,U)$, $$\label{eqa1}
\alpha_{n}=\dim\frac{T_{n+1}(\f,U)}{U}-\dim\frac{T_n(\f,U)}{U}.$$ As $\{\alpha_n\}_{n}$ is stationary, there exist $n_0>0$ and $\alpha\geq 0$ such that $\alpha_n=\alpha$ for every $n\geq n_0$. If $\alpha=0$, equivalently $\dim\frac{T_{n+1}(\f,U)}{U}=\dim\frac{T_n(\f,U)}{U}$ for every $n\geq n_0$, and hence $H(\f,U)=0$. If $\alpha>0$, by we have that for every $n\in{\mathbb N}$ $$\dim\frac{T_{n_0+n}(\phi,U)}{U}=\dim\frac{T_{n_0}(\phi,U)}{U}+n\alpha.$$ Thus, $$H(\f,U)=\lim_{n\to \infty} \frac{1}{n+n_0} \dim \frac{T_{n_0+n}(\f,U)}{U}=\lim_{n\to\infty}\frac{\dim\frac{T_{n_0}(\f,U)}{U}+n \alpha}{n+n_0}=\alpha.$$ This concludes the proof.
Proposition \[entvalue\] yields that the value of $\ent(\phi)$ is either a non-negative integer or $\infty$. Moreover, Example \[bernoulli\] below witnesses that $\ent$ takes all values in ${\mathbb N}\cup\{\infty\}$.
We see now that the algebraic entropy $\ent$ coincides with $\ent_{\dim}$ on discrete vector spaces.
\[entdim\] Let $V$ be a discrete vector space and $\f\colon V\to V$ an endomorphism. Then $$\ent(\phi)=\ent_{\dim}(\phi).$$
Note that $\mathcal B(V)=\{F\leq V:\dim F<\infty\}$. Let now $F\in{{\mathcal{B}}(V)}$. Then $$\begin{split}
H(\phi,F)=\lim_{n\to\infty}\frac{1}{n}\dim\frac{T_n(\phi,F)}{F}=\lim_{n\to\infty}\frac{1}{n}\left(\dim{T_n(\phi,F)}-\dim{F}\right)=\\=\lim_{n\to\infty}\frac{1}{n}\dim T_n(\phi,F)=H_{\dim}(\phi,F).
\end{split}$$ It follows from the definitions that $\ent(\phi)=\ent_{\dim}(\phi)$.
We compute now the algebraic entropy in the easiest case of the identity automorphism.
\[ex:id\]
- Let $\f\colon V\to V$ be a continuous endomorphism of an l.l.c. vector space $V$. Then $H(\f,U)=0$ for every $U\in{\mathcal{B}}(V)$ which is $\phi$-invariant.
- Let $\f=\mathrm{id}_V$. Since every element of ${{\mathcal{B}}(V)}$ is $\f$-invariant, (a) easily implies $\ent(\mathrm{id}_V)=0$.
Inspired by the above example we provide now the general case of when the algebraic entropy is zero.
Let $V$ be an l.l.c. vector space, $\f:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Then the following conditions are equivalent:
(a) $H(\phi,U)=0$;
(b) there exists $n\in{\mathbb N}_+$ such that $T(\phi,U)=T_n(\phi,U)$;
(c) $T(\phi,U)$ is linearly compact.
In particular, $\ent(\phi)=0$ if and only if $T(\phi,U)$ is linearly compact for all $U\in{{\mathcal{B}}(V)}$.
(a)$\Rightarrow$(b) If $H(\phi,U)=0$, then $\dim\frac{T_{n+1}(\phi,U)}{T_n(\phi,U)}=0$ eventually by Proposition \[entvalue\]. Therefore, the chain of linearly compact open linear subspaces $\left\{ T_n(\phi,U) \right\}_{n\in{\mathbb N}}$ is stationary. (b)$\Rightarrow$(c) is clear from the definition. (c)$\Rightarrow$(a) If $T(\phi,U)$ is linearly compact, by Proposition \[prop:lc properties\](d,e) we have that $\frac{T(\phi,U)}{U}$ is finite-dimensional. Since $T(\phi,U)=\bigcup_{n\in {\mathbb N}_+}T_n(\phi,U)$, it follows that $$\frac{T(\phi,U)}{U}=\bigcup_{n\in {\mathbb N}_+}\frac{T_n(\phi,U)}{U}$$ and so the chain $\left\{\frac{T_{n}(\phi,U)}{U}\right\}_{n\in{\mathbb N}}$ is stationary. Therefore, $H(\phi,U)=0$.
As a consequence we see that $\ent$ always vanishes on linearly compact vector spaces.
\[lc0\] If $V$ is a linearly compact vector space and $\phi\colon V\to V$ a continuous endomorphism, then $\ent(\f)=0$. In particular, if $V$ is a finite dimensional vector space, then $\ent(\f)=0$.
The next result shows that when $\ent(\phi)$ is finite, this value is realized on some $U\in{{\mathcal{B}}(V)}$.
\[ent=H\] Let $V$ be an l.l.c. vector space and $\phi:V\to V$ a continuous endomorphism. If $\ent(\phi)$ is finite, then there exists $U\in{{\mathcal{B}}(V)}$ such that $\ent(\phi)=H(\phi,U)$.
Since $\ent(\phi)$ is finite and $H(\phi,U)\in{\mathbb N}$ for every $U\in{{\mathcal{B}}(V)}$ by Proposition \[entvalue\], the subset $\{H(\phi,U):U\in{{\mathcal{B}}(V)}\}$ of ${\mathbb N}$ is bounded, hence finite. Therefore, $$\ent(\phi)=\sup\{H(\phi,U)\mid U\in{{\mathcal{B}}(V)}\}=\max\{H(\phi,U)\mid U\in{{\mathcal{B}}(V)}\};$$ in other words, $\ent(\phi)=H(\phi,U)$ for some $U\in{{\mathcal{B}}(V)}$ as required.
We prove now the monotonicity of $H(\f,-)$ on the family ${{\mathcal{B}}(V)}$ ordered by inclusion.
\[lem:mono\] Let $V$ be an l.l.c. vector space and $\f:V\to V$ a continuous endomorphism. If $U,U'\in{{\mathcal{B}}(V)}$ are such that $U'\leq U$, then $H(\f,U')\leq H(\f,U)$.
For $n\in{\mathbb N}_+$, since $T_n(\f,U')+U$ is a linear subspace of $T_n(\f,U)$, we have $$\frac{T_n(\f,U')/U'}{(T_n(\f,U')\cap U)/U'}\cong \frac{T_n(\f,U')}{T_n(\f,U')\cap U}\cong\frac{T_n(\f,U')+U}{U}\leq \frac{T_n(\f,U)}{U}.$$ Thus, $$\dim\frac{T_n(\f,U')}{U'}\leq\dim\frac{T_n(\f,U)}{U}+\dim\frac{T_n(\f,U')\cap U}{U'}.$$ Finally, since $\dim\frac{T_n(\f,U')\cap U}{U'}\leq \dim\frac{U}{U'}$, which is constant, for $n\to \infty$ we obtain the thesis.
Let $(I,\leq)$ be a poset. A subset $J\subseteq I$ is said to be *cofinal* in $I$ if for every $i\in I$ there exists $j\in J$ such that $i\leq j$. The following consequence of Lemma \[lem:mono\] permits to compute the algebraic entropy on a cofinal subset of ${{\mathcal{B}}(V)}$ ordered by inclusion.
\[cor:base\] Let $V$ be an l.l.c. vector space and $\phi: V \to V$ a continuous endomorphism.
(a) If ${\mathcal{B}}$ is a cofinal subset of ${{\mathcal{B}}(V)}$, then $\ent(\phi)=\sup\{H(\phi,U)\mid U\in{\mathcal{B}}\}$.
(b) If $U_0\in{{\mathcal{B}}(V)}$ and $\mathcal B=\{U\in{{\mathcal{B}}(V)}:U_0\leq U\}$, then $\ent(\phi)=\sup\{H(\phi,U)\mid U\in{\mathcal{B}}\}$.
\(a) follows immediately from Lemma \[lem:mono\] and the definition.
\(b) Since $U_0+U\in\mathcal B$ for every $U\in{{\mathcal{B}}(V)}$, it follows that that $\mathcal B$ is cofinal in ${{\mathcal{B}}(V)}$, so item (a) gives the thesis.
The following result simplifies the computation of the algebraic entropy in several cases.
\[lem:finite part\] Let $V$ be an l.l.c. vector space, $\f\colon V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Then there exists a finite-dimensional linear subspace $F$ of $U$ such that, for every $n\in{\mathbb N}_+$, $$T_n(\phi,U)=U+T_n(\phi,F).$$
We proceed by induction on $n\in{\mathbb N}_+$. For $n=1$ it is obvious. Since $U$ has finite codimension in $T_2(\phi,U)=U+\phi U$, there exists a finite-dimensional linear subspace $F$ of $V$ contained in $U$ and such that $T_2(\phi,U)=U+\phi F=U+T_2(\phi,F)$. Assume now that $T_n(\phi,U)=U+T_n(\phi,F)$ for some $n\in{\mathbb N}_+$, $n\geq 2$. Then $$T_{n+1}(\phi,U)=U+\phi T_n(\phi,U)=U+\phi(U)+\phi T_n(\phi,F)=U+\phi F+\phi T_n(\phi,F)=U+T_{n+1}(\phi,F).$$ This concludes the proof.
We end this section by discussing the relation of $\ent$ with $h_{alg}$. Recall that a topological abelian group $G$ is *compactly covered* if each element of $G$ is contained in some compact subgroup of $G$ (equivalently, the Pontryagin dual of $G$ is totally disconnected). If $G$ is a compactly covered locally compact abelian group, $\f\colon G\to G$ a continuous endomorphism and $U\in{\mathcal{B}}_{gr}(V)=\{U\leq G\mid\text{$U$ compact open}\}$, then (see [@GBD Theorem 2.3]) $$h_{alg}(\f)=\sup\{ H_{alg}(\f,U)\mid U\in\mathcal B_{gr}(V)\}$$ where $$H_{alg}(\f,U)=\lim_{n\to\infty}\frac{1}{n}\log\left|\frac{T_n(\f,U)}{U}\right|.$$
If $V$ is an l.l.c. vector space over a finite field ${\mathbb{F}}$, by Corollary \[cor:tdlc\] it is a totally disconnected locally compact abelian group. In particular $V$ is compactly covered, since $V$ is a torsion abelian group for ${\mathbb{F}}$ is finite.
\[halg\] Let $V$ be an l.l.c. vector space over a finite field ${\mathbb{F}}$ and let $\phi:V\to V$ be a continuous endomorphism. Then $$h_{alg}(\f)=\ent(\f)\cdot\log|{\mathbb{F}}|.$$
Let ${\mathbb{F}}=\{f_1,\ldots,f_{|{\mathbb{F}}|}\}$. Since every $U\in{{\mathcal{B}}(V)}$ is compact by Proposition \[prop:compact\], we have that $U\in{\mathcal{B}}_{gr}(V)$; hence, ${{\mathcal{B}}(V)}\subseteq {\mathcal{B}}_{gr}(V)$.
We show that ${{\mathcal{B}}(V)}$ is cofinal in $B_{gr}(V)$. To this end, let $U\in {\mathcal{B}}_{gr}(V)$ and $U'=\sum_{i=1}^{|{\mathbb{F}}|} f_i U$. Since $V$ is a topological vector space, $f_i U$ is compact for all $i=1,\ldots,|{\mathbb{F}}|$, so $U'$ is compact as well. Clearly, $U'$ is contained in the linear subspace $\langle U\rangle$ of $V$ generated by $U$. We see that actually $U'=\langle U\rangle$. Indeed, let $$x=f_{i_1} u_1+\ldots+f_{i_k} u_k,\quad u_1,\ldots, u_k\in U,\quad f_{i_1},\ldots,f_{i_k}\in{\mathbb{F}},$$ be an arbitrary element in $\langle U\rangle$. Rearranging the summands, we obtain $x=\sum_{j=1}^{|{\mathbb{F}}|} f_j u^j_{l_1\ldots l_j}\in U'$, where for every $j\in\{1,\ldots,|{\mathbb{F}}|\}$, we let $u^j_{l_1\ldots l_j}=u_{l_1}+\ldots+u_{l_j}\in U$ for $l_1,\ldots,l_j\in\{1,\ldots,k\}$ such that $f_{i_{l_1}}=\ldots=f_{i_{l_j}}=f_j$. Hence $U'=\langle U\rangle$. Therefore, $U'\in{{\mathcal{B}}(V)}$ and $U'$ contains $U$, ${{\mathcal{B}}(V)}$ is cofinal in ${\mathcal{B}}_{gr}(V)$ as claimed.
Thus, it follows by [@Virili Corollary 2.3] that $h_{alg}(\f)=\sup\{H_{alg}(\f,U)\mid U\in{{\mathcal{B}}(V)}\}$. Since for every $U\in{{\mathcal{B}}(V)}$, $$\left|\frac{T_n(\f,U)}{U}\right|=|{\mathbb{F}}|^{\dim\frac{T_n(\f,U)}{U}}$$ for all $n\in{\mathbb N}_+$, we obtain $$H_{alg}(\phi,U)=\lim_{n\to \infty}\frac{1}{n}\log\left|\frac{T_n(\phi,U)}{U}\right|=\lim_{n\to\infty}\frac{1}{n}\left(\dim\frac{T_n(\phi,U)}{U}\log|{\mathbb{F}}|\right)=H(\phi,U)\log|{\mathbb{F}}|,$$ and so the thesis follows.
General properties and examples {#ss:Properties}
===============================
In this section we prove the general basic properties of the algebraic entropy. These properties extend their counterparts for discrete vector spaces proved for $\ent_{\dim}$ in [@GBSalce]. Moreover, our proofs follow those of the same properties for the intrinsic algebraic entropy given in [@intrinsic].
We start by proving the invariance of $\ent$ under conjugation by a topological isomorphism.
\[conj\] Let $V$ be an l.l.c. vector space and $\phi:V\to V$ a continuous endomorphism. If $\alpha\colon V\to W$ is topological isomorphism of l.l.c. vector spaces, then $\ent(\phi)=\ent(\alpha\phi\alpha^{-1})$.
Let $U\in\mathcal{B}(W)$; then $\alpha^{-1}U\in{{\mathcal{B}}(V)}$. For $n\in{\mathbb N}_+$ we have $\alpha T_n(\phi,\alpha^{-1}U)=T_n(\alpha\phi\alpha^{-1},U)$. As $\alpha$ induces an isomorphism $\frac{V}{\alpha^{-1}U}\to \frac{W}{U}$, and furthermore through this isomorphism $\frac{T_n(\phi,\alpha^{-1}U)}{\alpha^{-1}U}$ is isomorphic to $\frac{T_n(\alpha\phi\alpha^{-1},U)}{U}$, by applying the definition we have $H(\phi,\alpha^{-1}U)=H(\alpha\phi\alpha^{-1},U)$. Now the thesis follows, since $\alpha$ induces a bijection between ${{\mathcal{B}}(V)}$ and ${\mathcal{B}}(W)$.
The next lemma is useful to prove the monotonicity of the algebraic entropy in Proposition \[monotonicity\].
\[lem:basis\] Let $V$ be an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $W$ a closed $\phi$-invariant linear subspace of $V$. Then: $$\begin{aligned}
&&\ent(\f\restriction_W)=\sup\{H(\f\restriction_W,U\cap W)\mid U\in{{\mathcal{B}}(V)}\}\ \text{and}\\
&&\ent(\overline\f)=\sup\{H(\overline\f,(U+W)/W)\mid U\in{{\mathcal{B}}(V)}\},\end{aligned}$$ where $\overline\f:V/W\to V/W$ is the continuous endomorphism induced by $\phi$.
Apply Proposition \[prop:basis\].
Next we see that the algebraic entropy is monotone under taking invariant linear subspaces and quotient vector spaces.
\[monotonicity\] Let $V$ be an l.l.c. vector space, $\phi: V \to V$ a continuous endomorphism, $W$ a $\f$-invariant closed linear subspace of $V$ and $\overline\f:V/W\to V/W$ is the continuous endomorphism induced by $\phi$. Then:
- $\ent(\phi)\geq\ent(\phi\restriction_W)$;
- $\ent(\phi)\geq\ent(\overline\f)$.
\(a) Let $U\in{{\mathcal{B}}(V)}$ and $n\in{\mathbb N}_+$. Since $$\frac{T_n(\phi,U)}{U}\geq \frac{U+T_n(\phi\restriction_W,U\cap W)}{U}\cong \frac{T_n(\phi\restriction_W,U\cap W)}{T_n(\phi\restriction_W,U\cap W)\cap U},$$ and $T_n(\phi\restriction_W,U\cap W)\cap U=U\cap W$, it follows that $$\dim\frac{T_n(\phi\restriction_W,U\cap W)}{U\cap W}\leq \dim \frac{T_n(\f,U)}{U}.$$ Hence, $H(\f\restriction_W,U\cap W)\leq H(\f,U)\leq\ent(\f)$. Finally, Lemma \[lem:basis\] yields the thesis.
\(b) For $U\in{{\mathcal{B}}(V)}$ and $n\in{\mathbb N}_+$, we have that $$\label{eq:iso1}
{{\raisebox{.3em}{$T_n\big(\overline\f,\frac{U+W}{W}\big)$}\left/\raisebox{-.3em}{$\frac{U+W}{W}$}\right.}}\cong \frac{T_n(\f, U+W)}{U+W}=\frac{T_n(\f,U)+W}{U+W}\cong \frac{T_n(\f,U)}{T_n(\f,U)\cap (U+W)},$$ where the latter vector space is clearly a quotient of $\frac{T_n(\phi,U)}{U}$. Therefore, $$\label{eq:monotonia}
H\left(\overline\f,\frac{U+W}{W}\right)\leq H(\f,U)\leq\ent(\phi).$$ Now Lemma \[lem:basis\] concludes the proof.
Note that equality holds in item (b) of the above proposition if $W$ is also linearly compact. In fact, in this case for every $U\in{{\mathcal{B}}(V)}$ we have $U+W\in{{\mathcal{B}}(V)}$ by Proposition \[prop:lc properties\](c), and hence Lemma \[lem:mono\] and the first isomorphism in yield $H(\f,U)\leq H(\f,U+W)=H\left(\overline{\f},\frac{U+W}{W}\right)$; therefore, $\ent(\f)\leq\ent(\overline{\f})$ and so $\ent(\phi)=\ent(\overline\phi)$ by Lemma \[monotonicity\](b).
Let $V$ be an l.l.c. vector space and $\phi:V\to V$ a continuous endomorphism. Then $\ent(\phi^k)=k \cdot \ent(\phi)$ for every $k\in{\mathbb N}$.
For $k=0$, it is enough to note that $\ent(\mathrm{id}_V)=0$ by Example \[ex:id\]. So let $k\in{\mathbb N}_+$ and $U\in{{\mathcal{B}}(V)}$. For every $n\in{\mathbb N}_+$, $$T_{nk}(\f,U)=T_n(\f^k,T_k(\f,U))\quad\text{and}\quad T_{n}(\f,T_k(\f,U))=T_{n+k-1}(\f,U).$$ Let $E=T_k(\f,U)\in{{\mathcal{B}}(V)}$. By Lemma \[lem:mono\], $$\begin{aligned}
k\cdot H(\f,U)&\leq& k\cdot H(\f,E)=k\cdot \lim_{n\to\infty}\frac{1}{nk}\dim \frac{T_{nk}(\f,E)}{E}=\lim_{n\to \infty}\frac{1}{n}\dim \frac{T_{(n+1)k-1}(\f,U)}{E}\\
&\leq& \lim_{n\to \infty}\frac{1}{n}\dim \frac{T_{(n+1)k}(\f,U)}{E} = \lim_{n\to \infty}\frac{1}{n}\dim \frac{T_{n+1}(\f^k,E)}{E} =H(\f^k,E);\end{aligned}$$ consequently, $k\cdot \ent(\f)\leq \ent(\f^k)$.
Conversely, as $U\leq E\leq T_{nk}(\f,U)$, $$\begin{aligned}
\ent(\f)\geq H(\f,U)&=&\lim_{n\to\infty}\frac{1}{nk}\dim\frac{T_{nk}(\f,U)}{U}=\lim_{n\to\infty}\frac{1}{nk}\dim \frac{T_n(\f^k,E)}{U}\\
&\geq&\lim_{n\to\infty}\frac{1}{nk}\dim\frac{T_n(\f^k,E))}{E}=\frac{1}{k}\cdot H(\f^k,E).\end{aligned}$$ By Lemma \[lem:mono\], it follows that $H(\f^k,E)\geq H(\f^k,U)$, and so $k\cdot \ent(\f)\geq\ent(\f^k)$.
The next property shows that the algebraic entropy behaves well with respect to direct limits.
Let $V$ be an l.l.c. vector space and $\phi\colon V\to V$ a continuous endomorphism. Assume that $V$ is the direct limit of a family $\{V_i\mid i\in I\}$ of closed $\phi$-invariant linear subspaces of $V$, and let $\phi_i=\phi\restriction_{V_i}$ for all $i\in I$. Then $\ent(\phi)=\sup_{i\in I}\ent(\phi_i).$
By Proposition \[monotonicity\](a), $\ent(\phi)\geq \ent(\phi_i)$ for every $i\in I$ and so $\ent(\phi)\geq \sup_{i\in I}\ent(\phi_i)$.
Conversely, let $U\in{{\mathcal{B}}(V)}$. By Lemma \[lem:finite part\], there exists a finite dimensional subspace $F$ of $U$ such that for all $n\in{\mathbb N}_+$ $$\label{eqa}
T_n(\phi,U)=U+T_n(\phi,F).$$ As $F$ is finite dimensional, $F\leq V_i$ for some $i\in I$. In particular, $$\label{eqb}
T_n(\phi_i,U\cap V_i)=(U\cap V_i)+T_n(\phi,F).$$ Indeed, since $F\leq U\cap V_i$, the inclusion $(U\cap V_i)+T_n(\phi,F)\leq T_n(\phi_i,U\cap V_i)$ follows easily. On the other hand, since $T_n(\phi,F)\leq V_i$, $$T_n(\phi_i, U\cap V_i)\leq T_n(\phi,U)\cap V_i=(U+T_n(\phi,F))\cap V_i=(U\cap V_i)+T_n(\phi,F).$$ Therefore, yields $$\frac{T_n(\phi_i,U\cap V_i)}{U\cap V_i}\cong \frac{(U\cap V_i)+T_n(\phi,F)}{U\cap V_i}\cong \frac{T_n(\phi,F)}{U\cap T_n(\phi,F)}.$$ At the same time, implies $$\frac{T_n(\phi,U)}{U}\cong \frac{U+T_n(\phi,F)}{U}\cong \frac{T_n(\phi,F)}{U\cap T_n(\phi,F)}.$$ Hence, $H(\phi,U)=H(\phi_i, U\cap V_i)\leq \sup_{i\in I}\ent(\phi_i)$, and so $\ent(\phi)\leq \sup_{i\in I}\ent(\phi_i)$.
We end this list of properties of the algebraic entropy with the following simple case of the Addition Theorem.
For $i=1,2$, let $V_i$ be an l.l.c. vector space and $\phi_i:V_1\to V_1$ a continuous endomorphism. Let $V = V_1 \times V_2$ and $\phi= \phi_1 \times \phi_2: V \to V$. Then $\ent(\phi) = \ent(\phi_1) + \ent(\phi_2)$.
Notice that $\mathcal B=\{U_1\times U_2\mid U_i\in\mathcal{B}(V_i),i=1,2\}$ is cofinal in ${{\mathcal{B}}(V)}$. Indeed, let $U\in{{\mathcal{B}}(V)}$; for $i=1,2$, since the canonical projection $\pi_i\colon V\to V_i$ is an open continuous map, $U_i=\pi_i U\in\mathcal B(V_i)$, and $U\leq U_1\times U_2$.
Now, for $U_1\times U_2\in\mathcal B$ and for every $n\in{\mathbb N}_+$, $$\frac{T_n(\phi,U_1\times U_2)}{U_1\times U_2}\cong \frac{T_n(\phi_1,U_1)}{U_1}\times \frac{T_n(\phi_2,U_2)}{U_2};$$ hence, $$\label{Heq}
H(\phi,U_1\times U_2)=H(\phi_1,U_1)+H(\phi_2,U_2).$$ By Corollary \[cor:base\](a) we conclude that $\ent(\phi)\leq \ent(\phi_1)+\ent(\phi_2)$.
If $\ent(\phi)=\infty$, the thesis holds true. So assume that $\ent(\phi)$ is finite; then $\ent(\phi_1)$ and $\ent(\phi_2)$ are finite as well by Proposition \[monotonicity\](a). Hence, for $i=1,2$ by Lemma \[ent=H\] there exists $U_i\in\mathcal B(V_i)$ such that $\ent(\phi_i)=H(\phi_i,U_i)$. By we obtain $$\ent(\phi_1)+\ent(\phi_2)=H(\phi_1,U_1)+H(\phi_2,U_2)=H(\phi,U_1\times U_2)\leq \ent(\phi),$$ where the latter inequality holds because $U_1\times U_2\in{{\mathcal{B}}(V)}$. Therefore, $\ent(\phi_1)+\ent(\phi_2)\leq \ent(\phi)$ and this concludes the proof.
In the case of a discrete vector space $V$ and an automorphism $\phi:V\to V$, we have that $\ent_{\dim}(\phi^{-1})=\ent_{\dim}(\phi)$ (see [@GBSalce]). This property does not extend to the general case of an l.l.c. vector space $V$ and a topological automorphism $\phi:V\to V$; in fact, the next example shows that $\ent(\phi)$ could not coincide with $\ent(\phi^{-1})$. Let $F$ be a finite dimensional vector space and let $V=V_c\oplus V_d$, with $$V_c=\prod_{n=-\infty}^0 F\quad\mbox{and}\quad V_d=\bigoplus_{n=1}^\infty F,$$ be endowed with the topology inherited from the product topology of $\prod_{n\in{\mathbb Z}}F$, so $V_c$ is linearly compact and $V_d$ is discrete.
The *left (two-sided) Bernoulli shift* is $${}_F\beta\colon V\to V, \quad (x_n)_{n\in{\mathbb Z}}\mapsto (x_{n+1})_{n\in{\mathbb Z}},$$ while the *right (two-sided) Bernoulli shift* is $$\beta_F\colon V\to V, \quad (x_n)_{n\in{\mathbb Z}}\mapsto(x_{n-1})_{n\in{\mathbb Z}}.$$ Clearly, $\beta_F$ and ${}_F\beta$ are topological automorphisms such that ${}_F\beta^{-1}=\beta_F.$
Let us compute their algebraic entropies.
\[ex:bs\]\[bernoulli\]
(a) Let $\f\in\{{}_{\mathbb K}\beta,\beta_{\mathbb K}\}$. By Corollary \[cor:base\](b), $$\ent(\f)=\sup\{H(\f,U)\mid U\in{{\mathcal{B}}(V)},\ V_c\leq U\}.$$ Let $U\in{{\mathcal{B}}(V)}$ such that $V_c\leq U$. Since $V_c$ has finite codimension in $U$ by Proposition \[prop:lc properties\](d,e), there exists $k\in{\mathbb N}_+$ such that $$U\leq U':=\prod_{n=-\infty}^0{\mathbb K}\times\bigoplus_{n=1}^k {\mathbb K}\in{{\mathcal{B}}(V)},$$ hence $H(\f,U)\leq H(\f,U')$ by Lemma \[lem:mono\]. Clearly, $$\ldots\leq {}_{\mathbb K}\beta^n(U')\leq\ldots\leq{}_{\mathbb K}\beta(U')\leq U'\leq\beta_{\mathbb K}(U')\leq\ldots\leq\beta_{\mathbb K}^n(U')\leq\ldots.$$ So, for all $n\in{\mathbb N}_+$, $T_n({}_{\mathbb K}\beta,U')=U'$, while $$\dim\frac{T_{n+1}(\beta_{\mathbb K},U')}{T_n(\beta_{\mathbb K},U')}=\dim\frac{\beta^{n+1}_{\mathbb K}(U')}{\beta^n_{\mathbb K}U'}=\dim\frac{\beta_{\mathbb K}(U')}{U'}=1.$$ By Corollary \[cor:base\](a), we can conclude that $$\ent({}_{\mathbb K}\beta)=0\quad\mbox{and}\quad\ent(\beta_{\mathbb K})=1.$$ In particular, $\ent(\f)\neq\ent(\f^{-1})$ for $\f\in\{{}_{\mathbb K}\beta,\beta_{\mathbb K}\}$.
(b) It is possible, slightly modifying the computations in item (a), to find that, for $F$ a finite dimensional vector space, $$\ent({}_F\beta)=0\quad \text{and}\quad \ent(\beta_F)=\dim F.$$
Limit-free Formula {#lff-sec}
==================
The aim of this subsection is to prove Proposition \[lf-ent\], that provides a formula for the computation of the algebraic entropy avoiding the limit in the definition. This formula is a fundamental ingredient in the proof of the Addition Theorem presented in the last section.
\[def:uminus\] Let $V$ be an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Let:
1. $U^{(0)}=U$;
2. $U^{(n+1)}=U+\phi^{-1} U^{(n)}$ for every $n\in{\mathbb N}$;
3. $U^-=\bigcup_{n\in{\mathbb N}} U^{(n)}$.
It can be proved by induction that $U^{(n)}\leq U^{(n+1)}$ for every $n\in{\mathbb N}$. Since $U$ is open, clearly every $U^{(n)}$ is open as well, so also $U^-$ and $\phi^{-1}U^-$ are open linear subspaces of $V$.
We see now that $U^-$ is the smallest linear subspace of $V$ containing $U$ and inversely $\phi$-invariant (i.e., $\phi^{-1}U^-\leq U^-$). Note that $U^-$ coincides with $T(\phi^{-1},U)$ when $\phi$ is an automorphism, otherwise it could be strictly smaller.
Let $V$ be an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Then:
(a) $\phi^{-1}U^-\leq U^-$;
(b) if $W$ is a linear subspace of $V$ such that $U\leq W$ and $\phi^{-1}W\leq W$, then $U^-\leq W$.
\(a) follows from the fact that $\phi^{-1}U^{(n)}\leq U^{(n+1)}$ for every $n\in{\mathbb N}$.
\(b) By the hypothesis, one can prove by induction that $U^{(n)}\leq W$ for every $n\in{\mathbb N}$; hence, $U^-\leq W$.
In the next lemma we collect some other properties that we use in the sequel.
\[Umeno\] Let $V$ be an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Then:
(a) $U^-=U+\phi^{-1}U^-$;
(b) $\frac{U^-}{\phi^{-1}U^-}$ has finite dimension.
\(a) follows from the equalities $$U+\phi^{-1}U^-=U+\phi^{-1} \bigcup_{n\in{\mathbb N}}U^{(n)}=U+\bigcup_{n\in{\mathbb N}}\phi^{-1}U^{(n)}=\bigcup_{n\in{\mathbb N}}(U+\phi^{-1}U^{(n)})=U^-.$$
\(b) The quotient $\frac{U}{U\cap\phi^{-1}U^-}$ has finite dimension by Proposition \[prop:lc properties\](d,e), since $U\cap\phi^{-1}U^-\leq U$ is open and $U$ is linearly compact. In view of item (a) we have the isomorphism $$\frac{U^-}{\phi^{-1}U^-}=\frac{U+\phi^{-1}U^-}{\phi^{-1}U^-}\cong \frac{U}{U\cap\phi^{-1}U^-},$$ so we conclude that also $\frac{U^-}{\phi^{-1}U^-}$ has finite dimension.
The next lemma is used in the proof of Proposition \[lf-ent\].
\[T-U\] Let $V$ be an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Then, for every $n\in{\mathbb N}_+$, $$\phi^{-n}T_{n}(\phi,U)=\phi^{-1}U^{(n-1)}.$$
We proceed by induction on $n\in{\mathbb N}_+$. We write simply $T_n=T_n(\phi,U)$.
If $n=1$ we have $\phi^{-1}T_1=\phi^{-1}U=\phi^{-1}U^{(0)}$. Assume now that the property holds for $n\in{\mathbb N}_+$, we prove it for $n+1$, that is, we verify that $$\label{n+1}
\phi^{-(n+1)}T_{n+1}=\phi^{-1}U^{(n)}.$$ Let $x\in \phi^{-1}U^{(n)}$. Then, by inductive hypothesis, $$\phi(x)\in U^{(n)}=U+\phi^{-1}U^{(n-1)}=U+\phi^{-n}T_n.$$ Consequently, $\phi^{n+1}(x)=\phi^n(\phi(x))\in \phi^n U+ T_n=T_{n+1}$; this shows that $x\in\phi^{-(n+1)}T_{n+1}$. Therefore, $\phi^{-1}U^{(n)}\leq \phi^{-(n+1)}T_{n+1}$.
To verify the converse inclusion, let $x\in \phi^{-(n+1)}T_{n+1}$. Then $\phi^{n+1}(x)\in T_{n+1}=T_n+\phi^nU$, so $\phi^{n+1}(x)=y+\phi^n(u)$, for some $y\in T_n$ and $u\in U$. Therefore, $\phi^n(\phi(x)-u)=y\in T_n$, that is, $\phi(x)-u\in\phi^{-n}T_n=\phi^{-1}U^{(n-1)}$ by inductive hypothesis. Hence, $\phi(x)\in U+\phi^{-1}U^{(n-1)}=U^{(n)}$, and we can conclude that $x\in\phi^{-1}U^{(n)}$. Thus, is verified. So, the induction principle gives the thesis.
We are now in position to prove the Limit-free Formula, where clearly we use that $\dim \frac{U^-}{\phi^{-1}U^-}$ has finite dimension by Lemma \[Umeno\](b).
\[lf-ent\] Let $V$ be an l.l.c. vector space, $\phi:V\to V$ a continuous endomorphism and $U\in{{\mathcal{B}}(V)}$. Then $$H(\phi,U)=\dim \frac{U^-}{\phi^{-1}U^-}.$$
We write simply $T_n=T_n(\phi,U)$ for every $n\in{\mathbb N}_+$. By Proposition \[entvalue\], there exist $n_0\in{\mathbb N}_+$ and $\alpha\in{\mathbb N}$, such that for every $n\geq n_0$, $H(\phi,U)=\alpha$, where $\alpha=\dim \frac{T_{n+1}}{T_n}$. So, our aim is to prove that $\alpha=\dim \frac{U^-}{\phi^{-1}U^-}$.
For every $n\in{\mathbb N}$, since $U\cap \phi^{-1}U^{(n)}\leq U$ is open and $U$ is linearly compact, by Proposition \[prop:lc properties\](d,e) the quotient $\frac{U}{U\cap \phi^{-1}U^{(n)}}$ has finite dimension; moreover, $U\cap \phi^{-1}U^{(n)}\leq U\cap \phi^{-1}U^{(n+1)}$ so $\frac{U}{U\cap \phi^{-1}U^{(n+1)}}$ is a quotient of $\frac{U}{U\cap \phi^{-1}U^{n}}$. The decreasing sequence of finite-dimensional vector spaces $\left\{\frac{U}{U\cap \phi^{-1}U^{n}}\mid n\in{\mathbb N}\right\}$ must stabilize; this means that there exists $n_1\in{\mathbb N}$ such that $U\cap \phi^{-1}U^{(n)}=U\cap \phi^{-1}U^{(n_1)}$ for every $n\geq n_1$. Hence, for every $m\geq n_1$, $$U\cap \phi^{-1}U^{(m)}=\bigcup_{n\in{\mathbb N}}(U\cap \phi^{-1}U^{(n)})=U\cap \bigcup_{n\in{\mathbb N}}\phi^{-1}U^{(n)}=U\cap\phi^{-1}\bigcup_{n\in{\mathbb N}}U^{(n)}=U\cap \phi^{-1}U^-.$$ Fix now $m\geq\max\{n_0,n_1\}$; since $\frac{U^-}{\phi^{-1}U^-}=\frac{U+\phi^{-1}U}{\phi^{-1}U}\cong\frac{U}{U\cap \phi^{-1}U^-}$ by Lemma \[Umeno\](a), we have $$\begin{aligned}
\nonumber
\dim \frac{U^-}{\phi^{-1}U^-}&=&\dim \frac{U}{U\cap \phi^{-1}U^-}=\dim\frac{U}{U\cap \phi^{-1}U^{(m)}}\\ \nonumber
&=&\dim\frac{U+\phi^{-1}U^{(m)}}{\phi^{-1}U^{(m)}}=\dim \frac{U^{(m+1)}}{\phi^{-1}U^{(m)}}.\end{aligned}$$ We see now that $$\dim\frac{U^{(m)}}{\phi^{-1}U^{(m-1)}}=\dim\frac{T_{m+1}}{T_m}=\alpha$$ and this concludes the proof. To this end, noting that $$\frac{U^{(m)}}{\phi^{-1}U^{(m-1)}}=\frac{U+\phi^{-1}U^{(m-1)}}{\phi^{-1}U^{(m-1)}}\quad\text{and}\quad\frac{T_{m+1}}{T_m}=\frac{\phi^{m+1}U+T_m}{T_m},$$ define $$\begin{aligned}
\Phi:&\frac{U+\phi^{-1}U^{(m-1)}}{\phi^{-1}U^{(m-1)}}\longrightarrow \frac{\phi^{m+1}U+T_m}{T_m} \\
& x+\phi^{-1}U^{(m-1)}\mapsto\phi^m(x)+T_m.\end{aligned}$$ Then $\Phi$ is a surjective homomorphism by construction and it is well-defined and injective since $\phi^{-m}T_m=\phi^{-1}U^{(m-1)}$ by Lemma \[T-U\].
Addition Theorem {#AT-sec}
================
This section is devoted to the proof of the Addition Theorem for the algebraic entropy $\ent$ for l.l.c. vector spaces (see Theorem \[thm:ATllc\]).
Let $V$ be an l.l.c. vector space and $\f:V\to V$ a continuous endomorphism. Theorem \[thm:dec\] allows us to decompose $V$ into the direct sum of a linearly compact open subspace $V_c$ and a discrete linear subspace $V_d$ of $V$, namely, $V\cong V_c\oplus V_d$ topologically. So, assume that $V=V_c\oplus V_d$ and let $$\label{eq:injproj}
\iota_*\colon V_*\to V,\quad p_*\colon V\to V_*,\quad *\in\{c,d\},$$ be respectively the canonical embeddings and projections. Accordingly, we may associate to $\f$ the following decomposition $$\label{eq:fdec}
\f=
\begin{pmatrix}
\f_{cc} & \f_{dc} \\
\f_{cd} & \f_{dd}
\end{pmatrix},$$ where $\phi_{\bullet*}:V_\bullet\to V_*$ is the composition $\phi_{\bullet*}=p_*\circ\phi\circ\iota_\bullet$ for $\bullet, *\in\{c,d\}$. Therefore, each $\phi_{\bullet*}$ is continuous as it is composition of continuous homomorphisms.
\[lem:kerim\] In the above notations, consider $\f_{cd}\colon V_c\to V_d$. Then:
(a) $\Im(\f_{cd})\in\mathcal B(V_d)$;
(b) $\ker(\f_{cd})\in\mathcal B(V_c)\subseteq{{\mathcal{B}}(V)}$.
\(a) Since $V_d$ is discrete, by Proposition \[prop:lc properties\](c,d) we have that $\Im(\f_{cd})\leq V_d$ has finite dimension, hence $\Im(\f_{cd})\in\mathcal B(V_d)=\{F\leq V_d\mid \dim F<\infty\}$.
\(b) As $\ker(\f_{cd})$ is a closed linear subspace of $V_c$, which is linearly compact, then $\ker(\f_{cd})$ is linearly compact as well by Proposition \[prop:lc properties\](b). Since $V_c/\ker(\f_{cd})\cong\Im(\f_{cd})$ is finite dimensional by item (a), $V_c/\ker(\f_{cd})$ is discrete and so $\ker(\f_{cd})$ is open in $V_c$; therefore, $\ker(\phi_{cd})\in\mathcal B(V_c)$.
We show now that the only positive contribution to the algebraic entropy of $\f$ comes from its “discrete component” $\phi_{dd}$.
\[prop:resdd\] In the above notations, $\ent(\f)=\ent(\f_{dd})$.
By Lemma \[lem:kerim\](a), $\Im(\f_{cd})\in{\mathcal{B}}(V_d)$; hence, letting $${\mathcal{B}}_d=\{F\leq V_d\mid \Im(\f_{cd})\leq F,\dim F<\infty\}\subseteq{\mathcal{B}}(V_d),$$ Corollary \[cor:base\](b) implies $$\label{entfdd}
\ent(\f_{dd})=\sup\{H(\f_{dd},F)\mid F\in{\mathcal{B}}_d\}.$$ Let $\mathcal B=\{U\in{{\mathcal{B}}(V)}\mid V_c\leq U\}$, which is cofinal in ${{\mathcal{B}}(V)}$. For $U\in{\mathcal{B}}$, since $V_c$ has finite codimension in $U$ by Proposition \[prop:lc properties\](d,e), there exists a finite dimensional linear subspace $F\leq V_d$ such that $U=V_c\oplus F$. Conversely, $V_c\oplus F\in{\mathcal{B}}$ for every finite dimensional linear subspace $F\leq V_d$. Hence, $\mathcal B=\{V_c\oplus F\mid F\in{\mathcal{B}}(V_d)\}$. Moreover, $\mathcal B'=\{V_c\oplus F\mid F\in{\mathcal{B}}_d\}$ is cofinal in ${\mathcal{B}}$ and so in ${{\mathcal{B}}(V)}$. Thus, Corollary \[cor:base\](b) yields $$\label{entf}
\ent(\f)=\sup\{H(\f,U)\mid U\in{\mathcal{B}}'\}.$$
For $U=V_c\oplus F\in{\mathcal{B}}'$, as in Definition \[def:uminus\] let, for every $n\in{\mathbb N}$, $$\begin{aligned}
U^{(0)}=U\quad&\mbox{and}&\quad F^{(0)}=F, \nonumber\\
U^{(n)}=U+\f^{-1}U^{(n-1)}\quad&\mbox{and}&\quad F^{(n)}=F+\f_{dd}^{-1}F^{(n-1)}, \nonumber\\
U^-=\bigcup_{n\in{\mathbb N}}U^{(n)}\quad&\mbox{and}&\quad F^-=\bigcup_{n\in{\mathbb N}} F^{(n)}.\nonumber\end{aligned}$$ Proposition \[lf-ent\], together with and respectively, implies that $$\begin{aligned}
\ent(\f)&=&\sup\left\{\dim\frac{U^-}{\f^{-1}U^-}\mid U\in{\mathcal{B}}'\right\},\label{1}\\
\ent(\f_{dd})&=&\sup\left\{\dim\frac{F^-}{\f_{dd}^{-1}F^-}\mid F\in{\mathcal{B}}_d\right\}.\label{2}\end{aligned}$$
Let $U=V_c\oplus F\in{\mathcal{B}}'$. We show by induction on $n\in{\mathbb N}$ that $$\label{eq:EQ}
U^{(n)}= V_c\oplus F^{(n)}\quad \text{for every}\ n\in{\mathbb N}.$$ For $n=0$, we have $U^{(0)}=U=V_c\oplus F=V_c\oplus F^{(0)}$. Assume now that $n\in{\mathbb N}$ and that $U^{(n)}=V_c\oplus F^{(n)}$. First note that $U^{(n+1)}=U+\f^{-1}U^{(n)}=U+\f^{-1}(V_c\oplus F^{(n)})$. Moreover, since $\Im(\f_{cd})\leq F\leq F^{(n)}$, $$\begin{aligned}
\f^{-1}(V_c\oplus F^{(n)})&=&\{(x,y)\in V_c\oplus V_d\mid \f_{cd}(x)+\f_{dd}(y)\in F^{(n)}\}\nonumber\\
&=&\{(x,y)\in V_c\oplus V_d\mid \f_{dd}(y)\in F^{(n)}\}\nonumber\\
&=&V_c\oplus \f_{dd}^{-1} F^{(n)}.\nonumber\end{aligned}$$ Thus, $U^{(n+1)}= V_c\oplus F^{(n+1)}$ as required in .
Now implies that $U^-= V_c\oplus F^-$; moreover, since $\Im(\phi_{cd})\leq F\leq F^-$, $$\f^{-1} U^-=\{(x,y)\in V_c\oplus V_d\mid\phi_{dd}(y)\in F^-\}=V_c\oplus \f_{dd}^{-1}F^-.$$ Therefore, $\frac{U^-}{\phi^{-1}U^-}=\frac{V_c\oplus F^-}{V_c\oplus\phi_{dd}^{-1}F^-}=\frac{F^-}{\phi_{dd}^{-1}F^-}$, so the thesis follows from and .
We are now in position to prove the Addition Theorem.
\[thm:ATllc\] Let $V$ be an l.l.c. vector space, $\f\colon V\to V$ a continuous endomorphism and $W$ a closed $\phi$-invariant linear subspace of $V$. Then $$\ent(\phi)=\ent(\phi\restriction_W)+\ent(\overline\phi),$$ where $\overline\phi:V/W\to V/W$ is the continuous endomorphism induced by $\phi$.
Let $V_c\in{{\mathcal{B}}(V)}$ and $W_c=W\cap V_c\in{\mathcal{B}}(W)$. By Theorem \[thm:dec\], there exists a discrete linear subspace $W_d$ of $W$ such that $W= W_c\oplus W_d$. Let $V_d$ be a linear subspace of $V$ such that $V=V_c\oplus V_d$ and $W_d\leq V_d$. Clearly, $V_d$ is discrete, since $V_c$ is open and $V_c\cap V_d=0$. By construction, the diagram $$\xymatrix{0\ar[r]&W_d\ar@/^2pc/[rrr]^{(\phi\restriction_{W})_{dd}} \ar[r]^{\iota^W_d}\ar@{^{(}->}[d]&W\ar[r]^{\f\restriction_W}&W\ar[r]^{p^W_d}&W_d\ar@{^{(}->}[d]\ar[r]&0\\
0\ar[r]&V_d\ar@/_2pc/[rrr]_{\phi_{dd}} \ar[r]^{\iota^V_d}&V\ar[r]^{\f}&V\ar[r]^{p^V_d}&V_d\ar[r]&0}$$ commutes, where $\iota^W_d, \iota^V_d,p^W_d,p^V_d$ are the canonical embeddings and projections of $W$ and $V$, respectively. This yields that $W_d$ is a $\f_{dd}$-invariant linear subspace of $V_d$ and that $$(\f\restriction_W)_{dd}=\f_{dd}\restriction_{W_d}.$$
Now, let $\pi\colon V\to V/W$ be the canonical projection and let $\overline V=V/W$. Let $\overline V_c=\pi(V_c)$ and $\overline V_d=\pi(V_d)$; then $\overline V_c$ is linearly compact and open, while $\overline V_d$ is discrete. Since $\overline V_c$ is open in $\overline V$, we have $\overline V=\overline V_c\oplus\overline V_d$. Clearly, there exists a canonical isomorphism $\alpha\colon \overline V_d\to V_d/W_d$ of discrete vector spaces making the following diagram $$\xymatrix{
\overline V_d\ar@/^2pc/[rrr]^{\overline\f_{dd}} \ar[r]^{\iota^{\overline V}_d}\ar[d]^\alpha&\overline V\ar[r]^{\overline\f}&\overline V\ar[r]^{p^{\overline V}_d}&\overline V_d\ar[d]^\alpha\\
V_d/W_d \ar[rrr]^{\overline{\phi_{dd}}}&&&V_d/W_d
}$$ commute, where $\overline{\f_{dd}}$ is the endomorphism induced by $\phi_{dd}$. Then, by Propositions \[prop:resdd\] and \[conj\], $$\ent(\phi)=\ent(\phi_{dd}),\quad \ent(\f\restriction_W)=\ent(\f_{dd}\restriction_{W_d})\quad \text{and}\quad \ent(\overline{\f})=\ent(\overline{\f_{dd}}).$$ Since $\ent(\phi_{dd})=\ent(\f_{dd}\restriction_{W_d})+\ent(\overline{\f_{dd}})$, in view of the Addition Theorem for $\ent_{\dim}$ (see [@GBSalce Theorem 5.1]) and Lemma \[entdim\], we can conclude that $\ent(\phi)=\ent(\phi\restriction_W)+\ent(\overline\phi)$.
[AAAA]{}
R. L. Adler, A. G. Konheim, and M. H. McAndrew, *Topological entropy*, Transactions of the American Mathematical Society 114 (1965): 309–319.
R. Bowen, *Entropy for group endomorphisms and homogeneous spaces*, Trans. Amer. Math. Soc. 153 (1971): 401–414.
I. Castellano, A. Giordano Bruno, *Topological entropy for locally linearly compact vector spaces*, in preparation (2016).
D. Dikranjan, A. Giordano Bruno, *The connection between topological and algebraic entropy*, Topology Appl. 159 (2012): 2980–2989.
D. Dikranjan, A. Giordano Bruno, *Limit free computation of entropy*, Rend. Istit. Mat. Univ. Trieste 44 (2012): 297–312.
D. Dikranjan, A. Giordano Bruno, *The Bridge Theorem for totally disconnected LCA groups*, Topology Appl. 169 (2014): 21–32.
D. Dikranjan, A. Giordano Bruno, *Entropy on abelian groups*, Adv. Math. 298 (2016) 612–653.
D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, *Intrinsic algebraic entropy*, J. Pure Appl. Algebra 219 (2015): 2933–2961.
D. Dikranjan, B. Goldsmith, L. Salce, P. Zanardo, *Algebraic entropy for abelian groups*, Trans. Amer. Math. Soc. 361 (2009): 3401–3434.
A. Giordano Bruno, L. Salce, *A soft introduction to algebraic entropy*, Arab. J. Math. 1 (2012): 69–87.
A. Giordano Bruno, S. Virili, *On the Algebraic Yuzvinski Formula*, Topol. Algebra and its Appl. 3 (2015): 86–103.
A. Giordano Bruno, S. Virili, *Topological entropy in totally disconnected locally compact groups*, Ergodic Theory and Dynamical Systems (2016): to appear, doi:10.1017/etds.2015.139.
B. M. Hood, *Topological entropy and uniform spaces*, J. London Math. Soc. 8 (1974): 633–641.
E. Matlis, *Injective modules over Noetherian rings*, Pacific J. Math. 8 (1958): 511–528.
G. K[ö]{}the, *Topological vector spaces*, Springer Berlin Heidelberg, 1983.
S. Lefschetz, *Algebraic topology*, Vol. 27. American Mathematical Soc., 1942.
J. Peters, *Entropy on discrete abelian groups*, Adv. Math. 33 (1979): 1–13.
J. Peters, *Entropy of automorphisms on LCA groups*, Pacific J. Math. 96 (1981): 475–488.
L. Salce, P. Zanardo, *A general notion of algebraic entropy and the rank-entropy*, Forum Math. 21 (2009): 579–599.
L. Salce, P. Vámos, S. Virili, *Length functions, multiplicities and algebraic entropy*, Forum Math. 25 (2013): 255–282.
L. Salce, S. Virili, *The Addition Theorem for algebraic entropies induced by non-discrete length functions*, Forum Math. 28 (2016): 1143–1157.
S. Virili, *Algebraic and topological entropy of group actions*, preprint.
S. Virili, *Entropy for endomorphisms of LCA groups*, Topology Appl. 159 (2012): 2546–2556.
M. D. Weiss, *Algebraic and other entropies of group endomorphisms*, Theory of Computing Systems 8 (1974): 243–248.
G. A. Willis, *The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups*, Math. Ann. 361 (2015): 403–442.
S. Yuzvinski, *Metric properties of endomorphisms of compact groups*, Izv. Acad. Nauk SSSR, Ser. Mat. 29 (1965): 1295–1328 (in Russian). English Translation: Amer. Math. Soc. Transl. 66 (1968): 63–98.
|
---
abstract: 'We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewdith, the scramble number is not minor monotone, but it is subgraph monotone and invariant under refinement. We compute the scramble number and gonality of several families of graphs for which these invariants are strictly greater than the treewidth.'
author:
- Michael Harp
- Elijah Jackson
- David Jensen
- Noah Speeter
bibliography:
- 'ref.bib'
title: A New Lower Bound on Graph Gonality
---
Introduction
============
In [@BakerNorine], Baker and Norine define the theory of divisors on graphs and a new graph invariant known as the *gonality*. Due to its connection to algebraic geometry, this invariant has received a great deal of interest [@dbG; @NPHard; @RandomGonality; @ExpanderGonality; @AidunMorrison; @Morrison]. Computing the gonality $\mathrm{gon}(G)$ of a graph $G$ is NP-hard [@NPHard]. To find an upper bound, one only has to produce an example of a divisor with positive rank, so much of the difficulty comes from finding lower bounds. A significant step in this direction was obtained in [@dbG], in which the authors show that the gonality of a graph $G$ is bounded below by a much-studied graph invariant known as the *treewidth*, $\mathrm{tw}(G)$.
In this paper we define a new graph invariant, which we call the *scramble number* of $G$ and denote $\mathrm{sn}(G)$. We refer the reader to § \[Sec:Scramble\] for a definition. Our main result is the following.
\[Thm:MainThm\] For any graph $G$, we have $$\mathrm{tw} (G) \leq \mathrm{sn} (G) \leq \mathrm{gon} (G) .$$
Theorem \[Thm:MainThm\] is proved in two parts. The left inequality is proved in Corollary \[Cor:TW\] and the right inequality in Theorem \[Thm:Bound\]. The proof of Theorem \[Thm:Bound\] follows closely that of [@dbG]. Indeed, the scramble number is defined in such a way as to generalize the statement of [@dbG Theorem 2.1] without significantly altering its proof.
After establishing Theorem \[Thm:MainThm\], we then examine properties of the scramble number. We show that it is subgraph monotone (Proposition \[Prop:Subgraph\]) and invariant under refinement (Proposition \[Prop:Refinement\]) but not minor montone (Example \[Ex:Minor\]).
Finally, in § \[Sec:Examples\] we use Theorem \[Thm:MainThm\] to compute the gonality of several families of graphs. In [@Morrison], the authors compute the treewidth of the *grid graphs* $G_{m,n}$, the *stacked prisms* $Y_{m,n}$, and the *toroidal grid graphs* $T_{m,n}$. (See § \[Sec:Examples\] for precise definitions of these graphs.) Combining their results with the bound from [@dbG], they compute the gonalities of all these graphs except for $T_{n,n}$, $T_{n+1,n}$, and $Y_{2n,n}$. Using Theorem \[Thm:MainThm\], we complete this project, computing the gonalities of the graphs in these families and some minor generalizations. Even in cases where the gonality was already known, our constructions are quite a bit simpler than those that arise in computations of treewidth. For this reason, we suspect that scrambles may be useful in further computations of graph gonality.
**Acknowledgments.** This research was conducted as a project with the University of Kentucky Math Lab. We would like to thank Ralph Morrison for some discussions on this material.
Preliminiaries
==============
In this section, we fix notation and review some basic definitions. For simplicity, we will assume throughout that all graphs are connected, possibly with multiple edges, but no loops. Given a graph $G$, we write $V(G)$ for its vertex set and $E(G)$ for its edge set. If $A$ is a subset of $V(G)$, we write $A^c$ for its complement. If $A$ and $B$ are subsets of $V(G)$, we write $E(A,B)$ for the set of edges with one endpoint in $A$ and the other endpoint in $B$. A set of vertices $B \subseteq V(G)$ is *connected* if, for every proper subset $A \subsetneq B$, the set $E(A, B \smallsetminus A)$ is nonempty.
A *subgraph* of a graph $G$ is a graph that can be obtained from $G$ by deleting edges and deleting isolated vertices. A *minor* of a graph $G$ is a graph that can be obtained from $G$ by contracting edges, deleting edges, and deleting isolated vertices. If $e \in E(G)$ is an edge with endpoints $v$ and $w$, the *subdivision* of $G$ at $e$ is obtained by introducing a new vertex $u$ in the middle of $e$. In other words, the vertex set of the subdivision is $V(G) \cup \{ u \}$, and the edge set of the subdivision is $E(G) \cup \{ uv, uw \} \smallsetminus \{ e \}$. A *refinement* of a graph $G$ is one that can be obtained from $G$ by finitely many subdivisions.
We briefly describe the theory of divisors on graphs. For a more detailed treatment, we refer the reader to [@BakerNorine] or [@BakerJensen]. A *divisor* on a graph $G$ is a formal $\mathbb{Z}$-linear combination of vertices of $G$. It is standard to think of a divisor as a stack of poker chips on each of the vertices, with negative coefficients corresponding to vertices that are “in debt”. For this reason, divisors are sometimes referred to as *chip configurations*. (We use the term “divisor” to emphasize the connection with algebraic geometry.) The *degree* of a divisor is the sum of the coefficients: $$\mathrm{deg} \left( \sum_{v \in V(G)} a_v \cdot v \right) := \sum_{v \in V(G)} a_v .$$ In other words, the degree is the total number of poker chips. A divisor is said to be *effective* if all of the coefficients are nonnegative – that is, none of the vertices are in debt. The *support* of an effective divisor $D$, denoted $\mathrm{Supp}(D)$, is the set of vertices with nonzero coefficient.
Given a divisor $D$ on $G$ and a vertex $v \in V(G)$, we may *fire* the vertex $v$ to obtain a new divisor. For each edge $e$ with $v$ as an endpoint, this new divisor has 1 fewer chip at $v$, and 1 more chip at the other endpoint of $e$. If we fire every vertex in a subset $A \subseteq V(G)$, we say that we fire the subset $A$. (The resulting divisor is independent of the order in which one fires the vertices in $A$.) We say that two divisors are *equivalent* if one can be obtained from the other by a sequence of chip-firing moves. Given a vertex $v \in V(G)$, an effective divisor $D$ is *$v$-reduced* if, for every subset $A \subseteq V(G) \smallsetminus \{ v \}$, the divisor obtained by firing $A$ is not effective. Every effective divisor is equivalent to a unique $v$-reduced divisor. We say that a divisor $D$ on $G$ has *positive rank* if its $v$-reduced representative contains $v$ in its support, for every vertex $v \in V(G)$. The *gonality* of $G$ is the minimum degree of a divisor of positive rank on $G$.
Brambles and Scrambles {#Sec:Scramble}
======================
We make the following definition.
A *scramble* in a graph $G$ is a set $\mathscr{S} = \{ E_1 , \ldots E_n \}$ of connected subsets of $V(G)$.
We will often refer to the subsets $E_i$ as *eggs*. Scrambles with certain properties have been studied extensively in the graph theory literature.
A *bramble* is a scramble $\mathscr{S}$ with the property that $E \cup E'$ is connected for every pair $E, E' \in \mathscr{S}$. It is called a *strict bramble* if every pair of elements $E, E' \in \mathscr{S}$ has nonempty intersection.
A set $C \subset V(G)$ is said to *cover* a scramble $\mathscr{S}$ if $C \cap E \neq \emptyset$ for all $E \in \mathscr{S}$.
The *order* of a bramble $\mathscr{B}$ is the minimum size of a set that covers $\mathscr{S}$. The *bramble number* of a graph $G$ is the maximum order of a bramble in $G$, and is denoted $\mathrm{bn}(G)$. A result of Seymour and Thomas shows that the bramble number of a graph is closely related to another well-known graph invariant, known as the *treewidth* $\mathrm{tw}(G)$. In particular, $\mathrm{tw}(G) = \mathrm{bn}(G) - 1$ for any graph $G$ [@SeymourThomas]. Here, we define some related notions for more general scrambles.
\[Def:ScrambleOrder\] The *scramble order* of a scramble $\mathscr{S}$ is the maximum integer $k$ such that:
1. no set $C \subset V(G)$ of size less than $k$ covers $\mathscr{S}$, and
2. if $A \subset V(G)$ is a set such that there exists $E, E' \in \mathscr{S}$ with $E \subseteq A$ and $E' \subseteq A^c$, then $\vert E(A,A^c) \vert \geq k$.
The scramble order of a scramble $\mathscr{S}$ is denoted $\vert \vert \mathscr{S} \vert \vert$. The *scramble number* of a graph $G$, denoted $\mathrm{sn}(G)$, is the maximum scramble order of a scramble in $G$.
We note the following observations about the scramble order of brambles.
The order of a strict bramble is equal to its scramble order.
Let $\mathscr{B}$ be a strict bramble of order $k$. By definition, there is a set $C \subset V(G)$ of size $k$ that covers $\mathscr{B}$, and no such set of size less than $k$. The scramble order of $\mathscr{B}$ is therefore at most $k$. Since $\mathscr{B}$ is a strict bramble, any two sets $E, E' \in \mathscr{B}$ have nonempty intersection. It follows that there is no set $A \subset V(G)$ that contains $E$ and whose complement contains $E'$, so property (2) of Definition \[Def:ScrambleOrder\] is satisfied vacuously.
\[Lem:BrambleScramble\] Let $\mathscr{B}$ be a bramble of order $k$. Then the scramble order of $\mathscr{B}$ is either $k$ or $k-1$.
By definition, there is a set $C \subset V(G)$ of size $k$ that covers $\mathscr{B}$, and no such set of size less than $k$. The scramble order of $\mathscr{B}$ is therefore at most $k$. By [@dbG Lemma 2.3], if $E, E' \in \mathscr{B}$ and $A \subset V(G)$ is a subset such that $E \subseteq A$ and $E' \subseteq A^c$, then $\vert E(A,A^c) \vert \geq k-1$. It follows that the scramble order of $\mathscr{B}$ is at least $k-1$.
\[Cor:TW\] For any graph $G$, we have $\mathrm{tw}(G) \leq \mathrm{sn}(G)$.
Let $\mathscr{B}$ be a bramble of maximum order $k$ in $G$. By [@SeymourThomas], we have $\mathrm{tw}(G) = k-1$. By Lemma \[Lem:BrambleScramble\], the scramble order of $\mathscr{B}$ is at least $k-1$, hence $\mathrm{sn}(G) \geq k-1$.
Properties of the Scramble Number
=================================
We now prove our main result about the scramble number. Namely, that the scramble number of a graph is a lower bound for the graph’s gonality. Our argument follows closely that of [@dbG Theorem 2.1], which shows that the treewidth of a graph is a lower bound for the graph’s gonality. Indeed, we defined the scramble number with the specific goal of stating [@dbG Theorem 2.1] in its maximum generality.
\[Thm:Bound\] For any graph $G$, we have $\mathrm{sn}(G) \leq \mathrm{gon}(G)$.
Let $\mathscr{S}$ be a scramble on $G$, and let $D'$ be an effective divisor of positive rank on $G$. We will show that $\mathrm{deg}(D') \geq \vert \vert \mathscr{S} \vert \vert$. Among the effective divisors equivalent to $D'$, we choose $D$ such that $\mathrm{Supp}(D)$ intersects a maximum number of eggs in $\mathscr{S}$. If $\mathrm{Supp}(D)$ is a hitting set for $\mathscr{S}$ then, by definition, $$\mathrm{deg}(D) \geq \vert \mathrm{Supp}(D) \vert \geq \vert \vert \mathscr{S} \vert \vert .$$
Conversely, suppose that there is some egg $E \in \mathcal{S}$ that does not intersect $\mathrm{Supp}(D)$, and let $v\in E$. Since $D$ has positive rank and $v \notin \mathrm{Supp}(D)$, it follows that $D$ is not $v-$reduced. Therefore there exists a chain $$\emptyset \subsetneq U_1 \subseteq \cdots \subseteq U_k \subset V(G) \smallsetminus \{v \}$$ and a sequence of effective divisors $D_0 , D_1, \ldots, D_k$ such that:
1. $D_0 = D$,
2. $D_k$ is $v$-reduced, and
3. $D_i$ is obtained from $D_{i-1}$ by firing the set $U_i$, for all $i$.
Since $D$ has positive rank, we see that $v \in \mathrm{Supp}(D_k)$ and hence $\mathrm{Supp}(D_k)$ intersects $E$. By assumption, $\mathrm{Supp}(D_k)$ does not intersect more eggs than $\mathrm{Supp}(D)$, so there is at least one egg $E'$ that intersects $\mathrm{Supp}(D)$ but not $\mathrm{Supp}(D_k)$. Let $i \leq k$ be the smallest index such that there is some $E' \in \mathscr{S}$ that intersects $\mathrm{Supp}(D)$ but not $\mathrm{Supp}(D_i)$. Then $E' \cap \mathrm{Supp}(D_{i-1}) \neq \emptyset$ and $E' \cap \mathrm{Supp}(D_i) = \emptyset$. By [@dbG Lemma 2.2], it follows that $E' \subseteq U_i$.
Again, by assumption, $\mathrm{Supp}(D_{i-1})$ does not intersect more eggs than $\mathrm{Supp}(D)$, so $\mathrm{Supp}(D_{i-1})$ does not intersect $E$. Let $j \geq i$ be the smallest index such that $E \cap \mathrm{Supp}(D_{j-1}) = \emptyset$ and $E \cap \mathrm{Supp}(D_j) \neq \emptyset$. Since $D_{j-1}$ can be obtained from $D_j$ by firing $U_j^c$, we see that $E \subseteq U_j^c \subseteq U_i^c$. Since $E \subseteq U_i^c$ and $E' \subseteq U_i$, it follows by the definition of a scramble that $\vert E(U_i , U_i^c) \vert \geq \vert \vert \mathscr{S} \vert \vert$. Since $$\mathrm{deg}(D_{i-1}) \geq \sum_{u \in U_i} D_{i-1}(u) \geq \vert E(U_i, U_i^c) \vert,$$ we have $$\mathrm{deg}(D_{i-1}) \geq \vert \vert \mathscr{S} \vert \vert .$$
One of the major advantages of the treewidth bound from [@dbG] is that the treewidth is minor monotone. In other words, if $G'$ is a graph minor of a graph $G$, then $\mathrm{tw}(G') \leq \mathrm{tw}(G)$. This is not true of the scramble number, as the following example shows.
\[Ex:Minor\] Let $G$ be the graph depicted in Figure \[Fig:Gon3\]. If $v$ is the green vertex, then the divisor $3v$ has positive rank. It follows that the gonality of $G$ is at most 3, and thus the scramble number of $G$ is at most 3 by Theorem \[Thm:Bound\].
(0,1) circle (3pt); (0,1) circle (3pt); (1,0) circle (2pt); (1,1) circle (2pt); (1,2) circle (2pt); (2,0) circle (2pt); (2,1) circle (2pt); (2,2) circle (2pt); (3,1) circle (2pt); (0,1)–(1,0); (0,1)–(1,1); (0,1)–(1,2); (1,0)–(1,1); (1,0)–(2,0); (1,1)–(1,2); (1,1)–(2,1); (1,2)–(2,2); (2,0)–(2,1); (2,0)–(3,1); (2,1)–(2,2); (2,1)–(3,1); (2,2)–(3,1);
Now, let $G'$ be the graph pictured in Figure \[Fig:S4\], obtained by contracting the red edge in $G$. The 4 colored subsets are the eggs of a scramble $\mathscr{S}$, which we now show has scramble order 4. Because the 4 eggs are disjoint, there is no hitting set of size less than 4. Now, let $A \subset V(G')$ be a set with the property that both $A$ and $A^c$ contain an egg. By exchanging the roles of $A$ and $A^c$, we may assume that $A$ contains the center red vertex. If $A$ consists solely of this vertex, then $\vert E(A,A^c) \vert = 6$. Otherwise, $A$ contains some, but not all, of the vertices on the hexagonal outer ring. We then see that $E(A,A^c)$ contains at least two edges in the hexagonal outer ring, and at least two edges that have the center red vertex as an endpoint. Thus, $\vert E(A,A^c) \vert \geq 4$.
(0,1) circle (3pt); (1,0) circle (3pt); (1.5,1) circle (3pt); (1,2) circle (3pt); (2,0) circle (3pt); (3,1) circle (3pt); (2,2) circle (3pt); (0,1) circle (3pt); (1,0) circle (3pt); (1.5,1) circle (3pt); (1,2) circle (3pt); (2,0) circle (3pt); (3,1) circle (3pt); (2,2) circle (3pt); (0,1)–(1,0); (0,1)–(3,1); (0,1)–(1,2); (1,0)–(2,2); (1,0)–(2,0); (1,2)–(2,0); (1,2)–(2,2); (2,2)–(3,1); (2,0)–(3,1);
While the scramble number is not minor monotone, it is subgraph monotone.
\[Prop:Subgraph\] If $G'$ is a subgraph of $G$, then $\mathrm{sn}(G') \leq \mathrm{sn}(G)$.
Let $\mathscr{S}'$ be a scramble on $G'$, and let $\mathscr{S}$ be the scramble on $G$ with the same eggs as $\mathscr{S}'$ on $G'$. We will show that $\vert \vert \mathscr{S} \vert \vert \geq \vert \vert \mathscr{S}' \vert \vert$. If $C \subset V(G)$ is a hitting set for $\mathscr{S}$, then $C \cap V(G')$ is a hitting set for $\mathscr{S}'$. Now, let $A$ be a subset of $V(G)$ such that $A$ and $A^c$ both contain eggs of $\mathscr{S}$. Then $A \cap V(G')$ is a subset of $V(G')$ with the property that both it and its complement contain eggs of $\mathscr{S}'$, and $\vert E(A,A^c) \vert \geq \vert E(A \cap V(G'), A^c \cap V(G')) \vert$. It follows that $\vert \vert \mathscr{S} \vert \vert \geq \vert \vert \mathscr{S}' \vert \vert$.
The scramble number is also invariant under refinement.
\[Prop:Refinement\] If $G'$ is a refinement of $G$, then $\mathrm{sn} (G) = \mathrm{sn} (G')$.
By induction, it suffices to consider the case where $G$ has one fewer vertex than $G'$. Let $v$ and $w$ be adjacent vertices in $G$, and let $G'$ be the graph obtained by subdividing an edge between $v$ and $w$, introducing a vertex $u$ between them.
First, we will show that $\mathrm{sn} (G) \leq \mathrm{sn} (G')$. To see this, let $\mathscr{S}$ be a scramble on $G$. For each egg $E \in \mathscr{S}$, we define a connected subset $E' \subset V(G')$ as follows. If $v \notin E$, then $E' = E$, and if $v \in E$, then $E' = E \cup \{ u \}$. Let $$\mathscr{S}' = \{ E' \vert E \in \mathscr{S} \} .$$ We will show that $\vert \vert \mathscr{S}' \vert \vert \geq \vert \vert \mathscr{S} \vert \vert$.
Let $C \subset V(G')$ be a hitting set for $\mathscr{S}'$. If $u \notin C$, then $C$ is also a hitting set for $\mathscr{S}$. On the other hand, if $u \in C$, then since every egg in $\mathscr{S}'$ that contains $u$ also contains $v$, the set $C' = C \cup \{ v \} \smallsetminus \{ u \}$ is a hitting set for $\mathscr{S}'$ with the property that $u \notin C'$ and $\vert C' \vert \leq \vert C \vert$. Now, let $A$ be a subset of $V(G')$ such that both $A$ and $A^c$ contain eggs of $\mathscr{S}'$. By exchanging $A$ and $A^c$, we may assume that $u \notin A$. We may then think of $A$ also as a subset of $V(G)$ with the property that both $A$ and $A^c$ contain eggs of $\mathscr{S}$. If both $v$ and $w$ are contained in $A$, then the number of edges leaving $A$ in $V(G)$ is 1 fewer than the number of edges leaving $A$ in $V(G')$. Otherwise, these two numbers are equal. It follows that $\vert \vert \mathscr{S}' \vert \vert \geq \vert \vert \mathscr{S} \vert \vert$.
We now show that $\mathrm{sn} (G) \geq \mathrm{sn} (G')$. To see this, let $\mathscr{S}'$ be a scramble on $G'$ of maximal scramble order. If $\mathrm{sn} (G) = 1$, then by Corollary \[Cor:Tree\] below, we see that $G$ is a tree. It follows that $G'$ is a tree as well, and $\mathrm{sn}(G') = 1$ by another application of Corollary \[Cor:Tree\]. We may therefore assume that $\mathrm{sn} (G) \geq 2$, and for contradiction that $\vert \vert \mathscr{S}' \vert \vert \geq 3$.
If every egg in $\mathscr{S}'$ contains $u$, then $\mathscr{S}'$ has a hitting set of size 1, a contradiction. It follows that if $\{ u \} \in \mathscr{S}'$, then the set $A = \{ u \}$ has the property that both $A$ and $A^c$ contain eggs of $\mathscr{S}'$. Thus, $\vert \vert \mathscr{S}' \vert \vert \leq \vert E (A,A^c) \vert = 2$, another contradiction. We may therefore assume that $\{ u \} \notin \mathscr{S}'$. Let $$\mathscr{S} = \{ E' \cap V(G) \vert E' \in \mathscr{S}' \}.$$ We will show that $\vert \vert \mathscr{S} \vert \vert \geq \vert \vert \mathscr{S}' \vert \vert$.
Let $C \subset V(G)$ be a hitting set for $\mathscr{S}$. Since $\{ u \} \notin \mathscr{S}'$, we see that $C$ is also a hitting set for $\mathscr{S}'$. Now, let $A$ be a subset of $V(G)$ with the property that both $A$ and $A^c$ contain eggs of $\mathscr{S}$. As above, define the set $A'$ as follows. If $v \notin A$, then $A' = A$, and if $v \in A$, then $A' = A \cup \{ u \}$. We see that $\vert E(A,A^c) \vert = \vert E(A',A'^c) \vert$. It follows that $\vert \vert \mathscr{S} \vert \vert \geq \vert \vert \mathscr{S}' \vert \vert$.
The graph on the left in Figure \[Fig:Refinement\] has gonality 2. By Theorem \[Thm:Bound\], its scramble number is at most 2. Since it is not a tree, by Corollary \[Cor:Tree\] below, its scramble number is exactly 2.
On the other hand, the graph on the right has gonality 3. Since it is a refinement of the graph on the left, however, by Proposition \[Prop:Refinement\] the two graphs have the same scramble number. Thus, the graph on the right is an example where the gonality and scramble number disagree.
(0,1) circle (2pt); (1,0) circle (2pt); (1,2) circle (2pt); (2,0) circle (2pt); (2,2) circle (2pt); (3,1) circle (2pt); (0,1)–(1,0); (0,1)–(1,2); (1,0)–(1,1); (1,0)–(2,0); (1,1)–(1,2); (1,2)–(2,2); (2,0)–(2,1); (2,0)–(3,1); (2,1)–(2,2); (2,2)–(3,1);
(4,1) circle (2pt); (5,0) circle (2pt); (5,2) circle (2pt); (6,0) circle (2pt); (6,2) circle (2pt); (7,1) circle (2pt); (5.5,0) circle (2pt); (4,1)–(5,0); (4,1)–(5,2); (5,0)–(5,1); (5,0)–(6,0); (5,1)–(5,2); (5,2)–(6,2); (6,0)–(6,1); (6,0)–(7,1); (6,1)–(6,2); (6,2)–(7,1);
We close out this section with some observations about graphs of low scramble number.
\[Lem:Cycle\] If $G$ is a cycle, the $\mathrm{sn}(G) = 2$.
For any $v\in V(G)$ consider the scramble $\mathscr{S} = \Big\{ \{ v \}, V(G) \smallsetminus \{ v \} \Big\}$. Because the two eggs are disjoint, any hitting set has size at least two. If $A$ is a subset of the vertices such that both $A$ and $A^c$ contain eggs, then either $A$ or $A^c$ is equal to $\{ v \}$. Since $\vert E(A,A^c) \vert = 2$, we see that $\vert \vert \mathscr{S} \vert \vert = 2$. There can be no scramble of higher order because, if $A \subsetneq V(G)$ is a connected subset, then $\vert E(A,A^c) \vert = 2$.
\[Cor:Tree\] The scramble number of a graph $G$ is 1 if and only if $G$ is a tree.
If $G$ is a tree, then $$1 = \mathrm{tw}(G) \leq \mathrm{sn}(G) \leq \mathrm{gon}(G) = 1,$$ so $\mathrm{sn}(G) = 1$. On the other hand, if $G$ is not a tree, then it contains a cycle. By Proposition \[Prop:Subgraph\], the scramble number of $G$ is at least that of the cycle, and by Lemma \[Lem:Cycle\], the scramble number of the cycle is 2.
Examples {#Sec:Examples}
========
In this section, we compute the scramble numbers and gonalities of several well-known families of graphs. Our hope is that these examples illustrate the advantages of the scramble number as a tool for computing gonality, as our constructions are relatively simple in comparison to the preexisting literature.
Our examples all arise as Cartesian products of graphs. Recall that the Cartesian product of two graphs $G_1$ and $G_2$, denoted $G_1 \square G_2$, is the graph with vertex set $V(G_1) \times V(G_2)$ and an edge between $(u_1, u_2)$ and $(v_1, v_2)$ if either $u_1 = v_1$ and there is an edge between $u_2$ and $v_2$, or $u_2 = v_2$ and there is an edge between $u_1$ and $v_1$. For a fixed vertex $v \in G_1$, we refer to the set $$C_v = \Big\{ (v,w) \in V(G_1 \square G_2) \vert w \in G_2 \Big\}$$ as a *column*. Similarly, for $w \in G_2$, we refer to the set $$R_w = \Big\{ (v,w) \in V(G_1 \square G_2) \vert v \in G_1 \Big\}$$ as a *row*. A bound on the gonality of Cartesian products can be found in [@AidunMorrison].
[@AidunMorrison Proposition 1.3] \[Prop:ProductBound\] For any two graphs $G_1$ and $G_2$, $$\mathrm{gon}(G_1 \square G_2) \leq \min \Big\{ \mathrm{gon}(G_1) \vert V(G_2) \vert, \mathrm{gon}(G_2) \vert V(G_1) \vert \Big\} .$$
We provide several examples where this bound is achieved. It is a standard result that the $m \times n$ grid graph has treewidth $\min \{ m,n \}$, and it is shown in [@dbG] that such graphs have gonality $\min \{ m,n \}$ as well. A grid graph is an example of the product of two trees, a family of graphs whose gonality is computed in [@AidunMorrison]. We reproduce this result here using the scramble number.
[@AidunMorrison Proposition 3.2] If $T_1$ and $T_2$ are trees, then $$\mathrm{gon} (T_1 \square T_2) = \mathrm{sn} (T_1 \square T_2) = \min \Big\{ \vert V(T_1) \vert, \vert V(T_2) \vert \Big\} .$$
By Proposition \[Prop:ProductBound\], the gonality of $T_1 \square T_2$ is at most $\min \{ \vert V(T_1) \vert, \vert V(T_2) \vert \}$. We therefore seek to bound the gonality from below. By Theorem \[Thm:Bound\], it suffices to construct a scramble of scramble order $\min \{ \vert V(T_1) \vert, \vert V(T_2) \vert \}$.
Let $\mathscr{S}$ be the set of columns in $T_1 \square T_2$. Any row $R_w$ is a hitting set for $\mathscr{S}$, and $\vert R_w \vert = \vert V(T_1) \vert$. Moreover, if $v \in T_1$ is a leaf, then $\vert E( C_v , C_v^c ) \vert = \vert V(T_2) \vert$. It follows that $$\vert \vert \mathscr{S} \vert \vert \leq \min \Big\{ \vert V(T_1) \vert, \vert V(T_2) \vert \Big\} .$$
Since the number of columns is $\vert V(T_1) \vert$ and they are disjoint, there is no hitting set of size less than $\vert V(T_1) \vert$. Now, let $A$ be a subset of $V(T_1 \square T_2)$ with the property that both $A$ and $A^c$ contain a column. Then every row of $T_1 \square T_2$ contains a vertex in $A$ and a vertex in $A^c$, so every row contains an edge in $E(A,A^c)$. It follows that $\vert E(A,A^c) \vert$ is greater than or equal to the number of rows, which is $\vert V(T_2) \vert$. It follows that $$\vert \vert \mathscr{S} \vert \vert \geq \min \Big\{ \vert V(T_1) \vert, \vert V(T_2) \vert \Big\} .$$
In [@Morrison], the authors compute the treewidth of the *stacked prism graphs* $Y_{m,n}$, the product of a cycle with $m$ vertices and a path with $n$ vertices. They show that the gonality of $Y_{m,n}$ is equal to its treewdith, except in the special case where $m = 2n$. We prove the following generalization, which holds even in this special case. Even in the cases where the gonality has been previously computed, we believe that our constructions, using scrambles rather than brambles, are much simpler. For this reason, we have treated these graphs for all $m$ and $n$ uniformly.
\[Prop:StackedPrism\] If $C$ is a cycle and $T$ is a tree, then $$\mathrm{gon} (C \square T) = \mathrm{sn} (C \square T) = \min \Big\{ \vert V(C) \vert, 2 \vert V(T) \vert \Big\} .$$
By Proposition \[Prop:ProductBound\], we have $\mathrm{gon} (C \square T) \leq \min \{ \vert V(C) \vert, 2\vert V(T) \vert \}$. We now compute a lower bound. By Theorem \[Thm:Bound\], it suffices to construct a scramble of scramble order $\min \{ \vert V(C) \vert, 2 \vert V(T) \vert \}$.
Again, we let $\mathscr{S}$ be the set of columns in $C \square T$. (See, for example, Figure \[Fig:StackedPrism\].) Any row $R_w$ is a hitting set for $\mathscr{S}$, and $\vert R_W \vert = \vert V(C) \vert$. Moreover, for any $v \in C$ we have $\vert E( C_v , C_v^c ) \vert = 2 \vert V(T) \vert$. It follows that $$\vert \vert \mathscr{S} \vert \vert \leq \min \Big\{ \vert V(C) \vert, 2 \vert V(T) \vert \Big\} .$$
Since the number of columns is $\vert V(C) \vert$ and they are disjoint, there is no hitting set of size less than $\vert V(C) \vert$. Now, let $A$ be a subset of $V(C \square T)$ with the property that both $A$ and $A^c$ contain a column. Then every row of $C \square T$ contains a vertex in $A$ and a vertex in $A^c$, so every row contains at least two edges in $E(A,A^c)$. It follows that $\vert E(A,A^c) \vert$ is greater than or equal to twice the number of rows, which is $\vert V(T) \vert$. It follows that $$\vert \vert \mathscr{S} \vert \vert \geq \min \Big\{ \vert V(C) \vert, 2 \vert V(T) \vert \Big\} .$$
(0,1) circle (3pt); (1,0) circle (3pt); (1,1) circle (3pt); (0,0) circle (3pt); (2,2) circle (3pt); (2,-1) circle (3pt); (-1,2) circle (3pt); (-1,-1) circle (3pt); (0,1) circle (3pt); (1,0) circle (3pt); (1,1) circle (3pt); (0,0) circle (3pt); (2,2) circle (3pt); (2,-1) circle (3pt); (-1,2) circle (3pt); (-1,-1) circle (3pt); (0,0)–(1,0); (0,0)–(0,1); (0,1)–(1,1); (1,0)–(1,1); (2,2)–(2,-1); (2,2)–(-1,2); (-1,-1)–(2,-1); (-1,-1)–(-1,2); (-1,-1)–(0,0); (2,-1)–(1,0); (-1,2)–(0,1); (2,2)–(1,1);
Note that in the special case where $m = 2n$, Proposition \[Prop:StackedPrism\] shows that the scramble number of the stacked prism graph $Y_{m,n}$ can be strictly greater than the treewidth. In [@Morrison], the authors also compute the treewidth of the *toroidal grid graphs* $T_{m,n}$, the product of a cycle with $m$ vertices and a cycle with $n$ vertices. They further show that the gonality of $T_{m,n}$ is equal to its treewidth, except in the special cases where $m = n $ or $m = n \pm 1$. As with the stacked prism graphs, we compute the gonality of these graphs for all $m$ and $n$ uniformly, including the cases not covered in [@Morrison].
We have $$\mathrm{gon}(T_{m,n}) = \mathrm{sn}(T_{m,n}) = \min \{ 2m, 2n \} .$$
By Proposition \[Prop:ProductBound\], $\mathrm{gon}(T_{m,n}) \leq \min \{ 2m, 2n \}$, so we will compute a lower bound. By Theorem \[Thm:Bound\], it suffices to construct a scramble of scramble order $\min \{ 2m, 2n \}$.
Let $\mathscr{S}$ be the set of columns in $T_{m,n}$ with one vertex removed. (See, for example, Figure \[Fig:Torus\].) The union of any two rows is a hitting set for $\mathscr{S}$ of size $2m$. Moreover, for any vertex $v$ in the cycle of length $m$, we see that both $C_v$ and $C_v^c$ contain an egg, and we have $\vert E( C_v , C_v^c ) \vert = 2n$. It follows that $$\vert \vert \mathscr{S} \vert \vert \leq \min \{ 2m, 2n \} .$$
If $C$ is a subset of the vertices of size less than $2m$, then some column contains at most 1 vertex of $C$, hence $C$ is not a hitting set for $\mathscr{S}$. Now, let $A$ be a subset of $V(T_{m,n})$ with the property that both $A$ and $A^c$ contain eggs. Specifically, suppose that $A$ contains every vertex in column $C_v$ except for possibly $(v,w)$, and that $A^c$ contains every vertex in column $C_{v'}$ except for possibly $(v',w')$. If $w'' \neq w, w'$ is a vertex in the cycle of length $n$, then the row $R_{w''}$ contains a vertex in $A$ and a vertex in $A^c$, so at least two edges in $R_{w''}$ are contained in $E(A,A^c)$. if $(v,w) \notin A$, then the two edges in column $C_v$ with endpoints $(v,w)$ are contained in $E(A,A^c)$, and similarly, if $(v',w') \notin A^c$, then the two edges in column $C_{v'}$ with endpoints $(v',w')$ are contained in $E(A,A^c)$. On the other hand, if $(v,w) \in A$ and $(v',w) \in A^c$, then at least two edges in $R_w$ are contained in $E(A,A^c)$, and similarly, if $(v',w') \in A^c$ and $(v,w') \in A$, then at least two edges in $R_{w'}$ are contained in $E(A,A^c)$. It follows that $$\vert \vert \mathscr{S} \vert \vert \geq \min \{ 2m , 2n \} .$$
(0,0) circle (3pt); (0,2) circle (3pt); (0,3) circle (3pt); (2,0) circle (3pt); (2,1) circle (3pt); (2,3) circle (3pt); (0,0) circle (3pt); (0,1) circle (3pt); (0,2) circle (3pt); (0,3) circle (3pt); (1,0) circle (3pt); (1,1) circle (3pt); (1,2) circle (3pt); (1,3) circle (3pt); (2,0) circle (3pt); (2,1) circle (3pt); (2,2) circle (3pt); (2,3) circle (3pt); (3,0) circle (3pt); (3,1) circle (3pt); (3,2) circle (3pt); (3,3) circle (3pt);
|
---
abstract: 'This is the first of a series of papers, where we introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions to the Ricci flow in the nonsmooth setting. In this first paper, we prove various new estimates for the Ricci flow, and show that they in fact characterize solutions of the Ricci flow. Namely, given a family $(M,g_t)_{t\in I}$ of Riemannian manifolds, we consider the path space $P\mathcal{M}$ of its space time $\mathcal{M}=M\times I$. Our first characterization says that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if an infinite dimensional gradient estimate holds for all functions on $P\mathcal{M}$. We prove additional characterizations in terms of the $C^{1/2}$-regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on $(M,g_t)_{t\in I}$, which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. We will develop the structure theory of these weak solutions in subsequent papers.'
author:
- 'Robert Haslhofer and Aaron Naber[^1]'
bibliography:
- 'HN\_weakricciflow.bib'
title: Weak solutions for the Ricci flow I
---
Introduction
============
Background and overview
-----------------------
The Ricci flow, introduced by Richard Hamilton [@Ham82], evolves Riemannian manifolds in time and is given by the equation $$\label{eq_ricci_flow}
\partial_t g_t=-2{{\rm Ric}}_{g_t}.$$ As with all geometric equations, the key to the analysis of is to prove estimates that are strong enough to capture the analytic and geometric behavior. Many of the known estimates for the Ricci flow are similar in nature to – but often have been harder to develop than – the corresponding estimates for other geometric equations. Since the geometry itself is evolving, even the most basic geometric quantities, like the heat kernel, can behave quite badly. Furthermore, many techniques from geometric analysis that rely on the presence of an ambient space (or a fixed underlying manifold) are not available for the Ricci flow. In particular, it has been a longstanding open problem to find a notion of weak solutions for the Ricci flow.\
The goal of this paper, the first in a series, is to introduce a new class of estimates for the Ricci flow. Our new estimates not only give new information about solutions of the Ricci flow, but are designed to be sufficiently powerful that they give analytic criteria for determining when a family of Riemannian manifolds solves the Ricci flow. That is, we will see that if a family $(M,g_t)_{t\in I}$ of Riemannian manifolds satisfies the analytic estimates of this paper, then in fact this family solves . Such analytic criteria can be used to define weak solutions and have become of increasing importance in other areas of Ricci curvature, see for instance [@LottVillani; @Sturm; @AGS; @Naber], but have not been available up to now for the Ricci flow itself.\
We start with the comparably simple task of characterizing supersoluions of the Ricci flow, i.e. families $(M,g_t)_{t\in I}$ such that $\partial_t g_t\geq -2{{\rm Ric}}_{g_t}$, see Section \[sec\_introsuper\] and Section \[sec\_supersol\]. As summarized in Theorem \[thm\_supersol\], supersoluions can be characterized in terms of various estimates for the linear heat equation on $(M,g_t)_{t\in I}$. These estimates generalize the Bakry-Emery-Ledoux estimates for manifolds with lower Ricci curvature bounds [@BakryEmery; @BakryLedoux], see also McCann-Topping [@McCannTopping]. In particular, one can characterize supersolutions in terms of a log-Sobolev inequality, and a Poincare-inequality. The log-Sobolev inequality is not the one discovered by Perelman [@Per1], but the more recent one from Hein-Naber [@HN_eps].\
To characterize solutions of the Ricci flow, and not just supersolutions, we prove infinite dimensional generalizations of the above estimates. Motivated by work in stochastic analysis [@Malliavin_icm; @Driver_ibp; @Fang; @AE_logsob; @Hsu_logsob] and prior work of the second author [@Naber], our approach to finding such infinite dimensional generalizations is to do analysis on path space. More precisely, it turns out that the right path space to consider, is the space $P\mathcal{M}$ of continuous curves in the space-time $\mathcal{M}=M\times I$, which are allowed to move arbitrarily along the manifold $M$ but are required to move backwards along the $I$ factor with unit speed. To be able to do analysis on $P\mathcal{M}$ we have to set up quite a bit of machinery from stochastic analysis, notably the notions of Wiener measure, stochastic parallel transport, parallel gradient and Malliavin gradient, adapted to our space-time setting. We describe this briefly in Section \[sec\_introprelim\] and give a comprehensive treatment in Section \[sec\_prelim\]. For example, the construction of parallel transport is quite subtle, since almost no curve of Brownian motion is $C^1$. Nevertheless, using ideas from Eells-Elworthy-Malliavin [@Elworthy; @Malliavin], we can make sense of parallel transport on space-time for almost every curve of Brownian motion, see Section \[sec\_brownian\].\
Having set the stage, let us now discuss our infinite dimensional estimates. Our first characterization in Section \[sec\_introgradient\] directly relates solutions of the Ricci flow to gradient estimates on path space. Specifically, we will see that a family $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if a certain gradient inequality (R2) holds for all functions on $P\mathcal{M}$. We will see how this directly generalizes the gradient estimate (S2) proved in Theorem \[thm\_supersol\] for supersolutions. Our second characterization in Section \[sec\_intromartingales\] is in terms of the time regularity of martingales on path space. Specifically, we will see that martingales $F^\tau$ on path space satisfy a precise $C^{1/2}$-Hölder estimate (R3) if and only if the family $(M,g_t)_{t\in I}$ evolves by Ricci flow. Our third characterization in Section \[sec\_intrologsob\] is in terms of an infinite dimensional log-Sobolev inequality (R4), and our final characterization in Section \[sec\_introgap\] is in terms of the corresponding spectral gap (R5). Our characterizations of solutions of the Ricci flow can be thought of as infinite dimensional generalizations of the estimates for supersolutions. Namely, if we evaluate our infinite dimensional estimates for the simplest possible test functions, i.e. functions on path space that only depend on the value of the curve at a single time, then we actually recover the finite dimensional estimates from Theorem \[thm\_supersol\]. Of course, there are many more sophisticated test functions that we can plug in our estimates, and this is one of the reasons why our estimates are actually strong enough to characterize solutions, and not just supersolutions. Our characterizations of solutions of the Ricci flow constitute the main results of this article and are summarized in Theorem \[main\_thm\_intro\].\
Let us also emphasize that Theorem \[main\_thm\_intro\] truly relies on ideas from stochastic analysis, i.e. doing analysis on path space $P{\mathcal{M}}$, as it seems that analysis on $(M,g_t)_{t\in I}$ can only be used to characterize supersolutions but not solutions. In fact, some indications that stochastic analysis might be useful in the study of Ricci flow have already appeared previously in the literature: Arnoundon-Coulibaly-Thalmaier proved the existence of Brownian motion in a time dependent setting [@ACT] (see also [@Coulibaly]), and used this to prove a Bismut type formula for the Ricci flow. Kuwada-Philipowski studied the relationship between time dependent Brownian motion and Perelman’s $\mathcal{L}$-geodesics and obtained a nice nonexplosion result [@KuwadaPhilip; @KuwadaPhilip2] (see also [@Cheng]), and Guo-Philipowski-Thalmaier found some applications of stochastic analysis to ancient solutions [@GPT]. Based on our new estimates there are many more directions to explore.\
In future papers of this series we will use our estimates to investigate singularities in the Ricci flow. In most situations, the Ricci flow develops singularities in finite time. Typically, the curvature blows up in certain regions but remains bounded on the remaining parts of the manifold [@Hamilton_survey]. One would then like to understand these singularities and find ways to continue the flow beyond the first singular time.\
The formation of singularities is of course an ubiquitous phenomenon in the study of nonlinear PDEs. For other geometric evolution equations there are good notions of weak solutions that allow one to continue the flow through any singularity, e.g. Brakke and level set solutions for the mean curvature flow [@Brakke; @EvansSpruck; @CGG], and Chen-Struwe solutions for the harmonic map heat flow [@ChenStruwe]. For the Ricci flow however, it is only known in a few special - albeit very important - cases, how to continue the flow through singularities. Most notably, Perelman’s Ricci flow with surgery [@Per1; @Per2] provides a highly successful way to deal with the formation of singularities in dimension three. Surgery has also been implemented in the case of four-manifolds with positive isotropic curvature [@Ham_pic; @CZ_pic]. Recently, Kleiner-Lott proved the beautiful result that as the surgery parameters degenerate it is possible to pass to certain limits, called singular Ricci flows [@KL_singular]. Also, there has been a lot of progress in the Kähler case, see e.g. Song-Tian [@SongTian] and Eyssidieux-Guedj-Zeriahi [@EGZ_kahler]. In most other cases however, it is a widely open problem how to deal with the formation of singularities.\
In the second paper of this series we will use the estimates of this first paper to give a notion of the Ricci flow for a family of metric-measure spaces. Using analytic characterizations to define weak solutions is a well developed tool in the context of lower Ricci curvature [@LottVillani; @Sturm; @AGS], and more recently in the context of bounded Ricci curvature [@Naber]. Similarly, based on the characterizations of Theorem \[main\_thm\_intro\] we will define a notion of weak solutions for the Ricci flow and develop their theory. We will discuss this in subsequent papers, but let us briefly describe the idea. We consider metric-measure spaces $\mathcal{M}$ equipped with a time function and a linear heat flow. We call $\mathcal{M}$ a weak solution of the Ricci flow if and only if the gradient estimate (R2) holds on $P\mathcal{M}$. We then establish various geometric and analytic estimates for these weak solutions. One of our applications concerns a question of Perelman about limits of Ricci flows with surgery [@Per1]. Namely, the metric completion of the space-time of Kleiner-Lott [@KL_singular], which they obtained as a limit of Ricci flows with surgery where the neck radius is sent to zero, is a weak solution in our sense.
Characterization of supersolutions of the Ricci flow {#sec_introsuper}
----------------------------------------------------
As a motivation for our approach to characterize solutions of the Ricci flow, let us first characterize supersolutions, i.e. smooth families of Riemannian manifolds such that $$\partial_t g_t\geq -2{{\rm Ric}}_{g_t}.$$
To fix notation, let $(M,g_t)_{t\in I}$ be a smooth family of Riemannian manifolds, where $I=[0,T_1]$. To avoid technicalities, we assume throughout the paper that all manifolds are complete and that $$\label{tech_ass}
\sup_{M\times I}\left({\lvert{\text{Rm}}\rvert}+{\lvert\partial_t g_t\rvert}+{\lvert\nabla \partial_t g_t\rvert}\right)<\infty.$$ However, all our estimates are independent of the implicit constant in . We consider the heat equation $(\partial_t-{\Delta}_{g_t})w=0$ on our evolving manifolds $(M,g_t)_{t\in I}$. For every $s,T\in I$ with $ s\leq T$, and every smooth function $u$ with compact support, we write $P_{sT}u$ for the solution at time $T$ with initial condition $u$ at time $s$. In other words, $$(P_{sT}u)(x)=\int_M u(y)\, H(x,T\,|\,y,s) d{{\rm vol}}_{g(s)}(y),$$ where $H(x,T\,|\,y,s)$ is the heat kernel with pole at $(y,s)$. We write $d \nu_{(x,T)}(y,s)=H(x,T\,|\,y,s) d{{\rm vol}}_{g(s)}(y)$. It is often useful to think of $d \nu_{(x,T)}$ as the adjoint heat kernel measure based at $(x,T)$.
The following theorem summarizes our characterizations of supersolutions of the Ricci flow.
\[thm\_supersol\] For every smooth family $(M,g_t)_{t\in I}$ of Riemannian manifolds (complete, satisfying ), the following conditions are equivalent:
1. \[S1\] The family $(M,g_t)_{t\in I}$ is a supersolution of the Ricci flow, $$\partial_t g_t\geq -2{{\rm Ric}}_{g_t}.$$
2. \[S2\] For all test functions $u$, the heat equation on $(M,g_t)_{t\in I}$ satisfies the gradient estimate $${\lvert{\nabla}P_{sT} u\rvert}\leq P_{sT}{\lvert{\nabla}u\rvert}.$$
3. \[S3\] For all test functions $u$, the heat equation on $(M,g_t)_{t\in I}$ satisfies the estimate $${\lvert{\nabla}P_{sT} u\rvert}^2\leq P_{sT}{\lvert{\nabla}u\rvert}^2.$$
4. \[S4\] For all functions $u:M\to {\mathds{R}}$ with $\int_M u^2(y) \, d\nu_{(x,T)}(y,s)=1$, we have the log-Sobolev inequality $$\int_M u^2(y)\log u^2(y) \, d\nu_{(x,T)}(y,s)\leq 4(T-s)\!\! \int_M{\lvert{\nabla}u\rvert}_{g_s}^2\! (y)\,d\nu_{(x,T)}(y,s).$$
5. \[S5\] For all functions $u:M\to {\mathds{R}}$ with $\int_M u(y) \, d\nu_{(x,T)}(y,s)=0$, we have the Poincare-inequality $$\int_M u^2 \, d\nu_{(x,T)}(y,s)\leq 2 (T-s)\!\! \int_M {\lvert{\nabla}u\rvert}^2_{g_s}\, d\nu_{(x,T)}(y,s).$$
In essence, this all follows from the Bochner-formula for the heat operator $\Box=\partial_t-{\Delta}_{g_t}$, $$\label{parboch}
\Box {\lvert{\nabla}u\rvert}^2=2{\left\langle{\nabla}u,{\nabla}\Box u\right\rangle}-2{\lvert{\nabla}^2 u\rvert}^2-(\partial_t g+2{{\rm Ric}})({{\rm grad}}u,{{\rm grad}}u),$$ see Section \[sec\_supersol\] for the (easy) proof of Theorem \[thm\_supersol\]. The reader can also view this as a good toy model for the more sophisticated infinite-dimensional computations on path space that we carry out in later sections.
Theorem \[thm\_supersol\] can be thought of as parabolic version of the Bakry-Emery characterization of nonnegative Ricci curvature [@BakryEmery; @BakryLedoux]. Another interesting characterization of supersolutions of the Ricci flow, in terms of the Wasserstein distance, has been given by McCann-Topping [@McCannTopping].
Characterization of solutions of the Ricci flow {#sec_introsol}
-----------------------------------------------
In this section we describe our main estimates on path space, and use them to characterize solutions of the Ricci flow.
### Stochastic analysis on evolving manifolds {#sec_introprelim}
Our estimates require quite some machinery from stochastic analysis, notably the notions of Wiener measure, stochastic parallel transport, parallel gradient and Malliavin gradient, adapted to our time-dependent setting. We will now briefly describe these notions, and refer to Section \[sec\_prelim\] for a more complete treatment.\
Let $(M,g_t)_{t\in I}$ be a smooth family of Riemannian manifolds, where $I=[0,T_1]$. We recall that we always assume that our manifolds are complete and that is satisfied, though the second assumption is for convenience. Throughout this work we will think of the evolving manifolds in terms of the **space-time** ${\mathcal{M}}=M\times I$. As observed by Hamilton [@Hamilton_Harnack] there is a natural **space-time connection** defined by $$\label{eq_spacetime_connection}
\nabla_X Y=\nabla^{g_t}_X Y,\qquad\qquad \nabla_t Y=\partial_t Y+\frac{1}{2}\partial_t g_t(Y,\cdot)^{\sharp_{g_t}}.$$ The point is that this connection is compatible with the metric, i.e. $\tfrac{d}{dt}{\lvertY\rvert}_{g_t}^2=2\langle Y,\nabla_t Y\rangle$.
It is useful to consider **space-time curves** going *backwards in time*, c.f. [@LiYau; @Per1]. Namely, for each $(x,T)\in {\mathcal{M}}$, we consider the **based path space** $P_{(x,T)}{\mathcal{M}}$ consisting of all space-time curves of the form $\{\gamma_\tau=(x_\tau,T-\tau)\}_{\tau\in [0,T]}$, where $\{x_\tau\}_{\tau\in [0,T]}$ is a continuous curve in $M$ with $x_0=x$.
We equip the path space $P_{(x,T)}{\mathcal{M}}$ with a probability measure $\Gamma_{(x,T)}$, that we call the **Wiener measure** of **Brownian motion** on our evolving family of manifolds, based at $(x,T)$. The measure $\Gamma_{(x,T)}$ is uniquely characterized by the following property. If $e_{\bf{\sigma}}:P_{(x,T)}{\mathcal{M}}\to M^{k}$, $\gamma\mapsto (x_{\sigma_1},\ldots, x_{\sigma_k})$, is the **evaluation map** at ${\bf{\sigma}}=\{0\leq \sigma_1\leq\ldots\leq \sigma_k\leq T\}$, and if we write $s_i=T-\sigma_i$, then $$\label{intro_pushforwardwiener}
e_{\bf{\sigma},\ast}d\Gamma_{(x,T)}(y_1,\ldots,y_k)=H(x,T|y_1,s_1)d{{\rm vol}}_{g_{s_1}}(y_1)\cdots H(y_{k-1},s_{k-1}|y_k,s_k)d{{\rm vol}}_{g_{s_k}}(y_k),$$ where $H$ is the heat kernel of $\partial_t-{\Delta}_{g_t}$; see Section \[sec\_brownian\] for the construction of Brownian motion.
It is often convenient to consider the **total path space** $P_T{\mathcal{M}}=\cup_{x\in M} P_{(x,T)}{\mathcal{M}}$. Note that we can identify $(P_T{\mathcal{M}},\Gamma_{(x,T)})$ with $(P_{(x,T)}{\mathcal{M}},\Gamma_{(x,T)})$, since the measure $\Gamma_{(x,T)}$ concentrates on curves starting at $(x,T)$. Sometimes it is also useful to equip the total path space $P_T{\mathcal{M}}$ with the measure $\Gamma_T=\int \Gamma_{(x,T)} d{{\rm vol}}_{g_T}(x)$.\
The space $(P_T{\mathcal{M}},\Gamma_{(x,T)})$ can be equipped with a notion of **stochastic parallel transport**, a family of isometries $P_\tau(\gamma):(T_{x_\tau}M,g_{T-\tau})\to (T_xM,g_T)$. If the curves $\gamma$ were $C^1$, then $P_\tau(\gamma)$ would just be the parallel transport from differential geometry, with respect to the natural space-time connection defined in . Of course, almost no curve of Brownian motion is $C^1$. Nevertheless, using deep ideas from Eells-Elworthy-Malliavin we can still make sense of $P_\tau(\gamma)$ for almost every curve $\gamma$, see Section \[sec\_brownian\] for the construction.\
The space $(P_T{\mathcal{M}},\Gamma_{(x,T)})$ can be equipped with two natural notions of gradient. Suppose first that $F:P_{(x,T)}{\mathcal{M}}\to {\mathds{R}}$ is a **cylinder function**, i.e. a function of the form $F=u\circ e_\sigma$, where $e_\sigma: P_{(x,T)}{\mathcal{M}}\to M^k$ is an evaluation map and $u:M^k\to {\mathds{R}}$ is a smooth function with compact support. If $v\in (T_xM,g_T)$, then for almost every (a.e.) curve $\gamma$, we can consider the continuous vector field $V=\{V_\tau=P_\tau^{-1}v\}_{\tau\in [0,T]}$ along $\gamma$, where $P_\tau=P_\tau(\gamma)$ denotes stochastic parallel transport as in the previous paragraph. Note that the directional derivative $D_VF(\gamma)$ is well defined, as a limit of difference quotients as usual.
The **parallel gradient** ${\nabla}^{\parallel}F(\gamma)\in (T_xM, g_T)$ is then defined by the condition that $$\label{intro_pargrad}
D_V F(\gamma)=\langle {\nabla}^{\parallel}F(\gamma),v\rangle_{(T_xM, g_T)}$$ for all $v\in (T_xM,g_T)$, where $V=\{V_\tau=P_\tau^{-1}v\}_{\tau\in [0,T]}$ is the parallel vector field associated to $v$, as above. More generally, there is a one parameter family of parallel gradients ${\nabla}^{\parallel}_\sigma$ ($0\leq \sigma\leq T$), which captures the part of the gradient coming from the time interval $[\sigma,T]$. In particular, ${\nabla}^{\parallel}={\nabla}^{\parallel}_0$.
The **Malliavin gradient** ${\nabla}^{\mathcal{H}}F$ is defined along similar lines, but takes values in an infinite dimensional Hilbert space. Namely, let $\mathcal{H}$ be the Hilbert-space of ${H}^1$-curves $\{v_\tau\}_{\tau\in[0,T]}$ in $(T_xM,g_T)$ with $v_0=0$, equipped with the inner product $ \langle v,w \rangle_{\mathcal{H}}=\int_0^T \left\langle \dot{v}_\tau,\dot{w}_\tau\right\rangle_{(T_xM,g_T)} d\tau$. Then $\nabla^{\mathcal{H}}F:P_{(x,T)}{\mathcal{M}}\to \mathcal{H}$ is the unique almost everywhere defined function such that $$\label{intro_mall_grad}
D_V F(\gamma)=\langle \nabla^{\mathcal{H}}F(\gamma),v\rangle_{\mathcal{H}},$$ for a.e. curve $\gamma$, and every $v\in \mathcal{H}$, where $V=\{P_\tau^{-1} v_\tau\}_{\tau\in [0,T]}$.
Having defined them on cylinder functions, the ($\sigma$-)parallel gradient and the Malliavin gradient can be extended to closed unbounded operators on $L^2$, see Section \[sec\_gradients\] for details.
Finally, the **Ornstein-Uhlenbeck operator** $\mathcal{L}=\nabla^{{\mathcal{H}}\ast}\nabla^{\mathcal{H}}$ is defined by composing the Malliavin gradient with its adjoint. More generally, there is a family of Ornstein-Uhlenbeck operators $\mathcal{L}_{\tau_1,\tau_2}$ ($0\leq \tau_1<\tau_2\leq T$), which captures the part of the Laplacian coming from the time interval $[\tau_1,\tau_2]$. In particular, $\mathcal{L}=\mathcal{L}_{0,T}$.
### Ricci flow and the gradient estimate {#sec_introgradient}
Our first characterization of solutions of the Ricci flow is in terms of an infinite dimensional gradient estimate on the associated path space. Let $(M,g_t)_{t\in I}$ be smooth family of Riemannian manifolds and let $P_T{\mathcal{M}}$ be its path space, equipped with the Wiener measure and the parallel gradient. If $F:P_T{\mathcal{M}}\to{\mathds{R}}$ is a sufficiently nice function, for instance a cylinder function, one can ask whether one can control the gradient of $\int_{P_T{\mathcal{M}}}Fd\Gamma_{(x,T)}$ viewed as a function of $x\in M$, in terms of some natural gradient of $F$ viewed as a function on path space. In fact, the answer to this question turns out to be highly relevant, in that it yields our first characterization of solutions of the Ricci flow. Namely, we prove that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if the gradient estimate $$(R2)\qquad {\lvert\nabla_x\int_{P_T{\mathcal{M}}} F d\Gamma_{(x,T)}\rvert}\leq \int_{P_T{\mathcal{M}}}{\lvert\nabla^{\parallel}F\rvert}d\Gamma_{(x,T)} \, ,$$ holds for all function $F\in L^2(P_T{\mathcal{M}},\Gamma_T)$ (for a.e. $(x,T)\in{\mathcal{M}}$).
\[rem\_gradest\] The infinite dimensional gradient estimate (R2) can be thought of as (vast) generalization of the finite dimensional gradient estimate (S2) for the heat equation. Namely, let $F=u\circ e_\sigma:P_T{\mathcal{M}}\to M\to {\mathds{R}}$ be a $1$-point cylinder function, and write $s=T-\sigma$. By equation the pushforward measure $$\label{intro_push_heat}
e_{\sigma,\ast} d\Gamma_{(x,T)}=d\nu_{(x,T)}(\cdot,s)$$ is given by the heat kernel measure $d \nu_{(x,T)}(y,s)=H(x,T\,|\,y,s) d{{\rm vol}}_{g(s)}(y)$, and thus $$\int_{P_T{\mathcal{M}}} F d\Gamma_{(x,T)}= \int_{M} u\, e_{\sigma,\ast} d\Gamma_{(x,T)} =(P_{sT}u)(x).$$ Moreover, using on sees that ${\lvert\nabla^{\parallel}F\rvert}(\gamma)={\lvert\nabla u\rvert}_{g_s}\!(e_\sigma(\gamma))$, which together with implies that $$\int_{P_T{\mathcal{M}}} {\lvert{\nabla}^{\parallel}F\rvert} d\Gamma_{(x,T)}= \int_{M} {\lvert{\nabla}u\rvert}\, e_{\sigma,\ast} d\Gamma_{(x,T)} =(P_{sT}{\lvert{\nabla}u\rvert})(x).$$ Thus, in the special case of $1$-point cylinder function the estimate (R2) reduces to the finite dimensional heat equation estimate $$(\textrm{S2})\qquad {\lvert\nabla P_{sT}u\rvert}\leq P_{sT}{\lvert{\nabla}u\rvert}.$$ Of course, there are many more test functions on path space than just $1$-point cylinder function. This is one of the reasons why our infinite dimensional estimate (R2) is strong enough to characterize solutions of the Ricci flow, while the finite dimensional heat equation estimate (S2) just characterizes supersolutions.
### Ricci flow and the regularity of martingales {#sec_intromartingales}
Our second characterization of solutions of the Ricci flow is in terms of the regularity of martingales on its path space. Let $(M,g_t)_{t\in I}$ be a smooth family of Riemannian manifolds, and let $P_T{\mathcal{M}}$ be its path space. For every function $F\in L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$, we can consider the **induced martingale** $\{F^\tau\}_{\tau\in [0,T]}$, $$\label{intro_martingale}
F^{\tau}(\gamma)=\int_{P_{T-\tau}{\mathcal{M}}} F(\gamma|_{[0,\tau]}\ast \gamma') d\Gamma_{\gamma_\tau}(\gamma'),$$ where the integral is over all Brownian curves $\gamma'$ based at $\gamma_\tau$, and $\ast$ denotes concatenation. The family $\{F^\tau\}_{\tau\in [0,T]}$ indeed has the martingale property $(F^{\tau'})^\tau=F^\tau$ ($\tau'\geq \tau$) and captures how $F$ depends on the $[0,\tau]$-part of the curves, see Section \[sec\_condexp\]. The **quadratic variation** $[F^\bullet]_\tau$ of the martingale $\{F^\tau\}_{\tau\in [0,T]}$ is defined by $[F^\bullet]_\tau=\lim_{{\lvert\lvert\{\tau_j\}\rvert\rvert}\to 0} \sum_k (F^{\tau_{k}}-F^{\tau_{k-1}})^2$, where the limit is taken in probability, over all partions $\{\tau_j\}$ of $[0,\tau]$ with mesh going to zero, see Section \[sec\_condexp\]. It turns out that solutions of the Ricci flow can be characterized in terms of certain bounds for $\frac{d[F^\bullet]_\tau}{d\tau}$. Namely, we prove that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if the estimate $$(R3)\qquad \int_{P_T{\mathcal{M}}}\frac{d[F^\bullet]_\tau}{d\tau} d\Gamma_{(x,T)}\leq 2\int_{P_T{\mathcal{M}}}{\lvert{\nabla}^{\parallel}_\tau F\rvert}^2 d\Gamma_{(x,T)}$$ holds for every $F\in L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$ (for all $(x,T)\in{\mathcal{M}}$).
The estimate (R3) is a (vast) generalization of (S3). Namely, let $F=u\circ e_\sigma:P_T{\mathcal{M}}\to M\to {\mathds{R}}$ be a $1$-point cylinder function, and write $s=T-\sigma$. If ${\varepsilon}>0$, then by and we have $$F^{\varepsilon}(\gamma)=\int_M u(y) d\nu_{\gamma_{\varepsilon}}(y,s)=(P_{s,T-{\varepsilon}}u)(x_{\varepsilon}).$$ Appying this twice and using the short time asymptotics of the heat kernel, one can compute that $$\begin{gathered}
\int_{P_T{\mathcal{M}}}\frac{d[F^\bullet]_\tau}{d\tau}|_{\tau=0}\, d\Gamma_{(x,T)}=\lim_{{\varepsilon}\to 0}\frac{1}{{\varepsilon}}\int_{P_T{\mathcal{M}}}\left(F^{\varepsilon}-(F^{\varepsilon})^0\right)^2 d\Gamma_{(x,T)}\\
=\lim_{{\varepsilon}\to 0}\frac{1}{{\varepsilon}}\int_{M}\left((P_{s,T-{\varepsilon}}u)(z)-\int_M (P_{s,T-{\varepsilon}}u)(\hat{z})\, d\nu_{(x,T)}(\hat{z}, T-{\varepsilon}) \right)^2 d\nu_{(x,T)}(z,T-{\varepsilon})
=2{\lvert\nabla P_{sT}u\rvert}^2(x).\nonumber\end{gathered}$$ Thus, in the special case of 1-point cylinder functions, (R3) for $\tau=0$ reduces to the estimate[^2] $$(\textrm{S3})\qquad {\lvert\nabla P_{sT}u\rvert}^2\leq P_{sT}{\lvert{\nabla}u\rvert}^2.$$
### Ricci flow and the log-Sobolev inequality {#sec_intrologsob}
Our third characterization of solutions of the Ricci flow is in terms of a log-Sobolev inequality on its path space. Log-Sobolev inequalities have a long history, going back to Gross [@Gross]. In the context of Ricci flow, they appear in Perelman’s monotonicity formula [@Per1] and also in the inequality (S4) of Hein-Naber [@HN_eps]. We characterize solutions of the Ricci flow via an infinite dimensional generalization of the inequality (S4). Namely, we prove that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if the log-Sobolev inequality $$(R4)\qquad \int_{P_T{\mathcal{M}}} \left((F^2)^{\tau_2} \log\, (F^2)^{\tau_2} - (F^2)^{\tau_1} \log\, (F^2)^{\tau_1} \right)\, d\Gamma_{(x,T)}\leq 4 \int_{P_T{\mathcal{M}}} \langle F,\mathcal{L}_{\tau_1,\tau_2} F\rangle d\Gamma_{(x,T)},$$ holds for every $F$ in the domain of the Ornstein-Uhlenbeck operator $\mathcal{L}_{\tau_1,\tau_2}$ (for all $(x,T)\in{\mathcal{M}}$ and all $0\leq \tau_1<\tau_2\leq T$). Here, $(F^2)^\tau$ denotes the martingale induced by $F^2$.
If $\tau_1=0$ and $\tau_2=T$ the inequality (R4) takes the somewhat simpler form $$\int_{P_T{\mathcal{M}}} F^2 \log\, F^2\, d\Gamma_{(x,T)}\leq 4 \int_{P_T{\mathcal{M}}} {\lvert\nabla^{\mathcal{H}}F\rvert}^2 d\Gamma_{(x,T)}$$ for all $F$ with $\int_{P_T{\mathcal{M}}}F^2=1$. Specializing further, for a 1-point cylinder function $F=u\circ e_\sigma:P_T{\mathcal{M}}\to M\to {\mathds{R}}$ ($s=T-\sigma$), using one can see that ${\lvert\nabla^{{\mathcal{H}}} F\rvert}_{\mathcal{H}}^2(\gamma)=(T-s){\lvert{\nabla}u\rvert}_{g_s}\!(e_\sigma(\gamma))$, c.f. Proposition \[prop\_malliavin\]. Together with this shows that (R4) then reduces to (S4).
### Ricci flow and the spectral gap {#sec_introgap}
Our final characterization of solutions of the Ricci flow is in terms of the spectral gap of the Ornstein-Uhlenbeck operator on its path space.[^3] We prove that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if the Ornstein-Uhlenbeck operator $\mathcal{L}_{\tau_1,\tau_2}$ (for all $(x,T)\in {\mathcal{M}}$ and all $0\leq \tau_1<\tau_2\leq T$) satisfy the spectral gap estimate $$(R5)\qquad \int_{P_T{\mathcal{M}}}(F^{\tau_2}-F^{\tau_1})^2 d\Gamma_{(x,T)}\leq 2 \int_{P_T{\mathcal{M}}}\langle F,\mathcal{L}_{\tau_1,\tau_2} F\rangle d\Gamma_{(x,T)}.$$
\[rem\_spectgap\_red\] In the special case of 1-point cylinder functions, the estimate (R5) again reduces to (S5).
### Summary of main results
Our main results are summarized in the following theorem.
\[main\_thm\_intro\] For every smooth family $(M,g_t)_{t\in I}$ of Riemannian manifolds (complete, satisfying ), the following conditions are equivalent:
1. \[R1\] The family $(M,g_t)_{t\in I}$ evolves by Ricci flow, $$\partial_t g_t= -2{{\rm Ric}}_{g_t}.$$
2. \[MT2\] For every $F\in L^2(P_T{\mathcal{M}},\Gamma_T)$, we have the gradient estimate $${\lvert\nabla_x\int_{P_T{\mathcal{M}}} F d\Gamma_{(x,T)}\rvert}\leq \int_{P_T{\mathcal{M}}}{\lvert\nabla^{\parallel}F\rvert}d\Gamma_{(x,T)}.$$
3. \[R3\] For every $F\in L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$, the induced martingale $\{F^\tau\}_{\tau\in [0,T]}$ satisfies the estimate $$\int_{P_T{\mathcal{M}}}{\frac{d[F^\bullet]_\tau}{d\tau}} d\Gamma_{(x,T)}\leq 2\int_{P_T{\mathcal{M}}}{\lvert{\nabla}^{\parallel}_\tau F\rvert}^2 d\Gamma_{(x,T)}.$$
4. \[R4\] The Ornstein-Uhlenbeck operator $\mathcal{L}_{\tau_1,\tau_2}$ on based path space $L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$ satisfies the log-Sobolev inequality $$\int_{P_T{\mathcal{M}}} \left((F^2)^{\tau_2} \log\, (F^2)^{\tau_2} - (F^2)^{\tau_1} \log\, (F^2)^{\tau_1} \right)\, d\Gamma_{(x,T)}\leq 4 \int_{P_T{\mathcal{M}}} \langle F,\mathcal{L}_{\tau_1,\tau_2} F\rangle \, d\Gamma_{(x,T)}.$$
5. \[R5\] The Ornstein-Uhlenbeck operator $\mathcal{L}_{\tau_1,\tau_2}$ on based path space $L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$ satisfies the spectral gap estimate $$\int_{P_T{\mathcal{M}}}(F^{\tau_2}-F^{\tau_1})^2 d\Gamma_{(x,T)}\leq 2 \int_{P_T{\mathcal{M}}}\langle F,\mathcal{L}_{\tau_1,\tau_2} F\rangle d\Gamma_{(x,T)}.$$
As explained above, in the special case of 1-point cylinder functions the estimates (R2)–(R5) reduce to the estimates (S2)–(S5), respectively.
Further characterizations are possible. In particular, we have an $L^2$-version of the gradient estimate, and a pointwise $L^1$-version of the martingale estimate, see (R2’) and (R3’) in Section \[sec\_proof\_main\].
**Outline.** This article is organized as follows. In Section \[sec\_supersol\], as a warmup for the proof of the main theorem, we prove Theorem \[thm\_supersol\] characterizing supersolutions of the Ricci flow. In Section \[sec\_prelim\], we set up the machinery of stochastic analysis in our setting of evolving manifolds. In Section \[sec\_proof\_main\], we prove the main theorem (Theorem \[main\_thm\_intro\]) characterizing solutions of the Ricci flow.
Supersolutions of the Ricci flow {#sec_supersol}
================================
In this short section we prove Theorem \[thm\_supersol\], characterizing supersolutions of the Ricci flow
We will prove the implications (S3)$\Leftrightarrow$(S1)$\Leftrightarrow$(S2) and (S1)$\Rightarrow$(S4)$\Rightarrow$(S5)$\Rightarrow$(S3).
(S1)$\Leftrightarrow$(S3): If $g_t$ is a supersolution of the Ricci flow, then the Bochner formula (\[parboch\]) implies $$\Box {\lvert{\nabla}P_{st}u\rvert}^2\leq 0.$$ Thus, ${\lvert{\nabla}P_{st} u\rvert}^2- P_{st}{\lvert{\nabla}u\rvert}^2$ is a subsolution of the heat equation. Since it is zero for $t=s$, it stays nonpositive for all $t>s$, in particular ${\lvert{\nabla}P_{sT} u\rvert}^2\leq P_{sT}{\lvert{\nabla}u\rvert}^2$. To prove the converse implication, assume that $(\partial_tg+2{{\rm Ric}})(X,X)<0$ for some unit tangent vector $X\in T_xM$ at some time $s$. Choose a test function $u$ with ${\nabla}u(p)=X$ and ${\nabla}^2 u(p)=0$. Then by (\[parboch\]) we have $\partial_t{\lvert{\nabla}P_{st}u\rvert}^2>{\Delta}{\lvert{\nabla}u\rvert}^2$ at $p$ at $t=s$; this contradicts (S3).
(S1)$\Leftrightarrow$(S2): If $g_t$ is a supersolution of the Ricci flow, then using the Bochner formula (\[parboch\]) and the Cauchy-Schwarz inequality we obtain $$\begin{aligned}
\Box {\lvert{\nabla}P_{st} u\rvert} &= \frac{1}{{\lvert{\nabla}P_{st} u\rvert}}\left(\frac{1}{2}\Box {\lvert{\nabla}P_{st} u\rvert}^2+\frac{1}{4}\frac{{\lvert{\nabla}{\lvert{\nabla}P_{st} u\rvert}^2\rvert}^2}{{\lvert{\nabla}P_{st} u\rvert}^2}\right)\leq 0.\end{aligned}$$ Thus, ${\lvert{\nabla}P_{st} u\rvert}- P_{st}{\lvert{\nabla}u\rvert}$ is a subsolution of the heat equation. Since it is zero for $t=s$, it stays nonpositive for all $t>s$, in particular ${\lvert{\nabla}P_{sT} u\rvert}\leq P_{sT}{\lvert{\nabla}u\rvert}$. The converse implication follows by considering a test function as above.
(S1)$\Rightarrow$(S4): Let $w>0$. We start by deriving another estimate for the heat equation. Using the Bochner formula (\[parboch\]) and the Peter-Paul inequality we compute $$\begin{aligned}
\Box \left(\frac{{\lvert{\nabla}P_{st} w\rvert}^2}{P_{st}w}\right) = \frac{\Box{\lvert{\nabla}P_{st}w\rvert}^2}{P_{st}w}+2\frac{{\left\langle{\nabla}{\lvert{\nabla}P_{st}w\rvert}^2,{\nabla}P_{st}w\right\rangle}}{(P_{st}w)^2}-2\frac{{\lvert{\nabla}P_{st}w\rvert}^4}{(P_{st}w)^3}\leq 0.\end{aligned}$$ Thus, $\frac{{\lvert{\nabla}P_{st} w\rvert}^2}{P_{st}w}-P_{st}\frac{{\lvert{\nabla}w\rvert}^2}{w}$ is a subsolution of the heat equation. Since it is zero for $t=s$, this implies the estimate $$\label{app_frac}
\frac{{\lvert{\nabla}P_{sr} w\rvert}^2}{P_{sr}w}\leq P_{sr}\frac{{\lvert{\nabla}w\rvert}^2}{w}.$$ Now, using the heat kernel homotopy principle [@HN_eps (3.7)] and we compute $$\int w\log w \, d\nu-\left(\int w\, d\nu\right) \log\left( \int w\, d\nu\right)=\int_s^T \left(P_{rT}\frac{{\lvert{\nabla}P_{sr} w\rvert}^2}{P_{sr}w}\right) (x)\, dr\leq (T-s)\int \frac{{\lvert{\nabla}w\rvert}^2}{w}\, d\nu.$$ Substituting $w=u^2$ this implies the log-Sobolev inequality (S4).
(S4)$\Rightarrow$(S5): This follows by evaluating (S4) for $w^2=1+\varepsilon u$ with $\int u\, d\nu=0$.
(S5)$\Rightarrow$(S3) By the heat kernel homotopy principle [@HN_eps (3.7)] we have $$\int u^2 \, d\nu-\left(\int u d\nu\right)^2 =2\int_s^T \left( P_{rT}{\lvert{\nabla}P_{sr}u\rvert}^2\right) (x)\, dr.$$ Thus, if (S3) fails at some $(x,T)$, then (S5) fails for $d\nu_{(x,T)}$ with ${\lvertT-s\rvert}$ small enough.
Stochastic calculus on evolving manifolds {#sec_prelim}
=========================================
We will now discuss in more detail the required background from stochastic analysis, adapted to our time-dependent setting. There are numerous excellent references for stochastic analysis on manfolds, e.g. [@Elworthy; @Emery; @Hsu; @IkedaWatanabe; @Malliavin; @Stroock]. For readers who wish to focus on one single reference which is particularly close in spirit to the content of the present section we recommend the book by Hsu [@Hsu].
Frame bundle on evolving manifolds {#sec_diffgeo}
----------------------------------
To set things up efficiently, we will first explain how to formulate the differential geometry of evolving manifolds in terms of the frame bundle. For the frame bundle formalism in the time-independent case, see e.g. Kobayashi-Nomizu [@KobNom], for the frame bundle formalism for the Ricci flow, see Hamilton [@Hamilton_Harnack].\
Let $(M,g_t)_{t\in I}$, $I=[0,T_1]$, be a smooth family of Riemannian manifolds, and write ${\mathcal{M}}=M\times I$. Let $Y$ be a time dependent vector field. For each $X\in (T_{x}M,g_t)$ we can compute the covariant spatial derivative $\nabla_X Y=\nabla^{g_t}_X Y$ using the Levi-Civita connection of the metric $g_t$. The covariant time derivative is defined as $\nabla_t Y=\partial_t Y+\frac{1}{2}\partial_t g_t(Y,\cdot)^{\sharp_{g_t}}$. The point is that this gives metric compatibility, namely $\frac{d}{dt}{\lvertY\rvert}_{g_t}^2=2\langle Y,\nabla_t Y\rangle$.
Consider the $O_n$-bundle $\pi: {\mathcal{F}}\to {\mathcal{M}}$, where the fibres ${\mathcal{F}}_{(x,t)}$ are given by the orthogonal maps $u: {\mathds{R}}^n\to (T_xM,g_t)$, and $g\in O_n$ acts from the right via composition. The horizontal lift of a curve $\gamma_t$ in ${\mathcal{M}}$ is a curve $u_t$ in ${\mathcal{F}}$ with $\pi u_t=\gamma_t$ such that $\nabla_{\dot \gamma_t}(u_te)=0$ for all $e\in{\mathds{R}}^n$. Given a vector $\alpha X+\beta \partial_t\in T_{(x,t)}{\mathcal{M}}$ and a frame $u\in {\mathcal{F}}_{(x,t)}$ there is a unique horizontal lift $\alpha X^*+\beta D_t$ with $\pi_* (\alpha X^*+\beta D_t)=X$. Here, $X^*$ is just the horizontal lift of $X\in T_xM$ with respect to the fixed metric $g_t$, and $D_t=\frac{d}{ds}|_0 u_s$, where $u_s$ is the horizontal lift based at $u$ of the curve $s\mapsto (x,t+s)$ with $x$ constant. Most of the time we only consider curves of the form $\gamma_\tau=(x_\tau,T-\tau)$. We denote space-time parallel transport by $P_{\tau_1,\tau_2}=u_{\tau_2}u_{\tau_1}^{-1}:(T_{x_{\tau_1}}M,g_{T-\tau_1})\to (T_{x_{\tau_2}}M,g_{T-\tau_2})$, and observe that this induces parallel translation maps for arbitrary tensor fields. We write $D_\tau=-D_t$.
Given a representation $\rho$ of $O_n$ on some vector space $V$ and an equivariant map from ${\mathcal{F}}$ to $V$, we get a section of the associated vector bundle ${\mathcal{F}}\times_\rho V$, and vice versa. For example, a time-dependent function $f$ corresponds to the invariant function $\tilde{f}=f\pi:{\mathcal{F}}\to{\mathds{R}}$, and a time-dependent vector field $Y$ corresponds to a function $\tilde{Y}:{\mathcal{F}}\to {\mathds{R}}^n$ via $\tilde{Y}(u)=u^{-1}Y_{\pi u}$, which is equivariant in the sense that $\tilde{Y}(ug)=g^{-1}\tilde{Y}(u)$. The following lemma shows how to compute derivatives in terms of the frame bundle.
\[lemma\_firstder\] $\widetilde{Xf}=X^*\tilde f$, $\widetilde{\partial_t f}=D_t\tilde{f}$, $\widetilde{\nabla_X Y}=X^*\tilde{Y}$, and $\widetilde{\nabla_t Y}=D_t\tilde{Y}$.
The first two formulas are obvious, since the horizontal lift of a function is constant in fibre direction. To prove the last formula, let $u_t$ be a horizontal curve with $\pi u_t=\gamma_t=(x,t)$, where $x$ is fixed. Then $$(D_t\tilde{Y})_{u_{t_1}}=\frac{d}{dt}|_{t_1}\tilde{Y}(u_t)=\frac{d}{dt}|_{t_1} u_t^{-1}Y_{\pi u_t}= u_{t_1}^{-1}\frac{d}{ds}|_0 P_{t_1,t_1+s}^{-1}Y_{(x,t_1+s)}=u_{t_1}^{-1}(\nabla_t Y)_{(x,t_1)}=(\widetilde{\nabla_t Y})_{u_1}.$$ The third formula follows from a similar computation. In fact, it is a well known formula from differential geometry with respect to a fixed metric $g_t$.
Let $e_1,\ldots,e_n$ be the standard basis of ${\mathds{R}}^n$. We write $H_i$ for the horizontal vector fields $H_i(u)=(ue_i)^*$, where $*$ denotes the horizontal lift, as before. The horizontal Laplacian is defined by ${\Delta}_H=\sum_{i=1}^n H_i^2$.
\[lemma\_laplacian\] $\widetilde{{\Delta}f}={\Delta}_H\tilde{f}$, $\widetilde{{\Delta}Y}={\Delta}_H\tilde{Y}$.
This is a classical fact from differential geometry with respect to a fixed metric $g_t$.
We also need the notion of the antidevelopment of a horizontal curve (this concept is also known as Cartan’s rolling without slipping), see e.g. [@KobNom], generalized to the time-dependent setting. The point is that the horizontal vector fields provide a way to identify curves in ${\mathds{R}}^n$ with horizontal curves in $\mathcal{F}$.
If $\{u_\tau\}_{\tau\in[0,T]}$ is a horizontal curve in $\mathcal{F}$ with $\pi(u_\tau)=(x_\tau,T-\tau)$, its **antidevelopment** $\{w_\tau\}_{\tau\in[0,T]}$ is the curve in ${\mathds{R}}^n$ that satisfies $$\label{antidevelop}
\frac{du_\tau}{d\tau} =D_\tau+ H_i(u_\tau)\frac{d{w}_\tau^i}{d\tau},\qquad w_0=0.$$
Brownian motion and stochastic parallel transport {#sec_brownian}
-------------------------------------------------
The goal of this section is to generalize the Eells-Elworthy-Malliavin construction of Brownian motion and stochastic parallel translation, see e.g. [@Hsu], to our setting of evolving manifolds. We note that a related construction in the time-dependent setting has been given by Arnoudon-Coulibaly-Thalmaier [@ACT].\
The idea is to solve in a stochastic setting. This provides a way to identify Brownian curves $\{w_\tau\}_{\tau\in[0,T]}$ in ${\mathds{R}}^n$ with horizontal Brownian curves $\{u_\tau\}_{\tau\in[0,T]}$ in $\mathcal{F}$. The virtue of this approach is that it yields both Brownian motion on $M$, via projecting, and stochastic parallel transport, via $P_{\tau_1,\tau_2}=u_{\tau_2}u_{\tau_1}^{-1}$.\
Let $(M,g_t)_{t\in I}$, $I=[0,T_1]$, be a one-parameter family of Riemannian manifolds, and let $\pi: {\mathcal{F}}\to M\times I$ be the time dependent $O_n$-bundle introduced in the previous section. We fix a frame $u\in{\mathcal{F}}$, write $\pi(u)=(x,T)$, and denote the projections to space and time by $\pi_1:{\mathcal{F}}\to M$ and $\pi_2:{\mathcal{F}}\to I$, respectively. It will be convenient to work with the backwards time $\tau$, defined by $t=T-\tau$. As before, we write $D_\tau=-D_t$.
Motivated by (\[antidevelop\]), we consider the following stochastic differential equation (SDE) on ${\mathcal{F}}$: $$\label{SDE}
dU_\tau=D_\tau d\tau+ H_i(U_\tau)\circ {dW}_\tau^i\, ,\qquad\qquad U_0=u.$$ Here, $W_\tau$ is Brownian motion on ${\mathds{R}}^n$, and $\circ$ indicates that the equation is in the Stratonovich sense. To keep the factor $2$ in Hamilton’s Ricci flow, $\partial_t g_t=-2{{\rm Ric}}_{g_t}$, we use the convention that $dW_\tau$ doesn’t have the standard normalization from stochastic calculus, but is scaled by a factor $\sqrt{2}$, i.e. $dW_\tau^i dW_\tau^j=2\delta_{ij}d\tau$.
\[prop\_SDE\] The SDE has a unique solution $\{U_\tau\}_{\tau\in [0,T]}$. The solution satisfies $\pi_2(U_\tau)=T-\tau$, and does not explode. Moreover, $U_\tau(\omega)$ is continuous in $\tau$ for almost every Brownian path $\omega\in C([0,T],{\mathds{R}}^n)$, and for any $C^2$-function $f:{\mathcal{F}}\to {\mathds{R}}$ we have the Ito formula $$\label{ito}
df(U_\tau)=H_if(U_\tau)dW^i_\tau+D_\tau f(U_\tau)d\tau+ H_iH_if(U_\tau)d\tau.$$
We recall that SDEs on manifolds can be reduced to SDEs on Euclidean space, see e.g. [@Hsu Sec. 1.2]. Choose an embedding ${\mathcal{F}}\subset {\mathds{R}}^N$ and suitable extensions of all functions to ${\mathds{R}}^N$. By the standard theory of SDEs on Euclidean space, there is a unique solution of the system ($a=1,\ldots, N$): $$\label{SDE_eucl}
dU^a_\tau=D^a_\tau d\tau+ H^a_i(U_\tau)\circ {dW}_\tau^i\, ,\qquad\qquad U_0=u.$$ It follows from a Gronwall type argument that the solution actually stays inside ${\mathcal{F}}$, see e.g. [@Hsu Prop. 1.2.8]. This proves existence of a solution of . Moreover, it is also easy to derive a uniqueness result for solutions of from the standard uniqueness result for SDEs on Euclidean space, see e.g. [@Hsu Thm. 1.2.9]. In particular, the solution is independent of the choices of embedding and extensions. Since Brownian motion in $\mathbb{R}^n$ is continuous in $\tau$ for almost every path, the same is true for $U_\tau$.
To prove , we first convert into a SDE in the Ito sense. Computationally this is done by dropping the $\circ$ and adding one half times the quadratic variation of $H(U_\tau)$ and $W_\tau$: $$\label{SDE_ito}
dU^a_\tau=D^a_\tau d\tau+ H^a_i(U_\tau) {dW}_\tau^i+\tfrac{1}{2}dH_{i}^a(U_\tau)dW_\tau^i\, ,\qquad\qquad U_0=u.$$ Now, using Ito calculus in Euclidean space we compute $$dH_{i}^a(U_\tau)dW_\tau^i=\partial_bH_{i}^a(U_\tau)dU_\tau^bdW_\tau^i=2\partial_bH_{i}^a(U_\tau)H_i^b(U_\tau)d\tau,$$ and $$\begin{aligned}
df(U_\tau)=&\partial_af(U_\tau)dU_\tau^a+\tfrac{1}{2}\partial_a\partial_bf(U_\tau)dU_\tau^adU_\tau^b\nonumber\\
=&\partial_af(U_\tau)D^a_\tau d\tau+\partial_af(U_\tau)H^a_i(U_\tau) {dW}_\tau^i\\
&+\left(\partial_af(U_\tau)\partial_bH_{i}^a(U_\tau)H_i^b(U_\tau)+\partial_a\partial_bf(U_\tau)H^a_i(U_\tau)H^b_i(U_\tau)\right)d\tau.\nonumber\end{aligned}$$ Observing that the term in brackets is equal to $H_iH_if(U_\tau)$, this proves .
By assumption the metrics are equivalent at all times and there exists a distance-like function, i.e. a smooth function $r:M\to \mathbb{R}$ such that, after fixing an arbitrary point and $o\in M$, $$C^{-1}(1+d_t(x,o))\leq r(x)\leq C(1+d_t(x,o)),\qquad {\lvert{\nabla}r\rvert} \leq C,\qquad {\nabla}{\nabla}r\leq C$$ for some $C<\infty$. Let $\tilde{r}:{\mathcal{F}}\to \mathbb{R}$ be the extension of $r$, that is independent of time and the fibre coordinates. Applying the Ito formula to $\tilde{r}$, we see that the solution of does not explode, i.e. that $U_\tau$ does not escape to spatial infinity. Finally, for $f=\pi_2$ the Ito formula takes the simple form $d\pi_2(U_\tau)=-d\tau$. Together with $\pi_2(U_0)=T$, this implies that $\pi_2(U_\tau)=T-\tau$.
Using Propositon \[prop\_SDE\] we can now define Brownian motion and stochastic parallel transport on our evolving family of Riemannian manifolds.
\[def\_brownian\_motion\] We call $\pi(U_\tau)=(X_\tau,T-\tau)$ **Brownian motion** based at $(x,T)$.
\[def\_stoch\_par\] The family of isometries $P_\tau=U_0U_\tau^{-1}: (T_{X_{\tau}}M,g_{T-\tau})\to (T_{x}M,g_T)$, depending on $\tau$ and the Brownian curve, is called **stochastic parallel transport**.
Brownian motion comes naturally with its path space, diffusion measure, and filtered $\sigma$-algebra.
We let $P_0{\mathds{R}}^n$ be the space of continuous curves $\{\omega_\tau\}_{\tau\in[0,T]}$ in ${\mathds{R}}^n$ with $\omega_0=0$, let $P_u{\mathcal{F}}$ be the space of continuous curves $\{u_\tau\}_{\tau\in[0,T]}$ in ${\mathcal{F}}$ with $u_0=u$ and $\pi_2(u_\tau)=T-\tau$, and let $P_{(x,T)}{\mathcal{M}}$ be the space of continuous curves $\{\gamma_\tau=(x_\tau,T-\tau)\}_{\tau\in[0,T]}$ in ${\mathcal{M}}$ with $\gamma_0=(x,T)$.
To introduce the diffusion measure, note that Proposition \[prop\_SDE\] defines a map $U:P_0{\mathds{R}}^n\to P_u{\mathcal{F}}$, $U(\omega)(\tau)=U_\tau(\omega)$. We also have a natural map $\Pi:P_u{\mathcal{F}}\to P_{(x,T)}{\mathcal{M}}$, induced by the projection $\pi:{\mathcal{F}}\to M\times I$.
Let $\Gamma_0$ be the Wiener measure on $P_0{\mathds{R}}^n$, let $\Gamma_u=U_\ast \Gamma_0$ be the probability measure on $P_u{\mathcal{F}}$ obtained by pushing forward via $U$, and let $\Gamma_{(x,T)}=(\Pi\circ U)_\ast \Gamma_0$ be the probability measure on $P_{(x,T)}{\mathcal{M}}$ obtained by pushing forward via $\Pi\circ U$.
Finally, recall the Wiener space $P_0{\mathds{R}}^n$ comes naturally equipped with a filtered family of $\sigma$-algebras $\Sigma^\tau=\Sigma^\tau({P_0{\mathds{R}}^n})$, which is generated by the evaluation maps $e_{\tau_1}:P_0{\mathds{R}}^n\to {\mathds{R}}^n$, $e_{\tau_1}(\omega)=\omega_{\tau_1}$ with $\tau_1\leq \tau$.
\[def\_filtered\] We denote by $\Sigma^\tau(P_u{\mathcal{F}})$ and $\Sigma^\tau(P_{(x,T)}{\mathcal{M}})$ (or simply by $\Sigma^\tau$ if there is no risk of confusion) the pushforward of $\Sigma^\tau({P_0{\mathds{R}}^n})$ under the maps $U$ and $\Pi \circ U$, respectively.
Conditional expectation and martingales {#sec_condexp}
---------------------------------------
If $F:P_u{\mathcal{F}}\to{\mathds{R}}$ is integrable, we write $E_{u}[F]=\int_{P_u{\mathcal{F}}} F d\Gamma_u$ for its **expectation**. More generally, if $\sigma\in[0,T]$, we write $F^\sigma=E_{u}[F|\Sigma^\sigma]$ for the **conditional expectation** given the $\sigma$-algebra $\Sigma^\sigma$ (see Definition \[def\_filtered\]). We recall that the conditional expectation $F^\sigma$ is the unique $\Sigma^\sigma$-measurable function such that $\int_\Omega F^\sigma d\Gamma_u =\int_\Omega F d\Gamma_u$ for all $\Sigma^\sigma$-measurable sets $\Omega$. Similarly, if $F$ is an integrable function on $P_{(x,T)}{\mathcal{M}}$, we also write $E_{(x,T)}[F]$ and $F^\sigma=E_{(x,T)}[F|\Sigma^\sigma]$ for its expectation and conditional expectation, respectively.
\[prop\_condexp\] If $F:P_{(x,T)}{\mathcal{M}}\to{\mathds{R}}$ is integrable and $\sigma\in[0,T]$, then for a.e. Brownian curve $\{\gamma_\tau\}_{\tau\in [0,T]}$ the conditional expectation $F^\sigma=E_{(x,T)}[F|\Sigma^\sigma]$ is given by the formula $$\label{eq_condexp}
F^\sigma(\gamma)=\int_{P_{T-\sigma}{\mathcal{M}}}F(\gamma|_{[0,\sigma]}\ast \gamma')\, d\Gamma_{\gamma_\sigma}(\gamma'),$$ where the integral is over all Brownian curves $\{\gamma'_\tau=(x'_\tau,T-\sigma-\tau)\}_{\tau\in [0,T-\sigma]}$ based at $\gamma_\sigma=(x_\sigma,T-\sigma)$ with respect to the measure $\Gamma_{\gamma_\sigma}$, and $\gamma|_{[0,\sigma]}\ast \gamma'\in P_{(x,T)}{\mathcal{M}}$ denotes the concatenation of $\gamma|_{[0,\sigma]}$ and $\gamma'$.
Using Proposition \[prop\_SDE\] we see that the martingale problem for is well posed. Thus, by the Stroock-Varadhan principle, c.f. [@StroockVaradhan Thm. 10.1.1], we have the strong Markov-property $$\label{eq_strong_markov}
E_{u}[f(U_{\sigma+\tau}^u)|\Sigma^\sigma]=E_{U_\sigma^u}[f(U^{U_\sigma^u}_\tau)]$$ for all test functions $f:\mathcal{F}\to {\mathds{R}}$ and all stopping times $\sigma\leq T$, where $\{U_\tau^{u_0}\}_{\tau\in[0,\pi_2(u_0)]}$ denotes the solution of with initial condition $u_0$. Pushing forward via $\pi:\mathcal{F}\to{\mathcal{M}}$, and choosing $\sigma$ constant, equation implies $$\label{eq_strong_markov2}
E_{(x,T)}\left[f\left(X_{\sigma+\tau}^{(x,T)}\right)\left|\right.\Sigma^\sigma\right]=E_{\left(X^{(x,T)}_{\sigma},T-\sigma\right)}\left[f\left(X^{\left({X^{(x,T)}_{\sigma}},T-\sigma\right)}_\tau\right)\right]$$ for all test functions $f:M\to {\mathds{R}}$. Note that equation is exactly equation for the case that $F$ is the $1$-point cylinder function $f\circ u_{\sigma+\tau}$.[^4] Now, if $F$ is a $k$-point cylinder function, then by conditioning at the first evaluation time we can split up the computation of its (conditional) expectation to computing an expectation of a $1$-point cylinder function and of a $(k-1)$-point cylinder function. Arguing by induction, we infer that holds for all cylinder functions. Since the cylinder functions are dense in the space of all integrable functions, c.f. Definition \[def\_filtered\], this proves the proposition.
For any $F\in L^1(P_T{\mathcal{M}},\Gamma_{(x,T)})$, the **induced martingale** $F^\tau=E_{(x,T)}[F|\Sigma^\tau]$ is defined by taking the conditional expectation with respect to the $\sigma$-algebras $\Sigma^\tau$ for every $\tau\in[0,T]$. It indeed has the martingale property $$E_{(x,T)}[F^{\tau'}|\Sigma^\tau]=F^\tau\qquad (\tau'\geq \tau).$$
The **quadratic variation** of the martingale $F^\bullet=\{F^\tau\}_{\tau\in[0,T]}$ (and more generally of any stochastic process where the following limit exists) is defined by $$\label{eq_qaudrvar}
[F^\bullet]_\tau=\lim_{{\lvert\lvert\{\tau_j\}\rvert\rvert}\to 0} \sum_k (F^{\tau_{k}}-F^{\tau_{k-1}})^2,$$ where the limit is taken in probability, over all partions $\{\tau_j\}$ of $[0,\tau]$ with mesh going to zero.
Assume now that $F\in L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$. Then the convergence in is not just in probability but also in $L^1$. Moreover, we have the Ito isometry $$\label{ito_isom}
E\left[[F^\bullet]_{\tau'}-[F^\bullet]_\tau\left|\Sigma^\tau\right.\right]=E\left[(F^{\tau'}-F^\tau)^2\left|\Sigma^\tau\right.\right].$$ The differential of $[F^\bullet]_\tau$ takes the form $d[F^\bullet]_\tau=Y_\tau\, d\tau$ for some nonnegative $\Sigma^\tau$-adapted stochastic process $Y$, which we denote by $Y_\tau=\tfrac{d[F^\bullet]_\tau}{d\tau}$. Using Fatou’s lemma and equation it can be estimated by $$\label{eq_fatou_est}
\frac{d[F^\bullet]_\tau}{d\tau}\leq\liminf_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}E\left[[F^\bullet]_{\tau+{\varepsilon}}-[F^\bullet]_\tau\left|\Sigma^\tau\right.\right]=\liminf_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}E\left[(F^{\tau+{\varepsilon}}-F^\tau)^2\left|\Sigma^\tau\right.\right],$$ for almost every $\tau$ for almost every $\gamma$.
Heat equation and Wiener measure
--------------------------------
The goal of this section is to explain the relationship between the Wiener measure and the heat equation on our evolving manifolds. In particular, we will see that the Wiener measure is indeed characterized by equation . We start with the following representation formula for solutions of the heat equation.
\[prop\_repformula\] If $s\in[0,T]$, and $w$ is a solution of the heat equation, $\partial_t w={\Delta}_{g_t}w$, with $w|_{s}=f\in C^\infty_c(M)$, then $w(x,T)=E_{(x,T)}[f(X_{T-s})]$.
By Definition \[def\_brownian\_motion\] we have $w(X_\tau,T-\tau)=\tilde{w}(U_\tau)$, where $\tilde{w}$ denotes the lift of $w$ to the frame bundle, which is constant in fibre directions. By the Ito formula (Proposition \[prop\_SDE\]) we have $$\begin{aligned}
d\tilde{w}(U_\tau)=H_i\tilde{w}(U_\tau)\, dW_\tau^i+D_\tau\tilde{w}(U_\tau)\, d\tau+{\Delta}_H\tilde{w}(U_\tau)\,d\tau,\end{aligned}$$ where ${\Delta}_H=\sum_{i=1}^n H_i^2$ is the horizontal Laplacian. Since $w$ solves the heat equation, the sum of the last two terms vanishes (see Lemma \[lemma\_firstder\] and Lemma \[lemma\_laplacian\]), and by integration we obtain $$\begin{aligned}
\label{repform_integrand}
\tilde{w}(U_{T-s})-\tilde{w}(U_0)=\int_0^{T-s} H_i \tilde{w}(U_\tau)\, dW_\tau^i.\end{aligned}$$ Note that $\tilde{w}(U_0)=\tilde{w}(u)=w(x,T)$, and that $\tilde{w}(U_{T-s})=w(X_{T-s},s)=f(X_{T-s})=f(\pi_1 U_{T-s})$. Moreover, after taking expectations the term on the right hand side of disappears by the martingale property, i.e. since the integrand is $\Sigma^\tau$-adapted (c.f. Definition \[def\_filtered\]), and since Brownian motion has zero expectation. Thus, $$w(x,T)=E_u[f(\pi_1 U_{T-s})]=E_{(x,T)}[f(X_{T-s})],$$ as claimed.
\[prop\_wiener\]
If $e_{\bf{\sigma}}:P_{(x,T)}{\mathcal{M}}\to M^{k}$ is the evaluation map at ${\bf{\sigma}}=\{0\leq \sigma_1\leq\ldots\leq \sigma_k\leq T\}$, given by $e_\sigma(\gamma)=(\pi_1\gamma_{\sigma_1},\ldots, \pi_1\gamma_{\sigma_k})$, and if we write $s_i=T-\sigma_i$, then $$\label{cor_pushwiener2}
e_{\bf{\sigma},\ast}d\Gamma_{(x,T)}(y_1,\ldots,y_k)=H(x,T|y_1,s_1)d{{\rm vol}}_{g_{s_1}}(y_1)\cdots H(y_{k-1},s_{k-1}|y_k,s_k)d{{\rm vol}}_{g_{s_k}}(y_k).$$ Moreover, equation uniquely characterizes the Wiener measure on $P_{(x,T)}{\mathcal{M}}$.
By Propositon \[prop\_repformula\] we have the equality $$\int_M H(x,T|y,s)f(y)d{{\rm vol}}_{g(s)}(y)=\int_{P_{(x,T)}{\mathcal{M}}}f(\pi_1\gamma_\sigma)d\Gamma_{(x,T)}(\gamma)$$ for every test function $f$, say smooth with compact support. Since these functions are dense in the space of all integrable functions on $M$, this proves for $k=1$.
Now, if $f:M^k\to {\mathds{R}}$ and ${\bf{\sigma}}=\{0\leq \sigma_1\leq\ldots\leq \sigma_k\leq T\}$, then using Proposition \[prop\_condexp\] and what we just proved, the conditional expectation $(e_{\sigma}^\ast f)^{\sigma_{k-1}}=E_{(x,T)} [e_{\sigma}^\ast f | \Sigma^{\sigma_{k-1}}]$ is given by $$(e_{\sigma}^\ast f)^{\sigma_{k-1}}(\gamma)=\int_M f(\pi_1\gamma_{\sigma_1},\ldots,\pi_1\gamma_{\sigma_{k-1}},y_k)H(\pi_1\gamma_{\sigma_{k-1}},s_{k-1}|y_k,s_k)d{{\rm vol}}_{g_{s_k}}(y_k).$$ Using the formula $E_{(x,T)}[e_{\sigma}^\ast f]=E_{(x,T)}[E_{(x,T)}[e_{\sigma}^\ast f | \Sigma^{\sigma_{k-1}}]]$ and induction, this proves .
Finally, by the density of cylinder functions in the space of measurable functions (c.f. Definition \[def\_filtered\]), equation uniquely characterizes the Wiener measure on $P_{(x,T)}{\mathcal{M}}$.
Feynman-Kac formula
-------------------
We will now prove a Feynman-Kac type formula for vector valued solutions of the heat equation with potential $$\label{eq_vvhwp}
{\nabla}_t Y={\Delta}_{g_t} Y+A_t Y,\qquad Y|_{s}=Z,$$ where $A_t\in \textrm{End}(TM)$ is a smooth family of endomorphisms, and $Z$ is say smooth with compact support.
The idea is to generalizes the representation formula for solutions of the heat equation (Proposition \[prop\_repformula\]) in two ways by: i) using stochastic parallel translation (Definition \[def\_stoch\_par\]) to transport everything to $T_xM$, and ii) multiplication by an endomorphism $R_{T-s}=R_{T-s}(\gamma):T_xM\to T_xM$, which is obtained by solving an ODE along every Brownian curve $\gamma$, to capture how the potential $A_t$ effects the solution.
\[prop\_feynman\_kac\] If $s\in[0,T]$, $A_t\in \textrm{End}(TM)$, and $Y$ is a vector valued solution of the heat equation with potential, ${\nabla}_t Y={\Delta}_{g_t} Y+A_t Y$, with $Y|_{s}=Z\in C^\infty_c(TM)$, then $$\label{eq_feynman_kac}
Y(x,T)=E_{(x,T)}[R_{T-s} P_{T-s}Z(X_{T-s})],$$ where $R_\tau=R_\tau(\gamma):T_xM\to T_xM$ is the solution of the ODE $\tfrac{d}{d\tau} R_{\tau}=R_\tau P_\tau A_{T-\tau} P_\tau^{-1}$ with $R_0=\textrm{id}$.
Similar formulas hold for tensor valued solutions of the heat equation with potential.
Let $\tilde{Y}:{\mathcal{F}}\to \mathbb{R}^n$, $\tilde{Y}(u)=u^{-1}Y_{\pi u}$, be the equivariant function associated to $Y$. Applying the Ito formula (Proposition \[prop\_SDE\]) to each component, we obtain $$d\tilde{Y}(U_\tau)=H_i\tilde{Y}(U_\tau)dW^i_\tau+D_\tau\tilde{Y}(U_\tau)d\tau+{\Delta}_H\tilde{Y}(U_\tau)d\tau=H_i\tilde{Y}(U_\tau)dW^i_\tau-\tilde{A}_{T-\tau}\tilde{Y}(U_\tau)d\tau,$$ where we lifted equation to ${\mathcal{F}}$ using Lemma \[lemma\_firstder\] and Lemma \[lemma\_laplacian\]. Let $\tilde{R}_\tau:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be the solution of the ODE $\tfrac{d}{d\tau} \tilde{R}_{\tau}=\tilde{R}_\tau \tilde{A}_{T-\tau}$ with $R_0=\textrm{id}$. Then $$\label{eq_proof_of_feynkac}
d\left(\tilde{R}_\tau \tilde{Y}(U_\tau)\right)=\tilde{R}_\tau H_i\tilde{Y}(U_\tau)dW^i_\tau.$$ The right hand side disappears after taking expectations, by the martingale property, as in the proof of Proposition \[prop\_repformula\]. Thus, $$\tilde{Y}(u)=E_u[\tilde{R}_{T-s}\tilde{Y}_{T-s}(U_{T-s})].$$ Finally, we can translate from $\tilde{Y}$ to $Y$ by computing $$Y(x,T)=u\tilde{Y}(u)=E_u[U_0\tilde{R}_{T-s}U_0^{-1}U_0U^{-1}_{T-s}U_{T-s}\tilde{Y}_{T-s}(U_{T-s})]=E_{(x,T)}[R_{T-s}P_{T-s}Z(X_{T-s})].$$ Here, we used that ${{ \rm R}}_\tau= U_0 \tilde{R}_\tau U_0^{-1}$, which can be checked by computing $$\tfrac{d}{d\tau} (U_0 \tilde{R}_\tau U_0^{-1})=U_0\tilde{R}_\tau \tilde{A}_{T-\tau}U_0^{-1}
=U_0\tilde{R}_\tau U_0^{-1} U_0 U_\tau^{-1} U_\tau \tilde{A}_{T-\tau}U_\tau^{-1}U_\tau U_0^{-1}
=U_0\tilde{R}_\tau U_0^{-1} P_\tau A_{T-\tau} P_\tau^{-1},\nonumber$$ which shows that ${{ \rm R}}_\tau$ and $U_0\tilde{R}_\tau U_0^{-1}$ solve the same ODE, and thus must be equal.
Parallel gradient and Malliavin gradient {#sec_gradients}
----------------------------------------
Let $F:P_{(x,T)}{\mathcal{M}}\to {\mathds{R}}$ be a cylinder function. If $\gamma\in P_{(x,T)}{\mathcal{M}}$ is a continuous curve and $V$ is a right continuous vector field along $\gamma$, then the directional derivative $D_VF(\gamma)$ is well defined as a limit of difference quotients, namely $$D_V F(\gamma)=\lim_{\varepsilon\to 0} \frac{F(\gamma^{V,{\varepsilon}})-F(\gamma)}{\varepsilon},$$ where $\gamma^{V,{\varepsilon}}=\{(x^{V,{\varepsilon}}_\tau,T-\tau)\}_{\tau\in [0,T]}$ is the curve in $P_{(x,T)}{\mathcal{M}}$ defined by $x^{V,{\varepsilon}}_\tau=\exp^{g_\tau}_{x_\tau}({\varepsilon}V_\tau)$.
\[def\_par\_grad\] Let $\sigma\in[0,T]$. If $F:P_{(x,T)}{\mathcal{M}}\to {\mathds{R}}$ is a cylinder function, then its **$\sigma$-parallel gradient** is the unique almost everywhere defined function $\nabla^{\parallel}_\sigma F:P_{(x,T)}{\mathcal{M}}\to (T_{x}M,g_T)$, such that $$D_{V^\sigma} F(\gamma)=\langle \nabla^{\parallel}_\sigma F(\gamma),v\rangle_{(T_xM,g_T)}$$ for almost every Brownian curve $\gamma$ and every $v\in (T_{x}M,g_T)$, where $V^\sigma=\{V^\sigma_\tau\}_{\tau\in [0,T]}$ is the vector field along $\gamma$ given by $V^\sigma_\tau=0$ if $\tau\in[0,\sigma)$ and $V^\sigma_\tau=P_\tau^{-1} v$ if $\tau\in[\sigma,T]$.
Explicitly, if $F=u\circ e_\sigma: P_{(x,T)}{\mathcal{M}}\to M^k\to {\mathds{R}}$, and if we write $s_j=T-\sigma_j$, then it is straightforward to check that $$\label{eqn_par_grad}
\nabla^{\parallel}_\sigma F=e_\sigma^\ast\left(\sum_{\sigma_j\geq \sigma} P_{\sigma_j}{{\rm grad}}_{g_{s_j}}^{(j)}u\right),$$ where ${{\rm grad}}^{(j)}$ denotes the gradient with respect to the $j$-th variable, and $P_{\sigma_j}$ is stochastic parallel transport.\
Let $\mathcal{H}$ be the Hilbert-space of $H^1$-curves $\{v_\tau\}_{\tau\in[0,T]}$ in $(T_xM,g_T)$ with $v_{0}=0$, equipped with the inner product $$\langle v,w \rangle_{\mathcal{H}}=\int_{0}^{T} \langle \dot{v}_\tau,\dot{w}_\tau \rangle_{(T_xM,g_T)}\, d\tau.$$
\[def\_mall\_grad\] If $F:P_{(x,T)}{\mathcal{M}}\to {\mathds{R}}$ is a cylinder function, then its **Malliavin gradient** is the unique almost everywhere defined function $\nabla^{\mathcal{H}}F:P_{(x,T)}{\mathcal{M}}\to \mathcal{H}$, such that $$D_V F(\gamma)=\langle \nabla^{\mathcal{H}}F(\gamma),v\rangle_{\mathcal{H}}$$ for every $v\in \mathcal{H}$ for almost every Brownian curve $\gamma$, where $V=\{P_\tau^{-1} v_\tau\}_{\tau\in [0,T]}$.
Let us now explain the extension to operators on $L^2$. This is based on the integration by parts formula from the appendix (Theorem \[thm\_intbyparts\]), which says that the formal adjoint of $D_V$ is given by $$\label{eq_adj1}
D_V^\ast G=-D_V G + \tfrac12 G \int_0^T\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t{g})P_\tau^{-1} v_\tau,dW_\tau\rangle.$$ By the Ito isometry and we have the estimate $$\label{eq_adj2}
E_{(x,T)}\left|\int_0^T\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t{g})P_\tau^{-1} v_\tau,dW_\tau\rangle\right|^2\leq C{\lvertv\rvert}_{\mathcal{H}}^2.$$ Using , , and the definition of the formal adjoint, we see that if $F_n$ is a sequence of cylinder functions with $F_n\to 0$ and $D_V F_n\to K$ in $L^2(P_{(x,T)}{\mathcal{M}})$, then $(K,G)=0$ for all cylinder functions $G$, and thus $K=0$. It follows that $\nabla^{\mathcal{H}}$ can be extended to a closed unbounded operator from $L^2(P_{(x,T)}{\mathcal{M}})$ to $L^2(P_{(x,T)}{\mathcal{M}},\mathcal{H})$, with the cylinder functions being a dense subset of the domain. Similarly, $\nabla^{\parallel}_\sigma$ can be extended to a closed unbounded operator from $L^2(P_{(x,T)}{\mathcal{M}})$ to $L^2(P_{(x,T)}{\mathcal{M}},T_xM)$, again with the cylinder functions being a dense subset of the domain.
Ornstein-Uhlenbeck operator
---------------------------
The **Ornstein-Uhlenbeck operator** $\mathcal{L}=\nabla^{{\mathcal{H}}\ast}\nabla^{\mathcal{H}}$ is an unbounded operator on $L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$ defined by composing the Malliavin gradient with its adjoint. More generally, there is a family of Ornstein-Uhlenbeck operators $\mathcal{L}_{\tau_1,\tau_2}$ on $L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$ defined by the formula $\mathcal{L}_{\tau_1,\tau_2}=\int_{\tau_1}^{\tau_2} \nabla^{{\parallel}\ast}_\tau\nabla^{\parallel}_\tau\, d\tau$, which captures the part of the Laplacian coming form the time range $[\tau_1,\tau_2]$. The next proposition shows in particular that $\mathcal{L}=\mathcal{L}_{0,T}$.
\[prop\_malliavin\] If $F:P_T{\mathcal{M}}\to {\mathds{R}}$ is a cylinder function, then for almost every curve $\gamma\in (P_T{\mathcal{M}},\Gamma_{(x,T)})$ we have the formula $${\lvert{\nabla}^{\mathcal{H}}F\rvert}^2(\gamma)=\int_{0}^{T}{\lvert{\nabla}^{\parallel}_\tau F\rvert}^2(\gamma)\, d\tau.$$
The cylinder function has the form $F=u\circ e_\sigma:P_{T}{\mathcal{M}}\to M^k\to{\mathds{R}}$. By the definition of the Malliavin gradient (Definition \[def\_mall\_grad\]), for almost every $\gamma\in (P_T{\mathcal{M}},\Gamma_{(x,T)})$ we have $$\sum_{j=1}^k \langle v_{\sigma_j},P_{\sigma_j}{{\rm grad}}^{(j)}_{g_{s_j}}u(e_{\sigma_j}\gamma)\rangle
=D_{V}F(\gamma)
=\langle {\nabla}^{\mathcal{H}}F(\gamma),v\rangle_{\mathcal{H}}=\int_0^T \langle \tfrac{d}{d\tau} {\nabla}^{\mathcal{H}}F(\gamma),\tfrac{d}{d\tau} v\rangle\, d\tau.$$ It follows that $$\tfrac{d}{d\tau} {\nabla}^{\mathcal{H}}F(\gamma)=\sum_{j=1}^k 1_{\{\tau\leq \sigma_j\}}P_{\sigma_j}{{\rm grad}}^{(j)}_{g_{s_j}}u(e_{\sigma_j}\gamma).$$ Based on this, writing $\sigma_0=0$, we compute $${\lvert{\nabla}^{\mathcal{H}}F\rvert}^2_{\mathcal{H}}(\gamma)=\int_0^T {\lvert\tfrac{d}{d\tau} {\nabla}^{\mathcal{H}}F(\gamma)\rvert}^2\,d\tau
=\sum_{j=1}^k(\sigma_j-\sigma_{j-1})\,\Big|\!\sum_{\ell=j}^k P_{\sigma_\ell}{{\rm grad}}^{(\ell)}_{g_{s_\ell}}u(e_{\sigma_\ell}\gamma)\Big|^2
=\int_0^T{\lvert{\nabla}^{\parallel}_\tau F\rvert}^2(\gamma)\, d\tau,$$ where we used that the integrands are piecewise constant. This proves the proposition.
Proof of the main theorem {#sec_proof_main}
=========================
In this section, we prove our main theorem (Theorem \[main\_thm\_intro\]) characterizing solutions of the Ricci flow.\
We will prove the implications (R1)$\Rightarrow$(R2)$\Rightarrow$(R3’)$\Rightarrow$(R4)$\Rightarrow$(R5)$\Rightarrow$(R3)$\Rightarrow$(R2’)$\Rightarrow$(R1). Here, (R3’) denotes the (seemingly stronger) statement that for every $F\in L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$ we have the pointwise estimate $$(R3')\qquad \sqrt{{\frac{d[F^\bullet]_\tau}{d\tau}}}(\gamma)\leq \sqrt{2}\, E_{(x,T)}\left[ |{{\nabla}^{\parallel}_\tau F}|\left| \Sigma^\tau\right.\right](\gamma)$$ for almost every $\gamma\in P_{(x,T)}{\mathcal{M}}$ for almost every $\tau\in[0,T]$, and (R2’) denotes the (seemingly weaker) statement that for every $F\in L^2(P_T{\mathcal{M}},\Gamma_T)$, we have the gradient estimate $$(R2')\qquad {\lvert\nabla_x\int_{P_T{\mathcal{M}}} F d\Gamma_{(x,T)}\rvert}^2\leq \int_{P_T{\mathcal{M}}}{\lvert\nabla^{\parallel}F\rvert}^2 d\Gamma_{(x,T)}.$$
Before delving into the proof, we observe that it suffices to prove the estimates for cylinder functions, since this implies the general case by approximation. For illustration, let us spell out the approximation argument for (R2): Let $F\in L^2(P_T{\mathcal{M}},\Gamma_T)$. Let $F_j$ be a sequence of cylinder functions that converges to $F$ in $L^2(P_T{\mathcal{M}},\Gamma_T)$ and pointwise almost everywhere. By Fubini’s theorem and the dominated convergence theorem, for a.e. $x\in M$ we obtain that $\lim_{j\to\infty} E_{(x,T)}F_j^2=E_{(x,T)}F^2<\infty$. We can assume that for a.e. $x\in M$ the function $F$ is in the domain of the parallel gradient based at $(x,T)$ (since otherwise the right hand side of (R2) is infinite by convention and the estimate holds trivially). Thus, $\lim_{j\to\infty} E_{(x,T)}{\lvert\nabla^{\parallel}F_j\rvert}=E_{(x,T)}{\lvert\nabla^{\parallel}F\rvert}<\infty$ for a.e. $x\in M$. If we know that (R3) holds for cylinder functions, then we can infer that $$\label{eq_loclip}
\limsup_{j\to\infty}\,\big|\nabla_x \!\int_{P_T{\mathcal{M}}} \!\!\!\!F_j\, d\Gamma_{(x,T)}\big|\leq \int_{P_T{\mathcal{M}}} {\lvert\nabla^{\parallel}F\rvert}\, d\Gamma_{(x,T)}$$ for a.e. $x\in M$. Once we know that the local Lipschitz-bounds holds, then passing to a subsequential limit we can conclude that (R2) holds for $F$ for a.e. $x\in M$.
The gradient estimate {#subsec_gradest}
---------------------
The goal of this section is to prove the implication (R1)$\Rightarrow$(R2). We start with the following theorem for the gradient of the expectation value.
\[thm\_gradient\_formula\] If $(M,g_t)_{t\in I}$ is an evolving family of Riemannian manifolds and $F:P_T{\mathcal{M}}\to {\mathds{R}}$ is a cylinder function, then $$\label{eq_bismut_form}
{{\rm grad}}_{g_T} E_{(x,T)}F=E_{(x,T)}\left[{\nabla}^{\parallel}F+\int_0^T \tfrac{d}{d\tau}R_\tau\, {\nabla}^{\parallel}_\tau F\,d\tau\right],$$ where $R_\tau=R_\tau(\gamma):T_xM\to T_xM$ is the solution of the ODE $\tfrac{d}{d\tau}R_\tau=-R_\tau P_\tau({{\rm Ric}}+\tfrac12 \partial_t g)P_\tau^{-1}$ with $R_0=\textrm{id}$.
Our proof of Theorem \[thm\_gradient\_formula\] is by induction on the order of the cylinder function. The main ingredients are the Feynman-Kac formula for vector valued solutions of the heat equation (Proposition \[prop\_feynman\_kac\]), the formula for the conditional expectation value (Proposition \[prop\_condexp\]), and the following evolution equation for the gradient.
\[prop\_evol\_grad\] If $(M,g_t)_{t\in I}$ is an evolving family of Riemannian manifolds, and $u$ solves the heat equation, $\partial_t u = {\Delta}_{g_t} u$, then its gradient, ${{\rm grad}}_{g_t} u$, solves the equation $$\nabla_t\, {{\rm grad}}_{g_t} u={\Delta}_{g_t}{{\rm grad}}_{g_t} u-({{\rm Ric}}+\tfrac12 \partial_tg_t)({{\rm grad}}_{g_t} u,\cdot)^{\sharp_{g_t}}.$$
Using the formula $\partial_t (g^{-1})=-g^{-1}(\partial_tg)g^{-1}$ and the definitions of ${{\rm grad}}_{g_t} (u)$ and ${\nabla}_t$, we compute $$\begin{aligned}
\nabla_t\, {{\rm grad}}_{g_t} u&={{\rm grad}}_{g_t} (\partial_t u)-\partial_tg_t({{\rm grad}}_{g_t} u, \cdot )^{\sharp_{g_t}}+\tfrac{1}{2}\partial_tg_t({{\rm grad}}_{g_t} u, \cdot )^{\sharp_{g_t}}\nonumber\\
&={\Delta}_{g_t}{{\rm grad}}_{g_t} u-({{\rm Ric}}+\tfrac12 \partial_tg_t)({{\rm grad}}_{g_t} u,\cdot)^{\sharp_{g_t}},\end{aligned}$$ where we used the equation $\partial_t u = {\Delta}_{g_t} u$ and commuted the Laplacian and the gradient.
We argue by induction on the order $k={\lvert\sigma\rvert}$ of the cylinder function $F=e_\sigma^\ast u$.
If $k=1$, then by equation the expectation $E_{(x,T)}F$ is given by integration with respect to the heat kernel, namely $$E_{(x,T)}F=\int_M u(y)H(x,T|y,s)d{{\rm vol}}_{g(s)}(y)=(P_{sT}u)(x),$$ where $s=T-\sigma$. On the other hand, by Proposition \[prop\_evol\_grad\] we have the evolution equation $$\nabla_t\, {{\rm grad}}_{g_t} P_{st}u={\Delta}_{g_t}{{\rm grad}}_{g_t} P_{st}u-({{\rm Ric}}+\tfrac12 \partial_tg_t)({{\rm grad}}_{g_t} P_{st}u),$$ where we view $({{\rm Ric}}+\tfrac12 \partial_tg_t)$ as endomorphism (using the metric $g_t$). We can thus apply the Feynman-Kac formula (Proposition \[prop\_feynman\_kac\]), and obtain $$({{\rm grad}}_{g_T}P_{sT}u)(x)=E_{(x,T)}[R_\sigma P_{\sigma}({{\rm grad}}_{g_s} u)(X_{\sigma})],$$ where $R_\tau=R_\tau(\gamma):T_xM\to T_xM$ is the solution of the ODE $\tfrac{d}{d\tau}R_\tau=-R_\tau P_\tau({{\rm Ric}}+\tfrac12 \partial_t g)P_\tau^{-1}$ with $R_0=\textrm{id}$. Using the fundamental theorem of calculus and equation , we can rewrite this as $$\begin{aligned}
({{\rm grad}}_{g_T}P_{sT}u)(x)=E_{(x,T)}\left[\left(\textrm{id}+\int_0^\sigma \tfrac{d}{d\tau} R_\tau\, d\tau\right) P_{\sigma}({{\rm grad}}_{g_s} u)(X_{\sigma})\right]=E_{(x,T)}\left[{\nabla}^{\parallel}F+\int_0^T \tfrac{d}{d\tau}R_\tau\, {\nabla}^{\parallel}_\tau F\,d\tau\right].\end{aligned}$$ Thus, the gradient formula holds true for $1$-point cylinder functions.
Now, arguing by induction, let $F=e_\sigma^\ast u$ be a $k$-point cylinder function and let $s_i=T-\sigma_i$. Note that $$\label{eq_doubleexp}
E_{(x,T)}F=E_{(x,T)}E_{(x,T)}[F|\Sigma^{\sigma_1}],$$ Using Proposition \[prop\_condexp\] we see that $G:=E_{(x,T)}[F|\Sigma^{\sigma_1}]$ is a $1$-point cylinder function given by $G=e_{\sigma_1}^\ast w$, $$w(y)=E_{(y,s_1)}[u(y,X'_{\sigma_2-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1})],$$ where the expectation is over all Brownian curves starting at $(y,T-\sigma_1)$. Note that by equation and the case $k=1$ of the gradient formula we have $${{\rm grad}}_{g_T} E_{(x,T)}F={{\rm grad}}_{g_T} E_{(x,T)}G=E_{(x,T)}R_{\sigma_1} P_{\sigma_1} ({{\rm grad}}_{g_{s_1}} w)(X_{\sigma_1}),$$ where $R_\tau=R_\tau(\gamma):T_xM\to T_xM$ is the solution of the ODE $\tfrac{d}{d\tau}R_\tau=-R_\tau P_\tau({{\rm Ric}}+\tfrac12 \partial_t g)P_\tau^{-1}$ with $R_0=\textrm{id}$. Using the product rule and induction, we compute $$\begin{aligned}
({{\rm grad}}_{g_{s_1}} w)(y)=&E_{(y,s_1)}{{\rm grad}}_{g_s^1}^{(1)}u(y,X'_{\sigma_2-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1})\nonumber\\
&+E_{(y,s_1)}\left[{\nabla}'^{\parallel}u(y,X'_{\sigma_2-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1})+\int_0^{T-\sigma_1}\!\!\!\!\!\!\!\!\! \tfrac{d}{d\tau}R'_{\tau}\,{\nabla}'^{\parallel}_\tau u(y,X'_{\sigma_2-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1})\,d\tau\right]\end{aligned}$$ where $X'$ and ${\nabla}'^{\parallel}$ denotes Brownian motion and the parallel gradient based at $(y,T-\sigma_1)$, and $R'_\tau=R'_\tau(\gamma'):T_yM\to T_yM$ is the solution of the ODE $\tfrac{d}{d\tau}R'_\tau=-R'_\tau P'_\tau({{\rm Ric}}+\tfrac12 \partial_t g)P_\tau'^{-1}$ with $R'_0=\textrm{id}$. Note that $$\begin{gathered}
E_{(y,s_1)}{{\rm grad}}_{g_s^1}^{(1)}u(y,X'_{\sigma_2-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1})
+E_{(y,s_1)}{\nabla}'^{\parallel}u(y,X'_{\sigma_2-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1})\\
=\sum_{j=1}^k E_{(y,s_1)} P'_{\sigma_j-\sigma_1}({{\rm grad}}_{g_{s_j}}^j u)(X'_{\sigma_1-\sigma_1},\ldots,X'_{\sigma_k-\sigma_1}).\end{gathered}$$ Moreover, if $\gamma=\gamma|_{[0,\sigma_1]}\ast \gamma'$ then $P_{\tau}(\gamma|_{[0,\sigma_1]}\ast \gamma')=P_{\sigma_1}(\gamma)\circ P'_{\tau-\sigma_1}(\gamma')$ and thus $$P_{\sigma_1}R'_{\tau-\sigma_1}P_{\sigma_1}^{-1}=R^{-1}_{\sigma_1}R_\tau$$ for $\tau\geq \sigma_1$, since both sides solve the same ODE with the same initial condition at time $\sigma_1$. Putting everything together, we conclude that $$\begin{aligned}
{{\rm grad}}_{g_T} E_{(x,T)}F=E_{(x,T)}\left[R_{\sigma_1}{\nabla}^{\parallel}F+\int_{\sigma_1}^T\tfrac{d}{d\tau}R_\tau\, \nabla_\tau^{\parallel}F\, d\tau\right]
=E_{(x,T)}\left[{\nabla}^{\parallel}F+\int_{0}^T\tfrac{d}{d\tau}R_\tau\, \nabla_\tau^{\parallel}F\, d\tau\right],\end{aligned}$$ where we also used Proposition \[prop\_condexp\], the formula $P_{\sigma_j}(\gamma|_{[0,\sigma_1]}\ast \gamma')=P_{\sigma_1}(\gamma)\circ P'_{\sigma_j-\sigma_1}(\gamma')$, and .
The gradient formula (Theorem \[thm\_gradient\_formula\]), together with the above approximation argument, immediately establishes the implication (R1) $\Rightarrow$ (R2). To see this, just observe that for families of Riemannian manifolds evolving by Ricci flow the time integral in vanishes, that ${\lvert\nabla_x \int_{P_T\mathcal{M}} F\,d\Gamma_x\rvert}$ and ${\lvert{{\rm grad}}_{g_T} E_{(x,T)}F\rvert}$ are the same (just in different notation), and that ${\lvertE_{(x,T)}{\nabla}^{\parallel}F\rvert}\leq \int_{P_T\mathcal{M}}{\lvert{\nabla}^{\parallel}F\rvert}\,d\Gamma_{(x,T)}$.
Regularity of martingales
-------------------------
The goal of this section is to establish the implication (R2)$\Rightarrow$(R3’). For convenience of the reader, we also prove the (obvious and logically not needed) implication (R3’)$\Rightarrow$(R3). We start with the following formula for the quadratic variation of a martingale on path space.
\[thm\_quadrvar\] If $(M,g_t)_{t\in I}$ is an evolving family of Riemannian manifolds and $F:P_{(x,T)}{\mathcal{M}}\to {\mathds{R}}$ is a cylinder function, then $$\frac{d[F^\bullet]_\tau}{d\tau}(\gamma)=2{\lvert{\nabla}_y E_{(y,T-\tau)}F_{\gamma[0,\tau]}\rvert}^2(\pi_1\gamma_\tau)$$ for almost every $\gamma\in P_{(x,T)}{\mathcal{M}}$, where $F_{\gamma[0,\tau]}:P_{T-\tau}{\mathcal{M}}\to{\mathds{R}}$ is defined by $F_{\gamma[0,\tau]}(\gamma')=F(\gamma|_{[0,\tau]}\ast \gamma')$.
Given a cylinder function $F=u\circ e_\sigma:P_{(x,T)}{\mathcal{M}}\to M^k\to {\mathds{R}}$, and a number $\tau\in[0,T]$, let $j$ be the largest integer such that $\sigma_j\leq \tau$. By the formula for the conditional expectation (Proposition \[prop\_condexp\]) and the characterization of the Wiener measure (Propositon \[prop\_wiener\]), for ${\varepsilon}>0$ small enough, $F^{\tau+{\varepsilon}}$ is given by $$F^{\tau+{\varepsilon}}(\gamma)=\int_{M^{k-j}}u(\pi_1\gamma_{\sigma_1},\ldots,\pi_1\gamma_{\sigma_j},y_{j+1},\ldots,y_k) d\nu_{\gamma_{\tau+{\varepsilon}}}(y_{j+1},s_{j+1})\ldots d\nu_{(y_{k-1},s_{k-1})}(y_k,s_k).$$ We can write this as $F^{\tau+{\varepsilon}}=e_{\tau+{\varepsilon}}^\ast w_{\varepsilon}$, where we define $w_{\varepsilon}=w_{{\varepsilon},\gamma_{\sigma_1},\ldots,\gamma_{\sigma_j}}$ by $$w_{\varepsilon}(z)=\int_{M^{k-j}}u(\pi_1\gamma_{\sigma_1},\ldots,\pi_1\gamma_{\sigma_j},y_{j+1},\ldots,y_k) d\nu_{(z,T-\tau-{\varepsilon})}(y_{j+1},s_{j+1})\ldots d\nu_{(y_{k-1},s_{k-1})}(y_k,s_k).$$ Now, since the function $\frac{d[F^\bullet]_\tau}{d\tau}$ is $\Sigma^\tau$-measurable, we can compute $$\frac{d[F^\bullet]_\tau}{d\tau}(\gamma)=E_{(x,T)}\left[\frac{d[F^\bullet]_\tau}{d\tau}\left|\,\Sigma^\tau\right.\right]=\lim_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}E_{(x,T)}\left[(F^{\tau+{\varepsilon}}-(F^{\tau+{\varepsilon}})^\tau)^2\left|\,\Sigma^\tau\right.\right],$$ where we also used the martingale property $(F^{\tau+\varepsilon})^{\tau}=F^\tau$ and the definition of the quadratic variation, c.f. Section \[sec\_condexp\]. Using again Proposition \[prop\_condexp\] and Propositon \[prop\_wiener\], as well as some rough short time asymptotics for the heat kernel, we conclude that $$\frac{d[F^\bullet]_\tau}{d\tau}(\gamma) =\lim_{{\varepsilon}\to 0^+}\frac{1}{{\varepsilon}}\int_{M}\left(w_{\varepsilon}(z)-\int_M w_{\varepsilon}(\hat{z})\, d\nu_{\gamma_\tau}(\hat{z}, T-\tau-{\varepsilon}) \right)^2 d\nu_{\gamma_\tau}(z,T-\tau-{\varepsilon})
=2{\lvert\nabla w_0\rvert}^2\!(\pi_1\gamma_\tau).$$ Observing that $w_0(y)=E_{(y,T-\tau)}F_{\gamma[0,\tau]}$, this proves the theorem.
Let $(M,g_t)_{t\in I}$ be a smooth family of Riemannian manifolds such that the gradient estimate (R2) holds, and let $F:P_{(x,T)}{\mathcal{M}}\to {\mathds{R}}$ be a cylinder function. Observe that $$\label{eq_observe_that}
|{{\nabla}^{\parallel}_\tau F}|(\gamma|_{[0,\tau]}\ast \gamma')=|{\nabla}_0^{\parallel}F_{\gamma[0,\tau]}|(\gamma').$$ Now, using Theorem \[thm\_quadrvar\], the gradient estimate (R2), and , we compute (for a.e. $\gamma$ for a.e. $\tau$) $$\begin{gathered}
\label{eq_proofmarting}
\sqrt{\frac{d[F^\bullet]_\tau}{d\tau}}(\gamma)=E_{(x,T)}\left[\sqrt{\frac{d[F^\bullet]_\tau}{d\tau}}\left|\Sigma^\tau\right.\right]=\sqrt{2}E_{(x,T)}\left[{\lvert{\nabla}_y E_{(y,T-\tau)}F_{\gamma[0,\tau]}\rvert}(\pi_1\gamma_\tau)\left|\Sigma^\tau\right.\right]\\
\leq \sqrt{2}E_{(x,T)}\left[E_{\gamma_\tau}|{\nabla}_0^{\parallel}F_{\gamma[0,\tau]}|\left|\Sigma^\tau\right.\right]
=\sqrt{2}E_{(x,T)}\left[|{{\nabla}^{\parallel}_\tau F}|\left|\Sigma^\tau\right.\right],\end{gathered}$$ where we also used Proposition \[prop\_condexp\] in the last step. This proves (R3’).
Let $F\in L^2(P_T{\mathcal{M}},\Gamma_{(x,T)})$. Using the assumption (R3’), the Cauchy-Schwarz inequality, and the definition of the conditional expectation, we compute $$\label{eq_r3pr3}
E_{(x,T)}\frac{d[F^\bullet]_\tau}{d\tau}
\leq 2 E_{(x,T)}\left(E_{(x,T)}\left[|{{\nabla}^{\parallel}_\tau F}|\left|\Sigma^\tau\right.\right]\right)^2\leq 2 E_{(x,T)}|{{\nabla}^{\parallel}_\tau F}|^2.$$ This proves the martingale estimate (R3).
Log-Sobolev inequality and spectral gap
---------------------------------------
In this section, we prove the implications (R3’)$\Rightarrow$(R4)$\Rightarrow$(R5).
Let $F:P_T{\mathcal{M}}\to {\mathds{R}}$ be a cylinder function, and let $\{G^\tau\}_{\tau\in [0,T]}$ be the martingale induced by the function $G=F^2$, i.e. $G^\tau=E_{(x,T)}[F^2|\Sigma^\tau]$. Using the Ito formula and the martingale property we compute $$\label{logsob1}
E_{(x,T)}[G^{\tau_2}\log G^{\tau_2}-G^{\tau_1}\log G^{\tau_1}]=E_{(x,T)}\int_{\tau_1}^{\tau_2} d (G^\tau\log G^\tau)=E_{(x,T)}\int_{\tau_1}^{\tau_2} \frac{1}{2G^\tau}\frac{d[G^\bullet]_\tau}{d\tau}d\tau.$$ By assumption (R3’), the Cauchy-Schwarz inequality, and the definition of $G^\tau$, we have the estimate $$\label{logsob2}
\frac{d[G^\bullet]_\tau}{d\tau}\leq 2 \left(E_{(x,T)}\left[|{2F {\nabla}^{\parallel}_\tau F}|\left|\Sigma^\tau\right.\right]\right)^2\leq 8 G^\tau\, E_{(x,T)}\left[|{{\nabla}^{\parallel}_\tau F}|^2\left|\Sigma^\tau\right.\right].$$ Combining and we conclude that $$E_{(x,T)}[G^{\tau_2}\log G^{\tau_2}-G^{\tau_1}\log G^{\tau_1}]\leq 4 E_{(x,T)}\int_{\tau_1}^{\tau_2} E_{(x,T)}\left[|{{\nabla}^{\parallel}_\tau F}|^2\left|\Sigma^\tau\right.\right]\, d\tau=4E_{(x,T)}\langle F,\mathcal{L}_{\tau_1,\tau_2}F\rangle,$$ where we used Propositon \[prop\_malliavin\] in the last step. This proves the log-Sobolev inequality (R4).
Applying the log-Sobolev inequality for $F^2=1+\varepsilon G$ and using approximation, we obtain $$E_{(x,T)}[(G^{\tau_2})^2-(G^{\tau_1})^2]\leq 2E_{(x,T)}\langle G,\mathcal{L}_{\tau_1,\tau_2}G\rangle.$$ Observing that $E_{(x,T)}[(G^{\tau_2})^2-(G^{\tau_1})^2]=E_{(x,T)}[(G^{\tau_2}-G^{\tau_1})^2]$, this proves the spectral gap.
Conclusion of the argument
--------------------------
The goal of this final section is to prove the remaining implications (R5)$\Rightarrow$(R3)$\Rightarrow$(R2’)$\Rightarrow$(R1).
Using the formula for the Malliavin gradient (Proposition \[prop\_malliavin\]) we can rewrite the spectral gap estimate (R5) in the form $$E_{(x,T)}(F^{\tau_2}-F^{\tau_1})^2\leq 2 E_{(x,T)}\int_{\tau_1}^{\tau_2}{\lvert\nabla_{\tau}^{\parallel}F\rvert}^2 d\tau.$$ Dividing both sides by $\tau_2-\tau_1$ and limiting $\tau_2\to\tau_1$ we obtain $$E_{(x,T)} \frac{d[F^\bullet]_\tau}{d\tau} \leq 2 E_{(x,T)}{\lvert\nabla_{\tau}^{\parallel}F\rvert}^2,$$ which is exactly the martingale estimate (R3).
The quadratic variation formula (Theorem \[thm\_quadrvar\]) at $\tau=0$ reads $${\lvert\nabla_x E_{(x,T)}F\rvert}^2=\tfrac12 E_{(x,T)} \frac{d[F^\bullet]_\tau}{d\tau}|_{\tau=0}.$$ Together with the martingale estimate (R3) at $\tau=0$ this implies $${\lvert\nabla_x E_{(x,T)}F\rvert}^2 \leq E_{(x,T)}{\lvert\nabla^{\parallel}F\rvert}^2,$$ which is exactly the gradient estimate (R2’).
Let $(M,g_t)_{t\in I}$ be an evolving family of Riemannian manifolds satisfying the gradient estimate (R2’). Plugging in a $1$-point cylinder function $F=u\circ e_\sigma:P_T{\mathcal{M}}\to M\to {\mathds{R}}$, the estimate (R2’) reduces to the estimate $${\lvert{\nabla}P_{sT} u\rvert}^2\leq P_{sT}{\lvert{\nabla}u\rvert}^2,$$ c.f. Remark \[rem\_gradest\]. Thus, by Theorem \[thm\_supersol\] (only the implication (S3)$\Rightarrow$(S1) is needed), $(M,g_t)_{t\in I}$ is a supersolution of the Ricci flow. To show that $(M,g_t)_{t\in I}$ is also a subsolution, we will analyze the gradient estimate (R2’) for a carefully chosen family of 2-point cylinder functions. Namely, given a point $(x,T)\in\mathcal{M}$ in space-time ($T>0$) and a unit tangent vector $v\in(T_xM,g_T)$ we choose a test function $u:M\times M\to{\mathds{R}}$ such that $$\textrm{grad}_{g_T}^{(1)}u=2v, \qquad \textrm{grad}_{g_T}^{(2)} u=- v,\qquad {{\rm Hess}}_{g_T} u=0\qquad\qquad \textrm{at} \,\, (x,x).$$ We consider the 1-parameter family of test functions $$F^\sigma(\gamma)=u(e_0(\gamma),e_\sigma(\gamma)),$$ where $\sigma\in[0,T]$. We will now analyze the asymptotics for $\sigma\to 0$. We start with the rough estimate $$\label{eq_roughassym}
E_{(x,T)}{\lvert\nabla^{\parallel}F^\sigma -v\rvert} = O(\sigma).$$ Together with the gradient formula (Theorem \[thm\_gradient\_formula\]) this implies that $$\label{eq_limitisone}
\lim_{\sigma\to 0}{\lvert\textrm{grad}_{g_T} E_{(x,T)}F^\sigma\rvert}^2=1=\lim_{\sigma\to 0} E_{(x,T)}{\lvert\nabla^{\parallel}F^\sigma\rvert}^2.$$ To compute the next order term, we first note that the gradient formula (Theorem \[thm\_gradient\_formula\]) yields the estimate $$\label{eq_asym1}
{{\rm grad}}_{g_T} E_{(x,T)}F^\sigma=E_{(x,T)}[{\nabla}^{\parallel}F^\sigma]+\sigma({{\rm Ric}}+\tfrac12\partial_t g)_{(x,T)}(v)+o(\sigma).$$ Using this, we compute $$\begin{aligned}
\label{eq_asym2}
\tfrac{1}{2} \tfrac{d}{d\sigma}|_{\sigma=0}\left(\left|{{\rm grad}}_{g_T} E_{(x,T)}F^\sigma\right|^2-E_{(x,T)}{\lvert{\nabla}^{\parallel}F^\sigma\rvert}^2\right)
&=\left\langle v, \tfrac{d}{d\sigma}|_{\sigma=0}\left({{\rm grad}}_{g_T} E_{(x,T)}F^\sigma-E_{(x,T)}[{\nabla}^{\parallel}F^\sigma]\right)\right\rangle\nonumber\\
&=({{\rm Ric}}+\tfrac12\partial_t g)_{(x,T)}(v,v).\end{aligned}$$ Together with , since the gradient estimate (R2’) holds by assumption, we conclude that $$({{\rm Ric}}+\tfrac12\partial_t g)_{(x,T)}(v,v)\leq 0.$$ Since $(x,T)$ and $v$ are arbitrary, this proves that $(M,g_t)_{t\in I}$ is a subsolution of the Ricci flow. Recalling that we already know that $(M,g_t)_{t\in I}$ is a supersolution of the Ricci flow, this finishes the proof.
A variant of Driver’s integration by parts formula
==================================================
The purpose of this appendix is to prove Theorem \[thm\_intbyparts\], a variant of Driver’s integration by parts formula [@Driver_ibp]. We write $(F,G)=E_{(x,T)}FG$. Moreover, if $v_\tau\in T_xM$ we use the notation $\langle v_\tau,dW_\tau\rangle=(U_0^{-1}v_\tau)_i\, dW_\tau^i$.
\[thm\_intbyparts\] Let $F,G:P_T{\mathcal{M}}\to{\mathds{R}}$ be cylinder functions, let $\{v_\tau\}_{\tau\in [0,T]}\in{\mathcal{H}}$, and write $V=\{P_\tau^{-1}v_\tau\}_{\tau\in[0,T]}$. Then $$D_V^\ast G=-D_V G + \tfrac12 G\!\! \int_0^T\!\!\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t{g})P_\tau^{-1} v_\tau,dW_\tau\rangle$$ satisfies $(D_V F,G)=(F,D_V^\ast G)$.
We adapt the proof from [@Hsu Sec. 8] to our setting of evolving manifolds.
Since $D_V$ satisfies the product rule it is enough to show that $$\label{eq_to_prove}
E_{(x,T)}[D_V F]=\tfrac12 E_{(x,T)}[F\!\! \int_0^T\!\!\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t g)P_\tau^{-1} v_\tau,dW_\tau\rangle]$$ for all cylinder functions $F$. We prove this by induction on the order $k$ of the cylinder function $F$.\
Let $F=e_\sigma^\ast u$ be a 1-point cylinder function, and let $s=T-\sigma$. Since $w(x,t):=P_{st}u(x)$ satisfies the heat equation, its gradient satisfies the equation $$\nabla_t\, {{\rm grad}}_{g_t}\!w={\Delta}_{g_t}\,{{\rm grad}}_{g_t}\!w-({{\rm Ric}}+\tfrac12 \partial_t g)({{\rm grad}}_{g_t}\!w,\cdot)^{\sharp_{g_t}},$$ c.f. the proof of Proposition \[prop\_evol\_grad\]. By the Feynman-Kac formula (Proposition \[prop\_feynman\_kac\]) we have $$\label{app_gradient_formula}
{{\rm grad}}_{g_T}\!w\,(x,T)=E_{(x,T)}[R_\sigma P_\sigma {{\rm grad}}_{g_s}\!u\, (X_\sigma)],$$ where $R_\tau=R_\tau(\gamma):(T_xM,g_T)\to (T_xM,g_T)$ solves the ODE $\tfrac{d}{d\tau} R_\tau=R_{\tau}P_\tau ({{\rm Ric}}+\tfrac12 \partial_t{g})_{T-\tau}P_\tau^{-1}$ with $R_0=\textrm{id}$, and where we view $({{\rm Ric}}+\tfrac12 \partial_tg)_{T-\tau}$ as endomorphism of $TM$ (using the metric $g_{T-\tau}$).
By equation we have $$\begin{aligned}
\label{repform_integrand_app}
u(X_{\sigma})=w(x,T)+\int_0^{\sigma} \nabla^H \tilde{w}\,(U_\tau)\cdot dW_\tau,\end{aligned}$$ where $\tilde{w}$ is the invariant lift of $w$ and $\nabla^H \tilde{w}=(H_1\tilde{w},\ldots,H_n\tilde{w})$ is its horizontal gradient.
Let $\{z_\tau\}_{\tau\in [0,T]}\in {\mathcal{H}}$. Using the above and the Ito isometry, we compute the following expectation value: $$\begin{aligned}
E_{(x,T)}\,\, u(X_\sigma)\!\!\int_0^\sigma\! \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle
&=E_{(x,T)}\int_0^{\sigma}\! \nabla^H \tilde{w}\,(U_\tau)\cdot dW_\tau \int_0^\sigma \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle\\
&=2E_{(x,T)}\int_0^{\sigma}\! \langle R_\tau^\dag \dot{z}_\tau, \nabla^H\tilde{w}(U_\tau)\rangle \, d\tau\\
&=2E_{(x,T)}\int_0^{\sigma}\! \langle \dot{z}_\tau, R_\tau U_0\nabla^H\tilde{w}(U_\tau)\rangle_{g_T} \, d\tau.\end{aligned}$$ Let $N_\tau :=R_\tau U_0\nabla^H\tilde{w}(U_\tau)=R_\tau P_\tau\,{{\rm grad}}_{g_{T-\tau}}\!\!\!\! w(X_\tau,T-\tau)$. Integration by parts gives $$E_{(x,T)}\int_0^{\sigma}\! \langle \dot{z}_\tau, N_\tau\rangle_{g_T} \, d\tau=E_{(x,T)}[\langle z_\sigma, N_\sigma\rangle_{g_T}-\int_0^{\sigma}\! \langle {z}_\tau, dN_\tau\rangle_{g_T}]= E_{(x,T)}\langle z_\sigma, N_\sigma\rangle_{g_T},$$ where in the last step we used that $N_\tau$ is a martingale, c.f. equation . Putting things together, and taking also into account that $$E_{(x,T)}\,\, u(X_\sigma)\!\!\int_\sigma^T\! \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle
=E_{(x,T)} u(X_\sigma)\,\,E_{(x,T)}\!\!\int_\sigma^T\! \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle=0,$$ we obtain $$E_{(x,T)}\,\, u(X_\sigma)\!\!\int_0^T\! \langle R_\tau^\dag \dot{z}_\tau,dW_\tau\rangle
=2E_{(x,T)}\langle R_\sigma^\dag z_\sigma, P_\sigma{{\rm grad}}_{g_s}\! u(X_\sigma) \rangle_{g_T}.$$ Finally, we let $v_\tau=R_\tau^\dag z_\tau$. Then $$R_\tau^\dag \dot{z}_\tau=\dot{v}_\tau-P_\tau({{\rm Ric}}+\tfrac{1}{2}\partial_t g)P_\tau^{-1}v_{\tau},$$ and equation follows.\
Let $F = e_\sigma^\ast f$ be a k-point cylinder function and let $s_i = T -\sigma_i$. Define a new function of $k-1$ variables by $$g(x_1,\ldots, x_{k-1})=E_{(x_{k-1},s_{k-1})}f(x_1,\ldots,x_{k-1},X'_{\sigma_k-\sigma_{k-1}}),$$ where $X'$ is based at $x_{k-1}$. Let $G:P_{(x,T)}{\mathcal{M}}\to{\mathds{R}}$ be the $(k-1)$-point cylinder function $$G(\gamma)=g(e_{\sigma_1}\gamma,\ldots,e_{\sigma_{k-1}}\gamma).$$ In belows computation we will frequently use the Markov property (Proposition \[prop\_condexp\]).
The first step is to express $$E_{(x,T)}D_VF=\sum_{j=1}^k E_{(x,T)}\langle v_{\sigma_j},P_{\sigma_j}{{\rm grad}}_{g_{s_j}}^{(j)} f(X_{\sigma_1},\ldots,X_{\sigma_k})\rangle_{g_T}$$ in terms of $G$. To this end, note that for $j=1,\ldots, k-2$ we simply have $${{\rm grad}}^{(j)}_{g_{s_j}}g(x_1,\ldots,x_{k-1})=E_{(x_{k-1},s_{k-1})}{{\rm grad}}^{(j)}_{g_{s_j}}f(x_1,\ldots,x_{k-1},X'_{\sigma_k-\sigma_{k-1}}).$$ For $j=k-1$ using the product rule and the gradient formula we have $$\begin{gathered}
{{\rm grad}}^{(k-1)}_{g_{s_{k-1}}}g(x_1,\ldots,x_{k-1})=
E_{(x_{k-1},s_{k-1})}{{\rm grad}}^{(k-1)}_{g_{s_{k-1}}}f(x_1,\ldots,x_{k-1},X'_{\sigma_k-\sigma_{k-1}})\\
+E_{(x_{k-1},s_{k-1})} R'_{\sigma_k-\sigma_{k-1}} P'_{\sigma_k-\sigma_{k-1}} {{\rm grad}}_{g_{s_k}}\!f\, (x_1,\ldots,x_{k-1},X'_{\sigma_k-\sigma_{k-1}}),\end{gathered}$$ where $R'_\tau=R'_\tau(\gamma)\!:\!(T_{x_{k-1}}M,g_{s_{k-1}})\to (T_{x_{k-1}}M,g_{s_{k-1}})$ solves the ODE $\tfrac{d}{d\tau} R'_\tau=R'_{\tau}P'_\tau ({{\rm Ric}}+\tfrac12 \partial_t{g})_{s_{k-1}-\tau}P'^{-1}_\tau$ with $R_0=\textrm{id}$. Taking expectations, we thus obtain $$\begin{gathered}
E_{(x,T)}D_VF=E_{(x,T)}D_VG+E_{(x,T)}\langle v_{\sigma_k},P_{\sigma_k}{{\rm grad}}_{g_{s_k}}^{(k)} f(X_{\sigma_1},\ldots,X_{\sigma_k})\rangle_{g_T}\\
-E_{(x,T)}E_{(X_{\sigma_{k-1}},s_{k-1})}\langle v_{\sigma_{k-1}},P_{\sigma_{k-1}} R'_{\sigma_k-\sigma_{k-1}} P'_{\sigma_k-\sigma_{k-1}} {{\rm grad}}_{g_{s_k}}^{(k)}\!f\, (X_{\sigma_1},\ldots,X_{\sigma_{k-1}},X'_{\sigma_k-\sigma_{k-1}})\rangle_{g_T}.\end{gathered}$$ By the induction hypothesis we have $$\label{app_term1}
E_{(x,T)}D_VG=\tfrac12 E_{(x,T)}[G \int_0^{\sigma_{k-1}}\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t g)P_\tau^{-1} v_\tau,dW_\tau\rangle].$$ Conditioning, using the induction hypothesis for $1$-point functions, and unconditioning again, we compute $$\begin{aligned}
\label{app_term2}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!E_{(x,T)}\langle v_{\sigma_k}-v_{\sigma_{k-1}},P_{\sigma_k}{{\rm grad}}_{g_{s_k}}^{(k)} f(X_{\sigma_1},\ldots,X_{\sigma_k})\rangle_{g_T}\nonumber\\
&=E_{(x,T)}E_{(X_{\sigma_{k-1}},s_{k-1})}\langle P_{\sigma_{k-1}}^{-1}( v_{\sigma_k}-v_{\sigma_{k-1}}),P'_{\sigma_k-\sigma_{k-1}}{{\rm grad}}_{g_{s_k}}^{(k)} f(X_{\sigma_1},\ldots,X_{\sigma_{k-1}},X'_{\sigma_k})\rangle_{g_{s_{k-1}}}\nonumber\\
&=\tfrac12 E_{(x,T)}[F \int_{s_{k-1}}^T\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t g)P_\tau^{-1} (v_\tau-v_{\sigma_{k-1}}),dW_\tau\rangle].\end{aligned}$$ Finally, using the induction hypothesis for $1$-point functions and the ODE for $R'$ we compute $$\begin{aligned}
\label{app_term3}
&\!\!\!\!\!\!E_{(x,T)}E_{(X_{\sigma_{k-1}},s_{k-1})}\langle v_{\sigma_{k-1}},(P_{\sigma_k}-P_{\sigma_{k-1}} R'_{\sigma_k-\sigma_{k-1}} P'_{\sigma_k-\sigma_{k-1}}) {{\rm grad}}_{g_{s_k}}^{(k)}\!f\, (X_{\sigma_1},\ldots,X_{\sigma_{k-1}},X'_{\sigma_k-\sigma_{k-1}})\rangle_{g_T}\nonumber\\
&=E_{(x,T)}E_{(X_{\sigma_{k-1}},s_{k-1})}\langle (I-R_{\sigma_{k}-\sigma_{k-1}}'^\dag) P_{\sigma_{k-1}}^{-1} v_{\sigma_{k-1}}, P'_{\sigma_k-\sigma_{k-1}} {{\rm grad}}_{g_{s_k}}^{(k)}\!f\, (X_{\sigma_1},\ldots,X_{\sigma_{k-1}},X'_{\sigma_k-\sigma_{k-1}})\rangle_{g_{s_{k-1}}}\nonumber\\
&= E_{(x,T)}[F \int_{s_{k-1}}^{s_k}\langle P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t g)P_\tau^{-1} v_{\sigma_{k-1}},dW_\tau\rangle].\end{aligned}$$ Adding , and we conclude that $$E_{(x,T)}D_VF=\tfrac12 E_{(x,T)}[F\!\! \int_0^T\!\!\langle \tfrac{d}{d\tau} v_\tau-P_{\tau}({{\rm Ric}}+\tfrac{1}{2}\partial_t g)P_\tau^{-1} v_\tau,dW_\tau\rangle].$$ This proves the theorem.
\
[Aaron Naber, Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA]{}\
*E-mail:* robert.haslhofer@cims.nyu.edu, anaber@math.northwestern.edu
[^1]: R.H. has been supported by NSF grant DMS-1406394, A.N. has been supported by NSF grant DMS-1406259.
[^2]: For $\tau\neq 0$, one gets the estimate $P_{tT}{\lvert\nabla P_{st}u\rvert}^2\leq P_{sT}{\lvert{\nabla}u\rvert}^2$, which is easily seen to be equivalent to (S3).
[^3]: It is of course well known that a log-Sobolev inequality implies a spectral gap. However, the important point we prove is that the spectral gap is in fact strong enough to characterize solutions of the Ricci flow.
[^4]: If $F=f\circ u_{\sigma'}$ is a 1-point cylinder function with $\sigma'\leq\sigma$, then holds true trivially.
|
---
abstract: 'In continual learning settings, deep neural networks are prone to catastrophic forgetting. Orthogonal Gradient Descent [@farajtabar2019orthogonal] achieves state-of-the-art results in practice for continual learning, although no theoretical guarantees have been proven yet. We derive the first generalisation guarantees for the algorithm OGD for continual learning, for overparameterized neural networks. We find that OGD is only provably robust to catastrophic forgetting across a single task. We propose OGD+, prove that it is robust to catastrophic forgetting across an arbitrary number of tasks, and that it verifies tighter generalisation bounds. Our experiments show that OGD+ achieves state-of-the-art results on settings with a large number of tasks, even though the models are not overparameterized. Also, we derive a closed form expression of the learned models through tasks, as a recursive kernel regression relation, which captures the transferability of knowledge through tasks. Finally, we quantify theoretically the impact of task ordering on the generalisation error, which highlights the importance of the curriculum for lifelong learning.'
bibliography:
- 'icml.bib'
nocite: '[@*]'
---
Introduction
============
Continual learning is a setting in which an agent is exposed to multiples tasks sequentially [@Kirkpatrick2016EWC]. The core challenge lies in the ability of the agent to learn the new tasks while retaining the knowledge acquired from previous tasks. Too much plasticity will lead to catastrophic forgetting, which means the degradation of the ability of the agent to perform the past tasks (@MCCLOSKEY1989109, @Ratcliff1990ConnectionistMO, @Goodfellow2013Forgetting). On the other hand, too much stability will hinder the agent from adapting to new tasks. Recent works on the Neural Tangent Kernel [@Jacot2018NTK] and on the convergence of Stochastic Gradient Descent for overparameterized neural networks [@Arora:2019:FineGrained] have unlocked powerful tools to analyze the training dynamics of over-parameterized neural networks. We leverage these theoretical findings in order to to prove guarantees on the convergence and the generalisation of the algorithm, Orthogonal Gradient Descent for Continual Learning [@farajtabar2019orthogonal].
Our contributions are summarized as follows:
1. We provide closed form expressions of the functions learned across tasks. We find that they can be expressed as a linear combination of kernel regressors, over the previously seen tasks. The relationship also captures task similarity and the transferability of knowledge across tasks (Sec. \[sec:convergence\], Theorem \[thm:conv\]).
2. We prove the first generalisation bound for continual learning with OGD, to our knowledge. We derive bounds for within-task and outside-task generalisation. We find that generalisation through time depends on task similarity, which we quantify rigorously (Sec. \[sec:generalisation\], Theorem \[thm:gen-ogd\]).
3. We prove that OGD is robust to forgetting with respect to the previous task only (Sec. \[sec:generalisation\], Lemma \[lem:train-unchanged\]).
4. We build-up on this insight to propose OGD+ (Sec. \[sec:ogd-plus\], Alg. \[alg:ogd-plus-train\]), an extension of OGD, which we prove robust to catastrophic forgetting across an arbitrary number of tasks (Sec. \[sec:ogd-plus\], Lemma \[lem:train-unchanged-ogd-plus\]). We also prove tighter generalisation bounds than OGD (Sec. \[sec:ogd-plus\], Theorem \[thm:gen-ogd-plus\]).
5. As a side result, we find that Lemma \[lem:rademacher-tasks\] also quantifies the impact of the learning curriculum on the generalisation error. We find that task dissimilarity impacts negatively generalisation, and that an ordering of tasks that minimises dissimilarity between neighbouring tasks leads to a tighter generalisation bound. (Sec. \[thm:gen-ogd\], Lemma \[lem:rademacher-tasks\]).
Even though the analysis relies on the assumption that the neural network is overparametrised, the analysis leads to practical insights to improve OGD, which led us to OGD+. We run experiments in the non-overparametrised setting, and show that OGD+ achieves state-of-the-art results in settings with large number of tasks (Sec. \[sec:experiments\]).
Preliminaries {#sec:ogd}
=============
#### Notation
We use bold-faced characters for vectors and matrices. We use $\norm{\cdot}$ to denote the Euclidian norm of a vector or the spectral norm of a matrix, and $\|\cdot\|_\mathrm{F}$ to denote the Frobenius norm of a matrix. We use $\langle\cdot,
\cdot\rangle$ for the Euclidian dot product, and $\langle\cdot,\cdot\rangle_{{\mbox{$\mathcal{H}$}}}$ the dot product in the Hilbert space ${\mbox{$\mathcal{H}$}}$. We index the the task ID by $\tau$. The $ \leq $ operator if used with matrices, corresponds to the partial ordering over symetric matrices. We denote ${\mbox{$\mathbb{N}$}}$ the set of natural numbers, ${\mbox{$\mathbb{R}$}}$ the space of real numbers and ${\mbox{$\mathbb{N}$}}^{\star}$ for the set ${\mbox{$\mathbb{N}$}}\smallsetminus\{0\}$. We use $\oplus$ to refer to the direct sum over Euclidian spaces.
Continual Learning
------------------
Continual learning considers a series of tasks $\{{\mbox{$\mathcal{T}$}}_1, {\mbox{$\mathcal{T}$}}_2, \ldots\}$, where each task can be viewed as a separate supervised learning problem. Similarly to online learning, data from each task is revealed only once. The goal of continual learning is to model each task accurately with a single model. The challenge is to achieve a good performance on the new tasks, while retaining knowledge from the previous tasks [@v2018variational].
We assume the data from each task ${\mbox{$\mathcal{T}$}}_\tau$, $\tau \in {\mbox{$\mathbb{N}$}}^\star$, is drawn from a distribution ${\mbox{$\mathcal{D}$}}_\tau$. Individual samples are denoted $({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}, y_{\tau, i})$, where $i \in [n_\tau]$. Also, we only consider the binary classification setting for the sake of simplicity: ${\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau,i} \in {\mbox{$\mathbb{R}$}}^d$ and $y_{\tau, i} \in \{ -1, +1 \}$. We note that it does not restrict the scope of the analysis, which can be easily extended to multiclass settings.
OGD for Continual Learning {#subsec:ogd-for-continual-learning}
--------------------------
Let ${\mbox{$\mathcal{T}$}}_T$ the current task, where $T \in {\mbox{$\mathbb{N}$}}^\star$. For all $i \in [n_T]$, let ${\mbox{${\mbox{$\mathbf{v}$}}$}}_{T, i} = \nabla_\theta f^\star_{T-1}({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1,i})$, which is the Jacobian of task ${\mbox{$\mathcal{T}$}}_T$. We define ${\mbox{$\mathbb{E}$}}_\tau = \mathrm{vec}( \{{\mbox{${\mbox{$\mathbf{v}$}}$}}_{\tau, i}, i \in [n_\tau] \} )$, which is the subspace induced by the Jacobian. The idea behind OGD is to update the weights along the projection of the gradient on the orthogonal space induced by the Jacobians over the previous tasks ${\mbox{$\mathbb{E}$}}_1 \oplus \ldots \oplus {\mbox{$\mathbb{E}$}}_{\tau-1} $. The update rule for the task ${\mbox{$\mathcal{T}$}}_T$ is as follows [@farajtabar2019orthogonal]: $${\mbox{${\mbox{$\mathbf{w}$}}$}}_T(t+1) = {\mbox{${\mbox{$\mathbf{w}$}}$}}_T(t) - \eta \Pi_{{\mbox{$\mathbb{E}$}}_{T-1}^\bot} \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} {\mbox{$\mathcal{L}$}}^T_{\lambda}({\mbox{${\mbox{$\mathbf{w}$}}$}}_T(t)).$$
The intuition behind OGD is to [@farajtabar2019orthogonal].
To prevent over-fitting and guarantee the uniqueness of the global minimum in the Neural Tangent Kernel (NTK) regime, we apply a ridge regularization with a parameter $\lambda \in {\mbox{$\mathbb{R}$}}^{+}$. For a task ${\mbox{$\mathcal{T}$}}_\tau$, we write the corresponding loss as follows: $${\mbox{$\mathcal{L}$}}_\lambda^\tau({\mbox{${\mbox{$\mathbf{w}$}}$}}) = \sum_{i=1}^{n_\tau} (f_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}) - y_{\tau, i})^2 + \lambda
\norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}-
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau-1}}^2.$$
Generalisation for Continual Learning
-------------------------------------
We define within-task generalisation as the ability of the agent to acquire new knowlege and outside-task generalisation as its ability to preserve the acquired knowledge.
Consider a loss function $l: {\mbox{$\mathbb{R}$}}\times {\mbox{$\mathbb{R}$}}\rightarrow {\mbox{$\mathbb{R}$}}$. The population loss over the distribution ${\mbox{$\mathcal{D}$}}$, and the empirical loss over $n$ samples $D = \{({\mbox{${\mbox{$\mathbf{x}$}}$}}_i, y_i), i \in [n] \}$ from the same distribution ${\mbox{$\mathcal{D}$}}$ are defined as: $$\begin{aligned}
L_D(f) &= {\mathbb{E}}_{({\mbox{${\mbox{$\mathbf{x}$}}$}}, y) \sim {\mbox{$\mathcal{D}$}}} [l(f({\mbox{${\mbox{$\mathbf{x}$}}$}}), y)], &\\
L_S(f) &= \frac{1}{n} \sum_{i=1}^n l(f({\mbox{${\mbox{$\mathbf{x}$}}$}}_i), y_i).
\end{aligned}$$
Let ${\mbox{$\mathcal{T}$}}_1, \ldots {\mbox{$\mathcal{T}$}}_T$ a sequence of tasks, and ${\mbox{$\mathcal{D}$}}_1, \ldots {\mbox{$\mathcal{D}$}}_T$ their corresponding distributions.\
Let $f_1^\star, \ldots f_T^\star$ the trained models at each task. Let $\tau \in [T]$ fixed.\
We define:
- within-task generalisation of the task ${\mbox{$\mathcal{T}$}}_\tau$ as $L_{D_\tau}(f_\tau^\star)$.
- outside-task generalisation of the task ${\mbox{$\mathcal{T}$}}_\tau$ with respect to a task ${\mbox{$\mathcal{T}$}}_{\tau^\prime}$, where $\tau^\prime < \tau$ as $L_{D_\tau}(f_{\tau^\prime}^\star)$.
In practice, several works also tracked these metrics in their experiments (@Kirkpatrick2016EWC, @farajtabar2019orthogonal).
Neural Tangent Kernel
---------------------
In their seminal paper, @Jacot2018NTK established the connection between deep networks and kernel methods by introducing the Neural Tangent Kernel (NTK). They showed that at the infinite width limit, the kernel remains constant throughout training. @lee2019wide also showed that a network evolves as a linear model in the infinite width limit when trained on certain losses under gradient descent.
Throughout our analysis, we make the assumption that the neural network is overparameterized, and consider the linear approximation of the neural network around its initialisation: $$f^{(t)}({\mbox{${\mbox{$\mathbf{x}$}}$}}) \approx f^{(0)}({\mbox{${\mbox{$\mathbf{x}$}}$}}) + \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f^{(0)}({\mbox{${\mbox{$\mathbf{x}$}}$}})^T ({\mbox{${\mbox{$\mathbf{w}$}}$}}(t) - {\mbox{${\mbox{$\mathbf{w}$}}$}}(0)).$$
Convergence of OGD for Continual Learning {#sec:convergence}
=========================================
In this section, we derive a closed form expression for the learned models across tasks. We find a recursive kernel ridge regression relationship between the models across tasks. The result is presented in Theorem \[thm:conv\], a stepping stone towards proving the generalisation bound for OGD in Sec. \[sec:generalisation\].
Convergence Theorem {#subsec:conv}
-------------------
Now, we state the main result of this section:
\[thm:conv\] Let ${\mbox{$\mathcal{T}$}}_1,\ldots, {\mbox{$\mathcal{T}$}}_T$ be a sequence of tasks. Fix a learning rate sequence $(\eta_\tau)_{\tau \in [T]}$. If, for all $\tau$, the learning rate satisfies $$\eta_\tau < \frac{1}{\norm{k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)} +\lambda_\tau^2},$$ $$\begin{aligned}
\intertext{then for all $\tau$, ${\mbox{${\mbox{$\mathbf{w}$}}$}}_\tau(t)$ converges linearly to a limit solution ${\mbox{${\mbox{$\mathbf{w}$}}$}}_\tau^\star$ such
that}
f_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})
&= f_{\tau-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + k_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T {\mbox{${\mbox{$\mathbf{H}$}}$}}_{\tau, \lambda_\tau}^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau,
\intertext{where}
k_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime) &= {\widetilde{\phi}}_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}})^T {\widetilde{\phi}}_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime), &\\
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau &= {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau}, &\\
{\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau} &= f_{\tau-1}^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau), &\\
\phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{\tau-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}), &\\
{\mbox{${\mbox{$\mathbf{H}$}}$}}_{\tau, \lambda} &= k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}}, &\\
{\widetilde{\phi}}_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) &=
\begin{cases}
\phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) \text{\quad $\mathrm{for SGD}$,}\\
{\mbox{${\mbox{$\mathbf{T}$}}$}}_\tau \phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) \text{\quad $\mathrm{for OGD}$.}
\end{cases}
\intertext{and $\{{\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau}, \tau \in [T]\}$ are proxy matrices for the analysis.}
\end{aligned}$$
The theorem describes how the model $f_\tau^\star$ evolves across tasks. The theorem is recursive because the learning is incremental. For a given task ${\mbox{$\mathcal{T}$}}_\tau$, $f_{\tau-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})$ is the knowledge acquired by the agent up to the task ${\mbox{$\mathcal{T}$}}_{\tau-1}$. At this stage, the model only fits the residual ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau ={\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau}$, which complements the knowledge acquired through previous tasks. This residual is also a proxy for task similarity. If the tasks are identical, the residual is equal to zero. The knowledge increment is captured by the term: $k_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T (k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda_\tau^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau$. Finally, task similarity is computed with respect to the most recent feature map ${\widetilde{\phi}}_\tau$, and $k_\tau$ is the NTK with respect to the feature map ${\widetilde{\phi}}_\tau$.
The recursive relation from Theorem \[thm:conv\] can also be written as a linear combination of kernel regressors as follows: $$\begin{aligned}
f_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= \sum_{k=1}^\tau {\mbox{$\tilde{f}$}}_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}),
\intertext{where}
{\mbox{$\tilde{f}$}}_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= k_k({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^T (k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k , {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) + \lambda_k^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k.
\end{aligned}$$
#### Proof Sketch:
We prove Theorem \[thm:conv\] by induction. We rewrite the loss function as a regression on the residual ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau$ instead of ${\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau$. Then, we rewrite the optimisation objective as an unconstrained strongly convex optimisation problem. Finally, we compute the unique solution in a closed form. The full proof is presented in App. \[subsec:conv-multiple-proof\].
Distance from Initialisation {#subsec:distance-from-initialisation}
----------------------------
As described in Sec. \[subsec:conv\], ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau$ is a residual. It is equal to zero if the model makes perfect predictions on the next task ${\mbox{$\mathcal{T}$}}_\tau$. The more the next task ${\mbox{$\mathcal{T}$}}_\tau$ is different, the further the neural network needs to move from its previous state in order to fit it. Corollary \[cor:weight-dist\] tracks the distance from initialisation as a function of task similarity.
\[cor:weight-dist\] For SGD, and for OGD under the additional assumption that $\{{\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau}, \tau \in [T]\}$ are orthonormal, $$\begin{aligned}
\norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau + 1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_\tau^\star}_\mathrm{F} &= \sqrt{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^T {\mbox{${\mbox{$\mathbf{H}$}}$}}_{T, \lambda}^{-1} {\mbox{${\mbox{$\mathbf{H}$}}$}}_{\tau, 0}
{\mbox{${\mbox{$\mathbf{H}$}}$}}_{T,
\lambda}^{-1}
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau},
\intertext{where}
{\mbox{${\mbox{$\mathbf{H}$}}$}}_{\tau, \lambda} &= k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}}.
\end{aligned}$$
The proof is presented in App. \[subsec:proof-weight-dist\]. The orthonormality assumption is not restrictive, since the set $\{{\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau}, \tau \in [T]\}$ is only a proxy for the analysis; indeed we can choose any convenient basis to work with.
\[rem:weight-dist\] Corollary \[cor:weight-dist\] can be used to get a similar result to Theorem 3 by @liu2019understanding. In this remark, we consider mostly their notations. Their theorem states that under some conditions, for 2-layer neural networks with a RELU activation function, with probability no less than $1 - \delta$ over random initialisation, $$\begin{aligned}
\norm{{\mbox{${\mbox{$\mathbf{W}$}}$}}(P) - {\mbox{${\mbox{$\mathbf{W}$}}$}}(Q)}_\mathrm{F} &\leq \sqrt{ {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{P \rightarrow Q}^T H_P^{\infty \space -1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{P \rightarrow Q}}
+ \epsilon, &\\
\intertext{where, in their work:}
{\mbox{${\mbox{$\mathbf{y}$}}$}}_{P \rightarrow Q} &= H_{PQ}^{\infty, T} H_P^{\infty \space -1} {\mbox{${\mbox{$\mathbf{y}$}}$}}_P,\\
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{P \rightarrow Q} &= {\mbox{${\mbox{$\mathbf{y}$}}$}}_Q - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{P \rightarrow Q}.
\end{aligned}$$
Note that $H_P^{\infty}$ is a Gram matrix, which also corresponds to the NTK of the neural network they consider. We see an analogy with our result, where we work directly with the NTK, with no assumptions on the neural network. One important observation is that, to our knowledge, since there are no guarantees for the invertibility of our Gram matrix, we add a ridge regularisation to work with a regularised matrix, which is then invertible. In our setting, by considering $\lambda \rightarrow 0$, and with the additional assumption of invertibility of ${\mbox{${\mbox{$\mathbf{H}$}}$}}_{\tau, 0}$,which is valid in the two-layer overparametrised RELU neural network considered in the setting of @liu2019understanding, we can recover a similar approximation.
Generalisation of OGD for Continual Learning {#sec:generalisation}
============================================
In this section, we study the generalisation properties of OGD. First, we prove that OGD is robust to catastrophic forgetting with respect to the previous task (Lemma \[lem:train-unchanged\]). Then, we present the the main generalisation theorem for OGD (Thm. \[thm:gen-ogd\]). The theorem provides several insights on the relation between task similarity and generalisation. Finally, we present how the Rademacher complexity relates to task similarity across a large number of tasks (Lemma \[lem:rademacher-tasks\]). The lemma states that the more dissimilar tasks are, the larger the class of functions explored by the neural network, with high probability.
Memorisation property of OGD {#subsec:memorisation-property-of-ogd}
----------------------------
The key to obtaining tight generalisation bounds for OGD is Lemma \[lem:train-unchanged\].
\[lem:train-unchanged\] Given a task ${\mbox{$\mathcal{T}$}}_\tau$, for all ${\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i} \in D_\tau$, a sample from the training data of the task ${\mbox{$\mathcal{T}$}}_\tau$, it holds that $$f_{\tau + 1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}) = f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}).$$
As motivated by @farajtabar2019orthogonal, the orthogonality of the gradient updates aims to preserve the acquired knowledge, by not altering the weights along relevant dimensions when learning new tasks. Lemma \[lem:train-unchanged\] states that the training error on the previous task is unchanged, when training with OGD. However, there are no guarantees that the knowledge from the tasks before the previous task is preserved.
The proof of Lemma \[lem:train-unchanged\] is presented in App. \[subsubsec:proof-train-unchanged\]
Generalisation of OGD for Continual Learning {#subsec:generalisation-ogd}
--------------------------------------------
Now, we state the main generalisation theorem for OGD, which provides within-task and outside-task generalisation bounds.
\[thm:gen-ogd\] Let $\{{\mbox{$\mathcal{T}$}}_1, \ldots {\mbox{$\mathcal{T}$}}_T\}$ be a sequence of tasks. Let be $\{{\mbox{$\mathcal{D}$}}_1, \ldots, {\mbox{$\mathcal{D}$}}_T\}$ the respective distributions over ${\mbox{$\mathbb{R}$}}^d \times \{-1, 1\}$. Let $\{({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}, y_{\tau, i}), i \in [n_t], \tau \in [T] \}$ be i.i.d. samples from ${\mbox{$\mathcal{D}$}}_\tau, \tau \in
[T] $. Denote ${\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau = ({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, 1}, \ldots, {\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, n_\tau})$, ${\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau = (y_{\tau, 1}, \ldots, y_{\tau,
n_\tau})$. Consider the kernel ridge regression solution $f_T^\star$. Suppose that the kernel matrices satisfy
(k\_(\_, \_)) &= (n\_), \_[D\_T]{}(f\_T\^) & + &\
&\_[k=1]{}\^T (( ) + \_k), &\
\_[D\_[T-1]{}]{}(f\_T\^) & + &\
&\_[k=1]{}\^T (( ) + \_k), &\
\_[D\_]{}(f\_T\^) & + &\
&\_[k=1]{}\^T (( ) + \_k), &\
\_k &= () + 3 c , &\
\_[, ]{} &= k\_(\_, \_) + \^2 , &\
A\_&= &\
\_() &= \_\_ f\_[-1]{}\^(), &\
\_&= \_- \_[- 1 ]{}, &\
\_[- 1 ]{} &= f\_[-1]{}\^(\_), &\
\_[i,j,k]{} &= k\_i(\_j, \_k), &\
\_[k, ]{} &= \_[k,k,k]{}\^[-1]{}\_[k,,k]{}\_[k,,k]{}\^T \_[k,k,k]{}\^[-1]{}.
The intution behind Theorem \[thm:gen-ogd\] is as follows:
- Within-task generalisation: The generalisation error on the most recent task leverages the information learned during training on the previous tasks. The bound is tighter compared to learning from scratch, since it depends on the residual ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau}$. Therefore it captures the transferability of knowledge across tasks.
- Outside-task generalisation (${\mbox{$\mathcal{T}$}}_{T - 1}$): The generalisation bound increases only with respect to the Rademacher complexity when training with OGD. The tightness of this bound for OGD is due to Lemma \[lem:train-unchanged\]. This lemma is valid for OGD and not for SGD, which implies that tighter generalisation is guaranteed compared to SGD.
- Outside-task generalisation (${\mbox{$\mathcal{T}$}}_{\tau}, \tau \leq T- 2$): The upper bound depends on the similarity between the outside task and the latest task; the more dissimilar the subsequent tasks are, the more the upper bound diverges from the initial upper bound. This bound captures catastrophic forgetting as a function of the tasks dissimilarity. This bound is the same for OGD and SGD.
These bounds share some similarities with the bounds derived by @Arora:2019:FineGrained, @liu2019understanding and @hu2019simple, where in these works, the bounds were derived for supervised learning settings, and in some cases for two-layer RELU neural networks. Similarly, the bounds depend on the Gram matrix of the data, with the feature map corresponding to the NTK.
#### Proof Sketch:
The proof is presented in App. \[subsec:proof-gen-ogd\]. One challenge is that the function class is the set of linear combinations of kernel regressors (Theorem \[thm:conv\]). We state Lemma \[lem:rademacher-kernels\] to bound the Rademacher complexity for this function class. Then we derive bounds for the training error for each case in Theorem \[thm:gen-ogd\]. The first case is straightforward. For the second case, we use Lemma \[lem:train-unchanged\], then derive a similar proof to the first case. The third case presents some additional technical challenges. In order to derive the upper bounds, we draw a strong inspiration from @hu2019simple, and leverage several of their proof techniques and mathematical tools.
Distance from Initialisation through Tasks {#subsec:distance-from-initialisation-through-tasks}
------------------------------------------
Now, we state Lemma \[lem:rademacher-tasks\], which tracks the Rademacher complexity through tasks.
\[lem:rademacher-tasks\] Keeping the same notations and setting as Theorem \[thm:gen-ogd\], the Rademacher Complexity can be bounded as follows: $$\begin{aligned}
\hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T}) &\leq \sum_{\tau=1}^T {\mathcal{O}}\left( \sqrt{\frac{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^T (k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau }{n_\tau}} \right) &\\
&+ \sum_{\tau=1}^T {\mathcal{O}}(\frac{1}{\sqrt{n_\tau}}).
\end{aligned}$$
The intuition behind Lemma \[lem:rademacher-tasks\] is that the upper bound on the Rademacher complexity increases when the tasks are dissimilar. The dissimilarity between two subsequent tasks is measured through ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^T (k_t({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau$. The knowledge from the previous tasks is implicitly encoded in the kernel $k_\tau$, which is based on the feature map $\phi_\tau$. This feature map encodes the acquired knowledge. As an edge case, if two successive tasks are identical, the residual ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau = {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau -
{\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau} = 0$, therefore, the upper bound on the Rademacher complexity does not increase.
#### Implications for Curriculum Learning
We also observe that the upper bound depends on the task ordering, which may provide a theoretical explanation on the importance of learning with a curriculum [@Bengio2009curriculum]. In the following, we present an edge case which provided an intuition on how the bound captures the importance of the order. Consider two dissimilar tasks ${\mbox{$\mathcal{T}$}}_1$ and ${\mbox{$\mathcal{T}$}}_2$. A sequence of tasks alternating between ${\mbox{$\mathcal{T}$}}_1$ and ${\mbox{$\mathcal{T}$}}_2$ will lead to a large upper bound, as explained in the first paragraph. While, a sequence of tasks concatenating two sequences of ${\mbox{$\mathcal{T}$}}_1$ then ${\mbox{$\mathcal{T}$}}_2$ will lead to a lower upper bound.
#### Proof Sketch:
The proof techniques for Lemma \[lem:rademacher-tasks\] are exactly the same as the ones for Theorem \[thm:gen-ogd\]. The full proof is presented in Sec. \[subsubsec:bound-rademacher\].
OGD+: Learning without Forgetting {#sec:ogd-plus}
=================================
In the previous section, we demonstrated the limits of OGD, in terms of robustness to catastrophic forgetting on the long run. Now, we present OGD+, an extension of OGD, which we prove robust to catastrophic forgetting, across an arbitrary number of tasks (Lemma \[lem:train-unchanged-ogd-plus\]). Then, we prove tighter generalisation bounds compared to OGD (Theorem \[thm:gen-ogd-plus\]).
The OGD+ Algorithm {#subsec:ogd+-:-the-algorithm}
------------------
Algorithm \[alg:ogd-plus-train\] presents the OGD+ algorithm, we highlight the differences with OGD in red. The main difference is that OGD+ stores the feature maps with respect to the samples from previous tasks, in addition to the feature maps with respect to the samples from the current task, as opposed to OGD. This small change unlocks the proof of Lemma \[lem:train-unchanged-ogd-plus\] given below, which implies tighter bounds for Theorem \[thm:gen-ogd-plus\].
The idea behind OGD+ comes from the convergence Theorem (Sec. \[sec:convergence\], Thm. \[thm:conv\]). After training on a task ${\mbox{$\mathcal{T}$}}_\tau$, the learned model is a linear combination of the previous models. For a given sample ${\mbox{${\mbox{$\mathbf{x}$}}$}}$ from a task ${\mbox{$\mathcal{T}$}}_k$ where $k < \tau$, in order to keep the training error identical, the weights need to be updated along the directions that are orthogonal to *all the subsequent feature maps* of ${\mbox{${\mbox{$\mathbf{x}$}}$}}$. OGD only considers the feature map of the source task of the sample. Storing all the feature maps implies that the learned model back from task ${\mbox{$\mathcal{T}$}}_k$, can be recovered even after training on an arbitrary number of tasks.
In order to compute the feature maps with respect to the previous samples, OGD+ saves these samples in a dedicated memory, we call this storage the *samples memory*. This memory comes in addition to the orthonormal *feature maps memory*. The only role of the *samples memory* is to compute the feature maps. While the proofs below are under the assumption that the memory size is infinite, in the experiments, we keep a limited size for both memories.
1. Initialize $S_{J}\leftarrow \{\}$ ; [ $S_{D}\leftarrow \{\}$]{}; ${\mbox{${\mbox{$\mathbf{w}$}}$}}\leftarrow {\mbox{${\mbox{$\mathbf{w}$}}$}}_0$
2.
Memorisation Property of OGD+ {#subsec:memorisation-ogd-plus}
-----------------------------
The key to obtaining tight generalisation bounds for OGD+ is the Lemma \[lem:train-unchanged-ogd-plus\] below. It states that the training error across all previous tasks is unchanged, when training with OGD+.
\[lem:train-unchanged-ogd-plus\] Given a task ${\mbox{$\mathcal{T}$}}_\tau$, for all ${\mbox{${\mbox{$\mathbf{x}$}}$}}_{k, i} \in D_k$, a sample from the training data of a previous task, it holds that: $$f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{k, i}) = f_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{k, i}).$$
The full proof of Lemma \[lem:train-unchanged-ogd-plus\] is presented in App . \[subsubsec:proof-train-unchanged-ogd-plus\].
Generalisation Guarantees for OGD+ {#subsec:generalisation-theorem}
----------------------------------
Now, we state the generalisation theorem for OGD+, which provides tighter generalisation bounds in comparison with Theorem \[thm:gen-ogd\], for OGD.
\[thm:gen-ogd-plus\] Under the same conditions as Theorem \[thm:gen-ogd\], for OGD+, it holds that, for all tasks ${\mbox{$\mathcal{T}$}}_\tau$, within-task and outside-task generalisation error can be bounded as follows $$\begin{aligned}
{\mbox{$\mathcal{L}$}}_{D_\tau}(f_T^\star) &\leq \frac{\lambda}{2} \sqrt{ \frac{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^T k_{T}
({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau}{n_\tau} } + &\\
&\sum_{k=1}^T {\mathcal{O}}\left( \sqrt{\frac{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T (k_t({\mbox{${\mbox{$\mathbf{X}$}}$}}_k , {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) +
\lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k }{n_k}} \right) + \Delta_k, &\\
\intertext{where}
\Delta_k &= {\mathcal{O}}(\frac{1}{\sqrt{n_k}}) + 3 c \sqrt{\frac{\log(2/\delta) }{2n_k}}, &\\
\phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= {\mbox{${\mbox{$\mathbf{T}$}}$}}_\tau \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{\tau-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}), &\\
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau &= {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau}, &\\
{\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau} &= f_{\tau-1}^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau).
\end{aligned}$$
The generalisation bounds of Theorem \[thm:gen-ogd-plus\] are tighter than the generalisation bounds for OGD. The tightness of the bounds is a consequence of Lemma \[lem:train-unchanged-ogd-plus\]. The term that corresponds to the Rademacher complexity is unchanged, while the term that bounds the training error is tighter. It is also tighter than a standard supervised learning bound, because it captures the transferability of knowledge across tasks through the residual ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau$, as opposed to the Supervised Learning only bounds, which would depend on ${\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau$ instead.
#### Proof Sketch
The full proof is presented in App. \[subsec:proof-gen-ogd-plus\]. The proof is based on Lemma \[lem:train-unchanged-ogd-plus\] and Lemma \[lem:rademacher-kernels\]. The proof techniques are the same as the ones for Theorem \[thm:gen-ogd\].
Experiments {#sec:experiments}
===========
We performed experiments on the continual learning benchmark Permuted Mnist. The setup is the same as the one described in (@Goodfellow2013Forgetting, @kirkpatrick2017forgetting, @chaudhry2018efficient, @farajtabar2019orthogonal). However, in order to assess the robustness to catastrophic forgetting over a long sequence of tasks, we increase the size of the task sequence to 15. We generated 15 i.i.d. permutations of the MNIST pixels, then defined each task as a MNIST supervised learning problem with respect to each permutation, respectively. We trained the neural network sequentially on all tasks and tracked the validation loss on all previous tasks. We considered the same neural network architecture and mostly similar hyperparameters as [@farajtabar2019orthogonal]. The reproducibility details are presented in Appendix \[subsec:reproductibility\].
We report the results in Figure \[fig:ogd-plus-acc\], extended results are presented in App . \[subsec:app-exp-gen-extended\]. The plot shows that OGD+ is more robust to catastrophic forgetting than OGD when the tasks occurrence difference is large. It also shows that OGD+ is equivalent to OGD when the tasks occurrence difference is small. These results concur with Lemma \[lem:ogd-train-gen-bounds\] which states that OGD is robust to catastrophic forgetting up to a single task ahead. forgetting up to a single task ahead. They also concur with Lemma \[lem:train-unchanged-ogd-plus\], which states that OGD+ is robust to catastrophic forgetting across any number of tasks. Two probable reasons that OGD+ is not perfectly prone to catastrophic forgetting in the experiment are the memory limit and the non-overparameterization of the neural network, while in Lemma \[lem:train-unchanged-ogd-plus\], we assumed that the memory is unlimited and that the neural network is overparameterized. The results in the Appendix \[subsec:app-exp-gen-extended\] concur with these hypotheses. Fig. \[fig:app:overparam\] shows that the test accuracy through time increases with overparameterization, in which case our approximation is more valid. Fig. \[fig:app:memory\] shows that the test accuracy also increases uniformly with the size of the memory.
We also observe that OGD outperforms OGD+ for shorter term tasks, on the long run (App . \[subsec:app-exp-gen-extended\], Fig. \[fig:app:gen\]). One probable reason is that OGD+ performs a uniform sampling across samples from all past tasks, considering its limited memory budget, and that the memory requirements increase quadratically through tasks, we expect an information loss with respect to the most recent tasks, since the corresponding storage is used by OGD+ to older tasks. OGD uses the equivalent storage for the most recent task.
![Test accuracy on the first 3 tasks of permuted MNIST, for SGD, OGD and OGD+. We report the dynamics of additional tasks in App. \[subsec:app-exp-gen-extended\], Fig. \[fig:app:gen\]. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks. []{data-label="fig:ogd-plus-acc"}](figs/out/gen/out.png){width="40.00000%"}
Related works
=============
#### Continual Learning
Approaches to Continual Learning can be categorised into: regularization methods, memory based methods, and dynamic architectural methods. We refer the reader to the survey [@PARISI201954] for an extensive overview on the existing methods. The idea behind memory-based methods is to store data from previous tasks in a buffer of fixed size, which can then be reused during training on the current task (@chaudhry2018efficient, @vandeven2018generative). Dynamic architectural methods rely on growing architectures which keep the past knowledge fixed and store new knowledge in new components, such as new nodes, layers ... (@Lee2017Lifelong, @Schwarz2018ProgressCompress) The idea behind regularization methods is to regularize the objective in order to preserve the knowledge acquired from the previous tasks (@Kirkpatrick2016EWC, @Aljundi2017MAS, @farajtabar2019orthogonal, @Zenke2017SI).
#### Catastrophic Forgetting
Catastrophic Forgetting refers to the tendency of agents to “forget” the previous tasks over the course of training. It has proven to be a challenging problem, several heuristics were developed in order to characterise it (@ANS1997989, @Ans2000Refreshing, @Goodfellow2013Forgetting, @Robert1999Connectionist, @MCCLOSKEY1989109, @Robins1995Rehearsal, @Nguyen2019UnderstandForgetting).
#### Deep Learning Theory
Recent work have started to provide explanations about the mechanics of overparametrised Neural Networks. In their seminal work, @Du2018GD prove that Gradient Descent on multilayer overparametrised RELU neural networks achieve zero training error at the limit. These works have unlocked the analysis of several properties of Deep Neural Networks, in the context of various applications, such as Transfer Learning [@liu2019understanding], Noisy Supervision [@hu2019simple], Reinforcement Learning [@wang2020long] ... Another line of works provide closed form expressions of the training dynamics of overparameterized neural networks, leveraging tools from statistical physics (@Goldt2019Dynamics, @goldt2020modelling).
Also, Corrolary \[cor:weight-dist\] in Sec. \[sec:convergence\] can be seen as a generalisation of the Theorem 3 by @liu2019understanding. While we don’t make any assumptions on the neural network as opposed to @liu2019understanding, which studies two-layer RELU neural networks, our analysis is based on less fine-grained approximations. Finally, as explained in Sec. \[subsec:generalisation-ogd\], we used several proof techniques from @hu2019simple to prove the Theorems \[thm:conv\], \[thm:gen-ogd\] and \[thm:gen-ogd-plus\].
#### Transfer Learning
Several works have also recently started studying Transfer Learning from a theoretical perspective (@yu2020gradient, @liu2019understanding, @Nguyen2019UnderstandForgetting, @Achille2019task2vec).
#### Statistical Learning Theory
@pmlr-v54-alquier17a define a compound regret for lifelong learning, as the regret with respect to the oracle who would have known the best common representation $g$ for all tasks in advance. Another line of works addresses lifelong learning and meta-learning from a statistical learning theory perspective, they define and provide regret bounds for lifelong learning [@pmlr-v54-alquier17a], or study the sample complexity and convergence of meta-learning algorithms (@denevi2018incremental, @du2020fewshot, @ji2020multistep, @saunshi2020sample, @ltl_common_mean).
Discussion {#sec:discussion-limits}
==========
We discuss some assumptions and limits of our analysis.
#### Linearization
We considered a linear approximation of the neural network, which may not be valid in the non overparameterized regime. We find that the approximation leads to multiple insights on how to improve OGD, generalisation, knowledge transferability, curriculum learning and convergence. Also, our analysis is based solely on convex optimisation, which surprisingly leads us to results from other works, which considered other approximations for overparameterized neural networks. Finally, the experiments with non-overparametrized neural networks concur with our theoretical findings.
#### How are the NTKs different between SGD and OGD ?
We note that the key difference comes from the feature map, which is projected with $T$ to a feature space orthogonal to ${\mbox{$\mathbb{E}$}}$. In practice, the size of the data is much smaller than the number of the parameters, therefore, the dimension of ${\mbox{$\mathbb{E}$}}$ is negligible in comparison with the $d$, the dimension of the parameter space of the neural net, independently of the NTK regime. It could be interesting to understand more in depth the difference between the kernels of SGD and OGD. Their formulation looks similar to Von Neumann Kernels, which may be a possible direction to investigate.
Conclusion {#sec:conclusion-and-outlook}
==========
In this paper, we present an approach to study the properties of OGD for Continual Learning theoretically. Through this approach, we present in a closed-form how the model evolves through tasks. This result leads to a generalisation theorem for OGD and provides insights on the transferability of knowledge across tasks and on the importance of task ordering and how the curriculum impacts generalisation. We also present OGD+, an extension of OGD, for which we derive stronger guarantees in terms of robustness to catastrophic forgetting and generalisation error. Finally, we observe that OGD+ achieves state-of-the-art performance for settings with a large number of tasks.
There are multiple avenues for future investigation. First, similarly to @Arora:2019:FineGrained and @liu2019understanding, a lower bound over the number of parameters to fall in the overparameterization regime is an important question, it would clarify the cases for which the theory is valid.
Another possible direction to investigate is the theoretical properties of the other Continual Learning training methods and the catastrophic forgetting heuristics, such as the ones proposed by @Nguyen2019UnderstandForgetting. A theroetical understanding of these algorithms and heuristics, even for asymptotic cases, may provide insights on their limits and directions of improvements, similarly to OGD+.
We also observed in the experiments that uniform sampling across all tasks for OGD+ would be the optimal sampling strategy for an average performance balance across all past tasks. In order to favor remembering specific tasks, a possible direction is adapting the sampling distribution with respect to the past tasks’ importance.
Finally, we found multiple connections to other fields such as Transfer Learning and Curriculum Learning. A promising direction for further investigation is to investigate how the theory could apply to neighbouring fields, such as meta-learning, multi-task learning ... We hope this work provides new keys to address these challenges.
#### Acknowledgements
We would like to thank Mohammad Emtiyaz Khan, Michael Przystupa, Pierre Alquier, Pierre Orenstein and Bo Han for their feedback and the helpful discussions.\
MS was supported by KAKENHI 17H00757.
Missing proofs of section \[sec:convergence\] - Convergence
===========================================================
Proof of Theorem \[thm:conv\] {#subsec:conv-multiple-proof}
-----------------------------
#### Orthogonal Gradient Descent
$$\begin{aligned}
\intertext{We prove the Theorem \ref{thm:conv} by induction. Our induction hypothesis $H_\tau$ is the
following :}
\intertext{$H_\tau$ : For all $k \leq \tau$, Theorem \ref{thm:conv} holds.}
\intertext{First, we prove that $H_1$ holds.}
\intertext{The proof is straightforward. For the first task, since there were no previous tasks ${\mbox{$\mathbb{E}$}}_1 =
\emptyset$. Therefore, OGD on this task is equivalent to SGD.}
\intertext{Therefore, it is equivalent to minimising the following objective, where $\tau = 1$ : }
\operatorname*{arg\,min}_{{\mbox{${\mbox{$\mathbf{w}$}}$}}\in {\mbox{$\mathbb{R}$}}^d} &\norm{\phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T ({\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}(t) - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star) -
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}}_2^2 + \lambda_\tau \norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}- {\mbox{${\mbox{$\mathbf{w}$}}$}}_0}^2 &\\
\intertext{The objective is quadratic and the Hessian is positive definite, therefore the minimum exists
and is unique :}
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{0} &= \phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) (\phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T \phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda_{\tau}^2 I)^{-1}
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} &\\
\intertext{For $\tau = 1$, since there are no previous tasks ${\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} = {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau}$. Therefore :}
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{0} &= k_{\tau}({\mbox{${\mbox{$\mathbf{x}$}}$}},{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) (k_{\tau}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau}^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} &\\
\intertext{Which completes the proof of $H_1$.}
\intertext{Let $\tau \in {\mbox{$\mathbb{N}$}}^\star$, assume $H_\tau$ is true, we show $H_{\tau+1}$ }
\intertext{On the task $\tau + 1$, we can write the loss ${\mbox{$\mathcal{L}$}}^{\tau+1}$ as :}
{\mbox{$\mathcal{L}$}}^{\tau+1}({\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}(t)) &= \norm{\phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T ({\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}(t) - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star) -
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}}_2^2 + \lambda_{\tau + 1} \norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1} - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}}^2 &\\
\intertext{We recall that the optimisation problem at time $(\tau + 1)$ :}
\operatorname*{arg\,min}_{{\mbox{${\mbox{$\mathbf{w}$}}$}}\in {\mbox{$\mathbb{R}$}}^d} &\norm{\phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T ({\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}(t) - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star) -
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}}_2^2 + \lambda_{\tau + 1} \norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1} - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}}^2 &\\
\text{u.c.} \quad & {\mbox{${\mbox{$\mathbf{V}$}}$}}_{\tau+1} ({\mbox{${\mbox{$\mathbf{w}$}}$}}- {\mbox{${\mbox{$\mathbf{w}$}}$}}^\star_\tau) = 0 &\\
\intertext{Let ${\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} \in {\mbox{$\mathbb{R}$}}^{d \times (d - K_{\tau+1})}$ and ${\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}_{\tau+1} \in
{\mbox{$\mathbb{R}$}}^{d-K_{\tau+1}}$ such as :}
{\mbox{${\mbox{$\mathbf{w}$}}$}}- {\mbox{${\mbox{$\mathbf{w}$}}$}}^\star_\tau &= {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}_{\tau+1} &\\
K_{\tau+1} &= \dim({\mbox{$\mathbb{E}$}}_{\tau+1}) &\\
\intertext{We rewrite the objective by plugging in the variables we just defined. The two objectives are
equivalent : }
\operatorname*{arg\,min}_{{\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}\in {\mbox{$\mathbb{R}$}}^{d - K_{\tau+1}}} &\norm{\phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}-
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}}_2^2 + \lambda_{\tau + 1}^2 \norm{{\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}}_2^2 &\\
\intertext{For clarity, we define ${\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1} {\mbox{$\mathbb{R}$}}^{n_{\tau + 1} \times (d - K_{\tau+1})}$ as :}
{\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1} &= \phi_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} &\\
\intertext{By plugging in ${\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}$, we rewrite the objective as :}
\operatorname*{arg\,min}_{{\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}\in {\mbox{$\mathbb{R}$}}^{d - K_{\tau+1}} } &\norm{{\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}-
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}}_2^2 + \lambda_{\tau + 1}^2 \norm{{\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}}_2^2 &\\
\intertext{The optimisation objective is quadratic, unconstrainted, with a positive definite hessian.
Therefore, an optimum exists and is unique : }
{\mbox{${\mbox{$\mathbf{\tilde{w}}$}}$}}^\star_{\tau+1} &= {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T ({\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T + \lambda_{\tau+1}^2 I)^{-1}
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
\intertext{We recover the expression of the optimum in the original space :}
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star &= {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T ({\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T + \lambda_{\tau+1}^2 I)^{-1}
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
\intertext{We define the kernel $k_{\tau+1} : {\mbox{$\mathbb{R}$}}^d \times {\mbox{$\mathbb{R}$}}^d \rightarrow {\mbox{$\mathbb{R}$}}$ as :}
k_{\tau+1}({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime) &= \phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}})^T {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1}^T \phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime) \quad \text{for all ${\mbox{${\mbox{$\mathbf{x}$}}$}},
{\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime \in {\mbox{$\mathbb{R}$}}^d$} &\\
\intertext{Now we rewrite ${\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star$ :}
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star &= {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 I)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
\intertext{Finally, we recover a closed form expression for $f_{\tau +1}^\star$ :}
\intertext{First, we use the induction hypothesis $H_\tau$ :}
f_{\tau +1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + \langle \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) ,{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star
- {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star \rangle &\\
&= f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + \phi_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 I)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
&= f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + k_{\tau+1}({\mbox{${\mbox{$\mathbf{x}$}}$}},{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 I)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
\intertext{At this stage, we have proven $H_{t+1}$.}
\intertext{We conclude.}
\end{aligned}$$
#### Stochastic Gradient Descent
The proof is exactly the same as the proof for Orthogonal Gradient Descent, except that there are no equalities constraints.
Proof of the Corollary \[cor:weight-dist\] {#subsec:proof-weight-dist}
------------------------------------------
#### Orthogonal Gradient Descent
$$\begin{aligned}
\intertext{In the proof of Theorem \ref{thm:conv} (App.~\ref{subsec:conv-multiple-proof}), we proved that :}
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star &= {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 I)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
\intertext{Therefore :}
\norm{{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star}^2 &=
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}^T (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1}^T {\mbox{${\mbox{$\mathbf{T}$}}$}}_{\tau+1} {\mbox{${\mbox{$\mathbf{Z}$}}$}}_{\tau+1}^T (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 I)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
&={\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1}^T (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_{\tau+1}^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) (k_{\tau+1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda_{\tau+1}^2 I)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau+1} &\\
\end{aligned}$$
#### Stochastic Gradient Descent
The proof is exactly the same as for Orthogonal Gradient Descent.
Missing proofs of section \[sec:generalisation\] - Generalisation
=================================================================
Proof of Theorem \[thm:gen-ogd\] {#subsec:proof-gen-ogd}
--------------------------------
### Notations
$$\begin{aligned}
\intertext{We recall that :}
f_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})
&= \sum_{k=1}^{\tau-1} f_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + k_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T (k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)
+ \lambda_\tau^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau
\intertext{We define :}
{\mbox{$\tilde{f}$}}_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= k_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)^T {\mbox{${\mbox{$\boldsymbol{\alpha}$}}$}}_\tau
\intertext{where :}
{\mbox{${\mbox{$\boldsymbol{\alpha}$}}$}}_\tau &= (k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda_\tau^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau &\\
\intertext{Then :}
f_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= \sum_{k=1}^\tau {\mbox{$\tilde{f}$}}_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})\end{aligned}$$
#### Reminder on RKHS norm
$$\begin{aligned}
\intertext{Let $k$ a kernel, and ${\mbox{$\mathcal{H}$}}$ the reproducing kernel Hilbert space (RKHS) corresponding to the kernel
$k$.}
\intertext{Recall that the RKHS norm of a function $f({\mbox{${\mbox{$\mathbf{x}$}}$}}) = \alpha^T k({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}})$ is :}
\norm{f}_{{\mbox{$\mathcal{H}$}}} &= \sqrt{\alpha^T k({\mbox{${\mbox{$\mathbf{X}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}) \alpha } &\\\end{aligned}$$
#### Reminder on Generalization and Rademacher Complexity {#par:recap-rademacher}
Consider a loss function $l : {\mbox{$\mathbb{R}$}}\times {\mbox{$\mathbb{R}$}}\rightarrow {\mbox{$\mathbb{R}$}}$. The population loss over the distribution ${\mbox{$\mathcal{D}$}}$, and the empirical loss over $n$ samples $D = \{({\mbox{${\mbox{$\mathbf{x}$}}$}}_i, y_i), i \in [n] \}$ from the same distribution ${\mbox{$\mathcal{D}$}}$ are defined as : $$\begin{aligned}
L_D(f) &= {\mathbb{E}}_{({\mbox{${\mbox{$\mathbf{x}$}}$}}, y) \sim {\mbox{$\mathcal{D}$}}} [l(f({\mbox{${\mbox{$\mathbf{x}$}}$}}), y)] &\\
L_S(f) &= \frac{1}{n} \sum_{i=1}^n l(f({\mbox{${\mbox{$\mathbf{x}$}}$}}_i), y_i)\end{aligned}$$
Suppose the loss function is bounded in $[0, c]$ and is $\rho-$Lipchitz in the first argument. Then, with probability at least $1 - \delta$ over sample $S$ of size n : $$\sup_{f \in {\mbox{$\mathcal{F}$}}} \{L_D(f) - L_S(f)\} \leq 2 \rho \hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}) + 3 c \sqrt{\frac{\log(2/\delta)}{2n}}$$
### Bounding the Rademacher Complexity
\[lem:rademacher-kernels\] Let $k_t : {\mbox{$\mathcal{X}$}}\cross {\mbox{$\mathcal{X}$}}\rightarrow {\mbox{$\mathbb{R}$}}, t \in [T]$ kernels such that : $$\sup_{{\mbox{${\mbox{$\mathbf{x}$}}$}}\in {\mbox{$\mathcal{X}$}}} \norm{k_t({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{x}$}}$}})} < \infty$$ To every kernel $k_t$, we associate a feature map $\phi_t : {\mbox{$\mathcal{X}$}}\rightarrow {\mbox{$\mathcal{H}$}}_t$, where ${\mbox{$\mathcal{H}$}}_t$ is a Hilbert space with inner product $\langle\cdot,\cdot\rangle_{{\mbox{$\mathcal{H}$}}_t}$, and for all ${\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime \in {\mbox{$\mathcal{X}$}}$, $k_t({\mbox{${\mbox{$\mathbf{x}$}}$}},
{\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime) = \langle \phi_t({\mbox{${\mbox{$\mathbf{x}$}}$}}), \phi_t({\mbox{${\mbox{$\mathbf{x}$}}$}}^\prime) \rangle_{{\mbox{$\mathcal{H}$}}_t}$
We define ${\mbox{$\mathcal{F}$}}$ as follows : $${\mbox{$\mathcal{F}$}}= \{ {\mbox{${\mbox{$\mathbf{x}$}}$}}\rightarrow \sum_{t=1}^T f_t({\mbox{${\mbox{$\mathbf{x}$}}$}}), \quad f_t({\mbox{${\mbox{$\mathbf{x}$}}$}}) = \alpha_t^T k_t({\mbox{${\mbox{$\mathbf{x}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_t) \quad \forall t \in
[T], \norm{f_t}_{{\mbox{$\mathcal{H}$}}_t} \leq B_t
\}$$ Let $X_1, ..., X_n$ be random elements of ${\mbox{$\mathcal{X}$}}$. Then for the class ${\mbox{$\mathcal{F}$}}$, we have : $$\hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}) \leq \sum_{t=1}^T \frac{2 B_t}{n_t} (Tr(k_t({\mbox{${\mbox{$\mathbf{X}$}}$}}_t, {\mbox{${\mbox{$\mathbf{X}$}}$}}_t)))^{1/2}$$
f() &= \_[t=1]{}\^T \_[i=1]{}\^[n\_t]{} \_i\^t k\_t(, \_i\^t) & , \^k\_t(, \^) &= \_t(), \_t(\^) \_[\_t]{} & f() &= \_[t=1]{}\^T \_[i=1]{}\^[n\_t]{} \_i\^t \_t(\_i\^t), \_t() \_[\_t]{} &\
&= \_[t=1]{}\^T \_[i=1]{}\^[n\_t]{} \_i\^t \_t(\_i\^t), \_t() \_[\_t]{} &\
\_[\_t]{}\^2 &= \_[i, j]{} \_i\^t \_j\^t k\_t(\_i\^t, \_j\^t) B\_t\^2 &\
&{ \_[t=1]{}\^T \_t, \_t() \_[\_t]{}, \_2 B\_t t } := & () &() &\
&= \[ \_[\_2 B\_t, t ]{} \_[t=1]{}\^T \_t, \_[i=1]{}\^[n\_t]{} \_i \_t(\_i\^t) \_[\_t]{} (\_t) \] &\
&= \_[t=1]{}\^T \[ \_[\_2 B\_t ]{} \_t, \_[i=1]{}\^[n\_t]{} \_i \_t(\_i\^t) \_[\_t]{} (\_t) \] &\
&\_[t=1]{}\^T (Tr(k\_t(\_t, \_t)))\^[1/2]{} &\
### Bounding $\norm{{\mbox{$\tilde{f}$}}_\tau^\star}_{{\mbox{$\mathcal{H}$}}_{\tau}}$ :
Let ${\mbox{$\mathcal{H}$}}_\tau$ the Hilbert space associated to the kernel $k_\tau$ .
\_\^() &= k\_(, \_)\^T \_&\
\_&= (k\_(\_, \_) + \_\^2 )\^[-1]{} \_&\
\_[\_]{}\^2 &\_\^T (k\_(\_, \_) + \^2 )\^[-1]{} \_&\
$$\begin{aligned}
\norm{{\mbox{$\tilde{f}$}}_\tau^\star}_{{\mbox{$\mathcal{H}$}}_{\tau}}^2 &= {\mbox{${\mbox{$\boldsymbol{\alpha}$}}$}}_\tau^T k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) {\mbox{${\mbox{$\boldsymbol{\alpha}$}}$}}_\tau &\\
&= {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau^T (k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) (k_\tau ({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})
^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau &\\ &\\
\intertext{Since $(k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}}) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} \leq k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}, {\mbox{${\mbox{$\mathbf{X}$}}$}})^{-1}$, we get :}
\norm{{\mbox{$\tilde{f}$}}_\tau^\star}_{{\mbox{$\mathcal{H}$}}_{\tau}}^2 &\leq {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^T k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) (k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau
\end{aligned}$$
### Proof of Lemma \[lem:train-unchanged\] {#subsubsec:proof-train-unchanged}
The intuition behind the proof is : since the gradient updates were performed orthogonally to the feature maps of the training data of the source task, the parameters in this space are unchanged, while the remaining space, which was changed, is orthogonal to these features maps, therefore, the inference is the same and the training error remanis the same as at the end of training on the source task.
$$\begin{aligned}
f_{T}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}) &= f_{T-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}) + \langle \phi_T({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}), {\mbox{${\mbox{$\mathbf{w}$}}$}}_T^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{T-1}^\star
\rangle &\\
&= f_{T-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}) + \langle \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f({\mbox{${\mbox{$\mathbf{w}$}}$}}_{T-1}^\star, {\mbox{${\mbox{$\mathbf{x}$}}$}}), {\mbox{${\mbox{$\mathbf{w}$}}$}}_T^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{T-1}^\star
\rangle &\\
\intertext{Since we the training on task ${\mbox{$\mathcal{T}$}}_T$ is perfomed with OGD, we have : }
\Pi_{{\mbox{$\mathbb{E}$}}_{T-1}}({\mbox{${\mbox{$\mathbf{w}$}}$}}_T^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{T-1}^\star) &= 0 &\\
\intertext{Since $\nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f({\mbox{${\mbox{$\mathbf{w}$}}$}}_{T-1}^\star, {\mbox{${\mbox{$\mathbf{x}$}}$}}) \in {\mbox{$\mathbb{E}$}}_{T-1}$ by definition, it follows that :}
f_{T}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}) &= f_{T-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}) &\\\end{aligned}$$
### Bounding the training error
\[lem:ogd-train-gen-bounds\] The training errors on the source and target tasks can be bounded as follows : $$\begin{aligned}
\intertext{Let $T \in {\mbox{$\mathbb{N}$}}$ fixed. Then, for all $\tau \in [T]$ }
\frac{1}{n_T}\sum_{i=1}^{n_T} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T,i}) - y_{T,i})^2 &\leq \frac{1}{n_T} \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T^T
(k_T({\mbox{${\mbox{$\mathbf{X}$}}$}}_T, {\mbox{${\mbox{$\mathbf{X}$}}$}}_T) )^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T &\\
\frac{1}{n_{T-1}} \sum_{i=1}^{n_{T-1}} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1,i}) - y_{T-1,i})^2 &\leq \frac{1}{n_{T-1}} \frac{\lambda^2}{4}
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}^T (k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1} &\\
\intertext{For all $\tau \in [T-2]$}
\frac{1}{n_{\tau}} \norm{f_T^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2 &\leq \frac{1}{n_{\tau}} (\frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau}^T (k_{\tau}
({\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} + &\\
&\sum_{k=\tau+1}^T \frac{1}{4 \lambda^2} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)
^{-1} k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k)
\end{aligned}$$
#### Task ${\mbox{$\mathcal{T}$}}_{T}$
$$\begin{aligned}
\intertext{We start from the definition of the training error :}
\sum_{i=1}^{n_T} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T,i}) - y_{T,i})^2
&= \norm{ (k_T({\mbox{${\mbox{$\mathbf{X}$}}$}}_T, {\mbox{${\mbox{$\mathbf{X}$}}$}}_T)^T (k_T({\mbox{${\mbox{$\mathbf{X}$}}$}}_T , {\mbox{${\mbox{$\mathbf{X}$}}$}}_T) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} - {\mbox{${\mbox{$\mathbf{I}$}}$}}) {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T }_2^2 &\\
\intertext{The expression is very similar to the previous norm, we can derive the same analysis as above to derive
the following bound :}
\sum_{i=1}^{n_T} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T,i}) - y_{T,i})^2 &\leq \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T^T (k_T({\mbox{${\mbox{$\mathbf{X}$}}$}}_T, {\mbox{${\mbox{$\mathbf{X}$}}$}}_T) )^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T &\\
\intertext{Therefore : }
\frac{1}{n_T}\sum_{i=1}^{n_T} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T,i}) - y_{T,i})^2 &\leq \frac{1}{n_T} \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T^T
(k_T({\mbox{${\mbox{$\mathbf{X}$}}$}}_T, {\mbox{${\mbox{$\mathbf{X}$}}$}}_T) )^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_T &\\\end{aligned}$$
#### Task ${\mbox{$\mathcal{T}$}}_{T-1}$
$$\begin{aligned}
\intertext{We start with the definition of the training error, then applying Lemma \ref{lem:train-unchanged} :}
\sum_{i=1}^{n_{T-1}} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1,i}) - y_{T-1,i})^2
&= \sum_{i=1}^{n_{T-1}} (f_{T-1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1, i}) - y_{T-1,i})^2 &\\
&= \norm{f_{T-1}^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{T-1}}_2^2 &\\
&= \norm{ k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1})^T (k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} - {\mbox{${\mbox{$\mathbf{I}$}}$}})
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}}_2^2 &\\
&= \norm{ - \lambda^2 (k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}}_2^2 &\\
&= \lambda^4 \norm{ (k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}}_2^2 &\\
&= \lambda^4 {\mbox{${\mbox{$\mathbf{y}$}}$}}_{T-1}^T (k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-2}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1} &\\
\intertext{Since : }
(k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) + \lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-2} &\leq \frac{1}{4 \lambda^2} k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1})^{-1}
\intertext{We get :}
\sum_{i=1}^{n_{T-1}} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1,i}) - y_{T-1,i})^2 &\leq \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}^T (k_{T-1}
({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1} &\\
\intertext{Therefore :}
\frac{1}{n_{T-1}} \sum_{i=1}^{n_{T-1}} (f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{T-1,i}) - y_{T-1,i})^2 &\leq \frac{1}{n_{T-1}} \frac{\lambda^2}{4}
{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}^T (k_{T-1}({\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{T-1}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{T-1}
\end{aligned}$$
#### Task ${\mbox{$\mathcal{T}$}}_1, ..., {\mbox{$\mathcal{T}$}}_{T-2}$
$$\begin{aligned}
\intertext{Let $\tau \in [T-2]$ fixed.}
\intertext{We recall that :}
f_T^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})
&= f_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + \sum_{k=\tau+1}^T {\mbox{$\tilde{f}$}}_k^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})
\intertext{Then :}
\norm{f_T^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2
&= \norm{f_\tau^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \sum_{k=\tau+1}^T {\mbox{$\tilde{f}$}}_k^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2 &\\
&\leq \norm{f_\tau^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2 + \norm{ \sum_{k=\tau+1}^T {\mbox{$\tilde{f}$}}_k^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) }^2_2 &\\
&\leq \underbrace{\norm{f_\tau^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2}_{{{ \tikz[baseline=(X.base)]
\node (X) [draw, shape=circle, inner sep=0] {\strut A};}}} + \underbrace{\sum_{k=\tau+1}^T \norm{
{\mbox{$\tilde{f}$}}_k^\star ({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) }^2_2}_{{{ \tikz[baseline=(X.base)]
\node (X) [draw, shape=circle, inner sep=0] {\strut B};}}}
&\\
\intertext{We can upper bound {{ \tikz[baseline=(X.base)]
\node (X) [draw, shape=circle, inner sep=0] {\strut A};}}similarly to the previous paragraphs, therefore we get :}
\norm{f_\tau^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2 &\leq \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau}^T (k_{\tau}
({\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} &\\
\intertext{Now, we upper bound {{ \tikz[baseline=(X.base)]
\node (X) [draw, shape=circle, inner sep=0] {\strut B};}}. Let $k \in [\tau+1, \space T]$ :}
\norm{{\mbox{$\tilde{f}$}}_k^\star ({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) }^2_2
&= {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T (k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) + \lambda_k^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^T (k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)
+ \lambda_k^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k &\\
&= \frac{1}{4 \lambda^2} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^{-1} \underbrace{k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)
^T}_{\text{Captures the similarity between the tasks ${\mbox{$\mathcal{T}$}}_\tau$ and ${\mbox{$\mathcal{T}$}}_k$}}
k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k &\\
\intertext{We conclude by plugging back the upper bounds of {{ \tikz[baseline=(X.base)]
\node (X) [draw, shape=circle, inner sep=0] {\strut A};}}and {{ \tikz[baseline=(X.base)]
\node (X) [draw, shape=circle, inner sep=0] {\strut B};}}}
\norm{f_T^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2 &\leq \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau}^T (k_{\tau}
({\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} +
\sum_{k=\tau+1}^T \frac{1}{4 \lambda^2} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)
^{-1} k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k
\intertext{Therefore :}
\frac{1}{n_{\tau}} \norm{f_T^\star({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) - {\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau}^2_2 &\leq \frac{1}{n_{\tau}} (\frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau}^T (k_{\tau}
({\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}, {\mbox{${\mbox{$\mathbf{X}$}}$}}_{\tau}) )^{-1}{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_{\tau} + \sum_{k=\tau+1}^T \frac{1}{4 \lambda^2} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)
^{-1} k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k) k({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^T k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k, {\mbox{${\mbox{$\mathbf{X}$}}$}}_k)^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k)\end{aligned}$$
### Bounding the Rademacher Complexity {#subsubsec:bound-rademacher}
$$\begin{aligned}
\intertext{The proof strategy is exactly the same as sec. \ref{subsec:proof-gen-ogd}. We generalize the previous
proof, by applying the same lemmas.}
{\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T} &= \{ {\mbox{${\mbox{$\mathbf{x}$}}$}}\rightarrow \sum_{\tau=1}^T f_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}), \quad f_\tau({\mbox{${\mbox{$\mathbf{x}$}}$}}) = {\mbox{${\mbox{$\boldsymbol{\alpha}$}}$}}_\tau^T k_\tau
({\mbox{${\mbox{$\mathbf{x}$}}$}},
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) \quad \forall \tau
\in [T], \norm{f_\tau}_{{\mbox{$\mathcal{H}$}}_\tau} \leq B_\tau\} &\\
\intertext{It holds that :}
f_T^\star &\in {\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T} &\\
\intertext{where, for all $\tau \in [T]$ :}
B_{\tau} &= \sqrt{({\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau})^T ( k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda^2_\tau {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} ({\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau})} &\\
\hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T}) &\leq \sum_{\tau=1}^T \frac{2(B_\tau + \epsilon) }{n_\tau} (Tr(k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)))^{1/2} &\\
\intertext{We made the assumption that for all $\tau \in [T]$ $\trace(k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau)) = {\mathcal{O}}(n_\tau)
$, also, by setting $\epsilon=1$ :}
\hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T}) &\leq \sum_{\tau=1}^T \frac{2(B_\tau + 1) }{n_\tau} {\mathcal{O}}(\sqrt{n_\tau}) &\\
&\leq \sum_{\tau=1}^T {\mathcal{O}}(\frac{B_\tau }{\sqrt{n_\tau}}) + \sum_{\tau=1}^T {\mathcal{O}}(
\frac{1}{\sqrt{n_\tau}}) &\\
&\leq \sum_{\tau=1}^T {\mathcal{O}}\left( \sqrt{\frac{({\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{\tau - 1 \rightarrow \tau})^T (
k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau , {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) + \lambda^2
{\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} ({\mbox{${\mbox{$\mathbf{y}$}}$}}_\tau - {\mbox{${\mbox{$\mathbf{y}$}}$}}_{ \tau - 1 \rightarrow \tau}) }{n_\tau}} \right) + \sum_{\tau=1}^T {\mathcal{O}}(
\frac{1}{\sqrt{n_\tau}}) &\\
\end{aligned}$$
### Proof of the Theorem \[thm:gen-ogd\]
$$\begin{aligned}
\intertext{With probability $1 - \delta$ we have :}
\sup_{f \in {\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T}} \{L_D(f) - L_S(f)\} &\leq 2 \rho \hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T}) + 3 c \sqrt{\frac{\log
(2/\delta)}{2n}} &\\
{\mbox{$\mathcal{L}$}}_{D_\tau}(f_T^\star) &\leq {\mbox{$\mathcal{L}$}}_{S_\tau}(f_T^\star) + 2 \rho \hat{{\mbox{$\mathcal{R}$}}}({\mbox{$\mathcal{F}$}}_{{\mbox{${\mbox{$\mathbf{B}$}}$}}_T}) + 3 c \sqrt{\frac{\log
(2/\delta) }{2n_T}}
&\\
{\mbox{$\mathcal{L}$}}_{D_\tau}(f_T^\star) &\leq {\mbox{$\mathcal{L}$}}_{S_\tau}(f_T^\star) + \sum_{\tau=1}^T {\mathcal{O}}\left( \sqrt{\frac{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^T
(k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau ,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) +
\lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau }{n_\tau}} \right) + \sum_{k=1}^T {\mathcal{O}}(\frac{1}{\sqrt{n_k}}) + 3 c
\sqrt{\frac{\log(2/\delta) }{2n_\tau}}
\intertext{We get Theorem \ref{thm:gen-ogd} by replacing into ${\mbox{$\mathcal{L}$}}_{S_\tau}(f_T^\star)$ using the
inequalities from Lemma \ref{lem:ogd-train-gen-bounds}}
\end{aligned}$$
Missing proofs of section \[sec:ogd-plus\] - OGD+ : Learning without forgetting {#sec:missing-proofs-of-sectionref--ogd+-:-learning-without-forgetting}
===============================================================================
Memorisation property of OGD+ - Proof {#subsubsec:proof-train-unchanged-ogd-plus}
-------------------------------------
$$\begin{aligned}
\intertext{In the proof of Theorem \ref{thm:conv}, App.~\ref{subsec:conv-multiple-proof}, we showed that, for
${\mbox{$\mathcal{T}$}}_\tau$ a fixed task:}
f_{\tau +1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) + \langle \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) ,{\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau+1}^\star
- {\mbox{${\mbox{$\mathbf{w}$}}$}}_{\tau}^\star \rangle. &\\
\intertext{We rewrite the recursive relation into a sum: }
f_{\tau +1}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= \sum_{k=1}^\tau \langle \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) ,{\mbox{${\mbox{$\mathbf{w}$}}$}}_{k+1}^\star
- {\mbox{${\mbox{$\mathbf{w}$}}$}}_{k}^\star \rangle. &\\
\intertext{We observe that, for all $k \in [T]$: }
{\mbox{${\mbox{$\mathbf{w}$}}$}}_{k+1}^\star - {\mbox{${\mbox{$\mathbf{w}$}}$}}_{k}^\star &\in {\mbox{$\mathbb{E}$}}_{k^\prime}.
\intertext{On the other hand, for OGD+, given a sample ${\mbox{${\mbox{$\mathbf{x}$}}$}}$ from ${\mbox{$\mathcal{D}$}}_\tau$, for all $k^\prime \in [\tau + 1, T]$ :}
\nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) \in {\mbox{$\mathbb{E}$}}_{k^\prime}
\intertext{Therefore, for all $k^\prime \in [k + 1, \tau]$ :}
\langle \nabla_{{\mbox{${\mbox{$\mathbf{w}$}}$}}} f_{k^\prime}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) ,{\mbox{${\mbox{$\mathbf{w}$}}$}}_{k^\prime+1}^\star- {\mbox{${\mbox{$\mathbf{w}$}}$}}_{k^\prime}^\star \rangle = 0
\intertext{Therefore :}
f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}) &= f_{k}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}})
\intertext{We conclude.}
\end{aligned}$$
OGD+ Generalisation Theorem - Proof {#subsec:proof-gen-ogd-plus}
-----------------------------------
$$\begin{aligned}
\intertext{The proof is very similar to the proof of Theorem \ref{thm:gen-ogd}.}
\intertext{Let ${\mbox{$\mathcal{T}$}}_\tau$ a given task and $T \in {\mbox{$\mathbb{N}$}}^\star$ fixed }
\intertext{We start from the following result in Appendix \ref{subsec:proof-gen-ogd}.}
{\mbox{$\mathcal{L}$}}_{D_\tau}(f_T^\star) &\leq {\mbox{$\mathcal{L}$}}_{S_\tau}(f_T^\star) + \sum_{k=1}^T {\mathcal{O}}\left( \sqrt{\frac{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T
(k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k ,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_k) +
\lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k }{n_k}} \right) + \sum_{k=1}^T {\mathcal{O}}(\frac{1}{\sqrt{n_k}}) + 3 c
\sqrt{\frac{\log(2/\delta) }{2n_\tau}}
\intertext{We apply Lemma \ref{lem:train-unchanged-ogd-plus} for tasks ${\mbox{$\mathcal{T}$}}_\tau$ and ${\mbox{$\mathcal{T}$}}_T$ :}
f_{T}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}) &= f_{\tau}^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau, i}).
\intertext{Therefore :}
{\mbox{$\mathcal{L}$}}_{S_\tau}(f_T^\star) &= {\mbox{$\mathcal{L}$}}_{S_\tau}(f_\tau^\star)
\intertext{We replace into the first inequality :}
{\mbox{$\mathcal{L}$}}_{D_\tau}(f_T^\star) &\leq {\mbox{$\mathcal{L}$}}_{S_\tau}(f_\tau^\star) + \sum_{k=1}^T {\mathcal{O}}\left( \sqrt{\frac{{\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k^T
(k_k({\mbox{${\mbox{$\mathbf{X}$}}$}}_k ,
{\mbox{${\mbox{$\mathbf{X}$}}$}}_k) +
\lambda^2 {\mbox{${\mbox{$\mathbf{I}$}}$}})^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_k }{n_k}} \right) + \sum_{k=1}^T {\mathcal{O}}(\frac{1}{\sqrt{n_k}}) + 3 c
\sqrt{\frac{\log(2/\delta) }{2n_\tau}}
\intertext{We recall the folowwing result from the proof of Theorem \ref{thm:gen-ogd} in App.
\ref{subsec:proof-gen-ogd} :}
\frac{1}{n_\tau}\sum_{i=1}^{n_\tau} (f_\tau^\star({\mbox{${\mbox{$\mathbf{x}$}}$}}_{\tau,i}) - y_{\tau,i})^2 &\leq \frac{1}{n_\tau} \frac{\lambda^2}{4} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau^\tau
(k_\tau({\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau, {\mbox{${\mbox{$\mathbf{X}$}}$}}_\tau) )^{-1} {\mbox{${\mbox{$\mathbf{\tilde{y}}$}}$}}_\tau &\\
\intertext{By replacing into the previous inequality, we conclude the proof of Theorem \ref{thm:gen-ogd-plus}.}\end{aligned}$$
Experiments : {#experiments}
=============
Reproducibility {#subsec:reproductibility}
---------------
### Code details
We implemented the code for the experiments in PyTorch. We initially forked the code from [https://github .com/GMvandeVen/continual-learning](https://github .com/GMvandeVen/continual-learning) . This source code is related to the works @vandeven2018generative and @vandeven2019three.
### Hyperparameters {#subsec:hyperparameters}
We use the same architecture and mostly the same hyperparameters as @farajtabar2019orthogonal. We also keep a small learning rate, in order to preserve the locality assumption of OGD, and in order to verify the conditions of the theorems.
The neural network is a three-layer MLP with 100 hidden units in two layers, each layer uses RELU activation function. The model has 10 logit outputs, which do not use any activation function. The optimiser is either SGD or OGD and the loss is Softmax cross-entropy. We report the hyperparameters in detail in Table \[tab:hyperparam\].
---------------- --------
Hyperparameter Value
Learning rate 1e-02
Batch size 256
Epochs 5
Torch seeds 0 to 4
Memory size 1000
Activation RELU
---------------- --------
: Hyperparameters used across experiments[]{data-label="tab:hyperparam"}
### Experiment setup {#subsec:experiment-setup}
We run each experiment 5 times, the seeds set is the same across all experiments sets. We report the mean and standard deviation of the measurements. The test error is measured every 50 mini-batch interval.
### OGD+ Implementation Details
In practice, we split the memories uniformly across tasks. Also, we construct ${\mbox{$\mathcal{S}$}}$ from ${\mbox{$\mathcal{S}$}}_{D}$ by sampling uniformly without replacement. Finally, for the memory reduction step, we truncate the last elements of the storage to free-up the space for the next task’s data.
Additional experiments
----------------------
### OGD+ Generalisation {#subsec:app-exp-gen-extended}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_1.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_5.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_2.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_6.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_3.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_7.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_4.png "fig:"){width="\textwidth"}
[.32]{} ![Test accuracy on the 10 first tasks of permuted MNIST, for SGD, OGD and OGD+. 15 different permutations are used, and the model is trained to classify MNIST digits for 5 epochs for each permutation. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:gen"}](figs/out/gen/val_acc_8.png "fig:"){width="\textwidth"}
### OGD+ Overparameterization
[.50]{} ![Test accuracy on the 3 first tasks of permuted MNIST, for the MLP’s hidden dimensions 100, 250 and 400. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:overparam"}](figs/out/hidden/val_acc_1.png "fig:"){width="\textwidth"}
[.50]{} ![Test accuracy on the 3 first tasks of permuted MNIST, for the MLP’s hidden dimensions 100, 250 and 400. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:overparam"}](figs/out/hidden/val_acc_2.png "fig:"){width="\textwidth"}
[.50]{} ![Test accuracy on the 3 first tasks of permuted MNIST, for the MLP’s hidden dimensions 100, 250 and 400. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:overparam"}](figs/out/hidden/val_acc_3.png "fig:"){width="\textwidth"}
### OGD+ Memory size
[.50]{} ![Test accuracy on the 3 first tasks of permuted MNIST, for the memory sizes 1.000, 1.500 and 2.000. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:memory"}](figs/out/mem/val_acc_1.png "fig:"){width="\textwidth"}
[.50]{} ![Test accuracy on the 3 first tasks of permuted MNIST, for the memory sizes 1.000, 1.500 and 2.000. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:memory"}](figs/out/mem/val_acc_2.png "fig:"){width="\textwidth"}
[.50]{} ![Test accuracy on the 3 first tasks of permuted MNIST, for the memory sizes 1.000, 1.500 and 2.000. The y-axis is truncated for clarity. We report the mean and standard deviation over 5 independent runs. The test error is measured for every 50 mini-batch interval. The vertical dashed lines separate the 15 tasks.[]{data-label="fig:app:memory"}](figs/out/mem/val_acc_3.png "fig:"){width="\textwidth"}
|
---
bibliography:
- '../../references/references.bib'
---
|
---
abstract: 'We make a detailed measurement of the metal abundance profiles and metal abundance ratios of the inner core of M87/Virgo observed by *XMM-Newton* during the PV phase. We use multi temperature models for the inner regions and we compare the plasma codes APEC and MEKAL. We confirm the strong heavy elements gradient previously found by *ASCA* and *BeppoSAX*, but also find a significant increase in light elements, in particular O. This fact together with the constant O/Fe ratio in the inner 9 arcmin indicates an enhancement of contribution in the core of the cluster not only by SNIa but also by SNII.'
author:
- Fabio Gastaldello
- Silvano Molendi
title: Abundance gradients and the role of SNe in M87
---
Introduction
============
The X-ray emitting hot intra-cluster medium (ICM) of clusters of galaxies is known to contain a large amount of metals: for rich clusters between red-shift 0.3-0.4 and the present day the observed metallicity is about 1/3 the solar value [@mush97; @fuka98; @allen98; @ceca00; @ettori01], suggesting that a significant fraction of the ICM has been processed into stars already at intermediate red-shifts.
While the origin of the metals observed in the ICM is clear (they are produced by supernovae), less clear is the transfer mechanism of these metals to the ICM. The main mechanisms that have been proposed for the metal enrichment in clusters are: enrichment of gas during the formation of the proto-cluster [@kauff98]; ram pressure stripping of metal enriched gas from cluster galaxies [@gunn72; @toni01]; stellar winds AGN- or SN-induced in Early-type galaxies [@matteucci88; @renzini97].
Spatially resolved abundance measurement in galaxy clusters are of great importance because they can be used to measure the precise amounts of metals in the ICM and to constrain the origin of metals both spatially and in terms of the different contributions of the two different type of SNe (SNII and SNIa) as a function of the position in the cluster. The first two satellites able to perform spatially resolved spectroscopy, *ASCA* and *BeppoSAX* have revealed abundance gradients in cD clusters [@dupke00b; @grandi01], in particular M87/Virgo [@matsumoto96; @guainazzi00], and variations in Si/Fe within a cluster [@fino00] and among clusters [@fuka98]. Since the SNIa products are iron enriched, while the SNII products are rich in $\alpha$ elements, such as O, Ne, Mg and Si, the variations in Si/Fe suggest that the metals in the ICM have been produced by a mix of the two types of SNe. The exact amount of these mix still remain controversial: @mush96 and @mush97 showed a dominance of SNII ejecta, while other works on *ASCA* data [@ishimaru97; @fuka98; @fino00; @dupke00a], still indicating a predominance of SNII enrichment at large radii in clusters, do not exclude that as much as 50% of the iron in clusters come from SNIa ejecta in the inner part of clusters.
M87 is the cD galaxy of the nearest cluster and its high flux and close location allows, using the unprecedented combination of spectral and spatial resolution and high throughput of the EPIC experiment on board *XMM-Newton*, a detailed study of the ICM abundance down to the scale of the kpc. Throughout this paper, we assume $H_{0}\,=\, 50\, \rm{km\, s^{-1} Mpc^{-1}}$, $q_{0} = 0.5$ and at the distance of M87 $1^{\prime}$ corresponds to 5 kpc.
Recently, when we were finishing the writing of the paper, we have learned of the @fino01 analysis of the same data. Our work is complementary with theirs in the sense that we use 10 annular bins instead of 2, fully exploiting the quality of the *XMM* data and taking into account the multi temperature appearance of spectra in the inner regions with adequate spectral modeling.
The outline of the paper is as follows. In section 2 we give information about the *XMM* observation and on the data preparation. In section 3 we suggest the use of a unique set of “standard” abundances. In section 4 we describe our spectral modeling. In section 5 we present spatially resolved measurements of metal abundances and abundance ratios and in section 6 we discuss our results. A summary of our conclusions is given in section 7.
Observation and Data Preparation
================================
M87/Virgo was observed with *XMM-Newton* [@jansen01] during the PV phase with the MOS detector in Full Frame Mode for an effective exposure time of about 39 ks. Details on the observation have been published in @boh01 and @belsole01. We have obtained calibrated event files for the MOS1 and MOS2 cameras with SASv5.0. Data were manually screened to remove any remaining bright pixels or hot column. Periods in which the background is increased by soft proton flares have been excluded using an intensity filter: we rejected all events accumulated when the count rates exceeds 15 cts/100s in the \[10 – 12\] keV band for the two MOS cameras.
We have accumulated spectra in 10 concentric annular regions centered on the emission peak extending our analysis out to 14 arcmin from the emission peak, thus exploiting the entire *XMM* field of view. We have removed point sources and the substructures which are clearly visible from the X-ray image [@belsole01] except in the innermost region, where we have kept the nucleus and knot A, because on angular scales so small it is not possible to exclude completely their emission. We prefer to fit the spectrum of this region with a model which includes a power law component to fit the two point like sources. We include only one power law component due to the similarity of the two sources spectra [@boh01]. The bounding radii are 0$^{\prime}$-0.5$^{\prime}$, 0.5$^{\prime}$-1$^{\prime}$, 1$^{\prime}$-2$^{\prime}$, 2$^{\prime}$-3$^{\prime}$, 3$^{\prime}$-4$^{\prime}$, 4$^{\prime}$-5$^{\prime}$, 5$^{\prime}$-7$^{\prime}$, 7$^{\prime}$-9$^{\prime}$, 9$^{\prime}$-11$^{\prime}$ and 11$^{\prime}$-14$^{\prime}$. The analysis of the 4 central regions within 3 arcmin was already discussed in @molegasta01.
Spectra have been accumulated for MOS1 and MOS2 independently. The Lockman Hole observations have been used for the background. Background spectra have been accumulated from the same detector regions as the source spectra.
The vignetting correction has been applied to the spectra rather than to the effective area, as is customary in the analysis of EPIC data [@arnaud01]. Spectral fits were performed in the 0.5-4.0 keV band. Data below 0.5 keV were excluded to avoid residual calibration problems in the MOS response matrices at soft energies. Data above 4 keV were excluded because of substantial contamination of the spectra by hotter gas emitting further out in the cluster, on the same line of sight.
As discussed in @molendi01 there are cross-calibration uncertainties between the spectral response of the two EPIC instruments, MOS and PN. In particular for what concern the soft energy band (0.5-1.0 keV) fitting six extra-galactic spectra for which no excess absorption is expected, MOS recovered the $\rm{N_{H}}$ galactic value, while PN gives smaller $\rm{N_{H}}$ by $1-2\times10^{20}\rm{cm^{-2}}$. Thus we think that at the moment the MOS results are more reliable than the PN ones in this energy band, which is crucial for the O abundance measure. For this reason and for the better spectral resolutions of MOS, which is again important in deriving the O abundance, we limit our analysis to MOS data.
Solar Abundances
================
The elemental abundances of astrophysical objects are usually expressed by the relative values to the solar abundances. The so-called solar abundances can be either “meteoritic” or “photospheric”.
This distinction between “meteoritic” and “photospheric” solar abundances was made in the review by @anders89. Significant discrepancies exist between the two sets of abundances quoted in that paper, particularly for iron and this has caused in the past some controversy in the discussion of the results of cluster abundances [@ishimaru97; @gibson97]. However recent photospheric models of the sun indicate that photospheric and meteoritic abundances agree perfectly and the community has converged toward a “standard solar composition” [@grevesse98], with suggestions to the astrophysical community to accept this new state of the art [@brighenti99]. For the above reason, in this paper we shall adopt the @grevesse98 values. Since the solar abundance table used by default in XSPEC is based on photospheric values of @anders89, we have switched to a table taken from the data by @grevesse98 by means of the XSPEC command ABUND. In general a simple scaling allows to switch from one set of abundances to the other.
Spectral modeling and plasma codes
==================================
All spectral fitting has been performed using version 11.0.1 of the XSPEC package.
All models discussed below include a multiplicative component to account for the galactic absorption on the line of sight of M87. The column density is always fixed at a value of $1.8\times 10^{20}\,\rm{cm^{-2}}$, which is derived from 21cm measurements [@lieu96]. Leaving $\rm{N_{H}}$ to freely vary does not improve the fit and does not affect the measure of the oxygen abundance, which could have been the more sensitive to the presence of excess absorption. The $\rm{N_{H}}$ value obtained is consistent within the errors with the 21cm value.
The temperature profile for M87 [@boh01] shows a small gradient for radii larger than $\sim 2$ arcmin and a rapid decrease for smaller radii. Moreover, as pointed out in @molepizzo01 all spectra at radii larger than 2 arcmin are characterized by being substantially isothermal (although the spectra of the regions between 2 and 7 arcmin are multi temperature spectra with a narrow temperature range rather than single temperature spectra), while at radii smaller than 2 arcmin we need models which can reproduce the broad temperature distribution of the inner regions.
We therefore apply to the central regions (inside 3 arcmin) three different spectral models.
A two temperature model (vmekal + vmekal in XSPEC and model II in @molegasta01 using the plasma code MEKAL [@mewe85; @liedahl95]. This model has 15 free parameters: the temperature and the normalization of the two components and the abundance of O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Fe and Ni, all expressed in solar units. The metal abundance of each element of the second thermal component is bound to be equal to the same parameter of the first thermal component. This model is used (e.g. @makishima01 and refs. therein) as an alternative to cooling-flow models in fitting the central regions of galaxy clusters.
A “fake multi-phase” model (vmekal + vmcflow in XSPEC and model III in @molegasta01. This model has 15 free parameters, as the two temperature model, because the maximum temperature $T_{max}$ is tied to the vmekal component temperature. As indicated in recent papers [@molepizzo01; @molegasta01] this model is used to describe a scenario different from a multi-phase gas, for which it was written for: the gas is all at one temperature and the multi-phase appearance of the spectrum comes from projection of emission from many different physical radii. A more correct description will be given by a real deprojection of the spectrum (Pizzolato et al., in preparation).
The third model is the analogue of the vmekal two temperature model using the plasma code APEC [@smith01]. This model has 14 free parameters, one less than the corresponding model using vmekal because APEC misses the Na parameter.
We can’t adopt an APEC analogue of the fake multi-phase model because the cooling flow model calculating its emission using APEC is still under development. Given the substantial agreement between 2T and fake multi-phase model [@molegasta01], we can regard the 2T APEC results as indicative also for a fake multi-phase model.
For the spectrum accumulated in the innermost region we included also a power law component to model the emission of the nucleus and of knot A.
For the outer regions (from 3 arcmin outwards) we apply single temperature models: vmekal using the MEKAL code, with 13 free parameters and vapec using the APEC code, with 12 free parameters.
As pointed out by the authors of the new code, cross-checking is very important, since each plasma emission code requires choosing from a large overlapping but incomplete set of atomic data and the results obtained by using independent models allows critical comparison and evaluation of errors in the code and in the atomic database.
Results
=======
Abundance measurements and modeling concerns
--------------------------------------------
The X-ray emission in cluster of galaxies originates from the hot gas permeating the cluster potential well. The continuum emission is dominated by thermal bremsstrahlung, which is proportional to the square of the gas density times the cooling function. From the shape and the normalization of the spectrum we derive the gas temperature and density. In addition the X-ray spectra of clusters of galaxies are rich in emission lines due to K-shell transitions from O, Ne, Mg, Si, S, Ar and Ca and K- and L-shell transitions from Fe and Ni, from which we can measure the relative abundance of a given element.
In Figure \[lines\] we show the data of the 3$^{\prime}$-4$^{\prime}$ bin together with the best fitting model calculated using the MEKAL code. The model has been plotted nine times, each time all element abundances, except one, are set to zero. In this way the contribution of the various elements to the observed lines and line blends become apparent. In the energy band (0.5-4 keV) we have adopted for the spectral fitting, the abundance measurements based on K-lines for all the elements except for Fe and Ni, for which the measure is based on L-lines. The K-lines of O, Si, S, Ar and Ca are well isolated from other emission features and clearly separated from the continuum emission, which are the requirements for a robust measure of the equivalent width of the lines and consequently of the abundances of these elements. The Fe-L lines are known to be problematic, because the atomic physics involved is more complicated than K-shell transitions [@liedahl95], but from the very good signal of *XMM* spectra and from the experience of *ASCA* data [@mush96; @hwang97; @fuka98] we can conclude that the Fe-L determination is reliable. Some of the stronger Fe-L lines due to Fe XXII and Fe XXIV are close to the K-lines of Ne and Mg, respectively and blending can lead to errors in the Ne abundance and, to a smaller extent, to the Mg abundance [@liedahl95; @mush96]. Also the Ni measure is difficult due to the possible confusion of its L-lines with the continuum and Fe-L blend.
Abundance profiles
------------------
In Figure \[z1\] we report the MOS radial abundance profiles for O (top panel), Si (middle panel) and Fe (bottom panel), in Figure \[z2\] those for Mg (top panel), Ar (middle panel) and S (bottom panel), in Figure \[z3\] those for Ne (top panel), Ca (middle panel) and Ni (bottom panel). We note that the measurements obtained using the two different plasma codes agree for what concerns Fe, Ar and Ca; they are somewhat different for what concerns O, S and Ni and in complete disagreement for Mg and Ne.
The temperature profile obtained with the two codes is showed in Figure \[temperature\]: there are some differences in the inner regions, while in the outer isothermal bins there is substantially agreement.
The models using the APEC code give a systematically worse description than the ones using MEKAL code: for the multi temperature models with APEC the $\chi^{2}$ range from 403 to 498 for $\nu = 218$ (216 in the central bin due to the two additional degree of freedom of the power-law component), while for MEKAL models (2T and fake multi-phase give the same results) the $\chi^{2}$ range from 323 to 382 for $\nu = 217$ (215 in the central bin due to the two additional degree of freedom of the power-law component); for single temperature models with APEC the $\chi^{2}$ range from 546 to 886 for $\nu = 222$, while for MEKAL models the $\chi^{2}$ range from 326 to 731 for $\nu = 221$. In Figure \[confapecmekal\] we compare the residuals in the form of $\Delta\chi^{2}$ between a 2T model using the MEKAL code and the APEC code for the inner bin 1$^{\prime}$-2$^{\prime}$, as well as a 1T model using the MEKAL code and the APEC code for the “isothermal” bins 3$^{\prime}$-4$^{\prime}$ and 11$^{\prime}$-14$^{\prime}$. It’s evident that in the external bins the differences in the fit between the two codes are due to APEC over-prediction of the flux of Fe-L lines from high ionization states, considering the fact that the temperature obtained by the two codes are nearly coincident. For the inner regions the differences between the two codes are further complicated by the different temperature range they find for the best fit. In general, where the temperature structure is very similar, as in the innermost bin, the difference is as in the outer bins in the high energy part of the Fe-L blend, while where the temperature structure is different, as in the 1$^{\prime}$-2$^{\prime}$ shown in Figure \[confapecmekal\], the differences between the two codes are primary due to different estimates of the flux of He-like Si-K line.
We therefore choose as our best abundance profiles those obtained with a 2T vmekal fit for the central regions and with a 1T vmekal fit for the outer regions. In Table \[abundances\] we report the abundance profiles for O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Fe and Ni obtained in this way. Abundance gradients are clearly evident for Fe, Si, S, Ar and Ca; O, Mg and Ni show evidence for an enhancement in the central regions while only Ne is substantially flat.
Comparison with Finoguenov et al. (2001) results
------------------------------------------------
We made a direct comparison of our results with the abundances and abundance ratios derived by @fino01. Their values are consistent within the errors with ours except for the oxygen abundance, which is roughly two times higher in Finoguenov’s analysis. To better understand the origin of this discrepancy we have extracted spectra from the same radial bins, 1$^{\prime}$-3$^{\prime}$ and 8$^{\prime}$-14$^{\prime}$, and fitted a single temperature model (vmekal) in the 0.5-10 keV band, excising the 0.7-1.6 keV energy range, in order to avoid the dependence from the Fe L-shell peak, as done by @fino01. Our results for the O abundance, given in units of the solar values from @anders89 to make a direct comparison with the results of Table 1 of @fino01, are $0.32\pm0.03$ for the 1$^{\prime}$-3$^{\prime}$ bin and $0.20\pm0.02$ for the 8$^{\prime}$-14$^{\prime}$ bin, to be compared with $0.535^{+0.019}_{-0.021}$ and $0.386^{+0.025}_{-0.021}$. Also interesting are the results for the Ni abundance: $2.55\pm0.33$ for the 1$^{\prime}$-3$^{\prime}$ bin and $2.34\pm0.26$ for the 8$^{\prime}$-14$^{\prime}$ bin, to be compared with $2.573^{+0.924}_{-0.918}$ and $0.800^{+1.732}_{-0.800}$. We also note that a 2T modeling of the inner bin 1$^{\prime}$-3$^{\prime}$ gives a statistically better fit over a single temperature model ( $\chi^{2}$/d.o.f of 919/440 respect to 1083/442, using the 0.5-10 keV band) breaking down the assumption of near isothermality.
We also cross compare the K- and L-shell results for Ni and Fe in our analysis, performing the spectral fits for all the radial bins in the 0.5-10 keV band, but excising the 0.7-1.6 keV energy range. The derived abundances for the two elements are consistent within $1\sigma$, although the K-shell Ni abundance is 20$\%$ higher than the L-shell abundance in the bins fitted with a single temperature model. It should be borne in mind that particularly for the outer bins the K-shell Ni abundance measure is very sensitive to the background estimate.
Abundance ratios and SNIa Fe mass fraction
------------------------------------------
From the abundance measurements we obtain the abundance ratios between all the elements relative to Fe, normalized to the solar value. They are shown in Figure \[ratio1\], Figure \[ratio2\], Figure \[ratio3\] and in Table \[ratios\], together with the abundance ratios obtained by models of supernovae taken by @nomoto97 and rescaled to the solar abundances reported in @grevesse98. We use those abundance ratios to estimate the relative contributions of SNIa and SNII to the metal enrichment of the intra-cluster gas. Such estimates are complicated by uncertainties both in the observations and in the theoretical yields. Our approach is to use the complete set of ratios trying to find the best fit of the function
$$\label{fit}
\Big(\frac{X/Fe}{X_{\odot}/Fe_{\odot}}\Big)_{observed}\,=\,f\,\Big(\frac{X/Fe}{X_{\odot}/Fe_{\odot}}\Big)_{SNIa}+(1-f)\,\Big(\frac{X/Fe}{X_{\odot}/Fe_{\odot}}\Big)_{SNII}$$
where $\Big(\frac{X/Fe}{X_{\odot}/Fe_{\odot}}\Big)_{observed}$ is the measured abundance ratio of the $X$ element to $Fe$, given in solar units, $\Big(\frac{X/Fe}{X_{\odot}/Fe_{\odot}}\Big)_{SNIa}$ and $\Big(\frac{X/Fe}{X_{\odot}/Fe_{\odot}}\Big)_{SNII}$ are the theoretical abundance ratio by the two types of supernovae, also given in solar units and $f$ is directly the SNIa Fe mass fraction. The result of the simultaneous fit of the eight ratios is presented as circles in Figure \[sniamassfractio\], using the SNII model by @nomoto97 and W7, WDD1 and WDD2 models for SNIa respectively for the three panels. Due to the large uncertainties in the yields for the SNII model, the results are strongly SNII model dependent. For comparison we use the range of SNII yields calculated by @gibson97, also listed in Table \[ratios\], which involve only ratios for O, Ne, Mg, Si and S and as before finding the best fit for eq.(\[fit\]). The results are shown as squares for the lower end of the range and triangles for the upper part of the range.
The best fits are obtained in the inner bins with a combination of the WDD2 model for SNIa and the Nomoto model for SNII, with reduced $\chi^{2}$ which ranges from 2 in the inner bin up to 10 in the 4$^{\prime}$-5$^{\prime}$ bin. In the outer bins the fit is slightly better (reduced $\chi^{2}$ of 8-10 instead of 12-13) with a combination of the W7 model for SNIa and the Nomoto model for SNII. This is shown in Figure \[fitsnia\] where the fits with the three SNIa models together with the Nomoto SNII model are reported for the 0.5$^{\prime}$-1$^{\prime}$ bin and for the 11$^{\prime}$-14$^{\prime}$ bin. It’s clear from the inspection of the residuals in terms of $\Delta\chi^{2}$ that the combination of W7 model and Nomoto SNII model fails in the inner bins because it predicts a higher Ni/Fe and O/Fe ratio, compared to the delayed detonation models. The situation is the opposite for the outer bins where a higher Ni/Fe and a lower S/Fe favors the W7 model. However the preference for the W7 model in combination with the Nomoto SNII model in the outer bins is strongly dependent on the Ni/Fe ratio: if we exclude it from the fit the WDD2 model provides the better fit to the data.
Discussion
==========
The model emerging from the *ASCA* and *BeppoSAX* data for the explanation of abundance gradients in galaxy clusters was that of a homogeneous enrichment by SNII, the main source of $\alpha$ elements, maybe in the form of strong galactic winds in the proto-cluster phase and the central increase in the heavy element distribution due to an enhanced contribution by SNIa, strongly related to the presence of a cD galaxy [@fuka98; @dupke00b; @fino00; @grandi01; @makishima01]. As a textbook example we can consider the case of A496 observed by *XMM-Newton* [@tamura01]. The O-Ne-Mg abundance is radially constant over the cluster, while the excess of heavy elements as Fe, Ar, Ca and Ni in the core is consistent with the assumption that the metal excess is solely produced by SNIa in the cD galaxy. The crucial ratio for the discrimination of the enrichment by the two types of supernovae, O/Fe, is then decreasing towards the center.
The *XMM* results for M87/Virgo question this picture. They confirm and improve the accuracy of the measure of heavy elements gradients previously found by *ASCA* and *BeppoSAX*, but they also show a a statistically significant enhancement of $\alpha$ elements O and Mg in the core. If we consider the inner 9 arcmin the ratio O/Fe is constant with $\chi^{2}=6.9$ for 7 d.of. and adding a linear component does not improve the fit ($\chi^{2}=5.4$ for 6 d.o.f). These facts points toward an increase in contribution also of SNII, since O is basically produced only by this kind of supernovae. Although there is little or no evidence of current star formation in the core of M87, the O excess could be related to a recent past episode of star formation triggered by the passage of the radio jet, as we see in cD galaxies with a radio source (A1795 cD: van Breugel et al. 1984 and A2597 cD: Koekemoer et al. 1999), nearby (Cen A: Graham 1998) and distant radio galaxies [@vanbreugel85; @vanbreugel93; @bicknell00] (for a comprehensive discussion see McNamara 1999). To put the above idea quantitatively, waiting for a true deprojection of our data, we use previous *ROSAT* estimate of the deprojected electron density in the center of the Virgo cluster [@nulsen95] to calculate the excess mass of oxygen. To estimate the excess abundance we fit the inner bins, where we see the stronger increase in the O abundance, with a constant obtaining an abundance of 0.32, while for the outer bins we obtain an abundance of 0.21 so the excess is 0.11. Then we estimate the oxygen mass, $M_{\rm{O}}$ to be $M_{\rm{O}} = A_{\rm{O}}\,y_{\rm{O},\odot}\,Z_{\rm{O}}^{excess}\,M_{H}$, where $M_{H} = 0.82\,n_{e}\,\frac{4\pi}{3}\,(R_{out}-R_{in})^{3}$, $R_{in}$ and $R_{out}$ being the bounding radii in kpc of the bins, $A_{\rm{O}}= 16$ and $y_{\rm{O},\odot} = 6.76 \times 10^{-5}$ from @grevesse98. We obtain a rough estimate of $10^{7}\,\rm{M_{\odot}}$ which, assuming that about $80\,\rm{M_{\odot}}$ of star formation are required to generate a SNII [@thomas90] and Nomoto SNII oxygen yield, requires a cumulative star formation of $4-5 \times 10^{7}\,\rm{M_{\odot}}$. This star formation is in agreement with a burst mode of star formation ($\lesssim 10^{7}$ yr) at rates of $\sim 10-40\,\rm{M_{\odot}\,yr^{-1}}$, as it is observed in the CD galaxies of A1795 and A2597 [@mcnamara99].
For what concern heavy element gradients and the contribution of SNIa, M87 data suggests an agreement with delayed detonation models (in particular for the inner bins), as stressed by @fino01, in contrast with the preference of W7 model set by the high Ni/Fe ratios found by @dupke00a. This is particular evident if we consider the S/Fe ratio in Figure \[ratio1\]. If we consider the set of theoretical values for SNII and W7 SNIa models the behavior of these ratio would indicate an increasing contribution by SNIa going *outward* to the center. We recover the correct behavior if we choose the WDD1 yield and we reduce the S SNII yield of @nomoto97 by a factor of two to three, as was already indicated by *ASCA* data [@dupke00a]. We caution however that this is a substantial contribution larger than that allowed by the SNII models choosed by @gibson97. The use of delayed detonation model for SNIa could also explain the over abundance of S and also Si (respect to the W7 model) found by @tamura01 for the core of A496. With increasing radius the W7 model gives a better fit to the data respect to delayed detonation models. This fact indicates a SNIa abundance pattern change with radius and could be taken as an independent X-ray confirmation of the conclusions of @hatano00 on the optical spectroscopic diversity of SNIa, as suggested by @fino01. However we stress that the preference of W7 over WDD models in the outer bins is entirely due to the Ni/Fe ratio which could be affected by systematic uncertainties, as discussed in section 5.1 and 5.3.
In Figure \[sniamassfractio\], we show the relative importance of SNIa, measured by the Fe mass fraction provided by this kind of supernovae. This is substantially constant through the 14 arcmin analyzed. This fraction is considerable and ranges between the 50% and 80% and it depends only slightly on the SNIa model used. Instead the uncertainties involved in using different SNII models are large and a definitive answer cannot be reached until further convergence of SNII models is achieved.
Summary
=======
We have performed a spatially resolved measurement of the element abundances in M87, the Virgo cluster cD galaxy. The main conclusion of our work are:
- the APEC code gives a systematically worse description than the MEKAL code in modeling M87 spectra;
- we confirm the increase of Fe and other heavy elements towards the core indicating that the SNIa contribution increases;
- the increase in O abundance and a constant O/Fe ratio in the inner 9 arcmin indicates an increase also in SNII ejecta possibly from star-burst in the recent past;
- Si/Fe and S/Fe profiles favor WDD models over W7, also requiring substantial reduction of the SNII yield of S;
- the indication of a change of the SNIa abundance pattern, provided by a preference of the W7 model over delayed detonation models in the outer bins, is entirely due to the Ni/Fe ratio. Since the Ni measurement is difficult and uncertain, this indication should be taken with some caution.
S. Ettori and S. Ghizzardi are thanked for useful discussions and suggestions. We thank the referee for several suggestions that improved the presentation of this work. This work is based on observations obtained with *XMM-Newton*, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA).
Allen, S. W. & Fabian, A. C. 1998, , 297, L63
Anders, E. & Grevesse, N. 1989, Geochimica et Cosmochimica Acta, 53, 197
Arnaud, M., Neumann, D. M., Aghanim, N., Gastaud, R., Majerowicz, S., Hughes, J. P. 2001, A&A, 365, L80
Belsole, E., Sauvageot, J. L., Böhringer, H., Worral, D. M., Matsushita, K., Mushotzky, R. F., Sakelliou, I., Molendi, S., Ehle, M., Kennea, J., Stewart, G., Vestrand, W. T. 2001, A&A, 365, L188
Bicknell, G., Sutherland, R., van Breugel, W.J.M., Dopita, M.A., Dey, A, Miley, G.K. 2000, ApJ, 540, 678
Böhringer, H., Belsole, E., Kennea, J., Matsushita, K., Molendi, S., Worral, D. M., Mushotzky, R. F., Ehle, M., Guainazzi, M., Sakelliou, I., Stewart, G., Vestrand, W. T., Dos Santos, S. 2001, A&A, 365 L181
Brighenti, F. & Mathews, W. G. 1999, ApJ, 515, 542
De Grandi, S. & Molendi, S. 2001, ApJ, 551, 153
Della Ceca, R., Scaramella, R., Gioia, I. M., Rosati, P., Fiore, F., Squires, G. 2000, A&A, 353, 498
Dupke, R. A. & White R. 2000a, ApJ, 528, 139
Dupke, R. A. & White R. 2000b, ApJ, 537, 123
Ettori, S., Allen, S. W., Fabian, A. C. 2001, MNRAS, 322, 187
Finoguenov, A., David, L. P., Ponman, T. J. 2000, ApJ, 544, 188
Finoguenov, A., Matsushita, K., Böhringer, H., Ikebe, Y., Arnaud, M. 2001, A&A in press, (astro-ph/0110516)
Fukazawa, Y., Makishima, K., Tamura, T., Ezawa, H., Xu, H., Ikebe, Y., kikuchi, K., Ohashi, T. 1998, PASJ, 50, 187
Gibson, B. K., Lowenstein, M., Mushotzky, R. F. 1997, MNRAS, 290, 623
Graham, J.A., 1998, ApJ, 502, 245
Grevesse, N. & Sauval, A. J., 1998, Space Science Reviews, 85, 161
Guainazzi, M. & Molendi, S. 2000, A&A, 351, L19
Gunn, J. E. & Gott, J. R. 1972, ApJ, 176, 1
Hatano, K., Branch, D., Lents, E.J., Baron, E., Filippenko, A.V., Garnavich, P.M., 2000, ApJ, 543, L9
Hwang, U., Mushotzky R., Loewenstein, M., Markert, T. H., Fukazawa, Y., Matsumoto, H. 1997, ApJ, 476, 560
Kauffmann, G. & Charlot, S. 1998, MNRAS, 294, 705
Koekemoer, A.M., O’Dea, C.P., Sarazin, C.L., McNamara, B.R., Donahue, M., Voit, G.M., Baum, S.A., Gallimore, J.F., ApJ, 525, 621
Jansen, F., Lumb, D., Altieri, B., Clavel, J., Ehle, M., Erd, C., Gabriel, C., Guainazzi, M., Gondoin, P., Much, R., Munoz, R., Santos, M., Schartel, N., Texier, D., Vacanti, G. 2001, A&A, 365, L1
Ishimaru, Y. & Arimoto, N. 1997, PASJ, 49,1
Liedahl, D. A., Osterheld A. L., Goldstein, W. H. 1995, ApJ, 438, L115
Lieu, R., Mittaz, J. P. D., Bowyer, S., Lockman, F. J., Hwang,, C. Y., Schmitt, J. H. M. M. 1996, ApJ, 458, L5
Makishima, K., Ezawa, H., Fukazawa, Y., Honda, H., Ikebe, Y., Kamae, T., Kikuchi, K., Matsushita, K., Nakazawa, K., Ohashi, T., Takahashi, T., Tamura, T., Xu, H. 2001, PASJ, 53, 401
Matsumoto, H., Koyama, K., Awaki, H., Tomida, H., Tsuru, T., Mushotzky, R., Hatsukade, I. 1996, PASJ, 48, 201
Matteucci, F. & Vettolani, G. 1988, A&A, 202, 21
McNamara 1999, presented at “Life cycles of Radio Galaxies”, July 15-17, 1999, STScI, Baltimore (astro-ph/9911129)
Mewe, R., Gronenschild, E. H. B. M., van den Oord, G. H. J., 1985, A&AS, 62, 197
Molendi, S. [*[Report on MOS PN cross-calibration presented at Leicester EPIC calibration meeting held in June 2001]{}*]{}
Molendi, S. & Gastaldello, F. 2001, A&A, 375, L14
Molendi, S. & Pizzolato, F. 2001, ApJ, 560, 194
Mushotzky, R., Loewenstein, M., Arnaud, K. A., Tamura, T., Fukazawa, Y., Matsushita, K., Kikuchi, K., Hatsukade, I. 1996, ApJ, 466, 686
Mushotzky, R. & Lowenstein, M. 1997, , 481, L63
Nomoto, K., Iwamoto, K., Nakasato, N., Thielemann, F. K., Brachwitz, F., Tsujimoto, T., Kubo, Y., Kishimoto, N. 1997, Nucl.Phys. A, 621, 467
Nulsen, P.E.J. & Böhringer, H., 1995, MNRAS, 274, 1093
Renzini, A., 1997, ApJ, 488, 35
Smith, R. K., Brickhouse, N. S., Liedahl, D. A., Raymond, J. C. 2001, ApJ, 556, L91
Tamura, T., Bleeker, A.M., Kaastra, J.S., Ferrigno, C., Molendi, S. 2001, A&A, 379, 107
Thomas, P.A. & Fabian, A.C., 1990, MNRAS, 246, 156
Toniazzo, T. & Schindler, S. 2001, MNRAS, 325, 509
van Breugel, W.J.M., Heckman, T., Miley, G. 1984, ApJ, 276, 79
van Breugel, W.J.M., Filippenko, A.V., Heckman, T., Miley, G. 1985, ApJ, 293, 83
van Breugel, W.J.M. & Dey, A. 1993, ApJ, 414, 563
-3.5truecm
-2.0truecm -2.0truecm
-2.0truecm -1.0truecm
[cccccccccc]{} 0$^{\prime}$-0.5$^{\prime}$ & $0.31^{+0.05}_{-0.04}$ & $0.30^{+0.14}_{-0.10}$ & $0.64^{+0.08}_{-0.05}$ & $1.03^{+0.08}_{-0.07}$ & $0.86^{+0.07}_{-0.06}$ & $1.41^{+0.19}_{-0.19}$ & $1.16^{+0.23}_{-0.22}$ & $1.03^{+0.08}_{-0.07}$ & $1.46^{+0.23}_{-0.18}$\
0.5$^{\prime}$-1$^{\prime}$ & $0.48^{+0.04}_{-0.05}$ & $0.44^{+0.12}_{-0.13}$ & $0.69^{+0.05}_{-0.06}$ & $1.32^{+0.04}_{-0.04}$ & $0.95^{+0.03}_{-0.04}$ & $1.88^{+0.12}_{-0.18}$ & $1.60^{+0.20}_{-0.20}$ & $1.24^{+0.03}_{-0.04}$ & $2.00^{+0.19}_{-0.14}$\
1$^{\prime}$-2$^{\prime}$ & $0.36^{+0.04}_{-0.03}$ & $0.48^{+0.10}_{-0.12}$ & $0.64^{+0.04}_{-0.04}$ & $1.20^{+0.03}_{-0.03}$ & $0.78^{+0.03}_{-0.02}$ & $1.58^{+0.12}_{-0.13}$ & $1.71^{+0.14}_{-0.14}$ & $1.07^{+0.03}_{-0.02}$ & $1.58^{+0.11}_{-0.11}$\
2$^{\prime}$-3$^{\prime}$ & $0.36^{+0.03}_{-0.03}$ & $0.37^{+0.09}_{-0.06}$ & $0.57^{+0.03}_{-0.04}$ & $0.96^{+0.03}_{-0.03}$ & $0.68^{+0.03}_{-0.02}$ & $1.42^{+0.11}_{-0.11}$ & $1.26^{+0.11}_{-0.11}$ & $0.92^{+0.02}_{-0.02}$ & $1.46^{+0.10}_{-0.09}$\
3$^{\prime}$-4$^{\prime}$ & $0.24^{+0.03}_{-0.03}$ & $0.34^{+0.05}_{-0.06}$ & $0.49^{+0.03}_{-0.04}$ & $0.87^{+0.02}_{-0.02}$ & $0.57^{+0.02}_{-0.02}$ & $1.22^{+0.11}_{-0.11}$ & $1.30^{+0.11}_{-0.12}$ & $0.77^{+0.01}_{-0.02}$ & $1.16^{+0.09}_{-0.08}$\
4$^{\prime}$-5$^{\prime}$ & $0.27^{+0.03}_{-0.03}$ & $0.47^{+0.05}_{-0.05}$ & $0.50^{+0.03}_{-0.03}$ & $0.75^{+0.02}_{-0.02}$ & $0.45^{+0.02}_{-0.02}$ & $1.15^{+0.10}_{-0.11}$ & $1.24^{+0.11}_{-0.11}$ & $0.68^{+0.01}_{-0.01}$ & $1.15^{+0.09}_{-0.08}$\
5$^{\prime}$-7$^{\prime}$ & $0.23^{+0.02}_{-0.02}$ & $0.44^{+0.04}_{-0.04}$ & $0.53^{+0.03}_{-0.03}$ & $0.63^{+0.02}_{-0.01}$ & $0.38^{+0.01}_{-0.01}$ & $0.93^{+0.07}_{-0.07}$ & $1.00^{+0.08}_{-0.07}$ & $0.60^{+0.01}_{-0.01}$ & $1.04^{+0.07}_{-0.06}$\
7$^{\prime}$-9$^{\prime}$ & $0.22^{+0.02}_{-0.02}$ & $0.30^{+0.03}_{-0.03}$ & $0.48^{+0.03}_{-0.03}$ & $0.53^{+0.02}_{-0.02}$ & $0.27^{+0.01}_{-0.01}$ & $0.57^{+0.08}_{-0.08}$ & $0.68^{+0.08}_{-0.09}$ & $0.53^{+0.02}_{-0.02}$ & $1.16^{+0.08}_{-0.06}$\
9$^{\prime}$-11$^{\prime}$ & $0.21^{+0.02}_{-0.03}$ & $0.28^{+0.03}_{-0.03}$ & $0.25^{+0.03}_{-0.03}$ & $0.49^{+0.02}_{-0.02}$ & $0.20^{+0.02}_{-0.02}$ & $0.60^{+0.09}_{-0.10}$ & $0.70^{+0.09}_{-0.10}$ & $0.46^{+0.01}_{-0.01}$ & $1.10^{+0.09}_{-0.07}$\
11$^{\prime}$-14$^{\prime}$ & $0.20^{+0.02}_{-0.02}$ & $0.21^{+0.03}_{-0.03}$ & $0.20^{+0.03}_{-0.03}$ & $0.48^{+0.02}_{-0.01}$ & $0.13^{+0.02}_{-0.02}$ & $0.59^{+0.08}_{-0.09}$ & $0.57^{+0.09}_{-0.09}$ & $0.38^{+0.01}_{-0.01}$ & $1.06^{+0.08}_{-0.07}$\
[|c|c|c|c|c|c|c|c|c|c|]{} & O & Ne & Mg & Si & S & Ar & Ca & Ni 0$^{\prime}$-0.5$^{\prime}$ & $0.30^{+0.06}_{-0.04}$ & $0.29^{+0.13}_{-0.10}$ & $0.62^{+0.09}_{-0.06}$ & $1.00^{+0.12}_{-0.10}$ & $0.83^{+0.10}_{-0.08}$ & $1.36^{+0.21}_{-0.20}$ & $1.12^{+0.24}_{-0.23}$ & $1.41^{+0.25}_{-0.20}$ 0.5$^{\prime}$-1$^{\prime}$ & $0.39^{+0.03}_{-0.04}$ & $0.35^{+0.10}_{-0.10}$ & $0.56^{+0.04}_{-0.05}$ & $1.06^{+0.04}_{-0.05}$ & $0.76^{+0.03}_{-0.04}$ & $1.51^{+0.10}_{-0.15}$ & $1.29^{+0.16}_{-0.17}$ & $1.61^{+0.15}_{-0.12}$ 1$^{\prime}$-2$^{\prime}$ & $0.34^{+0.04}_{-0.03}$ & $0.45^{+0.09}_{-0.11}$ & $0.60^{+0.04}_{-0.04}$ & $1.12^{+0.04}_{-0.05}$ & $0.73^{+0.03}_{-0.03}$ & $1.48^{+0.12}_{-0.12}$ & $1.60^{+0.14}_{-0.14}$ & $1.48^{+0.11}_{-0.10}$ 2$^{\prime}$-3$^{\prime}$ & $0.39^{+0.04}_{-0.03}$ & $0.40^{+0.10}_{-0.07}$ & $0.62^{+0.04}_{-0.04}$ & $1.04^{+0.04}_{-0.04}$ & $0.75^{+0.03}_{-0.03}$ & $1.55^{+0.12}_{-0.12}$ & $1.37^{+0.13}_{-0.13}$ & $1.60^{+0.12}_{-0.11}$ 3$^{\prime}$-4$^{\prime}$ & $0.31^{+0.04}_{-0.04}$ & $0.45^{+0.07}_{-0.07}$ & $0.64^{+0.04}_{-0.06}$ & $1.13^{+0.04}_{-0.04}$ & $0.74^{+0.03}_{-0.03}$ & $1.59^{+0.15}_{-0.14}$ & $1.69^{+0.15}_{-0.16}$ & $1.51^{+0.12}_{-0.11}$ 4$^{\prime}$-5$^{\prime}$ & $0.40^{+0.04}_{-0.04}$ & $0.69^{+0.08}_{-0.08}$ & $0.74^{+0.05}_{-0.05}$ & $1.10^{+0.04}_{-0.04}$ & $0.65^{+0.03}_{-0.03}$ & $1.69^{+0.15}_{-0.16}$ & $1.82^{+0.16}_{-0.16}$ & $1.69^{+0.14}_{-0.12}$ 5$^{\prime}$-7$^{\prime}$ & $0.38^{+0.04}_{-0.04}$ & $0.74^{+0.07}_{-0.06}$ & $0.88^{+0.05}_{-0.05}$ & $1.06^{+0.03}_{-0.03}$ & $0.63^{+0.03}_{-0.03}$ & $1.55^{+0.13}_{-0.13}$ & $1.68^{+0.13}_{-0.13}$ & $1.74^{+0.12}_{-0.11}$ 7$^{\prime}$-9$^{\prime}$ & $0.41^{+0.04}_{-0.04}$ & $0.57^{+0.06}_{-0.06}$ & $0.89^{+0.07}_{-0.06}$ & $1.00^{+0.05}_{-0.04}$ & $0.50^{+0.03}_{-0.04}$ & $1.07^{+0.15}_{-0.16}$ & $1.27^{+0.16}_{-0.17}$ & $2.17^{+0.17}_{-0.14}$ 9$^{\prime}$-11$^{\prime}$ & $0.45^{+0.05}_{-0.06}$ & $0.61^{+0.08}_{-0.08}$ & $0.56^{+0.08}_{-0.07}$ & $1.08^{+0.04}_{-0.04}$ & $0.45^{+0.04}_{-0.04}$ & $1.31^{+0.19}_{-0.21}$ & $1.54^{+0.20}_{-0.21}$ & $2.41^{+0.21}_{-0.16}$ 11$^{\prime}$-14$^{\prime}$ & $0.52^{+0.05}_{-0.06}$ & $0.55^{+0.07}_{-0.08}$ & $0.52^{+0.09}_{-0.09}$ & $1.26^{+0.05}_{-0.04}$ & $0.33^{+0.04}_{-0.05}$ & $1.54^{+0.22}_{-0.24}$ & $1.50^{+0.23}_{-0.25}$ & $2.78^{+0.22}_{-0.18}$
W7$^{a}$ & 0.031 & 0.004 & 0.022 & 0.36 & 0.30 & 0.42 & 0.33 & 3.23 WDD1$^{a}$ & 0.030 & 0.002 & 0.036 & 1.15 & 1.01 & 1.56 & 1.48 & 1.12 WDD2$^{a}$ & 0.016 & 0.001 & 0.013 & 0.69 & 0.62 & 0.98 & 0.97 & 0.95 SNII$^{b}$ & 3.23 & 1.90 & 2.58 & 2.37 & 1.17 & 1.72 & 1.09 & 1.27 SNII$^{c}$ & 1.25-2.80 & 0.96-1.93 & 0.96-1.89 & 1.81-2.54 & 0.85-2.08 & - & - & -
1.0truecm $^{a}$ Different models of SNIa taken by @nomoto97.
$^{b}$ Yields of SNII taken by @nomoto97.
$^{c}$ @gibson97 who choose a representative sample of SNII yields in literature.
|
---
abstract: 'For infinite machines which are free from the classical Thompson’s lamp paradox we show that they are not free from its inverted version. We provide a program for infinite machines and an infinite mechanism which simulate this paradox. While their finite analogs work predictably, the program and the infinite mechanism demonstrate an undefined behavior. As in the case of infinite Davies’s machines, our examples are free from infinite masses, infinite velocities, infinite forces, etc. Only infinite divisibility of space and timeis assumed. Thus, the considered infinite devices are possible in a continuous Newtonian Universe and they do not conflict with continuous Newtonian mechanics. Some possible applications to the analysis of the Navier-Stokes equations are discussed.'
author:
- 'Anton A. Kutsenko'
bibliography:
- 'infinite.bib'
title: Programming infinite machines
---
Introduction
============
The classical Thompson’s lamp paradox appears in [@T1]. Let us provide its computer interpretation. Suppose that we have one byte $a$ of memory and some CPU which can carry out an infinite number of operations within a finite length of time. Consider the following set of instructions (so-called Zeno process)
$t=0:$ $a:=0;$
$t=1/2:$ $a:=1;$
$t=3/4:$ $a:=0;$
$t=7/8:$ $a:=1;$
....,
where $t$ is time. Assuming that CPU time of each next operation is twice faster than the CPU time of a previous operation, we can write a pascal code for the Thompson’s program
$
a:=0;
$
$
\rm REPEAT
$
$
a:= {\rm not}\ a;
$
$
\rm UNTIL\ FALSE;
$
The paradox is that, we can not predict or determine the value of $a$ after the time $t$ when all operations are completed. For the first example, this time is $t=1$.
The theoretical description of infinite machines appears in [@BBJ1; @H1]. The possibility of producing of such machines in certain exotic relativistic spacetimes (sometimes called Hogarth-Malament spacetimes) is demonstrated in [@EN1]. The construction of infinite machines in a continuous Newtonian Universe is discussed in [@D1]. In [@D1], it is mentioned that the proposed infinite machine is free from the Thomson paradox. In this paper we show that such infinite machine is not free from the inverted Thompson’s paradox. The rough idea of this paradox consists of changing the order of operations in the classical Thompson’s paradox.
This topic is also closely related to the physical Church-Turing thesis which is the conjecture that no computing device that is physically realizable can exceed the computational barriers of a Turing machine, see, e.g., [@W; @ND; @S]. The result of the paper confirms this thesis since the infinite Davies’s machine, which allows a hypercomputation, demonstrates also an unpredictable behaviour. This raises doubts about the fundamental possibility of constructing this machine and other hypercomputers (even without taking into account the quantum nature of the real world). Moreover, it indicates some fundamental difficulties in a continuous Newtonian Universe itself. In particular, this observation may be helpful in analysis of Newtonian fluid dynamics, e.g. in analysis of the Navier-Stokes equations. For example, if a fluid analog of the mechanism considered in Section \[S3\] exists in a continuous Newtonian Universe then the Navier-Stokes equations do not have a unique solution since the mechanism demonstrates an undefined behavior. It is known that fluid motions can be very complex, see, e.g. [@FS]. They can create arbitrary small eddies and turbulent vortices with bizarre shapes. All this allows us to hope for the possibility of constructing the fluid analogs that will be close in some properties to the mechanism depicted on Fig. \[fig4\]. Then, it can be perspective for presenting a negative answer to the “millenium problem”.
Another useful information about the physical Church-Turing thesis along with hypercomputation and supertask can be found in [@CR; @C1; @HK; @C2; @N1]. It is also useful to note the paradox called “paradox of predictability” or “second oracle paradox”, see, e.g. [@RC]. The infinite analog of the paradox of predictability has some similar features with the inverted Thompson paradox. The corresponding analysis will be presented elsewhere.
The paper is organized as follows. Sections \[S1\],\[S2\] contain the description of the infinite machine and the program “puzzle” which demonstrates unpredictable behavior. Section \[S3\] contains the description of a pure mechanical device which demonstrates the same undefined behavior as the program “puzzle”. We conclude in Section \[S4\].
The construction
================
We consider a simplified version of infinite machine from [@D1]. The machine $\mM=\cup_{n=1}^{\infty}\cM_n$ consists of infinite number of finite machines $\cM_n$, $n\in\N$, see Fig. \[fig1\]. The machine $\cM_{n+1}$ is a small copy of the machine $\cM_n$ for all $n$. The machine $\cM_{n+1}$ is also twice faster than the machine $\cM_n$ for all $n$. For instance, we assume that CPU time $\t_n$ of $\cM_n$ is equal to $1/2^n$ for all $n$. We do not assume that the memory of $\cM_{n+1}$ is large than the memory of $\cM_{n}$. All machines have the same memory size, say $1$ byte for data and $1$ Kbyte for a program code and for built-in variables. Single-threaded CPU (interpreter) of each $\cM_n$ can perform integer and logic operations and simple data manipulations. Each $\cM_n$ can interact directly with adjacent $\cM_{n+1}$ only.
{width="100.00000%"}
Let us describe some commands of the machine $\mM$. If CPU of the machine $\cM_n$ gets the instruction $${\rm COPY\_PROGRAM\_NEXT}\ something;$$ then it copies the code placed between “${\rm PROGRAM}\ something:$” and “${\rm END}\ something;$” to the program memory of $\cM_{n+1}$ and runs the copy there. The instruction $${\rm IDLE}\ m;$$ says that CPU should skip $m$ CPU time’s $\t_n$ before executing next instructions. CPU time $\t_n(=1/2^n)$ depends on the machine $\cM_n$, where the instruction ${\rm IDLE}$ is performed. Any $\cM_n$ has built-in variables: $${\rm VALUE}$$ which refers to the byte data memory of $\cM_n$, and $${\rm VALUE\_NEXT}$$ which refers to the byte data memory of $\cM_{n+1}$. At the beginning of a program, all values are initialized to $0$. This remark is very important in the program “puzzle” considered below. The instructions $${\rm NOT},\ \ \ :=$$ mean the bitwise “not” and the assignment operation respectively. In particular, $({\rm NOT}\ 1)=0$ and $({\rm NOT}\ 0)=1$. We assume that all instructions described above (except ${\rm IDLE}$) take one CPU time $\t_n$ for performing. The CPU time $\t_n(=1/2^n)$ depends on the machine $\cM_n$, where the instruction is performed.
The machine $\mM$ is free from the classical Thompson’s paradox because the CPU-s can not manipulate with the fixed memory cell an infinite number of times. Nevertheless, $\mM$ is not free from the inverted Thompson’s paradox.
The puzzle
==========
The following program emulates the inverted Thompson’s paradox. The code is written in a pascal-based programming language. The comments are placed in parentheses $\{...\}$. $$\begin{array}{ll}
{\rm PROGRAM}\ puzzle: & \{entry\ point\} \\
{\rm COPY\_PROGRAM\_NEXT}\ puzzle; & \{instruction\ 1\} \\
{\rm IDLE}\ 2; & \{instruction\ 2\} \\
{\rm VALUE}:={\rm NOT}\ {\rm VALUE\_NEXT}; & \{instruction\ 3\ (two\ instructions)\} \\
{\rm END}\ puzzle; & \{exit\}
\end{array}$$ The program starts on $\cM_1$, copies to $\cM_2$ and starts there, waits for some time, takes the “inverted” value from $\cM_2$ and stops. The same happens in $\cM_2$, $\cM_3$, and so on. In fact, the program works like a shader for a multiprocessor system. The corresponding time diagram is plotted in Fig. \[fig2\]. Let us denote the time when $i$-th instruction ($i=1,2,3$) starts on $\cM_n$ ($n\in\N$) by $t_{ni}$. Let us denote the exit time on $\cM_n$ by $t_{n4}$, $n\in\N$. Then $$\begin{gathered}
\lb{001}
t_{n1}=\sum_{m=1}^{n-1}\t_m=1-2^{1-n},\ \ \ t_{n2}=t_{n1}+\t_n=1-2^{-n},
\\
t_{n3}=t_{n2}+2\t_n=1+2^{-n},\ \ \ t_{n4}=t_{n3}+2\t_n=1+2^{-n}+2^{1- n}.\end{gathered}$$ Due to , all values will be initialized and there are no conflicts between parallel programs working on different $\cM_n$, since only adjacent machines can interact. Nevertheless, we can not determine the value on $\cM_1$ at the end of the program “puzzle”. The reason is similar as in the inverted Thomson paradox. Both values ${\rm VALUE}_1=0$ and ${\rm VALUE}_1=1$ are possible (and impossible) at the end of the program. More precisely, if we execute “puzzle” on the finite machine $\mM_N=\cup_{n=1}^N\cM_n$ (the cascade stops in $\cM_N$) then ${\rm VALUE}_1=1$ for even $N$ and ${\rm VALUE}_1=0$ for odd $N$. But for $N=\infty$ we can not say: is $N$ even or odd?
{width="70.00000%"}
The mechanical interpretation
=============================
Let us consider another variant of the inverted Thompson’s lamp paradox. Consider the mechanism “mousetrap” depicted in Fig. \[fig3\]. The mechanism consists of the beam on the spring. The beam in tension (vertical position) is fixed with a thread. When ball is tearing the tread, the beam latches horizontally and it does not let through another ball.
{width="99.00000%"}
Consider infinite number of the finite mechanisms depicted in Fig. \[fig4\]. Each next finite mechanism is a small (half of the size) replica of the previous finite mechanism. To avoid various (e.g. centrifugal) effects, we can tune material properties of the spring and the beam of the next mechanism. We suppose also that there are infinite number of balls that move with a same constant velocity to the threads of the finite mechanisms. The size of each next ball is twice smaller than the size of the previous ball. The distances between the balls and the corresponding threads are chosen such that the smaller ball can tear the thread before the larger ball can reach the clipped horizontal beam corresponding to the smaller ball. Thus, the larger ball can not tear its thread since the beam is latched.
Note that any fixed constant value can be added to the distances between the balls and the threads. It is useful if we want that the smallest (limit) distance between the balls and the threads or beams is non-zero. Thus, there is a non-zero time interval between the start and the time when the balls reach the threads or latched beams.
The behavior of the infinite mechanism from Fig. \[fig4\] is indeterminate. We can not predict: will the largest beam be in a vertical or horizontal (latched) position after the balls fall down? The reason is the same as in the programm “puzzle”. If the number of balls is a finite number, say $N$ then the largest beam is in a horizontal position for odd $N$ and in a vertical position for even $N$. But we can not say $N=\infty$ is even or odd number. Note that in our example we do not assume infinite masses, velocities, densities. So, the unpredictable infinite mechanism may well exist in a Newtonian Universe. Of course, such mechanism is not possible in our world because of the principles of quantum mechanics.
{width="99.00000%"}
Conclusion
==========
Perhaps, any machine which uses the actual infinity is not free from Thompson-type paradoxes. Even physically reasonable assumptions may not be helpful. Probably, the main problem lies in our understanding of infinity. Nevertheless, a part of our mind can successfully develop infinite theories such as Peano arithmetic. Hence, there is a natural question which, however, can not be formulated rigorously: Is that part of our mind is an infinite machine and how it works?
|
---
abstract: 'We have succeeded in obtaining magnetized star models that have extremely strong magnetic fields in the interior of the stars. In our formulation, arbitrary functions of the magnetic flux function appear in the expression of the current density. By appropriately choosing the functional form for one of the arbitrary functions which corresponds to the distribution of the [*toroidal*]{} current density, we have obtained configurations with magnetic field distributions that are highly localized within the central part and near the magnetic axis region. The absolute values of the central magnetic fields are stronger than those of the surface region by two orders of magnitude. By applying our results to magnetars, the internal magnetic [*poloidal*]{} fields could be $10^{17}$ G, although the surface magnetic fields are about $10^{15}$ G in the case of magnetars. For white dwarfs, the internal magnetic [*poloidal*]{} fields could be $10^{12}$ G, when the surface magnetic fields are $10^{9} - 10^{10}$ G .'
author:
- |
Kotaro Fujisawa [^1], Shin’ichirou Yoshida and Yoshiharu Eriguchi\
Department of Earth Science and Astronomy, Graduate School of Arts and Sciences, University of Tokyo,\
Komaba, Meguro-ku, Tokyo 153-8902, Japan
date: 'Accepted 2012 January 23. Received 2012 January 23; in original form 2010 October 30'
title: Axisymmetric and stationary structures of magnetized barotropic stars with extremely strong magnetic fields deep inside
---
stars: magnetic field – stars: neutron – stars: white dwarf
Introduction
============
The magnetic field inside a star is scarcely detectable by direct observations but has been considered to affect stellar evolutions and activities in many aspects. For instance, if strong magnetic fields are hidden inside degenerate stars such as white dwarfs or neutron stars, they may significantly affect the cooling process of the stars by providing an energy reservoir or by modifying heat conduction. Highly localized, anisotropic and relatively strong magnetic field configurations, on the other hand, may affect accretion modes onto degenerate stars in close binary systems by providing a well-focused channel of accretion to their magnetic poles. In order to know the possible distributions and strengths of the magnetic fields inside the stars, we have to rely on theoretical studies. Until very recently, however, theoretical investigations could give us few hints about the interior magnetic fields. The reason for that may be twofold: one is related to the difficulty of the evolutionary computations of stellar magnetic fields and the other is related to the lack of methods to obtain stationary configurations of the magnetized stars.
Concerning the evolution of the stellar magnetic fields, since it has been very difficult to pursue evolutionary computations of the global magnetic fields for both interiors and exteriors of stars, few results have been obtained. Recently, however, Braithwaite and his collaborators have succeeded in following the evolution of global stellar magnetic fields (@Braithwaite_Spruit_2004; @Braithwaite_Nordlund_2006; @Braithwaite_Spruit_2006; @Braithwaite_2006; @Braithwaite_2007; @Braithwaite_2008; @Braithwaite_2009; @Duez_Braithwaite_Mathis_2010). They found that the twisted-torus configurations of the magnetic fields inside stars seem to be stable across the dynamical timescale.
On the other hand, to investigate possible structures of the interior and exterior magnetic fields by imposing stationarity is a different theoretical approach. Concerning this problem, many attempts have been made but it has also been difficult to obtain stellar structures with both [*poloidal*]{} and [*toroidal*]{} non force-free magnetic fields self-consistently, not only in the Newtonian gravity but also in general relativity (see e.g. @Chandrasekhar_Fermi_1953; @Ferraro_1954; @Chandrasekhar_1956; @Chandrasekhar_Prendergast_1956; @Prendergast_1956; @Woltjer_1959a; @Woltjer_1959b; @Woltjer_1960; @Wentzel_1961; @Ostriker_Hartwick_1968; @Miketinac_1973; @Miketinac_1975; @Bocquet_et_al_1995; @Ioka_Sasaki_2004; @Kiuchi_Yoshida_2008; @Haskell_Samuelsson_Glampedakis_2008 ; @Duez_Mathis_2010). It is only recently that axisymmetric and stationary barotropic stellar structures have been successfully solved for configurations with both [*poloidal*]{} and [*toroidal*]{} magnetic components (@Tomimura_Eriguchi_2005; @Yoshida_Eriguchi_2006; @Yoshida_Yoshida_Eriguchi_2006; @Lander_Jones_2009; @Otani_Takahashi_Eriguchi_2009) in a non-perturbative manner.
It should be noted that the twisted-torus magnetic configuration that appears during the evolutionary computations by [@Braithwaite_Spruit_2004] is qualitatively the same as one of the exact axisymmetric and stationary solutions obtained in [@Yoshida_Yoshida_Eriguchi_2006]. Moreover, stable configurations of stellar magnetic fields must have a twisted-torus structure according to [@Braithwaite_2009]. Concerning the stability analysis, this type of configurations is expected to be stable, while magnetic fields with purely [*poloidal*]{} configurations or purely [*toroidal*]{} configurations have been shown to be unstable (see e.g. @Tayler_1973_mnras; @Wright_1973_mnras; @Markey_Tayler_1973_mnras; @Flowers_Ruderman_1977_apj).
In this paper, we apply the formulation developed by [@Tomimura_Eriguchi_2005], [@Yoshida_Eriguchi_2006] and [@Yoshida_Yoshida_Eriguchi_2006] in order to find out how strong and localized [*poloidal*]{} magnetic fields can exist inside stars, as far as equilibrium configurations are concerned. In this formulation, the electric current density consists of several terms with different physical significances which contain arbitrary functionals of the magnetic flux function. These arbitrary functions correspond to the degrees of freedom in magnetized equilibria. One of the arbitrary functionals in the expression for the electric current density corresponds to the current in the [*toroidal*]{} direction. By choosing this functional form properly, we would be able to obtain equilibrium configurations of axisymmetric barotropic stars with highly localized and extremely strong [*poloidal*]{} magnetic fields.
Formulation and numerical method
================================
Since we employ the formulation developed by [@Tomimura_Eriguchi_2005], [@Yoshida_Eriguchi_2006], and [@Yoshida_Yoshida_Eriguchi_2006], here we summarize the main scheme briefly and explain newly introduced parts in detail.
Assumptions and basic equations
-------------------------------
We make the following assumptions for the magnetized stars.
1. The system is in a stationary state, i.e. $ \P{}{t} = 0$ .
2. When stars are rotating and have magnetic fields, the rotational axis and the magnetic axis coincide.
3. The rotation is rigid.
4. The configurations are axisymmetric about the magnetic or the rotational axis, i.e. $ \P{}{\varphi} = 0 $, where we use the spherical coordinates $(r, \theta, \varphi)$.
5. The configurations are symmetric with respect to the equator.
6. There are no meridional flows.
7. The star is self-gravitating.
8. The systems are treated in the framework of non-relativistic physics.
9. The conductivity of the stellar matter is infinite, i.e. the ideal magnetohydrodynamics (MHD) approximation is employed.
10. No electric current is assumed in the vacuum region.
11. The barotropic equation of state is assumed : $$\begin{aligned}
p = p(\rho) \ .\end{aligned}$$
Here $p$ and $\rho$ are the pressure and the mass density, respectively. Assumptions of axisymmetric and equatorial symmetries as well as rigid rotation are adopted here in order to simplify our investigations.
In a rotating star under a radiative equilibrium, there appears to be meridional flow in special cases. However, we neglect it because the time scale is many orders of magnitude larger than the (magneto)hydrodynamic one (@Tassoul_2000). Also, there is a suggestion that gradual diffusion of the internal magnetic fields drives a meridional flow (@Urpin_Ray_1994). The time scale of this, again, is much larger than the (magneto)hydrodynamic one. Thus it is also neglected.
Under these assumptions, the basic equations are written as follows. The continuity equation is expressed as $$\begin{aligned}
\nabla \cdot (\rho \Vec{v}) = 0 \ ,\end{aligned}$$ where $\Vec{v}$ is the fluid velocity. The equations of motion in the stationary state are written as: $$\begin{aligned}
\frac{1}{\rho} \nabla p = -\nabla \phi_g +
R \Omega^2 \Vec{e}_R+ \frac{1}{\rho} \left( \frac{\Vec{j}}{c} \times
\Vec{H} \right),
\label{Eq:eular}\end{aligned}$$ where $\phi_g$, $\Omega$, $\Vec{j}$, $c$ and $\Vec{H}$ are gravitational potential, angular velocity, electric current density, speed of light and magnetic field, respectively. Here we use the cylindrical coordinates $(R, \varphi, z)$ and $\Vec{e}_R$ is the unit vector in the $R$-direction. The gravitational potential satisfies Poisson equation: $$\begin{aligned}
\Delta \phi_g = 4 \pi G \rho \ ,
\label{Eq:Poisson}\end{aligned}$$ where $G$ is the gravitational constant. Maxwell’s equations are written as, $$\begin{aligned}
\nabla \cdot \Vec{E} = 4\pi \rho_e \ ,\end{aligned}$$ $$\begin{aligned}
\nabla \cdot \Vec{H} = 0 \ ,
\label{Eq:divH}\end{aligned}$$ $$\begin{aligned}
\nabla \times \Vec{E} = 0 \ ,
\label{Eq:rotE}\end{aligned}$$ $$\begin{aligned}
\nabla \times \Vec{H} = 4 \pi \frac{\Vec{j}}{c} \ ,
\label{Eq:rotH}\end{aligned}$$ where $\rho_e$ and $\Vec{E}$ are the electric charge density and the electric field, respectively. Notice that we neglect the displacement current term in Eq.(\[Eq:rotE\]) as is common in MHD approximation. The ideal MHD condition, or the generalized Ohm’s equation, can be expressed as: $$\begin{aligned}
\Vec{E} = - \frac{\Vec{v}}{c} \times \Vec{H} \ .\end{aligned}$$ We choose two kinds of barotropic equations of state. One is the polytropic equation of state: $$\begin{aligned}
p = K_0 \rho^{1+1/N} \ ,\end{aligned}$$ where $N$ and $K_0$ are the polytropic index and the polytropic constant, respectively. The other is the degenerated Fermi gas at zero temperature, defined as $$\begin{aligned}
p = a[x(2x^2 - 3)\sqrt{x^2 + 1} + 3 \ln(x + \sqrt{x^2 + 1}) ],\end{aligned}$$ where $$\begin{aligned}
\rho = bx,\end{aligned}$$ $$\begin{aligned}
a = 6.00 \times 10^{22} \hspace{10pt} \mathrm{dyn/cm}^2,\end{aligned}$$ $$\begin{aligned}
b = 9.825 \times 10^5 \mu_e \hspace{10pt} \mathrm{g/cm}^3.\end{aligned}$$ Here $\mu_e$ is the mean molecular weight. We fix $\mu_e=2$ in all our computations here, which corresponds to a fully ionized pure hydrogen gas. This choice of parameters is same as that in [@Hachisu_1986].
The form of the current density and the boundary condition {#sec:bc}
----------------------------------------------------------
From the assumptions of axisymmetry and stationarity, we introduce magnetic flux function $\Psi$ as follows: $$\begin{aligned}
H_R \equiv -\frac{1}{R}\P{\Psi}{z} , \hspace{10pt}
H_z \equiv \frac{1}{R}\P{\Psi}{R} \ ,\end{aligned}$$ where $H_R$ and $H_z$ are magnetic field components in the $R$-direction and $z$-direction, respectively. We assume this flux function is positive in the entire space. By introducing this magnetic flux function, equation (\[Eq:divH\]) can be automatically satisfied. It should be noted that the magnetic flux function $\Psi$ can be expressed as: $$\begin{aligned}
\Psi = r \sin \theta A_{\varphi} \ , \end{aligned}$$ where $A_{\varphi}$ is the $\varphi$-component of the vector potential $\Vec{A} = (A_R, A_{\varphi}, A_z)$.
As shown in [@Tomimura_Eriguchi_2005], for axisymmetric and stationary barotropes with rigid rotation we can constrain the form of the electric current density by using an integrability condition of the equations of motion, equation (\[Eq:eular\]): $$\begin{aligned}
\frac{\Vec{j}}{c} = \frac{1}{4\pi} \D{\kappa(\Psi)}{\Psi} \Vec{H} + r \sin \theta \rho
\mu(\Psi) \Vec{e}_\varphi \ ,
\label{Eq:current}\end{aligned}$$ where $\kappa(\Psi)$ and $\mu(\Psi)$ are arbitrary functions of the magnetic flux function $\Psi$. Notice, in particular, that the [*toroidal*]{} component of magnetic field is given as $$H_\varphi = \frac{\kappa (\Psi)}{r \sin \theta}
\label{Eq:H_phi}$$ which can be derived from equations (\[Eq:eular\]), (\[Eq:rotH\]) and (\[Eq:current\]). It should be noted that these two arbitrary functions are conserved along the [*poloidal*]{} magnetic field lines. Although the meanings of these two functions are described in previous works (see, e.g., @Lovelace_et_al_1986), in this paper we will explain their meanings differently from our point of view.
Since we have assumed that there is no electric current in the vacuum region, in other words that there is no [*toroidal*]{} magnetic field outside the star (see equation \[Eq:current\]), the form for $\kappa$ needs to be a special one. The simplest form can be $\kappa=$ constant (@Ioka_Sasaki_2004, @Haskell_Samuelsson_Glampedakis_2008), but for this choice of $\kappa$ the [*toroidal*]{} magnetic field would extend to the vacuum region. In order to avoid this possibility, we choose the functional form of $\kappa$ as follows: $$\begin{aligned}
\kappa (\Psi) =
\left\{
\begin{array}{lr}
0 \ , & \mathrm{for} \hspace{10pt} \Psi \leq \Psi_{\max} \ , \\
\dfrac{\kappa_0}{k+1}(\Psi - \Psi_{\max})^{k+1} \ ,
& \mathrm{for} \hspace{10pt} \Psi \geq \Psi_{\max} \ ,
\end{array}
\right.
\label{Eq:kappa}\end{aligned}$$ This choice of $\kappa$ is the same as that in [@Yoshida_Eriguchi_2006] and [@Lander_Jones_2009]. In this paper we fix $k=0.1$. Equations (\[Eq:H\_phi\]) and (\[Eq:kappa\]) ensure that the [*toroidal*]{} magnetic field vanishes smoothly at the stellar surface. Incidentally, using these functionals, we obtain the first integral of equation (\[Eq:eular\]) as follows: $$\begin{aligned}
\int \frac{dp}{\rho} = -\phi_g + \frac{1}{2}(r \sin \theta)^2\Omega^2_0
+ \int \mu(\Psi) \, d\Psi + C \ ,
\label{Eq:first_int}\end{aligned}$$ where $C$ is an integration constant. The first term of the right-hand side is the gravitational potential. The second term on the right hand side is related to rotation. We can consider it as a rotational potential. Similarly, the third term means the potential of Lorentz force. We can regard this term as the magnetic force potential. Therefore, $\int \mu \, d\Psi$ is considered to be non-force-free contribution from the current density, as is seen in equation (\[Eq:current\]). Since the Lorentz force is given by the cross product $\Vec{j} / c \times \Vec{H}$, the first term of equation (\[Eq:current\]) has no effect on the equation of motion, i.e, it is force-free, and only the second term contributes to the Lorentz force, i.e., non force-free. The distribution of Lorentz force could be changed by adopting different functional forms for $\mu$. All previous works (@Tomimura_Eriguchi_2005, @Yoshida_Eriguchi_2006, @Yoshida_Yoshida_Eriguchi_2006, @Lander_Jones_2009, @Otani_Takahashi_Eriguchi_2009) fixed $\mu = \mu_0 $ (constant). We choose a different functional form for $\mu$ in this paper as follows: $$\begin{aligned}
\mu(\Psi) & = & \mu_0 (\Psi + \epsilon )^m \ , \\
\int \mu(\Psi) \, d \Psi & = & \frac{\mu_0}{m+1}(\Psi + \epsilon )^{m+1} \ ,
\label{eq:muuu}\end{aligned}$$ where $m$ and $\epsilon$ are two constant parameters. In order to avoid singular behavior, we fix $\epsilon = 1.0 \times 10^{-6}$ in all calculations. As we shall see below, the parameter $m$ determines a degree of localization of the interior [*poloidal*]{} magnetic field. We assume that [*poloidal*]{} magnetic fields extend throughout the whole space and that there are no discontinuities even at the stellar surface. The global magnetic field configurations of our models are nearly dipole-like because of the requirement of the functional form for $\kappa$ at the stellar surface. These configurations contain closed [*poloidal*]{} magnetic field lines inside the star. The flux function $\Psi$ attains its maximum at the central parts of these closed field lines and it takes its minimum on the symmetric axis and at infinity. The minimum value is zero because of $\Psi = r \sin \theta A_\varphi$ and the boundary condition for $A_\varphi = 0 $ at infinity. The magnetic potential $\left(\int\mu d\Psi\right)$ changes its qualitative behavior in its spatial distribution when $m=-1$. If we adopt $m < -1$, as $\Psi$ decreases from its maximum to zero on the axis of the star the value of the magnetic potential increases unboundedly if $\epsilon\to 0$. As a result, the [*poloidal*]{} magnetic field lines are concentrated near the axis in order to fulfill such magnetic potential distributions. On the other hand, if we choose $m > - 1$, the value of the magnetic potential decreases as $\Psi$ decreases from its maximum to zero, which is realized on the axis. Then the [*poloidal*]{} magnetic field lines are distributed more uniformly than those for configurations with $m < -1$. If we choose $m=0$, we obtain $\mu = $ constant configurations. They are the same as those investigated by other authors.
It is remarkable that the only freedom that we can take in our formulation is related to the choices of functional forms and the values of the parameters which appear in those functions. It implies that degrees of freedom for choices for these functions and parameters correspond to degrees of freedom for many kinds of stationary axisymmetric magnetic field configurations. In fact, as we see from our results, different values for $m$ result in qualitatively different distributions for the magnetic potentials and the [*poloidal*]{} magnetic fields. In other words, we can control the magnetic field distributions to a certain extent by adjusting the value for $m$. This is the reason why we use this functional form of $\mu$ in this paper.
After we choose the functional form of the current density, by using Eq.(\[Eq:current\]) and the definition of the vector potential, we obtain the following partial differential equation of the elliptic type: $$\begin{aligned}
\Delta (A_\varphi \sin\varphi)
= 4 \pi S_A(r,\theta) \sin\varphi \ , \hspace{10pt}
S_A \equiv - \frac{j_\varphi}{c}.
\label{Eq:Poisson_A}\end{aligned}$$ As we have seen in the previous paragraph, all the physical quantities related to the vector potential can be expressed solely by $\Psi$. Therefore we need not solve for $A_R$ and $A_z$. It implies that our present formulation does not depend on the gauge condition for the vector potential $\Vec{A}$. Next we impose the boundary conditions for the gravitational potential and the vector potential, chosen as follows: $$\begin{aligned}
\phi_g \sim {\cal O} \left(\frac{1}{r}\right) \ ,
\hspace{10pt}(r \rightarrow \infty) \ ,
\label{Eq:BC_gphi}\end{aligned}$$ $$\begin{aligned}
A_\varphi \sim {\cal O} \left(\frac{1}{r}\right) \ ,
\hspace{10pt}(r \rightarrow \infty) \ .
\label{Eq:BC_A}\end{aligned}$$ This boundary condition for $A_\varphi$ results in $$\begin{aligned}
H_p \sim {\cal O} \left(\frac{1}{r^2}\right) \ ,
\hspace{10pt}(r \rightarrow \infty) \ .
\label{Eq:BC_H}\end{aligned}$$ where $H_p$ is the [*poloidal*]{} magnetic field. From these boundary conditions and using a proper Green’s function for the Laplacian, we have the integral representations of equation (\[Eq:Poisson\]) and equation (\[Eq:Poisson\_A\]) as follows: $$\phi_g(\Vec{r}) = -G \int \frac{\rho(\Vec{r}')}{|\Vec{r} - \Vec{r}'|}
\, d^3 \Vec{r}' \ ,$$ $$A_\varphi(\Vec{r}) \sin \varphi
= - \int \frac{S_A(\Vec{r}')\sin\varphi'}
{|\Vec{r} - \Vec{r}'|} \, d^3 \Vec{r}' \ .$$ Therefore, we can obtain smooth potentials, $\phi_g$ and $A_\varphi$ by integrating these equations. Since we have chosen the functional form of the current density which decreases near the surface and vanishes at the stellar surface sufficiently smoothly, we obtain continuous [*poloidal*]{} magnetic fields from $A_\varphi$.
Global characteristics of equilibria
------------------------------------
To see the global characteristic of magnetized equilibria, we define some integrated quantities as follows: $$\begin{aligned}
W \equiv \frac{1}{2}\int \phi_g \rho \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
T \equiv \frac{1}{2} \int \rho (R \Omega)^2 \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
\Pi \equiv \int p \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
U \equiv N \Pi \ ,\end{aligned}$$ for polytropic models and $$\begin{aligned}
U \equiv \int g(x) \, d^3 \Vec{r} \ ,\end{aligned}$$ $$\begin{aligned}
g(x) = a\{ 8x^3 [(x^2 + 1)^\frac{1}{2} - 1]\} - p, \end{aligned}$$ for the Fermi gas configurations (see. @Chandrasekhar_stellar_structure), $$\begin{aligned}
{\cal H} \equiv \int r \cdot \left(\frac{\Vec{j}}{c} \times \Vec{H} \right) \, d^3 \Vec{r} \ .\end{aligned}$$ $$\begin{aligned}
K = \int (\nabla \times \Vec{A}) \cdot \Vec{A} \, d^3 \Vec{r}
= \int \Vec{H} \cdot \Vec{A} \, d^3 \Vec{r} \ ,\end{aligned}$$ where $W$, $T$, $\Pi$, $U$, ${\cal H}$ and $K$ are the gravitational energy, rotational energy, total pressure, internal energy, magnetic field energy and magnetic helicity, respectively. In order to evaluate the structures of magnetic fields, we define some physical quantities related to the magnetic fields as follows: $$\begin{aligned}
H_{sur} = \dfrac{\int_0^{2\pi}
\int_0^{\pi} \,
r_s^2(\theta) \,
\sin \theta
|\Vec{H}(r_s, \theta)|
\, d\theta
d\varphi
}{S} \ ,\end{aligned}$$ where $r_s(\theta)$ and $|H_{sur}|$ are the stellar radius in the direction of $\theta$ and the surface magnetic field strength, respectively, and the surface area of the star is defined as: $$\begin{aligned}
S = \int_0^{2\pi}
\int_0^{\pi} \, r_s^2(\theta) \sin \theta \, d\theta
\, d \varphi .\end{aligned}$$ The volume-averaged magnetic field strength in the central region of the star is defined as $$\begin{aligned}
H_\mathrm{c} =
\dfrac{\int_0^{2\pi}
\int_0^\pi \,
\int_0^{r_c} \,
r^2 \sin \theta |\Vec{H}(r,\theta)| dr d\theta d\varphi}
{V},
\label{Eq:H_c}\end{aligned}$$ where we choose $r_c = 0.01 r_e$ and $V$ is the volume of the central region with $r \le r_c$, defined as $$\begin{aligned}
V = \int_0^{2\pi} \,
\int_0^\pi
\int_0^{r_c} r^2\,
\sin \theta dr
\, d\theta
d\varphi \ .\end{aligned}$$ This central region seems to be very small, but we can resolve it sufficiently because we use non-uniform and centrally concentrated meshes (see. Fig. \[fig:mesh\] and Eq. \[Eq:r\_mesh\] in Appendix). We have $77$ meshes to resolve the region in actual numerical computations.
In order to know the contributions of the [*poloidal*]{} magnetic field and the [*toroidal*]{} magnetic field separately, we define the [*poloidal*]{} magnetic energy ${\cal H}_p$ and the [*toroidal*]{} magnetic energy ${\cal H}_t$ as $$\begin{aligned}
{\cal H}_p = \frac{1}{8\pi}
\int_0^{2 \pi} \int_0^{\pi} \int_0^{\infty}
r^2 \, \sin \theta \,
|H_r(r,\theta)^2 + H_\theta(r,\theta)^2| dr d\theta d\varphi \ ,\end{aligned}$$ $$\begin{aligned}
{\cal H}_t = \frac{1}{8\pi}
\int_0^{2 \pi} \int_0^{\pi} \int_0^{\infty}
r^2 \, \sin \theta \,
|H_\varphi(r,\theta)^2| dr d\theta d\varphi \ ,\end{aligned}$$ As for the magnetic multipole moment seen outside a star, we compute each multipole component by solving the following equation in a vacuum: $$\begin{aligned}
\Delta \left(A_\varphi \sin \varphi\right) = 0 \ .\end{aligned}$$ Considering the boundary conditions at infinity and the symmetry of the magnetized stars, the solution of the above equation can be expressed as $$\begin{aligned}
A_\varphi \sin \varphi
\equiv \sum_{n=1}^{\infty} A_{\varphi, n} \sin \varphi
= \sum_{n=1}^{\infty} b_{n,1} r^{-n-1} Y_{n,1}(\theta,\varphi) \ ,
\label{Eq:bn}\end{aligned}$$ where $Y_{n,1}(\theta,\varphi)$ is the spherical harmonics of degree $n$ and order $m = 1$. The coefficients $b_{n,1}$ correspond to the magnetic multipoles.
Setting for Numerical Computations
----------------------------------
For numerical computations, the physical quantities are transformed into dimensionless ones using the maximum density $\rho_{\max}$, the maximum pressure $p_{\max}$ and the equatorial radius $r_e$ as follows: $$\begin{aligned}
\hat{r} \equiv \frac{r}{r_e} = \frac{r}{
\sqrt{\frac{1}{\alpha}\frac{p_{\mathrm{max}}}{4\pi G
\rho_{\mathrm{max}}^2}}} \ ,\end{aligned}$$ for polytropic configurations and $$\begin{aligned}
\hat{r} \equiv \frac{r}{r_e} = \frac{r}{
\sqrt{\frac{1}{\alpha}\frac{8a}{b} \frac{1}{4\pi G
\rho_{\mathrm{max}}^2}}} \ ,\end{aligned}$$ for the Fermi gas models, and $$\begin{aligned}
\hat{\rho} \equiv \frac{\rho}{\rho_\mathrm{max}} \ .\end{aligned}$$ Here $\alpha$ is introduced so as to make the distance from the center to the equatorial surface of the star to be unity. Arbitrary functions are also transformed into dimensionless ones. Quantities with $\hat{}$ are dimensionless. For example, the dimensionless length is $\hat{r}$ and the dimensionless arbitrary functions are $\hat{\mu}$ and $\hat{\kappa}$, respectively. Dimensionless forms of other quantities are collected in Appendix \[App:dimensionless\].
The computational domain is defined as $0\leq\theta\leq\frac{\pi}{2}$ in the angular direction and $0\leq \hat{r}\leq 2$ in the radial direction. Since the equation of magnetohydrostationary equilibrium is defined only inside the star and the source terms of the elliptic equations for the gravitational potential and the magnetic flux function vanish outside the star, our computational domain covers a region of the space that is sufficient for obtaining equilibria. In order to resolve the region near the axis sufficiently, we use a special coordinate in actual numerical computations. Total mesh numbers in $r$-direction and in $\theta$-direction are 1025 and 1025, respectively. We describe details of the computational grid points in Appendix \[App:grid\].
Numerical method
----------------
We use the scheme of [@Tomimura_Eriguchi_2005]. This scheme is based on the Hachisu Self-Consistent Field (HSCF) scheme (@Hachisu_1986), which is the method for obtaining equilibrium configurations of rotating stars. We define the ratio of the equatorial radius to the polar radius as the axis ratio $q$. This quantity $q$ characterizes how distorted the stars are due to non-spherical forces. The stronger the non-spherical force becomes, the more distorted the stellar shape is. The non-spherical force can be the centrifugal force, the magnetic force or both of them. We fix the value of $q$ in order to obtain the magnetized equilibria. We also fix one of $\hat{\mu}_0$ and $\hat{\Omega}_0$. If we fix $\hat{\mu}_0$, we will obtain the value of $\hat{\Omega}_0$ after the relaxation and iteration. If we fix $\hat{\Omega}_0$, we will obtain $\hat{\mu}_0$. Then, we will obtain one magnetized equilibrium state.
Numerical accuracy check
------------------------
![The virial quantity VC, plotted against the number of grid points in the $r$-direction.[]{data-label="fig:VC"}](img/fig1.eps)
In order to check the accuracy of converged solutions, we compute a relative value of the virial relation as follows: $$\begin{aligned}
\mathrm{VC} \equiv \frac{|2T + W + 3\Pi +{\cal H}|}{|W|}.\end{aligned}$$ Since this quantity VC must vanish for exact equilibrium configurations, we can check the global accuracies of the numerically obtained models as a whole (see e.g. @Hachisu_1986). Since the numerical results depend on mesh size, we have computed the same model by changing the number of grid points in the $r$-coordinate but fixing the number of grid points in the $\theta$-direction as $n_\theta = 513$. Fig. \[fig:VC\] shows VC as a function of the number of grid points in the $r$-coordinate for polytropic models. Since we use schemes of second-order accuracy, VC decreases as the square inverse of the number of grid points (see also @Lander_Jones_2009; @Otani_Takahashi_Eriguchi_2009).
Numerical Results
=================
We give a brief summary of our numerical results here. First we show the basic features for negative $m$ models and the dependences of the magnetic field configurations on the values of $m$ for barotropes. We also show rotating and magnetized polytropic models in order to examine the effect of rotation on magnetic fields. The influence of the equation of state on the interior magnetic field is also displayed. We have computed $N=0.5, 1, 1.5$ polytropic models and four white dwarf models with $\rho_c = 1.0\times 10^7, 1.0 \times 10^8,$ $1.0 \times 10^9,$ and $1.0 \times 10^{10} \mathrm{g cm^{-3}}$.
Effect of the distribution of the [*toroidal*]{} current density on the distribution of the magnetic field {#Sec:current}
----------------------------------------------------------------------------------------------------------
$m$ $1-q$ $H_\mathrm{c} / H_\mathrm{sur}$ ${\cal H}_p/{\cal H}$ ${\cal H}/|W|$ $\Pi/|W|$ $\alpha$ $\hat{\mu}_0$ $\hat{K}$ VC
------ -------- --------------------------------- ----------------------- ---------------- ----------- ---------- --------------- ----------- ---------- -- -- -- -- -- --
$N = 1.0$
-2.0 2.2E-2 1.03E+2 9.987E-1 3.74E-5 3.33E-1 5.07E-2 2.28E-9 7.76E-7 5.132E-8
-1.5 1.9E-3 4.44E+1 9.982E-1 3.02E-5 3.33E-1 5.07E-2 9.65E-8 8.27E-7 2.646E-6
-1.1 4.2E-4 2.19E+1 9.978E-1 2.60E-5 3.33E-1 5.07E-2 1.80E-6 8.33E-7 2.775E-6
-0.9 2.5E-4 1.62E+1 9.976E-1 2.45E-5 3.33E-1 5.07E-2 7.74E-6 8.35E-7 2.785E-6
-0.5 1.3E-4 1.02E+1 9.972E-1 2.21E-5 3.33E-1 5.07E-2 1.41E-4 8.35E-7 2.788E-6
0.0 8.8E-5 7.17E+0 9.968E-1 1.99E-5 3.33E-1 5.07E-2 5.25E-3 8.31E-7 2.789E-6
0.5 7.0E-5 5.69E+0 9.963E-1 1.83E-5 3.33E-1 5.07E-2 1.92E-1 8.23E-7 2.789E-6
1.0 6.1E-5 4.78E+0 9.959E-1 1.70E-5 3.33E-1 5.07E-2 6.90E+0 8.11E-7 2.789E-6
 
 
 
  
  
We show the results for the distributions of the magnetic fields for different values of $m$. In particular, in order to examine the effect of magnetic fields alone, we consider configurations without rotation. The effect of stellar rotation is discussed in Sec. \[Sec:rotation\]. Thus we set $\hat{\Omega}_0 = 0$ and compute $N=1$ polytropic equilibrium models with different values of $ m$ and appropriate values of $q$ so that the surface magnetic field becomes roughly $H_{sur} = 10^{15}$ G when $\rho_c = 1.0 \times 10^{15} \mathrm{g cm^{-3}}$ and mass $M = 1.4 M_\odot$. By setting $N=1$ and an appropriate choice of polytropic constant $K$ of $p=K\rho^{2}$, we obtain models with $M=1.4M_\odot$. It should be noted that these models have the typical mass and radius for neutron stars. We choose $N=1$ as a simple approximation of neutron stars here. We searched and found the value of $q$ by calculating many equilibrium states.
Physical quantities of these models are shown in Table \[tab:m-mu\]. It can be seen that values of $\Pi / |W|$ and $\alpha$ are almost the same among these models. Although the strength of the averaged surface magnetic field is $H_{sur} = 1.5 \times10^{15}$ G, the values of ${\cal H} / |W|$ are much smaller than those of $\Pi / |W|$. It implies that the effect of the magnetic fields in these configurations on their global structures is very small. On the other hand, values of $H_c / H_{sur}$ and ${\cal H}/ |W|$ vary rather considerably for different values of $m$. As the value of $m$ is decreased, values of $H_c / H_{sur}$ and ${\cal H}/ |W|$ increase. In Fig. \[fig:magnetic\_structure\] the structure and strength of magnetic fields are shown for three different values of $m$, i.e. $m = -2.0$ (negative $m$ model), $m = 0.0$ ($\hat{\mu}$ = constant model) and $m = 1.0$ (positive $m$ model). The left-hand panels show the [*poloidal*]{} magnetic field lines and the regions where the [*toroidal*]{} magnetic field exists. The right-habd panels display the strength of the magnetic field $|\Vec{H}|$ normalized by the averaged surface magnetic field $H_{sur}$.
As seen from these figures, there are no discontinuities of the magnetic fields at the stellar surfaces. Due to the choice of the functional form of the arbitrary function $\hat{\kappa}(\hat{\Psi})$ and the distribution of the magnetic flux function, [*toroidal*]{} magnetic fields appear only in the region that is bounded by the outermost closed [*poloidal*]{} magnetic field line inside the star (thick line). Thus the toroidal magnetic fields exist inside the torus region.
As the value of $m$ is increased, i.e. from top panels to bottom panels, the ratio of $H_c / H_{sur}$ decreases (see left panels in Fig. \[fig:magnetic\_structure\]) because the [*poloidal*]{} magnetic field becomes weaker. This is also related to the fact that the interior [*poloidal*]{} magnetic field lines are much more localized near the axis for negative $m$ models. The contours of magnetic field strength also display the same tendency. For the $m = 1.0$ model, the contour of $|\Vec{H}| = H_{sur}$ (thick line) shows the stellar surface and the shapes of contours are nearly spherical. By contrast, the contours of the $m=-2.0$ model are highly distorted near the axis. The strength of the [*poloidal*]{} magnetic fields for the negative $m$ models could exceed $10^{17}$G near the central region. Fig. \[fig:beta\] shows the profiles of the plasma $\beta$ on the $\theta = \pi / 2$ plane, i.e. on the equatorial plane. Here, the plasma $\beta$ is defined as follows: $$\begin{aligned}
\beta = 8 \pi p / |\Vec{H}|^2 .\end{aligned}$$ This quantity denotes the contribution of the gas pressure effect compared with the magnetic pressure effect. Fig. \[fig:beta\] shows profiles of $\beta$ for models with $m = -2.0, 0.0, 1.0$. As seen from Fig.\[fig:beta\], the profiles of $\beta$ are very similar to each other near the stellar surface regions. For the region around $\hat{r} \sim 0.6$, however, the value of $\beta$ for the $m=-2.0$ model is larger than those for the $m=0$ and $m=1.0$ models. Since these models have almost the same mass density distributions, this difference means a difference of magnetic pressure distribution. In this region the magnetic field of the $m=-2.0$ configuration is weaker and thus the $\beta$ becomes larger. However, [*it should be noted that*]{} these contours for the model with $m=-2.0$ are rather confined to the very narrow region near the central part. In other words, the gradient of the magnetic field distribution for the model with $m = -2$ is much steeper than the gradient of the gas pressure distribution compared with the models with $m=1.0$ and $m = 0.0$. Thus the value of $\log \beta$ becomes dramatically small within the $\hat{r} [0:0.1]$ region and the minimum value of $\beta$ can reach about $\sim$ 20 in the central part. Therefore, in the central region of the model with $m=-2.0$ the influence of magnetic field on the local structure of the star is no longer negligible.
Here we explain the reason why this kind of highly localized [*poloidal*]{} magnetic field configuration can be realized. We need to note the distribution of the [*toroidal*]{} current density $\hat{j}_\varphi$ in order to analyse our models properly, because the current density is related to the magnetic field closely by the two equations (\[Eq:rotH\]) and (\[Eq:current\]). In Fig. \[fig:j\_phi\] we show the distributions of the [*toroidal*]{} current density for models with different values of $m$. As seen from Fig. \[fig:j\_phi\], the distribution of the [*toroidal*]{} current density is concentrated toward the magnetic axis for the configuration with negative values of $m$. This is due to the dependence of $\hat{\mu}$ on the value of $m$. The current density distribution spreads over a large region inside the star as the value of $m$ [*increases*]{} (from left panel to right panel). In other words, the distribution of the magnetic flux function becomes more and more concentrated toward the magnetic axis as the value of $m$ [*decreases*]{}. It implies that the strengths of magnetic fields for models with negative values of $m$ become very great near the magnetic axis. Our results show one possibility that a strong [*poloidal*]{} magnetic field can exist deep inside a star. If such a strong [*poloidal*]{} magnetic field is sustained deep inside a star, the contours of the magnetic field strength are no longer nearly spherical as in the bottom right panel of Fig. \[fig:magnetic\_structure\]. Although this feature might be modified by dropping the assumption of the axisymmetry, it would give us one possibility for the presence of a strong [*poloidal*]{} magnetic field configuration deep inside a star.
Finally, to characterize the magnetic structure we show the magnetic multipole moments of magnetized stars. In Fig. \[fig:A\_phi\] the values of $|b_{n,1} / b_{1,1}|$ (equation \[Eq:bn\]) are plotted. There appear to be only multipolar magnetic moments with odd degree ($n=1, 3, 5$), because we have assumed the equatorial symmetry. As seen from these figures, in configurations with negative values of $m$ the higher order magnetic multipole moments contribute ($|b_{n,1}/b_{1,1}|$) significantly to the total magnetic field, while in configurations with positive values of $m$ the magnetic dipole moment is the dominant component of the total magnetic field. These figures show that the external magnetic field is nearly dipole when we adopt $m=0$ but it is not simple dipole when $m > 0$ and $m < 0$. From the left panel, we see that the $n=3$ (octupole) component reaches about a few tens of per cent of the dipole component when $m=-2.0$.
Effect of stellar rotation {#Sec:rotation}
--------------------------
We calculate two sequences with rotation for different values of $m$ in order to examine the influence of rotation. We choose the value of $\hat{\mu}_0$ by obtaining a configuration with $\hat{\Omega}_0 = 0$ and $q = 0.99$ as a non-rotating limit of our equilibrium sequence. We choose $q = 0.99$ here for simplicity. The value of $q = 0.99$ corresponds to an equilibrium configuration with $H_{sur} \sim 10^{15}$ G when we consider a typical neutron star model with negative $m$. We have obtained sequences of stationary configurations by fixing the parameters $m$ and $\hat{\mu}_0$ and changing the value of $q$. By changing the value of $q$ for a fixed value of $\hat{\mu}_0$, we have equilibrium configurations with shapes that are deformed from spheres by rotational effect in addition to the magnetic force. Since we fix the magnetic potential parameter $\hat{\mu}_0$ and $m$ along one sequence, the equilibrium sequence is the one with approximately constant magnetic effect. If the values of $m$ and $\hat{\mu}_0$ are changed, we will be able to solve another stationary sequence. We have calculated two stationary sequences with negative $m$ ($m = -1.5$) and with $m = 0.0$, i.e. $\hat{\mu} = $ constant .
$q$ $H_\mathrm{c} / H_\mathrm{sur}$ ${\cal H}_p /{\cal H}$ $|\hat{W}|$ ${\cal H}/|W|$ $\Pi/|W|$ $T/|W|$ $\alpha$ $\hat{\Omega}_0^2$ $\hat{K}$ VC
------ --------------------------------- ------------------------ ------------- ---------------- ----------- -------------------------- ---------- -------------------- ----------- --------- --
$m=-1.5$ $\hat{\mu}_0 =$ 5.070E-7
0.99 4.75E+1 0.9979 9.71E-2 1.14E-4 3.33E-1 0.00E+0 5.06E-2 0.00E+0 3.33E-6 1.80E-6
0.9 4.54E+1 0.9972 8.00E-2 1.26E-4 3.19E-1 2.14E-2 4.51E-2 1.17E-2 3.61E-6 3.41E-5
0.8 4.36E+1 0.9960 6.16E-2 1.45E-4 3.02E-1 4.67E-2 3.87E-2 2.36E-2 3.95E-6 1.34E-5
0.7 4.19E+1 0.9940 4.40E-2 1.74E-4 2.85E-1 7.30E-2 3.22E-2 3.37E-2 4.32E-6 5.70E-6
$m= 0.0$ $\hat{\mu}_0 =$ 5.520E-2
0.99 6.98E+0 0.9947 9.60E-2 2.31E-3 3.33E-1 0.00E+0 5.03E-2 0.00E+0 1.20E-4 1.85E-6
0.9 6.63E+0 0.9934 7.90E-2 2.35E-3 3.18E-1 2.14E-2 4.48E-2 1.16E-2 1.15E-4 1.24E-6
0.8 6.29E+0 0.9913 6.05E-2 2.39E-3 3.01E-1 4.73E-2 3.83E-2 2.37E-2 1.06E-4 1.36E-6
0.7 5.99E+0 0.9878 4.32E-2 2.39E-3 2.83E-1 7.40E-2 3.18E-2 3.37E-2 9.24E-5 1.51E-6
Physical quantities of stationary configurations are tabulated in Table \[tab:q-mu\]. As seen from this table, the quantities $|\hat{W}|$ and $\alpha$ or the ratio $\Pi / |W|$ and $T/|W|$ depend on the strength of the rotation. By contrast, magnetic quantities are almost unaffected by rotation. The dependence of the ratio $H_{c} / H_{sur}$ on rotation is relatively small. The equilibrium configurations with highly localized magnetic fields that we have obtained in this paper are almost unchanged even by rapid rotation. Therefore, we do not consider the effect of rotation any longer in this paper.
Effect of equations of state {#Sec:EOS}
----------------------------
Thus far, we have discussed our magnetized configurations by showing the results for $N = 1$ polytropic models. The distribution of the [*toroidal*]{} current density, however, depends on the mass density profile through equation (\[Eq:current\]). Thus we show other polytropic models, i.e. $N=0.5$ and $N=1.5$ polytropes, as well as configurations for degenerate gases, i.e. white dwarf models, in order to examine the influence of equations of state on configurations with highly localized magnetic fields.
We set $q=0.99$ for polytropes and $q = 0.999$ for degenerate gases. The degenerate model with $q=0.999$ corresponds to a configuration with a $H_{sur} \sim 1.0
\times10^{9}$G magnetized white dwarf with $m = -3.0$, the central density is $1.0 \times 10^{8} \mathrm{g cm^{-3}}$. This central density results in a white dwarf of about $1.16
M_\odot$. Neither models rotates. We calculate 11 models with fixed values for $q$ by setting $m = $ $-3.0, -2.5, -2.0, -1.5, -1.1, -0.9, -0.5, 0.0, 0.5, 1.0,
1,3$ and examine the dependence of $H_c / H_{sur}$ on the equation of state.
 
 
 
 
 
 
Fig. \[fig:m\_ratio\] displays the ratio $H_c / H_{sur}$ against the value of $m$ for different equations of state. The dependency of this ratio on the value of $m$ is qualitatively similar for these equations of state. Whichever equation of state we choose, we obtain configurations with highly localized magnetic fields, for which $H_c / H_{sur}$ can exceed 100. The same is true for white dwarfs with highly localized magnetic fields. However, $H_c / H_{sur} $ tends to become smaller for stiffer equations of state, as seen from Fig. \[fig:m\_ratio\].
Fig. \[fig:rho1\] and Fig. \[fig:rho2\] display the distribution of mass density, current density and the contour of $\log_{10}|\Vec{H}| / H_{sur}$ of $m = -0.99$ configurations. Fig. \[fig:rho1\] shows results for polytropes $N=0.5$ and $N=1.5$ (stiffest and softest equations of state among the polytropic models considered here) and Fig. \[fig:rho2\] shows results for white dwarfs with $\rho_c = 1.0 \times 10^{7} \mathrm{g cm^{-3}}$ and $\rho_c = 1.0 \times 10^{10} \mathrm{g cm^{-3}}$ (stiffest and softest among the white dwarf models considered here). As seen from top panels in each figure, the mass density distributions of the softer equation of state ($N=1.5$ and $\rho_c = 1.0 \times 10^{10} \rm{g cm^{-3}}$) are more centrally concentrated than those of the stiffer equation of state ($N=0.5$ and $\rho_c = 1.0 \times 10^{7} \rm{g cm^{-3}}$). The current density distributions are also more centrally concentrated compared with the mass density distribution (middle panels). As a result, the [*poloidal*]{} magnetic fields become more highly localized for the softer equation of state (bottom panels). The mass of the white dwarf becomes higher for the higher central density. This implies that higher mass white dwarfs can have stronger interior magnetic fields deep inside if the magnetic field structure is fixed as in the present study.
Discussion and conclusions
==========================
In this paper we have constructed axisymmetric and stationary magnetized barotropes that have extremely strong [*poloidal*]{} magnetic fields around the central region near the magnetic axis. The strength of the magnetic field in that region could be two orders of magnitude larger than that of the surface magnetic field. In the context of the neutron star physics, this would imply that there might be magnetars whose interior magnetic fields amounting to $10^{17}$ G if we assume the surface field to be order of $10^{15}$G and that there might be magnetized white dwarfs with interior magnetic fields that reach $10^{12}$ G when the mass is nearly the Chandrasekhar limit and the surface field is of the order of $10^9$ G.
Moreover, it should be noted that highly localized magnetized stars could have higher order magnetic multipole moments in addition to the dipole moment. Although in most astrophysical situations magnetic dipole fields have been assumed, we may need to consider configurations with contributions from higher multipole magnetic moments for some situations. In those cases, configurations with negative values of $m$ might be used to analyze such systems.
Higher order magnetic multipole moments with even $n$
------------------------------------------------------
It should be noted that in the analysis of this paper only higher magnetic multipole moments with odd $n = 2 \ell + 1$ where $\ell$ is an integer, i.e. $2^{2 \ell + 1}$ moments, appear and that there are no higher magnetic multipole moments with even $n = 2 \ell$. This is due to the choice of the current density. Our choice of the arbitrary function $\mu(\Psi)$ and the assumption of the symmetry of $\Psi$ about the equator necessarily result in magnetic field distributions that are symmetric about the equator. It implies that the magnetic field should penetrate the equator and that $2^{2 \ell}$ type distributions that are confined the upper or lower half of the space of the equator are excluded. In order to obtain closed magnetic field distributions in the half plane above or below the equator, the current density must be chosen so as to flow in opposite directions above and below the equatorial plane. It also implies that we need to set the current density on the equator in the $\varphi$-direction to vanish.
Concerning $2^{2 \ell}$ multipole magnetic moments distributions, [@Ciolfi_et_al_2009] have obtained such configurations. Their solutions correspond to the choice of the current density distributions that are antisymmetric about the equator.
Forms of arbitrary functions
----------------------------
One might think it curious that functions appear in the formulation and that there is no physical principle specifying how to choose those arbitrary functional forms. The same situation appears for the problem of calculating equilibrium structures or stationary structures of rotating and axisymmetric [*barotropes*]{}. For that problem, the three component equations of the equations of motion do not remain independent but come to depend on each other. This implies that one could not solve for all the three components of the flow velocity completely. Assumptions of the [*stationarity*]{} and [*barotropy*]{} reduce the problem to a degenerate problem concerning the components of the flow velocity. Although there are [*three*]{} component equations for the [*three*]{} components of the flow velocity, those three component equations are no more independent. They become dependent each other due to the [*barotropic nature*]{} of the assumption for the gas. Therefore, one needs to [*specify*]{} the [*rotation law or corresponding relation*]{} in order to find stationary or equlibrium configurations for axisymmetric barotropes. The form of the rotation law is [*arbitrary*]{}.
The only requirement for the functional form regarding the rotation law comes from the nature of the stability of the system. However, one needs to know the stability of the system beforehand. If one does not have any information about the system to be solved, one has no principle by which to choose the form of the rotation law.
The situation is the same for the [*stationary*]{} problem for axisymmetric magnetized [*barotropes*]{}. For the stationary states of axisymmetric magnetized barotropes, the situation is more complicated than that for rotating barotropes, because not only the flow velocity but also the magnetic field appears in the problem. That also leads to the appearance of a greater number of arbitrary functions in the problem. Thus it is very hard to specify the forms of arbitrary functions [*physically meaningfully*]{}. In such situations the only thing one can might be to explore many kinds of arbitrary functions to find out the general consequences of the resulting magnetic fields.
Of course, if one could obtain a lot of information of the magnetic characteristics about the equilibrium states at hand, one could constrain the arbitrary functions more appropriately and more physically meaningfully. One possibility is to rely on the stability nature of the equilibrium, as in the rotating barotropic stars. Since there is no useful stability criterion for the field configuration with both [*poloidal*]{} and [*toroidal*]{} fields and linear stability analysis of the equilibrium is beyond our scope, we leave this issue of constraining the functional form for a future study.
Application to magnetars
------------------------
The typical strength of the surface magnetic field of anomalous X-ray pulsar (AXP) and soft gamma-ray repeater (SGR) is considered to be $10^{14}-10^{15}$G by assuming the magnetic dipole spin down (see e.g. @Kouveliotou_et_al_1998; @Kouveliotou_et_al_1999; @Murakami_et_al_1999; @Esposito_et_al_2009_mn; @Enoto_et_al_2009_apj; @Enoto_et_al_2010_pasj). According to recent observational evidences, some types of AXP and SGR are regarded as similar kinds of isolated neutron star and are categorized as magnetars, although they were first considered to belong to two different types of neutron star. (see e.g. @Duncan_Thompson_1992_ApJ; @Duncan_Thompson_1996_AIPConf; @Woods_Thompson_2006; @Mereghetti_2008_aar).
For neutron stars with a strong magnetic field, such as magnetars, the strength of the maximum [*toroidal*]{} magnetic field inside has been estimated to be $10^{17}$ G (see e.g. @Thompson_Duncan_1995_mn; @Kluzniak_Ruderman_1998_apjl; @Spruit_1999; @Spruit_2009_IAUsymp). Many authors have considered that only [*toroidal*]{} magnetic fields could become extremely strong and be hidden below the surfaces of the stars. Concerning [*poloidal*]{} magnetic fields, a very strong field is not considered because it would be observed as a strong surface field since it is dipole-dominated. However, as shown in this paper, extremely strong [*poloidal*]{} magnetic fields can exist in the very central region at $r_c \sim 0.01 r_e$, as seen from Tables \[tab:m-mu\] and \[tab:q-mu\] and Fig. \[fig:m\_ratio\] and the definition of $H_c$, Equation(\[Eq:H\_c\]). If we apply our equilibrium models with negative values of $m$ to magnetars with mass $1.4M_\odot$, central density $\rho_{\max} = 1.0\times10^{15}
\mathrm{g cm^{-3}}$ and average strength of the surface magnetic fields $10^{15}$ G, the strengths of the [*poloidal*]{} magnetic fields could be $10^{16} - 10^{17}$G. We also consider weak magnetized magnetars with average strength of the surface magnetic fields $10^{13}$ G (@Rea_et_al_2010). If we apply our equilibrium models, the strengths of the [*poloidal*]{} magnetic fields could be $10^{14} - 10^{15}$G. Since these strong [*poloidal*]{} magnetic fields located nearly along the magnetic axis in the central core region, the magnetic structures in the core region are highly anisotropic. If extremely strong magnetic [*poloidal*]{} fields are hidden within the core region, there could be magnetic fields with higher order multipole moments.
If the neutron star shape is deformed by a strong magnetic field and the magnetic axis is not aligned the rotational axis, gravitational waves will be emitted (@Cutler_2002; @Haskell_Samuelsson_Glampedakis_2008; @Mastrano_et_al_2011). Gravitational wave emission tends to become stronger as the ellipticity of the meridional plane of the star becomes larger. For our models, decreasing $m$ increases the value of $1-q$ in the $H_{sur}$ constant sequence (see the value of $1-q$ in Table \[tab:m-mu\]). Thus those models with highly localized magnetic field here may be efficient emitters of gravitational wave.
Some features of highly magnetized white dwarfs
-----------------------------------------------
It is widely believed that the effect of the stellar magnetic fields play a significant role in astrophysics. For example, isolated magnetized white dwarfs tend to have a higher mass than non-magnetic white dwarfs (@Wickramasinghe_Ferrario_2000). According to observations, the surface magnetic field strength of white dwarfs varies from very little to $10^9$ G (@Wickramasinghe_Ferrario_2000). Therefore, there are some strongly magnetized white dwarfs whose surface magnetic field about $10^8$-$10^9$G. For example, [@Jordan_et_al_1998] estimated the field range $3.0 \times 10^8$-$7.0 \times 10^8$ G in GD 299. EUVE J0317-855 is a massive high-field magnetic white dwarf with rapid rotation. Its magnetic field was calculated by an offset dipole model with $4.5 \times 10^8$G and period of 725 s. PG 1031+234 is a high-field magnetized white dwarf. [@Schmidt_et_al_1986] and [@Latter_Schmidt_Green_1987] estimated its rotation period 3.4 h and its magnetic field as $5.0\times 10^{8}$ - $1.0 \times 10^9$ G. The observed spectral variations cannot be fitted well by a simple dipole magnetic or offset dipole model, so they have proposed a two-component model composed of a nearly centered dipole and a strongly off-centered dipole. In other words, the magnetic field structures of several strongly magnetized white dwarfs could not be explained by applying simple dipole structures.
We have obtained strongly magnetized white dwarfs with higher order magnetic multipole moments in this paper. If we apply our configurations with negative $m$, some strongly magnetized star such as PG 1031+234 may have strong interior magnetic fields. According to our numerical results, $H_{c}$ could reach as high as $10^{12}$ G when $H_{sur} \sim 3.0 \times 10^{9}$ G for a highly localized ($m = -3.0$) and high mass ($\rho_c = 1.0 \times 10^9$, $M \sim 1.34 M_\odot$) model (see Fig. \[fig:m\_ratio\]). Since the central magnetic field strength $H_c$ depends on the equation of state as we have shown in Sec.\[Sec:EOS\], it becomes higher as the central density increases. Thus high mass white dwarfs could have strong [*poloidal*]{} magnetic fields according to our models with negative $m$. As we have displayed in Sec. \[Sec:current\], $N=1.5$ polytropes with negative values of $m$ have rather large higher order magnetic multipole moments. The same is the case for magnetized white dwarf models, i.e. they have rather large higher order magnetic multipole moments. Therefore, the magnetic fields outside of such stars are far from simple dipole fields if the magnetized white dwarfs have highly localized strong [*poloidal*]{} magnetic fields deep inside the stars.
Comments on stability of magnetized barotropes
----------------------------------------------
Once equilibrium configurations are obtained, it would be desirable to investigate their stability. However, a satisfactory formulation for the linear stability analysis for general magnetic configurations has not been fully developed, although there is a stability criterion only for purely [*toroidal*]{} magnetic configurations (@Tayler_1973_mnras). For purely [*poloidal*]{} or mixed [*poloidal-toroidal*]{} magnetic configurations, magnetic configurations with rotation or other general situations, no authors have ever succeeded in obtaining a clear stability criterion (see e.g. @Markey_Tayler_1973_mnras; @Wright_1973_mnras; @Markey_Tayler_1974_mn; @Tayler_1980_mn ; @BonannoUrpin_2008_AA). Therefore the stability of the configurations obtained in this paper contain both [*poloidal*]{} and [*toroidal*]{} magnetic fields has not been investigated.
By contrast, the stability of magnetized stars may be investigated through that time-dependent evolutionary computations of the magnetic configurations. Thanks to powerful computers, some authors have recently employed magnetohydrodynamical codes to follow the time evolutions of magnetized configurations and find out whether these configurations would settle down to certain ’stable equilibrium states’. Such investigations concerning the magnetic configurations have been carried out by Braithwaite and his coworkers as mentioned in Introduction (see e.g. @Braithwaite_Spruit_2004; @Braithwaite_Nordlund_2006; @Braithwaite_Spruit_2006; @Braithwaite_2006; @Braithwaite_2007; @Braithwaite_2009; @Duez_Braithwaite_Mathis_2010). According to their results, purely [*toroidal*]{} configurations and purely [*poloidal*]{} configurations are shown to be all unstable, as previously shown or expected (e.g. @Tayler_1973_mnras; @Markey_Tayler_1973_mnras; @Flowers_Ruderman_1977_apj. However, see @Geppert_Rheinhardt_2006 for some results about stability). Concerning the mixed [*poloidal-toroidal*]{} magnetic configurations, recent numerical studies (@Braithwaite_2009; @Duez_Braithwaite_Mathis_2010) have shown that they are stable as long as the following condition is satisfied: $$\begin{aligned}
\alpha_0 \frac{{\cal H}}{|W|} \le \frac{{\cal H}_p}{{\cal H}} \le 0.8 \ ,
\label{criterion_by_braithwaite}\end{aligned}$$ where $\alpha_0$ is a numerical factor of $10 - 10^3$ depending on the stellar structures. By performing 3D MHD simulations, it has been shown that non-axisymmetric perturbations to equilibrium stars grow when this condition is not satisfied. Stars with mixed magnetic fields whose dominant component is [*poloidal*]{} field seem to evolve toward non-axisymmetric configurations until the amplitude of the perturbations reach nonlinear regime and saturate. As can be seen from tables in this paper, we havefound no models that satisfy that criterion (equation \[criterion\_by\_braithwaite\]) for our particular choice of functional forms presented above (see Sec.\[sec:bc\]), because the energy stored in the [*toroidal*]{} magnetic field is at most a few per cent for all of our models. In order to obtain configurations that satisfy the criterion, we need to choose different functional forms from those used in this paper. We should be careful to apply the criterion, however, to general configurations of magnetic fields. The class of solutions with both [*toroidal*]{} and [*poloidal*]{} magnetic fields obtained here may be rather different from the ones studied by Braithwaite and his collaborators, even if they share the obvious characteristics of twisted-torus structures of magnetic fields. As is seen in completely different stability natures of seemingly similar configurations in [@Geppert_Rheinhardt_2006] and [@Braithwaite_2007], it is quite uncertain at this moment that failure to satisfy the criterion (equation \[criterion\_by\_braithwaite\]) for our models here means unstable nature of them. It would be interesting to study the stability nature of our configurations thorough either linear perturbation analysis or direct MHD simulations.
Conclusions
-----------
In this paper, we have presented an extended formulation for obtaining axisymmetric and stationary barotropic configurations with both the [*poloidal*]{} and [*toroidal*]{} magnetic fields. We have shown the possibility that magnetized stars have strong [*poloidal*]{} magnetic fields inside the star. Our findings and conjectures can be summarized as follows.
1. By choosing the functional form for one of the arbitrary functions that appear in the basic formulation for the configurations under the assumptions mentioned before, we have obtained magnetized configurations in which extremely strong [*poloidal*]{} fields are confined within the central part of the near axis region. When we apply our models to magnetars, the interior magnetic strength would be $10^{17}$ G while the surface magnetic strength is $10^{14}$ - $10^{15}$ G. On the other hand, if we apply our models to magnetized white dwarfs with mass $\sim 1.34 M_\odot$, the surface field strength would be $10^{9}$ G and $H_c$ reaches $10^{12}$ G.
2. If stars have extremely strong [*poloidal*]{} magnetic fields deep inside, the contours of magnetic field strengths are not spherical but rather column-like shapes as shown in the figures.
3. If stars have extremely strong magnetic fields deep inside, contributions from higher order magnetic multipole moments to the outer fields around the stars cannot be neglected. This implies that if stars have highly localized and extremely strong magnetic fields deep inside, then observations of magnetic fields around the stars could not be explained by the simple dipole models that have been used in most situations.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
---------------
We would like to thank Dr. R. Takahashi for his discussion while we were extending the formulation of the present paper. KF would like to thank Dr. K. Taniguchi for discussion and comments on this paper. We would also like to thank the anonymous reviewer for useful comments and suggestions that help us to improve this paper. This research was partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (20540225) and by Grand-in-Aid for JSPS Fellows.
[65]{} natexlab\#1[\#1]{}
M., [Bonazzola]{} S., [Gourgoulhon]{} E., [Novak]{} J., 1995, , 301, 757
A., [Urpin]{} V., 2008, , 477, 35
J., 2006, , 453, 687
—, 2007, , 469, 275
—, 2008, , 386, 1947
—, 2009, , 397, 763
J., [Nordlund]{} [Å]{}., 2006, , 450, 1077
J., [Spruit]{} H. C., 2004, , 431, 819
—, 2006, , 450, 1097
S., 1939, [An Introduction to the Study of Stellar Structure]{}., Chicago Univ. Press, Chicago, IL
—, 1956, Proceedings of the National Academy of Science, 42, 1
S., [Fermi]{} E., 1953, , 118, 116
S., [Prendergast]{} K. H., 1956, Proceedings of the National Academy of Science, 42, 5
R., [Ferrari]{} V., [Gualtieri]{} L., [Pons]{} J. A., 2009, , 397, 913
C., 2002, , 66, 084025
V., [Braithwaite]{} J., [Mathis]{} S., 2010, , 724, L34
V., [Mathis]{} S., 2010, , 517, A58+
R. C., [Thompson]{} C., 1992, , 392, L9
—, 1996, in [R. E. Rothschild & R. E. Lingenfelter]{}, eds., AIP Conf. Proc., 366, High Velocity Neutron Stars and Gamma-Ray Bursts. Am. Inst. Phys., New York, p. 111
T. et al., 2009, , 693, L122
T. et al., 2010, , 62, 475
P. et al., 2009, , 399, L44
V. C. A., 1954, , 119, 407
E., [Ruderman]{} M. A., 1977, , 215, 302
U., [Rheinhardt]{} M., 2006, , 456, 639
I., 1986, , 61, 479
B., [Samuelsson]{} L., [Glampedakis]{} K., [Andersson]{} N., 2008, , 385, 531
K., [Sasaki]{} M., 2004, , 600, 296
S., [Schmelcher]{} P., [Becken]{} W., [Schweizer]{} W., 1998, , 336, L33
K., [Yoshida]{} S., 2008, , 78, 044045
W., [Ruderman]{} M. A., 1998, , 505, L113
C., [Dieters]{} S., [Strohmayer]{} T., [van Paradijs]{} J., [Fishman]{} G. J., [Meegan]{} C. A., [Hurley]{} K., [Kommers]{} J., [Smith]{} I., [Frail]{} D., [Murakami]{} T., 1998, , 393, 235
C., [Strohmayer]{} T., [Hurley]{} K., [van Paradijs]{} J., [Finger]{} M. H., [Dieters]{} S., [Woods]{} P., [Thompson]{} C., [Duncan]{} R. C., 1999, , 510, L115
S. K., [Jones]{} D. I., 2009, , 395, 2162
W. B., [Schmidt]{} G. D., [Green]{} R. F., 1987, , 320, 308
R. V. E., [Mehanian]{} C., [Mobarry]{} C. M., [Sulkanen]{} M. E., 1986, , 62, 1
P., [Tayler]{} R. J., 1973, , 163, 77
—, 1974, , 168, 505
A., [Melatos]{} A., [Reisenegger]{} A., [Akg[ü]{}n]{} T., 2011, , 417, 2288
S., 2008, Astron Astrophys Rev, 15, 225
M. J., 1973, , 22, 413
—, 1975, , 35, 349
T., [Kubo]{} S., [Shibazaki]{} N., [Takeshima]{} T., [Yoshida]{} A., [Kawai]{} N., 1999, , 510, L119
J. P., [Hartwick]{} F. D. A., 1968, , 153, 797
J., [Takahashi]{} R., [Eriguchi]{} Y., 2009, , 396, 2152
K. H., 1956, , 123, 498
N., [Esposito]{} P., [Turolla]{} R., [Israel]{} G. L., [Zane]{} S., [Stella]{} L., [Mereghetti]{} S., [Tiengo]{} A., [G[ö]{}tz]{} D., [G[ö]{}[ğ]{}[ü]{}[ş]{}]{} E., [Kouveliotou]{} C., 2010, Science, 330, 944
G. D., [Stockman]{} H. S., [Grandi]{} S. A., 1986, , 300, 804
H. C., 1999, , 341, L1
—, 2009, [K. G. Strassmeier, A. G. Kosovichev & J. E. Beckman]{}, eds. Proc. IAU Symp. 259, Cosmic Magnetic Fields: From Planets, to Stars and Galaxies. Cambridge Univ. Press, Cambridge, p.61
J. L., 2000, Stellar Rotation. Cambridge Univ. Press, New York, p.18
R. J., 1973, , 161, 365
—, 1980, , 191, 151
C., [Duncan]{} R. C., 1995, , 275, 255
Y., [Eriguchi]{} Y., 2005, , 359, 1117
V. A., [Ray]{} A., 1994, , 267, 1000
D. G., 1961, , 133, 170
D. T., [Ferrario]{} L., 2000, , 112, 873
L., 1959, , 130, 400
—, 1959, , 130, 405
—, 1960, , 131, 227
P. M., [Thompson]{} C., 2006, in Lewin W., van der Kils M., eds, Compact Stellar X-ray Sources, Cambridge Univ. Press, New York, p.547
G. A. E., 1973, , 162, 339
S., [Eriguchi]{} Y., 2006, , 164, 156
S., [Yoshida]{} S., [Eriguchi]{} Y., 2006, , 651, 462
Numerical method
================
Dimensionless quantities {#App:dimensionless}
------------------------
In this paper, physical quantities are used in their dimensionless forms as follows: $$\begin{aligned}
\hat{\phi}_g \equiv \frac{\phi_g}{4 \pi G r_e^2 \rho_{\mathrm{max}}} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{\Omega} \equiv \frac{\Omega}{\sqrt{{4 \pi G \rho_{\mathrm{max}}}}} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{\kappa} \equiv \frac{\kappa}{\sqrt{4\pi G} r_e^2 \rho_{\max}} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{\mu} \equiv \frac{\mu}{\sqrt{4 \pi G}/r_e} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{H}_{suffix} \equiv \frac{H_{suffix}}{\sqrt{4 \pi G }
r_e \rho_{\max} } \ ,\end{aligned}$$ $$\begin{aligned}
\hat{A}_\varphi \equiv \frac{A_\varphi}{\sqrt{4 \pi G }r_e^2 \rho_{\max} } \ ,\end{aligned}$$ $$\begin{aligned}
\hat{\Psi} \equiv \frac{\Psi}{\sqrt{4 \pi G }r_e^3 \rho_{\max} } \ ,\end{aligned}$$ $$\begin{aligned}
\hat{K} \equiv \frac{K}{4 \pi G r_e^6 \rho_{\max}^2 } \ ,\end{aligned}$$ $$\begin{aligned}
\hat{j}_\varphi \equiv \frac{j_\varphi}{\sqrt{4\pi G}\rho_{\max} c} \ .\end{aligned}$$ $$\begin{aligned}
\hat{C} &\equiv& \frac{C}{4 \pi G r_e^2 \rho_{\max}} \ .\end{aligned}$$ Here $H_{suffix}$ is the component of the magnetic field where $suffix$ may be $c$ (center), $sur$ (surface), $p$ ([*poloidal*]{}) and $t$ ([*toroidal*]{}). Similarly we define normalized global quantities as follows: $$\begin{aligned}
\hat{M} = \frac{M}{r_e^3 \rho_{\max}} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{W} = \frac{W}{4\pi G r_e^5 \rho_{\max}^2} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{T} = \frac{T}{4\pi G r_e^5 \rho_{\max}^2} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{\Pi} = \frac{\Pi}{4\pi G r_e^5 \rho_{\max}^2} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{U} = \frac{U}{4\pi G r_e^5 \rho_{\max}^2} \ ,\end{aligned}$$ $$\begin{aligned}
\hat{{\cal H}} = \frac{{\cal H}}{4\pi G r_e^5 \rho_{\max}^2} \ .\end{aligned}$$ We also define dimensionless forms of arbitrary functions as follows: $$\begin{aligned}
\hat{\kappa}(\hat{\Psi}) =
\left\{
\begin{array}{lr}
0 \ , & \mathrm{for} \hspace{10pt} \hat{\Psi} \leq \hat{\Psi}_{\max} \ , \\
\dfrac{\hat{\kappa}_0}{k+1}(\hat{\Psi} - \hat{\Psi}_{\max})^{k+1} \ ,
& \mathrm{for} \hspace{10pt} \hat{\Psi} \geq \hat{\Psi}_{\max} \ ,
\end{array}
\right.\end{aligned}$$ $$\begin{aligned}
\D{\hat{\kappa}(\hat{\Psi})}{\hat{\Psi}} =
\left\{
\begin{array}{lr}
0 \ , & \mathrm{for} \hspace{10pt} \hat{\Psi} \leq \hat{\Psi}_{\max} \ , \\
\hat{\kappa}_0(\hat{\Psi} - \hat{\Psi}_{\max})^{k} \ , & \mathrm{for} \hspace{10pt} \hat{\Psi} \geq \hat{\Psi}_{\max} \ .
\end{array}
\right.\end{aligned}$$ for $\kappa$ and $$\begin{aligned}
\hat{\mu}(\hat{\Psi}) & = & \hat{\mu}_0 (\hat{\Psi} + \hat{\epsilon} )^m \ , \\
\int \hat{\mu}(\hat{\Psi}) \, d \hat{\Psi} & = & \frac{\hat{\mu}_0}{m+1}(\hat{\Psi} + \hat{\epsilon} )^{m+1} \ .\end{aligned}$$ for $\mu$. We choose $k=0.1$, $\hat{\kappa}_0 = 10$ and $\hat{\epsilon}= 1.0 \times 10^{-6}$ and keep their values fixed during all calculations in this paper.
Computational grids {#App:grid}
-------------------
We describe the details of our numerical grid points. In order to resolve the distributions of the source term of the vector potential equation without loss of accuracy, we choose the following non-uniformly distributed grid points in the actual numerical computations. In the $\hat{r}$-direction, we divide the whole space into two distinct regions: $[0, 1.0]$ (region 1), and $[1.0, 2.0]$ (region 2). In each region, the following mesh points are defined: $$\begin{aligned}
\hat{r}_i = w_i^2 \
\left\{
\begin{array}{lll}
\ w_i = (i-1) \Delta w_1 \ ,\
& \Delta w_1 \equiv \dfrac{\sqrt{1} - \sqrt{0}}{n_1-1} \ ,
& \mathrm{for} \hspace{10pt} 1 \leq i \leq n_1 \ , \\
%
\ w_i = 1.0 + (i - n_1) \Delta w_2 ,\
& \Delta w_2 \equiv \dfrac{\sqrt{2} - \sqrt{1}}{n_2-1} ,\
& \mathrm{for} \hspace{10pt} n_1 \leq i \ .
\end{array}
\right.
\label{Eq:r_mesh}\end{aligned}$$ where $n_1$ and $n_2$ are the mesh numbers defined as follows: $$\begin{aligned}
n_1 & \equiv & \frac{3}{4} (n_r - 1) +1 \ , \\
n_2 & \equiv & \frac{1}{4} (n_r - 1) +1 \ .\end{aligned}$$ Here $n_r$ is the total mesh number in the $r$-direction. In practice, since we use a difference scheme of the second-order accuracy for the derivative and Simpson’s integration formula, we divide each mesh interval defined above further into two equal size intervals in the $r$ coordinate. We use $n_r = 513$ and thus the actual total number of the mesh points is $(2 n_r - 1) = 1025$.
Concerning the $\theta$-direction, we have to resolve the region near the axis, because for $m<0$ values the magnetic fields seem to be highly localized to the axis region. In order to treat such magnetic fields near the axis region, we introduce the following mesh in the $\theta$-direction: $$\begin{aligned}
\theta_j = \lambda_j^2 \ , \ \lambda_j = (j-1) \ \Delta \lambda \ ,
\ 1 \le j \le n_\theta \ , \ \Delta \lambda = \frac{ \sqrt{\pi /2}}{n_\theta-1} \ ,\end{aligned}$$ where $n_\theta$ is the total mesh number in the $\theta$-direction. We also divide each mesh interval defined above further into two equal size intervals. Then, we use $n_\theta = 513$ and thus the actual total number of the mesh points is $1025$. Fig. \[fig:mesh\] shows the relations between the order of the grid points and the $r$- or $\theta$-coordinate value.
 
[^1]: E-mail: fujisawa@ea.c.u-tokyo.ac.jp
|
---
abstract: 'We investigate numerically the relaxation dynamics of an elastic string in two-dimensional random media by thermal fluctuations starting from a flat configuration. Measuring spatial fluctuations of its mean position, we find that the correlation length grows in time asymptotically as $\xi \sim (\ln t)^{1/\tilde\chi}$. This implies that the relaxation dynamics is driven by thermal activations over random energy barriers which scale as $E_B(\ell) \sim \ell^{\tilde\chi}$ with a length scale $\ell$. Numerical data strongly suggest that the energy barrier exponent $\tilde{\chi}$ is identical to the energy fluctuation exponent $\chi=1/3$. We also find that there exists a long transient regime, where the correlation length follows a power-law dynamics as $\xi \sim t^{1/z}$ with a nonuniversal dynamic exponent $z$. The origin of the transient scaling behavior is discussed in the context of the relaxation dynamics on finite ramified clusters of disorder.'
author:
- Jae Dong Noh
- Hyunggyu Park
title: Relaxation dynamics of an elastic string in random media
---
Interaction and quenched disorder are essential ingredients in condensed matter physics. Many-body systems may undergo a phase transition as a cooperative phenomenon mediated by interaction. When quenched disorder comes into play, there may emerge a glass phase in which degrees of freedom are pinned by random impurities and a slow relaxation dynamics appears. An elastic string in random media is one of the simplest systems where the interplay between interaction and quenched disorder yields a nontrivial effect [@Kardar87; @Fisher91]. This has been studied extensively in literatures since it is relevant to many interesting physical systems such as a growing interface [@KPZ86], a domain wall in random magnets [@Huse85; @Bray9402], and a magnetic flux line in superconductors [@Kardar97].
In low temperatures, one may approximate an elastic string as an elastically coupled directed polymer where no overhang is allowed. Then it can be described by a single valued function $\bm{x}(u)$, where $\bm{x} \in \mathcal{R}^d$ and $u \in \mathcal{R}$ are the transverse and the longitudinal coordinates to the polymer direction, respectively, in a $(d+1)$ dimensional space. The energy of a polymer of length $L$ in a configuration $\bm{x}(u)$ with $0\leq u \leq L$ is given by the Hamiltonian $$\mathcal{H} = \int_0^L du \left[ \frac{1}{2}
\left| \frac{\partial \bm{x}}{\partial u}\right|^2 +
V(u,\bm{x}(u))\right] \ .$$ The first term accounts for an elastic tension and the second term $V(u,\bm{x})$ is a random pinning potential with short-range correlations.
Equilibrium properties of the directed polymer in random media (DPRM) are rather well understood. The tension favors a flat state, while thermal fluctuations and the disorder potential favor a rough state. The competition between them leads to the scaling law $|\Delta {\bf x}| \sim L^{\zeta}$ for the transverse fluctuation (interface roughness) and $\Delta E \sim L^{\chi}$ for the (free) energy fluctuation. The quenched disorder is relevant for $d\leq 2$, and the polymer is in a super-rough phase ($\zeta > 1/2$) at all temperatures. Especially for $d=1$, the scaling exponents are known exactly as $\zeta_{1D} = 2/3$ and $\chi_{1D} = 1/3$ [@Huse85]. For $d>2$, it is believed that there is a transition from a super-rough phase into a normal-rough phase ($\zeta = 1/2$) as the temperature $T$ increases [@Fisher91]. In the latter, the thermal fluctuations dominate over the disorder fluctuations, while vice versa in the former.
When a polymer is in a nonequilibrium state, e.g., a flat configuration, it will relax to the equilibrium rough state. Without disorder or in the normal-rough phase with disorder for $d>2$, the elastic polymer equilibrates diffusively. The correlation length $\xi$ in the longitudinal $u$ direction grows algebraically in time as $\xi \sim t^{1/z_o}$ with the dynamic exponent $z_o=2$. In the presence of the quenched disorder, one expects a slower relaxation because random impurities tend to trap the polymer into metastable states in local energy valleys. Upon equilibration, the polymer has to overcome energy barriers $E_B$ separating those valleys through thermal fluctuations to approach the true equilibrium state. It is believed that the energy barrier height scales as $E_B(\ell) \sim \ell^{\tilde\chi}$ in a region with linear size $\ell$.
Thermal activations allow the correlated polymer segment of length $\xi$ to overcome the energy barriers in a time scale $t_\xi \sim e^{E_B(\xi)/T}$. Then, it follows that the correlation length $\xi$ grows as $$\label{eq:corr_length}
\xi(t) \sim (T\ln t)^{1/\tilde\chi}, \$$ with the universal energy barrier exponent $\tilde\chi$, independent of the disorder strength and the temperature [@Huse85]. Assuming that there is only a single relevant energy scale in this system, the exponent $\tilde\chi$ should be equal to the energy fluctuation exponent $\chi$ [@Huse85]. This conjecture is supported at least in low dimensions [@Drossel95].
However, even for $d=1$, there is a long-standing controversy on the scaling law of Eq. (\[eq:corr\_length\]). Numerical simulation study [@Kolton05] reports a signature of the expected logarithmic scaling, but only after a long and clean intermediate power-law scaling regime where $\xi\sim t^{1/z}$ with a nonuniversal dynamic exponent $z$, whose origin is not clear. Moreover, the scaling exponent associated with the logarithmic scaling seems different from the conjectured value of $\tilde\chi=\chi=1/3$ [@Kolton05]. There is also a recent claim of $\tilde\chi=d/2$ based on the droplet theory [@Monthus08]. Besides, there are many numerical works in the context of domain wall coarsening dynamics in two dimensional random ferromagnets [@Bray9402], which seem to support the nonuniversal power-law scaling without any signature of the asymptotic logarithmic scaling [@OC86; @PPR04]. These intriguing results are also left unexplained.
In this work, we study the relaxation dynamics of the DPRM in $(1+1)$ dimensional lattices with extensive numerical simulations and a scaling theory approach. Measured are the spatial fluctuations of the mean position of the polymer, from which we derive the dynamic scaling behavior of the correlation length $\xi(t)$ through a simple scaling hypothesis. The purpose of this study is to settle down the controversy by providing a decisive numerical evidence on the asymptotic relaxation dynamics of the DPRM for $d=1$. In addition, we suggest a reasonable scenario for the origin of the transient power-law scaling regime.
We consider a discrete model for the DPRM in the 45-degree-rotated square lattices of size $L\times M$. Each lattice site is represented as $(i,x)$ with the longitudinal coordinate $i=0,\cdots,L-1$ and the transverse coordinate $x=-M/2+1,\cdots,M/2$ with the constraint $i=x$ in modulo 2. Assigned to bonds are quenched disorder variables $J$ which are distributed independently and randomly according to a probability density function $p(J)$. The polymer of length $L$ is placed along the bonds and directed in the longitudinal direction without any back bending. Then its configuration is described by the fluctuating variables $\{x(i)\}$ with the solid-on-solid (SOS) constraint of $|x(i)-x(i\pm 1)|=1$ for all $i$. We adopt the periodic boundary condition in the longitudinal direction, i.e., $x(L) = x(0)$. The transverse size $M$ is taken to be large enough ($M=4096 \sim 8192$) to avoid any possible interference.
The polymer energy is given by the lattice Hamiltonian $$\label{Hamiltonian}
\mathcal{H} = \sum_{i=1}^L J(i,x(i);i+1,x(i+1)) \ ,$$ where $J(i,x;i+1,x')$ denotes the disorder strength of the bond between neighboring sites $(i,x)$ and $(i+1,x')$. In this study, we consider the uniform distribution in the range $-1\leq J\leq 1$ and the bimodal distribution $p(J) = f \delta(J+1/2) + (1-f) \delta (J-1/2)$ with a model parameter $f$. It turns out that both cases lead to the same conclusion.
We start with the flat configuration with $\{x(i) = i\mod{2}\}$ at $t=0$ and study its relaxation dynamics toward the equilibrium state at temperature $T$. We adopt the Glauber Monte Carlo dynamics: First select a site $i$ at random and try to flip $x(i)
\rightarrow x(i)\pm 2$ with probability $1/2$, respectively. Unless the trial violates the SOS constraint, it is accepted with the probability of $\min[1,e^{-\Delta E/T}]$ where $\Delta E$ is the energy change. The time is incremented by one unit after $L$ such trials.
We focus on the spatial dispersion of the mean position $\overline{x} \equiv \sum_{i} x(i)/L$ as function of elapsed time $t$, which is given as $$\label{eq:disp_x}
(\Delta x)^2 (t) \equiv \left[ \left\langle \left(\overline{x}(t) -
\overline{x}(0)\right)^2 \right\rangle_T\right]_D \ .$$ Here $\langle \cdot\rangle_T$ and $[ \cdot ]_D$ denote the thermal and disorder average, respectively. Let $\xi(t)$ be the characteristic correlation length of the polymer in the longitudinal direction. At time $t$, each segment of size $\xi(t)$ equilibrates with transverse displacement of the order $\delta x \sim \xi^\zeta$ with the roughness exponent $\zeta$. When $\xi \ll L$, each segment is independent and the total displacement is given by $$\label{eq:scaling_form}
(\Delta x)^2 \sim \frac{ \xi^{2\zeta} }{ (L/\xi) }
\sim \frac{\xi^{1+2\zeta}}{L} \ .$$ Utilizing this relation, we can derive the correlation length $\xi(t)$ from the ensemble-averaged global quantity $(\Delta x)^2(t)$, which usually bears a better statistics than the distance-dependent correlation function of the transverse displacement.
When the polymer fully equilibrates, i.e., $\xi(t=\tau) \simeq L$ with the relaxation time $\tau=\tau(L)$, the polymer as a whole (the mean position) starts to diffuse normally in the transverse direction. One expects that $(\Delta x)^2$ grows linearly in time scaled by $\tau$ as $$\label{eq:scaling_form_s}
(\Delta x)^2 \sim L^{2\zeta} \left(\frac{t}{\tau}\right) \ .$$ Without disorder, the motion of the polymer is governed by the linear Edward-Wilkinson (EW) equation [@EW82]. The EW class is characterized by $\zeta=1/2$ and $\xi \sim t^{1/2}$, i.e., $\tau\sim L^2$. In this case, the dynamic behaviors before and after equilibration, Eqs. (\[eq:scaling\_form\]) and (\[eq:scaling\_form\_s\]), follow the same scaling law $(\Delta x)^2 \sim t/L$ at all $t$. Indeed, this coincides with the exact solution of the EW equation. However, in general with disorder, these two scaling laws are distinct.
![Monte Carlo simulation data for the system with the bimodal disorder distribution with $f=0.25$ and at $T=1.0$. Different symbols represent data for different polymer lengths $L$. The data are averaged over $N_S=5000$ disorder samples.[]{data-label="fig1"}](fig1.eps){width="\columnwidth"}
![Effective exponent plots for the data given in Fig. \[fig1\]. Our estimates are $\alpha=0.63\pm 0.03$ in regime II and $\phi=6.8\pm 0.5$ in regime III.[]{data-label="fig2"}](fig2.eps){width="\columnwidth"}
We have performed extensive Monte Carlo simulations to examine the scaling property of $(\Delta x)^2$. Figure \[fig1\] shows a plot of the numerical data with the bimodal disorder distribution with $f=0.25$ at $T=1.0$. As the scaling form predicts in Eq. (\[eq:scaling\_form\]), $(\Delta x)^2$ is inversely proportional to $L$ for $t<\tau(L)$, so it is convenient to plot $L (\Delta x)^2$ versus $t$, where all curves with different $L$ collapse into one scaling curve for $t<\tau(L)$ and then start to deviate and show the finite size effects given by Eq. (\[eq:scaling\_form\_s\]).
We find that there exist four distinct regimes: (I) For $t<t_0 (\sim 10^1)$, the polymer moves diffusively as $L (\Delta x)^2 \sim t$. In this regime, the polymer behaves as in the EW class since it does not feel the disorder pinning as yet. (II) For $t_0 < t < t_c(\sim 10^{4 \sim 5}) $, the polymer is affected by the disorder and exhibits a power-law scaling behavior as $L (\Delta x)^2 \sim t^{\alpha}$ with a nonuniversal exponent $\alpha$. The crossover time $t_c$ is very large but finite and independent of $L$, which implies that this power-law scaling is transient. (III) For $t_c < t < \tau(L)$, there is a continuous downward curvature in the plot suggesting a possible logarithmic scaling with $L (\Delta x)^2
\sim (\ln t)^\phi$. The crossover time $\tau(L)$ increases indefinitely with $L$, which implies that this regime should be the true asymptotic scaling regime. (IV) For $t> \tau(L)$, the polymer displays a diffusive motion with a size-dependent diffusion amplitude as in Eq. (\[eq:scaling\_form\_s\]).
We investigate the scaling behavior in each regime quantitatively. Useful are the effective exponents defined as $\alpha_{eff}(t) \equiv d \ln (L (\Delta x)^2 ) / d \ln t$ and $\phi_{eff}(t) \equiv d
\ln (L (\Delta x)^2 ) / d \ln \ln t$. If $L (\Delta x)^2 \sim t^{\alpha}$ as in the regime II, one would obtain that $\alpha_{eff}(t) =
\alpha$ and $\phi_{eff}(t) = \alpha \ln t$. On the other hand, if $L (\Delta x)^2 \sim (\ln t)^\phi$ as in the regime III, one would obtain that $\alpha_{eff}(t) = \phi / \ln t$ and $\phi_{eff}(t) =
\phi$.
We plot the effective exponents in Figs. \[fig2\] (a) and (b). In (a), one can clearly see a plateau at $\alpha_{eff} = 0.63\pm 0.03$ for $10^1 \lesssim
t \lesssim 10^4(=t_c)$ (regime II). For $t>t_c$, it continuously decreases in the regime III before hiking up in the regime IV as expected. In (b), there appears a plateau at $\phi_{eff} = 6.8 \pm 0.5$ in the regime III, which widens as $L$ increases. These numerical evidences lead to a definitive conclusion that the asymptotic motion of the polymer follows the logarithmic (not power-law) scaling as $$\label{eq:log_scaling_Dx}
L (\Delta x)^2 (t) \sim (\ln t)^\phi \ ,$$ with the exponent $\phi=6.8\pm 0.5$. In terms of the correlation length, we find, using Eq. (\[eq:scaling\_form\]), $$\label{eq:log_scaling_xi}
\xi (t) \sim (\ln t)^{1/\tilde\chi} \ ,$$ where $1/\tilde\chi=\phi/(1+2 \zeta)=2.9\pm0.2$. The estimated value of $\tilde\chi=0.34\pm 0.03$ clearly favors the conjecture value of $\tilde\chi=\chi=1/3$ and invalidates the recent claim of $\tilde\chi=1/2$ (or equivalently $\phi=14/3$).
![Effective exponent plots with the uniform disorder distributions with the temperature $T=0.25$ in (a) and $T=0.75$ in (b), and the bimodal disorder distribution with $f=0.1$ and $T=1$ in (c) and $f=0.5$ and $T=1$ in (d). The symbols have the same attribute as in Fig. \[fig1\]. The dashed lines are drawn at $\phi_{eff}=7$.[]{data-label="fig3"}](fig3.eps){width="\columnwidth"}
In order to examine the universality of the logarithmic scaling behavior, we have performed the simulations with different values of $f$ in the case of the bimodal disorder distribution and also in the case of the uniform disorder distribution. The effective exponents are then presented in Fig. \[fig3\]. The plateaus around $\phi_{eff}=7.0$ shown in Figs. \[fig3\](a), (c), and (d) confirm the universality of the logarithmic scaling as well as the universality of its exponent $\tilde\chi$. The plot in Fig. \[fig3\](b) seems to be incompatible with $\phi=7$. Comparing Figs. \[fig3\](a) and (b), one can notice that the finite-size behavior (regime IV) sets in earlier at the higher temperature. So the logarithmic scaling regime is observed only when $L\ge 256$ in (b), while it is already evident at $L= 64$ in the other cases. This suggests that one would need larger polymers at higher temperatures.
It is puzzling why there exists the extremely long transient regime II where the polymer relaxation seems to follow a power-law scaling such as $L (\Delta x)^2 \sim t^\alpha$ or $\xi \sim t^{1/z}$ with $z=(1+2\zeta)/\alpha$. Moreover the exponent $\alpha$ is nonuniversal and varies with the disorder strength and the temperature. Such a transient behavior was also reported in Ref. [@Kolton05; @power-law-literatures], but its origin has never been explored. We suggest one reasonable scenario as below.
It is convenient to consider the bimodal disorder distribution. When $f< 1/2$, the energetically favorable bonds with $J=-1$ may play the role of local pinning centers for the polymer [@comment2]. As the polymer considered here is directed, relevant are the directed percolation clusters of the pinning bonds. These clusters are ramified but finite in size, as the bond density $f$ is smaller than the directed percolation threshold $f_c \simeq 0.6449$ in the square lattice [@Jensen99]. The characteristic size and the mean distance between them are denoted by $l_0$ and $l_1$, respectively. After the initial diffusive motion, polymer segments are trapped by those clusters independently as long as the correlation length is smaller than the cluster size ($\xi <l_0$).
The pinning mechanism in fractal-like ramified lattices is different from that in bulks. The energy barrier height in such lattices is shown to scale logarithmically with a length scale $\ell$ as $E_B(\ell) \simeq E_0 \ln \ell$ with an universal constant $E_0$ depending only on the ramification degree [@Henley85]. Then, the time scale associated with the thermal activation of the correlated segment of length $\xi$ is given by $t_\xi \sim e^{E_B(\xi)/T}\sim \xi^{E_0/T}$. This yields the power-law growth of the correlation length as $\xi \sim \sqrt{t/t_\xi} \sim
t^{1/z}$ with the nonuniversal dynamic exponent $z=2+E_0/T$. The temperature dependence seems consistent with our numerical estimates for $z$ (not shown here). When $\xi$ exceeds $l_0$, the polymer segments are pinned by a few pinning clusters. If $\xi\ge l_1$, then the polymer starts to be pinned collectively, and the transient power-law scaling behavior crosses over to the asymptotic logarithmic scaling behavior.
Finally, we add one remark on the domain coarsening dynamics in the two-dimensional random ferromagnets. When the system is quenched well below an ordering temperature from a disordered state, the characteristic size $R$ of ordered domains increases and the domain wall motion may be described by the DPRM. Hence, it is natural to expect that $R(t) \sim \xi^{2-\zeta}\sim
(\ln t)^{(2-\zeta)/\tilde\chi}$ [@Huse85]. Surprisingly, recent high accuracy numerical simulation studies report that $R(t) \sim t^{1/z}$ with a nonuniversal exponent $z$ [@PPR04]. Our result suggests that those behaviors may be due to the pinning of domain walls by finite pinning clusters in the transient regime.
In summary, we have investigated numerically the relaxation dynamics of the DPRM. The numerical data show unambiguously that the correlation length grows as $\xi \sim t^{1/z}$ in the transient regime and then $\xi\sim (\ln t)^{1/\tilde\chi}$ in the asymptotic regime. The transient behavior is originated from the pinning independently by local ramified impurity clusters. The asymptotic logarithmic scaling is compatible with the scaling picture that the energy barrier height scales in the same way as the energy fluctuations with $\tilde\chi=\chi=1/3$. Implication on the domain coarsening dynamics is also discussed.
We thank Doochul Kim, Malte Henkel, and Heiko Rieger for useful discussions. This work was supported by KOSEF grant Acceleration Research (CNRC) (Grant No. R17-2007-073-01001-0).
M. Kardar and Y.-C. Zhang, Phys. Rev. Lett. [**58**]{}, 2087 (1987); J.M. Kim, M.A. Moore, and A.J. Bray, Phys. Rev. A [**44**]{}, 2345 (1991). D.S. Fisher and D.A. Huse, Phys. Rev. B [**43**]{}, 10728 (1991). M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. [**56**]{}, 889 (1986); T. Halpin-Healey and Y.-C. Zhang, Phys. Rep. [**254**]{}, 215 (1995). D.A. Huse and C.L. Henley, Phys. Rev. Lett. [**54**]{}, 2708 (1985); M. Kardar, [*ibid.*]{} [**55**]{}, 2923 (1985); D.A. Huse, C.L. Henley, and D.S. Fisher, [*ibid.*]{}, [**55**]{}, 2924 (1985). A.J. Bray, Adv. Phys. [**43**]{}, 357 (1994); [**51**]{}, 481 (2002). M. Kardar, in [*Dynamics of fluctuating interfaces and related phenomena*]{}, edited by D. Kim, H. Park, and B. Kahng (World Scientific, Singapore, 1997); T. Nattermann and S. Scheidl, Adv. Phys. [**49**]{}, 607 (2000). L.V. Mikheev, B. Drossel, and M. Kardar, Phys. Rev. Lett. [**75**]{}, 1170 (1995); B. Drossel and M. Kardar, Phys. Rev. E [**52**]{}, 4841 (1995). A.B. Kolton, A. Rosso, and T. Giamarchi, Phys. Rev. Lett. [**95**]{}, 180604 (2005). C. Monthus and T. Garel, J. Phys. A: Math. Theor. [**41**]{}, 115002 (2008). J.H. Oh and D.-I. Choi, Phys. Rev. B [**33**]{}, 3448 (1986). R. Paul, S. Puri, and H. Rieger, Europhys. Lett. [**68**]{}, 881 (2004); M. Henkel and M. Pleimling, Phys. Rev. B [**78**]{}, 224419 (2008). S.F. Edwards and D. Wilkinson, Proc. R. Soc. A [**381**]{}, 17 (1982). H. Yoshino, J. Phys. A: Math. Gen. [**29**]{}, 1421 (1996); A. Barrat, Phys. Rev. E [**55**]{}, 5651 (1997); H. Yoshino, Phys. Rev. Lett. [**81**]{}, 1493 (1998); S. Bustingory [*et al.*]{}, Europhys. Lett. [**76**]{}, 856 (2006). When $f>1/2$, the energetically favorable bonds become the majority and the unfavorable bonds play the role of local pinning centers. Although the correspondence is not exact, we expect that the whole arguments developed for small $f$ are also valid for large $f$. J. Jensen, J. Phys. A [**32**]{}, 5233 (1999). C.L. Henley, Phys. Rev. Lett. [**54**]{}, 2030 (1985); R. Rammal and A. Benoit, [*ibid.*]{} [**55**]{}, 649 (1985).
|
---
abstract: 'Let $p$ be a prime number, and $F$ a nonarchimedean local field of residual characteristic $p$. We explore the interaction between the pro-$p$-Iwahori-Hecke algebras of the group $\textnormal{GL}_n(F)$ and its derived subgroup $\textnormal{SL}_n(F)$. Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic $p$ over the pro-$p$-Iwahori-Hecke algebra of $\textnormal{SL}_2(\mathbb{Q}_p)$ and smooth mod-$p$ representations of $\textnormal{SL}_2(\mathbb{Q}_p)$ generated by their pro-$p$-Iwahori-invariants. The second is a “numerical correspondence” between packets of supersingular Hecke modules in characteristic $p$ over the pro-$p$-Iwahori-Hecke algebra of $\textnormal{SL}_n(F)$, and irreducible, $n$-dimensional projective Galois representations.'
address: 'Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada'
author:
- Karol Kozioł
title: 'Pro-$p$-Iwahori invariants for $\textnormal{SL}_2$ and $L$-packets of Hecke modules'
---
Introduction
============
In recent years, there has been a great deal of interest and activity surrounding the (still nebulous) mod-$p$ version of the Local Langlands Program. This situation is best understood for the group $\textnormal{GL}_2({\mathbb{Q}_p})$: there exists a correspondence between isomorphism classes of semisimple mod-$p$ representations of $\textnormal{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}_p})$ of dimension 2 and (certain) smooth, finite length, semisimple mod-$p$ representations of $\textnormal{GL}_2({\mathbb{Q}_p})$, due to Breuil ([@Br03]). This correspondence is compatible with a $p$-adic version of the Local Langlands Correspondence, and, even better, both the mod-$p$ and $p$-adic versions are induced by a *functor* (see [@Br032], [@Br04], [@Em10], [@Ki09], [@Ki10], and especially [@Co10], [@Pas10]).
Serious difficulties arise when one considers groups other than $\textnormal{GL}_2({\mathbb{Q}_p})$, however. For example, Breuil and Paškūnas have shown in [@BP12] that, for $F$ a nontrivial unramified extension of ${\mathbb{Q}_p}$, there is an infinite family of representations of $\textnormal{GL}_2(F)$ associated to a “generic” Galois representation. Therefore, it is not clear what the “shape” of a mod-$p$ correspondence should be for a general reductive group. Nevertheless, Breuil and Herzig have given a construction of the “ordinary part” of such a correspondence, and shown that it appears in certain spaces of mod-$p$ automorphic forms ([@BH12]).
An alternative viewpoint for examining these difficulties comes through the study of Hecke modules. From this point onwards, we let $F$ be a nonarchimedean local field with residue field of size $q$ and characteristic $p$. Let $G$ denote the group of $F$-rational points of a connected reductive group $\mathbf{G}$, which we assume to be split over $F$. We let $G_{\textnormal{S}}$ be the $F$-rational points of the derived subgroup of $\mathbf{G}$, let $I(1)$ denote a pro-$p$-Iwahori subgroup of $G$, and set $I_{\textnormal{S}}(1) := I(1)\cap G_{\textnormal{S}}$. Letting $\bullet$ represent either the empty symbol or “${\textnormal{S}}$,” we define the *pro-$p$-Iwahori-Hecke algebra* ${{\mathcal{H}}}_\bullet$ as the convolution algebra of compactly supported, ${{\overline{\mathbb{F}}}_p}$-valued functions on the double coset space $I_\bullet(1)\backslash G_\bullet/I_\bullet(1)$.
Taking the $I_\bullet(1)$-invariants of a smooth representation of $G_\bullet$ over ${{\overline{\mathbb{F}}}_p}$ yields a functor $$\mathcal{I}_\bullet:\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}(G_\bullet)\longrightarrow \mathfrak{Mod}-{{\mathcal{H}}}_\bullet$$ from the category of smooth, mod-$p$ representations of $G_\bullet$ to the category of right ${{\mathcal{H}}}_\bullet$-modules. When $G = \textnormal{GL}_2({\mathbb{Q}_p})$, the functor ${{\mathcal{I}}}$ induces an equivalence between the subcategory of $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}(\textnormal{GL}_2({\mathbb{Q}_p}))$ consisting of representations generated by their $I(1)$-invariants and $\mathfrak{Mod}-{{\mathcal{H}}}$. Thus one hopes to glean information about the category of representations by examining ${{\mathcal{H}}}$-modules. This doesn’t give an equivalence of categories in general, however (cf. [@Oll09]).
One can also ask to what extent the classical Local Langlands Correspondence is reflected by modules over ${{\mathcal{H}}}_\bullet$. For example, when $G = \textnormal{GL}_n(F)$, results of Vignéras and Ollivier ([@Vig05] and [@Oll10]) show that we have an equality $$\label{glnnum}
\begin{gathered}
\xymatrix{
\#\left\{\txt{\textnormal{simple, supersingular}\\ ${{\mathcal{H}}}$\textnormal{-modules of dimension}~$n$\\ \textnormal{with fixed action}\\ \textnormal{of a uniformizer}}\right\} = \#\left\{\txt{\textnormal{irreducible, mod-}$p$\\ \textnormal{Galois representations}\\ \textnormal{of dimension}~$n$~\textnormal{with fixed}\\ \textnormal{determinant of Frobenius}}\right\},
}
\end{gathered}$$ where we consider all objects up to isomorphism, and where a supersingular module is an object intended to mirror a supercuspidal representation of $G$. By recent work of Große-Klönne ([@GK13]), we now know that this numerical bijection is induced by a functor, at least in the case when $F = {\mathbb{Q}_p}$.
The goal of the present article is to analyze the situations of the previous two paragraphs for the group $\textnormal{SL}_n(F)$. We begin with a general split ${{\mathbf{G}}}$, and recall in Section \[algs\] the presentations of the algebras ${{\mathcal{H}}}$ and ${{\mathcal{H}}}_{\textnormal{S}}$, due to Vignéras. Proposition \[embedding\] shows that we have an injection ${{\mathcal{H}}}_{\textnormal{S}}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}{{\mathcal{H}}}$, making ${{\mathcal{H}}}$ into a free ${{\mathcal{H}}}_{\textnormal{S}}$-module. We then use a descent argument to deduce when certain universal modules corresponding to ${{\mathcal{H}}}$ and ${{\mathcal{H}}}_{\textnormal{S}}$ are flat (resp. faithfully flat, resp. projective) (Corollaries \[cflatiffcsflat\] and \[flatcor\]).
From Section \[equivs\] onwards, we assume $G = \textnormal{GL}_n(F)$ and $G_{\textnormal{S}}= \textnormal{SL}_n(F)$. In Section \[equivs\], we examine more closely the functor $\mathcal{I}_\bullet$. In order to accurately speak of an equivalence of categories, we must consider the full subcategory of $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}(G_\bullet)$ consisting of representations generated by their space of $I_\bullet(1)$-invariant vectors, which we denote $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_\bullet(1)}(G_\bullet)$. We continue to denote by $\mathcal{I}_\bullet$ the functor of invariants restricted to this subcategory. Our main result in this section is the following:
Assume $(n,p) = 1$. Then the functor $\mathcal{I}$ induces an equivalence of categories between $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)$ and $\mathfrak{Mod}-{{\mathcal{H}}}$ if and only if the functor $\mathcal{I}_{\textnormal{S}}$ induces an equivalence of categories between $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_{\textnormal{S}}(1)}(G_{\textnormal{S}})$ and $\mathfrak{Mod}-{{\mathcal{H}}}_{\textnormal{S}}$.
Using results of Ollivier ([@Oll09]) on the functor of $I(1)$-invariants for $\textnormal{GL}_2(F)$ (and a slight extension of Theorem \[equiv\]), we obtain the following corollary.
The functor $\mathcal{I}_{\textnormal{S}}$ induces an equivalence between $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_{\textnormal{S}}(1)}(\textnormal{SL}_2({\mathbb{Q}_p}))$ and $\mathfrak{Mod}-{{\mathcal{H}}}_{\textnormal{S}}$ when $p>2$.
As a second application of the interaction between the algebras ${{\mathcal{H}}}$ and ${{\mathcal{H}}}_{\textnormal{S}}$, we investigate the supersingular modules for ${{\mathcal{H}}}_{\textnormal{S}}$ in Section \[packets\]. The precise notion of supersingularity may be found in [@Vig05], and a complete description of such modules may be found in [@Oll12]. There is a natural conjugation action on ${{\mathcal{H}}}$ by the multiplicative subgroup of elements of length $0$, which preserves the subspace ${{\mathcal{H}}}_{\textnormal{S}}$. Therefore, given any supersingular character $\chi$ of ${{\mathcal{H}}}_{\textnormal{S}}$ and an element ${\textnormal{T}}_\omega$ of ${{\mathcal{H}}}$ of length $0$, we may define a new character ${\textnormal{T}}_\omega\cdot\chi$, given on ${{\mathcal{H}}}_{\textnormal{S}}$ by first conjugating the argument by ${\textnormal{T}}_\omega$ and then applying $\chi$. Mimicking the classical case of complex representations of $G_{\textnormal{S}}$, we make the following definition:
An *L-packet of supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-modules* is an orbit of the subgroup of ${{\mathcal{H}}}$ of length $0$ elements on the set of supersingular characters of ${{\mathcal{H}}}_{\textnormal{S}}$.
In order to proceed further, we impose an additional “regularity” condition on $L$-packets. Given this, we are able to count the number of regular, supersingular $L$-packets of size $d$, for $d$ a divisor of $n$ (Corollary \[numorbits\]). Our next goal is to relate the $L$-packets thus constructed to projective, mod-$p$ Galois representations, which we take up in Section \[galois\]. Our main theorem is as follows.
Let $d$ be a divisor of $n$. The number of regular supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules of size $d$ is equal to the number of isomorphism classes of irreducible projective Galois representations $$\sigma:\textnormal{Gal}(\overline{F}/F)\longrightarrow \textnormal{PGL}_n({{\overline{\mathbb{F}}}_p})$$ having exactly $d\frac{q - 1}{n}$ isomorphism classes of lifts to genuine Galois representations. This number is equal to $$h(d) = \frac{1}{d}\sum_{e|d}\mu\left(\frac{d}{e}\right)g(e),$$ where $\mu$ denotes the Möbius function, and $$g(e) = \sum_{f|n} \mu\left(\frac{n}{f}\right)\left(\frac{f}{(e,f)}, q - 1\right)\frac{q^{(e,f)} - 1}{q - 1}.$$ In particular, the number of regular supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules is equal to the number of isomorphism classes of irreducible projective Galois representations of dimension $n$.
Some remarks are in order. Firstly, we note the function $g(e)$ can be computed quite explicitly in terms of Euler’s phi function. With some work, one can show that $h(d)\neq 0$ implies $d\frac{q - 1}{n}\in \mathbb{Z}$ (Lemma \[g(d)neq0\]), so that the statement of the above theorem makes sense. Secondly, when $n = 2$ and $F = {\mathbb{Q}_p}$, we recover the bijection contained in work of Abdellatif ([@Ab11]).
Finally, let us remark that the work of [@GK13] shows how to construct a functor from the category of finite length modules over the pro-$p$-Iwahori-Hecke algebra of a general connected reductive group ${{\mathbf{G}}}$, defined and split over ${\mathbb{Q}_p}$, to the category of étale $(\varphi^r, \Gamma_0)$-modules. When ${{\mathbf{G}}}= \mathbf{GL}_n$, this functor (along with Fontaine’s equivalence of categories) induces the numerical correspondence . We compute this functor for supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules in Subsection \[gk\], and show how to define a map from the set of regular supersingular $L$-packets to the set of irreducible, projective mod-$p$ Galois representations. Moreover, we show in Corollary \[Wbij\] that this map realizes the numerical correspondence of Corollary \[finalcor\].
**Acknowledgements.** I would like to thank Rachel Ollivier for her support and guidance throughout the course of working on this article, and for many extremely useful comments. Several parts of this paper were written during the conference “Modular Representation Theory of Finite and $p$-adic Groups” at the Institute for Mathematical Sciences, National University of Singapore, and I would like to thank the institution for its support. During the preparation of this article, support was also provided by NSF Grant DMS-0739400.
Notation
========
Fix a prime number $p$, and let $F$ be a nonarchimedean local field of residual characteristic $p$. Denote by ${\mathfrak{o}}$ its ring of integers, and by ${\mathfrak{p}}$ the unique maximal ideal of ${\mathfrak{o}}$. Fix a uniformizer $\varpi$ and let $k = {\mathfrak{o}}/{\mathfrak{p}}$ denote the (finite) residue field. The field $k$ is a finite extension of $\mathbb{F}_p$ of size $q$. We fix also a separable closure $\overline{F}$ of $F$, and let $k_{{\overline{F}}}$ denote its residue field. Let $\iota:k_{{\overline{F}}}\stackrel{\sim}{\longrightarrow} {{\overline{\mathbb{F}}}_p}$ denote a fixed isomorphism, and assume that every ${{\overline{\mathbb{F}}}_p}^\times$-valued character factors through $\iota$.
Let ${{\mathbf{G}}}$ denote a connected, reductive group, with derived subgroup ${{\mathbf{G}}}_{\textnormal{S}}$. We assume that ${{\mathbf{G}}}$ is split over $F$. For any algebraic subgroup ${{\mathbf{J}}}$ of ${{\mathbf{G}}}$, we denote by ${{\mathbf{J}}}_{\textnormal{S}}$ its intersection with ${{\mathbf{G}}}_{\textnormal{S}}$. We let ${{\mathbf{T}}}$ denote a fixed split maximal torus of ${{\mathbf{G}}}$, so that ${{\mathbf{T}}}_{\textnormal{S}}$ is a maximal torus of ${{\mathbf{G}}}_{\textnormal{S}}$. Let ${{\mathbf{Z}}}$ denote the connected center of ${{\mathbf{G}}}$; note that ${{\mathbf{Z}}}_{\textnormal{S}}$ is not necessarily connected. Let ${{\mathbf{N}}}$ denote the normalizer of ${{\mathbf{T}}}$ in ${{\mathbf{G}}}$, so that ${{\mathbf{N}}}_{\textnormal{S}}$ is the normalizer of ${{\mathbf{T}}}_{\textnormal{S}}$ in ${{\mathbf{G}}}_{\textnormal{S}}$. We will generally denote algebraic groups by boldface letters, and their groups of $F$-rational points by the corresponding italicized Roman letter (e.g., $G = {{\mathbf{G}}}(F), T = {{\mathbf{T}}}(F)$, etc.).
Weyl Groups
===========
In order to ease notation in the following discussion (and throughout the remainder of the article), we let $\bullet$ denote either the empty symbol or ${\textnormal{S}}$. For an algebraic group ${{\mathbf{J}}}$, we let $X^*({{\mathbf{J}}})$ (resp. $X_*({{\mathbf{J}}})$) denote the group of algebraic characters (resp. cocharacters) of ${{\mathbf{J}}}$. We let $\Phi_\bullet\subset X^*({{\mathbf{T}}}_\bullet)$ denote the set of roots of ${{\mathbf{T}}}_\bullet$ acting on $\textnormal{Lie}({{\mathbf{G}}}_\bullet)$ by conjugation. Restriction to ${{\mathbf{T}}}_{\textnormal{S}}$ gives a bijection between $\Phi$ and $\Phi_{\textnormal{S}}$ ([@Bo91], Section 21.1).
Let $A_\bullet := (X_*({{\mathbf{T}}}_\bullet)/X_*({{\mathbf{Z}}}_\bullet^\circ))\otimes_{{\mathbb{Z}}}{{\mathbb{R}}}$ be the standard apartment corresponding to ${{\mathbf{T}}}_\bullet$ in the (adjoint) Bruhat–Tits building $X_\bullet$ of $G_\bullet$ (see [@SS97] for an overview). Since $X_*({{\mathbf{T}}}_{\textnormal{S}})$ is of finite index in $X_*({{\mathbf{T}}})/X_*({{\mathbf{Z}}})$, the apartment $A_{\textnormal{S}}$ identifies canonically with $A$. We fix a hyperspecial point $x_0\in A_{\textnormal{S}}$ and a chamber $C\subset A_{\textnormal{S}}$ containing $x_0$ (we view both in either $A_{\textnormal{S}}$ or $A$). Since $x_0$ is hyperspecial, the set of roots $\Phi_\bullet$ identifies with the subset of affine roots which are zero on $x_0$, and we let $\Phi_\bullet^+$ denote the set of those roots which are positive on $C$ (see Section 1.9 of [@Ti79]). As above, restriction to ${{\mathbf{T}}}_{\textnormal{S}}$ gives a bijection between $\Phi^+$ and $\Phi_{\textnormal{S}}^+$.
Let $I_\bullet$ denote the Iwahori subgroup in $G_\bullet$ corresponding to $C$, and $I_\bullet(1)$ its pro-$p$ radical. We have $I\cap G_{\textnormal{S}}= I_{\textnormal{S}}$ and $I(1)\cap G_{\textnormal{S}}= I_{\textnormal{S}}(1)$. The group $N_\bullet$ acts on $A_\bullet$ by affine transformations; the group $T_\bullet\cap I_\bullet$ acts trivially. Moreover, the group $(T_\bullet\cap I_\bullet)/(T_\bullet\cap I_\bullet(1))$ identifies with the group of $k$-points of a torus, which we denote $T_\bullet(k)$. The Iwahori decomposition implies $I_\bullet = T_\bullet(k)\ltimes I_\bullet(1)$ (cf. *loc. cit.*, Section 3.7).
We define the following (Iwahori–)Weyl groups: $$\begin{aligned}
W_{0,\bullet} & := & N_\bullet/T_\bullet\\
W_\bullet & := & N_\bullet/(T_\bullet \cap I_\bullet)\\
W_\bullet(1) & := & N_\bullet/(T_\bullet \cap I_\bullet(1))\end{aligned}$$ Theorem 21.2 in [@Bo91] implies $W_{0,\bullet} \cong {{\mathbf{N}}}_\bullet/{{\mathbf{T}}}_\bullet$, and by the discussion in Section 21.1 of *loc. cit.* we have $W_0 \cong W_{0,{\textnormal{S}}}$. Section 3.3 of [@Ti79] shows that we have the following Bruhat decomposition: $$G_\bullet = \bigsqcup_{w\in W_\bullet} I_\bullet wI_\bullet.$$ Here $I_\bullet wI_\bullet$ denotes the double coset $I_\bullet n_w I_\bullet$ for any lift $n_w$ in $N_\bullet$ of $w$. Using this, one easily obtains the following double coset decomposition ([@Vig05], Theorem 6): $$\label{bruhat}
G_\bullet = \bigsqcup_{w\in W_\bullet(1)} I_\bullet(1) wI_\bullet(1).$$
The affine Weyl group $W_{{\textnormal{aff}},\bullet}$ is defined as the subgroup of $W_\bullet$ generated by the reflections in the hyperplanes corresponding to the affine roots of ${{\mathbf{T}}}_\bullet$. We let $S_\bullet$ denote the set of reflections in the hyperplanes containing a facet of $C$; the pair $(W_{{\textnormal{aff}},\bullet}, S_\bullet)$ is then a Coxeter system (Théorème 1 of [@Bo81] V §3.2). We denote by $\ell:W_{{\textnormal{aff}},\bullet}\longrightarrow {{\mathbb{N}}}$ the length function on $W_{{\textnormal{aff}},\bullet}$ with respect to $S_\bullet$. Sections 1.4 and 1.5 of [@Lu89] imply that the length function inflates to $W_\bullet$, and we have a decomposition $$W_\bullet \cong \Omega_\bullet\ltimes W_{{\textnormal{aff}},\bullet},$$ where $\Omega_\bullet$ denotes the elements of length 0 in $W_\bullet$. Alternatively, $\Omega_\bullet$ is the subgroup of elements $\omega\in W_\bullet$ for which $\omega.C = C$. The length function further inflates to $W_\bullet(1)$, being trivial on $(T_\bullet\cap I_\bullet)/(T_\bullet\cap I_\bullet(1))$.
The injection $N_{\textnormal{S}}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}N$ induces an injection $W_{\textnormal{S}}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}W$, which fits into a diagram $$\label{diag}
\begin{gathered}
\xymatrix{
1 \ar[r] & X_*({{\mathbf{T}}}_{\textnormal{S}}) \ar[r]\ar@{^{(}->}[d] & W_{\textnormal{S}}\ar[r]\ar@{^{(}->}[d] & W_{0,{\textnormal{S}}} \ar[r]\ar[d] & 1\\
1 \ar[r] & X_*({{\mathbf{T}}}) \ar[r] & W \ar[r] & W_0 \ar[r] & 1
}
\end{gathered}$$ with exact rows and commuting squares. Here we identify $X_*({{\mathbf{T}}}_\bullet)$ with $T_\bullet/(T_\bullet\cap I_\bullet)$ by sending the cocharacter $\xi$ to the class of $\xi(\varpi)$.
\[sequalsaff\] The diagram above induces an isomorphism $W_{{\textnormal{aff}},{\textnormal{S}}}\cong W_{\textnormal{aff}}$, and an injection $\Omega_{\textnormal{S}}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}\Omega$.
Let $\imath$ denote the injection $W_{\textnormal{S}}{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}W$, and note firstly that $\imath$ induces a bijection between the coroots of ${{\mathbf{T}}}_{\textnormal{S}}$ and the coroots of ${{\mathbf{T}}}$. Let ${{\mathbb{Z}}}(\Phi_\bullet^\vee)\subset X_*({{\mathbf{T}}}_\bullet)$ denote the ${{\mathbb{Z}}}$-module generated by the coroots of ${{\mathbf{T}}}_\bullet$. Since the chosen vertex $x_0$ is hyperspecial, we have $\textnormal{stab}_{W_\bullet}(x_0)\cong W_{0,\bullet}$, which induces splittings of the short exact sequences above. The discussion contained in Section 1.5 of [@Lu89] implies that the map $$W_{{\textnormal{aff}},{\textnormal{S}}} = {{\mathbb{Z}}}(\Phi_{\textnormal{S}}^\vee)\rtimes W_{0,{\textnormal{S}}}\stackrel{\imath}{\longrightarrow}{{\mathbb{Z}}}(\Phi^\vee)\rtimes W_0 = W_{{\textnormal{aff}}}$$ is an isomorphism. One easily checks that the injection $\imath$ is compatible with the length function (using formula 1.4(a) in *loc. cit.*, for example), which shows that the image of $\Omega_{\textnormal{S}}$ lies in $\Omega$.
The group $W_{\textnormal{S}}$ is normal in $W$, and $W_{\textnormal{S}}\backslash W$ admits coset representatives of length $0$.
We now consider the following diagram
with exact rows and commuting squares. A quick diagram chase verifies the following.
\[length0\] The group $W_{\textnormal{S}}(1)$ is normal in $W(1)$, and $W_{\textnormal{S}}(1)\backslash W(1)$ admits coset representatives of length $0$.
\[WGcoset\] Let ${{\mathcal{Z}}}$ denote a closed subgroup of the connected center $Z$ of $G$ which satisfies ${{\mathcal{Z}}}\cap I(1)G_{\textnormal{S}}= \{1\}$ (this implies that ${{\mathcal{Z}}}\cap W_{\textnormal{S}}(1) = \{1\}$, the intersection taken inside $W(1)$). Then the map $$\begin{aligned}
W_{\textnormal{S}}(1)\backslash W(1)/{{\mathcal{Z}}}& \longrightarrow & {{\mathcal{Z}}}I(1)\backslash G/G_{\textnormal{S}}\\
W_{\textnormal{S}}(1)w{{\mathcal{Z}}}& \longmapsto & {{\mathcal{Z}}}I(1)n_wG_{\textnormal{S}}\end{aligned}$$ induces a bijection of sets. Indeed, it is an isomorphism of groups.
It suffices to prove the claim with ${{\mathcal{Z}}}= \{1\}$, since the assumption ${{\mathcal{Z}}}\cap I(1)G_{\textnormal{S}}= \{1\}$ guarantees that $W_{\textnormal{S}}(1)\backslash W(1)$ fibers over $W_{\textnormal{S}}(1)\backslash W(1)/{{\mathcal{Z}}}$ (resp. $I(1)\backslash G/G_{\textnormal{S}}$ fibers over ${{\mathcal{Z}}}I(1)\backslash G/G_{\textnormal{S}}$) with fiber ${{\mathcal{Z}}}$.
Let $B\subset G$ denote the Borel subgroup defined by $\Phi^+$ containing $T$, with unipotent radical $U$, and let $U^-$ denote the opposite unipotent subgroup. The Iwahori decomposition ([@Ti79], Section 3.1.1) then implies $$I(1) = (I(1)\cap T)(I(1)\cap U)(I(1)\cap U^-).$$ Since the group $G_{\textnormal{S}}$ is normal in $G$, we get $I(1)\backslash G/G_{\textnormal{S}}= I(1)G_{\textnormal{S}}\backslash G = (I(1)\cap T)G_{\textnormal{S}}\backslash G$, and a straightforward check shows that the group $(I(1)\cap T)G_{\textnormal{S}}$ is normal in $G$.
One easily checks that the map $W_{\textnormal{S}}(1)\backslash W(1)\longrightarrow (I(1)\cap T)G_{\textnormal{S}}\backslash G$ is well-defined, and in fact defines a group homomorphism. Since $N$ normalizes $I(1)\cap T$, the Bruhat and Iwahori decompositions imply $$G = \bigsqcup_{w\in W(1)}I(1)n_w(I(1)\cap U)(I(1)\cap U^-),$$ which shows that the map is surjective. Finally, assume that for $w\in W(1)$, we have $n_w = tg'$, $t\in I(1)\cap T, g'\in G_{\textnormal{S}}$. This implies $t^{-1}n_w = g'\in G_{\textnormal{S}}\cap N = N_{\textnormal{S}}$, and therefore (by projecting to $W(1)$) we get $w\in W_{\textnormal{S}}(1)$. This shows injectivity.
Pro-$p$-Iwahori-Hecke Algebras {#algs}
==============================
Structure of the algebras
-------------------------
Let $\bullet$ denote the empty symbol or ${\textnormal{S}}$, and let $R$ denote an arbitrary commutative unital ring. Given a smooth $R$-representation $\sigma$ of an open subgroup $J_\bullet$ of $G_\bullet$, we let $\textnormal{c-ind}_{J_\bullet}^{G_\bullet}(\sigma)$ denote the compact induction of $\sigma$ from $J_\bullet$ to $G_\bullet$. That is, $\textnormal{c-ind}_{J_\bullet}^{G_\bullet}(\sigma)$ is the space of all functions $f:G_\bullet\longrightarrow\sigma$ for which $f(jg) = \sigma(j).f(g)$ for all $j\in J_\bullet, g\in G_\bullet$, such that the support of $f$ in $J_\bullet\backslash G_\bullet$ is compact. The group $G_\bullet$ acts by right translation: if $f\in \textnormal{c-ind}_{J_\bullet}^{G_\bullet}(\sigma)$ and $g,g'\in G_\bullet$, we have $(g.f)(g') = f(g'g)$.
We consider the $R$-representation of $G_\bullet$ afforded by the *pro-$p$ universal module*, defined by $${{\mathcal{C}}}_{\bullet} := \textnormal{c-ind}_{I_\bullet(1)}^{G_\bullet}(1),$$ where $1$ denotes the free $R$-module of rank 1, with $I_\bullet(1)$ acting trivially. For any element $g\in G_\bullet$, we denote by $\mathbf{1}_{I_\bullet(1)g}\in {{\mathcal{C}}}_\bullet$ the characteristic function of the coset $I_\bullet(1)g$.
We define the *pro-$p$-Iwahori-Hecke algebra* as $${{\mathcal{H}}}_\bullet := \textnormal{End}_{G_\bullet}({{\mathcal{C}}}_\bullet)$$ with product given by composition. The space ${{\mathcal{C}}}_\bullet$ then becomes a left module over ${{\mathcal{H}}}_\bullet$. By Frobenius Reciprocity we have $${{\mathcal{H}}}_\bullet = \textnormal{End}_{G_\bullet}({{\mathcal{C}}}_\bullet)\cong \textnormal{Hom}_{I_\bullet(1)}(1,{{\mathcal{C}}}_\bullet|_{I_\bullet(1)}) \cong {{\mathcal{C}}}_\bullet^{I_\bullet(1)};$$ we therefore identify ${{\mathcal{H}}}_\bullet$ with the $R$-module $R[I_\bullet(1)\backslash G_\bullet /I_\bullet(1)]$, with the product given by convolution. For an element $g\in G_\bullet$, we let ${\textnormal{T}}_g^\bullet$ denote the characteristic function of the coset $I_\bullet(1)gI_\bullet(1)$. By abuse of notation, we shall often speak of elements ${\textnormal{T}}_w^\bullet$, where $w\in W_\bullet(1)$; by the Bruhat decomposition (equation ), this is independent of the choice of lift of $w$ to $N_\bullet$.
We shall need one more algebra. Let ${{\mathcal{Z}}}$ be a closed subgroup of the connected center $Z$ of $G$ which satisfies ${{\mathcal{Z}}}\cap I(1)G_{\textnormal{S}}= \{1\}$ (we allow the case ${{\mathcal{Z}}}= \{1\}$). Note that, when viewed as a subgroup of $W$, the elements of ${{\mathcal{Z}}}$ have length 0. We define $$\underline{{{\mathcal{C}}}} := \textnormal{c-ind}_{{{\mathcal{Z}}}I(1)}^{G}(1),$$ where $1$ denotes the free $R$-module of rank 1, with ${{\mathcal{Z}}}I(1)$ acting trivially. We set $$\underline{{{\mathcal{H}}}} := \textnormal{End}_G(\underline{{{\mathcal{C}}}});$$ by Frobenius Reciprocity, the algebra $\underline{{{\mathcal{H}}}}$ identifies with $\underline{{{\mathcal{C}}}}^{{{\mathcal{Z}}}I(1)} = \underline{{{\mathcal{C}}}}^{I(1)}$, with the product given by convolution of functions. When ${{\mathcal{Z}}}= \{1\}$, we have $\underline{{{\mathcal{C}}}} = {{\mathcal{C}}}$ and $\underline{{{\mathcal{H}}}} = {{\mathcal{H}}}$ as special cases.
Recall that we have an identification of $W_{{\textnormal{aff}},{\textnormal{S}}}$ with $W_{\textnormal{aff}}$; we therefore identify the sets $S_{\textnormal{S}}$ and $S$. For $s\in S$, there is an associated affine root $\alpha_{\textnormal{aff}}$ which is positive on $C$; we let $\alpha_s^\vee:{{\mathbf{G}}}_m\longrightarrow{{\mathbf{T}}}_{\textnormal{S}}\subset{{\mathbf{T}}}$ denote the coroot associated to the “root part” of $\alpha_{\textnormal{aff}}$. We set $$\tau_s^\bullet := \sum_{a\in k^\times}{\textnormal{T}}_{\alpha_s^\vee(a)}^\bullet.$$
The structures of ${{\mathcal{H}}}_\bullet$ and $\underline{{{\mathcal{H}}}}$ are summarized in the following theorem.
\[strthm\] Let $\bullet$ denote either the empty symbol or ${\textnormal{S}}$.
1. As an $R$-module, ${{\mathcal{H}}}_\bullet$ is free with basis $\{{\textnormal{T}}_w^\bullet\}_{w\in W_\bullet(1)}$.
2. (Braid relations) We have $${\textnormal{T}}_w^\bullet{\textnormal{T}}_{w'}^\bullet = {\textnormal{T}}_{ww'}^\bullet$$ for any $w,w'\in W_\bullet(1)$ satisfying $\ell(ww') = \ell(w) + \ell(w')$.
3. (Quadratic relations) For $s\in S$, we have $$({\textnormal{T}}_{n_s}^\bullet)^2 = q{\textnormal{T}}_{n_s^2}^\bullet + {\textnormal{T}}_{n_s}^\bullet\tau_s^\bullet.$$
4. Let $\Omega_\bullet(1)$ denote the preimage of $\Omega_\bullet$ under the natural projection $W_\bullet(1)\longrightarrow W_\bullet$. Then the algebra ${{\mathcal{H}}}_\bullet$ is generated by ${\textnormal{T}}_{n_s}^\bullet$ and ${\textnormal{T}}_\omega^\bullet$, where $s\in S$ and $\omega\in \Omega_\bullet(1)$.
5. We have an isomorphism of algebras $$\underline{{{\mathcal{H}}}} \cong {{\mathcal{H}}}/({\textnormal{T}}_z - 1)_{z\in{{\mathcal{Z}}}},$$ which sends the characteristic function of ${{\mathcal{Z}}}I(1)wI(1)$ to (the image of) the characteristic function of $I(1)wI(1)$.
For future applications, we will also need the affine subalgebra of ${{\mathcal{H}}}_\bullet$.
\[defaff\] Let $W_{{\textnormal{aff}},\bullet}(1)$ denote the preimage of $W_{{\textnormal{aff}},\bullet}$ under the natural projection $W_\bullet(1)\longrightarrow W_\bullet$. We denote by ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$ the $R$-submodule of ${{\mathcal{H}}}_\bullet$ generated by ${\textnormal{T}}_w$ for $w\in W_{{\textnormal{aff}},\bullet}(1)$. By Corollary 3 of [@Vig05], ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$ is a subalgebra of ${{\mathcal{H}}}_\bullet$, called the *affine pro-$p$-Iwahori-Hecke algebra*.
By Theorem \[strthm\], we see that ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$ is generated by the elements ${\textnormal{T}}_{n_s}$ and ${\textnormal{T}}_t$ for $s\in S$ and $t\in T_\bullet(k)$.
We now relate the various Hecke algebras. We are mainly interested in how ${{\mathcal{H}}}_{\textnormal{S}}$ is related to ${{\mathcal{H}}}$; however, we give the proofs for the algebras ${{\mathcal{H}}}_{\textnormal{S}}$ and $\underline{{{\mathcal{H}}}}$, noting that $\underline{{{\mathcal{H}}}} = {{\mathcal{H}}}$ in the case ${{\mathcal{Z}}}= \{1\}$.
\[embedding\] Let $\overline{{\textnormal{T}}_g}$ denote the image of ${\textnormal{T}}_g$ in ${{\mathcal{H}}}/({\textnormal{T}}_z - 1)_{z\in{{\mathcal{Z}}}}\cong \underline{{{\mathcal{H}}}}$. Then the linear map defined by
----------------- --------------------------------------- ------------------- ---------------------------------
$\mathfrak{f}:$ ${{\mathcal{H}}}_{\textnormal{S}}$ $\longrightarrow$ $\underline{{{\mathcal{H}}}}$
${\textnormal{T}}_g^{\textnormal{S}}$ $\longmapsto$ $\overline{{\textnormal{T}}_g}$
----------------- --------------------------------------- ------------------- ---------------------------------
where $g\in G_{\textnormal{S}}$, is an injective algebra homomorphism.
The $G_{\textnormal{S}}$-linear map defined by
----------------- -------------------------------------- ------------------- -----------------------------------------------------
$\mathfrak{f}$: ${{\mathcal{C}}}_{\textnormal{S}}$ $\longrightarrow$ $\underline{{{\mathcal{C}}}}|_{G_{\textnormal{S}}}$
$\mathbf{1}_{I_{\textnormal{S}}(1)}$ $\longmapsto$ $\mathbf{1}_{{{\mathcal{Z}}}I(1)}$
----------------- -------------------------------------- ------------------- -----------------------------------------------------
is easily seen to be injective. Taking $I_{\textnormal{S}}(1)$-invariants gives the injection $${{\mathcal{H}}}_{\textnormal{S}}\cong {{\mathcal{C}}}_{\textnormal{S}}^{I_{\textnormal{S}}(1)}\stackrel{\mathfrak{f}}{{\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}} \underline{{{\mathcal{C}}}}^{I_{\textnormal{S}}(1)} = \underline{{{\mathcal{C}}}}^{{{\mathcal{Z}}}I(1)} \cong \underline{{{\mathcal{H}}}},$$ which sends ${\textnormal{T}}_g^{\textnormal{S}}$ to $\overline{{\textnormal{T}}_g}$ for $g\in G_{\textnormal{S}}$.
It remains to check compatibility of $\mathfrak{f}$ with the algebra structures. The injection $W_{\textnormal{S}}(1){\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}W(1)$ is compatible with the length function $\ell$. Hence, if $w,w'\in W_{\textnormal{S}}(1)$ satisfy $\ell(ww') = \ell(w) + \ell(w')$, we get $$\mathfrak{f}({\textnormal{T}}_w^{\textnormal{S}})\mathfrak{f}({\textnormal{T}}_{w'}^{\textnormal{S}}) = \overline{{\textnormal{T}}_w{\textnormal{T}}_{w'}} = \overline{{\textnormal{T}}_{ww'}} = \mathfrak{f}({\textnormal{T}}_{ww'}^{\textnormal{S}}) = \mathfrak{f}({\textnormal{T}}_w^{\textnormal{S}}{\textnormal{T}}_{w'}^{\textnormal{S}}).$$ In addition, $$\mathfrak{f}(\tau_s^{\textnormal{S}}) = \overline{\tau_s}$$ for $s\in S$. Since $\ell(t) = 0$ for $t\in T_{\textnormal{S}}(k)$, we obtain $$\begin{aligned}
(\mathfrak{f}({\textnormal{T}}_{n_s}^{\textnormal{S}}))^2 & = & \overline{{\textnormal{T}}_{n_s}}^2\\
& = & q\overline{{\textnormal{T}}_{n_s^2}} + \overline{{\textnormal{T}}_{n_s}\tau_s}\\
& = & q\mathfrak{f}({\textnormal{T}}_{n_s^2}^{\textnormal{S}}) + \mathfrak{f}({\textnormal{T}}_{n_s}^{\textnormal{S}})\mathfrak{f}(\tau_s^{\textnormal{S}})\\
& = & \mathfrak{f}(q{\textnormal{T}}_{n_s^2}^{\textnormal{S}}+ {\textnormal{T}}_{n_s}^{\textnormal{S}}\tau_s^{\textnormal{S}})\\
& = & \mathfrak{f}(({\textnormal{T}}_{n_s}^{\textnormal{S}})^2).\end{aligned}$$
Using the proposition above, we shall henceforth identify ${{\mathcal{H}}}_{\textnormal{S}}$ with its images in ${{\mathcal{H}}}$ and $\underline{{{\mathcal{H}}}}$. We also fix a set of length 0 representatives ${{\mathcal{R}}}$ for $W_{\textnormal{S}}(1)\backslash W(1)/{{\mathcal{Z}}}$, which contains $1$ (cf. Lemmas \[length0\] and \[WGcoset\]).
Let $\omega\in {{\mathcal{R}}}$. Then the subspace ${{\mathcal{H}}}_{\textnormal{S}}\overline{{\textnormal{T}}_\omega}$ of $\underline{{{\mathcal{H}}}}$ is stable by left and right multiplication by ${{\mathcal{H}}}_{\textnormal{S}}$.
We prove the claim for right multiplication, the case of left multiplication being obvious. Let $w\in W_{\textnormal{S}}(1)\subset W(1)$. One easily checks that $\ell(\omega w\omega^{-1}) = \ell(w)$, and we obtain $$\overline{{\textnormal{T}}_{\omega}{\textnormal{T}}_w} = \overline{{\textnormal{T}}_{\omega w}} = \overline{{\textnormal{T}}_{\omega w\omega^{-1}}{\textnormal{T}}_{\omega}}\in {{\mathcal{H}}}_{\textnormal{S}}\overline{{\textnormal{T}}_{\omega}},$$ which suffices to prove the claim.
\[freeness\] As a left (resp. right) ${{\mathcal{H}}}_{\textnormal{S}}$-module, $\underline{{{\mathcal{H}}}}$ is free with basis $\{{\overline{{\textnormal{T}}_{\omega}}}\}_{\omega\in{{\mathcal{R}}}}$.
This follows immediately from Lemma \[length0\] and the braid relations of Theorem \[strthm\].
\[dirsumdecomp\] The algebra ${{\mathcal{H}}}_{\textnormal{S}}$ is a direct summand of $\underline{{{\mathcal{H}}}}$ as a left (resp. right) ${{\mathcal{H}}}_{\textnormal{S}}$ module.
We now use the above results to relate the modules ${{\mathcal{C}}}_{\textnormal{S}}$, ${{\mathcal{C}}}$, and $\underline{{{\mathcal{C}}}}$.
\[univmodres\] There exists an isomorphism $$\underline{{{\mathcal{H}}}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}\cong \underline{{{\mathcal{C}}}}|_{G_{\textnormal{S}}}$$ which is both $G_{\textnormal{S}}$-equivariant and $\underline{{{\mathcal{H}}}}$-equivariant.
Recall the map $\mathfrak{f}$ defined in the proof of Proposition \[embedding\]:
----------------- -------------------------------------- ------------------- -----------------------------------------------------
$\mathfrak{f}$: ${{\mathcal{C}}}_{{\textnormal{S}}}$ $\longrightarrow$ $\underline{{{\mathcal{C}}}}|_{G_{\textnormal{S}}}$
$\mathbf{1}_{I_{\textnormal{S}}(1)}$ $\longmapsto$ $\mathbf{1}_{{{\mathcal{Z}}}I(1)}$
----------------- -------------------------------------- ------------------- -----------------------------------------------------
It is obviously $G_{\textnormal{S}}$- and ${{\mathcal{H}}}_{\textnormal{S}}$-equivariant. By Frobenius Reciprocity, we obtain a map $\widetilde{\mathfrak{f}}$, defined by
----------------------------- ----------------------------------------------------------------------------------------------------------- ------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------
$\widetilde{\mathfrak{f}}$: $\underline{{{\mathcal{H}}}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{{\textnormal{S}}}$ $\longrightarrow$ $\underline{{{\mathcal{C}}}}|_{G_{\textnormal{S}}}$
$\overline{{\textnormal{T}}_{w}}\otimes g.\mathbf{1}_{I_{\textnormal{S}}(1)}$ $\longmapsto$ $\overline{{\textnormal{T}}_w}(\mathfrak{f}(g.\mathbf{1}_{I_{\textnormal{S}}(1)})) = g.\overline{{\textnormal{T}}_w}(\mathbf{1}_{{{\mathcal{Z}}}I(1)})$
----------------------------- ----------------------------------------------------------------------------------------------------------- ------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------
for $g\in G_{\textnormal{S}}$ and $w\in W(1)$. The map $\widetilde{\mathfrak{f}}$ is $G_{\textnormal{S}}$- and $\underline{{{\mathcal{H}}}}$-equivariant. It remains to show that it is an isomorphism.
Using the Mackey decomposition and Lemma \[WGcoset\], we obtain $$\underline{{{\mathcal{C}}}}|_{G_{\textnormal{S}}} \cong \bigoplus_{\omega\in {{\mathcal{R}}}} \textnormal{c-ind}_{{{\mathcal{Z}}}I(1)}^{{{\mathcal{Z}}}I(1)n_\omega G_{\textnormal{S}}}(1),$$ where $\textnormal{c-ind}_{{{\mathcal{Z}}}I(1)}^{{{\mathcal{Z}}}I(1)n_\omega G_{\textnormal{S}}}(1)$ denotes the subspace of $\underline{{{\mathcal{C}}}}$ with support contained in ${{\mathcal{Z}}}I(1)n_\omega G_{\textnormal{S}}$. Analogously, $\{\overline{{\textnormal{T}}_{\omega}}\}_{\omega\in{{\mathcal{R}}}}$ is a basis for $\underline{{{\mathcal{H}}}}$ over ${{\mathcal{H}}}_{\textnormal{S}}$, and we obtain $$\underline{{{\mathcal{H}}}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}\cong \bigoplus_{\omega\in{{\mathcal{R}}}}\overline{{\textnormal{T}}_{\omega}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}.$$ It is clear that $\widetilde{\mathfrak{f}}$ defines an isomorphism between $\overline{{\textnormal{T}}_{\omega}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}$ and $\textnormal{c-ind}_{{{\mathcal{Z}}}I(1)}^{{{\mathcal{Z}}}I(1)n_\omega G_{\textnormal{S}}}(1)$.
\[easyisom\] Let ${\mathfrak{M}}$ be a right $\underline{{{\mathcal{H}}}}$-module. We then have an isomorphism $${\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{S}}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}\cong {\mathfrak{M}}\otimes_{\underline{{{\mathcal{H}}}}} \underline{{{\mathcal{C}}}}|_{G_{\textnormal{S}}}$$ as $G_{\textnormal{S}}$-representations.
\[cflatiffcsflat\] The module ${{\mathcal{C}}}$ is flat (resp. projective) over ${{\mathcal{H}}}$ if and only if ${{\mathcal{C}}}_{\textnormal{S}}$ is flat (resp. projective) over ${{\mathcal{H}}}_{\textnormal{S}}$, if and only if $\underline{{{\mathcal{C}}}}$ is flat (resp. projective) over $\underline{{{\mathcal{H}}}}$.
Using the fact that ${{\mathcal{H}}}$ and $\underline{{{\mathcal{H}}}}$ are free over ${{\mathcal{H}}}_{\textnormal{S}}$, this follows from a straightforward descent argument.
\[flatcor\] Suppose that $R = {{\overline{\mathbb{F}}}_p}$. Let ${{\mathbf{G}}}= \mathbf{GL}_n$, so that $G = \textnormal{GL}_n(F)$ and $G_{\textnormal{S}}= \textnormal{SL}_n(F)$, and let ${{\mathcal{Z}}}$ denote the subgroup of $G$ consisting of central elements whose entries are powers of $\varpi$.
1. Let $n = 2$. Then ${{\mathcal{C}}}_{\textnormal{S}}$ is flat if and only if $q = p$, in which case it is projective and faithfully flat.
2. Let $n = 3$ and assume $q = p > 2$. Then ${{\mathcal{C}}}_{\textnormal{S}}$ is not flat.
Using the previous corollary, part (1) follows from [@Oll07], Théorème 1, while part (2) follows from [@OS11], Proposition 7.9.
Iwahori–Hecke Algebras
----------------------
We also have a variant of the above setup. Consider the universal module ${{\mathcal{C}}}'_\bullet := \textnormal{c-ind}_{I_\bullet}^{G_\bullet}(1)$, and its endomorphism algebra ${{\mathcal{H}}}'_\bullet := \textnormal{End}_{G_\bullet}({{\mathcal{C}}}'_\bullet)$, called the *Iwahori-Hecke algebra*. Its structure is well known (cf. [@Vig05]), and the proofs above apply mutatis mutandis to the Iwahori case. We remark that some of the results relating ${{\mathcal{H}}}_{\textnormal{S}}'$ and ${{\mathcal{H}}}'$ have also been obtained by Abdellatif in [@Ab11] (for ${{\mathbf{G}}}= \mathbf{GL}_n$). Translating Corollary \[flatcor\] to the present situation, we obtain the following.
Suppose $R = {{\overline{\mathbb{F}}}_p}$, and let ${{\mathbf{G}}}= \mathbf{GL}_2$, so that $G = \textnormal{GL}_2(F)$ and $G_{\textnormal{S}}= \textnormal{SL}_2(F)$. Then ${{\mathcal{C}}}'_{\textnormal{S}}$ is projective and faithfully flat.
We once again use [@Oll07], Théorèmes 1 and 2.
Equivalence of Categories between $G_{\textnormal{S}}$-representations and ${{\mathcal{H}}}_{\textnormal{S}}$-modules {#equivs}
=====================================================================================================================
We assume from this point onwards that $R = {{\overline{\mathbb{F}}}_p}$ and ${{\mathbf{G}}}= \mathbf{GL}_n$ with $n\geq 2$, so that ${{\mathbf{G}}}_{\textnormal{S}}= \mathbf{SL}_n, G = \textnormal{GL}_n(F)$, and $G_{\textnormal{S}}= \textnormal{SL}_n(F)$. We take ${{\mathbf{T}}}$ to be the diagonal maximal torus, $I$ the subgroup of $G$ with entries in ${\mathfrak{o}}$ which are upper-triangular modulo $\varpi$, and $I(1)$ those elements in $I$ which are unipotent modulo $\varpi$. We take ${{\mathcal{Z}}}$ to be the central subgroup consisting of diagonal matrices whose entries are a power of $\varpi$.
Let $\bullet$ denote either the empty symbol or ${\textnormal{S}}$. Let $\pi$ be a smooth ${{\overline{\mathbb{F}}}_p}$-representation of the group $G_\bullet$; Frobenius Reciprocity gives $$\pi^{I_\bullet(1)}\cong \textrm{Hom}_{I_\bullet(1)}(1,\pi|_{I_\bullet(1)})\cong \textnormal{Hom}_{G_\bullet}({{\mathcal{C}}}_{\bullet},\pi).$$ The algebra ${{\mathcal{H}}}_\bullet$ has a natural right action on $\textrm{Hom}_{G_\bullet}({{\mathcal{C}}}_\bullet,\pi)$ by pre-composition, which induces a right action on $\pi^{I_\bullet(1)}$. In this way, we obtain the functor of $I_\bullet(1)$-invariants
------------------------ ----------------------------------------------------------- ------------------- ------------------------------------------
$\mathcal{I}_\bullet:$ $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}(G_\bullet)$ $\longrightarrow$ $\mathfrak{Mod}-{{\mathcal{H}}}_\bullet$
$\pi$ $\longmapsto$ $\pi^{I_\bullet(1)}$
------------------------ ----------------------------------------------------------- ------------------- ------------------------------------------
from the category of smooth ${{\overline{\mathbb{F}}}_p}$-representations of $G_\bullet$ to the category of right ${{\mathcal{H}}}_\bullet$-modules.
Let $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_\bullet(1)}(G_\bullet)$ denote the full subcategory of $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}(G_\bullet)$ of objects generated by their space of $I_\bullet(1)$-invariants. We continue to denote by $\mathcal{I}_\bullet$ the functor above restricted to the subcategory $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_\bullet(1)}(G_\bullet)$. Lemma 3 Part (1) of [@BL94] implies that this functor is faithful.
Given a right ${{\mathcal{H}}}_\bullet$-module ${\mathfrak{m}}$, we may consider the $G_\bullet$-representation ${\mathfrak{m}}\otimes_{{{\mathcal{H}}}_\bullet}{{\mathcal{C}}}_\bullet$, with the action of $G_\bullet$ given on the right tensor factor. We thus obtain a functor
------------------------ ------------------------------------------ ------------------- ---------------------------------------------------------------------------
$\mathcal{T}_\bullet:$ $\mathfrak{Mod}-{{\mathcal{H}}}_\bullet$ $\longrightarrow$ $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_\bullet(1)}(G_\bullet)$
${\mathfrak{m}}$ $\longmapsto$ ${\mathfrak{m}}\otimes_{{{\mathcal{H}}}_\bullet}{{\mathcal{C}}}_\bullet$.
------------------------ ------------------------------------------ ------------------- ---------------------------------------------------------------------------
The functors $\mathcal{I}_\bullet$ and $\mathcal{T}_\bullet$ are adjoint to each other: $$\textnormal{Hom}_{\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_\bullet(1)}(G_\bullet)}(\mathcal{T}_\bullet({\mathfrak{m}}),\pi) \cong \textnormal{Hom}_{\mathfrak{Mod}-{{\mathcal{H}}}_\bullet}({\mathfrak{m}},\mathcal{I}_\bullet(\pi)).$$
We furthermore let $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)_{\varpi = 1}$ denote the full subcategory of $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)$ of representations on which the central element $$\begin{pmatrix}\varpi & & \\ & \ddots & \\ & & \varpi\end{pmatrix}$$ acts trivially. Taking $I(1)$-invariants of a representation in $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)_{\varpi = 1}$ yields a right ${{\mathcal{H}}}$-module on which the element ${\textnormal{T}}_{\textnormal{diag}(\varpi,\ldots,\varpi)}$ acts trivially. Therefore, we obtain a functor
---------------------------- ----------------------------------------------------------------------- ------------------- ----------------------------------------------
$\underline{\mathcal{I}}:$ $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)_{\varpi = 1}$ $\longrightarrow$ $\mathfrak{Mod}-\underline{{{\mathcal{H}}}}$
$\pi$ $\longmapsto$ $\pi^{I(1)}$.
---------------------------- ----------------------------------------------------------------------- ------------------- ----------------------------------------------
The adjoint functor $\underline{\mathcal{T}}$ is given by
---------------------------- ---------------------------------------------- ------------------- -----------------------------------------------------------------------------------
$\underline{\mathcal{T}}:$ $\mathfrak{Mod}-\underline{{{\mathcal{H}}}}$ $\longrightarrow$ $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)_{\varpi = 1}$
${\mathfrak{m}}$ $\longmapsto$ ${\mathfrak{m}}\otimes_{\underline{{{\mathcal{H}}}}}\underline{{{\mathcal{C}}}}$.
---------------------------- ---------------------------------------------- ------------------- -----------------------------------------------------------------------------------
It is natural to ask whether the functors defined above induce equivalences of categories. This shall be the main goal of this section.
\[I1Is1invts\] Assume that $(n,q) = 1$ and let $\pi\in \mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)$. We then have an equality of vector spaces $\pi^{I_{\textnormal{S}}(1)} = \pi^{I(1)}$.
Consider the short exact sequence $$1\longrightarrow I_{\textnormal{S}}(1)\longrightarrow I(1)\stackrel{\det}{\longrightarrow} 1 + {\mathfrak{p}}\longrightarrow 1.$$ Since $n$ and $p$ are coprime, the map $x\longmapsto x^n$ is an automorphism of $1 + {\mathfrak{p}}$. Hence, the map $$x\longmapsto\begin{pmatrix}x^{1/n} & & \\ & \ddots & \\ & & x^{1/n} \end{pmatrix}$$ gives an isomorphism $1 + {\mathfrak{p}}\cong I(1)\cap Z$, and a section to the surjection $I(1)\stackrel{\det}{\longrightarrow} 1 + {\mathfrak{p}}$. Thus, we have a decomposition $$I(1)\cong I_{\textnormal{S}}(1)\times (I(1)\cap Z).$$
Now, since $I(1)\cap Z$ is contained in $I(1)$, it acts trivially on $\pi^{I(1)}$. As $\pi$ is generated by its space of $I(1)$-invariants, $I(1)\cap Z$ acts trivially on the whole of $\pi$. Hence, by the above decomposition, we get $\pi^{I_{\textnormal{S}}(1)} \subset \pi^{I(1)}$. The opposite inclusion is obvious.
By considering the action of ${{\mathcal{H}}}$ on $\pi^{I(1)}$ and restricting to ${{\mathcal{H}}}_{\textnormal{S}}$, the above lemma actually yields an isomorphism of ${{\mathcal{H}}}_{\textnormal{S}}$-modules $$\label{HandHsmods}
\pi^{I_{\textnormal{S}}(1)}\cong \pi^{I(1)}|_{{{\mathcal{H}}}_{\textnormal{S}}}.$$
\[equiv\] Assume that $(n,q) = 1$. The functors $\mathcal{I}$ and $\mathcal{T}$ induce an equivalence of categories between $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)$ and $\mathfrak{Mod}-{{\mathcal{H}}}$ if and only if $\mathcal{I}_{\textnormal{S}}$ and $\mathcal{T}_{\textnormal{S}}$ induce and equivalence of categories between $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_{\textnormal{S}}(1)}(G_{\textnormal{S}})$ and $\mathfrak{Mod}-{{\mathcal{H}}}_{\textnormal{S}}$, if and only if $\underline{\mathcal{I}}$ and $\underline{\mathcal{T}}$ induce an equivalence of categories between $\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)_{\varpi = 1}$ and $\mathfrak{Mod}-\underline{{{\mathcal{H}}}}$.
The proofs of both equivalences are similar. We prove the first statement.
($\Longrightarrow$) Assume first that $\mathcal{I}$ and $\mathcal{T}$ induce an equivalence of categories, and let ${\mathfrak{m}}$ be a right ${{\mathcal{H}}}_{\textnormal{S}}$-module. We will show that the natural map
------------------ ------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
${\mathfrak{m}}$ $\longrightarrow$ $\mathcal{I}_{\textnormal{S}}\circ\mathcal{T}_{\textnormal{S}}({\mathfrak{m}}) = ({\mathfrak{m}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)}$
$m$ $\longmapsto$ $m\otimes\mathbf{1}_{I_{\textnormal{S}}(1)}$
------------------ ------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
is an isomorphism.
Let ${\widetilde{\mathfrak{m}}}$ denote the induced ${{\mathcal{H}}}$-module ${\mathfrak{m}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{H}}}$. Since $\mathcal{I}$ and $\mathcal{T}$ induce an equivalence, the homomorphism
------------------------------ ------------------- -----------------------------------------------------------------------------------------------------------------------------------------
${\widetilde{\mathfrak{m}}}$ $\longrightarrow$ $\mathcal{I}\circ\mathcal{T}({\widetilde{\mathfrak{m}}}) = ({\widetilde{\mathfrak{m}}}\otimes_{{{\mathcal{H}}}}{{\mathcal{C}}})^{I(1)}$
$m$ $\longmapsto$ $m\otimes\mathbf{1}_{I(1)}$
------------------------------ ------------------- -----------------------------------------------------------------------------------------------------------------------------------------
is bijective. On restricting to ${{\mathcal{H}}}_{\textnormal{S}}$, we get $$\begin{aligned}
{\widetilde{\mathfrak{m}}}|_{{{\mathcal{H}}}_{\textnormal{S}}} & \cong & ({\widetilde{\mathfrak{m}}}\otimes_{{\mathcal{H}}}{{\mathcal{C}}})^{I(1)}|_{{{\mathcal{H}}}_{\textnormal{S}}}\\
& \stackrel{\textnormal{eq.}~\eqref{HandHsmods}}{\cong} & ({\widetilde{\mathfrak{m}}}\otimes_{{{\mathcal{H}}}}{{\mathcal{C}}}|_{G_{\textnormal{S}}})^{I_{\textnormal{S}}(1)}\\
& \stackrel{\textnormal{Cor.}~\ref{easyisom}}{\cong} & ({\widetilde{\mathfrak{m}}}|_{{{\mathcal{H}}}_{\textnormal{S}}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)}.\end{aligned}$$ Since ${{\mathcal{H}}}_{\textnormal{S}}$ is a direct summand of ${{\mathcal{H}}}$, we have ${\widetilde{\mathfrak{m}}}|_{{{\mathcal{H}}}_{\textnormal{S}}}\cong {\mathfrak{m}}\oplus{\mathfrak{m}}',$ where ${\mathfrak{m}}'$ is the right ${{\mathcal{H}}}_{\textnormal{S}}$-module $\bigoplus_{\omega\in{{\mathcal{R}}}, \omega\neq 1}{\mathfrak{m}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}} {\textnormal{T}}_\omega$. Therefore, the above isomorphisms give $${\mathfrak{m}}\oplus{\mathfrak{m}}' \cong ({\mathfrak{m}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)} \oplus ({\mathfrak{m}}'\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)};$$ since the image of ${\mathfrak{m}}$ (resp. ${\mathfrak{m}}'$) under the natural map ${\widetilde{\mathfrak{m}}}\longrightarrow ({\widetilde{\mathfrak{m}}}\otimes_{{\mathcal{H}}}{{\mathcal{C}}})^{I(1)}$ must lie in the space $({\mathfrak{m}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)}$ (resp. $({\mathfrak{m}}'\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)}$), we conclude that $${\mathfrak{m}}\cong ({\mathfrak{m}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)} = \mathcal{I}_{{\textnormal{S}}}\circ\mathcal{T}_{\textnormal{S}}({\mathfrak{m}}).$$
Now let $\pi\in\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_{\textnormal{S}}(1)}(G_{\textnormal{S}})$, and consider the natural map
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------- --------
$\mathcal{T}_{\textnormal{S}}\circ\mathcal{I}_{\textnormal{S}}(\pi) = \pi^{I_{\textnormal{S}}(1)}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}$ $\longrightarrow$ $\pi$
$v\otimes \mathbf{1}_{I_{\textnormal{S}}(1)g^{-1}}$ $\longmapsto$ $g.v$,
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------- --------
where $v\in \pi^{I_{\textnormal{S}}(1)}$ and $g\in G_{\textnormal{S}}$. Since $\pi$ is generated by its space of $I_{\textnormal{S}}(1)$-invariant vectors, this map is surjective. Letting $\pi'$ denote its kernel, we obtain a short exact sequence $$0\longrightarrow \pi'\longrightarrow \pi^{I_{\textnormal{S}}(1)}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}\longrightarrow \pi \longrightarrow 0.$$ Taking $I_{\textnormal{S}}(1)$-invariants of this short exact sequence yields $$0\longrightarrow (\pi')^{I_{\textnormal{S}}(1)}\longrightarrow (\pi^{I_{\textnormal{S}}(1)}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)} \longrightarrow \pi^{I_{\textnormal{S}}(1)},$$ and the statement just proved shows that the third arrow is an isomorphism. Hence $(\pi')^{I_{\textnormal{S}}(1)} = 0$, and faithfulness of the functor $\mathcal{I}_{\textnormal{S}}$ shows $\pi' = 0$. We conclude that $$\mathcal{T}_{\textnormal{S}}\circ\mathcal{I}_{\textnormal{S}}(\pi) = \pi^{I_{\textnormal{S}}(1)}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}}\cong \pi.$$
($\Longleftarrow$) Assume now that $\mathcal{I}_{\textnormal{S}}$ and $\mathcal{T}_{\textnormal{S}}$ induce an equivalence of categories, and let ${\mathfrak{M}}\in \mathfrak{Mod}-{{\mathcal{H}}}$. Consider the natural map
------------------ ------------------- -----------------------------------------------------------------------------------------------------------------
${\mathfrak{M}}$ $\longrightarrow$ $\mathcal{I}\circ\mathcal{T}({\mathfrak{M}}) = ({\mathfrak{M}}\otimes_{{{\mathcal{H}}}}{{\mathcal{C}}})^{I(1)}$
$m$ $\longmapsto$ $m\otimes\mathbf{1}_{I(1)}$.
------------------ ------------------- -----------------------------------------------------------------------------------------------------------------
We claim that this map is an isomorphism. Indeed, if we restrict this morphism to ${{\mathcal{H}}}_{\textnormal{S}}$, we obtain $$\begin{aligned}
{\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{S}}} & \longrightarrow & ({\mathfrak{M}}\otimes_{{{\mathcal{H}}}}{{\mathcal{C}}})^{I(1)}|_{{{\mathcal{H}}}_{\textnormal{S}}}\\
& & \stackrel{\textnormal{eq.}~\eqref{HandHsmods}}{\cong} ({\mathfrak{M}}\otimes_{{\mathcal{H}}}{{\mathcal{C}}}|_{G_{\textnormal{S}}})^{I_{\textnormal{S}}(1)}\\
& & \stackrel{\textnormal{Cor.}~\ref{easyisom}}{\cong} ({\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{S}}}\otimes_{{{\mathcal{H}}}_{\textnormal{S}}}{{\mathcal{C}}}_{\textnormal{S}})^{I_{\textnormal{S}}(1)}.\end{aligned}$$ Since $\mathcal{I}_{\textnormal{S}}$ and $\mathcal{T}_{\textnormal{S}}$ induce an equivalence of categories, this map is an isomorphism, and we obtain $${\mathfrak{M}}\cong ({\mathfrak{M}}\otimes_{{{\mathcal{H}}}}{{\mathcal{C}}})^{I(1)} = \mathcal{I}\circ\mathcal{T}({\mathfrak{M}}).$$
Now let $\Pi\in\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)$. In order to show that $\mathcal{T}\circ\mathcal{I}$ is naturally isomorphic to $\textnormal{id}_{\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)}$, we proceed exactly as in the proof of $\mathcal{T}_{\textnormal{S}}\circ\mathcal{I}_{\textnormal{S}}\simeq\textnormal{id}_{\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_{\textnormal{S}}(1)}(G_{\textnormal{S}})}$ above. Hence, we conclude $$\mathcal{T}\circ\mathcal{I}(\Pi) = \Pi^{I(1)}\otimes_{{{\mathcal{H}}}}{{\mathcal{C}}}\cong \Pi.$$
\[equivcor\] Let $\mathcal{I}:\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I(1)}(G)\longrightarrow\mathfrak{Mod}-{{\mathcal{H}}}$ denote the functor of $I(1)$-invariants.
1. The functor $\mathcal{I}$ induces an equivalence of categories for $n = 2$ and $F = {\mathbb{Q}_p}$ with $p > 2$.
Let $\mathcal{I}_{\textnormal{S}}:\mathfrak{Rep}_{{{\overline{\mathbb{F}}}_p}}^{I_{\textnormal{S}}(1)}(G_{\textnormal{S}})\longrightarrow\mathfrak{Mod}-{{\mathcal{H}}}_{\textnormal{S}}$ denote the functor of $I_{\textnormal{S}}(1)$-invariants.
1. The functor $\mathcal{I}_{\textnormal{S}}$ induces an equivalence of categories for $n = 2$ and $F = {\mathbb{Q}_p}$ with $p > 2$.
2. The functor $\mathcal{I}_{\textnormal{S}}$ *does not* induce an equivalence of categories when $n = 2$ and $q > p > 2$.
3. The functor $\mathcal{I}_{\textnormal{S}}$ *does not* induce an equivalence of categories when $n = 2$ and $F = \mathbb{F}_p((T))$ with $p>2$.
4. The functor $\mathcal{I}_{\textnormal{S}}$ *does not* induce an equivalence of categories when $n = 3$ and $q = p > 3$.
Using Theorem \[equiv\] above, parts (1) - (4) follow from Théorème 1.3 of [@Oll09], and part (5) follows from Corollary \[flatcor\].
$L$-Packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules {#packets}
=========================================================
Once again, we let $\bullet$ denote either the empty symbol or ${\textnormal{S}}$ throughout. We introduce some additional notation. For $1\leq i \leq n - 1$, we define $$n_i := \begin{pmatrix}1 & & & & & \\ & \ddots & & & & \\ & & \phantom{-}0 & 1 & & \\ & & -1 & 0 & & \\ & & & & \ddots & \\ & & & & & 1 \end{pmatrix}\begin{array}{c} \phantom{-} \\ \phantom{-} \\ i^{\textnormal{th}}~\textnormal{row} \\ (i + 1)^{\textnormal{th}}~\textnormal{row}\\ \phantom{-} \\ \phantom{-} \end{array}\quad\textnormal{and}\quad n_0 := \begin{pmatrix} & & & & -\varpi^{-1}\\ & 1 & & & \\ & & \ddots & & \\ & & & 1 & \\ \varpi & & & & \end{pmatrix}.$$ If we choose for $x_0$ the hyperspecial vertex corresponding to the subgroup $\textnormal{GL}_n({\mathfrak{o}})$, then the set $S$ of affine reflections generating $W_{\textnormal{aff}}\cong W_{\textnormal{S}}$ is given by (the images of) the elements $n_i$, $0\leq i \leq n - 1$. Finally, we let $$\omega := \begin{pmatrix}0 & 1 & & \\ & \ddots & \ddots & \\ & & \ddots & 1\\ \varpi & & & 0 \end{pmatrix}.$$ The element $\omega$ satisfies $$\omega^{-1}n_i\omega = n_{i + 1}$$ (the index being considered modulo $n$), and is a generator for the group $\Omega$ of length $0$ elements of $W$.
Supersingular Hecke Modules
---------------------------
We recall the following definition of supersingular modules.
Let ${\mathfrak{m}}$ be a nonzero right ${{\mathcal{H}}}_\bullet$-module. Then ${\mathfrak{m}}$ is said to be *supersingular* if the center of ${{\mathcal{H}}}_\bullet$ acts by a character which is null.
For a more precise definition of a null character of the center of ${{\mathcal{H}}}_\bullet$, we refer to Definition 2 (*loc. cit.*). In order to describe the supersingular modules more explicitly, we recall the classification of characters of ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$.
Recall that to each $n_i\in S$, we have an associated coroot $\alpha_{n_i}^\vee:{{\mathbf{G}}}_m\longrightarrow {{\mathbf{T}}}_{\textnormal{S}}\subset {{\mathbf{T}}}$. Let $\lambda:T_\bullet(k)\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ be a character, and set $$S_\lambda := \{n_i\in S:~\lambda\circ\alpha_{n_i}^\vee:k^\times\longrightarrow {{\overline{\mathbb{F}}}_p}^\times ~\textnormal{is trivial}\}.$$
The characters of ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$ are parametrized by pairs $(\lambda,J)$, where $\lambda:T_\bullet(k)\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ is a character and $J\subset S_\lambda$. We denote the character associated to the pair $(\lambda,J)$ by $\chi_{\lambda,J}^\bullet$. This character is defined by
----- ---------------------------------------------------- ----- -------------- --------------------------
(1) $\chi_{\lambda,J}^\bullet({\textnormal{T}}_t)$ $=$ $\lambda(t)$ for $t\in T_\bullet(k)$,
(2) $\chi_{\lambda,J}^\bullet({\textnormal{T}}_{n_i})$ $=$ $0$ if $n_i\not\in J$,
(3) $\chi_{\lambda,J}^\bullet({\textnormal{T}}_{n_i})$ $=$ $-1$ if $n_i\in J$.
----- ---------------------------------------------------- ----- -------------- --------------------------
Let $1$ denote the trivial character of $T_\bullet(k)$. The proposition above shows, in particular, that the algebra ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$ possesses two distinguished characters: the *trivial character* $\chi_{1,\emptyset}^\bullet$, sending all ${\textnormal{T}}_{n_i}$ to 0, and the *sign character* $\chi_{1,S}^\bullet$, sending all ${\textnormal{T}}_{n_i}$ to $-1$.
We have the following classification of supersingular modules.
\[class\] Let ${\mathfrak{m}}$ be a simple right ${{\mathcal{H}}}_\bullet$-module. Then ${\mathfrak{m}}$ is supersingular if and only if it contains a character of ${{\mathcal{H}}}_{{\textnormal{aff}},\bullet}$, which is different from a twist of $\chi_{1,\emptyset}^\bullet$ or $\chi_{1,S}^\bullet$.
Since ${{\mathcal{H}}}_{{\textnormal{aff}},{\textnormal{S}}} = {{\mathcal{H}}}_{\textnormal{S}}$, this theorem implies in particular that every supersingular module of ${{\mathcal{H}}}_{\textnormal{S}}$ is a character, unequal to $\chi_{1,\emptyset}^{\textnormal{S}}$ or $\chi_{1,S}^{\textnormal{S}}$.
$L$-packets
-----------
Consider now a simple supersingular module ${\mathfrak{M}}$ for ${{\mathcal{H}}}$. By Theorem \[class\], ${\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{aff}}}$ contains a character. Restricting to ${{\mathcal{H}}}_{\textnormal{S}}$, we obtain a character $\chi$ of ${{\mathcal{H}}}_{\textnormal{S}}$, which furthermore must be supersingular. By the construction of the character $\chi$, the underlying vector space is stable by elements of the form ${\textnormal{T}}_{t}$ for $t\in T(k)$, and we obtain $${\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{S}}} = \sum_{j = 0}^{n - 1}\chi\cdot{\textnormal{T}}_{\omega^j}$$ by simplicity of ${\mathfrak{M}}$.
The element $\omega$ acts on the set $S$ by conjugation, and we see that $$\label{omorbit}
\chi_{\lambda,J}^\bullet\cdot{\textnormal{T}}_\omega\cong \chi_{\lambda^\omega,\omega^{-1}J\omega}^\bullet,$$ where $\lambda^\omega$ is defined by $\lambda^{\omega}(t) = \lambda(\omega t\omega^{-1})$ for $t\in T_\bullet(k)$. This leads to the following definition.
Let $\lambda:T_{\textnormal{S}}(k)\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ be a character, and $J\subset S_\lambda \subset S$. We define an action of $\omega^{\mathbb{Z}}$ on the characters of ${{\mathcal{H}}}_{{\textnormal{S}}}$ by $$\omega.\chi_{\lambda,J}^{\textnormal{S}}:= \chi_{\lambda^\omega,\omega^{-1} J\omega}^{\textnormal{S}}\cong \chi_{\lambda,J}^{\textnormal{S}}\cdot{\textnormal{T}}_\omega.$$ We define an *L-packet of ${{\mathcal{H}}}_{\textnormal{S}}$-modules* to be an orbit of $\omega^\mathbb{Z}$ acting on characters of ${{\mathcal{H}}}_{\textnormal{S}}$. We say an $L$-packet is *supersingular* if it consists entirely of supersingular characters, or, equivalently, if it contains a supersingular character.
In particular, we see that the size of an $L$-packet must divide $n$.
\[regular\] We say a supersingular character $\chi_{\lambda,J}^{\textnormal{S}}$ is *regular* if there exists a simple supersingular ${{\mathcal{H}}}$-module ${\mathfrak{M}}$ of dimension $n$ such that $\chi_{\lambda,J}^{\textnormal{S}}$ is a Jordan-Hölder factor of ${\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{S}}}$. We say an $L$-packet is *regular* if every character contained in the packet is regular, or, equivalently, if it contains a regular character.
It is an easy exercise to see that if ${\mathfrak{M}}$ is a simple $n$-dimensional supersingular ${{\mathcal{H}}}$-module, the restriction ${\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{aff}}}$ is a direct sum of $n$ distinct characters. This implies that $\chi_{\lambda,J}^{\textnormal{S}}$ is regular if and only if, for any character $\widetilde{\lambda}:T(k)\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ satisfying $\widetilde{\lambda}|_{T_{\textnormal{S}}(k)} = \lambda$, the orbit of the character $\chi_{\widetilde{\lambda},J}$ of ${{\mathcal{H}}}_{\textnormal{aff}}$ has size $n$ under the action of $\omega^\mathbb{Z}$ (where the action of $\omega^\mathbb{Z}$ on $\chi_{\widetilde{\lambda},J}$ is defined by equation ).
In what follows, we let $(e_1, \ldots, e_j)$ denote the greatest common divisor of integers $e_1,\ldots, e_j\in \mathbb{Z}$, with $(e_1) = |e_1|$. In addition, for any natural number $a\in{{\mathbb{N}}}$, we shall denote by $$[a] = \frac{q^a - 1}{q - 1}$$ the $q$-analog of $a$.
To proceed further, we need a combinatorial lemma.
\[comb\] Let $f:\mathbb{Z}_{>0}\longrightarrow \mathbb{C}$ be an arbitrary arithmetic function, let $\mu:\mathbb{Z}_{>0}\longrightarrow \{-1,0,1\}$ denote the Möbius function, and let $\sigma_0(m)$ denote the number of divisors of $m$. We then have $$f(m) - \sum_{j=1}^{\sigma_0(m) - 1}(-1)^{j + 1}\sum_{{\genfrac{}{}{0pt}{}{1\leq e_1 < \ldots < e_j < m}{e_i|m}}}f((e_1, \ldots, e_j)) = \sum_{e|m}\mu\left(\frac{m}{e}\right)f(e).$$
It suffices to show $$\mu(m') = \sum_{j = 1}^{\sigma_0(m') - 1}(-1)^j|\{1\leq e_1' < \ldots < e_j' < m': e_i'|m',~(e_1', \ldots, e_j') = 1\}|;$$ this follows from (the example following) Proposition 4.29 in [@Aig97].
\[orbdivd\] Let $d$ be a divisor of $n$, and let $g(d)$ denote the number of regular supersingular characters of ${{\mathcal{H}}}_{\textnormal{S}}$ whose orbit under $\omega^\mathbb{Z}$ has size dividing $d$. We then have $$g(d) = \sum_{e|n} \mu\left(\frac{n}{e}\right)[(d,e)]\left(\frac{e}{(d,e)}, q - 1\right).$$
Let $\chi_{\lambda,J}^{\textnormal{S}}$ be a supersingular character whose orbit under $\omega^\mathbb{Z}$ has size dividing $d$. This means that $\omega^{-d}J\omega^d = J$, that is, the set $J$ is stable under the map $n_i \longmapsto n_{i+d}$. Hence, the subsets $J$ of $S$ satisfying $\omega^{-d}J\omega^d = J$ correspond bijectively to subsets $J'$ of $\{n_1, \ldots, n_d\}$ in the obvious way.
Let $\lambda$ correspond to the equivalence class $$((a_1, a_2, \ldots, a_n))\in (\mathbb{Z}/(q-1)\mathbb{Z})^n/\langle(1,1,\ldots, 1)\rangle;$$ the correspondence is defined by $\lambda(\textnormal{diag}(t_1,t_2,\ldots, t_n)) = \prod_{i = 1}^nt_i^{a_i}$, where $t_i\in k^\times$ and $\prod_{i = 1}^n t_i = 1$. The condition $\lambda^{\omega^d} = \lambda$ implies that there exists $z\in\mathbb{Z}/(q-1)\mathbb{Z}$ such that $$a_{i + d} \equiv a_{i} + z~(\textnormal{mod}~ q - 1)$$ for every $0 < i \leq n$ (where we consider the indices modulo $n$). Summing gives $$\sum_{j = 0}^{n/d - 1} a_{i + jd} \equiv \frac{n}{d}z + \sum_{j = 0}^{n/d - 1} a_{i + jd}~(\textnormal{mod}~ q - 1)~ \Longleftrightarrow~ z \equiv 0~ \left(\textnormal{mod}\frac{q - 1}{(\frac{n}{d}, q - 1)}\right).$$ Hence, $\lambda$ is determined by $(a_1, \ldots, a_d)$ modulo the diagonal, and the element $z$. Now, let $J'$ be the subset of $\{n_1, \ldots, n_d\}$ to which $J$ corresponds. Note that $J'$ must be a *proper* subset, else we would have $J = S$ and $\lambda$ would be the trivial character. The number of characters $\chi_{\lambda,J}^{\textnormal{S}}$ which satisfy $\omega^d.\chi_{\lambda,J}^{\textnormal{S}}= \chi_{\lambda,J}^{\textnormal{S}}$ and for which $J$ corresponds to a fixed $J'$ is therefore equal to $$(q - 1)^{d - 1 - |J'|}\left(\frac{n}{d},q - 1\right).$$ Hence, the total number of supersingular characters satisfying $\omega^d.\chi_{\lambda,J}^{\textnormal{S}}= \chi_{\lambda,J}^{\textnormal{S}}$ is equal to $$\begin{aligned}
-1 + \sum_{J'\subsetneq\{n_1, \ldots, n_d\}}(q - 1)^{d - 1 - |J'|}\left(\frac{n}{d}, q - 1\right) & = & -1 + [d]\left(\frac{n}{d}, q - 1\right)\end{aligned}$$ (the $-1$ accounts for the contribution of the trivial character $\chi_{1,\emptyset}^{\textnormal{S}}$).
It remains to verify how many of these characters are regular. Let $\widetilde{\lambda}:T(k)\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ be a character whose restriction to $T_{\textnormal{S}}(k)$ is equal to $\lambda$, and let $e$ be a proper divisor of $n$. Denote by ${\chi}_{\widetilde{\lambda},J}:{{\mathcal{H}}}_{\textnormal{aff}}\longrightarrow{{\overline{\mathbb{F}}}_p}^\times$ the character of the affine Hecke algebra ${{\mathcal{H}}}_{\textnormal{aff}}$ associated to $\widetilde{\lambda}$ and $J$ (so that $\chi_{\widetilde{\lambda},J}|_{{{\mathcal{H}}}_{\textnormal{S}}} = \chi_{\lambda,J}^{\textnormal{S}}$), and assume $\omega^e.{\chi}_{\widetilde{\lambda},J} = {\chi}_{\widetilde{\lambda},J}$. This implies in particular that $\omega^{-e}J\omega^e = J$; hence, we obtain $\omega^{-(d,e)}J\omega^{(d,e)} = J$, and the set of such $J$ correspond bijectively to subsets $J'$ of $\{n_1, \ldots, n_{(d,e)}\}$.
We let $\widetilde{\lambda}$ correspond to $$(a_1, a_2, \ldots, a_n)\in (\mathbb{Z}/(q - 1)\mathbb{Z})^n$$ (lifting the class $((a_1, a_2, \ldots, a_n))\in(\mathbb{Z}/(q-1)\mathbb{Z})^n/\langle(1,1,\ldots, 1)\rangle$ above). By the above computation, the $n$-tuple corresponding to $\widetilde{\lambda}$ satisfies $$\label{modd}
a_i \equiv a_{{\overline{i}}} + \left\lfloor\frac{i - 1}{d}\right\rfloor z~(\textnormal{mod}~q - 1),$$ where $0< i \leq n$, and where $0 < \overline{i} \leq d$ is congruent to $i$ modulo $d$. The condition $\widetilde{\lambda}^{\omega^e} = \widetilde{\lambda}$ implies, in particular, that $$a_1 \equiv a_{1 + de/(d,e)} \equiv a_1 + \frac{e}{(d,e)}z~(\textnormal{mod}~q - 1),$$ which is equivalent to $$z \equiv 0~\left(\textnormal{mod}~\frac{q - 1}{\left(\frac{e}{(d,e)}, q - 1\right)}\right).$$
As above, the character $\widetilde{\lambda}$ is determined by the integers $a_1, \ldots, a_{(d,e)}$ and the element $z$. Proceeding as above, given a proper subset $J'$ of $\{n_1, \ldots, n_{(d,e)}\}$, we obtain $$(q - 1)^{(d,e) - 1 - |J'|}\left(\frac{e}{(d,e)},q - 1\right)$$ characters $\chi_{\lambda,J}^{\textnormal{S}}$ such that the lift ${\chi}_{\widetilde{\lambda},J}$ has an orbit of size dividing $e$, with $J$ corresponding to a fixed $J'$. Hence, the total number of supersingular characters $\chi_{\lambda,J}^{\textnormal{S}}$ such that the lift ${\chi}_{\widetilde{\lambda},J}$ has an orbit of size dividing $e$ is $$-1 + [(d,e)]\left(\frac{e}{(d,e)},q - 1\right).$$
By the inclusion-exclusion principle, the number of regular supersingular characters of ${{\mathcal{H}}}_{\textnormal{S}}$ of orbit size dividing $d$ is $$-1 + [d]\left(\frac{n}{d}, q - 1\right) - \sum_{j = 1}^{\sigma_0(n) - 1}(-1)^{j + 1}\sum_{{\genfrac{}{}{0pt}{}{1\leq e_1 < \ldots < e_j < n}{e_i|n}}}-1 + [(d, e_1, \ldots, e_j)]\left(\frac{(e_1, \ldots, e_j)}{(d,e_1, \ldots, e_j)}, q - 1\right).$$ Applying Lemma \[comb\] with $f(e) = -1 + [(d,e)]\left(\frac{e}{(d,e)},q - 1\right)$ and using the fact that $\sum_{e|n}\mu\left(\frac{n}{e}\right) = 0$ gives the result.
Evaluating the function $g$ at $1$, we obtain $$g(1) = \sum_{e|n}\mu\left(\frac{n}{e}\right)(e, q - 1).$$ As a function of $n$, the above expression is multiplicative, which implies $$g(1) = \begin{cases}\varphi(n) & \textnormal{if}~(n,q - 1) = n,\\ 0 & \textnormal{if}~(n,q - 1)\neq n,\end{cases}$$ where $\varphi$ denotes Euler’s phi function.
\[numorbits\] Let $d$ be a divisor of $n$, and let $h(d)$ denote the number of regular supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules of size $d$. We then have $$h(d) = \frac{1}{d}\sum_{e|d} \mu\left(\frac{d}{e}\right)g(e).$$
\[g(d)neq0\] Let $d$ be a divisor of $n$.
1. We have $g(d)\neq 0$ if and only if $\left(\frac{n}{d}, q - 1\right) = \frac{n}{d}$.
2. If $h(d)\neq 0$, then $\left(\frac{n}{d}, q - 1\right) = \frac{n}{d}$.
By Corollary \[numorbits\], it suffices to prove the first claim. The proof is a tedious (but straightforward) exercise in elementary number theory, and is left to the reader.
Galois Groups and Projective Galois Representations {#galois}
===================================================
Galois Groups
-------------
Let ${\mathcal{G}}_{F} := \textrm{Gal}({\overline{F}}/F)$ denote the absolute Galois group of $F$, and let ${\mathcal{I}}_{F}$ denote the inertia subgroup of elements which act trivially on the residue field $k_{{\overline{F}}}$. For any extension $L$ of $F$ contained in ${\overline{F}}$, we define ${\mathcal{G}}_L := \textrm{Gal}({\overline{F}}/L)$. We have ${\mathcal{G}}_{F}/{\mathcal{I}}_{F} \cong \textrm{Gal}(k_{\overline{F}}/k)\cong \widehat{\mathbb{Z}}$; we denote by $\textnormal{Fr}_q$ a fixed element of ${\mathcal{G}}_{F}$ whose image in $\textnormal{Gal}(k_{\overline{F}}/k)$ is the geometric Frobenius element. Finally, for $m\geq 1$, we let $F_{m}$ denote the unique unramified extension of $F$ of degree $m$ contained in ${\overline{F}}$.
We fix a compatible system $\{\sqrt[q^m-1]{\varpi}\}_{m\geq 1}$ of $(q^m-1)^{\textnormal{th}}$ roots of $\varpi$, and let $\omega_m:{\mathcal{I}}_{F}\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ denote the character given by $$\label{omegam}
\omega_m: h\longmapsto \iota\circ\mathfrak{r}_{{\overline{F}}}\left(\frac{h.\sqrt[q^m-1]{\varpi}}{\sqrt[q^m-1]{\varpi}}\right),$$ where $h\in {\mathcal{I}}_{F}$ and $\mathfrak{r}_{{\overline{F}}}:{\mathfrak{o}}_{{\overline{F}}}\longrightarrow k_{{\overline{F}}}$ denotes the reduction modulo the maximal ideal. Lemma 2.5 of [@Br07] shows that the character $\omega_m$ extends to a character of ${\mathcal{G}}_{F_{m}}$; we continue to denote by $\omega_m$ the extension which sends the element $\textnormal{Fr}_q^m$ to 1. Moreover, for $\lambda\in{{\overline{\mathbb{F}}}_p}^\times$, we let $\mu_{m,\lambda}:{\mathcal{G}}_{F_m}\longrightarrow {{\overline{\mathbb{F}}}_p}^\times$ denote the unramified character which sends $\textnormal{Fr}_q^m$ to $\lambda$. Lemma 2.2 of *loc. cit.* implies that every smooth ${{\overline{\mathbb{F}}}_p}$-character of ${\mathcal{G}}_{F_m}$ is of the form $\mu_{m,\lambda}\omega_m^r$ with $\lambda\in{{\overline{\mathbb{F}}}_p}^\times$ and $0\leq r < q^m - 1$.
Galois Representations
----------------------
We begin by recalling the classification of irreducible $n$-dimensional mod-$p$ representations of the group ${\mathcal{G}}_F$. Throughout, we assume that $\textnormal{GL}_n({{\overline{\mathbb{F}}}_p})$ and $\textnormal{PGL}_n({{\overline{\mathbb{F}}}_p})$ are given the discrete topology. We take [@Vig97], Sections 1.13 and 1.14, and [@Be10], Section 2, as our references.
An element $r$ of $\mathbb{Z}/(q^n - 1)\mathbb{Z}$ is said to be *primitive* if we have $$r\not\equiv 0~\left(\textnormal{mod}~\frac{[n]}{[d]}\right)$$ for every proper divisor $d$ of $n$. The necessary results are summarized in the following proposition.
\[galreps\]
(1) Any continuous irreducible $n$-dimensional mod-$p$ representation of ${\mathcal{G}}_F$ is isomorphic to $$\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_{F}}(\mu_{n,\lambda}\omega_n^r),$$ where $\lambda\in{{\overline{\mathbb{F}}}_p}^\times$, and $r\in\mathbb{Z}/(q^n - 1)\mathbb{Z}$ is primitive.
(2) We have an isomorphism $$\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_{F}}(\mu_{n,\lambda}\omega_n^r) \cong \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_{F}}(\mu_{n,\lambda'}\omega_n^{r'})$$ if and only if $\lambda' = \lambda$ and $r' \equiv q^ar~(\textnormal{mod}~q^n - 1)$ for some $a\in \mathbb{Z}$.
See Section 1.14 of [@Vig97], or Lemma 2.1.4 and the subsequent remarks in [@Be10].
We now consider projective representations.
By an *$n$-dimensional mod-$p$ projective Galois representation* we mean a continuous homomorphism from ${\mathcal{G}}_F$ to $\textnormal{PGL}_n({{\overline{\mathbb{F}}}_p})$. We say a projective Galois representation is *irreducible* if it does not factor through a proper parabolic subgroup of $\textnormal{PGL}_n({{\overline{\mathbb{F}}}_p})$. Moreover, we say two projective representations $\sigma$ and $\sigma'$ are *equivalent* if there exists an element $m\in\textnormal{PGL}_n({{\overline{\mathbb{F}}}_p})$ such that $\sigma(g) = m\sigma'(g)m^{-1}$ for all $g\in {\mathcal{G}}_F$. This equivalence relation will be denoted $\sigma\sim\sigma'$.
Given any continuous $n$-dimensional Galois representation $\rho$, we denote by $$\llbracket\rho\rrbracket: {\mathcal{G}}_F\stackrel{\rho}{\longrightarrow} \textnormal{GL}_n({{\overline{\mathbb{F}}}_p}) \stackrel{\llbracket-\rrbracket}{\longrightarrow} \textnormal{PGL}_n({{\overline{\mathbb{F}}}_p})$$ the projective representation obtained as the composition of $\rho$ with the natural quotient map. The extent to which these representations constitute all projective Galois representations is given by the following theorem.
\[tatesthm\] We have $$\textnormal{H}^2({\mathcal{G}}_F,{{\overline{\mathbb{F}}}_p}^\times) = 0.$$ Consequently, every irreducible $n$-dimensional projective Galois representation lifts to a genuine Galois representation, i.e., is of the form $\llbracket\rho\rrbracket$, where $\rho$ is a continuous irreducible $n$-dimensional Galois representation.
This follows from (the proof of) Theorem 4 (and its corollary) in [@Se77]; one simply uses the decomposition ${{\overline{\mathbb{F}}}_p}^\times \cong \bigoplus_{\ell\neq p} \mathbb{Q}_\ell/\mathbb{Z}_\ell$.
\[lambda=1\] We have an equivalence $$\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\mu_{n,\lambda}\omega_n^r)\rrbracket\sim \llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket.$$
Let $\sqrt[n]{\lambda}$ denote an $n^{\textnormal{th}}$ root of $\lambda$. We then have $$\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\mu_{n,\lambda}\omega_n^r)\rrbracket\sim\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\otimes \mu_{1,\sqrt[n]{\lambda}}\rrbracket\sim\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket.$$
Since we are interested in irreducible projective Galois representations, the above results imply we only need to consider representations of the form $\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket$ with $r$ primitive.
Let $\sigma$ be an irreducible projective Galois representation of dimension $n$. We will say a Galois representation $\rho$ is a *lift of* $\sigma$ if $\rho$ is of the form $$\rho = \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)$$ and $\llbracket\rho\rrbracket\sim \sigma$.
Note that by Theorem \[tatesthm\] and Lemma \[lambda=1\], such a lift always exists. Moreover, any lift of $\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket$ is of the form $$\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^{q^ar})\otimes\omega_1^m\cong \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^{q^ar + m[n]}) \cong \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^{q^a(r + m[n])}).$$ Hence, the representations $\{\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^{r + m[n]})\}_{0\leq m < q - 1}$ constitute all lifts of $\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket$, up to isomoprhism (and possibly with repetition). As the group ${{\mathbb{Z}}}/(q - 1){{\mathbb{Z}}}$ acts transitively on this set by cyclically permuting the $m$ parameter, we see that any irreducible projective Galois representation has $\frac{q - 1}{d'}$ isomorphism classes of lifts, where $d'$ divides $q - 1$. This discussion also shows we may always choose a lift satisfying $0 \leq r < [n]$.
\[rprim\] Let $d$ be a divisor of $n$ and let $r\in\mathbb{Z}/(q^n - 1)\mathbb{Z}$ with $0\leq r < [n]$. The number of such $r$ which are primitive and satisfy $q^dr\equiv r~(\textnormal{mod}~[n])$ is equal to $$g(d) = \sum_{e|n} \mu\left(\frac{n}{e}\right)[(d,e)]\left(\frac{e}{(d,e)}, q - 1\right).$$
Assume we have $(q^d - 1)r \equiv 0~(\textnormal{mod}~ [n])$; this easily implies $r \equiv s\frac{[n]}{[d]\left(\frac{n}{d}, q - 1\right)}~(\textnormal{mod}~q^n - 1)$ with $0\leq s < [d](\frac{n}{d}, q - 1)$. It remains to determine which of these elements are primitive. Let $e$ be a proper divisor of $n$, and assume $$r \equiv s\frac{[n]}{[d]\left(\frac{n}{d}, q - 1\right)} \equiv 0~\left(\textnormal{mod}~\frac{[n]}{[e]}\right);$$ again, one easily checks that this is equivalent to $$s \equiv 0~\left(\textnormal{mod}~\frac{[d]\left(\frac{n}{d},q - 1\right)}{[(d,e)]\left(\frac{e}{(d,e)},q - 1\right)}\right).$$ Hence, the number of $0 \leq r < [n]$ satisfying $q^dr \equiv r~(\textnormal{mod}~[n])$ and $r\equiv 0~\left(\textnormal{mod}~\frac{[n]}{[e]}\right)$ is $$[(d,e)]\left(\frac{e}{(d,e)}, q - 1\right).$$ We use the inclusion-exclusion principle and Lemma \[comb\] to conclude (cf. proof of Proposition \[orbdivd\]).
\[cormind\] Let $d$ be a divisor of $n$, and let $r\in\mathbb{Z}/(q^n - 1)\mathbb{Z}$ with $0\leq r < [n]$. The number of such $r$ which are primitive and such that $d$ is the minimal integer satisfying $q^dr \equiv r~(\textnormal{mod}~ [n])$ is $$\sum_{e|d}\mu\left(\frac{d}{e}\right)g(e).$$
We define a map from isomorphism classes of irreducible $n$-dimensional projective Galois representations $\sigma$ to ${{\mathbb{Z}}}_{>0}$ by the formula $$d_\sigma:=\min\{e\in {{\mathbb{Z}}}_{>0}: q^er\equiv r~(\textnormal{mod}~[n])\},$$ where $\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)$ is a fixed lift of $\sigma$. One easily checks that this definition is independent of the choice of lift.
\[rtoprojrep\] The integer $d_\sigma$ divides $n$ and $\frac{n}{d_\sigma}$ divides $q - 1$. Moreover, the projective representation $\sigma$ has exactly $d_\sigma\frac{q - 1}{n}$ isomorphism classes of lifts.
Let $\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)$ be a fixed lift of $\sigma$, and define ${{\mathcal{S}}}_\sigma := \{e\in {{\mathbb{Z}}}_{>0}:q^er \equiv r~(\textnormal{mod}~[n])\}$; this definition is independent of the choice of lift. Additionally, the following properties are easily verified: $n\in {{\mathcal{S}}}_\sigma$, and if $e_1, e_2\in {{\mathcal{S}}}_\sigma$, then $(e_1,e_2)\in {{\mathcal{S}}}_\sigma$. The properties above and minimality of $d_\sigma$ imply that if $e\in {{\mathcal{S}}}_\sigma$, then $d_\sigma\leq (d_\sigma,e)\leq d_\sigma$, so $d_\sigma$ divides $e$. In particular, $d_\sigma$ divides $n$, and we have ${{\mathcal{S}}}_\sigma = d_\sigma{{\mathbb{Z}}}_{>0}$.
Now, by definition of ${{\mathcal{S}}}_\sigma$, we have $q^{d_\sigma}r \equiv r + m[n]~(\textnormal{mod}~q^n - 1)$ for some $m\in {{\mathbb{Z}}}/(q - 1){{\mathbb{Z}}}$. Applying this equation recursively yields $$\label{rtoprojrepform}
q^{ad_\sigma}r \equiv r + am[n]~(\textnormal{mod}~q^n - 1),$$ for $a\in {{\mathbb{Z}}}$. Taking $a = q - 1$ in equation yields $$q^{(q - 1)d_\sigma}r \equiv r~(\textnormal{mod}~q^n - 1),$$ which implies that $n$ divides $(q - 1)d_\sigma$ by primitivity of $r$.
Taking $a = \frac{n}{d_\sigma}$ in equation gives $$\frac{n}{d_\sigma}m[n]\equiv 0~(\textnormal{mod}~q^n - 1),$$ which is equivalent to $m \equiv 0~\left(\textnormal{mod}~\frac{d_\sigma(q - 1)}{n}\right)$ (here we use that $\frac{n}{d_\sigma}$ divides $q - 1$ from above). Let us write $m\equiv m'\frac{d_\sigma(q - 1)}{n}$, with $m'\in {{\mathbb{Z}}}/(n/d_\sigma){{\mathbb{Z}}}$, so that equation becomes $$q^{ad_\sigma}r \equiv r + am'\frac{d_\sigma(q - 1)}{n}[n]~(\textnormal{mod}~q^n - 1).$$ We claim $m'$ and $\frac{n}{d_\sigma}$ are relatively prime. If not, then there would exist $0< a < \frac{n}{d_\sigma}$ such that $am'\equiv 0~(\textnormal{mod}~\frac{n}{d_\sigma})$, which implies $q^{ad_\sigma}r\equiv r~\left(\textnormal{mod}~q^n - 1\right)$, contradicting the primitivity of $r$. Therefore, we can choose $a'$ such that $a'm' \equiv 1~\left(\textnormal{mod}~\frac{n}{d_\sigma}\right)$, which gives $$q^{a'd_\sigma}r \equiv r + \frac{d_\sigma(q - 1)}{n}[n]~(\textnormal{mod}~q^n - 1).$$ In particular, this shows that $$\label{rtoprojrepcong}
\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\otimes \omega_1^{d_\sigma(q - 1)/n}\cong \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r).$$
The discussion preceding Proposition \[rprim\] shows that there exists an integer $d'$ dividing $q - 1$ such that $$\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r),~ \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\otimes\omega_1,\ldots,~ \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\otimes\omega_1^{((q - 1)/d') - 1}$$ are a full and pairwise nonisomorphic set of representatives of lifts of $\llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket$. This implies $\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\otimes\omega_1^{(q - 1)/d'}\cong \textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)$, meaning $$q^{bd_\sigma}r\equiv r + \frac{q - 1}{d'}[n]~(\textnormal{mod}~q^n - 1)$$ for some $b\in {{\mathbb{Z}}}$ (note that the exponent of $q$ on the left-hand side must be an element of ${{\mathcal{S}}}_\sigma$). Proceeding as above, we iterate this relation $\frac{n}{d_\sigma}$ times to obtain $$\frac{n}{d_\sigma}\frac{q - 1}{d'}[n]\equiv 0~(\textnormal{mod}~q^n - 1),$$ which is equivalent to saying that $d_\sigma d'$ divides $n$. On the other hand, the definition of $d'$ and equation show that $\frac{q - 1}{d'}$ divides $\frac{d_\sigma(q - 1)}{n}$. These two facts show that $d' = \frac{n}{d_\sigma}$.
\[numprojreps\] Let $d$ be a divisor of $n$. The number of isomorphism classes of irreducible projective Galois representations $\sigma$ of dimension $n$ for which $d_\sigma = d$ is equal to $$h(d) = \frac{1}{d}\sum_{e|d}\mu\left(\frac{d}{e}\right)g(e).$$
Let $\textnormal{pr}$ denote the surjective map from the set of primitive integers $r$ such that $0\leq r < [n]$, to the set of isomorphism classes of irreducible $n$-dimensional projective Galois representations given by $\textnormal{pr}(r) = \llbracket\textnormal{ind}_{{\mathcal{G}}_{F_n}}^{{\mathcal{G}}_F}(\omega_n^r)\rrbracket$. The sizes of the fibers of the map $r\longmapsto d_{\textnormal{pr}(r)}$ are given by Corollary \[cormind\]. Furthermore, by definition of the integer $d_\sigma$, the fiber of the map $\textnormal{pr}$ over the representation $\sigma$ has size $d_\sigma$. Combining these two facts gives the corollary.
\[finalcor\] Let $d$ be a divisor of $n$. The number of regular supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules of size $d$ is equal to the number of (isomorphism classes of) irreducible projective Galois representations $\sigma$ of dimension $n$ for which $d_\sigma = d$. In particular, the number of regular supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules is equal to the number of (isomorphism classes of) irreducible projective Galois representations of dimension $n$.
This follows from Corollaries \[numorbits\] and \[numprojreps\], and Lemma \[rtoprojrep\].
Lemma \[rtoprojrep\] shows that the condition “$d_\sigma = d$” is related to the number of lifts of $\sigma$.
Comparison with Große-Klönne’s Functor {#gk}
--------------------------------------
Große-Klönne has recently constructed a functor from the category of finite-length right ${{\mathcal{H}}}_\bullet$-modules to the category of étale $(\varphi^r,\Gamma_0)$-modules ([@GK13]). When applied to the group $\textnormal{GL}_n({\mathbb{Q}_p})$, his construction, composed with Fontaine’s equivalence of categories, yields a bijection between isomorphism classes of (absolutely) simple, supersingular right ${{\mathcal{H}}}$-modules of dimension $n$ and isomorphism classes of (absolutely) irreducible $n$-dimensional mod-$p$ representations of $\textnormal{Gal}(\overline{\mathbb{Q}}_p/{\mathbb{Q}_p})$. We now analyze these $(\varphi^r,\Gamma_0)$-modules for $\textnormal{SL}_n({\mathbb{Q}_p})$. For this subsection only, we adhere to the notation of *loc. cit.*; the reader should consult that article for precise statements and definitions.
We take $F = {\mathbb{Q}_p}, {\mathfrak{o}}= {\mathbb{Z}_p}$, with residue field ${\mathbb{F}_p}$, and uniformizer $\varpi = p$. We let $k$ denote the residue field in a fixed (sufficiently large) finite extension of ${\mathbb{Q}_p}$. Recall that we have identified the apartments $A$ and $A_{\textnormal{S}}$, and we let $C$ denote the chamber in $A_{\textnormal{S}}$ corresponding the the Iwahori subgroup $I_{\textnormal{S}}$. We choose a semiinfinite chamber gallery in $A_{\textnormal{S}}$ by setting, for $i\geq 0$, $$C^{(i)} := (n_{n - 1}^{-1}\omega)^i.C,$$ and note that the action on $A_{\textnormal{S}}$ of $$(n_{n - 1}^{-1}\omega)^n = \underbrace{n_{n - 1}^{-1}\omega\cdots n_{n - 1}^{-1}\omega}_{n~\textnormal{times}} = \omega^nn_{n - 1}^{-1}n_{n - 2}^{-1}\cdots n_1^{-1}n_0^{-1}$$ is the same as the action of $$\phi := n_{n - 1}^{-1}n_{n - 2}^{-1}\cdots n_1^{-1}n_0^{-1} \in G_{\textnormal{S}}.$$ We have $\phi.C^{(i)} = C^{(i + n)}$ by definition. The choice of a chamber gallery and such an element $\phi$ provides us with a half tree $Y_{\textnormal{S}}\subset X_{\textnormal{S}}$ and a simplicial isomorphism between $Y_{\textnormal{S}}$ and “the half tree of $\textnormal{PGL}_2({\mathbb{Q}_p})$” (cf. *loc. cit.*, Section 3).
To every simple supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-module $\chi_{\lambda,J}^{\textnormal{S}}$ we associate an $n$-tuple of integers $$(k_1,\ldots, k_{n - 1}, k_n)$$ as follows (cf. *loc. cit.*, Section 8). For $0\leq i \leq n - 1$, we let $0 \leq k_{i + 1} \leq p - 1$ satisfy $$\lambda\circ\alpha_{n_i}^\vee(x^{-1}) = x^{k_{i + 1}}.$$ If $\lambda\circ \alpha_{n_i}^\vee$ is not the trivial character, then $k_{i + 1}$ is uniquely determined. The condition of $\lambda\circ\alpha_{n_i}^\vee$ being equal to the trivial character is equivalent to $n_i\in S_\lambda$; in this case, we set $k_{i + 1} = p - 1$ if $n_i\in J$ and $k_{i + 1} = 0$ otherwise.
Tracing through the construction of *loc. cit.*, we arrive at the following proposition.
\[phingamma0\] Let $\chi_{\textnormal{cyc}}:\Gamma\stackrel{\sim}{\longrightarrow}{\mathbb{Z}_p}^\times$ denote the cyclotomic character, and let $\Gamma_0 = \chi_{\textnormal{cyc}}^{-1}(1 + p{\mathbb{Z}_p})$. The étale $(\varphi^n,\Gamma_0)$-module $\mathbf{D}_{\lambda,J}$ associated to $\chi_{\lambda,J}^{\textnormal{S}}$ is one-dimensional over $k_\mathcal{E} = k((t))$, spanned by a vector $\vec{e}$, with actions given by $$\begin{aligned}
\varphi^n(\vec{e}) & = & (-1)^n\left(\prod_{j = 1}^{n}k_j!\right)^{-1}t^{-\sum_{j = 0}^{n - 1}(p - 1 - k_{n - j})p^j}\vec{e},\\
\gamma(\vec{e}) & = & \left(\frac{t}{(1 + t)^{\chi_{\textnormal{cyc}}(\gamma)} - 1}\right)^{\frac{1}{p^n - 1}\sum_{j = 0}^{n - 1}(p - 1 - k_{n - j})p^j}\vec{e},\end{aligned}$$ where $\gamma\in \Gamma_0$. In particular, we see that distinct supersingular characters $\chi_{\lambda,J}^{\textnormal{S}}$ give rise to distinct $(\varphi^n,\Gamma_0)$-modules.
This is a straightforward computation using [@GK13].
The construction of $\mathbf{D}_{\lambda,J}$ depends on the choice of chamber gallery $C^{(0)}, C^{(1)}, C^{(2)}, \ldots$ and the element $\phi$.
We can push the construction of Proposition \[phingamma0\] a bit further. Given a one-dimensional $(\varphi^n,\Gamma_0)$-module $\mathbf{D}_{\lambda,J}$ as above, we construct an $n$-dimensional étale $(\varphi,\Gamma_0)$-module as follows. Let $\widetilde{\mathbf{D}_{\lambda,J}}$ denote the $k_{\mathcal{E}}$ vector space spanned by $\{\vec{e}_0,\ldots, \vec{e}_{n - 1}\}$, with actions given by $$\begin{aligned}
\varphi(\vec{e}_i) & = & \begin{cases}\vec{e}_{i + 1} & \textnormal{if}~ 0\leq i < n - 1,\\
\displaystyle{(-1)^n\left(\prod_{j = 1}^{n}k_j!\right)^{-1}}t^{-\sum_{j = 0}^{n - 1}(p - 1 - k_{n - j})p^j}\vec{e}_0 & \textnormal{if}~ i = n - 1, \end{cases}\\
\gamma(\vec{e}_i) & = & \left(\frac{t}{(1 + t)^{\chi_{\textnormal{cyc}}(\gamma)} - 1}\right)^{\frac{p^i}{p^n - 1}\sum_{j = 0}^{n - 1}(p - 1 - k_{n - j})p^j}\vec{e}_i,\end{aligned}$$ where $\gamma\in \Gamma_0$. It is clear that $\widetilde{\mathbf{D}_{\lambda,J}}$ is isomorphic to $\widetilde{\mathbf{D}_{\lambda^{\omega^{-1}},\omega J\omega^{-1}}}$. Moreover, we have $$\widetilde{\mathbf{D}_{\lambda,J}} \cong \bigoplus_{i = 0}^{n - 1}\mathbf{D}_{\lambda^{\omega^{-i}},\omega^{i}J\omega^{-i}}$$ as $(\varphi^n,\Gamma_0)$-modules, the isomorphism given by sending $k_{\mathcal{E}}.\vec{e}_i$ to $\mathbf{D}_{\lambda^{\omega^{-i}},\omega^{i}J\omega^{-i}}$. The discussion of Section 2.2 of [@Be10] shows that we may (nonuniquely) extend the action of $\Gamma_0$ to $\Gamma$, so that we obtain a bona fide $(\varphi,\Gamma)$-module. By abuse of notation, we shall also denote this module by $\widetilde{\mathbf{D}_{\lambda,J}}$.
We may now relate simple supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-modules and projective Galois representations more precisely. Let $\chi_{\lambda,J}^{\textnormal{S}}$ be as before, and let $(k_1,\ldots k_{n - 1}, k_n)$ denote the associated $n$-tuple. We define the rational number $r$ (depending on $\lambda$ and $J$) by $$\label{defofr}
r := \frac{1}{p - 1}\sum_{j = 0}^{n - 1}(p - 1 - k_{n - j})p^j.$$ By Theorems 8.5, 8.6(c) and 8.7 of [@GK13], $r$ is in fact an integer, and $\chi_{\lambda,J}^{\textnormal{S}}$ is regular if and only if $r$ is primitive.
Denote by $$\mathbf{D}\longmapsto W(\mathbf{D})$$ Fontaine’s equivalence of categories, from the category of étale $(\varphi,\Gamma)$-modules over $k_{\mathcal{E}}$ to the category of finite-dimensional representations of ${\mathcal{G}}_{{\mathbb{Q}_p}}$ over $k$ (see [@Fon90] for more details). Applying this functor to the $(\varphi,\Gamma)$-module $\widetilde{\mathbf{D}_{\lambda,J}}$ and using the computations contained in Section 2.2 of [@Be10], we obtain $$W(\widetilde{\mathbf{D}_{\lambda,J}}) = \textnormal{ind}_{{\mathcal{G}}_{\mathbb{Q}_{p^n}}}^{{\mathcal{G}}_{{\mathbb{Q}_p}}}(\omega_n^r)\otimes \mu_{1,\beta}\omega_1^s$$ for some $0\leq s < p - 1, \beta\in {{\overline{\mathbb{F}}}_p}^\times$, and $r$ as in equation . Here $\mathbb{Q}_{p^n}$ denotes the unique unramified extension of ${\mathbb{Q}_p}$ of degree $n$ contained in a fixed algebraic closure ${\overline{\mathbb{Q}}}_p$. Precomposing this with Große-Klönne’s functor (and the lifting map $\mathbf{D}_{\lambda,J}\longmapsto\widetilde{\mathbf{D}_{\lambda,J}}$) and postcomposing with the projectivization functor, we get a map $\mathcal{W}$ from the set of simple supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-modules to the category of projective Galois representations. Explicitly, it is given by $$\mathcal{W}(\chi_{\lambda,J}^{\textnormal{S}}) = \llbracket W(\widetilde{\mathbf{D}_{\lambda,J}})\rrbracket \sim \llbracket\textnormal{ind}_{{\mathcal{G}}_{\mathbb{Q}_{p^n}}}^{{\mathcal{G}}_{{\mathbb{Q}_p}}}(\omega_n^r)\rrbracket.$$
There are several immediate consequences of this definition. Firstly, we see that the isomorphism class of $\mathcal{W}(\chi_{\lambda,J}^{\textnormal{S}})$ is independent of the choice of action of $\Gamma$ on the $(\varphi,\Gamma_0)$-module $\widetilde{\mathbf{D}_{\lambda,J}}$, so that the map $\mathcal{W}$ is well-defined. Secondly, Proposition \[galreps\] and Theorem \[tatesthm\] show that $\chi_{\lambda,J}^{\textnormal{S}}$ is regular if and only if $\mathcal{W}(\chi_{\lambda,J}^{\textnormal{S}})$ is irreducible. Finally, it is clear that two simple supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-modules in the same orbit under $\omega^{\mathbb{Z}}$ will yield isomorphic projective Galois representations under $\mathcal{W}$, so we may view $\mathcal{W}$ as a map defined on supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules.
\[orblifts\] Let $\chi_{\lambda,J}^{\textnormal{S}}$ be a simple, regular supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-module, and let $d$ be a divisor of $n$. Then the orbit of $\chi_{\lambda,J}^{\textnormal{S}}$ under $\omega^{\mathbb{Z}}$ has size $d$ if and only if $d_{\mathcal{W}(\chi_{\lambda,J}^{\textnormal{S}})} = d$.
Let $(k_1,\ldots, k_{n - 1}, k_n)$ denote the $n$-tuple of integers associated to $\chi_{\lambda,J}^{\textnormal{S}}$, $r$ the integer given by equation , and $d$ a divisor of $n$. The group $\omega^\mathbb{Z}$ acts on the set of $n$-tuples by $$\omega.(k_1,\ldots, k_{n - 1}, k_n) := (k_n, k_1,\ldots, k_{n - 1}).$$ One easily verifies the following equivalences: $$\begin{aligned}
\omega^d.\chi_{\lambda, J}^{\textnormal{S}}= \chi_{\lambda, J}^{\textnormal{S}}& \Longleftrightarrow & \omega^d.(k_1,\ldots, k_{n - 1}, k_n) = (k_1, \ldots, k_{n - 1}, k_n)\\
& \Longleftrightarrow & r = \frac{1}{p - 1}\frac{p^n - 1}{p^d - 1}\sum_{j = 0}^{d - 1}(p - 1 - k_{n - j})p^j\\
& \Longleftrightarrow & p^dr = r + [n]\sum_{j = 0}^{d - 1}(p - 1 - k_{n - j})p^j\\
& \Longleftrightarrow & p^dr \equiv r~(\textnormal{mod}~[n]).\end{aligned}$$ By the definition of $d_\sigma$, we are done.
\[Wbij\] The map $\mathcal{W}$ realizes the numerical bijection of Corollary \[finalcor\]. More precisely, $\mathcal{W}$ induces a bijection between regular supersingular $L$-packets of ${{\mathcal{H}}}_{\textnormal{S}}$-modules of size $d$ and irreducible projective representations $\sigma$ of ${\mathcal{G}}_{{\mathbb{Q}_p}}$ of dimension $n$ for which $d_\sigma = d$.
Given a irreducible projective representation $\llbracket \textnormal{ind}_{{\mathcal{G}}_{\mathbb{Q}_{p^n}}}^{{\mathcal{G}}_{{\mathbb{Q}_p}}}(\omega_n^r)\rrbracket$, Theorem 8.7 of [@GK13] shows how to construct a simple, regular supersingular ${{\mathcal{H}}}_{\textnormal{S}}$-module $\chi_{\lambda,J}^{\textnormal{S}}$ such that $\mathcal{W}(\chi_{\lambda,J}^{\textnormal{S}}) \sim \llbracket \textnormal{ind}_{{\mathcal{G}}_{\mathbb{Q}_{p^n}}}^{{\mathcal{G}}_{{\mathbb{Q}_p}}}(\omega_n^r)\rrbracket$. Hence, the map $\mathcal{W}$ from regular supersingular $L$-packets of size $d$ to irreducible projective Galois representations with $d_\sigma = d$ is surjective. By Corollary \[finalcor\], these two sets have the same size, and $\mathcal{W}$ must be injective as well.
\[compat\] Let ${\mathfrak{M}}$ be an ${{\mathcal{H}}}$-module. We let ${\mathfrak{M}}\longmapsto\mathcal{GK}({\mathfrak{M}})$ denote the functor from the category of finite-length ${{\mathcal{H}}}$-modules over $k$ to the category of continuous ${\mathcal{G}}_{{\mathbb{Q}_p}}$-representations over $k$ constructed in Section 8 of [@GK13], and let ${\mathfrak{M}}\longmapsto\textnormal{JH}({\mathfrak{M}}|_{{{\mathcal{H}}}_{{\textnormal{S}}}})$ denote the functor obtained by taking the Jordan-Hölder constituents of the ${{\mathcal{H}}}_{\textnormal{S}}$-module ${\mathfrak{M}}|_{{{\mathcal{H}}}_{\textnormal{S}}}$ (without multiplicity). The following diagram of sets is commutative (where we consider all objects up to isomorphism and over $k$), and the horizontal arrows are bijections:
This follows from the explicit calculation of Theorem 8.5 in [@GK13], and the comments preceding Proposition \[orblifts\].
[0]{} Abdellatif, R. “Autour des répresenations modulo $p$ des groupes réductifs $p$-adiques de rang 1.” Thesis, Université Paris-Sud XI, (2011).
Aigner, M., “Combinatorial Theory.” Grundlehren der mathematischen Wissenschaften, 234, *Springer*, (1997).
Barthel, L. and Livné, R., “Irreducible Modular Represenations of $GL_2$ of a Local Field.” *Duke Mathematical Journal*, 75, (1994): 261–292.
Berger, L., “On some modular representations of the Borel subgroup of $\textnormal{GL}_2(\mathbf{Q}_p)$.” *Compositio Mathematica*, 146, No. 1, (2010): 58–80.
Borel, A., “Linear Algebraic Groups.” Graduate Texts in Mathematics, 126, *Springer*, (1991).
Bourbaki, N., “Éléments de mathématique, Groupes et alg[è]{}bres de Lie. Chapitres 4, 5 et 6.” *Masson*, (1981).
Breuil, C., “Sur quelques représentations modulaires et $p$-adiques de $\textrm{GL}_2(\mathbf{Q}_p)$ I.” *Compositio Mathematica*, 138, (2003): 165–188.
Breuil, C., “Sur quelques représentations modulaires et $p$-adiques de $\textrm{GL}_2(\mathbf{Q}_p)$ II.” *Journal of the Institute of Mathematics of Jussieu*, 2, (2003): 1–36.
Breuil, C. “Invariant $\mathcal{L}$ et série spéciale $p$-adique.” *Annales Scientifiques de l’École Normale Supérieure*, 37, (2004): 559–610.
Breuil, C., “Representations of Galois and $\textnormal{GL}_2$ in characteristic $p$.” Course given at Columbia University, available at [http://www.math.u-psud.fr/$\sim$breuil/PUBLICATIONS/New-York.pdf](http://www.math.u-psud.fr/~breuil/PUBLICATIONS/New-York.pdf)
Breuil, C. and Herzig, F., “Ordinary representations of $G(\mathbb{Q}_p)$ and fundamental algebraic representations.” To appear in *Duke Mathematical Journal*. Preprint available at [arXiv:1206.4413.](http://arxiv.org/abs/1206.4413)
Breuil, C. and V. Paškūnas. “Toward a modulo $p$ Langlands correspondence for $\textrm{GL}_2$.” *Memoirs of the AMS*, 216, (2012).
Colmez, P. “Représentations de $\mathbf{GL}_2(\mathbf{Q}_p)$ et $(\varphi,\Gamma)$-modules.” *Astérisque* 330, (2010): 281–509.
Emerton, M., “Local-global compatibility in the $p$-adic Langlands programme for $\textrm{GL}_{2/\mathbb{Q}}$.” Preprint (2010); available at [http://www.math.uchicago.edu/$\sim$emerton/pdffiles/lg.pdf](http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf).
Fontaine, J.-M., “Représentations $p$-adiques des corps locaux ($1^{\textnormal{\`{e}re}}$ partie).” The Grothendieck Festschrift, Volume II. *Birkhauser*, (1990): 249–309.
Große-Klönne, E., “From pro-$p$ Iwahori-Hecke modules to $(\varphi,\Gamma)$-modules I.” Preprint (2013); available at [http://www.math.hu-berlin.de/$\sim$zyska/Grosse-Kloenne/iwahecgal.pdf](http://www.math.hu-berlin.de/~zyska/Grosse-Kloenne/iwahecgal.pdf)
Kisin, M., “The Fontaine-Mazur conjecture for $\textrm{GL}_2$.” *Journal of the AMS*, 22, (2009): 641–690.
Kisin, M., “Deformations of $G_{\mathbb{Q}_p}$ and $\textrm{GL}_2(\mathbb{Q}_p)$ representations.” *Astérisque* 330, (2010): 511–528.
Lusztig, G., “Affine Hecke algebras and their graded version.” *Journal of the AMS*, vol. 2:3, (1989): 599–635.
Ollivier, R., “Platitude du pro-$p$-module universel de $\textnormal{GL}_2(F)$ en caractéristique $p$.” *Compositio Mathematica*, 143, No. 3, (2007): 703–720.
Ollivier, R., “Le foncteur des invariants sous l’action du pro-$p$-Iwahori de $\textrm{GL}_2(F)$.” *Journal für die reine und angewandte Mathematik.*, 635, (2009): 149–185.
Ollivier, R., “Parabolic induction and Hecke modules in characteristic $p$ for $p$-adic $\textnormal{GL}_n.$” *Algebra and Number Theory*, 4, No. 6, (2010): 701–742.
Ollivier, R., “Compatibility between Satake and Bernstein-type isomorphisms in characteristic $p$.” *Algebra and Number Theory*, 8, No. 5, (2014): 1071–1111.
Ollivier, R. and Sécherre, V., “Modules universels de $\textnormal{GL}_3$ sur un corps $p$-adique en caractéristique $p$.” Preprint (2011); available at [arXiv:1105.2957.](http://arxiv.org/abs/1105.2957)
Paškūnas, V., “The image of Colmez’s Montreal functor.” *Publications Mathématiques de l’IHÉS*, 118, (2013): 1–191.
Schneider, S. and Stuhler, U., “Representation Theory and Sheaves on the Bruhat-Tits Building.” *Publications Mathématiques de l’IHÉS*, 85, (1997): 97–191.
Serre, J.-P., “Modular forms of weight one and Galois representations.” Algebraic Number Fields ($L$-functions and Galois properties), *Academic Press*, (1977): 193–268.
Tits, J., “Reductive Groups over Local Fields.” Automorphic Forms, Representations and $L$-function, Proc. Symp. Pure Math. XXXIII, *American Mathematical Society*, (1979): 29–69.
Vignéras, M.-F., “A propos d’une conjecture de Langlands modulaire.” Finite Reductive Groups: Related Structures and Representations, *Birkhauser*, (1997): 415–452.
Vignéras, M.-F., “Pro-$p$-Iwahori Hecke algebra and supersingular $\overline{\mathbf{F}}_p$-representations.” *Mathematische Annalen*, 331, (2005): 523–556.
|
---
abstract: 'The detection of anomalies in real time is paramount to maintain performance and efficiency across a wide range of applications including web services and smart manufacturing. This paper presents a novel algorithm to detect anomalies in streaming time series data via statistical learning. We adapt the generalised extreme studentised deviate (ESD) test [@rosner1983] to streaming time series data by using time series decomposition and a sliding window approach. This is made computationally feasible by recursive updates of the ESD test statistic [@grubbs1950]. Our method is statistically principled and it outperforms the `AnomalyDetection` software package, recently released by Twitter Inc. (Twitter) [@twitterpackage] and used by multiple teams at Twitter as their state of the art on a daily basis [@twitter14]. The methodology is demonstrated using unlabelled data from the Twitter `AnomalyDetection` GitHub repository and using a real manufacturing example with labelled anomalies.'
author:
-
title: 'Real-Time Anomaly Detection for Advanced Manufacturing: Improving on Twitter’s State of the Art'
---
Anomaly detection; time series; streaming; real-time analytics; Grubb’s test; recursive extreme studentised deviate test (R-ESD); advanced manufacturing; Twitter
Introduction {#sec:intro}
============
The detection of anomalies (data that deviates from what is expected) is important to protect revenue, reputation and resources in many applications such as web services, smart manufacturing, telecommunications, fraud detection and biosurveillance. For example, exogenic factors such as bots, spams and sporting events can affect web services as can hardware problems and other endogenic factors [@twitter17]. In advanced manufacturing, the detection of anomalies in streaming machine data, for example from machine sensors that monitor processing conditions, can aid the identification of tool wear and tear and any problems in the structure or quality of a part in production [@konrad18].
The `AnomalyDetection` software package [@twitterpackage] was recently released by Twitter and is used daily to detect anomalies in their cloud infrastructure data, for example Tweets Per Second (TPS) and CPU utilisation. A conference publication [@twitter14], an article published on ArXiv [@twitter17] and a blogpost [@twitterblog15] have generated much interest with over $120$ citations since 2014, accepting Twitter’s challenge to the public and academic community to “evolve the package and learn from it as they have” [@twitterblog15].
The problem of anomaly detection in time series data of this nature is challenging due to its seasonal nature and its tendency to exhibit a trend. The approach taken by Twitter [@twitter14; @twitter17; @twitterblog15; @twitterpackage] is the seasonal hybrid extreme studentised deviate (SH-ESD) test. This is an adaptation of the generalised extreme studentised deviate (ESD) test [@rosner1983] which is itself a repeated application of the Grubbs hypothesis test [@grubbs1950] for a single outlier. These tests assume that the data is normally distributed. Thus it is necessary to decompose the time series, subtract the seasonal and trend components and perform the hypothesis tests on the resulting residuals. Twitter [@twitter14; @twitter17; @twitterblog15] use the median value of non-overlapping windows of data to estimate the trend as a stepwise function, which they argue is more robust to outliers. Whilst this is computationally fast and works well for some datasets, the results will be especially sensitive to the choice of window size and location. LOESS was used to determine the seasonal component via `stl` [@stlpackage]. However, the `AnomalyDetection` package requires the period to be specified by the user and a further requirement is that each non-overlapping window is assumed to capture at least one period of each seasonal pattern. In contrast, our approach avoids this restriction and uses a rolling window of streaming data.
A major statistical consideration arises from the implementation of SH-ESD. In order to increase robustness of the method against a large number of outliers, median and median absolute deviations (MAD) are used to studentise observations. This is not statistically appropriate for the generalised ESD test, as the resulting residuals may follow a heavier-tailed distribution than the adjusted t-distribution used in the appropriate tests of significance [@rosner1975]. Furthermore, SH-ESD has a number of limitations that our algorithm addresses; (i) its run time and nature is prohibitive for streaming data; (ii) it is prone to high levels of type one error (the detection of false positive anomalies), perhaps due to the incorrect use of median and MAD as described above and (iii) the period (seasonality) of the time series data must be specified by the user.
We present a novel algorithm entitled Recursive ESD (R-ESD) for fast anomaly detection in time series streaming data. Formulating the test statistic in a novel recursive way allows the test to be implemented while streaming data in real time. This is a key improvement over existing methods. First the seasonality and trend is estimated in an initial phase. Then the statistically principled generalised ESD test is implemented in a sliding window of time series data. Our approach results in fast identification of anomalies in each window which can be communicated to the end user while the data is being streamed. We address the problems in SH-ESD outlined above by; (i) formulating two recursive updates of the ESD test statistic, enabling anomaly detection while streaming in real time; (ii) using the mean to studentise observations, as it is statistically principled and appropriate for the Grubbs hypothesis test for outliers and (iii) estimating the period in the initial phase of our algorithm using a Fourier transformation via the `periodogram` function in TSA [@tsapackage].
The Numenta data repository provides annotated real-world and artificial time series data [@lavin15]. Figure \[fig:intromachines\] displays one such time series; temperature sensor data from an internal component of a large, industrial machine. The first anomaly is not explained or discussed in @lavin15. The second anomaly is a planned shutdown of the machine. The third anomaly is difficult to detect and directly led to the fourth anomaly, a catastrophic failure of the machine [@lavin15].
\[fig:intromachines\]
![Machine Temperature data stream with known anomalies marked in red[]{data-label="fig:intromachines"}](finalintromachines8aug.pdf)
The paper is organised as follows. Section \[sec:AD\] describes the problem of anomaly detection, including a short review of existing methods and outlines how we propose to measure the performance of our method. Section \[sec:grubbs\] formally defines our problem and presents our approach. In Section \[sec:results\] we present a set of experiments, followed by our results. We conclude with a discussion in Section \[sec:discussion\].
Anomaly detection {#sec:AD}
=================
Anomaly detection is the problem of identifying patterns in data that do not conform to typical behaviour. By definition, the anomaly detection problem depends on the data and or application in question. @chandolaetal09 provide a thorough review and compare a range of approaches to anomaly detection given in the scientific literature including examples from industrial damage detection, medical anomaly detection, cyber intrusion detection. and sensor networks. @gupta14 provide a survey of anomaly detection methods in the computer science literature for temporal data. The challenge of selecting a suitable algorithm is discussed in @kandanaarachchi18. @leighetal19 presents a framework to identify and compare suitable methods for a water-quality problem to detect anomalies in high frequency sensor data.
In this paper we focus on the problem of univariate time series streaming data such as those presented by Twitter [@twitter14; @twitter17; @twitterblog15] and the manufacturing problem displayed in Figure \[fig:intromachines\]. The typical behaviour of data of this nature is to exhibit trend and/or seasonality. The existence of a trend might itself be an anomaly. The research challenge is to detect anomalous data points as they arrive in streaming applications or soon after they arrive. The exact nature of the streaming capacity will be application dependent.
Consider a univariate time series data stream $\ldots,x_{t-1}, x_t, x_{t+1} \ldots$, where $x_t$ is an observation recorded at time $t$. We implement a rolling window approach to the problem of anomaly detection, where the window ${{\mathbf{x}}}= \{x_{t-w}, x_{t-w+1}, \ldots , x_{t} \}$ is the set of the $w$ observations of the data stream prior to and including time $t$. The goal is to detect anomalies $\tilde{{{\mathbf{x}}}} \in {{\mathbf{x}}}$ and $\tilde{{{\mathbf{t}}}} \in \{t-w,\ldots, t\}$, the anomalies and associated time-stamps of a subset of observations in each rolling window while streaming.
In order to assess an anomaly detection algorithm, it is useful to analyse datasets that are annotated with ground truth labels. In such cases, one can measure the precision; the proportion of true positive anomalies of all detected anomalies and recall; the ratio of true positive anomalies to the sum of true positive anomalies and false negative anomalies. We use these measures to compare our performance to the Twitter `AnomalyDetection` package in detecting known anomalies in manufacturing data.
Recursive ESD for streaming time series data {#sec:grubbs}
=============================================
@grubbs1950 provides a hypothesis test for a single outlier. This was generalised to the ESD test [@rosner1983], where a pre-specified number of $k$ anomalies can be detected. The ESD test statistics $R_1,\ldots,R_k$ are calculated from samples of size $n,n-1,\ldots,n-k+1$, successively reduced by the most extreme deviate (and potential anomaly) in the sample. For example, in the full sample of size $n$, the most extreme deviate would correspond to $x_i$, such that $\| x_i - \bar{x} \| \ge \| x_j- \bar{x} \| \; \;\; \forall i,j = 1, \ldots, n$, with equality only when $i=j$. $\bar{x}$ is the full sample mean. This is computed analogously for subsequent reduced samples.
In general, we denote $\tilde{x}_j$ as the dataset with the $j$th most extreme deviates removed and $\tilde{n}_j$ as the sample size of this set. So, for example $\tilde{n}_1$ will be equal $n - 1$ when the most extreme deviate is removed. The ESD test statistic is defined by: $$R_{j+1}=\frac{\max_i \left|x_i-\bar{\tilde x}_{j}\right|}{\tilde s_j}, \;\;\; i=1,\ldots, \tilde n _ {j}; \;\;\; j=1,\ldots, k;$$ where the reduced sample mean is $$\bar{\tilde x}_{j} = \frac{\sum_{i=1}^{\tilde n _ {j}}\tilde {x}_i}{\tilde n _ {j}},$$ and where the sum of squares with all the $j$th most extreme deviates removed is
$$\tilde s^2_j = \frac{\sum_{i=1}^{\tilde n _ {j}}\left(x_i-\bar {\tilde x}_{j}\right)^2}{\tilde n _ {j}-1}.$$
The critical values for this series of Student’s t-tests are $$\gamma_{l+1} = \frac{t_{n-l-2,p} (n-l-1)}{\sqrt{(n-l-2+t^2_{n-l-2,p}) (n-l)}}, \hspace{0.2cm} l=0,1,\ldots, k-1,$$ where $n$ is the number of data points in the dataset, $k$ is the maximum number of anomalies, $l$ is the order statistic and $p = 1 - (\alpha/2)(n-l)$. Further details can be found in Equation $2.5$ of @rosner1983.
In order to adopt the ESD test for streaming data, we note that the ESD test statistic $R_{j+1}$ can also be expressed as a function of the Grubb’s ratio $\frac{S_n^2}{S^2}$ used in @grubbs1950 such that, in our notation; $$R_{j+1} = \sqrt{\left(1-\frac{\tilde S^{2}_{j+1} }{\tilde S^{2}_{j}}\right)(\tilde n _ {j}-1)},$$ where $$\frac{\tilde S^2_{j+1}}{\tilde S^2_{j}}= \frac{\sum_{i=1}^{\tilde n _ {j+1}} \left(x_i-\bar {\tilde x} _{j+1} \right)}{\sum_{i=1}^{\tilde n _ {j}} \left(x_i-\bar{ \tilde x }_{j} \right)}.$$ This construction of the ESD test statistic is novel and useful in the context of our streaming anomaly detection problem as it permits recursive calculations. Having identified $x^* $ as the most extreme deviate in a sample $\tilde x$, the sums of squares can be reduced using the following recursive calculation; $$\tilde {S}_{j+1}^2=\tilde {S}_{j}^2 - \tilde{ n}_{j+1} (x^* - \bar {\tilde{x}}_j )^2 /\tilde{n}_j.$$ The recursive ESD test is outlined in Algorithm 1.
\[alg:grubbs\]
**INPUTS:** dataset ${{\mathbf{x}}}=(x_1,x_2, \ldots, x_n)$; Significance level $\alpha$; Maximum number of anomalies to be tested $k$; The initial full sample mean $\bar{\tilde x}_0 = \bar x$ and $\tilde S_0^2 = \sum_{i=1}^{n} \left(x_i-\bar x \right) $; Identify $x^*$, the maximum deviate in the dataset; Perform a recursive update of the sum of squares using Equation 3; Calculate the critical value $\gamma_{j-1}$ and test statistic $R_j$ using Equations $1$ and $2$; flag $x^*$ as an anomaly and add to the anomaly vector ${{\mathbf{x}}}_A$ ; Recursively update $\bar{\tilde x}_{j} $, the mean of the reduced dataset; Reduce the dataset by removing $x^*$; **OUTPUTS:** Anomaly vector ${{\mathbf{x}}}_A$.
Moreover, this formulation of the test statistic enables the ESD test to be used while streaming data by using a rolling window approach and recursively updating the test statistic as each data point arrives. Let $x_w$ be a newly streamed data point and let $x_0$ be the datapoint that is being removed as the window rolls forward by one at time $t$. Then the sum of squares and the sample mean can be calculated at time $t+1$ by the following recursive formulae; $$\label{eqn:ssrecstream}
\begin{array}{l}
S_{t+1}^2 = S_t^2 +\left(x_w-x_0\right)\left(x_w +x_0-2\bar x_t - \frac{x_w-x_0}{w}\right);
\\ \bar {x}_{t+1}=\bar{ x}_t+\frac{(x_w-x_0)}{w}.
\end{array}$$
The Recursive ESD (R-ESD) algorithm for anomaly detection is presented in Algorithm 2. It is a two stage approach. In the initial phase, a window of data ${{\mathbf{x}}}'$, of size $w'$, is decomposed into its seasonal $({{\mathbf{S}}})$, trend $({{\mathbf{T}}})$ and residual $({{\hbox{\boldmath$\epsilon$}}})$ components, such that $$x_t' = S_t + T_t + \epsilon_t,$$ where $\epsilon_t \sim N(0,\sigma^2)$, that is, the residuals at each time step are assumed normally distributed with zero mean and variance $\sigma^2$ to be estimated. We also assume that the errors in this general model are uncorrelated in time. These assumptions render the generalised ESD test appropriate to detect anomalies in the residuals ${{\hbox{\boldmath$\epsilon$}}}$. Note that the initial window size $w' \geq w$ to allow the fit of a useful statistical model. We assume that in this training period, no anomalies are detected. In practice for example, an engineer would monitor a manufacturing process carefully during this initial phase.
\[alg:recesd\]
**INPUTS:** Time series data ${{\mathbf{x}}}=(x_1,x_2, \ldots, x_t)$ observed at time $t$, with streaming new observations $(x_{t+1}, \ldots)$; Initial window size $w'$; Streaming window size $w$; Maximum number of anomalies $k$ in any given window; Define the initial training window of data by ${{\mathbf{x}}}'=(x_{t-w'+1},x_{t-w'+2}, \ldots, x_t)$; Perform trend and seasonal decomposition of ${{\mathbf{x}}}'$ for example by using methods described in Section \[sec:grubbs\];
Create forecasts for example using the `forecast` function [@forecast2] ${{\mathbf{x}}}^f=(x^f_{t+1},\ldots)$ as far as is required for the application and/or is computationally feasible; Define the current window of data to search for anomalies by ${{\mathbf{x}}}=(x_{t-w+1},x_{t-w+2}, \ldots, x_t)$ and denote the associated stationary residuals ${{\hbox{\boldmath$\epsilon$}}}= (\epsilon_{1},\epsilon_{2}\dots,\epsilon_{w})$ found in the model decomposition in line 2; Compute the initial sum of squares of the residuals $S_t^2$ using Equation \[eqn:ss\]; Compute the initial mean of the residuals $\bar\epsilon_t=\sum_{i=1}^{w} \epsilon_i/w$;
Let $\epsilon_0=\epsilon_1$; Slide the current window of data by one observation such that ${{\mathbf{x}}}=(x_{s-w+1},x_{s-w+2}, \ldots, x_s)$; Calculate $\epsilon_w = x_s - x^f_s$ using the forecasts found in line 2; Perform recursive updates of $S_{s}^2$ and $\bar {\epsilon}_{s}$ using Equation \[eqn:ssrecstream\];
Perform the sequential Grubbs test to detect anomalies ${{\mathbf{x}}}_{A,s}$ using Algorithm 2;
**OUTPUTS:** $\{ {{\mathbf{x}}}_{A,s},t_s\}_{s=(t+1)}^{\ldots } $, where ${{\mathbf{x}}}_A,s$ is a vector of anomalies found using the Grubbs algorithm in the $s^{th}$ window and where $t_s$ denotes the time stamp of the last observation in the relevant window.
Time series decomposition is a well-studied topic and commonly used methods are described in Chapter 6 of [@hyndmanbook]. A suitable period $p$ can be found in the initial phase for example, by using a Fourier transformation via `periodogram` from the `TSA` software package [@tsapackage]. We then use the `stlm` function of the `forecast` package [@forecast2] to model the trend using LOESS and to forecast the typical behaviour of future observations as far as is required. For example, in the context of smart manufacturing, this would be the length of the production process that is about to be performed. If this is not computationally feasible or if the typical behaviour of the process is expected to change across time, a suitable model can be re-fitted as often as is deemed necessary. The resulting model is used to calculate the residuals in the streaming window ${{\mathbf{x}}}$ and initialise the statistics required for the generalised ESD test namely, the mean of the residuals; $\bar {\epsilon}_t=\sum_{i=1}^{w} \epsilon_i/w$ and the sum of squares of the residuals; $$\label{eqn:ss}
S_t^2= \sum_{i=1}^w(\epsilon_i-\bar{ \epsilon}_t)^2.$$
In the streaming phase, the recursive generalised ESD test outlined in Algorithm 1 is applied to the residuals in each window. As the window slides forward by one datapoint at each iteration, fast recursive updates of $S_t^2$ and $\bar {\epsilon_t}$ at time $t$ are employed using Equation \[eqn:ssrecstream\] during streaming R-ESD (line 11, Algorithm 2).
Results {#sec:results}
=======
The R-ESD approach is demonstrated and compared with SH-ESD for the example presented in the `AnomalyDetection` package published on the GitHub repository [@twitterpackage] and for the machine temperature introduced in Figure \[fig:intromachines\], Section \[sec:intro\].
Figure $2$ displays the R-ESD and SH-ESD results for a single window of $4$ days of the first example. The significance level used for the generalised ESD test was $\alpha=0.05$, the default in the `AnomalyDetection` package. The number of anomalies tested by R-ESD in the single window is $k=288$. This is to coincide with max\_anoms $=0.02$ as is given in the `AnomalyDetection` package (since $k=w\times$ max\_anoms $=288$). R-ESD and SH-ESD agree in detecting $106$ anomalies with a further $24$ distinct anomalies detected by R-ESD and $8$ by SH-ESD. The R-ESD anomalies appear to be more convincing although there is no ground truth for this example. CPU time for SH-ESD was $0.17$ seconds, while R-ESD required $0.46$ seconds. However, while the R-ESD is computationally less efficient than SH-ESD, it is statistically principled and the algorithm is designed using recursive updates of the test statistic to allow real-time anomaly detection while streaming new datapoints using a sliding window.
\[fig:twitteronewindow\]
![A single window of Twitter data: Resulting anomalies detected by R-ESD and SH-ESD are shown in green and blue respectively, with $k=288$ (and equivalently max\_anoms$=0.02$) anomalies allowed in the window. A 4 day window is used from the example given in Twitter `AnomalyDetection` package on their GitHub repository with unlabelled anomalies.[]{data-label="fig"}](finaltwitterk288onewindow8aug.pdf)
Results for streaming windows of size $w=1440$, that is, one day of minutely data, are given in Figure $3$ and are compared to the non-overlapping window approach of SH-ESD described in [@twitter14] using the `longterm_period=TRUE` option in the `AnomalyDetection` package. As described in Section \[sec:AD\], the trend in each non-overlapping window is treated by SH-ESD as a flat line corresponding to the median of the values of the data in the window. Therefore, one might expect that R-ESD is less sensitive than SH-ESD to the choice of window size and certainly to the starting point of the algorithm. The CPU time for R-ESD to stream $7197$ data points in this example was $65$ seconds, that is only $0.02$ seconds per window, where the window size was $w=1440$. Thus streaming is highly feasible for many applications. Moreover, the R-ESD streaming approach seems to choose more sensible anomalies than the non-overlapping window approach of SH-ESD. Agreement occurs for $39$ anomalies with a further $118$ and $103$ anomalies detected by R-ESD and SH-ESD respectively.
\[fig:twitterstream\]
![Streaming Twitter data: Resulting anomalies detected by R-ESD and SH-ESD are shown in green and blue respectively. The number of anomalies tested per window is $k=288$ (and equivalently max\_anoms$=0.02$ for SH-ESD) anomalies allowed in the window.[]{data-label="fig"}](finaltwitterstreamingk2888aug.pdf)
The improved performance of R-ESD over SH-ESD is further demonstrated in the machine temperature example displayed in Figure $4$ for the data described in Section \[sec:intro\]. Here the number of anomalies per window $k=10$ was deemed appropriate in the context of manufacturing. There are $4$ known anomalies in this dataset and these are noted as being difficult to detect, the third in particular [@lavin15]. SH-ESD failed to detect any of the $4$ known anomalies when using the non-overlapping window approach to the anomaly detection. R-ESD performs better, providing anomaly detection in advance of the first labelled anomaly and thus allowing time to alert the engineer to an anomaly in advance of the problem. Furthermore it correctly detects one of the other anomalies. In practice, the former is more useful as the engineer can intervene in advance of a machine failure. Precision and recall are $0.004$ and $0.25$ for R-ESD. Both measures are $0$ for SH-ESD. This first anomaly is not explained or discussed in [@lavin15]. In fact their analysis does not utilise the first portion of the dataset although it is given and labelled on the `Numenta` GitHub repository. The third anomaly is notoriously difficult to detect as the lead up to this anomaly is a very gradual decline in machine temperature. In terms of CPU time, R-ESD requires approximately $10$ seconds to stream $20,000$ windows, that is only $0.0005$ seconds per window. This is slower than SH-ESD which takes $0.31$ seconds. However this is for non-overlapping windows of size $961$ rendering the two reasonably computationally similar (since $(20000/961)\times0.31\approx 6.5$ seconds).
\[fig:machinestream\]
![Streaming machine temperature example: Anomalies detected by R-ESD and SH-ESD are shown in green and blue respectively. The number of anomalies tested per window is $k=10$ for R-ESD (and equivalently max\_anoms$=0.0004$ for SH-ESD). Known anomalies are marked in red.[]{data-label="fig"}](finalmachinestreamingk10aug12.pdf)
Discussion {#sec:discussion}
==========
This paper presents a novel approach to anomaly detection for streaming time series data, which typically exhibits trend and or seasonality. The primary novelty of the R-ESD algorithm is the use of multiple recursive updates within the ESD test and across rolling windows of data. The major advantage is that this renders the approach computationally feasible for streaming data. In examples presented, computation times were as little as $0.02$ and $0.0005$ seconds per window i.e. per streamed datapoint. If required, computation times could be reduced further by implementing a priority queue [@knuth97] to reduce memory requirements.
Further studies are required to extend the comparison of R-ESD to alternative anomaly detection algorithms such as presented in @lavin15, to carry out these comparisons on many different types of datasets and to calculate the Numenta benchmark tests, which explicitly reward early detection. We suspect that R-ESD will perform well by this measure given the results presented for the machine temperature example.
In summary R-ESD is a fast, statistically principled and novel recursive approach to anomaly detection in time series data. It is highly feasible for streaming in real-time. It correctly studentises observations according to the theoretical distribution of the ESD test statistic and it outperforms the `AnomalyDetection` package, thus improving on Twitter’s approach.
Funding {#funding .unnumbered}
=======
This publication has emanated from research supported by a research grant from Science Foundation Ireland (SFI) under Grant Number 16/RC/3872 and is co-funded under the European Regional Development Fund.
B. Rosner. 1983. “Percentage points for a generalized ESD many-outlier procedure,” *Technometrics*, 25, 2, 165-172. F. E. Grubbs. 1950. “Sample Criteria for Testing Outlying Observations,” *Annals of Mathematical Statistics*, 21: 27-58. Twitter Inc. 2015. “AnomalyDetection R package,” GitHub repository: https://GitHub.com/twitter/AnomalyDetection. Accessed April 17, 2019. O. Vallis, J. Hochenbaum, A. and Kejariwal, “A Novel Technique for Long-Term Anomaly Detection in the Cloud,” *6th USENIX Workshop on Hot Topics in Cloud Computing*, Philadelphia, June 17–18. J. Hochenbaum, O. S. Vallis, A. Kejariwal. 2017. “Automatic anomaly detection in the cloud via statistical learning.’ arXiv:1704.07706v1. K. Mulrennan, J. Donovan, L. Creedon, I. Rogers, J. G. Lyons, M. McAfee. 2018. “A soft sensor for prediction of mechanical properties of extruded PLA sheet using an instrumented slit die and machine learning algorithms," *Polymer Testing*, 69, 462-469. A. Kejariwal. 2015. Introducing Practical and a Robust Anomaly Detection in a Time Series, (2015, 6 January) Retrieved from: https://blog.twitter.com/2015/introducing-practical-and-robust-anomaly-detection-in-a-time-series. B. Rosner. 1975 “On the detection of many outliers." *Technometrics*, 17(2), 221-227. K-S Chan and B. Ripley (2018). TSA: Time Series Analysis. R package version 1.2. https://CRAN.R-project.org/package=TSA. A. Lavin and S. Ahmad. 2015. “ Evaluating real-time anomaly detection algorithms - the Numenta anomaly benchmark.” In: *Machine Learning and Applications (ICMLA), 2015 IEEE 14th International Conference on*, 38-44 V. Chandola, A. Banerjee , V. Kumar. 2009. “Anomaly detection: A survey, ” *ACM Computing Surveys (CSUR)*, 41, 3, 1-58. M. Gupta, J. Gao, C. C. Aggarwal and J. Han. 2014. “Outlier detection for temporal data: a survey," *IEEE Transactions on Knowledge and Data Engineering*, 26, 9, 2250-2267. S. Kandanaarachchi, M. A. Muñoz, R. J. Hyndman and K. Smith-Miles “On normalization and algorithm selection for unsupervised outlier detection”, unpublished. P. D. Talagala, R. J. Hyndman, K. Smith-Miles, S. Kandanaarachchi and M. A. Muñoz “Anomaly detection in streaming nonstationary temporal data”, *Journal of Computational and Graphical Statistics*,“in press” C. Leigh, O. Alsibai, R. J. Hyndman, S. Kandanaarachchi, O. C. King, J. M. McGree, C. Neelamraju, J. Strauss, P. D. Talagala, R. Turner, K. Mengersen, and E. Peterson. 2019 “A framework for automated anomaly detection in high frequency water-quality data from in situ sensors". *Science of the Total Environment*, 664, 885-898. R. J. Hyndman, and G. Athanasopoulos, Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. OTexts.com/fpp2, 2018, Accessed on 17 April 2019 R. B. Cleveland, W. S. Cleveland, J. E. McRae and I. Terpenning, I., 1990. STL: A Seasonal-Trend Decomposition. *Journal of Official Statistics*, 6(1),3-73. S. Ramaswamy, R. Rastogi, K. Shim. 2000. “Efficient algorithms for mining outliers from large datasets. In: *Proceedings of the ACM international conference on management of data" (SIGMOD), Dallas*, 427-438. M. M. Breunig, H. P. Kriegel, R. Ng and J. Sander. 2000. “LOF: identifying density-based local outliers”. *In: Proceedings of the ACM international conference on management of data (SIGMOD), Dallas*, 93-104. R.J. Hyndman, Y. Khandakar. 2008. “Automatic time series forecasting: the forecast package for R.” *Journal of Statistical Software*, 26(3), 1-22. D. E. Knuth, *The art of computer programming: sorting and searching* (Vol. 3). Pearson Education, 1997.
|
---
abstract: 'Multi-output is essential in machine learning that it might suffer from nonconforming residual distributions, i.e., the multi-output residual distributions are not conforming to the expected distribution. In this paper, we propose “Wrapped Loss Function” to wrap the original loss function to alleviate the problem. This wrapped loss function acts just like the original loss function that its gradient can be used for backpropagation optimization. Empirical evaluations show wrapped loss function has advanced properties of faster convergence, better accuracy, and improving imbalanced data.'
author:
- 'Chun Ting Liu$^1$, Ming Chuan Yang$^2$, and Meng Chang Chen$^3$'
bibliography:
- 'example\_paper.bib'
title: Wrapped Loss Function for Regularizing Nonconforming Residual Distributions
---
Introduction
============
A deep learning model learns data representations and embeds the knowledge in its edge weights. A loss function is chosen to determine the goodness of training. Currently, backpropagation with gradient descent optimization algorithm is considered as the de facto standard of the learning process. As a deep learning model is learned to optimize its loss function, a loss function can determine the functionality and quality of the learned model, deal with nonconforming data set, and decide the convergence time. In this paper, we propose a wrapped loss function, which can enfold any a regular loss function so that to obtain many excellent properties.
Generally, the task of a deep learning model can be regression or classification. If the task is classification, it means the output variables have class labels; conversely, if the task is regression, the output variable takes continuous vector. The excellent performance of the model depends on an appropriate loss function. For example, the loss function using cross-entropy is more suitable for classification tasks than using least-squares error, because the cross-entropy allows finding a better local optimum for classification in a comparable environment and with randomly initialized weights [@pavel2013]. On the other hand, according to Gauss-Markov theorem, the least-squares estimator is the best linear unbiased estimators (BLUE) for linear regression tasks; however, the Gauss-Markov theorem establishes on the assumptions that finite variance of the error is constant for all variables and over time. However, this assumption is too strong for many real-world data.
Besides, multi-outputs is common in machine learning, such as multi-class classification and language translation, which could be trained either by combining several binary classifiers, or all-together training. In all-together deep learning model, loss function has to compute the probability of each class, and the weighted average is a frequently used method to alleviate nonconforming problems. For instance, if a task deals with imbalanced training data for multi-class classification give higher weights to rare classes can adjust the situation. Besides, least-square error having weights with appropriate distributions keeps the BLUE property even when data are heteroscedastic.
In this paper, we propose a wrapped loss function to wrap the original loss function by adding weight, estimated by Gauss maximum likelihood function, to each output residual. This approach can wrap up any loss function to become a “wrapped loss function,” and it is the namesake of the proposed approach. The wrapped loss function regularizes the loss of multi-output model outcomes and adjusts gradient accordingly during model training. In the later sections, we give a formal definition of the wrapped loss function and analyze its gradient and other properties.
In the empirical evaluations, we build both classification and regression multi-output cases to examine the performance of wrapped loss function, which is applied to the original loss functions. For the first case, we use a fully connected neural network to predict air pollution level, and for the second case, LeNet and AlexNet are used as the fundamental neural network for CIFAR100 test. Besides the accuracy rate, convergence rate, and using imbalanced and/or heteroscedastic data are also used for performance comparisons.
This paper is organized as follows. Section 2 reviews related work. Section 3 gives the problem definition and motivation. Section 4 formally introduces the properties of the wrapped loss function. Section 5 is the empirical evaluation. Finally, Section 6 is the conclusion.
Related Work
============
The work of introducing weights in the learning process has inspired this study.
- Weights on Multi-output Loss Function
The most similar concept with wrapped loss function is cost-sensitive learning and label re-balancing. The target of cost-sensitive learning is to find a minimal cost in the imbalanced label’s problems. In general, the errors are from the class with a rare label. In cost-sensitive learning, some methods add weights on loss function to learn its features. For example, there are the Prior Scaling [@steve1998] and Minimization of the misclassification costs method [@kukar1998]. Alternatively, just pre-set label weights before model training. like Median Frequency Balancing: [@bad2015; @farabet2012; @eigen2015]. ($\alpha_c = mdeian_{freq}/freq(c)$, where freq(c) is the number of data in the class c divided by the total number of data, and $median_{freq}$ is the median of the frequencies.
- Weights on Data Points
Besides add weights to multi-output loss, add weights to data points has been common practice on controlling outliers rejection. The outliers affect the performance in data mining and machine learning tasks. For example, if a data set includes outliers, the learned model would be misled. The method of Iteratively Reweighted Least Squares(IRLS) [@green1984] is used to find the maximum likelihood estimates of a generalized linear model to get the weights of data points to eliminate outliers.
Another noteworthy modify weights on data usage are AdaBoost [@freund1995boosting]. While AdaBoost boosts the performance of a collection of the weak learner, it also modifies the weights of data depending on their current classification performance so that the final classifier achieves the better performance.
- Weights on Multi-Task Learning
For the multi-task learning, there are two or more related tasks jointly learned, and their loss functions are combined into a loss function. Then the weights can be added to each task to generate a loss function. Huy [@huy2017] uses convolutional neural networks and deep neural network coupled with weighted loss function to solve audio event detection, and Sankaran [@sankaran2016] shows that the combination of 3 techniques: LSCSR-initialization, Multi-task training, and Class-weighted cross-entropy training gives the best results on keyword spotting. Unlike the above cases use pre-set weights, Kendall [@kendall2017] applies IRLS to find maximum likelihood estimates of a multi-task learning model for scene geometry and semantics.
Background and Motivation
=========================
In this section, we will introduce the problem when dealing with heteroscedasticity by using least square loss function as an example. By $D=(X,Y)$ we denote the given training data, and by $f(x)$ or $f_{W}(x)$ the learned function or neural network with the weight set $W$ of cardinality $d$. Recall $c$ is the number of class or the length of multi-output.
Least Square Loss Function
--------------------------
The least-square method is famous for finding a curve fitting for a given set of data with assumptions: expected zero error, uncorrelated, and constant variance in the errors. According to Gauss-Markov theorem, if all assumptions are satisfied, the least square error method is BLUE.
- Least Square Error $$\label{OriginalLoss}
\ell_{original} = \sum_{i=1}^c(y_i-f(x))^2$$
However, heteroscedasticity is a problem because the original least square method assumes that all residuals are drawn from the same population. This assumption may be violated in some real-world time series and cross-sectional data.
This problem can be solved by using a weighted least square estimation to obtain asymptotically efficient estimators. While some approaches add weights to data points, in this paper, we add weights to lose function.
- Weighted Least Square Error $$\label{HeteroscedasticityLoss}
\ell_{weight}=\sum_{i=1}^co_i(y_i-f(x))^2$$
where $o_i$’s are the weights for losses.
Gradient Descent of Loss Functions
----------------------------------
The gradient descent of loss function with back propagation is the most popular optimization algorithm for updating the weights of artificial intelligent network. In this subsection, we will show the difference of gradient descents between original and weighted losses.
### In The original Loss Function
Gradient descent is a first-order optimization algorithm to seek for the minimum loss and Eq (3) shows the loss function of i-th output $l_i$ and its gradient:
$$\begin{split}
\ell_i &= (y_i-f(x))^2\\
\frac{\partial{\ell_i}}{\partial{w}}&=\frac{\partial{\ell_i}}{\partial{f(x)}}\frac{\partial{f(x)}}{\partial{w}}\\
&=2(y_i-f(x))\frac{\partial{f(x)}}{\partial{w}}\\
\end{split}$$
The weighted loss function and gradient are defined as below.
$$\begin{split}
\ell_{o_i} &= o_i(y_i-f(x))^2\\
\frac{\partial{\ell_{o_i}}}{\partial{o_i}}&=(y_i-f(x))^2\\
\frac{\partial{\ell_{o_i}}}{\partial{w}}&=\frac{\partial{\ell_i}}{\partial{f(x)}}\frac{\partial{f(x)}}{\partial{w}}\\
&=2o_i\frac{\partial{f(x)}}{\partial{w}}(y_i-f(x))
\end{split}$$
Although the weighted least square error function can alleviate the heteroscedasticity problem, the weights might quickly approach to zero from the partial differentiation of the loss function by the weights. This problem can be solved by introducing a particular loss function that direct the learning process.
Wrapped Loss Function
======================
In this section, we propose the wrapped loss function and its gradient and generalization error. Formally at first, we define a residual vector by:
$$\label{Heteroscedasticity}
Y_i =f(X)+\epsilon_i \hspace{4mm} where \hspace{2mm} \epsilon_i {\sim} N(0,\sigma_i^2)$$
with three assumptions:
- Errors are mean zero: E\[$\epsilon_i$\]=0,
- Error are heteroscedastic, that is all have the same finite variance: Var($\epsilon_i$)=$\sigma_i^2$ $<$ $\infty$,
- Distinct error terms are uncorrelated: Cov($\epsilon_i$,$\epsilon_j$)=0, $\forall$i$\neq$j
Heteroscedasticity occurs when the error is correlated with an independent variable, for example, in a regression on household saving and income. Low-income people generally save a similar amount of money, while high-income people may have a significant variation in their savings. Also, heteroscedasticity is often the case with cross-sectional or time-series data that it makes estimation inefficient because the actual variance is underestimated. To solve the problem, we propose the wrapped loss function, unlike weighted loss function having zero weight problem in its gradient, the wrapped loss function is differentiable and learnable.
\[NegLogLikelihood\] If Eq. \[Heteroscedasticity\] holds, the negative log-likelihood is decided by $$\ell(\mathbf{\sigma},f)=\sum_{i=1}^c\left(\frac{\ell_i}{\sigma_i^2}+\log(\sigma_i^2)\right)$$ Furthermore, we have $$\aligned
\frac{\partial{\ell(\mathbf{\sigma},f)}}{\partial{\sigma_i}}&=\frac{2}{\sigma_i}-\frac{2(y_i-f(x))^2}{\sigma_i^3}\\
\frac{\partial{\ell(\mathbf{\sigma},f)}}{\partial{w_j}}&=\sum_{i=1}^c \frac{1}{\sigma_i^2}\frac{\partial{\ell_i}}{\partial{w_j}}
\endaligned$$
From Eq. \[Heteroscedasticity\] , $$p(y_i|x,\sigma_i)=\frac{1}{\sqrt{2\pi\sigma_i^2}}exp\left(\frac{-(y_i-f(x))^2}{2\sigma_i^2}\right)$$For the multi-output model, it specifies the joint density function for all outputs by the method of maximum likelihood function, derived as: $$\aligned
L(\mathbf{\sigma}|\mathbf{Y})
&=\prod_{i=1}^c p(y_i|x,\sigma_i)\\
\endaligned$$ where $\mathbf{\sigma}=(\sigma_1,\dots,\sigma_c)$ and $\mathbf{Y}=(Y_1,...,Y_c)$. The negative log-likelihood is as below: $$\aligned
-&\log{L(\mathbf{\sigma}|\mathbf{Y})}\\
=&\frac{c}{2}\log(2\pi)+\sum_{i=1}^c\frac{log(\sigma_i^2)}{2}+\sum_{i=1}^c\frac{(y_i-f(x))^2}{2\sigma_i^2}\\
\endaligned$$ Recall that the neural network $f(x)$ depends on the parameters $w_j$ and $\ell_i=(y_i-f(x))^2$. Hence we denote $\sum_{i=1}^c\left(\frac{\ell_i}{\sigma_i^2}+\log(\sigma_i^2)\right)$ as $\ell(\mathbf{\sigma},f)$. The partial derivatives of $\ell(\mathbf{\sigma},f)$ are easy to calculate.
Comparing Eq.\[HeteroscedasticityLoss\] with the loss in Lemma \[NegLogLikelihood\]: $$\aligned
\ell_{weight}=& \sum_{i=1}^c o_i\ell_i \\
\ell(\mathbf{\sigma},f)=&\sum_{i=1}^c\left(\frac{\ell_i}{\sigma_i^2}+\log(\sigma_i^2)\right)
\endaligned$$
We found that it is natural to consider $\sigma_i^{-2}$ as an estimator of $o_i$ and the term $\log(\sigma_i^2)$ as an regularizer. Hence, we heuristically assume $\forall i\in [c], o_i\approx\frac{k_i}{\sigma_i^2}$, where $k_i$’s are constants.
(The Wrapped Loss Function) $$\ell_{wrap} = \sum_{i=1}^c\left(o_i\ell_i +{\log o_i^{-1}}\right).$$
The function is called the “Wrapped Loss Function” since it can wrap up any loss function to become a new function. This wrapped loss function is differentiable that it can derive the weights $o_i$ to minimize the loss function.
The following Lemma and Theorem show the core idea in our training process, which combines similar concepts of the EM algorithm, (output loss) normalization, and regularization.
With the same assumptions (Eq.\[Heteroscedasticity\]),
1. The estimator $\hat{\sigma_i}^2=(y_i-f(x))^2$ maximizes the likelihood of Lemma \[NegLogLikelihood\].
2. If $o_i\leftarrow \hat{\sigma}^{-2}$, then $\frac{\partial{\ell_{wrap}}}{\partial{o_i}}=\ell_i-\frac{1}{o_i}$ and $\frac{\partial{\ell_{wrap}}}{\partial{w_j}}=\sum_{i=1}^{c}{o_i\frac{\partial{\ell_i}}{\partial{w_j}}}$.
By lemma \[NegLogLikelihood\], $ \frac{\partial{\ell(\mathbf{\sigma},f)}}{\partial{\sigma_i}}=0
\Rightarrow \sigma_i^2=(y_i-f(x))^2$. This means for the given training data, setting the the residual sum of squares $\hat{\sigma_i}^2$ as above can minimize the loss $\ell(\mathbf{\sigma},f)$, which maximizes the likelihood function in Lemma \[NegLogLikelihood\].
The second item is immediate by calculation.
It is easy to see that if $\forall i\in[c]$, $o_i=1$ then $\ell_{wrap}=\ell_{original}$. The following shows that when $o_i$’s are all near 1, the proposed $\ell_{wrap}$ approximates $\ell_{original}$ well.
\[Thm1\] For given data $D$, class number $c$ and neural network $f_{W}(x)$, let $L=max_{i\in[c],x\in X}\{(y_i-f_{W}(x))^2\}$. Assume $L>1$ and $\delta\in (0,(c(L+1))^{-2}]$. If $\forall i\in [c], |o_i-1|\leq \delta$, then $|\ell_{wrap}-\ell_{original}|\leq \frac{1}{c(L+1)}$.
Since $\log(1-\delta)=\delta-O(\delta^2)$, $$\aligned
\ell_{wrap}-\ell_{original}
&=\sum_{i=1}^{c}{(o_i-1)\ell_i+\log(o_i^{-1})}\\
&\leq \sum_{i=1}^{c}{\delta L+\log(o_i^{-1})}\\
&\leq c(\delta L+\delta -O(\delta^2))\\
&\leq c\delta (L+1)\\
&\leq \frac{1}{c(L+1)}.
\endaligned$$
Theorem \[Thm1\] implies the neural network $f(x)$ trained by the wrap loss $\ell_{wrap}$, if $o_i$’s are all near $1$, can be a approximation of the problem with the original loss Eq. \[OriginalLoss\].
wrapped loss function $\ell_w$; data input $x$; model parameters $w$; wrapped parameters $o$; number of epoch $t$; learning rate $\alpha$; initial $o=1$ and $w$ and $t=0$; compute $g_{o_i}\leftarrow\frac{\partial{\ell_{wrap}}}{\partial{o_i}}$ $\forall i\in \lbrace1,...,c\rbrace,o_{i,t+1}\leftarrow o_{i,t} -\alpha g_{o_i}$ compute $g_{w_j}\leftarrow\frac{\partial{\ell_{wrap}}}{\partial{w_j}}$ $\forall j\in \lbrace1,...,d\rbrace, w_{j,t+1}\leftarrow w_{j,t}-\alpha g_{w_j}$ $t\leftarrow t+1$
Note that Algorithm \[alg::WrappedLossGradientUpdate\] is used for training only. The real output is the original loss.
Properties of Wrap Error
------------------------
In the first glance at Wrap loss function $\ell_{wrap}$, it is not easy to see how $\ell_{wrap}$ effects the training. The following shows some advantages.
\[StepSize\] The setting of $o_i$ in Line 4 of Algorithm \[alg::WrappedLossGradientUpdate\] helps the convergence.
Observe that for $i\in[c]$,
- if $\ell_{i,t}<\ell_{i,t+1}$, i.e. $\frac{1}{\ell_{i,t}}>\frac{1}{\ell_{i,t+1}}$; this means the updating at time $t$ was in a bad direction (since the loss increases), so at time $t+1$, the updating should be cautious with a smaller ratio $o_{i,t+1}\leftarrow \frac{1}{\ell_{i,t+1}}$.
- if $\ell_{i,t}=\ell_{i,t+1}$, then ${o_i\frac{\partial{\ell_i}}{\partial{w_i}}}=0$. Furthermore, if for all $j\in[d]$ we have $\sum_{i=1}^{c}{o_i\frac{\partial{\ell_i}}{\partial{w_j}}}=0$, then Algorithm \[alg::WrappedLossGradientUpdate\] convergences.
- if $\ell_{i,t}>\ell_{i,t+1}$, i.e. $\frac{1}{\ell_{i,t}}<\frac{1}{\ell_{i,t+1}}$; this means the updating at time $t$ was in a good direction (since the loss decreases), so at time $t+1$, the updating can be vigorous with a larger ratio $o_{i,t+1}\leftarrow\frac{1}{\ell_{i,t+1}}$.
Now we can further discuss loss surface with wrapped; the wrapped loss surfaces displayed how the loss affected by the parameter $o_i$. The wrap loss $\ell_{wrap} = \sum_{i=1}^c\left(o_i\ell_i +{\log o_i^{-1}}\right)$ is complicated; to illustrate the main idea, we just consider one term in the summation and simplify as $WrapErr=oP^2+\log(o^{-1})$, where $P$ is the square error of the prediction and $o$ is the parameter in the wrap loss.
For instance, we assume predict target is 20, and the loss function is the least square. The wrapped loss has a parameter $o_i$ to control each outputs loss. As shown in Figure \[2DWrapSurface\], if the higher $o_i$, then the loss will be a sharper parabola; on the contrary, if the lower $o_i$, then the loss will be a flatter curve. Besides, if the $o_i$ equals to 1, the wrapped loss is the same with the original least square loss, and less than 1 will be more flat than original.
![Illustration of Wrap Error Surface 2D[]{data-label="2DWrapSurface"}](errorsurface2d.png)
In Figure \[3DWrapSurface\], we have a global view of the illustrated wrapped loss surface. As mentioned above, the loss and the gradient affected by the parameter $o_i$, and the updating step seems “jump” to the next location.
![Illustration of Error Surface 3D[]{data-label="3DWrapSurface"}](errorsurface3d.png)
\[HatSigma\] If $\hat{\sigma}^2=(y_i-f(x))^2=\ell_i$, then $E\left[\frac{\hat{\sigma_i}^2}{\sigma_i^2} \right]=DoF_i,$ where $DoF_i$ is the degree of freedom of the corresponding model.
Since $\hat{\sigma_i}^2=(y_i-f(x))^2=\ell_i$ and $y_i-f(x)=\epsilon_i {\sim} N(0,\sigma_i^2)$, it is well known in the chi-square test for variance that $\frac{o_i^2}{\sigma_i^2}\sim\chi_{DoF_i}^2$. Hence $E\left[\frac{\hat{\sigma_i}^2}{\sigma_i^2} \right]=DoF_i$.
In general, the degree of freedom can be a measure of the complexity of a statistical learning model.
\[Thm2\] Suppose Eq. \[Heteroscedasticity\], and the degree of freedom of the neural network $f(x)$ is fixed for multi-output, i.e. $\forall i\in[c]$, $DoF_i=DoF$. Then, on average and approximately, wrap loss function can be presented as $$E[\ell_{wrap}|ALGO\ref{alg::WrappedLossGradientUpdate} ]
\approx c(1+\log(DoF))+\sum_{i=1}^{c}{\log(E[\sigma_i^2])}.$$
Lemma \[HatSigma\] implies that on average $\hat{\sigma_i}^2$ equals to $DoF_i\cdot\sigma_i^2$ and by assumption $DoF_i=DoF$, which means $\hat{\sigma_i}^2\approx DoF\cdot\sigma_i^2$. On the other hand, in Algorithm \[alg::WrappedLossGradientUpdate\], we assign the residual sum of squares $\hat{\sigma_i}^2$, i.e. $\ell_i=(y_i-f(x))^2$, to $o_i^{-1}$. Hence if the expectation conditions on Algorithm \[alg::WrappedLossGradientUpdate\], then $$\aligned
\sum_{i=1}^{c}{o_i\ell_i+\log(o_i^{-1})}
&= \sum_{i=1}^{c}{\ell_i^{-1}\ell_i+\log(\hat{\sigma_i}^2)}\\
&\approx\sum_{i=1}^{c}{1+\log(DoF\cdot{\sigma_i}^2)}\\
&=c+c\log(DoF)+\sum_{i=1}^{c}{\log(\sigma_i^2)}.
\endaligned$$
![Illustration of Theorem \[Thm2\] []{data-label="3DWrapGE"}](wrap3d.png)
Figure \[3DWrapGE\] shows the simplified surface of Theorem \[Thm2\]. The term $\sum_{i=1}^clog(\sigma_i)$ is omitted to present the relation between the degree of freedom and the number of outputs. From the degree of freedom, the approximation surface is a logarithmic growth function when the number of outputs is fixed. On the contrary, the illustrated surface is a linear growth function of the number of outputs when a fixed degree of freedom.
Empirical Evaluations
=====================
In this section, we prepare two sets of experiments with distinct characteristics to examine the properties of the proposed wrapped loss function. The first set is fine atmospheric particulate matter (PM2.5) forecast, which is formulated as a multi-output regression. The second set is the image object recognition, which is a multi-class classification task. These show the wrapped loss functions has the following properties from experiments.
- Wrapped loss function improves the accuracy of the other models with the original loss function.
- Wrapped loss function converges faster than original loss function.
- Wrapped loss function alleviates imbalanced data problem.
PM2.5 Prediction
----------------
### Data Description
Fine Atmospheric particulate matter, known as PM2.5, is a collection particulates matter with a diameter of 2.5 $\mu m$ or less that they have significant impacts on environment and climate, and damages to human health. In this task, we predict PM2.5 in an urban area (Taipei, Taiwan, in this study) using open data provided by the Environmental Protection Administration. The data consists of 18 monitor stations with features including weather features (precipitation, pressure, temperature, wind speed, wind direction, etc.), air quality features (PM2.5, CO, NO2 , PM10, RH, SO2, etc.), and traffic-related features (average vehicle speed, traffic load, intersection numbers, etc.). The data of the year of 2015 is used as the training set and 2016 as the test set. Note that the PM2.5 level is heteroscedastic since the variances are different at a different time of day. For instance, at 3 am, the PM2.5 level is low and has a small variance since there is almost no traffic on the roads, while at 2 pm, the variance becomes large depending on the events of the city and traffic situation.
### Model Architecture
In this set of experiments, we design a neural network with 6 fully connected layers, namely, FC1024, FC512, FC256, FC128, FC64 and output layer (Figure \[DNNforPM25\]), where there is a dropout layer or batch normalization between each layer. The input vector is the information of a day and predicts target is a day after hours of PM2.5.
![Deep Learning Model[]{data-label="DNNforPM25"}](dnn.png)
The optimization algorithm of this model is AdaGrad [@john2011] with a fixed learning rate is set as 0.01 and mini-batch size is 10. Mean square error is the loss function and.
### Data Preprocessing
Generally, deep learning method automatically extracts useful knowledge to create a model without handcrafted feature engineering. In this experiment, we only normalize training data to Gaussian distributions with zero mean and unit variance.
### Results
The goal of the experiment is to predict the PM2.5 level of BangQiao and Cailiao in 6 hours, 12 hours, 18 hours and 24 hours with dropout (DO) [@srivastava2014], batch normalization (BN) [@sergey2015] and wrapped loss function (WR) and their combinations (WR+DO, WR+BN, and WR+BN+DO). Note that this task is a heteroscedasticity problem. For example, PM2.5 at 2 pm and 3 am has a different variance. The RMSE comparison between the original model with others techniques and the one with wrapped loss function are displayed in Table \[Table01\] for and the Banqiao monitoring station, and Table \[Table02\] for Cailiao monitoring station.
The results of the experiment show that if the training uses only dropout, batch normalization, or wrapped loss, then the proposed wrapped loss is usually the second-best. Although the wrapped loss function performs worse than the original model with dropout, the wrapped loss function works with dropout and/or batch normalization at the same time, and it shows the performance is best for wrapped loss combined with both dropout and batch normalization. Generally, the wrapped loss function usually improves the performance of the original methods.
---------------------------------------------- -- -- -- -- -- --
**\#Times &**DO & **BN &**WR &**W+D & **W+B\
6 & 11.06 &13.05 &13.2 &11.08 &13.42\
12 & 12.83 &14.77 &14.14 &12.19 &15.97\
18 & 12.96 &15.64 &15.62 &12.65 &15.07\
24 & 13.66 &18.29 &16.94 &13.30 &15.36\
************
---------------------------------------------- -- -- -- -- -- --
: DNN: PM2.5 Best RMSE of Banqiao Station[]{data-label="Table01"}
---------------------------------------------- -- -- -- -- -- --
**\#Times &**DO & **BN &**WR &**W+D & **W+B\
6 & 8.83 &14.42 &13.48 &9.13 &13.18\
12 & 9.65 &14.03 &13.73 &10.37 &12.13\
18 & 10.29 &15.02 &12.63 &11.02 &13.95\
24 & 10.93 &14.1 &12.74 &10.56 &15.18\
************
---------------------------------------------- -- -- -- -- -- --
: DNN: PM2.5 Best RMSE of Cailiao Station[]{data-label="Table02"}
We also compared the number of Epochs needed to complete the training process. According to Table \[Table03\], it can be seen that the wrapped loss function has fastest convergence rate.
-------------------------------------------------- -- -- -- -- -- --
**\#Times &**DO & **BN &**WR &**WR+DO & **WR+BN\
6 & 1789 &1 &1 &1665 &1\
12 & 1262 &1 &1 &1190 &1\
18 & 1427 &2 &1 &1163 &1\
24 & 641 &1 &1 &1294 &1\
************
-------------------------------------------------- -- -- -- -- -- --
: DNN: Epoch of Best PM2.5 RMSE of Banqiao Station[]{data-label="Table03"}
-------------------------------------------------- -- -- -- -- -- --
**\#Times &**DO & **BN &**WR &**WR+DO & **WR+BN\
6 & 1769 &3 &2 &953 &2\
12 & 1503 &1 &1 &802 &1\
18 & 1297 &2 &1 &986 &1\
24 & 1132 &1 &1 &787 &1\
************
-------------------------------------------------- -- -- -- -- -- --
: DNN: Epoch of Best PM2.5 RMSE of Cailiao Station[]{data-label="Table04"}
Image Object Recognition
------------------------
### Data Description
Different from the multi-output regression in the previous subsection, the second task is of multi-class classification to assign a class to the image object. The CIFAR-100 [@krizhevsky2009learning] data set consists of 60000 32$\times$32 RGB color images in 100 classes. In each class, there are 600 images with 500 training data and 100 testing data. In addition, the 100 classes are grouped into 20 super-classes. In this task, it is to classify color photographs of object into one of 100 classes.
### Model Architecture
Convolution neural networks (CNN) is one of most popular and successful deep learning models [@lecun1998] in image processing tasks. We use LeNet-5 and AlexNet [@alex2012] as the original functions for image recognition task. In the architecture of LeNet (as shown in the left part of Figure \[LeNet\_AlexNet\]), after two stacks of convolution with padding and max pooling sections, the activation function ReLU is computed. Then, two fully connected layers follow, and a softmax layer is at the end. The loss function is cross entropy function. The reason we choose LeNet and AlexNet is that their source codes were available from tensorflow and have credibility. The training parameters are set as: Fixed learning rate of 0.001, Mini-batch size of 100. And the AlexNet (as shown in the right part of Figure \[LeNet\_AlexNet\]) had a similar architecture as LeNet but is deeper and has more filters.
![LeNet and AlexNet Model[]{data-label="LeNet_AlexNet"}](la.png)
do you do data preprocessing? In image processing, normalization is a process that scaled the range of pixel values, then the pixels divide by 255 are interval\[0 ,1\].
### Results
First, We run both LeNet and AlexNet to classify pictures into one of 5, 10, 30, 50, 80, 100 classes. The results are shown in Table \[Table05\] and Table \[Table06\]. It can be observed that the wrapped loss function performs better than DO and BN in both LeNet and AlexNet, especially when the task is classified image objects into large number of classes.
---------------------------------------------------- -- -- -- -- -- --
**\#Classes &**DO & **BN &**WR &**WR+DO & **WR+BN\
5 & 75.6 &77.4 &78 &77.4 & 77.2\
10 & 66.2 &66.5 &68.2 &65.7 & 69.1\
30 & 46.7 &48.3 &50.7 &49.8 &50.7\
50 & 39.1 &40.1 &42.4 &42.7 &42.4\
80 & 36.2 &37.6 &38.8 &38.3 &38.2\
100 & 32.6 &34 &35.9 &35.8 &36\
************
---------------------------------------------------- -- -- -- -- -- --
: LeNet: Best Accuracy of CIFAR 100 []{data-label="Table05"}
---------------------------------------------------- -- -- -- -- -- --
**\#Classes &**DO & **BN &**WR &**WR+DO & **WR+BN\
5 & 78.4 &79.8 &77.4 &78.4 &78.8\
10 & 66 &67 &67.5 &66 &66.5\
30 & 40.7 &40.5 &44.4 &40.7 &46.3\
50 & 31.8 &31.4 &37.5 &31.8 &36.8\
80 & 27.6 &27.9 &33 &27.6 &33.2\
100 & 25.6 &25.2 &30.3 &25.6 &30.1\
************
---------------------------------------------------- -- -- -- -- -- --
: AlexNet: Best Accuracy of CIFAR 100 []{data-label="Table06"}
The numbers of Epochs needed to complete the training process are compared. From the results of Table \[Table07\] and Table \[Table08\], it can be seen that the wrapped loss function converged mush faster than the LeNet and AlexNet. It is also interesting to find out when the wrapped loss function works with dropout or batch normalization, the training process is completed in an even shorter time. The possible reason is that the wrapped loss function approach adopts idea similar to Gauss-Newton method [@wedderburn1974quasi] that fast convergence is a famous advantage of Newton’s optimization methods.
---------------------------------------------------- -- -- -- -- -- --
**\#Classes &**DO & **BN &**WR &**WR+DO & **WR+BN\
5 & 9k &26k &8k &3k &5k\
10 & 35k &29k &13k &11k &14k\
30 & 39k &42k &22k &18k &24k\
50 & 57k &65k &22k &20k &29k\
80 & 76k &140k &56k &21k &55k\
100 & 91k &104k &56k &28k &9k\
************
---------------------------------------------------- -- -- -- -- -- --
: LeNet: Epoch of Best Accuracy of CIFAR 100[]{data-label="Table07"}
---------------------------------------------------- -- -- -- -- -- --
**\#Classes &**DO & **BN &**WR &**WR+DO & **WR+BN\
5 & 29k &30k &4k &15k &5k\
10 & 50k &45k &7k &5k &6k\
30 & 36k &50k &10k &13k &14k\
50 & 39k &46k &16k &15k &17k\
80 & 58k &53k &18k &20k &24k\
100 & 59k &62k &26k &24k &27k\
************
---------------------------------------------------- -- -- -- -- -- --
: AlexNet: Epoch of Best Accuracy of CIFAR 100[]{data-label="Table08"}
### imbalancing data
Here we show the capability of wrapped loss function in improving the performance on imbalanced data set. 10 classes from CIFAR100 are selected randomly and the number of data a selected class is reduced to 2%, 5%, 10%, 20% and 30% respectively of the original size. For example, the adjusted class only has 50 images when the proportion is down to 10%. We compare the performances on accuracy of three settings: LeNet, LeNet with wrapped loss function and the frequency balancing method [@bad2015; @farabet2012; @eigen2015].
The experiment results of using imbalanced data as in Table \[Table09\] shows the wrapped loss function and the frequency balancing has similar accuracy rate, while both are better than the original LeNet. Table \[Table09\] has an ADJ column shows the accuracy rate of the adjusted class that the Frequency Balancing method has a slightly better performance than the wrapped loss function. However, the wrapped loss function still has the shortest time to complete the training process, as in Table \[Table10\]. The possible reason is if the class has larger total loss, $o_i$ will be smaller. So the rare class get the higher gradient that it affects the performance on imbalanced data.
----- ----- ------- ----- ------- ----- -------
% ADJ Total ADJ Total ADJ Total
2% 3 60.4 16 60.4 18 61.7
5% 25 61.2 35 62.5 30 63.6
10% 39 63.2 51 64.5 45 64.4
20% 68 65.3 74 65 68 65.7
30% 74 66.2 76 66.2 73 68.1
----- ----- ------- ----- ------- ----- -------
: LeNet: Accuracy of Imbalanced Data[]{data-label="Table09"}
% [**Original**]{} [**Freq Balancing**]{} [**Wrapped Loss**]{}
----- ------------------ ------------------------ ---------------------- --
2% 34k 26k 11k
5% 37k 38k 12k
10% 48k 37k 10k
20% 54k 29k 7k
30% 35k 30k 14k
: LeNet: Epoch of Best Accuracy of Imbalanced Cifar 100 with different Imbalanced Rate[]{data-label="Table10"}
Conclusions
===========
We have introduced the approach of wrapped loss function and shown it alleviates the problem of the heteroscedastic variances, it has fast convergence rate and improves accuracy for both multi-output regression task and multi-class classification on deep learning model.
In theoretical aspects, we analyze the wrapped loss function and derive its gradient, which leads to the training process in Algorithm 1, and obtain an approximation of expected error. The training process using wrapped loss iteratively modulates the updating ratio according to the corresponding residual loss, which boosts the convergence rate. The approximation of expected wrap error which conditions on the execution of our training process shows the effects of the length of multi-output (or the number of class) and the model complexity (measured by the degree of freedom). In the empirical evaluations, we compared original loss and wrapped loss on four experiments of multi-output regression and classification tasks. The experiment results are encouraging, which display wrapped loss can have better accuracy rate than batch normalization and dropout in the CIFAR100 case. In the air pollution case, if we combined wrapped loss function with batch normalization and dropout, it improves the performance too. Moreover, it has advantageous properties of faster convergence rate and alleviation the ill effect of imbalance labels. We believe the wrapped function is convenient to apply and its gain cannot be overlooked during the training phrase of deep learning model.
|
---
abstract: 'In logistic regression, separation occurs when a linear combination of the predictors can perfectly classify part or all of the observations in the sample, and as a result, finite maximum likelihood estimates of the regression coefficients do not exist. @Gelman_etal_2008 recommended independent Cauchy distributions as default priors for the regression coefficients in logistic regression, even in the case of separation, and reported posterior modes in their analyses. As the mean does not exist for the Cauchy prior, a natural question is whether the posterior means of the regression coefficients exist under separation. We prove theorems that provide necessary and sufficient conditions for the existence of posterior means under independent Cauchy priors for the logit link and a general family of link functions, including the probit link. We also study the existence of posterior means under multivariate Cauchy priors. For full Bayesian inference, we develop a Gibbs sampler based on Pólya-Gamma data augmentation to sample from the posterior distribution under independent Student-$t$ priors including Cauchy priors, and provide a companion R package in the supplement. We demonstrate empirically that even when the posterior means of the regression coefficients exist under separation, the magnitude of the posterior samples for Cauchy priors may be unusually large, and the corresponding Gibbs sampler shows extremely slow mixing. While alternative algorithms such as the No-U-Turn Sampler in Stan can greatly improve mixing, in order to resolve the issue of extremely heavy tailed posteriors for Cauchy priors under separation, one would need to consider lighter tailed priors such as normal priors or Student-$t$ priors with degrees of freedom larger than one.'
author:
- 'Joyee Ghosh[^1] ^,[fnsymbol[3]{}]{}^'
- 'Yingbo Li[^2] ^,^[^3]'
- 'Robin Mitra[^4]'
bibliography:
- 'Cauchy.bib'
title: On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression
---
Introduction
============
In Bayesian linear regression, the choice of prior distribution for the regression coefficients is a key component of the analysis. Noninformative priors are convenient when the analyst does not have much prior information, but these prior distributions are often improper which can lead to improper posterior distributions in certain situations. @Fernandez_Steel_2000 investigated the propriety of the posterior distribution and the existence of posterior moments of regression and scale parameters for a linear regression model, with errors distributed as scale mixtures of normals, under the independence Jeffreys prior. For a design matrix of full column rank, they showed that posterior propriety holds under mild conditions on the sample size; however, the existence of posterior moments is affected by the design matrix and the mixing distribution. Further, there is not always a unique choice of noninformative prior [@Yang_Berger_1996]. On the other hand, proper prior distributions for the regression coefficients guarantee the propriety of posterior distributions. Among them, normal priors are commonly used in normal linear regression models, as conjugacy permits efficient posterior computation. The normal priors are informative because the prior mean and covariance can be specified to reflect the analyst’s prior information, and the posterior mean of the regression coefficients is the weighted average of the maximum likelihood estimator and the prior mean, with the weight on the latter decreasing as the prior variance increases.
A natural alternative to the normal prior is the Student-$t$ prior distribution, which can be viewed as a scale mixture of normals. The Student-$t$ prior has tails heavier than the normal prior, and hence is more appealing in the case where weakly informative priors are desirable. The Student-$t$ prior is considered robust, because when it is used for location parameters, outliers have vanishing influence on posterior distributions [@Dawid_1973]. The Cauchy distribution is a special case of the Student-$t$ distribution with 1 degree of freedom. It has been recommended as a prior for normal mean parameters in a point null hypothesis testing [@Jeffreys_1961], because if the observations are overwhelmingly far from zero (the value of the mean specified under the point null hypothesis), the Bayes factor favoring the alternative hypothesis tends to infinity. Multivariate Cauchy priors have also been proposed for regression coefficients [@Zellner_Siow_1980].
While the choice of prior distributions has been extensively studied for normal linear regression, there has been comparatively less work for generalized linear models. Propriety of the posterior distribution and the existence of posterior moments for binary response models under different noninformative prior choices have been considered [@Ibrahim_Laud_1991; @Chen_Shao_2001].
Regression models for binary response variables may suffer from a particular problem known as separation, which is the focus of this paper. For example, complete separation occurs if there exists a linear function of the covariates for which positive values of the function correspond to those units with response values of 1, while negative values of the function correspond to units with response values of 0. Formal definitions of separation [@Albert_Anderson_1984], including complete separation and its closely related counterpart quasicomplete separation, are reviewed in Section 2. Separation is not a rare problem in practice, and has the potential to become increasingly common in the era of big data, with analysis often being made on data with a modest sample size but a large number of covariates. When separation is present in the data, @Albert_Anderson_1984 showed that the maximum likelihood estimates (MLEs) of the regression coefficients do not exist (i.e., are infinite). Removing certain covariates from the regression model may appear to be an easy remedy for the problem of separation, but this ad-hoc strategy has been shown to often result in the removal of covariates with strong relationships with the response [@Zorn_2005].
In the frequentist literature, various solutions based on penalized or modified likelihoods have been proposed to obtain finite parameter estimates [@Firth_1993; @Heinze_Schemper_2002; @Heinze_2006; @Rousseeuw_Christmann_2003]. The problem has also been noted when fitting Bayesian logistic regression models [@Clogg_etal_1991], where posterior inferences would be similarly affected by the problem of separation if using improper priors, with the possibility of improper posterior distributions [@Speckman_etal_2009].
@Gelman_etal_2008 recommended using independent Cauchy prior distributions as a default weakly informative choice for the regression coefficients in a logistic regression model, because these heavy tailed priors avoid over-shrinking large coefficients, but provide shrinkage (unlike improper uniform priors) that enables inferences even in the presence of complete separation. @Gelman_etal_2008 developed an approximate EM algorithm to obtain the posterior mode of regression coefficients with Cauchy priors. While inferences based on the posterior mode are convenient, often other summaries of the posterior distribution are also of interest. For example, posterior means under Cauchy priors estimated via Monte Carlo and other approximations have been reported in @Bardenet_etal_2014 [@Chopin_Ridgway_2015]. It is well-known that the mean does not exist for the Cauchy distribution, so clearly the prior means of the regression coefficients do not exist. In the presence of separation, where the maximum likelihood estimates are not finite, it is not clear whether the posterior means will exist. To the best of our knowledge, there has been no investigation considering the existence of the posterior mean under Cauchy priors and our research is filling this gap. We find a necessary and sufficient condition where the use of independent Cauchy priors will result in finite posterior means here. In doing so we provide further theoretical underpinning of the approach recommended by [@Gelman_etal_2008], and additionally provide further insights on their suggestion of centering the covariates before fitting the regression model, which can have an impact on the existence of posterior means.
When the conditions for existence of the posterior mean are satisfied, we also empirically compare different prior choices (including the Cauchy prior) through various simulated and real data examples. In general, posterior computation for logistic regression is known to be more challenging than probit regression. Several MCMC algorithms for logistic regression have been proposed [@OBrien_Dunson_2004; @Holmes_Held_2006; @Gramacy_Polson_2012], while the most recent P[ó]{}lya-Gamma data augmentation scheme of @Polson_etal_2013 emerged superior to the other methods. Thus we extend this P[ó]{}lya-Gamma Gibbs sampler for normal priors to accommodate independent Student-$t$ priors and provide an R package to implement the corresponding Gibbs sampler.
The remainder of this article is organized as follows. In Section 2 we derive the theoretical results: a necessary and sufficient condition for the existence of posterior means for coefficients under independent Cauchy priors in a logistic regression model in the presence of separation, and extend our investigation to binary regression models with other link functions such as the probit link, and multivariate Cauchy priors. In Section 3 we develop a Gibbs sampler for the logistic regression model under independent Student-$t$ prior distributions (of which the Cauchy distribution is a special case) and briefly describe the NUTS algorithm of @Hoff:Gelm:2014 which forms the basis of the software Stan. In Section 4 we illustrate via simulated data that Cauchy priors may lead to coefficients of extremely large magnitude under separation, accompanied by slow mixing Gibbs samplers, compared to lighter tailed priors such as Student-$t$ priors with degrees of freedom $7$ ($t_7$) or normal priors. In Section 5 we compare Cauchy, $t_7$, and normal priors based on two real datasets, the SPECT data with quasicomplete separation and the Pima Indian Diabetes data without separation. Overall, Cauchy priors exhibit slow mixing under the Gibbs sampler compared to the other two priors. Although mixing can be improved by the NUTS algorithm in Stan, normal priors seem to be the most preferable in terms of producing more reasonable scales for posterior samples of the regression coefficients accompanied by competitive predictive performance, under separation. In Section 6 we conclude with a discussion and our recommendations.
Existence of Posterior Means Under Cauchy Priors
================================================
In this section, we begin with a review of the concepts of complete and quasicomplete separation proposed by @Albert_Anderson_1984. Then based on a new concept of solitary separators, we introduce the main theoretical result of this paper, a necessary and sufficient condition for the existence of posterior means of regression coefficients under independent Cauchy priors in the case of separation. Finally, we extend our investigation to binary regression models with other link functions, and Cauchy priors with different scale parameter structures.
Let $\mathbf{y} = (y_1, y_2, \ldots,y_n)^T$ denote a vector of independent Bernoulli response variables with success probabilities $\pi_{1}, \pi_2, \ldots, \pi_{n}$. For each of the observations, $i = 1, 2, \ldots, n$, let $\mathbf{x}_i= (x_{i1}, x_{i2}, \ldots,x_{ip})^T$ denote a vector of $p$ covariates, whose first component is assumed to accommodate the intercept, i.e., $x_{i,1}=1$. Let $\mathbf{X}$ denote the $n \times p$ design matrix with $\mathbf{x}_i^T$ as its $i$th row. We assume that the column rank of $\mathbf{X}$ is greater than 1. In this paper, we mainly focus on the logistic regression model, which is expressed as: $$\log\left(\frac{\pi_{i}}{1-\pi_i}\right) = \mathbf{x}_i^T\boldsymbol\beta, \quad i = 1,
2, \ldots, n,
\label{eqn:logreg}$$ where $\boldsymbol\beta = (\beta_1, \beta_2, \ldots, \beta_p)^T$ is the vector of regression coefficients.
A Brief Review of Separation
----------------------------
We denote two disjoint subsets of sample points based on their response values: $ A_0 = \{i: y_i = 0\}$ and $A_1 = \{i: y_i = 1\}$. According to the definition of @Albert_Anderson_1984, complete separation occurs in the sample if there exists a vector $\boldsymbol\alpha = (\alpha_1, \alpha_2, \ldots,\alpha_p)^T$, such that for all $i = 1, 2, \ldots, n$, $$\mathbf{x}_i^T \boldsymbol\alpha > 0 \ \text{ if } i \in A_1, \quad
\mathbf{x}_i^T \boldsymbol\alpha < 0 \ \text{ if } i \in A_0. \label{eqn:compsep}$$ Consider a simple example in which we wish to predict whether subjects in a study have a certain kind of infection based on model . Let $ y_i = 1$ if the $i$th subject is infected and 0 otherwise. The model includes an intercept ($x_{i1}=1$) and the other covariates are age ($x_{i2}$), gender ($x_{i3}$), and previous records of being infected ($x_{i4}$). Suppose in the sample, all infected subjects are older than 25 ($x_{i2} > 25$), and all subjects who are not infected are younger than 25 ($x_{i2} < 25$). This is an example of complete separation because is satisfied for $\boldsymbol\alpha = (-25,1,0,0)^T$.
If the sample points cannot be completely separated, @Albert_Anderson_1984 introduced another notion of separation called quasicomplete separation. There is quasicomplete separation in the sample if there exists a non-null vector $\boldsymbol\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_p)^T$, such that for all $i = 1, 2, \ldots, n$, $$\mathbf{x}_i^T \boldsymbol\alpha \geq 0 \ \text{ if } i \in A_1, \quad
\mathbf{x}_i^T \boldsymbol\alpha \leq 0 \ \text{ if } i \in
A_0, \label{eqn:qcompsep}$$ and equality holds for at least one $i$. Consider the set up of the previous example where the goal is to predict whether a person is infected or not. Suppose we have the same model but there is a slight modification in the dataset: all infected subjects are at least 25 years old ($x_{i2} \geq 25$), all uninfected subjects are no more than 25 years old ($x_{i2} \leq 25$), and there are two subjects aged exactly 25, of whom one is infected but not the other. This is an example of quasicomplete separation because is satisfied for $\boldsymbol\alpha = (-25,1,0,0)^T$ and the equality holds for two observations with age exactly 25.
Let $\mathcal{C}$ and $\mathcal{Q}$ denote the set of all vectors $\bm{\alpha}$ that satisfy and , respectively. For any $\bm{\alpha} \in \mathcal{C}$, all sample points must satisfy , so $\bm{\alpha}$ cannot lead to quasicomplete separation which requires at least one equality in . This implies that $\mathcal{C}$ and $\mathcal{Q}$ are disjoint sets, while both can be non-empty for a certain dataset. Note that @Albert_Anderson_1984 define quasicomplete separation only when the sample points cannot be separated using complete separation. Thus according to their definition, only one of $\mathcal{C}$ and $\mathcal{Q}$ can be non-empty for a certain dataset. However, in our slightly modified definition of quasicomplete separation, the absence of complete separation is not required. This permits both $\mathcal{C}$ and $\mathcal{Q}$ to be non-empty for a dataset. In the remainder of the paper, for simplicity we use the term “separation” to refer to either complete or quasicomplete separation, so that $\mathcal{C} \cup \mathcal{Q}$ is non-empty.
Existence of Posterior Means Under Independent Cauchy Priors
------------------------------------------------------------
When Markov chain Monte Carlo (MCMC) is applied to sample from the posterior distribution, the posterior mean is a commonly used summary statistic. We aim to study whether the marginal posterior mean $E(\beta_j \mid \mathbf{y})$ exists under the independent Cauchy priors suggested by @Gelman_etal_2008. Let $C(\mu,\sigma)$ denote a Cauchy distribution with location parameter $\mu$ and scale parameter $\sigma$. The default prior suggested by @Gelman_etal_2008 corresponds to $\beta_j \stackrel{\text{ind}}{\sim} C(0,\sigma_j)$, for $j=1, 2, \dots, p$.
For a design matrix with full column rank, @Albert_Anderson_1984 showed that a finite maximum likelihood estimate of $\boldsymbol{\beta}$ does not exist when there is separation in the data. However, even in the case of separation and/or a rank deficient design matrix, the posterior means for some or all $\beta_j$’s may exist because they incorporate the information from the prior distribution. Following Definition 2.2.1 of @Casella_Berger_1990 [pp. 55], we say $E(\beta_j \mid \mathbf{y})$ exists if $E(|\beta_j| \mid \mathbf{y}) < \infty$, and in this case, $E(\beta_j \mid \mathbf{y})$ is given by $$\label{eqn:pm_betaj}
E(\beta_j \mid \mathbf{y})
= \int_0^{\infty} \beta_j ~p(\beta_j \mid \mathbf{y})~d\beta_j
+ \int_{-\infty}^0 \beta_j ~p(\beta_j \mid \mathbf{y})~d\beta_j.$$ Note that alternative definitions may require only one of the integrals in to be finite for the mean to exist, e.g., @Bickel_Doksum_2001 [pp. 455]. However, according to the definition used in this paper, both integrals in have to be finite for the posterior mean to exist. Our main result shows that for each $j = 1, 2, \ldots, p$, the existence of $E(\beta_j \mid \mathbf{y})$ depends on whether the predictor $\mathbf{X}_j$ is a solitary separator or not, which is defined as follows:
The predictor $\mathbf{X}_j$ is a solitary separator, if there exists an $\boldsymbol\alpha \in (\mathcal{C} \cup \mathcal{Q})$ such that $$\label{eqn:solitary}
\alpha_j \neq 0, \quad \alpha_r = 0 \text{ for all } r \neq j.$$
This definition implies that for a solitary separator $\mathbf{X}_j$, if $\alpha_j > 0$, then $x_{i,j} \geq 0$ for all $i\in A_1$, and $x_{i,j} \leq 0$ for all $i \in A_0$; if $\alpha_j < 0$, then $x_{i,j} \leq 0$ for all $i\in A_1$, and $x_{i,j} \geq 0$ for all $i \in A_0$. Therefore, the hyperplane $\{\mathbf{x}\in \mathbb{R}^p: x_{j} = 0\}$ in the predictor space separates the data into two groups $A_1$ and $A_0$ (except for the points located on the hyperplane). The following theorem provides a necessary and sufficient condition for the existence of marginal posterior means of regression coefficients in a logistic regression model.
\[theorem:existence\] In a logistic regression model, suppose the regression coefficients (including the intercept) $\beta_j \stackrel{ind}{\sim} C(0,\sigma_j)$ with $\sigma_j>0$ for $j=1, 2, \dots, p$, so that $$\label{eqn:prior}
p(\bm{\beta}) = \prod_{j=1}^{p} p(\beta_j)
= \prod_{j=1}^{p}\frac{1}{\pi \sigma_j (1 + \beta_j^2 / \sigma_j^2)}.$$ Then for each $j = 1, 2, \ldots, p$, the posterior mean $E(\beta_j \mid \mathbf{y})$ exists if and only if $\mathbf{X}_j$ is not a solitary separator.
A proof of Theorem \[theorem:existence\] is available in Appendices \[proof\_th1\_necessary\] and \[proof\_th1\_sufficient\].
Theorem \[theorem:existence\] implies that under independent Cauchy priors in logistic regression, the posterior means of all coefficients exist if there is no separation, or if there is separation with no solitary separators.
@Gelman_etal_2008 suggested centering all predictors (except interaction terms) in the pre-processing step. A consequence of Theorem \[theorem:existence\] is that centering may have a crucial role in the existence of the posterior mean $E(\beta_j \mid \mathbf{y})$.
We expand on the second remark with a toy example where a predictor is a solitary separator before centering but not after centering. Consider a dataset with $n=100$, $\mathbf{y}=(\underbrace{0, \dots 0}_{25}, \underbrace{1, \dots,
1}_{75})^T$ and a binary predictor $\mathbf{X}_j = (\underbrace{0, \dots 0}_{50}, \underbrace{1, \dots, 1}_{50})^T$. Here $\mathbf{X}_j$ is a solitary separator which leads to quasicomplete separation before centering. However, the centered predictor $\mathbf{X}_j = (\underbrace{-0.5, \dots -0.5}_{50}, \underbrace{0.5, \dots, 0.5}_{50})^T$ is no longer a solitary separator because after centering the hyperplane $\{\mathbf{x}: x_{j} = -0.5\}$ separates the data but $\{\mathbf{x}: x_{j} = 0\}$ does not. Consequently, the posterior mean $E(\beta_j \mid \mathbf{y})$ does not exist before centering but it exists after centering.
Extensions of the Theoretical Result
------------------------------------
So far we have mainly focused on the logistic regression model, which is one of the most widely used binary regression models because of the interpretability of its regression coefficients in terms of odds ratios. We now generalize Theorem \[theorem:existence\] to binary regression models with link functions other than the logit. Following the definition in @McCullagh_Nelder_1989 [pp. 27], we assume that for $i = 1, 2, \ldots, n$, the linear predictor $\mathbf{x}_i^T\boldsymbol\beta$ and the success probability $\pi_{i}$ are connected by a monotonic and differentiable link function $g(\cdot)$ such that $g(\pi_{i}) = \mathbf{x}_i^T\boldsymbol\beta$. We further assume that $g(.)$ is a one-to-one function, which means that $g(.)$ is strictly monotonic. This is satisfied by many commonly used link functions including the probit. Without loss of generality, we assume that $g(\cdot)$ is a strictly increasing function.
\[theorem:general\] In a binary regression model with link function $g(.)$ described above, suppose the regression coefficients have independent Cauchy priors in . Then for each $j =1, 2, \ldots, p$,
- a necessary condition for the existence of the posterior mean $E(\beta_j \mid \mathbf{y})$ is that $\mathbf{X}_j$ is not a solitary separator;
- a sufficient condition for the existence of $E(\beta_j \mid \mathbf{y})$ consists of the following:
- $\mathbf{X}_j$ is not a solitary separator, and
- $\forall \epsilon>0$, $$\label{eq:general_condition}
\int_{0}^{\infty} \beta_j p(\beta_j) g^{-1}(- \epsilon \beta_j) d \beta_j < \infty,
\quad
\int_{0}^{\infty} \beta_j p(\beta_j) \left[1 - g^{-1}(\epsilon \beta_j)\right] d \beta_j< \infty.$$
Note that in the sufficient condition of Theorem \[theorem:general\] imposes constraints on the link function $g(.)$, and hence the likelihood function. A proof of this theorem is given in Appendix \[proof\_th2\]. Moreover, it is shown that condition holds for the probit link function.
In certain applications, to incorporate available prior information, it may be desirable to use Cauchy priors with nonzero location parameters. The following corollary states that for both logistic and probit regression, the condition for existence of posterior means derived in Theorems \[theorem:existence\] and \[theorem:general\] continues to hold under independent Cauchy priors with nonzero location parameters.
\[corollary:nonzero\_prior\_mean\] In logistic and probit regression models, suppose the regression coefficients $\beta_j \stackrel{\text{ind}}{\sim} C(\mu_j,\sigma_j)$, for $j=1, 2, \dots, p$. Then a necessary and sufficient condition for the existence of the posterior mean $E(\beta_j \mid \mathbf{y})$ is that $\mathbf{X}_j$ is not a solitary separator, for $j=1, 2, \dots, p$.
A proof of Corollary \[corollary:nonzero\_prior\_mean\] is available in Appendix \[proof\_corollary:nonzero\_prior\_mean1\].
In some applications it could be more natural to allow the regression coefficients to be dependent, [*a priori*]{}. Thus in addition to independent Cauchy priors, we also study the existence of posterior means under a multivariate Cauchy prior, with the following density function: $$\label{eq:multi_Cauchy_prior}
p(\boldsymbol\beta) = \frac{\Gamma\left( \frac{1+p}{2} \right)}
{\Gamma\left( \frac{1}{2} \right) \pi^{\frac{p}{2}} |\boldsymbol\Sigma|^{\frac{1}{2}}
\left[ 1 + (\boldsymbol\beta - \boldsymbol\mu)^T \boldsymbol\Sigma^{-1}
(\boldsymbol\beta - \boldsymbol\mu) \right]^{\frac{1 + p}{2}}},$$ where $\boldsymbol\beta \in \mathbb{R}^p$, $\boldsymbol\mu$ is a $p \times 1$ location parameter and $\boldsymbol\Sigma$ is a $p \times p$ positive-definite scale matrix. A special case of the multivariate Cauchy prior is the Zellner-Siow prior [@Zellner_Siow_1980]. It can be viewed as a scale mixture of $g$-priors, where conditional on $g$, $\boldsymbol\beta$ has a multivariate normal prior with a covariance matrix proportional to $g(\mathbf{X}^T\mathbf{X})^{-1}$, and the hyperparameter $g$ has an inverse gamma prior, $\text{IG}(1/2, n/2)$. Based on generalizations of the $g$-prior to binary regression models [@Fouskakis_etal_2009; @Bove_Held_2011; @Hanson_etal_2014], the Zeller-Siow prior, which has a density with $\boldsymbol\Sigma \propto n(\mathbf{X}^T\mathbf{X})^{-1}$, can be a desirable objective prior as it preserves the covariance structure of the data and is free of tuning parameters.
\[theorem:multivariate\_Cauchy\] In logistic and probit regression models, suppose the vector of regression coefficients $\boldsymbol\beta$ has a multivariate Cauchy prior as in . If there is no separation, then all posterior means $E(\beta_j \mid \mathbf{y})$ exist, for $j = 1, 2, \ldots, p$. If there is complete separation, then none of the posterior means $E(\beta_j \mid \mathbf{y})$ exist, for $j = 1, 2, \ldots, p$.
A proof of Theorem \[theorem:multivariate\_Cauchy\] is available in Appendices \[proof\_theorem:multivariate\_Cauchy\_no\_separation\] and \[proof\_theorem:multivariate\_Cauchy\_complete\_separation\]. The study of existence of posterior means under multivariate Cauchy priors in the presence of quasicomplete separation has proved to be more challenging. We hope to study this problem in future work. Note that although under , the induced marginal prior of $\beta_j$ is a univariate Cauchy distribution for each $j = 1, 2, \ldots, p$, the multivariate Cauchy prior is different from independent Cauchy priors, even with a diagonal scale matrix $\boldsymbol\Sigma = \text{diag}(\sigma_1^2, \sigma_2^2,\ldots, \sigma_p^2)$. In fact, as a rotation invariant distribution, the multivariate Cauchy prior places less probability mass along axes than the independent Cauchy priors (see Figure \[fig:Cauchy\_contours\]). Therefore, it is not surprising that solitary separators no longer play an important role for existence of posterior means under multivariate Cauchy priors, as evident from Theorem \[theorem:multivariate\_Cauchy\].
![Contour plots of log-density functions of independent Cauchy distributions with both scale parameters being $1$ (left) and a bivariate Cauchy distribution with scale matrix $\mathbf{I}_2$ (right). These plots suggest that independent Cauchy priors place more probability mass along axes than a multivariate Cauchy prior, and thus impose stronger shrinkage. Hence, if complete separation occurs, $E(\beta_j \mid \mathbf{Y})$ may exist under independent Cauchy priors for some or all $j = 1,2, \ldots, p$ (Theorem \[theorem:existence\]), but does not exist under a multivariate Cauchy prior (Theorem \[theorem:multivariate\_Cauchy\]).[]{data-label="fig:Cauchy_contours"}](Cauchy_contours.pdf){height="2.15in" width="4in"}
So far we have considered Cauchy priors, which are $t$ distributions with 1 degree of freedom. We close this section with a remark on lighter tailed $t$ priors (with degrees of freedom greater than 1) and normal priors, for which the prior means exist.
In a binary regression model, suppose that the regression coefficients have independent Student-t priors with degrees of freedom greater than one, or independent normal priors. Then it is straightforward to show that the posterior means of the coefficients exist because the likelihood is bounded above by one and the prior means exist. The same result holds under multivariate $t$ priors with degrees of freedom greater than one, and multivariate normal priors.
MCMC Sampling for Logistic Regression {#sec:mcmc}
=====================================
In this section we discuss two algorithms for sampling from the posterior distribution for logistic regression coefficients under independent Student-$t$ priors. We first develop a Gibbs sampler and then briefly describe the No-U-Turn Sampler (NUTS) implemented in the freely available software Stan [@Carp:Gelm:Hoff:Lee:Good:Beta:Brub:Guo:Li:Ridd:2016].
P[ó]{}lya-Gamma Data Augmentation Gibbs Sampler {#sec:gibbs}
-----------------------------------------------
@Polson_etal_2013 showed that the likelihood for logistic regression can be written as a mixture of normals with respect to a P[ó]{}lya-Gamma (PG) distribution. Based on this result, they developed an efficient Gibbs sampler for logistic regression with a multivariate normal prior on $\boldsymbol\beta$. @Choi_Hobert_2013 showed that their Gibbs sampler is uniformly ergodic. This guarantees the existence of central limit theorems for Monte Carlo averages of functions of $\boldsymbol\beta$ which are square integrable with respect to the posterior distribution $p(\boldsymbol\beta \mid \mathbf{y})$. @Choi_Hobert_2013 developed a latent data model which also led to the Gibbs sampler of @Polson_etal_2013. We adopt their latent data formulation to develop a Gibbs sampler for logistic regression with independent Student-$t$ priors on $\boldsymbol\beta$.
Let $U = (2 / \pi^2)\sum_{l=1}^{\infty}W_{l}/{(2l-1)}^2$, where $W_{1},W_{2}, \dots$ is a sequence of i.i.d. Exponential random variables with rate parameters equal to 1. The density of $U$ is given by $$h(u) = \sum_{l=0}^{\infty}{(-1)}^l \frac{(2l+1)}{\sqrt{2 \pi u^3}}e^{-\frac{(2l+1)^2}{8u}}, \quad 0< u < \infty.$$ Then for $k \geq 0$, the P[ó]{}lya-Gamma (PG) distribution is constructed by exponential tilting of $h(u)$ as follows: $$p(u; k) = \cosh\left(\frac{k}{2}\right) e^{-\frac{k^2u}{2}}h(u), \quad 0< u <\infty.$$ A random variable with density $p(u; k)$ has a PG($1,k$) distribution.
Let $ t_{v}(0, \sigma_j)$ denote the Student-$t$ distribution with $v$ degrees of freedom, location parameter 0, and scale parameter $\sigma_j$. Since Student-$t$ distributions can be expressed as inverse-gamma (IG) scale mixtures of normal distributions, for $j = 1, 2, \ldots, p$, we have: $$\beta_j \sim t_{v}(0, \sigma_j) \Longleftrightarrow
\begin{cases}
\beta_j \mid \gamma_j \sim \text{N}(0, \gamma_j), \\
\gamma_j \sim \text{IG}\left(\frac{v}{2}, \frac{v\sigma_j^2}{2} \right).
\end{cases}$$ Conditional on $\boldsymbol\beta$ and $\boldsymbol\Gamma = \text{diag}(\gamma_1, \gamma_2, \ldots, \gamma_p)$, let $(y_1, z_1), (y_2, z_2), \ldots, (y_n, z_n)$ be $n$ independent random vectors such that $y_i$ has a Bernoulli distribution with success probability $\exp(\mathbf{x}_i^T\boldsymbol\beta)/
(1+\exp(\mathbf{x}_i^T\boldsymbol\beta))$, $z_i \sim PG(1,|\mathbf{x}_i^T\boldsymbol\beta|)$, and $y_i$ and $z_i$ are independent, for $i=1, 2, \ldots, n$. Let $\boldsymbol{Z}_{D}= \text{diag}(z_1, z_2, \ldots, z_n)$, then the augmented posterior density is $p(\boldsymbol\beta,\boldsymbol\Gamma,\boldsymbol{Z}_{D} \mid \mathbf{y})$. We develop a Gibbs sampler with target distribution $p(\boldsymbol\beta,
\boldsymbol\Gamma,\boldsymbol{Z}_{D} \mid \mathbf{y})$, which cycles through the following sequence of distributions iteratively:
1. $\boldsymbol\beta \mid \boldsymbol\Gamma, \boldsymbol{Z}_{D}, \mathbf{y}
\sim \text{N}\left( (\mathbf{X}^T\boldsymbol{Z}_{D}\mathbf{X} + \boldsymbol\Gamma^{-1})^{-1}
\mathbf{X}^T\mathbf{\tilde{y}},
(\mathbf{X}^T\boldsymbol{Z}_{D}\mathbf{X} + \boldsymbol\Gamma^{-1})^{-1} \right)$, where $\tilde{y}_i = y_i -1/2$ and $\mathbf{\tilde{y}} = (\tilde{y}_1, \tilde{y}_2, \ldots, \tilde{y}_n)^T$,
2. $\gamma_j \mid \boldsymbol\beta, \boldsymbol{Z}_{D}, \mathbf{y}
\stackrel{\text{ind}}{\sim} \text{IG}\left( \frac{v+1}{2}, \frac{\beta_j^2 + v\sigma_j^2}{2} \right)$, for $j=1, 2, \ldots, p$,
3. $z_i \mid \boldsymbol\Gamma,\boldsymbol\beta,\mathbf{y}
\stackrel{\text{ind}}{\sim} \text{PG}(1, |\mathbf{x}_i^T\boldsymbol\beta|)$, for $i=1, 2, \ldots, n$.
Steps 1 and 3 follow immediately from @Choi_Hobert_2013 [@Polson_etal_2013] and step 2 follows from straightforward algebra. In the next section, for comparison of posterior distributions under Student-$t$ priors with different degrees of freedom, we implement the above Gibbs sampler, and for normal priors we apply the Gibbs sampler of @Polson_etal_2013. Both Gibbs samplers can be implemented using the R package [tglm]{}, available in the supplement.
Stan {#sec:nuts}
----
Our empirical results in the next section suggest that the Gibbs sampler exhibits extremely slow mixing for posterior simulation under Cauchy priors for data with separation. Thus we consider alternative MCMC sampling algorithms in the hope of improving mixing. A random walk Metropolis algorithm shows some improvement over the Gibbs sampler in the $p=2$ case. However, it is not efficient for exploring higher dimensional spaces. Thus we have been motivated to use the software Stan [@Carp:Gelm:Hoff:Lee:Good:Beta:Brub:Guo:Li:Ridd:2016], which implements the No-U-Turn Sampler (NUTS) of @Hoff:Gelm:2014, a tuning free extension of the Hamiltonian Monte Carlo (HMC) algorithm [@Neal:2011].
It has been demonstrated that for continuous parameter spaces, HMC can improve over poorly mixing Gibbs samplers and random walk Metropolis algorithms. HMC is a Metropolis algorithm that generates proposals based on Hamiltonian dynamics, a concept borrowed from Physics. In HMC, the parameter of interest is referred to as the “position” variable, representing a particle’s position in a $p$-dimensional space. A $p$-dimensional auxiliary parameter, the “momentum” variable, is introduced to represent the particle’s momentum. In each iteration, the momentum variable is generated from a Gaussian distribution, and then a proposal of the position momentum pair is generated (approximately) along the trajectory of the Hamiltonian dynamics defined by the joint distribution of the position and momentum. Hamiltonian dynamics changing over time can be approximated by discretizing time via the “leapfrog” method. In practice, a proposal is generated by applying the leapfrog algorithm $L$ times, with stepsize $\epsilon$, to the the current state. The proposed state is accepted or rejected according to a Metropolis acceptance probability. Section 5.3.3 of the review paper by @Neal:2011 illustrates the practical benefits of HMC over random walk Metropolis algorithms. The examples in this section demonstrate that the momentum variable may change only slowly along certain directions during leapfrog steps, permitting the position variable to move consistently in this direction for many steps. In this way, proposed states using Hamiltonian dynamics can be far away from current states but still achieve high acceptance probabilities, making HMC more efficient than traditional algorithms such as random walk Metropolis.
In spite of its advantages, HMC has not been very widely used in the Statistics community until recently, because its performance can be sensitive to the choice of two tuning parameters: the leapfrog stepsize $\epsilon$ and the number of leapfrog steps $L$. Very small $\epsilon$ can lead to waste in computational power whereas large $\epsilon$ can yield large errors due to discretization. Regarding the number of leapfrog steps $L$, if it is too small, proposed states can be near current states and thus resemble random walk. On the other hand, if $L$ is too large, the Hamiltonian trajectory can retrace its path so that the proposal is brought closer to the current value, which again is a waste of computational power.
The NUTS algorithm tunes these two parameters automatically. To select $L$, the main idea is to run the leapfrog steps until the trajectory starts to retrace its path. More specifically, NUTS builds a binary tree based on a recursive doubling procedure, that is similar in flavor to the doubling procedure used for slice sampling by @Neal:2003, with nodes of the tree representing position momentum pairs visited by the leapfrog steps along the path. The doubling procedure is stopped if the trajectory starts retracing its path, that is making a “U-turn”, or if there is a large simulation error accumulated due to many steps of leapfrog discretization. NUTS consists of a carefully constructed transition kernel that leaves the target joint distribution invariant. It also proposes a way for adaptive tuning of the stepsize $\epsilon$.
We find that by implementing this tuning free NUTS algorithm, available in the freely available software Stan, substantially better mixing than the Gibbs sampler can be achieved in all of our examples in which posterior means exist. We still include the Gibbs sampler in this article for two main reasons. First, it illustrates that Stan can provide an incredible improvement in mixing over the Gibbs sampler in certain cases. Stan requires minimal coding effort, much less than developing a Gibbs sampler, which may be useful information for readers who are not yet familiar with Stan. Second, Stan currently works for continuous target distributions only, but discrete distributions for models and mixed distributions for regression coefficients frequently arise in Bayesian variable selection, for regression models with binary or categorical response variables [@Holmes_Held_2006; @Mitr:Duns:2010; @Ghos:Clyd:2011; @Ghos:Herr:Sieg:2011; @Ghos:Reit:2013; @Li:Clyd:2015]. Unlike HMC algorithms, Gibbs samplers can typically be extended via data augmentation to incorporate mixtures of a point mass and a continuous distribution, as priors for the regression coefficients, without much additional effort.
Simulated Data
==============
In this section, we use two simulation examples to empirically demonstrate that under independent Cauchy priors, the aforementioned MCMC algorithm for logistic regression may suffer from extremely slow mixing in the presence of separation in the dataset.
For each simulation scenario, we first standardize the predictors following the recommendation of @Gelman_etal_2008. Binary predictors (with 0/1 denoting the two categories) are centered to have mean 0, and other predictors are centered and scaled to have mean 0 and standard deviation 0.5. Their rationale is that such standardizing makes the scale of a continuous predictor comparable to that of a symmetric binary predictor, in the sense that they have the same sample mean and sample standard deviation. @Gelman_etal_2008 made a distinction between input variables and predictors, and they suggested standardizing the input variables only. For example, temperature and humidity may be input variables as well as predictors in a model; however, their interaction term is a predictor but not an input variable. In our examples, except for the constant term for the intercept, all other predictors are input variables and standardized appropriately.
We compare the posterior distributions under independent i) Cauchy, i.e., Student-$t$ with 1 degree of freedom, ii) Student-$t$ with 7 degrees of freedom ($t_{7}$), and iii) normal priors for the regression coefficients. In binary regression models, while the inverse cumulative distribution function (CDF) of the logistic distribution yields the logit link function, the inverse CDF of the Student-$t$ distribution yields the robit link function. @Liu_2004 showed that the logistic link can be well approximated by a robit link with 7 degrees of freedom. So a $t_{7}$ prior approximately matches the tail heaviness of the logistic likelihood underlying logistic regression. For Cauchy priors we use the default choice recommended by @Gelman_etal_2008: all location parameters are set to 0 and scale parameters are set to 10 and 2.5 for the intercept and other coefficients, respectively. To be consistent we use the same location and scale parameters for the other two priors. @Gelman_etal_2008 adopted a similar strategy in one of their analyses, to study the effect of tail heaviness of the priors. Among the priors considered here, the normal prior has the lightest tails, the Cauchy prior the heaviest, and the $t_{7}$ prior offers a compromise between the two extremes. For each simulated dataset, we run both the Gibbs sampler developed in Section \[sec:gibbs\] and Stan, for 1,000,000 iterations after a burn-in of 100,000 samples, under each of the three priors.
Complete Separation with a Solitary Separator
---------------------------------------------
First, we generate a dataset with $p=2$ (including the intercept) and $n=30$. The continuous predictor $\mathbf{X}_2$ is chosen to be a solitary separator (after standardizing), which leads to complete separation, whereas the constant term $\mathbf{X}_1$ contains all one’s and is not a solitary separator. A plot of $\mathbf{y}$ versus $\mathbf{X}_2$ in Figure \[fig:sim1scatter\] demonstrates this graphically. So by Theorem \[theorem:existence\], under independent Cauchy priors, $E(\beta_1 \mid \mathbf{y})$ exists but $E(\beta_2 \mid \mathbf{y})$ does not.
![Scatter plot of $\mathbf{y}$ versus $\mathbf{X}_2$ in the first simulated dataset, where $\mathbf{X}_2$ is a solitary separator which completely separates the samples (the vertical line at zero separates the points corresponding to $y = 1$ and $y = 0$).[]{data-label="fig:sim1scatter"}](scatter-sim1.pdf){height="3in" width="5.2in"}
The results from the Gibbs sampler are reported in Figures \[fig:sim1post\] and \[fig:sim1means\]. Figure \[fig:sim1post\] shows the posterior samples of $\boldsymbol{\beta}$ under the different priors. The scale of $\beta_2$, the coefficient corresponding to the solitary separator $\mathbf{X}_2$, is extremely large under Cauchy priors, less so under $t_7$ priors, and the smallest under normal priors. In particular, under Cauchy priors, the posterior distribution of $\beta_2$ seems to have an extremely long right tail. Moreover, although $\mathbf{X}_1$ is not a solitary separator, under Cauchy priors, the posterior samples of $\beta_1$ have a much larger spread. Figure \[fig:sim1means\] shows that the running means of both $\beta_1$ and $\beta_2$ converge rapidly under normal and $t_7$ priors, whereas under Cauchy priors, the running mean of $\beta_1$ does not converge after a million iterations and that of $\beta_2$ clearly diverges. We also ran Stan for this example but do not report the results here, because it gave warning messages about divergent transitions for Cauchy priors, after the burn-in period. Given that the posterior mean of $\beta_2$ does not exist in this case, the lack of convergence is not surprising.
![Scatter plots (top) and box plots (bottom) of posterior samples of $\beta_1$ and $\beta_2$ from the Gibbs sampler, under independent Cauchy, $t_7$, and normal priors for the first simulated dataset.[]{data-label="fig:sim1post"}](beta1-beta2-sim1.png){height="5in" width="5in"}
![Plots of running means of $\beta_1$ (top) and $\beta_2$ (bottom) sampled from the posterior distributions via the Gibbs sampler, under independent Cauchy, $t_7$, and normal priors for the first simulated dataset. Here $E(\beta_1 \mid \mathbf{y})$ exists under independent Cauchy priors but $E(\beta_2 \mid \mathbf{y})$ does not.[]{data-label="fig:sim1means"}](runningmeans-sim1.png){height="5in" width="5in"}
Complete Separation Without Solitary Separators {#section:sim2}
-----------------------------------------------
Now we generate a new dataset with $p=2$ and $n=30$ such that there is complete separation but there are no solitary separators (see Figure \[fig:sim2scatter\]). This guarantees the existence of both $E(\beta_1 \mid \mathbf{y})$ and $E(\beta_2 \mid \mathbf{y})$ under independent Cauchy priors. The difference in the existence of $E(\beta_2 \mid \mathbf{y})$ for the two simulated datasets is reflected by the posterior samples from the Gibbs sampler: under Cauchy priors, the samples of $\beta_2$ in Figure 1 in the Appendix are more stabilized than those in Figure \[fig:sim1post\] in the manuscript. However, when comparing across prior distributions, we find that the posterior samples of neither $\beta_1$ nor $\beta_2$ are as stable as those under $t_7$ and normal priors, which is not surprising because among the three priors, Cauchy priors have the heaviest tails and thus yield the least shrinkage. Figure 2 in the Appendix shows that the convergence of the running means under Cauchy priors is slow. Although we have not verified the existence of the second or higher order posterior moments under Cauchy priors, for exploratory purposes we examine sample autocorrelation plots of the draws from the Gibbs sampler. Figure \[fig:sim2acf\] shows that the autocorrelation decays extremely slowly for Cauchy priors, reasonably fast for $t_7$ priors, and rapidly for normal priors.
Some results from Stan are reported in Figures 3 and 4 in the Appendix. Figure 3 in the Appendix shows posterior distributions with nearly identical shapes as those obtained using Gibbs sampling in Figure 1 in the Appendix, with the only difference being that more extreme values appear under Stan. This is most likely due to faster mixing in Stan. As Stan traverses the parameter space more rapidly, values in the tails appear more quickly than under the Gibbs sampler. Figures 2 and 4 in the Appendix demonstrate that running means based on Stan are in good agreement with those based on the Gibbs sampler.
The autocorrelation plots for Stan in Figure \[fig:sim2acf:stan\] demonstrate a remarkable improvement over those for Gibbs in Figure \[fig:sim2acf\] for all priors, and the difference in mixing is the most prominent for Cauchy priors.
To summarize, all the plots unequivocally suggest that Cauchy priors lead to an extremely slow mixing Gibbs sampler and unusually large scales for the regression coefficients, even when all the marginal posterior means are guaranteed to exist. While mixing can be improved tremendously with Stan, the heavy tailed posteriors under Stan are in agreement with those obtained from the Gibbs samplers. One may argue that in spite of the unnaturally large regression coefficients, Cauchy priors could lead to superior predictions. Thus in the next two sections we compare predictions based on posteriors under the three priors for two real datasets. As Stan generates nearly independent samples, we use Stan for MCMC simulations for the real datasets.
![Scatter plot of $\mathbf{y}$ versus $\mathbf{X}_2$ for the second simulated dataset. The solid vertical line at $-0.3$ demonstrates complete separation of the samples. However, $\mathbf{X}_2$ is not a solitary separator, because the dashed vertical line at zero does not separate the points corresponding to $y = 1$ and $y = 0$. The other predictor $\mathbf{X}_1$ is a vector of ones corresponding to the intercept, which is not a solitary separator, either. []{data-label="fig:sim2scatter"}](scatter-sim2.pdf){height="3in" width="5.2in"}
![Autocorrelation plots of the posterior samples of $\beta_1$ (top) and $\beta_2$ (bottom) from the Gibbs sampler, under independent Cauchy, $t_7$, and normal priors for the second simulated dataset.[]{data-label="fig:sim2acf"}](acf-sim2.png){height="5in" width="5in"}
![Autocorrelation plots of the posterior samples of $\beta_1$ (top) and $\beta_2$ (bottom) from Stan, under independent Cauchy, $t_7$, and normal priors for the second simulated dataset.[]{data-label="fig:sim2acf:stan"}](acf-sim2-stan.png){height="5in" width="5in"}
Real Data
=========
SPECT Dataset
-------------
The “SPECT” dataset [@Kurg:Cios:Tade:Ogie:Good:2001] is available from the UCI Machine Learning Repository[^5]. The binary response variable is whether a patient’s cardiac image is normal or abnormal, according to the diagnosis of cardiologists. The predictors are 22 binary features obtained from the cardiac images using a machine learning algorithm. The goal of the study is to determine if the predictors can correctly predict the diagnoses of cardiologists, so that the process could be automated to some extent.
Prior to centering, two of the binary predictors are solitary quasicomplete separators: $x_{i, j}=0$ $\forall i \in A_{0}$ and $x_{i, j} \geq 0$ $\forall i \in A_{1}$, for $j=18, 19$, with $\mathbf{X}_1$ denoting the column of ones. @Ghos:Reit:2013 analyzed this dataset with a Bayesian probit regression model which incorporated variable selection. As some of their proposed methods relied on an approximation of the marginal likelihood based on the MLE of $\boldsymbol{\beta}$, they had to drop these potentially important predictors from the analysis. If one analyzed the dataset with the uncentered predictors, by Theorem \[theorem:existence\], the posterior means $E(\beta_{18} \mid \mathbf{y})$ and $E(\beta_{19} \mid \mathbf{y})$ would not exist under independent Cauchy priors. However, after centering there are no solitary separators, so the posterior means of all coefficients exist.
The SPECT dataset is split into a training set of 80 observations and a test set of 187 observations by @Kurg:Cios:Tade:Ogie:Good:2001. We use the former for model fitting and the latter for prediction. First, for each of the three priors (Cauchy, $t_7$, and normal), we run Stan on the training dataset, for 1,000,000 iterations after discarding 100,000 samples as burn-in.
As in the simulation study, MCMC draws from Stan show excellent mixing for all priors. However, the posterior means of the regression coefficients involved in separation are rather large under Cauchy priors compared to the other priors. For example, the posterior means of $(\beta_{18},\beta_{19})$ under Cauchy, $t_7$, and normal priors are $(10.02,5.57), (3.24,1.68),$ and $(2.73,1.43)$ respectively. These results suggest that Cauchy priors are too diffuse for datasets with separation.
Next for each $i=1,2,\dots, n_{\text{test}}$ in the test set, we estimate the corresponding success probability $\pi_{i}$ by the Monte Carlo average: $$\widehat{\pi}_i^{\text{MC}} = \frac{1}{S}
\sum_{s=1}^{S}\frac{e^{\mathbf{x}_i^{T} \boldsymbol{\beta}^{(s)}}}
{1+e^{\mathbf{x}_i^{T} \boldsymbol{\beta}^{(s)}}},
\label{eqn:probmc}$$ where $\boldsymbol{\beta}^{(s)}$ is the sampled value of $\boldsymbol{\beta}$ in iteration $s$, after burn-in. Recall that here $n_{\text{test}}=187$ and $S=10^6$. We calculate two different types of summary measures to assess predictive performance. We classify the $i$th observation in the test set as a success, if $\widehat{\pi}_i^{\text{MC}}\geq 0.5$ and as a failure otherwise, and compute the misclassification rates. Note that the misclassification rate does not fully take into account the magnitude of $\widehat{\pi}_i^{\text{MC}}$. For example, if $y_i=1$ both $\widehat{\pi}_i^{\text{MC}}=0.5$ and $\widehat{\pi}_i^{\text{MC}}=0.9$ would correctly classify the observation, while the latter may be more preferable. So we also consider the average squared difference between $y_i$ and $\widehat{\pi}_i^{\text{MC}}$: $$MSE^{\text{MC}} = \frac{1}{n_{\text{test}}}
\sum_{i=1}^{n_{\text{test}}}{(\widehat{\pi}_i^{\text{MC}}-y_i)}^2,
\label{eqn:msemc}$$ which is always between 0 and 1, with a value closer to 0 being more preferable. Note that the Brier score [@Brier_1950] equals $2 MSE^{\text{MC}}$, according to its original definition. Since in some modified definitions [@Blattenberger_Lad_1985], it is the same as $MSE^{\text{MC}}$, we refer to $MSE^{\text{MC}}$ as the Brier score.
Cauchy $t_7$ normal
------ -------- ------- --------
MCMC 0.273 0.257 0.251
EM 0.278 0.262 0.262
: Misclassification rates based on $\widehat{\pi}_i^{\text{MC}}$ and $\widehat{\pi}_i^{\text{EM}}$, under Cauchy, $t_7$, and normal priors for the SPECT data. Small values are preferable.[]{data-label="tb:SPECT_cr"}
Cauchy $t_7$ normal
------ -------- ------- --------
MCMC 0.172 0.165 0.163
EM 0.179 0.178 0.178
: Brier scores $MSE^{\text{MC}}$ and $MSE^{\text{EM}}$, under Cauchy, $t_7$, and normal priors for the SPECT data. Small values are preferable.[]{data-label="tb:SPECT_mse"}
To compare the Monte Carlo estimates with those based on the EM algorithm of @Gelman_etal_2008, we also estimate the posterior mode, denoted by $\widetilde{\boldsymbol{\beta}}$ under identical priors and hyperparameters, using the R package [arm]{} [@Gelman_etal_2015]. The EM estimator of $\pi_{i}$ is given by: $$\widehat{\pi}_i^{\text{EM}}= \frac{e^{\mathbf{x}_i^{T} \widetilde{\boldsymbol{\beta}}}}
{1+e^{\mathbf{x}_i^{T} \widetilde{\boldsymbol{\beta}}}},
\label{eqn:probem}$$ and $MSE^{\text{EM}}$ is calculated by replacing $\widehat{\pi}_i^{\text{MC}}$ by $\widehat{\pi}_i^{\text{EM}}$ in .
We report the misclassification rates in Table \[tb:SPECT\_cr\] and the Brier scores in Table \[tb:SPECT\_mse\]. MCMC achieves somewhat smaller misclassification rates and Brier scores than EM, especially under $t_{7}$ and normal priors. This suggests that a full Bayesian analysis using MCMC may produce estimates that are closer to the truth than modal estimates based on the EM algorithm. The predictions are similar across the three prior distributions with the normal and $t_7$ priors yielding slightly more accurate results than Cauchy priors.
Pima Indians Diabetes Dataset
-----------------------------
We now analyze the “Pima Indians Diabetes” dataset in the R package [MASS]{}. This is a classic dataset without separation that has been analyzed by many authors in the past. Using this dataset we aim to compare predictions under different priors, when there is no separation. Using the training data provided in the package we predict the class labels of the test data. In this case the difference between different priors is practically nil. The Brier scores are same up to three decimal places, across all priors and all methods (EM and MCMC). The misclassification rates reported in Table \[tb:pima\_cr\] also show negligible difference between priors and methods. Here Cauchy priors have a slightly better misclassification rate compared to normal and $t_7$ priors, and MCMC provides slightly more accurate results compared to those obtained from EM. These results suggest that when there is no separation and maximum likelihood estimates exist, Cauchy priors may be preferable as default weakly informative priors in the absence of real prior information.
Cauchy $t_7$ normal
------ -------- ------- --------
MCMC 0.196 0.199 0.199
EM 0.202 0.202 0.202
: Misclassification rates based on $\widehat{\pi}_i^{\text{MC}}$ and $\widehat{\pi}_i^{\text{EM}}$, under Cauchy, $t_7$, and normal priors for the Pima Indians data. Small values are preferable.[]{data-label="tb:pima_cr"}
Discussion
==========
We have proved that posterior means of regression coefficients in logistic regression are not always guaranteed to exist under the independent Cauchy priors recommended by @Gelman_etal_2008, if there is complete or quasicomplete separation in the data. In particular, we have introduced the notion of a solitary separator, which is a predictor capable of separating the samples on its own. Note that a solitary separator needs to be able to separate without the aid of any other predictor, not even the constant term corresponding to the intercept. We have proved that for independent Cauchy priors, the absence of solitary separators is a necessary condition for the existence of posterior means of all coefficients, for a general family of link functions in binary regression models. For logistic and probit regression, this has been shown to be a sufficient condition as well. In general, the sufficient condition depends on the form of the link function. We have also studied multivariate Cauchy priors, where the solitary separator no longer plays an important role. Instead, posterior means of all predictors exist if there is no separation, while none of them exist if there is complete separation. The result under quasicompelte separation is still unclear and will be studied in future work.
In practice, after centering the input variables it is straightforward to check if there are solitary separators in the dataset. The absence of solitary separators guarantees the existence of posterior means of all regression coefficients in logistic regression under independent Cauchy priors. However, our empirical results have shown that even when the posterior means for Cauchy priors exist under separation, the posterior samples of the regression coefficients may be extremely large in magnitude. Separation is usually considered to be a sample phenomenon, so even if the predictors involved in separation are potentially important, some shrinkage of their coefficients is desirable through the prior. Our empirical results based on real datasets have demonstrated that the default Cauchy priors can lead to posterior means as large as 10, which is considered to be unusually large on the logit scale. Our impression is that Cauchy priors are good default choices in general because they contain weak prior information and let the data speak. However, under separation, when there is little information in the data about the logistic regression coefficients (the MLE is not finite), it seems that lighter tailed priors, such as Student-$t$ priors with larger degrees of freedom or even normal priors, are more desirable in terms of producing more plausible posterior distributions.
From a computational perspective, we have observed very slow convergence of the Gibbs sampler under Cauchy priors in the presence of separation. Note that if the design matrix is not of full column rank, for example when $p>n$, the $p$ columns of $\mathbf{X}$ will be linearly dependent. This implies that the equation for quasicomplete separation will be satisfied with equality for all observations. Empirical results (not reported here for brevity) demonstrated that independent Cauchy priors show convergence of the Gibbs sampler in this case also compared to other lighter tailed priors. Out-of-sample predictive performance based on a real dataset with separation did not show the default Cauchy priors to be superior to $t_7$ or normal priors.
In logistic regression, under a multivariate normal prior for $\boldsymbol{\beta}$, @Choi_Hobert_2013 showed that the P[ó]{}lya-Gamma data augmentation Gibbs sampler of @Polson_etal_2013 is uniformly ergodic, and the moment generating function of the posterior distribution $p(\boldsymbol{\beta} \mid \mathbf{y})$ exists for all $\mathbf{X},\mathbf{y}$. In our examples of datasets with separation, the normal priors led to the fastest convergence of the Gibbs sampler, reasonable scales for the posterior draws of $\boldsymbol{\beta}$, and comparable or even better predictive performance than other priors. The results from Stan show no problem in mixing under any of the priors. However, the problematic issue of posteriors with extremely heavy tails under Cauchy priors cannot be resolved without altering the prior. Thus, after taking into account all the above considerations, for a full Bayesian analysis we recommend the use of normal priors as a default, when there is separation. Alternatively, heavier tailed priors such as the $t_7$ could also be used if robustness is a concern. On the other hand, if the goal of the analysis is to obtain point estimates rather than the entire posterior distribution, the posterior mode obtained from the EM algorithm of @Gelman_etal_2015 under default Cauchy priors [@Gelman_etal_2008] is a fast viable alternative.
Supplementary Material {#supplementary-material .unnumbered}
----------------------
In the supplementary material, we present additional simulation results for logistic and probit regression with complete separation, along with an appendix that contains the proofs of all theoretical results. The Gibbs sampler developed in the paper can be implemented with the R package [tglm]{}, available from the website: <https://cran.r-project.org/web/packages/tglm/index.html>.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The authors thank the Editor-in-Chief, Editor, Associate Editor and the reviewer for suggestions that led to a greatly improved paper. The authors also thank Drs. David Banks, James Berger, William Bridges, Merlise Clyde, Jon Forster, Jayanta Ghosh, Aixin Tan, and Shouqiang Wang for helpful discussions. The research of Joyee Ghosh was partially supported by the NSA Young Investigator grant H98230-14-1-0126.
[ *Supplementary Material: On the Use of Cauchy Prior Distributions for Bayesian Logistic Regression*]{}
In this supplement, we first present an appendix with additional simulation results for logit and probit link functions, and then include an appendix that contains proofs of the theoretical results.
Appendix: Simulation Results for Complete Separation Without Solitary Separators {#section:sim2:app}
================================================================================
In this section we present some supporting figures for the analysis of the simulated dataset described in Section \[section:sim2\] of the manuscript under logit and probit links.
Logistic Regression for Complete Separation Without Solitary Separators {#section:sim2:app:logit}
-----------------------------------------------------------------------
Figures \[fig:app:sim2post\] and \[fig:app:sim2means\] are based on the posterior samples from a Gibbs sampler under a logit link, whereas Figures \[fig:app:sim2post:stan\] and \[fig:app:sim2means:stan\] are corresponding results from Stan. A detailed discussion of the results is provided in Section \[section:sim2\] of the manuscript.
![Scatter plots (top) and box plots (bottom) of posterior samples of $\beta_1$ and $\beta_2$ for a logistic regression model, from the Gibbs sampler, under independent Cauchy, $t_7$, and normal priors for the second simulated dataset.[]{data-label="fig:app:sim2post"}](beta1-beta2-sim2.png){height="5in" width="5in"}
![Plots of running means of $\beta_1$ (top) and $\beta_2$ (bottom) sampled from the posterior distributions for a logistic regression model, via the Gibbs sampler, under independent Cauchy, $t_7$, and normal priors for the second simulated dataset. Here both $E(\beta_1 \mid \mathbf{y})$ and $E(\beta_2 \mid \mathbf{y})$ exist under independent Cauchy priors.[]{data-label="fig:app:sim2means"}](runningmeans-sim2.png){height="5in" width="5in"}
![Scatter plots (top) and box plots (bottom) of posterior samples of $\beta_1$ and $\beta_2$ for a logistic regression model, from Stan, under independent Cauchy, $t_7$, and normal priors for the second simulated dataset.[]{data-label="fig:app:sim2post:stan"}](beta1-beta2-sim2-stan.png){height="5in" width="5in"}
![Plots of running means of $\beta_1$ (top) and $\beta_2$ (bottom) sampled from the posterior distributions for a logistic regression model, via Stan, under independent Cauchy, $t_7$, and normal priors for the second simulated dataset. Here both $E(\beta_1 \mid \mathbf{y})$ and $E(\beta_2 \mid \mathbf{y})$ exist under independent Cauchy priors.[]{data-label="fig:app:sim2means:stan"}](runningmeans-sim2-stan.png){height="5in" width="5in"}
Probit Regression for Complete Separation Without Solitary Separators {#section:sim2probit}
---------------------------------------------------------------------
In this section we analyze the simulated dataset described in Section \[section:sim2\] of the manuscript under a probit link, while keeping everything else the same. We have shown in Theorem \[theorem:general\], that the theoretical results hold for a probit link. The goal of this analysis is to demonstrate that the empirical results are similar under the logit and probit link functions. For this dataset, Theorem \[theorem:general\] guarantees the existence of both $E(\beta_1 \mid \mathbf{y})$ and $E(\beta_2 \mid \mathbf{y})$ under independent Cauchy priors and a probit link function. As in the case of logistic regression the heavy tails of Cauchy priors translate into an extremely heavy right tail in the posterior distributions of $\beta_1$ and $\beta_2$, compared to the lighter tailed priors (see Figure \[fig:sim2post\_probit\] and \[fig:sim2means\_probit\] here). Thus in the case of separation, normal priors seem to be reasonable for probit regression also.
![Scatter plots (top) and box plots (bottom) of posterior samples of $\beta_1$ and $\beta_2$, for a probit regression model, under Cauchy, $t_7$, and normal priors for the second simulated dataset. Posterior sampling was generated via Stan.[]{data-label="fig:sim2post_probit"}](beta1-beta2-sim2-probit.png){height="5in" width="5in"}
![Plots of running means of $\beta_1$ (top) and $\beta_2$ (bottom) sampled from the posterior distributions for a probit regression model, under Cauchy, $t_7$, and normal priors for the second simulated datatset. Here both $E(\beta_1 \mid \mathbf{y})$ and $E(\beta_2 \mid \mathbf{y})$ exist under Cauchy priors. Posterior sampling was generated via Stan.[]{data-label="fig:sim2means_probit"}](runningmeans-sim2-probit.png){height="5in" width="5in"}
Appendix: Proofs
================
First, we decompose the proof of Theorem \[theorem:existence\] into two parts: in Appendix \[proof\_th1\_necessary\] we show that a necessary condition for the existence of $E(\beta_j \mid \mathbf{y})$ is that $\mathbf{X}_j$ is not a solitary separator; and in Appendix \[proof\_th1\_sufficient\] we show that it is also a sufficient condition. Then, we prove Theorem \[theorem:general\] in Appendix \[proof\_th2\], and Corollary \[corollary:nonzero\_prior\_mean\] in Appendix \[proof\_corollary:nonzero\_prior\_mean1\]. Finally, we decompose the proof of Theorem \[theorem:multivariate\_Cauchy\] into two parts: in Appendix \[proof\_theorem:multivariate\_Cauchy\_no\_separation\] we show that all posterior means exist if there is no separation; then in Appendix \[proof\_theorem:multivariate\_Cauchy\_complete\_separation\] we show that none of the posterior means exist if there is complete separation.
Proof of the Necessary Condition for Theorem \[theorem:existence\] {#proof_th1_necessary}
------------------------------------------------------------------
Here we show that if $\mathbf{X}_j$ is a solitary separator, then $E(\beta_j \mid \mathbf{y})$ does not exist, which is equivalent to the necessary condition for Theorem \[theorem:existence\].
For notational simplicity, we define the functional form of the success and failure probabilities in logistic regression as $$\label{eqn:f1f0}
f_1(t) = e^t / (1 + e^t), \quad f_0(t) =1 - f_1(t) = 1 / (1 + e^t),$$ which are strictly increasing and decreasing functions of $t$, respectively. In addition, both functions are bounded: $0 < f_1(t),f_0(t) < 1$ for $t \in \mathbb{R}$. Let $\boldsymbol\beta_{(-j)}$ and $\mathbf{x}_{i, (-j)}$ denote the vectors $\boldsymbol\beta$ and $\mathbf{x}_i$ after excluding their $j$th entries $\beta_{j}$ and $x_{i,j}$, respectively. Then the likelihood function can be written as $$\label{eq:likelihood}
p(\mathbf{y} \mid \boldsymbol\beta)
= \prod_{i =1}^n p(y_i \mid \boldsymbol\beta)
= \prod_{i\in A_1} f_1\left(x_{i,j}\beta_j + \mathbf{x}_{i, (-j)}^T \boldsymbol\beta_{(-j)} \right) \cdot
\prod_{k \in A_0} f_0\left(x_{k,j}\beta_j + \mathbf{x}_{k, (-j)}^T \boldsymbol\beta_{(-j)} \right).$$ The posterior mean of $\beta_j$ exists provided $E(|\beta_j| \mid \mathbf{y})< \infty$. When the posterior mean exists it is given by . Clearly if one of the two integrals in is not finite, then $E(|\beta_j| \mid \mathbf{y}) = \infty$. In this proof we will show that if $\alpha_j > 0$, the first integral in equals $\infty$. Similarly, it can be shown that if $\alpha_j < 0$, the second integral in equals $-\infty$.
If $\alpha_j > 0$, by -, $x_{i,j} \geq 0$ for all $i \in A_1$, and $x_{k,j} \leq 0$ for all $k \in A_0$. When $\beta_j > 0$, by the monotonic property of $f_1(t)$ and $f_0(t)$ we have, $p(\mathbf{y} \mid \boldsymbol\beta) \geq
\prod_{i\in A_1} f_1\left(\mathbf{x}_{i, (-j)}^T \boldsymbol\beta_{(-j)} \right)
\cdot \prod_{k \in A_0} f_0\left(\mathbf{x}_{k, (-j)}^T \boldsymbol\beta_{(-j)} \right)$, which is free of $\beta_j$. Therefore, $$\begin{aligned}
& \int_0^{\infty} \beta_j ~p(\beta_j \mid \mathbf{y})~d\beta_j
= \int_0^{\infty} \beta_j \left[ \int_{\mathbb{R}^{p-1}}
\frac{p(\mathbf{y} \mid \boldsymbol\beta) p(\beta_j) p(\boldsymbol\beta_{(-j)})}{p(\mathbf{y})}
d\boldsymbol\beta_{(-j)}\right] ~d\beta_j \nonumber \\
=~& \frac{1}{p(\mathbf{y})}\int_0^{\infty} \beta_j p(\beta_j) \left[ \int_{\mathbb{R}^{p-1}}
p(\mathbf{y} \mid \boldsymbol\beta) p(\boldsymbol\beta_{(-j)})
d\boldsymbol\beta_{(-j)}\right] ~d\beta_j \nonumber \\
\geq ~& \frac{1}{p(\mathbf{y})}\int_0^{\infty} \beta_j p(\beta_j) \left[ \int_{\mathbb{R}^{p-1}} \prod_{i\in A_1} f_1\left(\mathbf{x}_{i, (-j)}^T \boldsymbol\beta_{(-j)} \right)
\prod_{k \in A_0} f_0\left(\mathbf{x}_{k, (-j)}^T \boldsymbol\beta_{(-j)} \right)
p(\boldsymbol\beta_{(-j)})
d\boldsymbol\beta_{(-j)}\right] ~d\beta_j \nonumber \\
= ~ & \frac{\int_0^{\infty} \beta_j p(\beta_j)~d\beta_j}{p(\mathbf{y})}
\left[ \int_{\mathbb{R}^{p-1}}
\prod_{i\in A_1} f_1\left(\mathbf{x}_{i, (-j)}^T \boldsymbol\beta_{(-j)} \right)
\prod_{k \in A_0} f_0\left(\mathbf{x}_{k, (-j)}^T \boldsymbol\beta_{(-j)} \right)
p(\boldsymbol\beta_{(-j)})
d\boldsymbol\beta_{(-j)}\right].
\label{eqn:pm_betaj_plus} \end{aligned}$$ Here the first equation results from independent priors, i.e., $p(\bm{\beta}_{(-j)} \mid \beta_j) = p(\bm{\beta}_{(-j)})$. Since $ p(\mathbf{y} \mid \boldsymbol\beta) < 1$, $p(\mathbf{y}) = \int_{\mathbb{R}^p} p(\mathbf{y} \mid \boldsymbol\beta)
p(\boldsymbol\beta) d\boldsymbol\beta < \int_{\mathbb{R}^p}
p(\boldsymbol\beta) d\boldsymbol\beta=1$. Moreover, $p(\mathbf{y} \mid \boldsymbol\beta)
p(\boldsymbol\beta)> 0$ for all $\boldsymbol\beta \in \mathbb{R}^p$, so we also have $p(\mathbf{y}) > 0$, implying that $0<p(\mathbf{y}) <1$. For the independent Cauchy priors in , $\int_0^{\infty} \beta_j
p(\beta_j) ~d\beta_j = \infty$ and the second integral in is positive, hence equals $\infty$.
Proof of the Sufficient Condition for Theorem \[theorem:existence\] {#proof_th1_sufficient}
-------------------------------------------------------------------
Here we show that if $\mathbf{X}_j$ is not a solitary separator, then the posterior mean of $\beta_j$ exists.
When $E(|\beta_j| \mid \mathbf{y})< \infty$ the posterior mean of $\beta_j$ exists and is given by $$\begin{aligned}
\nonumber
E(\beta_j \mid \mathbf{y})
& = \int_{-\infty}^{\infty} \beta_j p(\beta_j \mid \mathbf{y}) d \beta_j\\ \label{eq:posterior_mean_beta_j}
& = \frac{1}{p(\mathbf{y})}
\underbrace{ \int_{0}^{\infty} \beta_j p(\beta_j) p(\mathbf{y} \mid \beta_j) d \beta_j
}_{\text{denoted by } E(\beta_j \mid \mathbf{y})^+}
+
\frac{1}{p(\mathbf{y})}
\underbrace{ \int_{-\infty}^{0} \beta_j p(\beta_j) p(\mathbf{y} \mid \beta_j) d \beta_j
}_{\text{denoted by } E(\beta_j \mid \mathbf{y})^-}.\end{aligned}$$ Since $0<p(\mathbf{y})<1$ it is enough to show that the positive term $E(\beta_j \mid \mathbf{y})^+$ has a finite upper bound, and the negative term $E(\beta_j \mid \mathbf{y})^-$ has a finite lower bound. For notational simplicity, in the remainder of the proof, we let $\bm{\alpha}_{(-j)}$, $ \mathbf{x}_{i, (-j)}$, $\bm{\beta}_{(-j)}$, and $\bm{\sigma}_{(-j)}$ denote the vectors $\bm{\alpha}$, $\mathbf{x}_i$, $\bm{\beta}$, and $\bm{\sigma} = (\sigma_1, \ldots, \sigma_p)^T$ after excluding their $j$th entries, respectively.
[**We first show that $E(\beta_j \mid \mathbf{y})^+$ has a finite upper bound.**]{} Because $\mathbf{X}_j$ is not a solitary separator, there exists either
1. \[condition\_a\] an $i' \in A_1$ such that $x_{i', j} < 0$, or
2. \[condition\_b\] a $k' \in A_0$, such that $x_{k', j} > 0$.
If both \[condition\_a\] and \[condition\_b\] are violated then $\mathbf{X}_j$ is a solitary separator and leads to a contradiction. Furthermore, for such $\mathbf{x}_{i', (-j)}$ or $\mathbf{x}_{k', (-j)}$, it contains at least one non-zero entry. This is because if $j \neq 1$, i.e., $\mathbf{X}_j$ does not correspond to the intercept (the column of all one’s), then the first entry in $\mathbf{x}_{i', (-j)}$ or $\mathbf{x}_{k', (-j)}$ equals $1$. If $j = 1$, due to the assumption that $\mathbf{X}$ has a column rank $>1$, there exists one row $i^{\diamond} \in \{1, 2, \ldots, n\}$ such that $\mathbf{x}_{i^{\diamond}, (-1)}$ contains at least one non-zero entry. If $i^{\diamond} \in A_0$, then we let $k' = i^{\diamond}$ and the condition \[condition\_b\] holds because $x_{i^{\diamond}, 1} = 1 > 0$. If $i^{\diamond} \in A_1$, we may first rescale $\mathbf{X}_1$ by $-1$, which transforms $\beta_1$ to $-\beta_1$. Since the Cauchy prior centered at zero is invariant to this rescaling, and $E(-\beta_1 \mid \mathbf{y})$ exists if and only if $E(\beta_1 \mid \mathbf{y})$ exists, we can just apply this rescaling, after which $x_{i^{\diamond}, 1} = -1 < 0$. Then we let $i' = i^{\diamond}$ and the condition \[condition\_a\] holds.
We first assume that condition \[condition\_a\] is true. We define a positive constant $\epsilon = |x_{i',j}| / 2 = -x_{i',j} / 2$. For any $\beta_j > 0$, we define a subset of the domain of $\bm{\beta}_{(-j)} $ as follows $$\label{eq:G}
G(\beta_j) \stackrel{\text{def}}{=} \left\{
\bm{\beta}_{(-j)} \in \mathbb{R}^{p-1}: \mathbf{x}_{i', (-j)}^T \bm{\beta}_{(-j)}
< \epsilon \beta_j \right\}.$$ Then for any $\bm{\beta}_{(-j)} \in G(\beta_j)$, $\mathbf{x}_{i'}^T \bm{\beta}
= x_{i',j}\beta_j+ \mathbf{x}_{i', (-j)}^T \bm{\beta}_{(-j)}
<(x_{i',j} + \epsilon) \beta_j= - \epsilon \beta_j$. Therefore, $$\label{eqn:G}
f_1(\mathbf{x}_{i'}^T \bm{\beta} ) < f_1(- \epsilon \beta_j), \quad \text{for all } \bm{\beta}_{(-j)} \in G(\beta_j).$$ As $f_1(\cdot)$ and $f_0(\cdot)$ are bounded above by 1, the likelihood function $p(\mathbf{y} \mid \bm{\beta})$ in is bounded above by $f_1(\mathbf{x}_{i'}^T \bm{\beta})$. Thus $$\begin{aligned}
& E(\beta_j \mid \mathbf{y})^+
= \int_{0}^{\infty} \beta_j p(\beta_j)
\left[ \int_{\mathbb{R}^{p-1}} p(\mathbf{y} \mid \bm{\beta})
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j \nonumber \\
<~& \int_{0}^{\infty} \beta_j p(\beta_j)
\left[ \int_{\mathbb{R}^{p-1}} f_1(\mathbf{x}_{i'}^T \bm{\beta})
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j \nonumber \\
=~& \int_{0}^{\infty} \beta_j p(\beta_j)
\left[ \int_{G(\beta_j)} f_1(\mathbf{x}_{i'}^T \bm{\beta})
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}
+ \int_{\mathbb{R}^{p-1}\backslash G(\beta_j)} f_1(\mathbf{x}_{i'}^T \bm{\beta})
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j\nonumber \\
<~& \int_{0}^{\infty} \beta_j p(\beta_j)
\left[ \int_{G(\beta_j)} f_1(- \epsilon \beta_j)
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}
+ \int_{\mathbb{R}^{p-1}\backslash G(\beta_j)}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j \label{eqn:pm_gamma1_plus}\end{aligned}$$ Here the last inequality results from and the fact that the function $f_1(\cdot)$ is bounded above by $1$. An upper bound is obtained for the first term in the bracket in using the fact that the integrand is non-negative as follows: $$\begin{aligned}
\nonumber
\int_{G(\beta_j)} f_1(- \epsilon \beta_j)
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}
&< \int_{\mathbb{R}^{p-1}} f_1(- \epsilon \beta_j)
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \\ \label{eqn:pm_gamma1_plus3}
&= f_1(- \epsilon \beta_j)
\underbrace{\int_{\mathbb{R}^{p-1}}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}}_{=1}
= \frac{e^{-\epsilon \beta_j}}{1 + e^{-\epsilon \beta_j}}
< e^{-\epsilon \beta_j}. \end{aligned}$$ Recall that $\mathbf{x}_{i', (-j)}$ contains at least one non-zero entry. We assume that $\mathbf{x}_{i', r} \neq 0$. Then to simplify the second term in the bracket in , we transform $\bm{\beta}_{(-j)}$ to $(\eta, \bm{\xi})$ via a linear transformation, such that $\eta = \mathbf{x}_{i', (-j)}^T \bm{\beta}_{(-j)}$, and $\bm{\xi}$ is the vector $\bm{\beta}_{(-j)}$ after excluding $\beta_r$. The characteristic function of a Cauchy distribution $C(\mu,\sigma)$ is $\varphi(t) = e^{it\mu - |t|\sigma}$, where $t \in \mathbb{R}$. Since [*a priori*]{}, $\eta$ is a linear combination of independent $C(0,\sigma_{\ell})$ random variables, $\beta_\ell$, for $1\leq \ell \leq p, \ell \neq j$, its characteristic function is $$\varphi_\eta(t) = E(e^{it \eta})
= \prod_{1\leq \ell \leq p, \ell \neq j} E\left[e^{i(t x_{i',\ell}) \beta_\ell}\right]
= \prod_{1\leq \ell \leq p, \ell \neq j} \varphi_{\beta_\ell}(t x_{i', \ell})
= e^{ - |t| \sum_{1\leq \ell \leq p, \ell \neq j} |x_{i', \ell}| \sigma_\ell}.$$ So the induced prior of $\eta$ is $C(0, \sum_{1\leq \ell \leq p, \ell \neq j} |x_{i', \ell}| \sigma_\ell)$. Let $|\mathbf{x}_{i', (-j)}|$ denote the vector obtained by taking absolute values of each element of $\mathbf{x}_{i', (-j)}$, then the above scale parameter $\sum_{1\leq \ell \leq p, \ell \neq j} |x_{i', \ell}| \sigma_\ell$= $|\mathbf{x}_{i', (-j)}|^T \bm{\sigma}_{(-j)}$. By , for any $\bm{\beta}_{(-j)} \not\in G(\beta_j)$, the corresponding $\eta \geq \epsilon \beta_j$ and $\bm{\xi} \in \mathbb{R}^{p-2}$. An upper bound is calculated for the second term in the bracket in . Note that $p(\eta)p(\bm{\xi} \mid \eta)$ is the joint density of $\eta$ and $\bm{\xi}$. Since it incorporates the Jacobian of transformation from $\bm{\beta}_{(-j)} $ to $\eta$ and $\bm{\xi}$, a separate Jacobian term is not needed in the first equality below. $$\begin{aligned}
\nonumber
\int_{\mathbb{R}^{p-1}\backslash G(\beta_j)}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}
&= \int_{\epsilon\beta_j}^{\infty} \int_{\mathbb{R}^{p-2}}
p(\eta)p(\bm{\xi} \mid \eta) d \bm{\xi} d \eta \\ \nonumber
&= \int_{\epsilon\beta_j}^{\infty}
\frac{\int_{\mathbb{R}^{p-2}} p(\bm{\xi} \mid \eta) d \bm{\xi} }
{\pi~ |\mathbf{x}_{i', (-j)}|^T \bm{\sigma}_{(-j)}
\left[ 1 + \eta^2 /
\left(|\mathbf{x}_{i', (-j)}|^T \bm{\sigma}_{(-j)}\right)^2
\right]} d \eta\\ \nonumber
&= \frac{1}{\pi}\left[\frac{\pi}{2} - \arctan \left( \frac{\epsilon\beta_j}
{|\mathbf{x}_{i', (-j)}|^T \bm{\sigma}_{(-j)}} \right) \right]\\
&= \frac{1}{\pi}\arctan \left( \frac{|\mathbf{x}_{i', (-j)}|^T \bm{\sigma}_{(-j)}}{\epsilon\beta_j}\right)
< \frac{|\mathbf{x}_{i', (-j)}|^T
\bm{\sigma}_{(-j)}}{\pi\epsilon\beta_j}.
\label{eqn:upperbound2}\end{aligned}$$ Here, the second equality holds because $\int_{\mathbb{R}^{p-2}} p(\bm{\xi} \mid \eta) d \bm{\xi} = 1$; the last inequality holds because $\epsilon$ and $\beta_j$ are both positive, and for any $t > 0$, $\arctan(t) < t$.
Then substituting the expression for $p(\beta_j)$ as in , we continue with to find an upper bound. $$\begin{aligned}
E(\beta_j \mid \mathbf{y})^+
& < \int_{0}^{\infty} \frac{\beta_j}{\pi \sigma_j (1 + \beta_j^2 / \sigma_j^2)} \left[ e^{-\epsilon \beta_j}
+ \frac{|\mathbf{x}_{i', (-j)}|^T
\bm{\sigma}_{(-j)}}{\pi\epsilon\beta_j} \right] d \beta_j
\nonumber \\
& < \int_{0}^{\infty} \frac{\beta_j e^{-\epsilon \beta_j}}{\pi \sigma_j } d \beta_j
+ \int_{0}^{\infty} \frac{|\mathbf{x}_{i', (-j)}|^T \bm{\sigma}_{(-j)}}
{\pi^2 \sigma_j \epsilon (1 + \beta_j^2 / \sigma_j^2)} d \beta_j
= \frac{1}{\pi \sigma_j \epsilon^2} + \frac{|\mathbf{x}_{i',
(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}.
\label{eqn:upperbound+}\end{aligned}$$ On the other hand, if condition \[condition\_b\] holds, then we just need to slightly modify the above proof. We define $\epsilon = |x_{k', j}|/2 = x_{k', j}/2$, and change to $$G(\beta_j) \stackrel{\text{def}}{=} \left\{
\bm{\beta}_{(-j)} \in \mathbb{R}^{p-1}: \mathbf{x}_{k', (-j)}^T \bm{\beta}_{(-j)}
> - \epsilon \beta_j \right\}.$$ Consequently, the terms $f_1(\mathbf{x}_{i'}^T \bm{\beta})$ and $f_1(- \epsilon \beta_j)$ in have to be changed to $f_0(\mathbf{x}_{k'}^T \bm{\beta})$ and $f_0(\epsilon \beta_j)$, respectively. For the logit link, $f_0(\epsilon \beta_j) = f_1(- \epsilon \beta_j)$. The range of the integral in with respect to $\eta$ is from $-\infty$ to $-\epsilon\beta_j$; however, because the density of $\eta$ is symmetric around 0, the value of the integral stays the same. So it can be shown an upper bound for $E(\beta_j \mid \mathbf{y})^+$ is $\left[1 / \left(\pi \sigma_j \epsilon^2\right)\right] + \left[|\mathbf{x}_{k',(-j)}|^T
\bm{\sigma}_{(-j)}/ \left(2\pi\epsilon \right)\right]$.
[**We now show that the negative term $E(\beta_j \mid \mathbf{y})^-$ has a finite lower bound.** ]{} For any $\beta_j < 0$, by expressing $\beta_j^* = - \beta_j$, we need to show that the positive term $-E(\beta_j \mid \mathbf{y})^- = - \int_{-\infty}^{0} \beta_j p(\beta_j) p(\mathbf{y} \mid \beta_j) d \beta_j
= \int_0^{\infty} \beta_j^* p(\beta_j^*) p(\mathbf{y} \mid -\beta_j^*) d \beta_j^*$ has a finite upper bound. As the idea is very similar to the proof of existence of $E(\beta_j \mid \mathbf{y})^+$, we present less details here.
Since the predictor $\mathbf{X}_j$ is not a solitary separator, there exists either
1. \[condition\_c\] an $i^* \in A_1$ such that $x_{i^*, j} > 0$, or
2. \[condition\_d\] a $k^* \in A_0$, such that $x_{k^*, j} < 0$.
If \[condition\_c\] and \[condition\_d\] are both violated $\mathbf{X}_j$ has to be a solitary separator, which leads to a contradiction. WLOG, we assume that condition \[condition\_c\] is true, and as before $\mathbf{x}_{i^*, (-j)}$ must contain at least one non-zero entry, say, $\mathbf{x}_{i^*, s} \neq 0$. If condition \[condition\_d\] is true, then we can adopt a modification similar to the one that is used to prove the existence under condition \[condition\_b\] based on the proof under condition \[condition\_a\].
We define a positive constant $\epsilon = x_{i^*, j}/2$. For any $\beta_j^* > 0$, we define a subset of $\mathbb{R}^{p-1}$ as $G(\beta_j^*)\stackrel{\text{def}}{=} \left\{
\bm{\beta}_{(-j)} \in \mathbb{R}^{p-1}: \mathbf{x}_{i^*, (-j)}^T \bm{\beta}_{(-j)}
< \epsilon \beta_j^* \right\}$. Then for all $\bm{\beta}_{(-j)} \in G(\beta_j^*)$, $\mathbf{x}_{i^*}^T\bm{\beta} = -x_{i^*, j}\beta_j^* + \mathbf{x}_{i^*, (-j)}^T \bm{\beta}_{(-j)}
< (-x_{i^*,j} + \epsilon) \beta_j^* = - \epsilon \beta_j^*$, hence $f_1(\mathbf{x}_{i^*}^T\bm{\beta}) < f_1\left(- \epsilon \beta_j^* \right)$. Since the likelihood function $p(\mathbf{y} \mid -\beta_j^*, \bm{\beta}_{(-j)}) <
f_1(x_{i^*, j}(-\beta_j^*) + \mathbf{x}_{i^*, (-j)}^T \bm{\beta}_{(-j)} ) < 1$, $$\begin{aligned}
& -E(\beta_j \mid \mathbf{y})^-
= \int_{0}^{\infty} \beta_j^* p(\beta_j^*)
\left[ \int_{\mathbb{R}^{p-1}} p(\mathbf{y} \mid -\beta_j^*, \bm{\beta}_{(-j)})
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j^* \nonumber \\
<~& \int_{0}^{\infty} \beta_j^* p(\beta_j^*)
\left[ \int_{G(\beta_j^*)} f_1(- \epsilon \beta_j^*)
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}
+ \int_{\mathbb{R}^{p-1}\backslash G(\beta_j^*)}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j^*. \label{eqn:pm_gamma1_plus2}\end{aligned}$$ The first term in the bracket in has an upper bound $\exp(-\epsilon \beta_j^*)$ as in . Recall that $\mathbf{x}_{i^*, s} \neq 0$. We now transform $\bm{\beta}_{(-j)}$ to $(\eta, \bm{\xi})$ via a linear transformation, such that $\eta = \mathbf{x}_{i^*, (-j)}^T \bm{\beta}_{(-j)}$ and $\bm{\xi}$ is the vector $\bm{\beta}_{(-j)}$ after excluding $\beta_s$. The prior of $\eta$ is $C(0,|\mathbf{x}_{i^*, (-j)}|^T
\bm{\sigma}_{(-j)})$. For any $\bm{\beta}_{(-j)} \not\in G(\beta_j^*)$, the corresponding $\eta \geq \epsilon \beta_j^*$. Therefore as in , we obtain an upper bound for the second term in the bracket in as $|\mathbf{x}_{i^*, (-j)}|^T \bm{\sigma}_{(-j)}/\left(\pi\epsilon\beta_j^*\right)$. Finally, following an upper bound for $-E(\beta_j \mid \mathbf{y})^-$ is $\left[1/\left(\pi \sigma_j \epsilon^2\right)\right] + \left[|\mathbf{x}_{i^*, (-j)}|^T \bm{\sigma}_{(-j)}/ \left(2\pi\epsilon\right)\right]$.
Proof of Theorem \[theorem:general\] {#proof_th2}
------------------------------------
Following the proof of Theorem \[theorem:existence\], we denote the success probability function by $\pi = f_1(\mathbf{x}^T \bm{\beta})$, where $f_1(\cdot)$ is the inverse link function, i.e., $f_1(\cdot) = g^{-1}(\cdot)$. Similarly, let the failure probability function be $f_0(\mathbf{x}^T \bm{\beta}) = 1 - f_1(\mathbf{x}^T \bm{\beta})$. Note that the proof of the necessary condition for Theorem \[theorem:existence\] given in Appendix \[proof\_th1\_necessary\] only relies on the fact the $f_1(\cdot)$ is increasing, continuous, and bounded between $0$ and $1$. Since the link function $g(\cdot)$ is assumed to be strictly increasing and differentiable, so is $f_1(\cdot)$. Moreover, the range of $f_1$ is $(0, 1)$. Therefore, the proof of the necessary condition for Theorem \[theorem:general\] follows immediately.
For the proof of the sufficient condition in Theorem \[theorem:general\], one can follow the proof in Appendix \[proof\_th1\_sufficient\] and proceed with the specific choice of $\epsilon$ used there, when condition \[condition\_a\] holds. The proof is identical until because the explicit form of the inverse link function is not used until that step. We re-write the final step in below and proceed from there: $$\begin{aligned}
\nonumber
& E(\beta_j \mid \mathbf{y})^+ \\
<& \int_{0}^{\infty} \beta_j p(\beta_j)
\left[ \int_{G(\beta_j)} f_1(- \epsilon \beta_j)
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)}
+ \int_{\mathbb{R}^{p-1}\backslash G(\beta_j)}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j
\nonumber \\
= & \int_{0}^{\infty} \beta_j p(\beta_j) f_1(- \epsilon \beta_j)
\underbrace{\left[ \int_{G(\beta_j)}p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right]
}_{< \int_{\mathbb{R}^{p-1}}p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} =1} d \beta_j
\nonumber \\
& + \int_{0}^{\infty} \beta_j p(\beta_j) \left[ \int_{\mathbb{R}^{p-1}\backslash G(\beta_j)}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j
\nonumber \\
<& \int_{0}^{\infty} \beta_j p(\beta_j) \underbrace{f_1(- \epsilon \beta_j)}_{=g^{-1}(- \epsilon \beta_j)}d \beta_j
+ \int_{0}^{\infty} \beta_j p(\beta_j) \left[ \int_{\mathbb{R}^{p-1}\backslash G(\beta_j)}
p(\bm{\beta}_{(-j)}) d\bm{\beta}_{(-j)} \right] d \beta_j.
\label{eqn:th2:sufficient:proof}
$$ The sufficient condition in Theorem \[theorem:general\] states that for every positive constant $\epsilon$, $$\int_{0}^{\infty} \beta_j
p(\beta_j) g^{-1}(- \epsilon \beta_j) d \beta_j < \infty.$$ This implies that the integral will be bounded for the specific choice of $\epsilon$ used in the above proof, and hence the first integral in is bounded above. The second integral does not depend on the link function and its bound can be obtained exactly as in Appendix \[proof\_th1\_sufficient\]. Thus $E(\beta_j \mid
\mathbf{y})^+ < \infty$ under condition (a). On the other hand if condition (b) holds, the proof follows similarly as in Appendix \[proof\_th1\_sufficient\] and now we need to use the sufficient condition in Theorem \[theorem:general\] that for every positive constant $\epsilon$, $\int_{0}^{\infty} \beta_j p(\beta_j)
\left[1-g^{-1}(\epsilon \beta_j)\right] d \beta_j< \infty$. A bound for $-E(\beta_j \mid \mathbf{y})^-$ can be obtained similarly, which completes the proof.
In probit regression, we first show that $$\label{eq:probit_logit}
g^{-1}_{\text{probit}}(t) = \Phi(t)
< e^t/(1 + e^t) = g^{-1}_{\text{logit}}(t),\quad \text{for any } t < 0,$$ where $\Phi(t)$ is the standard normal cdf. It is equivalent to show that the difference function $$\label{eq:u}
u(t) = \Phi(t) - \frac{e^t}{1 + e^t} < 0, \text{ for all } t < 0.$$ Note that $u(0) = 1/2 - 1/2 = 0$, and $\lim_{t\rightarrow -\infty} u(t) = 0$. Since $u(t)$ is differentiable, we have $$u'(t) = \frac{1}{\sqrt{2\pi}} e^{-\frac{t^2}{2}} - \frac{e^t}{(1 + e^t)^2}.$$ Now $u'(0) = 1/\sqrt{2\pi} - 1/4 >0$, and when $t$ is very small, $u'(t) < 0$ since $e^{-t^2/2}$ decays to zero at a faster speed than $e^t$, i.e., there exists a $\tilde{t}<0$ such that $u'(\tilde{t})<0$. Since $u'(t)$ is a continuous function, the intermediate value theorem applied to $[\tilde{t},0]$ shows that there exists a $t_1 < 0$ such that $u'(t_1) = 0$. Therefore, to show , it is sufficient to show that $u'(t)$ has a unique root on $\mathbb{R}^-$, which is proved by contradiction as follows.
If $u'(t)$ has two distinct roots $t_1, t_2 < 0$, i.e., for $i = 1, 2$, $u'(t_i) = 0$, then $$\begin{aligned}
\nonumber
& \frac{1}{\sqrt{2\pi}} e^{-\frac{t_i^2}{2}} = \frac{e^{t_i}}{(1 + e^{t_i})^2}, \ i = 1, 2
\Longleftrightarrow
\frac{e^{-\frac{t_1^2}{2}}}{e^{-\frac{t_2^2}{2}}}
= \frac{e^{t_1}}{e^{t_2}}\cdot \frac{(1 + e^{t_2})^2}{(1 + e^{t_1})^2}\\ \label{eq:u_prime}
\Longleftrightarrow &
\frac{e^{\frac{(t_2+1)^2}{2}}}{e^{\frac{(t_1+1)^2}{2}}}
= \left( \frac{1 + e^{t_2}}{1 + e^{t_1}} \right)^2
\Longleftrightarrow
\frac{(t_2 + 1)^2}{4} - \log(1 + e^{t_2}) = \frac{(t_1 + 1)^2}{4} - \log(1 + e^{t_1}).\end{aligned}$$ Note that the derivative of the function $(t + 1)^2/4 - \log(1 + e^{t})$ is $(t+1)/2 - e^t/(1+e^t)$. It is straightforward to show that this derivative is strictly less than 0 for all $t<0$, so $(t + 1)^2/4 - \log(1 + e^{t})$ is a strictly decreasing function. Thus holds only if $t_1 = t_2$, which leads to a contradiction.
Hence for any $\epsilon > 0$, $$\begin{aligned}
\int_{0}^{\infty} \beta_j p(\beta_j) g^{-1}_{\text{probit}}(- \epsilon \beta_j) d \beta_j
& < \int_{0}^{\infty} \beta_j p(\beta_j) g^{-1}_{\text{logit}}(-
\epsilon \beta_j) d \beta_j
< \int_{0}^{\infty} \beta_j p(\beta_j) {e^{-\epsilon
\beta_j}} d \beta_j \\
=\int_{0}^{\infty} \frac{\beta_j e^{-\epsilon
\beta_j}}{\pi \sigma_j (1 + \beta_j^2 / \sigma_j^2)} d \beta_j
& < \int_{0}^{\infty} \frac{\beta_j e^{-\epsilon
\beta_j} }{\pi \sigma_j} d \beta_j = \frac{1}{\pi \sigma_j \epsilon^2} < \infty.\end{aligned}$$
Since the probit link is symmetric, i.e., $1 - g^{-1}_{\text{probit}}(\epsilon \beta_j) = 1 - \Phi(\epsilon \beta_j)
= \Phi(-\epsilon \beta_j) = g^{-1}_{\text{probit}}(-\epsilon \beta_j)$, we also have $\int_{0}^{\infty} \beta_j p(\beta_j) \left[ 1 - g^{-1}_{\text{probit}}(\epsilon \beta_j)\right] d \beta_j
< \infty$.
Proof of Corollary \[corollary:nonzero\_prior\_mean\] {#proof_corollary:nonzero_prior_mean1}
-----------------------------------------------------
To prove Corollary \[corollary:nonzero\_prior\_mean\], we mainly use a similar strategy to the proof of Theorem \[theorem:existence\]. To show the necessary condition, we can use all of Appendix \[proof\_th1\_necessary\] without modification, for both logistic and probit regression models. To show the sufficient condition, we can follow the same proof outline as in Appendix \[proof\_th1\_sufficient\], with some minimal modification as described in the following proof.
First, we denote the vector of prior location parameters by $\boldsymbol\mu = (\mu_1, \mu_2, \ldots, \mu_p)^T$. If we shift the coefficients $\boldsymbol\beta$ by $\boldsymbol\mu$ units, then $$\boldsymbol\gamma \stackrel{\text{def}}{=} \boldsymbol\beta - \boldsymbol\mu ~ \Longrightarrow~
\gamma_j \stackrel{\text{ind}}{\sim} \text{C}(0, \sigma_j), \quad j = 1, 2, \ldots, p,$$ that is, the resulting parameters $\gamma_j$ have independent Cauchy priors with location parameters being zero. Since the original parameter $\beta_j = \gamma_j + \mu_j$ for each $j = 1, 2, \ldots, p$, the existence of $E(\beta_j \mid \mathbf{Y})$ is equivalent to the existence of $E(\gamma_j \mid \mathbf{Y})$. So we just need to show that if $\mathbf{X}_j$ is not a solitary separator, then $E(\gamma_j \mid \mathbf{Y})$ exists. For simplicity, here we just show that the positive term $$E(\gamma_j \mid \mathbf{y})^+ \stackrel{\text{def}}{=}
\int_{0}^{\infty} \gamma_j p(\gamma_j) p(\mathbf{y} \mid \gamma_j) d \gamma_j$$ has a finite upper bound. The other half of the result that the negative term $E(\gamma_j \mid \mathbf{y})^-$ has a finite lower bound will follow with a similar derivation.
As in Appendix \[proof\_th1\_sufficient\], we first assume that condition \[condition\_a\] is true, and define $\epsilon$ in the same way. For any $\gamma_j > 0$, we define a subset of the domain of $\bm{\gamma}_{(-j)} $ as follows $$G(\gamma_j) \stackrel{\text{def}}{=} \left\{
\bm{\gamma}_{(-j)} \in \mathbb{R}^{p-1}: \mathbf{x}_{i', (-j)}^T \bm{\gamma}_{(-j)}
< \epsilon \gamma_j \right\},$$ then for any $\bm{\gamma}_{(-j)} \in G(\gamma_j)$, $\mathbf{x}_{i'}^T \bm{\gamma} < - \epsilon \gamma_j$. Since $f_1(\cdot)$ is an increasing function, $$f_1(\mathbf{x}_{i'}^T \bm{\beta} ) = f_1(\mathbf{x}_{i'}^T \bm{\gamma} + \mathbf{x}_{i'}^T \bm{\mu})
< f_1(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}), \quad \text{for all } \bm{\gamma}_{(-j)} \in G(\gamma_j).$$ A similar derivation to gives $$\begin{aligned}
& E(\gamma_j \mid \mathbf{y})^+\\ <~& \int_{0}^{\infty} \gamma_j p(\gamma_j)
\left[ \int_{G(\gamma_j)} f_1(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu})
p(\bm{\gamma}_{(-j)}) d\bm{\gamma}_{(-j)}
+ \int_{\mathbb{R}^{p-1}\backslash G(\gamma_j)}
p(\bm{\gamma}_{(-j)}) d\bm{\gamma}_{(-j)} \right] d \gamma_j,\end{aligned}$$ where by the first integral inside the bracket has an upper bound $$\label{eq:f1_mu}
\int_{G(\gamma_j)} f_1(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu})
p(\bm{\gamma}_{(-j)}) d\bm{\gamma}_{(-j)}
< f_1(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}),
$$ and by the second integral inside the bracket also has an upper bound $$\int_{\mathbb{R}^{p-1}\backslash G(\gamma_j)}
p(\bm{\gamma}_{(-j)}) d\bm{\gamma}_{(-j)}
< \frac{|\mathbf{x}_{i', (-j)}|^T
\bm{\sigma}_{(-j)}}{\pi\epsilon\gamma_j}.$$
In logistic regression, the right hand side of is further bounded $$f_1(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu})
= \frac{e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}}}
{1 + e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}}}
< e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}},$$ and hence by , $$E(\gamma_j \mid \mathbf{y})^+
< \frac{e^{\mathbf{x}_{i'}^T \bm{\mu}}}{\pi \sigma_j \epsilon^2} + \frac{|\mathbf{x}_{i',
(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}.$$
In probit regression, the function $f_1(\cdot)$ in the above derivations equals the standard normal cdf $\Phi(\cdot)$. By , for any $\gamma_j > \mathbf{x}_{i'}^T \bm{\mu}/ \epsilon$, we have $$\Phi \left(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}\right)
< \frac{e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}}}{1 + e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}}}
< e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}}.$$ Hence for $\mathbf{x}_{i'}^T \bm{\mu}/\epsilon > 0$ we have an upper bound $$\begin{aligned}
& E(\gamma_j \mid \mathbf{y})^+
< \int_{0}^{\infty} \gamma_j p(\gamma_j)
\left[ \Phi \left(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}\right)
+ \frac{|\mathbf{x}_{i', (-j)}|^T
\bm{\sigma}_{(-j)}}{\pi\epsilon\gamma_j} \right] d \gamma_j \\
=& \int_{0}^{\infty} \gamma_j p(\gamma_j)
\Phi \left(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}\right) d \gamma_j
+ \frac{|\mathbf{x}_{i',(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}\\
=& \int_{0}^{ \mathbf{x}_{i'}^T \bm{\mu}/\epsilon} \gamma_j p(\gamma_j)
\Phi \left(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}\right) d \gamma_j
+ \int_{ \mathbf{x}_{i'}^T \bm{\mu}/ \epsilon}^{\infty} \gamma_j p(\gamma_j)
\Phi \left(- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}\right) d \gamma_j
+ \frac{|\mathbf{x}_{i',(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}\\
<& \int_{0}^{ \mathbf{x}_{i'}^T \bm{\mu}/\epsilon} \gamma_j p(\gamma_j) d \gamma_j
+ \int_{ \mathbf{x}_{i'}^T \bm{\mu}/ \epsilon}^{\infty} \gamma_j p(\gamma_j)
e^{- \epsilon \gamma_j + \mathbf{x}_{i'}^T \bm{\mu}} d \gamma_j
+ \frac{|\mathbf{x}_{i',(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}\\
<& \int_{0}^{ \mathbf{x}_{i'}^T \bm{\mu}/\epsilon}
\frac{\gamma_j}{\pi \sigma_j (1 + \gamma_j^2 / \sigma_j^2)} d \gamma_j
+ e^{\mathbf{x}_{i'}^T \bm{\mu}}\int_{ 0}^{\infty}
\frac{\gamma_j e^{- \epsilon \gamma_j }}{\pi \sigma_j (1 + \gamma_j^2 / \sigma_j^2)}
d \gamma_j
+ \frac{|\mathbf{x}_{i',(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}\\
<&~ \frac{\sigma_j}{2\pi} \log\left[1 + \left(\frac{\mathbf{x}_{i'}^T \bm{\mu}}{\epsilon\sigma_j} \right)^2 \right]
+ e^{\mathbf{x}_{i'}^T \bm{\mu}}\int_{ 0}^{\infty}
\frac{\gamma_j e^{- \epsilon \gamma_j }}{\pi \sigma_j }
d \gamma_j
+ \frac{|\mathbf{x}_{i',(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}\\
=&~ \frac{\sigma_j}{2\pi} \log\left[1 + \left(\frac{\mathbf{x}_{i'}^T \bm{\mu}}{\epsilon\sigma_j} \right)^2 \right]
+ \frac{e^{\mathbf{x}_{i'}^T \bm{\mu}}}{\pi \sigma_j \epsilon^2}
+ \frac{|\mathbf{x}_{i',(-j)}|^T \bm{\sigma}_{(-j)}}{2\pi\epsilon}.\end{aligned}$$ Note that a similar derivation also holds if $\mathbf{x}_{i'}^T \bm{\mu}/\epsilon < 0$.
On the other hand, if condition \[condition\_b\] is true, we can follow the same modification in Appendix \[proof\_th1\_sufficient\] to find upper bounds in a similar way.
To show Theorem \[theorem:multivariate\_Cauchy\], we decompose its proof into two parts: in Appendix \[proof\_theorem:multivariate\_Cauchy\_no\_separation\] we show that all posterior means exist if there is no separation; then in Appendix \[proof\_theorem:multivariate\_Cauchy\_complete\_separation\] we show that under a multivariate Cauchy prior, none of the posterior means exist if there is complete separation.
Proof of Theorem \[theorem:multivariate\_Cauchy\], in the case of no separation {#proof_theorem:multivariate_Cauchy_no_separation}
-------------------------------------------------------------------------------
For any $j = 1, 2, \ldots, p$, to show that $E(\beta_j \mid \mathbf{y})$ exists, we just need to show the positive term $E(\beta_j \mid \mathbf{y})^+$ in has an upper bound, because the negative term $E(\beta_j \mid \mathbf{y})^-$ in having a lower bound follows a similar derivation.
When working on $E(\beta_j \mid \mathbf{y})^+$, we only need to consider positive $\beta_j$. Denote a new $p-1$ dimensional variable $\bm{\gamma} = \bm{\beta}_{(-j)}/ \beta_j$, then for $i = 1, 2, \ldots, n$, $$\mathbf{x}_i^T \bm{\beta} = \beta_j\left( x_{i,j} + \mathbf{x}_{i,(-j)}^T \bm{\gamma} \right).$$ If there is no separation, for any $\bm{\gamma} \in \mathbb{R}^{p-1}$, there exists at least one $i \in \{1, 2, \ldots, n\}$, such that $$\label{eq:no_separtion_1}
x_{i,j} + \mathbf{x}_{i,(-j)}^T \bm{\gamma} < 0, \text{ if } i \in A_1, \text{ or }
x_{i,j} + \mathbf{x}_{i,(-j)}^T \bm{\gamma} > 0, \text{ if } i \in A_0.$$ For each $i = 1, 2, \ldots, n$, denote the vector $\mathbf{z}_i$ and the function $h_i(\cdot)$ as follows, $$\label{eq:z_h}
\mathbf{z}_i \stackrel{\text{def}}{=} \begin{cases}
\mathbf{x}_i & \text{ if } i \in A_1 \\
-\mathbf{x}_i & \text{ if } i \in A_0 \\
\end{cases}, \quad
h_i(\bm{\gamma}) \stackrel{\text{def}}{=} z_{i, j} + \mathbf{z}_{i, (-j)}^T \bm{\gamma},$$ then can be rewritten as $h_i (\bm{\gamma}) < 0$. Denote for $i = 1, 2, \ldots, n$, $$B_i \stackrel{\text{def}}{=} \{\bm{\gamma}: h_i(\bm{\gamma}) < 0\}.$$ Then each $B_i$ is a non-empty subset of $\mathbb{R}^{p-1}$, unless $z_{i,j} \geq 0$ and $\mathbf{z}_{i, (-j)} = \mathbf{0}$. Let $\mathcal{I} = \{i: B_i \neq \o\}$ denote the set of indices $i$, for which the corresponding $B_i$ are non-empty. Because there is no separation, $$\label{eq:B_i_union}
\bigcup_{i \in \mathcal{I}} B_i = \mathbb{R}^{p-1}.$$ Hence, the set $\mathcal{I}$ is non-empty. We denote its size by $q \stackrel{\text{def}}{=} |\mathcal{I}|$, and rewrite $\mathcal{I} = \{i_1, i_2, \ldots, i_q\}$.
Now we show that there exist positive constants $\epsilon_{i_1}, \epsilon_{i_2}, \ldots, \epsilon_{i_q}$, such that $$\label{eq:tildeB_i2}
\bigcup_{k=1}^q \tilde{B}_{i_k} = \mathbb{R}^{p-1},$$ where $$\label{eq:tildeB_i}
\tilde{B}_{i_k} \stackrel{\text{def}}{=} \{\bm{\gamma}: h_{i_k}(\bm{\gamma}) < -\epsilon_{i_k}\},$$ are subsets of the corresponding $B_{i_k}$, for all $k = 1, 2, \ldots, q$.
If there exists an $i_k \in \mathcal{I}$ such that $z_{i_k,j} < 0$ and $\mathbf{z}_{i_k, (-j)} = \mathbf{0}$, then $B_{i_k} = \mathbb{R}^{p-1}$. In this case, we just need to let $\epsilon_{i_k} = - z_{i_k,j} /2$, and $\epsilon_{i_r} = M$, for all $r \neq k$, where $M$ is an arbitrary positive number. Under this choice of $\epsilon_{i}$’s, the sets $\tilde{B}_i$’s defined by satisfy .
If, on the other hand, $\mathbf{z}_{i_k, (-j)} \neq \mathbf{0}$ for all $i_k \in \mathcal{I}$, i.e., all $B_{i_k}$ are open half spaces in $\mathbb{R}^{p-1}$, then we can find the constants $\epsilon_{i_1}, \epsilon_{i_2}, \ldots, \epsilon_{i_q}$ sequentially. For $i_1$, if $\bigcup_{k=2}^q B_{i_k} = \mathbb{R}^{p-1}$, we can set $\epsilon_{i_1} = M$. Then the resulting $\tilde{B}_{i_1}$ defined by satisfies $$\label{eq:B1_modified}
\tilde{B}_{i_1} \cup B_{i_2} \cup B_{i_3} \cup \cdots \cup B_{i_q} = \mathbb{R}^{p-1}.$$ If $\bigcup_{k=2}^q B_{i_k} \neq \mathbb{R}^{p-1}$, then suggests $$\label{eq:B1_supset}
B_{i_1} \supset \left(\bigcup_{k=2}^q B_{i_k} \right)^c = \bigcap_{k=2}^q B_{i_k}^c.$$ For to hold, we just need to find an positive $\epsilon_{i_1}$ such that the resulting $\tilde{B}_{i_1}$ has $\bigcap_{k=2}^q B_{i_k}^c$ as a subset, i.e., $-\epsilon_{i_1}$ should be larger than the maximum of $h_{i_1}(\bm{\gamma})$ over the polyhedral region $\bm{\gamma} \in \bigcap_{k=2}^q B_{i_k}^c$. Note that maximizing $h_{i_1}(\bm{\gamma})$ over the polyhedron can be represented as a linear programming question, $$\begin{aligned}
\label{eq:lp1}
\text{maximize} \quad & z_{i_1, j} + \mathbf{z}_{i_1, (-j)}^T\bm{\gamma}\\ \nonumber
\text{subject to} \quad & \mathbf{z}_{i_2, (-j)}^T\bm{\gamma} \geq - z_{i_2, j}\\ \nonumber
& \vdots\\ \nonumber
& \mathbf{z}_{i_q, (-j)}^T\bm{\gamma} \geq - z_{i_q, j}.\end{aligned}$$ By @Bertsimas_Tsitsiklis_1997 [pp. 67, Corollary 2.3], for any linear programming problem over a non-empty polyhedron, including the one in to maximize $h_{i_1}(\bm{\gamma}) = z_{i_1, j} + \mathbf{z}_{i_1, (-j)}^T\bm{\gamma}$, either the optimal $h_{i_1}(\bm{\gamma}) = \infty$, or there exists an optimal solution, $\bm{\gamma}^*$. Here, the latter case always occurs, because by , the maximum of $h_{i_1}(\bm{\gamma})$ over the polyhedron $\bigcap_{k=2}^q B_{i_k}^c$ does not exceed zero, so it does not go to infinity. Hence, we just need to let $$\epsilon_{i_1} = -\frac{1}{2}\left[
\max_{\bm{\gamma} \in \bigcap_{k=2}^q B_{i_k}^c} z_{i_1, j} + \mathbf{z}_{i_1, (-j)}^T\bm{\gamma} \right]
= -\frac{z_{i_1, j} + \mathbf{z}_{i_1, (-j)}^T\bm{\gamma}^*}{2},$$ so that the resulting $\tilde{B}_{i_1}= \{\bm{\gamma}: h_{i_1}(\bm{\gamma}) < -\epsilon_{i_1}\}$ contains $\bigcap_{k=2}^q B_{i_k}^c$ as a subset, which yields . After finding $\epsilon_{i_1}$, we can apply similar procedures sequentially, to find positive values $\epsilon_{i_k}$, for $k = 2, 3, \ldots,q$, such that $$\tilde{B}_{i_1} \cup \cdots \cup \tilde{B}_{i_k} \cup B_{i_{k+1}} \cup \cdots \cup B_{i_q} = \mathbb{R}^{p-1}.$$ After identifying all $\epsilon_{i_k}$’s, the resulting $\tilde{B}_{i_k}$’s satisfy . Note that the choice of $\epsilon_{i_k}$’s only depend on the data $\mathbf{z}_i$, $i = 1, 2, \ldots, n$, so they are constants given the observed data.
For each $k = 1, 2, \ldots, q$, next we show that for any $\bm{\gamma}\in \tilde{B}_{i_k}$, the likelihood function of the $i_k$th observation is bounded above by $(\beta_j \epsilon_{i_k} e)^{-1}$. This is because in a logistic regression, if $i_k \in A_1$, then $$\label{eq:f1_upper_bound1}
p(y_{i_k} \mid \beta_j, \bm{\gamma}) = f_1\left(\beta_j h_{i_k}(\bm{\gamma})\right)
= \frac{e^{\beta_j h_{i_k}(\bm{\gamma})}}{1 + e^{\beta_j h_{i_k}(\bm{\gamma})}} < e^{\beta_j h_{i_k}(\bm{\gamma})}
< e^{-\beta_j \epsilon_{i_k}} \leq \frac{1}{\beta_j \epsilon_{i_k}e},$$ if $i_k \in A_0$, then $$\label{eq:f1_upper_bound0}
p(y_{i_k} \mid \beta_j, \bm{\gamma}) = f_0\left(-\beta_j h_{i_k}(\bm{\gamma})\right)
= \frac{1}{1 + e^{-\beta_j h_{i_k}(\bm{\gamma})}} < e^{\beta_j h_{i_k}(\bm{\gamma})}
< e^{-\beta_j \epsilon_{i_k}} \leq \frac{1}{\beta_j \epsilon_{i_k}e}.$$ Here, the last inequality holds because $e^{-t} \leq \frac{e^{-1}}{t}$ for any $t > 0$. By , in a probit regression model, the inequalities and also hold.
Now we show that the positive term $E(\beta_j \mid \mathbf{y})^+$ has a finite upper bound. $$\begin{aligned}
E(\beta_j \mid \mathbf{y})^+
& = \int_{0}^{\infty} \beta_j \int_{\mathbb{R}^{p-1}}
p(\mathbf{y} \mid \beta_j, \bm{\gamma}) p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\
& \leq \int_{0}^{\infty} \beta_j \sum_{k=1}^q \int_{\tilde{B}_{i_k}}
p(\mathbf{y} \mid \beta_j, \bm{\gamma}) p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\
& < \int_{0}^{\infty} \beta_j \sum_{k=1}^q \int_{\tilde{B}_{i_k}}
p(y_{i_k} \mid \beta_j, \bm{\gamma}) p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\
& \leq \int_{0}^{\infty} \beta_j \sum_{k=1}^q \int_{\tilde{B}_{i_k}}
\frac{1}{\beta_j \epsilon_{i_k} e} p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\
& = \sum_{k=1}^q \frac{1}{\epsilon_{i_k} e} \int_{0}^{\infty} \int_{\tilde{B}_{i_k}}
p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\
& \leq \sum_{k=1}^q \frac{1}{\epsilon_{i_k} e} \int_{0}^{\infty} \int_{\mathbb{R}^{p-1}}
p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j
< \sum_{k=1}^q \frac{1}{\epsilon_{i_k} e}.\end{aligned}$$
Note that in Appendix \[proof\_theorem:multivariate\_Cauchy\_no\_separation\], the specific formula of the prior density of $\boldsymbol\beta$ is not used. Therefore, if there is no separation in logistic or probit regression, posterior means of all coefficients exist under all proper prior distributions.
Proof of Theorem \[theorem:multivariate\_Cauchy\], in the case of complete separation {#proof_theorem:multivariate_Cauchy_complete_separation}
-------------------------------------------------------------------------------------
Here we show that if there is complete separation, then none of the posterior means $E(\beta_j \mid \mathbf{y})$ exist, for $j = 1, 2, \ldots, p$. Using the notation $\mathbf{z}_i$, defined in , we rewrite the set of all vectors satisfying the complete separation condition as $$\mathcal{C} = \bigcap_{i=1}^n \left\{\bm{\beta}\in \mathbb{R}^{p}: \mathbf{z}_i^T \bm{\beta} > 0\right\}.$$ According to @Albert_Anderson_1984, $\mathcal{C}$ is a convex cone; moreover, if $\bm{\beta} \in \mathcal{C}$, then $\bm{\beta} + \bm{\delta} \in \mathcal{C}$ for any $\bm{\delta} \in \mathbb{R}^{p}$ that is small enough. Hence, the open set $\mathcal{C}$, as a subset of the $\mathbb{R}^p$ Euclidean space, has positive Lebesgue measure.
To show that $E(\beta_j \mid \mathbf{y})$ does not exist, if $\mathcal{C}$ projects on the positive half of the $\beta_j$ axis, we will show that $E(\beta_j \mid \mathbf{y})^+$ diverges, otherwise, we will show that $E(\beta_j \mid \mathbf{y})^-$ diverges (if both, then showing either is sufficient). Now we assume that the former is true, and denote the intersection $$\mathcal{C}_j^+ \stackrel{\text{def}}{=} \mathcal{C} \cap \{\bm{\beta}\in \mathbb{R}^{p}: \beta_j > 0\},$$ which is again an open convex cone. Since $\mathcal{C}_j^+$ has positive measure in $\mathbb{R}^p$, under the change of variable from $(\beta_j, \bm{\beta}_{(-j)})$ to $(\beta_j, \bm{\gamma})$, where $\bm{\beta}_{(-j)} = \beta_j\bm{\gamma}$, there exists an open set $\tilde{\mathcal{C}}_j^+ \in \mathbb{R}^{p-1}$ such that $\mathcal{C}_j^+$ can be written as $$\mathcal{C}_j^+ = \{(\beta_j, \beta_j \bm{\gamma}): \beta_j > 0, \bm{\gamma}\in \tilde{\mathcal{C}}_j^+ \}.$$ Suppose that $\bm{\gamma}$ can be written as $(\gamma_1, \ldots, \gamma_{j-1}, \gamma_{j+1}, \ldots, \gamma_p)^T$. We define a variant of it by $\tilde{\bm{\gamma}} \stackrel{\text{def}}{=}
(\gamma_1, \ldots, \gamma_{j-1}, 1, \gamma_{j+1}, \ldots, \gamma_p)^T$, such that $\bm{\beta} = \beta_j \tilde{\bm{\gamma}}$. Under the multivariate Cauchy prior , the induced prior distribution of $(\beta_j, \bm{\gamma})$ is $$\begin{aligned}
p(\beta_j, \bm{\gamma})
& \propto \frac{\beta_j^{p-1}}
{\left[1 + \left(\beta_j \tilde{\bm{\gamma}} - \bm{\mu}\right)^T
\boldsymbol\Sigma^{-1}
\left(\beta_j \tilde{\bm{\gamma}} - \bm{\mu}\right)\right]^{\frac{p+1}{2}}}\\
& = \frac{\beta_j^{p-1}}
{\left[\left(\tilde{\bm{\gamma}}^T \boldsymbol\Sigma^{-1} \tilde{\bm{\gamma}}\right) \beta_j^2
- 2 \left(\tilde{\bm{\gamma}}^T \boldsymbol\Sigma^{-1} \bm{\mu}\right) \beta_j
+ \left(\bm{\mu}^T \boldsymbol\Sigma^{-1} \bm{\mu} + 1\right)\right]^{\frac{p+1}{2}}}.\end{aligned}$$
Inside $\tilde{\mathcal{C}}_j^+$, there must exist a closed rectangular box, denoted by $\tilde{\mathcal{D}}_j^+ = \{\bm{\gamma}: \gamma_k \in [l_k, u_k], k = 1, \ldots, j-1, j+ 1, \ldots, p\}
\subset \tilde{\mathcal{C}}_j^+$. By @Browder_1996 [pp. 142, Corollary 6.57], a continuous function takes its maximum and minimum on a compact set. Since $\mathcal{D}_j^+$ is a compact set (closed and bounded in $\mathbb{R}^{p-1}$), $$a \stackrel{\text{def}}{=} \max_{\bm{\gamma} \in \tilde{\mathcal{D}}_j^+}
\tilde{\bm{\gamma}}^T \boldsymbol\Sigma^{-1} \tilde{\bm{\gamma}}, \quad
b \stackrel{\text{def}}{=} \min_{\bm{\gamma} \in \tilde{\mathcal{D}}_j^+}
\tilde{\bm{\gamma}}^T \boldsymbol\Sigma^{-1} \bm{\mu}
\label{eq:def_a_b}$$ both exist.
Recall that for all $\bm{\gamma}\in \tilde{\mathcal{C}}_j^+$ (hence including all elements in $\mathcal{D}_j^+$), $z_{i,j} + \mathbf{z}_{i, (-j)}^T \bm{\gamma} >0$; equivalently, if $i \in A_1$, then $x_{i,j} + \mathbf{x}_{i,(-j)}^T \bm{\gamma} > 0$, and if $i \in A_0$, then $x_{i,j} + \mathbf{x}_{i,(-j)}^T \bm{\gamma} < 0$. Now we show that in both logistic and probit regressions, the positive term $E(\beta_j \mid \mathbf{y})^+$ diverges. $$\begin{aligned}
\nonumber
& E(\beta_j \mid \mathbf{y})^+
= \int_{0}^{\infty} \beta_j \int_{\mathbb{R}^{p-1}} p(\beta_j, \bm{\gamma})
p(\mathbf{y} \mid \beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\ \nonumber
\geq& \int_{0}^{\infty} \beta_j \int_{\tilde{\mathcal{D}}_j^+} p(\beta_j, \bm{\gamma})
\prod_{i \in A_1}f_1(\beta_j (x_{i,j} + \mathbf{x}_{i,(-j)}^T \bm{\gamma}))
\prod_{k \in A_0}f_0(\beta_j (x_{k,j} + \mathbf{x}_{k,(-j)}^T \bm{\gamma}))
d \bm{\gamma} d \beta_j\\ \nonumber
\geq& \int_{0}^{\infty} \beta_j \int_{\tilde{\mathcal{D}}_j^+} p(\beta_j, \bm{\gamma})
\prod_{i \in A_1}f_1(0)
\prod_{k \in A_0}f_0(0)
d \bm{\gamma} d \beta_j\\ \nonumber
=& \left(\frac{1}{2}\right)^n \int_{0}^{\infty} \beta_j
\int_{\tilde{\mathcal{D}}_j^+} p(\beta_j, \bm{\gamma}) d \bm{\gamma} d \beta_j\\ \nonumber
=& ~C \int_{0}^{\infty} \beta_j
\int_{\tilde{\mathcal{D}}_j^+} \frac{\beta_j^{p-1}}
{\left[\left(\tilde{\bm{\gamma}}^T \boldsymbol\Sigma^{-1} \tilde{\bm{\gamma}}\right) \beta_j^2
- 2 \left(\tilde{\bm{\gamma}}^T \boldsymbol\Sigma^{-1} \bm{\mu}\right) \beta_j
+ \left(\bm{\mu}^T \boldsymbol\Sigma^{-1} \bm{\mu} + 1\right)\right]^{\frac{p+1}{2}}}
d \bm{\gamma} d \beta_j\\ \nonumber
\geq& ~C \int_{0}^{\infty}
\int_{\tilde{\mathcal{D}}_j^+} \frac{\beta_j^{p}}
{\left[a\beta_j^2 - 2b \beta_j + \left(\bm{\mu}^T \boldsymbol\Sigma^{-1} \bm{\mu} + 1\right) \right]^{\frac{p+1}{2}}}
d \bm{\gamma} d \beta_j\\ \label{eq:int_diverge}
=& ~C \left[\prod_{k=1}^{p-1}(u_k - l_k)\right]\int_{0}^{\infty}
\frac{\beta_j^{p}}
{\left[a(\beta_j - b/a)^2 + c\right]^{\frac{p+1}{2}}}
d \beta_j \\ \nonumber
=& ~ \infty,\end{aligned}$$ where $C$ is a positive constant, $c$ is a constant, and $a$ and $b$ have been defined previously in .
On the other hand, if the set $\mathcal{C}$ of complete separation vectors only projects on the negative half of the $\beta_j$ axis, following a similar deviation, we can show that $E(\beta_j \mid \mathbf{y})^-$ diverges to $-\infty$.
[^1]: The University of Iowa, Iowa City, IA. Email: [joyee-ghosh@uiowa.edu]{}
[^2]: Clemson University, Clemson, SC. Email: [ybli@clemson.edu]{}
[^3]: These authors contributed equally.
[^4]: University of Southampton, Southampton, UK. Email: [R.Mitra@soton.ac.uk]{}
[^5]: https://archive.ics.uci.edu/ml/datasets/SPECT+Heart
|
---
abstract: |
We construct and study a natural compactification $\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld ${{\mathbb F}}_q[T]$-modules with a structure of level $N \in {{\mathbb F}}_q[T]$. Namely, $\overline{M}^r(N) =
{\rm Proj}\,{\bf Eis}(N)$, the projective variety associated with the graded ring ${\bf Eis}(N)$ generated by the Eisenstein series of rank $r$ and level $N$. We use this to define the ring ${\bf Mod}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of ${\bf Eis}(N)$ in their common quotient field $\widetilde{{{\mathcal F}}}_r(N)$. Modular forms are characterized as those holomorphic functions on the Drinfeld space ${{\Omega}}^r$ with the right transformation behavior under the congruence subgroup ${{\Gamma}}(N)$ of ${{\Gamma}}= {\rm GL}(r,{{\mathbb F}}_q[T])$ (“weak modular forms”) which, along with all their conjugates under ${{\Gamma}}/{{\Gamma}}(N)$, are bounded on the natural fundamental domain ${{\boldsymbol{F}}}$ for ${{\Gamma}}$ on ${{\Omega}}^r$.
author:
- 'Ernst-Ulrich Gekeler'
title: 'On Drinfeld modular forms of higher rank IV: Modular forms with level'
---
[**0. Introduction.**]{}\
(0.1) This is the fourth of a series of papers (see [@19], [@20], [@21]) which aim to lay the foundations for a theory of Drinfeld modular forms of higher rank. These are modular forms for the modular group ${{\Gamma}}= {\rm GL}(r,{{\mathbb F}}_q[T])$ or its congruence subgroups, where “higher rank” refers to $r$ larger or equal to 2. The case of $r=2$, remarkably similar in some aspects but rather different in others to the theory of classical elliptic modular forms for ${\rm SL}(2,{{\mathbb Z}})$ or its congruence subgroups, is meanwhile well-established and the subject of several hundred publications since about 1980.
We leave aside to deal with more general Drinfeld coefficient rings $A$ than $A={{\mathbb F}}_q[T]$, as the amount of technical and notational efforts required would obscure the overall picture. The interested reader may consult [@15] to get an impression of the complications that - even for $r=2$ - result from class numbers $h(A) > 1$ for general $A$.
(0.2) While we developed some of the theory of modular forms “without level” in [@19] and [@21] and focussed on the connection with the geometry of the Bruhat-Tits building in [@19] and [@20], the current part IV is devoted to forms “with level”, i.e., forms for congruence subgroups of ${{\Gamma}}$. Again we restrict to the most simple case of full congruence subgroups ${{\Gamma}}(N)=\{{{\gamma}}\in {{\Gamma}}\,|\, {{\gamma}}\equiv 1 (\bmod N)\}$ for $N \in A$. Finer arithmetic/geometric properties of modular forms (or varieties) for other congruence subgroups ${{\Gamma}}' \supset {{\Gamma}}(N)$ may be derived in the course of the further development of the theory from those for ${{\Gamma}}(N)$, by taking invariants (or quotients) of the finite group ${{\Gamma}}'/{{\Gamma}}(N)$.
(0.3) Let us introduce a bit of notation: ${{\mathbb F}}= {{\mathbb F}}_q$ is the finite field with $q$ elements, $A = {{\mathbb F}}[T]$ the polynomial ring in an indeterminate $T$, with quotient field $K = {{\mathbb F}}(T)$, and its completion $K_{\infty} = {{\mathbb F}}((T^{-1}))$ at infinity, and $C_{\infty}$ the completed algebraic closure of $K_{\infty}$. The Drinfeld symmetric space ${{\Omega}}^r$ (where $r \geq 2$) is the complement in $\mathbb P^{r-1}(C_{\infty})$ of the $K_{\infty}$-rational hyperplanes. The modular group ${{\Gamma}}= {\rm GL}(r,A)$ acts in the usual fashion on ${{\Omega}}^r$, and we let $M^r(N)$ be the quotient analytic space ${{\Gamma}}(N)\setminus {{\Omega}}^r$ (which is also the set of $C_{\infty}$-points of an affine variety labelled by the same symbol, and which is smooth if $N \in A$ is non-constant).
The modular forms dealt with will be holomorphic functions on ${{\Omega}}^r$ with certain additional properties; so the theory is “over $C_{\infty}$”; we will only briefly touch on questions of rationality.
(0.4) Our approach is based on
- the use of the natural fundamental domain ${{\boldsymbol{F}}}$ for ${{\Gamma}}$ on ${{\Omega}}^r$ introduced in [@18]; it relies on the notion of successive minimum basis (SMB) of an $A$-lattice in $C_{\infty}$. On ${{\boldsymbol{F}}}$, one may perform explicit calculations;
- a natural compactification $\overline{M}^r(N)$ of $M^r(N)$, the [*Eisenstein compactification*]{}, whose construction is influenced by but different from Kapranov’s in [@28].
The obvious examples of modular forms-to-be for ${{\Gamma}}(N)$ are the Eisenstein series of level $N$. They generate a graded $C_{\infty}$-algebra ${\bf Eis}(N)$ (generated in dimension 1 if $N$ is non-constant), and $\overline{M}^r(N)$ will be the associated projective variety ${\rm Proj}({\bf Eis}(N))$, see Theorem 5.9. It is a closed subvariety of a certain projective space $\mathbb P^{c-1}$, where $c$ is the number of cusps of ${{\Gamma}}(N)$ (Corollary 4.7, Theorem 5.9), and is therefore supplied with a natural very ample line bundle $\mathfrak{M}$. We define strong modular forms of weight $k$ for ${{\Gamma}}(N)$ as sections of $\mathfrak M^{\otimes k}$, and thereby get the graded ring ${\bf Mod}^{\rm st}(N)$ of strong modular forms, which encompasses ${\bf Eis}(N)$.
(0.5) The Eisenstein compactification is natural and explicit, and has good functorial properties (see Remark 5.8; it is, e.g., compatible with level change), but unfortunately we presently cannot assure that it is normal. Correspondingly, strong modular forms are integral over ${\bf Eis}(N)$ (and in fact over ${\bf Mod} = {\bf Mod}(1)$, the ring of modular forms of type 0 for ${{\Gamma}}(1) = {{\Gamma}}$), but we don’t know whether ${\bf Mod}^{\rm st}(N)$ is integrally closed. We [*define*]{} the Satake compactification $M^r(N)^{\rm Sat}$ of $M^r(N)$ as the normalization of $\overline{M}^r(N)$ (as Kapranov does) and a modular form of weight $k$ for ${{\Gamma}}(N)$ as a section of the pull-back of $\mathfrak{M}^{\otimes k}$ to $M^r(N)^{\rm Sat}$. This yields the graded ring ${\bf Mod}(N)$ of all modular forms. Hence we have inclusions $${\bf Eis}(N) \subset {\bf Mod}^{\rm st}(N) \subset {\bf Mod}(N)\leqno{(0.6)}$$ of finitely generated graded integral $C_{\infty}$-algebras, where ${\bf Mod}(N)$ is the integral closure of ${\bf Eis}(N)$ in their common quotient field $\widetilde{\mathcal{F}}_r(N)$. Elements of ${\bf Mod}(N)$ have a nice characterization given by Theorem 7.9: A weak modular form $f$ of weight $k$ is modular if and only if $f$, together with all its conjugates $f_{[{{\gamma}}]_k}$ (${{\gamma}}\in {{\Gamma}}/{{\Gamma}}(N))$, is bounded on the fundamental domain ${{\boldsymbol{F}}}$. Further ${\bf Eis}(N)$ has always finite codimension in ${\bf Mod}^{\rm st}(N)$ (Corollary 7.11), while $\dim({\bf Mod}(N)/{\bf Mod}^{\rm st}(N))$ is either zero or infinite, according to whether $M^r(N)^{\rm Sat}$ agrees with $\overline{M}^r(N)$ or not (Corollary 7.14). Except for some examples presented in Section 8, where the two compactifications and also the three rings in (0.6) agree (these examples depend crucially on work of Cornelissen [@8] and Pink-Schieder [@32]), we don’t know what happens in general: more research is needed! At least $M^r(N)^{\rm Sat}$ is not very far from $\overline{M}^r(N)$: the normalization map $$\nu:\: M^r(N)^{\rm Sat} {{\longrightarrow}}\overline{M}^r(N)$$ is bijective on $C_{\infty}$-points (Corollary 7.6, see also Proposition 1.18 in [@28]), and is an isomorphism on the complement of a closed subvariety of codimension $\geq 2$ (Corollary 6.10).
(0.7) We now describe the plan of the paper. In the first section, we introduce the space $\overline{{{\Omega}}}^r$ with its strong topology, which upon dividing out the action of ${{\Gamma}}(N)$ will yield the underlying topological space for the Eisenstein compactification $\overline{M}^r(N)$. Its points correspond to homothety classes of pairs $(U,i)$, where $U \not= 0$ is a $K$-subspace of $K^r$ and $i$ is a discrete embedding of $U\cap A^r$ into $C_{\infty}$. For technical purposes we also consider the ${{\mathbb G}}_m$-torsor $\overline{\Psi}^r$ over $\overline{{{\Omega}}}^r$ whose points correspond to pairs (not homothety classes) $(U,i)$ as above. Further, the fundamental domains $\widetilde{{{\boldsymbol{F}}}}$ on $\Psi^r$ and ${{\boldsymbol{F}}}$ on ${{\Omega}}^r$ for ${{\Gamma}}$ are introduced. Although $\overline{{{\Omega}}}^r$ and $\overline{\Psi}^r$ come with the same information, it will sometimes be more convenient to work with $\overline{\Psi}^r$ and $\widetilde{{{\boldsymbol{F}}}}$ instead of $\overline{{{\Omega}}}^r$ and ${{\boldsymbol{F}}}$. We take particular care to give a consistent description of the group actions on $\overline{{{\Omega}}}^r$ and related objects.
In Section 2 the (well-known) relationship of ${{\Omega}}^r$ with the moduli of Drinfeld modules of rank $r$ is presented. We further show the crucial technical result Theorem 2.3, which asserts that the bijection $$j:\, {{\Gamma}}\setminus \overline{{{\Omega}}}^r\stackrel{\cong}{{{\longrightarrow}}} {\rm Proj}({\bf Mod})$$ is a homeomorphism for the strong topologies on both sides. We further introduce and describe the function fields of the analytic spaces $M^r(N) = {{\Gamma}}(N)\setminus {{\Omega}}^r$ and $\widetilde{M}^r(N) = {{\Gamma}}(N)\setminus \Psi^r$.
In Sections 3 and 4, the boundary components and the (non-)vanishing of Eisenstein series on them are studied. We find in Corollary 4.7 that the space ${\rm Eis}_k(N)$ of Eisenstein series of level $N$ and weight $k$ has dimension $c_r(N)$, the number of cuspidal divisors of ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$, independently of $k$. Further (Proposition 4.8), ${\rm Eis}_1(N)$ separates points of ${{\Gamma}}(N)\setminus\overline{{{\Omega}}}^r$, which will give rise to its projective embedding. This latter is defined and investigated in Section 5; we thereby interpret ${{\Gamma}}(N)\setminus
\overline{{{\Omega}}}^r$ as the Eisenstein compactification $\overline{M}^r(N)$ of $M^r(N)$.
Section 6 is of a more technical nature. There we construct tubular neighborhoods along the cuspidal divisiors of $\overline{M}^r(N)$, see Theorem 6.9.
In Section 7, the rings ${\bf Mod}^{\rm st}(N)$ and ¢${\bf Mod}(N)$ of modular forms are introduced and their relation with the Eisenstein ring ${\bf Eis}(N)$ and the compactifications $\overline{M}^r(N)$ and $M^r(N)^{\rm Sat}$ is discussed.
We conclude in Section 8 with the two classes of examples where our knowledge is more satisfactory than in the general situation, namely the special cases where either the rank $r$ equals $2$ or where the conductor $N$ has degree 1.
(0.8) The point of view (and the notation, see below) of this paper widely agrees with that of the preceding [@19], [@20], [@21], to which we often refer. As in these, our basic references for rigid analytic geometry are the books [@12] by Fresnel-van der Put and [@7] of Bosch-Güntzer-Remmert. The canonical topology on the set $X(C_{\infty})$ of $C_{\infty}$-points of an analytic space $X$ ([@7] Section 7.2) is labelled as the [*strong topology*]{}, so functions continuous with respect to it are strongly continuous, etc. In general, we don’t distinguish in notation between $X$ and $X(C_{\infty})$; ditto, a $C_{\infty}$-variety and its analytification are usually described by the same symbol. It will (hopefully) always be clear from the context whether e.g. the “algebraic” or the “analytic” local ring is intended.
(0.9) After this paper was largely completed, I got access to the recent preprints [@4], [@5], [@6] of Dirk Basson, Florian Breuer, and Richard Pink, which go about the same topic: providing a foundation for the theory of higher rank Drinfeld modular forms. As it turns out, the relative perspectives of Basson-Breuer-Pink’s work and of the current paper are rather different. While BBP deal with the most general Drinfeld coefficient rings $A$ and arithmetic subgroups of ${\rm GL}(r,A)$, for which they establish basic but sophisticated facts like e.g. the existence of expansions around infinity of weak modular forms, we restricted to the coefficient ring $A={{\mathbb F}}_q[T]$ and full congruence subgroups and focus on the role of Eisenstein series, their arithmetic properties, and their impact on compactifications of the moduli schemes. Apart from examples, there is little overlap between the two works; so the reader who wants to enter into the field might profit from studying the two of them.
Finally, I wish to point to the recent thesis [@245] of Simon Häberli, whose purpose is similar. In contrast with the present article, Häberli gives a direct construction of the Satake compactification, which he uses for the description of modular forms.
[**Notation.**]{}
${{\mathbb F}}= {{\mathbb F}}_q$ the finite field with $q$ elements;\
$A={{\mathbb F}}[T]$ the polynomial ring in an indeterminate $T$, with quotient field $K={{\mathbb F}}(T)$ and its completion $K_{\infty}=
{{\mathbb F}}((T^{-1}))$ at infinity;\
$C_{\infty}=$ completed algebraic closure of $K_{\infty}$, with absolute value $|~.~|$ and valuation $v:\: C_{\infty}^{\ast} {{\longrightarrow}}{{\mathbb Q}}$ normalized by $v(T) = -1$, $|T| = q$;\
$\Psi^r = \{{{\boldsymbol{\omega}}}= ({{\omega}}_1,\ldots,{{\omega}}_r) \in C_{\infty}^r~|~\mbox{the ${{\omega}}_i$ are $K_{\infty}$-linearly independent}\}$\
${{\Omega}}^r = \{{{\boldsymbol{\omega}}}= ({{\omega}}_1:\ldots: {{\omega}}_r) \in \mathbb P^{r-1} (C_{\infty})~|~{{\boldsymbol{\omega}}}\mbox{ represented by } ({{\omega}}_1,\ldots,{{\omega}}_r)
\in \Psi^r\}$\
${{\Gamma}}= {{\Gamma}}_r ={\rm GL}(r,A)$ with center $Z \cong {{\mathbb F}}^{\ast}$ of scalar matrices;\
${{\Gamma}}(N) = \{{{\gamma}}\in {{\Gamma}}~|~{{\gamma}}\equiv 1 (\bmod N)\}$, $N \in A$;\
$\mathfrak{U} = \mbox{set of $K$-subspaces $U \not= 0$ of $V = K^n$}$;\
$\Psi_U \cong \Psi^s$, ${{\Omega}}_U \cong {{\Omega}}^s$ attached to $U \in \mathfrak U$, where $\dim U = s$;\
$\overline{\Psi}^r = \underset{U\in \mathfrak U}{\stackrel{\bullet}{\bigcup}}\Psi_U$, $\overline{{{\Omega}}}^r = \underset{U\in \mathfrak U}{\stackrel{\bullet}{\bigcup}} {{\Omega}}_U.$
If the group $G$ acts on the space $X$ then $G_x$, $Gx$ and $G\setminus X$ denote the stabilizer of $x \in X$, its orbit, and the space of all orbits, respectively. Also, for $Y \subset X$, $G\setminus Y$ is the image of $Y$ in $G\setminus X$. The multiplicative group of the ring $R$ is $R^{\ast}$; the $R$-module generated by $x_1,\ldots,x_r$ is written either as $\sum Rx_i$ or as $\langle x_1,\ldots,x_r\rangle_R$. We use the convention ${{\mathbb N}}=\{1,2,3,\ldots\} $, ${{\mathbb N}}_0 =\{0,1,2,\ldots\}$.
[**1. The spaces $\overline{\Psi}^r$ and $\overline{{{\Omega}}}^r$.**]{}
(1.1) We let $V$ be the $K$-vector space $K^r$, where $r \geq 2$, and $\mathfrak U$ the set of $K$-subspaces $U \not= 0$ of $V$. An $A$-lattice in $U \in \mathfrak U$ is a free $A$-submodule $L$ of $U$ of full rank ${\rm rk}_A(L) = \dim_K(U)$, that is $K \otimes
L = KL = U$. A subset of $C_{\infty}$ is [*discrete*]{} if the intersection with each ball of finite radius in $C_{\infty}$ is finite. A [*discrete embedding*]{} of $U \in \mathfrak U$ (“embedding” for short) is some $K$-linear injective map $i:\: U {{\longrightarrow}}C_{\infty}$ such that $i(L)$ is discrete in $C_{\infty}$ for one fixed (or equivalently, for each) $A$-lattice $L$ in $U$. We put $$\begin{array}{lll}
\Psi_U & := & \mbox{set of discrete embeddings of $U$, and}\smallskip\\
{{\Omega}}_U & := & C_{\infty}^{\ast}\setminus \Psi_U, \mbox{the quotient of $\Psi_U$ modulo the}\\
& & \mbox{action of the multiplicative group $C_{\infty}^{\ast}$.}
\end{array}\leqno{(1.2)}$$ Further, $\Psi^r := \Psi_V = \Psi_{K^r}$, ${{\Omega}}^r := {{\Omega}}_V$, and $$\overline{\Psi}^r := \underset{U\in \mathfrak U}{\stackrel{\bullet}{\bigcup}} \Psi_U,\: \overline{{{\Omega}}}^r :=
\underset{U\in \mathfrak U}{\stackrel{\bullet}{\bigcup}} {{\Omega}}_U.$$ If $U \subset U' \in \mathfrak A$, restriction to $U$ defines canonical maps $$\Psi_{U'} {{\longrightarrow}}\Psi_U \mbox{ and } {{\Omega}}_{U'} {{\longrightarrow}}{{\Omega}}_U. \leqno{(1.2.1)}$$ (1.3) We let $L_V := A^r$ and $L_U := L_V \cap U$ be the standard lattices in $V$ and $U$, respectively. As a $K$-linear map $i:\: V {{\longrightarrow}}C_{\infty}$ is discrete if and only if the images ${{\omega}}_j := i(e_j)$ of the standard basis vectors $e_j$ ($1 \leq j \leq r$) are $K_{\infty}$-linearly independent (l.i.), we see that $$\Psi^r = \{{{\boldsymbol{\omega}}}= ({{\omega}}_1,\ldots,{{\omega}}_r) \in C_{\infty}^{\ast}~|~{{\omega}}_1,\ldots,{{\omega}}_r \mbox{ l.i.}\}.$$ After choosing bases of the subspaces $U$, we get similar descriptions for $\Psi_U$ and the quotients ${{\Omega}}^r$ and ${{\Omega}}_U$. In particular, we find for $r=2$ the familiar Drinfeld upper half-plane $$\begin{array}{rll}
{{\Omega}}^2 = C_{\infty}^{\ast}\setminus \{({{\omega}}_1,\omega_2)~|~{{\omega}}_1,{{\omega}}_2 \mbox{ l.i.}\} &\stackrel{\cong}{{{\longrightarrow}}}& C_{\infty}
\setminus K_{\infty}.\\
({{\omega}}_1,{{\omega}}_2) & \longmapsto & {{\omega}}_1/\omega_2 \end{array}$$ (1.4) The sets $\Psi^r$ and ${{\Omega}}^r$ (and therefore also $\Psi_U$ and ${{\Omega}}_U$) are equipped with structures of $C_{\infty}$-analytic spaces (actually defined over $K_{\infty}$), namely as admissible open subspaces of ${{\mathbb A}}^r(C_{\infty}) =
C_{\infty}^r$ or of $\mathbb P^{r-1}(C_{\infty})$, respectively, see [@11],[@10], or [@33].
(1.5) The group ${\rm GL}(r,K)$ acts as a matrix group from the right on $V$, which induces left actions on $\overline{\Psi}^r$ and $\overline{{{\Omega}}}^r$, viz.: For ${{\gamma}}\in {\rm GL}(r,K)$, let $r_{{{\gamma}}}:\: V {{\longrightarrow}}V$ be the map $x \longmapsto x{{\gamma}}$. Then ${{\gamma}}$ maps $(U,i:\: U \hookrightarrow C_{\infty}) \in \Psi_U$ to ${{\gamma}}(U,i) := (U{{\gamma}}^{-1},
i \circ r_{{{\gamma}}})$. The reader may verify that this, together with the description of $\Psi^r$ in (1.3), yields the standard left matrix action of ${{\gamma}}$ on $\Psi^r$, the elements of $\Psi^r$ being regarded as column vectors $({{\omega}}_1,\ldots,{{\omega}}_r)^t$.
(1.6) Since $A$ is a principal ideal domain, the theory of finitely generated modules over such (e.g. [@29] XV Sect. 2) shows that ${{\Gamma}}:= {\rm GL}(r,A)$ acts transitively on the set $\mathfrak U_s$ of $U \in \mathfrak U$ of fixed dimension $s$. We use as a standard representative for $\mathfrak U_s$ the space $$V_s := \{(0,\ldots,0,*,\ldots,*) \in V\}\leqno{(1.6.1)}$$ of vectors whose first $r-s$ entries vanish. The fixed group of $V_s$ ($1 \leq s <r$) in ${\rm GL}(r,K)$ is the maximal parabolic subgroup $$P_s := \left\{
\begin{array}{|c|c|}\hline
* & *\\ \hline
0 & *\\ \hline \end{array} \right\}
\leqno{(1.6.2)}$$ of matrices with an $(r-s,s)$-block structure whose lower left block vanishes. The action of $P_s$ on $V_s$ is via the group $$M_s := \left\{
\begin{array}{|c|c|}\hline
1 & 0\\ \hline
0 & *\\ \hline \end{array} \right\}
\leqno{(1.6.3)}$$ regarded as a factor group of $P_s$.
(1.7) As explained in (1.3), the choice of a $K$-basis of $U \in \mathfrak U$ yields an embedding of $\Psi_U$ into $C_{\infty}^{\dim U}$. The Haussdorff topology induced on $\Psi_U$ is independent of that choice, and is referred to as the [*strong topology*]{} on $\Psi_U$. Similarly, using embeddings into projective spaces, we define the strong topologies on the ${{\Omega}}_U$.
(1.8) Our next aim is to define reasonable strong topologies on $\overline{\Psi}^r$ and $\overline{{{\Omega}}}^r$ extending the topologies on the strata. For this we recall the concept of successive minimum bases. An [*$A$-lattice in*]{} $C_{\infty}$ is a discrete $A$-submodule ${{\Lambda}}$ of finite rank. A [*successive minimum basis*]{} (SMB) of ${{\Lambda}}$ is an ordered $A$-basis $\{{{\omega}}_1,\ldots,{{\omega}}_r\}$ of ${{\Lambda}}$ (note this differs from usual set-theoretic notation) subject to: For each $1 \leq j \leq r$, $|{{\omega}}_j|$ is minimal among $$\{|{{\omega}}|~|~{{\omega}}\in {{\Lambda}}\setminus (A{{\omega}}_1+ \cdots + A{{\omega}}_{j-1})\}.$$ (For $j = 1$ this means: ${{\omega}}_1$ is a lattice vector of minimal non-zero length.) It is shown in [@18] Proposition 3.1 that each $A$-lattice ${{\Lambda}}$ in $C_{\infty}$ possesses an SMB $\{{{\omega}}_1,\ldots,{{\omega}}_r\}$, and it has the following additional properties:
(1.8.1) The ${{\omega}}_i$ are orthogonal, that is, given $a_1,\ldots,a_r \in K_{\infty}$, $$|\sum_{1 \leq i \leq r} a_i{{\omega}}_i| = \underset{i}{\max} |a_i||{{\omega}}_i|;$$ (1.8.2) The series of positive real numbers $|{{\omega}}_1| \leq |{{\omega}}_2| \leq \ldots \leq |{{\omega}}_r|$ is an invariant of ${{\Lambda}}$, that is, independent of the choice of the SMB.
(1.9) We define the strong topology on $\overline{\Psi}^r$ as the unique Hausdorff topology which satisfies for each $U' \in \mathfrak A$:
(1.9.1) Restricted to $\Psi_{U'}$, it agrees with the strong topology given there by (1.7);
(1.9.2) The topological closure $\overline{\Psi}_{U'}$ of $\Psi_{U'}$ equals $\underset{U \subset U'}{\stackrel{\bullet}{\bigcup}}\Psi_U$;
(1.9.3) Assume $U' \supset U \in \mathfrak U$, and let $i:\: U {{\longrightarrow}}C_{\infty}$ and $i_k:\: U' {{\longrightarrow}}C_{\infty}$ ($k \in {{\mathbb N}}$) be discrete embeddings. Then $(U,i) = {\displaystyle \lim_{k\to \infty} (U',i_k)}$ if and only if
- for each ${{\lambda}}\in L_U$, $i({{\lambda}}) = {\displaystyle \lim_{k\to \infty} i_k({{\lambda}})}$ and
- for each ${{\lambda}}\in L_{U'} \setminus L_U$, ${\displaystyle \lim_{k\to \infty} |i_k({{\lambda}})| = \infty}$, uniformly in ${{\lambda}}$.
Note that it suffices to require (a) for the elements of a basis of $L_U$. In qualitative terms, $i_k:\: U' \hookrightarrow
C_{\infty}$ is very close to $i:\: U \hookrightarrow C_{\infty}$ iff
- $i_k({{\lambda}})$ is very close to $i({{\lambda}})$ for the elements ${{\lambda}}$ of an $A$-basis of $L_U$, and
- for each ${{\lambda}}\in L_{U'} \setminus L_U$, $|i_k({{\lambda}})|$ is very large compared to the $|{{\omega}}_j|$, where $\{{{\omega}}_j\}$ is an SMB of $i(L_U)$.
Furthermore, (b’) my be replaced by
- for each ${{\lambda}}\in L_{U'}\setminus L_U$, $i_k({{\lambda}})$ has very large distance $d(i_k({{\lambda}}),K_{\infty}i(U))$ to the $K_{\infty}$-space generated by $i(U)$.
The strong topology on $\overline{{{\Omega}}}^r = C_{\infty}^{\ast} \setminus \overline{\Psi}^r$ is the quotient topology; it has properties analogous to (1.9.1)–(1.9.3). Obviously, the action of ${\rm GL}(r,K)$ on both $\overline{\Psi}^r$ and $\overline{{{\Omega}}}^r$ is through homeomorphisms w.r.t. the so defined topologies.
(1.10) A continuous function $f:\: \overline{\Psi}^r {{\longrightarrow}}C_{\infty}$ has [*weight*]{} $k \in {{\mathbb Z}}$ if $$f(U,c \cdot i) = c^{-k} f(U,i)$$ holds for $c \in C_{\infty}^{\ast}$ and $(U,i) \in \overline{\Psi}^r$.
(1.11) The basic examples of functions with weight are the various types of [*Eisenstein series*]{} defined below. For $k \in {{\mathbb N}}$ put $$E_k(U,i) := \underset{{{\lambda}}\in L_U}{\sum{'}} \,i({{\lambda}})^{-k}.$$ (The prime $\sum'$ indicates that the sum is over the non-zero elements of the index set.) The following are obvious or easy to show:
- The sum converges and defines a continuous (even analytic) function $E_k$ on $\Psi_U$ ($U \in \mathfrak A$), which is non-trivial if and only if $k \equiv 0 (\bmod\, q-1)$;
- $E_k$ is continuous on the whole of $\overline{\Psi}^r$ with respect to the strong topology (due to the very definition of the latter);
- $E_k$ has weight $k$;
- $E_k$ is invariant under ${\rm GL}(r,A)$.
(1.12) Now let $N$ be a non-constant monic element of $A$ and ${{\Gamma}}(N) = \{{{\gamma}}\in {{\Gamma}}~|~{{\gamma}}\equiv 1 (\bmod N)\}$ be the [*full congruence subgroup*]{} of level $N$. Fix some vector ${{\boldsymbol{u}}}= (u_1,\ldots,u_r) \in V = K^r$ with $N{{\boldsymbol{u}}}\in L_V = A^r$, and put $$E_{k,{{\boldsymbol{u}}}}(U,i) := \underset{{{\lambda}}\in U \atop {{\lambda}}\equiv {{\boldsymbol{u}}}(\bmod L_V)}{\sum^{}{'}}i({{\lambda}})^{-k}.$$ The following hold:\
(i’) The sum converges and defines a continuous (even analytic) function $E_{k,{{\boldsymbol{u}}}}$ on $\Psi_U$; it depends only on the residue class of ${{\boldsymbol{u}}}$ modulo $L_V$, and is called the [*partial Eisenstein series*]{} with congruence condition ${{\boldsymbol{u}}}$;\
(ii), (iii) (see (1.11)), and\
(iv’) $E_{k,{{\boldsymbol{u}}}{{\gamma}}}(U,i) = E_{k,{{\boldsymbol{u}}}}({{\gamma}}(U,i))$, ${{\gamma}}\in {{\Gamma}}$. In particular, $E_{k,{{\boldsymbol{u}}}}$ is invariant under ${{\Gamma}}(N)$.
[**Remark.**]{} $$E_{k,{{\boldsymbol{u}}}}(U,i) = N^{-k}\underset{{{\lambda}}\in L_U \atop {{\lambda}}\equiv N{{\boldsymbol{u}}}(\bmod NL_V)}{\sum{'}} i({{\lambda}})^{-k},$$ which up to the factor $N^{-k}$ is a partial sum of $E_k(U,i)$. This explains the notation “partial Eisenstein series”.
(1.13) By definition, ${{\omega}}_r \not= 0$ for ${{\boldsymbol{\omega}}}= ({{\omega}}_1,\ldots,{{\omega}}_r) \in \Psi^r$. Therefore we can normalize projective coordinates on ${{\Omega}}^r \subset \mathbb P^{r-1}(C_{\infty})$ so that
(1.13.1) $ {{\omega}}_r=1$, i.e., $${{\Omega}}^r = \{({{\omega}}_1,\ldots,{{\omega}}_{r-1})=({{\omega}}_1:\ldots:{{\omega}}_{r-1}:1) ~|~
{{\omega}}_1,\ldots,{{\omega}}_{r-1},{{\omega}}_r=1 \mbox{ l.i.}\}.$$ Similarly we usually assume ${{\omega}}_r=1$ for ${{\boldsymbol{\omega}}}= ({{\omega}}_1:\ldots:{{\omega}}_r) \in {{\Omega}}_U$ if $U$ is one of the spaces $V_s$ of (1.6). With that convention, the Eisenstein series $E_k$ and $E_{k,{{\boldsymbol{u}}}}$ may be regarded as functions on $\underset{1 \leq s \leq r}{\bigcup} {{\Omega}}_{V_s}$. If ${{\boldsymbol{\omega}}}\in {{\Omega}}^r$ then (iii), (iv), (iv’) imply $$E_k({{\gamma}}{{\boldsymbol{\omega}}}) = {\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})^k E_k({{\boldsymbol{\omega}}}) \leqno{(1.13.2)}$$ and $$E_{k,{{\boldsymbol{u}}}}({{\gamma}}{{\boldsymbol{\omega}}}) = {\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})^k E_{k,{{\boldsymbol{u}}}{{\gamma}}}({{\boldsymbol{\omega}}}).\leqno{(1.13.3)}$$ Here ${{\gamma}}\in {{\Gamma}}$, ${{\boldsymbol{\omega}}}= ({{\omega}}_1:\ldots:{{\omega}}_r)$ with ${{\omega}}_r=1$, and ${\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})$ is the factor of automorphy $${\rm aut}({{\gamma}},{{\boldsymbol{\omega}}}) = \sum_{1 \leq i \leq r}{{\gamma}}_{r,i} {{\omega}}_i \not= 0.\leqno{(1.13.4)}$$ We assign no value to $E_k({{\boldsymbol{\omega}}})$ or $E_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})$ if ${{\boldsymbol{\omega}}}= C_{\infty}^*(U,i) \in \overline{{{\Omega}}}^r$ does not belong to $\bigcup {{\Omega}}_{V_s}$, but are content with the distinction (always well-defined) of whether $E_k$ (resp. $E_{k,{{\boldsymbol{u}}}}$) vanishes at ${{\boldsymbol{\omega}}}$ or not.
[**1.14 Remark**]{} (on notation). In order to avoid notational overflow, we use the same symbol $E_k$ for both occurrences: as a ${{\Gamma}}$-invariant function on $\overline{\Psi}^r$ of weight $k$, or as a function on $\bigcup
{{\Omega}}_{V_s}$ subject to (1.13.2). A similar remark applies to $E_{k,{{\boldsymbol{u}}}}$ and to other functions with weight.
(1.15) We finally define fundamental domains for the actions of ${{\Gamma}}$ on $\Psi^r$ and ${{\Omega}}^r$. To wit, put $$\begin{array}{lll}
\widetilde{{{\boldsymbol{F}}}} & := &\{{{\boldsymbol{\omega}}}= ({{\omega}}_1,\ldots,{{\omega}}_r)\in \Psi^r~|~\{{{\omega}}_r,{{\omega}}_{r-1},\ldots,{{\omega}}_1\} \mbox{ is an SMB}\\
&&\hspace*{4cm} \mbox{of its lattice } {{\Lambda}}_{{{\boldsymbol{\omega}}}} = \langle {{\omega}}_1,\ldots,{{\omega}}_r\rangle_A\}\\
{{\boldsymbol{F}}}&:=& C_{\infty}^*\setminus \tilde{{{\boldsymbol{F}}}}. \end{array}$$ (Note the reverse order of the ${{\omega}}_i$!) They have the following properties.
(1.15.1) As the condition for ${{\boldsymbol{\omega}}}\in \widetilde{{{\boldsymbol{F}}}}$ is stable under the multiplicative group, $\widetilde{{{\boldsymbol{F}}}}$ is the full cone above ${{\boldsymbol{F}}}$.
(1.15.2) Each ${{\boldsymbol{\omega}}}\in \Psi^r$ (resp. ${{\Omega}}^r$) is ${{\Gamma}}$-equivalent with at least one and at most a finite number of ${{\boldsymbol{\omega}}}' \in \widetilde{{{\boldsymbol{F}}}}$ (resp. ${{\boldsymbol{\omega}}}' \in {{\boldsymbol{F}}}$).
It suffices to treat the case $\widetilde{{{\boldsymbol{F}}}}$. As each $A$-lattice ${{\Lambda}}$ in $C_{\infty}$ has an SMB, the existence of a representative ${{\boldsymbol{\omega}}}' \in \tilde{{{\boldsymbol{F}}}}$ for ${{\boldsymbol{\omega}}}\in \Psi^r$ is obvious. Given ${{\boldsymbol{\omega}}}\in \widetilde{{{\boldsymbol{F}}}}$, the condition ${{\gamma}}{{\boldsymbol{\omega}}}\in \widetilde{{{\boldsymbol{F}}}}$ on ${{\gamma}}\in {{\Gamma}}$ together with (1.8.1) leads to bounds on the entries of ${{\gamma}}$, which can be satisfied for a finite number of ${{\gamma}}$’s only.
(1.15.3) $\widetilde{{{\boldsymbol{F}}}}$ resp. ${{\boldsymbol{F}}}$ is an admissible open subspace of $\Psi^r$ resp. ${{\Omega}}^r$.
The most intuitive way to see this comes from identifying ${{\boldsymbol{F}}}$ as the inverse image under the building map ${{\lambda}}:\: {{\Omega}}^r {{\longrightarrow}}{{\mathcal B}}{{\mathcal T}}({{\mathbb Q}})$ of a subcomplex $W$ of the Bruhat-Tits building ${{\mathcal B}}{{\mathcal T}}$ of ${\rm PGL}(r,K_{\infty})$: see [@19] Sect. 2. In fact, $W$ is a fundamental domain for ${{\Gamma}}$ on ${{\mathcal B}}{{\mathcal T}}$.
In view of the above, we refer to $\widetilde{{{\boldsymbol{F}}}}$ resp. ${{\boldsymbol{F}}}$ as the [*fundamental domain*]{} for ${{\Gamma}}$ on $\Psi^r$ resp. ${{\Omega}}^r$. As uniqueness of the representative in $\widetilde{{{\boldsymbol{F}}}}$ resp. ${{\boldsymbol{F}}}$ fails, this is weaker than the classical notion of fundamental domain, but is still useful. Property (1.8.1) turns out particularly valuable for explicit calculations with modular forms, as exemplified in [@19]. Also useful is the following observation, which is immediate from definitions. We formulate it for ${{\boldsymbol{F}}}$ only, but it holds true also for $\widetilde{{{\boldsymbol{F}}}}$.
(1.15.4) Let ${{\boldsymbol{F}}}_s$ be the fundamental domain for ${{\Gamma}}_s = {\rm GL}(s,A)$ in ${{\Omega}}_{V_s} \stackrel{\cong}{{{\longrightarrow}}} {{\Omega}}^s$ ($1 \leq s \leq r$). Then the strong closure of ${{\boldsymbol{F}}}$ in $\overline{{{\Omega}}}^r$ is $\overline{{{\boldsymbol{F}}}} = \underset{1\leq s \leq r}{\bigcup}
F_s$. Each point of $\overline{{{\Omega}}}^r$ is ${{\Gamma}}$-equivalent with at least one and at most a finite number of points of $\overline{{{\boldsymbol{F}}}}$.
Therefore, we can regard $\overline{{{\boldsymbol{F}}}}$ as a fundamental domain for ${{\Gamma}}$ on $\overline{{{\Omega}}}^r$.
[**2. Quotients by congruence subgroups and moduli schemes.**]{}
(2.1) Given an $A$-lattice ${{\Lambda}}$ in $C_{\infty}$ of rank $r \in {{\mathbb N}}$, we dispose of\
$\bullet$ the exponential function $e_{{\Lambda}}:\: C_{\infty} {{\longrightarrow}}C_{\infty}$ $$e_{{{\Lambda}}}(z) = z\underset{{{\lambda}}\in {{\Lambda}}}{\prod}'(1-z/{{\lambda}}) = \sum_{i\geq 0} \alpha_i({{\Lambda}})z^{q^{i}}; \leqno{(2.1.1)}$$ $\bullet$ the Drinfeld $A$-module $\phi^{{{\Lambda}}}$ of rank $r$, defined by the operator polynomial $$\phi_T^{{{\Lambda}}}(X) = TX + g_1({{\Lambda}}) X^q+\cdots + g_r({{\Lambda}})X^{q^r}, \mbox{ and}\leqno{(2.1.2)}$$ $\bullet$ the Eisenstein series $$E_k({{\Lambda}}) = \underset{{{\lambda}}\in {{\Lambda}}}{\sum}' {{\lambda}}^{-k} \quad (k \in {{\mathbb N}}).\leqno{(2.1.3)}$$ We further put $g_0({{\Lambda}}) = T$, $E_0({{\Lambda}}) = -1$. These are connected by $$e_{{{\Lambda}}}(Tz) = \phi_T^{{{\Lambda}}}(e_{{{\Lambda}}}(z)); \leqno{(2.1.4)}$$ $$\underset{i,j \geq 0 \atop i+j =k}{\sum} \alpha_i E_{q^j-1}^{q^{i}} = \underset{i+j=k}{\sum} \alpha_i^{q^{j}} E_{q^{j}-1} = 1
\mbox{ if } k = 0 \mbox{ and }
0 \mbox{ otherwise},\leqno{(2.1.5)}$$ which determines a number of further relations, see e.g. [@16] Sect. 2.
If ${{\Lambda}}= {{\Lambda}}_{{{\boldsymbol{\omega}}}} = \underset{1 \leq i \leq r}{\sum} A{{\omega}}_i$ with ${{\boldsymbol{\omega}}}= ({{\omega}}_1,\ldots,{{\omega}}_r) \in \Psi^r$, we use ${{\boldsymbol{\omega}}}$ instead of ${{\Lambda}}$ as the argument. Thus $\phi^{{{\boldsymbol{\omega}}}} = \phi^{{{\Lambda}}_{{{\boldsymbol{\omega}}}}}$, $e_{{{\boldsymbol{\omega}}}} = e_{{{\Lambda}}_{{{\boldsymbol{\omega}}}}}$, etc. As functions on $\Psi^r$, $g_i$, $\alpha_i$ are - like the Eisenstein series - holomorphic and ${{\Gamma}}$-invariant of weight $q^{i}-1$, while considered as functions on ${{\Omega}}^r$, $g_i$ (and $\alpha_i$) satisfies $$g_i({{\gamma}}{{\boldsymbol{\omega}}}) = {\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})^{q^{i}-1} g_i({{\boldsymbol{\omega}}})$$ (see Remark 1.14).
(2.1.6) The three systems of functions on $\Psi^r$: $\{g_1,\ldots,g_r\}$, $\{\alpha_1,\ldots,\alpha_r\}$, $\{E_{q^{i}-1}~|~~
1 \leq i \leq r\}$ are each algebraically independent, and the relations between them are such that the ring $${\bf Mod} = \bigoplus_{k\leq 0} {\bf Mod}_k = C_{\infty}[g_1,\ldots,g_r],$$ graded by the weight ${\rm wt}(g_i) := g^{i}-1$, may also be described as $$C_{\infty}[\alpha_1,\ldots \alpha_r] = C_{\infty}[\alpha_i~|~i\in {{\mathbb N}}] = C_{\infty}[E_{q^{i}-1}~|~1 \leq i \leq r] =
C_{\infty}[E_{q^{i}-1}~|~i\in {{\mathbb N}}].$$ (Actually [**Mod**]{} is the ring of modular forms of type 0 for ${{\Gamma}}$, see [@19].)
(2.1.7) As a consequence, since the $g_i$ and $\alpha_i$ may be expressed through Eisenstein series, they have strongly continuous extensions to $\overline{\Psi}^r$ and may therefore be evaluated on arbitrary points ${{\boldsymbol{\omega}}}= (U,i) \in \overline{\Psi}^r$.
(2.2) The Drinfeld modules $\phi^{{{\boldsymbol{\omega}}}}$ and $\phi^{{{\boldsymbol{\omega}}}'}$ (${{\boldsymbol{\omega}}}, {{\boldsymbol{\omega}}}' \in {{\Omega}}^r$) are isomorphic if and only if ${{\boldsymbol{\omega}}}' = {{\gamma}}{{\boldsymbol{\omega}}}$ with some ${{\gamma}}\in {{\Gamma}}$.
Hence the map $$\begin{array}{rll}
j:\: {{\Gamma}}\setminus {{\Omega}}^r & \hookrightarrow & {\rm Proj}\,{\bf Mod}\\
{{\boldsymbol{\omega}}}& \longmapsto & (g_1({{\boldsymbol{\omega}}}):\ldots: g_r({{\boldsymbol{\omega}}}))
\end{array}$$ identifies the quotient analytic space of ${{\Omega}}^r$ modulo ${{\Gamma}}$ with the complement of the vanishing locus of $\Delta := g_r$ in the weighted projective space $\overline{M}^r = {\rm Proj}\,{\bf Mod}$. (We remind the reader that we do not distinguish in notation between a $C_{\infty}$-variety, its associated analytic space, and the set of its $C_{\infty}$-points.) Here the $g_i$ are considered as formal variables of weight $q^{i}-1$, that is $(x'_1:\ldots:x'_r) = (x_1:\ldots:x_r)$ in ${\rm Proj}\,{\bf Mod}$ if and only if there exists $c \in C_{\infty}^*$ such that $x'_i = c^{q^{i}-1} x_i$ for all $i$. In other words, via $j$ $${{\Gamma}}\setminus {{\Omega}}^r \stackrel{\cong}{{{\longrightarrow}}} M^r := ({\rm Proj}\,{\bf Mod})_{(g_r \not= 0)}\leqno{(2.2.1)}$$ equals (the set of $C_{\infty}$-points of) the moduli scheme $M^r$ for Drinfeld $A$-modules of rank $r$ over $C_{\infty}$. The natural compactification of $M^r$ is $${\rm Proj}\,{\bf Mod} = \overline{M}^r = M^r \cup M^{r-1} \cup \cdots M^1, \leqno{(2.2.2)}$$ where for $1 \leq s \leq r$, $$({{\Gamma}}\cap P_s)\setminus {{\Omega}}_{V_s} = {\rm GL}(s,A)\setminus {{\Omega}}^s \stackrel{\cong}{{{\longrightarrow}}} M^s \quad
\mbox{(see (1.6))}$$ and ${{\Omega}}^1 = M^1 = \{\mbox{point}\}$. Hence the stratification of the variety $\overline{M}^r$ corresponds to that of $${{\Gamma}}\setminus \overline{{{\Omega}}}^r = {{\Gamma}}\setminus (\underset{1 \leq s \leq r \atop U \in \mathfrak U_s}
{\stackrel{\bullet}{\bigcup}} {{\Omega}}_U) = \underset{1 \leq s \leq r}{\stackrel{\bullet}{\bigcup}} {\rm GL}(s,A)\setminus {{\Omega}}^s
\leqno{(2.2.3)}$$ under the bijection $$\begin{array}{rll}
j:\: {{\Gamma}}\setminus \overline{{{\Omega}}}^r & \stackrel{\cong}{{{\longrightarrow}}} & \overline{M}^r \\
{{\boldsymbol{\omega}}}& \longmapsto & (g_1({{\boldsymbol{\omega}}}): \ldots : g_r({{\boldsymbol{\omega}}})),
\end{array} \leqno{(2.2.4)}$$ which is well-defined in view of (2.1.7).
In a similar way (although this looks a bit artificial), we may describe ${{\Gamma}}\setminus \Psi^r$ via
$$\begin{array}{rcl}
\tilde{j}:\: {{\Gamma}}\setminus \Psi^r & \hookrightarrow & {{\mathbb A}}^r (C_{\infty})\\
{{\boldsymbol{\omega}}}& \longmapsto & (g_1({{\boldsymbol{\omega}}}), \ldots , g_r({{\boldsymbol{\omega}}})),
\end{array} \leqno{(2.2.5)}$$ as the complement $\widetilde{M}^r$ of ($g_r=0$) in ${{\mathbb A}}^r$. It is the moduli scheme of rank-$r$ Drinfeld $A$-modules over $C_{\infty}$ with a “non-vanishing differential”, that is, with an identification of the underlying additive group with ${{\mathbb G}}_a$ or, what is the same, with explicit coefficients $g_i$ of its $T$-operator polynomial. The horizontal compactification ${{\Gamma}}\setminus \overline{\Psi}^r$ then becomes $$\begin{array}{lcl}
{{\Gamma}}\setminus \overline{\Psi}^r &= & {{\Gamma}}\setminus (\underset{1 \leq s \leq r \atop U \in \mathfrak U_s}
{\stackrel{\bullet}{\bigcup}} \Psi_U) = \underset{1 \leq s \leq r}{\stackrel{\bullet}{\bigcup}}
{\rm GL}(s,A)\setminus \Psi^s\\
& \underset{\widetilde{j}}{\stackrel{\cong}{{{\longrightarrow}}}} & \underset{1 \leq s \leq r}{\stackrel{\bullet}{\bigcup}}
\widetilde{M}^s =: \overline{\widetilde{M}}^r = C_{\infty}^r \setminus\{0\}, \end{array}
\leqno{(2.2.6)}$$ in analogy with (2.2.2), (2.2.3), (2.2.4).
(2.2.7) In the sequel, whenever writing ${{\Gamma}}\setminus {{\Omega}}^r = M^r$ or ${{\Gamma}}\setminus \Psi^r = \widetilde{M}^r$, the identification is via $j$ or $\widetilde{j}$, respectively.
[**2.3 Theorem.**]{} [*The map $j:\: {{\Gamma}}\setminus \overline{{{\Omega}}}^r \stackrel{\cong}{{{\longrightarrow}}} {\rm Proj}\,{\bf Mod} =
{\rm Proj}\,C_{\infty}[g_1,\ldots,g_r]$ of $(2.2.4)$ is a strong homeomorphism, i.e., with respect to the strong topologies on both sides. Similarly, $\widetilde{j}:\: {{\Gamma}}\setminus\overline{\Psi}^r \stackrel{\cong}{{{\longrightarrow}}} \overline{\widetilde{M}}^r$ is a strong homeomorphism.*]{}
The proof for $j$ will also show the statement for $\widetilde{j}$.
- By construction, $j$ is continuous as a map from $\overline{{{\Omega}}}^r$, and thus as a map from ${{\Gamma}}\setminus
\overline{{{\Omega}}}^r$ supplied with the quotient topology. Therefore we must show that $j^{-1}$ is continuous.
- Let $(\phi^{(n)})_{n\in {{\mathbb N}}}$ be a series of Drinfeld modules of rank $\leq r$, given by their $T$-division polynomials $\phi^{(n)}_T(X) = \underset{0 \leq i \leq r}{\sum} g_i^{(n)} X^{q^{i}}$ and converging to $\phi$ with $\phi_T(X) = \sum g_i X^{q^{i}}$. This means that ${{\boldsymbol{g}}}^{(n)} = (g_1^{(n)}: \ldots : g_r^{(n)})$ converges to ${{\boldsymbol{g}}}= (g_1:\ldots:g_r)$. Let $s$ be the rank of $\phi$, i.e., $g_s \not= 0$, $g_{s+1} = \cdots = g_r = 0$. We may suppose that $g_s = \underset{n\to \infty}{\lim} g_s^{(n)} = 1$. Let ${{\Lambda}}^{(n)}$ (resp. ${{\Lambda}}$) be the lattice associated to $\phi^{(n)}$ (resp. $\phi$), each provided with an SMB $\{{{\omega}}_r^{(n)},{{\omega}}_{r-1}^{(n)},\ldots,{{\omega}}_1^{(n)}\}$ (resp. $\{{{\omega}}_r,\ldots,{{\omega}}_{r-s+1}\}$), where we have put ${{\omega}}_i^{(n)} = 0$ for $i \leq r-{\rm rk}(\phi^{(n)})=r-{\rm rk}_A({{\Lambda}}^{(n)})$. Put ${{\boldsymbol{\omega}}}:= (0:\ldots:0:{{\omega}}_{r-s+1}:\ldots:{{\omega}}_r)$ and ${{\boldsymbol{\omega}}}^{(n)} := ({{\omega}}_1^{(n)}: \cdots: {{\omega}}_r^{(n)})$, and let $[{{\boldsymbol{\omega}}}]$ resp. $[{{\boldsymbol{\omega}}}^{(n)}]$ be the corresponding class in ${{\Gamma}}\setminus \overline{{{\Omega}}}^r$. Then we must show that $\underset{n\to \infty}{\lim}[{{\boldsymbol{\omega}}}^{(n)}] = [{{\boldsymbol{\omega}}}]$. Note that we suppress here our usual assumption ${{\omega}}_r= 1$, which would conflict with the normalization $g_s = \underset{n\to \infty}{\lim} g_s^{(n)} = 1$.
- If $s=r$, we are done. This follows from the fact that $j:\: {{\Gamma}}\setminus {{\Omega}}^r \stackrel{\cong}{{{\longrightarrow}}} M^r$ is an isomorphism of analytic spaces, thus a strong homeomorphism. Hence we may suppose that $s <r$.
- Consider the Newton polygon ${\rm NP}(\phi)$ of $\phi_T$, i.e., the lower convex hull of the vertices $(q^{i},v(g_i))$ with $0 \leq i \leq s$ in the plane (see [@30] II Sect. 6). If ${\rm rk}(\phi^{(n)})\leq s$ for $n\gg 0$, then in fact ${\rm rk}(\phi^{(n)}) = s$ for $n \gg 0$, and we are ready as in (iii). Therefore, possibly restricting to a subsequence, we may assume that ${\rm rk}(\phi^{(n)}) >s$ for $n \gg 0$. Then if $\phi^{(n)}$ is sufficiently close to $\phi$, the Newton polygon ${\rm NP}(\phi^{(n)})$ agrees with ${\rm NP}(\phi)$ from the leftmost vertex $(1,-1)$ up to $(q^s,0)$ and, since $g_{s+1}^{(n)}, \ldots,g_r^{(n)}$ tend to zero:\
(2.3.1) The slope of ${\rm NP}(\phi^{(n)})$ right to $(q^s,0)$ tends to infinity if $n \to \infty$.
- Considering [@18] (3.3), (3.4), (3.5), the assertion (2.3.1) implies that the quotient $|{{\omega}}_{r-s}^{(n)}|/|{{\omega}}_{r-s+1}^{(n)}|$ (i.e., the quotient of absolute values of the $(s+1)$-th divided by the $s$-the element of our SMB $\{{{\omega}}_r^{(n)},{{\omega}}_{r-1}^{(n)},\ldots\}$) tends to infinity with $n\to \infty$.
- Let ${^s\phi}^{(n)}$ be the rank-$s$ Drinfeld module that corresponds to the lattice ${^s{{\Lambda}}}^{(n)} = A{{\omega}}_r^{(n)} + \cdots+ A{{\omega}}_{r-s+1}^{(n)}$, with $${^s\phi}_T^{(n)}(X) = \sum_{0 \leq i \leq s} s_{g_i}^{(n)} X^{q^{i}},\: {^s{{\boldsymbol{g}}}}^{(n)} :=
({^sg_1}^{(n)}: \ldots:{^sg_s}^{(n)}:0: \ldots:0).$$ Then $\underset{n\to \infty}{\lim} (g_i^{(n)} - {^sg}_i^{(n)}) = 0$, as follows from (v). (The analogous statement for the Eisenstein series $E_{q^{i}-1}$, to wit $$\lim_{n\to \infty}(E_{q^{i}-1}({{\Lambda}}^{(n)})-E_{q^{i}-1}({^s{{\Lambda}}}^{(n)})) = 0,$$ is obvious; then we use the fact (2.1.6) that the $g_i$ are polynomials in the $E_k$.)
As $g_i^{(n)} {{\longrightarrow}}g_i$ for $1 \leq i \leq r$, we find ${^s{{\boldsymbol{g}}}}^{(n)} {{\longrightarrow}}{{\boldsymbol{g}}}$ and therefore ${^s\phi}^{(n)} {{\longrightarrow}}\phi$ in $\overline{M}^s = {\rm Proj}\, C_{\infty}[g_1,\ldots,g_s]$ with respect to the strong topology.
- Denote by ${^s{{\boldsymbol{\omega}}}}^{(n)}$ the point $(0:\ldots:0:{{\omega}}_{r-s+1}^{(n)}:\ldots:{{\omega}}_r^{(n)})$ in ${{\Omega}}_{V_s} \hookrightarrow \overline{{{\Omega}}}^r$. Applying (iii) with $r$ replaced by $s$ and using the identification ${{\Omega}}^s \stackrel{\cong}{{{\longrightarrow}}} {{\Omega}}_{V_s}$, $$\lim_{n\to \infty} [{^s{{\boldsymbol{\omega}}}}^{(n)}] = [{{\boldsymbol{\omega}}}]\leqno{(2.3.2)}$$ holds in ${\rm GL}(s,A)\setminus {{\Omega}}_{V_s} \hookrightarrow {{\Gamma}}\setminus \overline{{{\Omega}}}^r$, where $[~.~ ]$ is the class modulo ${{\Gamma}}$. But (2.3.2) together with (v) means that $[{{\boldsymbol{\omega}}}^{(n)}]$ tends strongly to $[{{\boldsymbol{\omega}}}]$ in ${{\Gamma}}\setminus \overline{{{\Omega}}}^r$.
(2.4) We want to give similar descriptions for the quotients ${{\Gamma}}'\setminus \overline{{{\Omega}}}^r$ and ${{\Gamma}}'\setminus \overline{\Psi}^r$, where ${{\Gamma}}' \subset {{\Gamma}}$ is a congruence subgroup and $r \geq 2$. This turns out, however, to be much more difficult. We restrict to deal with the case where ${{\Gamma}}' = {{\Gamma}}(N)$, the full congruence subgroup of level $N$.
Fix a monic $N\in A$ of degree $d \geq 1$, and write the $N$-th division polynomial of the Drinfeld module $\phi^{{{\boldsymbol{\omega}}}}$ (${{\boldsymbol{\omega}}}\in \Psi^r$) as $$\phi_N^{^{{\boldsymbol{\omega}}}}(X) = \sum_{0 \leq i \leq rd} \ell_i(N,{{\boldsymbol{\omega}}})X^{q^{i}}$$ with $\ell_0(N,{{\boldsymbol{\omega}}}) = N$, $\ell_{rd}(N,{{\boldsymbol{\omega}}}) = \Delta({{\boldsymbol{\omega}}})^{(q^{rd}-1)/(q^r-1)}$, where $\Delta({{\boldsymbol{\omega}}}) = g_r({{\boldsymbol{\omega}}})$ is the discriminant function and, more generally, all the coefficient functions $\ell_i(N,.)$ lie in ${\bf Mod} = C_{\infty}[g_1,\ldots,g_r]$. It satisfies $$\phi_N^{{{\boldsymbol{\omega}}}}(X) = \Delta({{\boldsymbol{\omega}}})^{(q^{rd}-1)/(q^r-1)} \prod_{({{\boldsymbol{u}}}\in N^{-1}A/A)^r}(X-e_{{{\boldsymbol{\omega}}}}({{\boldsymbol{u}}}{{\boldsymbol{\omega}}})).\leqno{(2.4.1)}$$ Here ${{\boldsymbol{u}}}$ runs through a system of representatives of the finite $A$-module $(N^{-1}A/A)^r$ and ${{\boldsymbol{u}}}{{\boldsymbol{\omega}}}=
\underset{1 \leq i \leq r}{\sum} u_i{{\omega}}_i$. That is, the $$d_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) := e_{{{\boldsymbol{\omega}}}} ({{\boldsymbol{u}}}{{\boldsymbol{\omega}}}) \leqno{(2.4.2)}$$ are the $N$-division points of $\phi^{{{\boldsymbol{\omega}}}}$. It is known (see [@23] Proposition 2.7 or [@13] 3.3.5) that for ${{\boldsymbol{u}}}\not= 0$, $$d_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) = E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})^{-1} \leqno{(2.4.3)}$$ with the partial Eisenstein series of weight 1 (see (1.12)) $$E_{{{\boldsymbol{u}}}} := E_{1,{{\boldsymbol{u}}}}.\leqno{(2.4.4)}$$ We draw the conclusions
(2.4.5) $E_{{{\boldsymbol{u}}}}$ never vanishes on $\Psi^r$ and ${{\Omega}}^r$;
(2.4.6) The coefficient $\ell_i(N,{{\boldsymbol{\omega}}})$ may be expressed as a homogeneous polynomial in the $E_{{{\boldsymbol{u}}}}$ (${{\boldsymbol{u}}}\not= 0$); more precisely, $$\ell_i(N,{{\boldsymbol{\omega}}}) = Ns_{q^{i}-1} (E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})~|~0 \not= {{\boldsymbol{u}}}\in (N^{-1}A/A)^r),$$ where $s_k$ is the $k$-th elementary symmetric polyomial.
We let ${{\mathcal T}}(N)$ be the index set $${{\mathcal T}}(N) := (N^{-1}A/A)^r \setminus \{0\}.\leqno{(2.4.7)}$$ (2.5) As the definition of fields of meromorphic functions on non-complete analytic spaces requires some boundary conditions, we make the following [*ad hoc*]{} definitions. They are motivated from the fact that the analytic spaces $\widetilde{M}^r$, $\widetilde{M}^r(N)$, $M^r$, $M^r(N)$ appearing below are actually $C_{\infty}$-varieties (see (2.2.1), (2.2.5) and Remark 2.7) and that by GAGA the algebraic and the analytic function fields of projective $C_{\infty}$-varieties agree.
(2.5.1) The function field of $\widetilde{M}^r = {{\Gamma}}\setminus \Psi^r$ is $$\widetilde{{{\mathcal F}}}_r = \widetilde{{{\mathcal F}}}(1) := C_{\infty}(g_1,\ldots,g_r);$$ (2.5.2) The function field of $\widetilde{M}^r(N):= {{\Gamma}}(N) \setminus \Psi^r$ is $\widetilde{{{\mathcal F}}}_r(N)$, the field of those meromorphic functions on $\widetilde{M}^r(N)$ which are algebraic over $\widetilde{{{\mathcal F}}}_r$;
(2.5.3) The function field of $M^r = {{\Gamma}}\setminus {{\Omega}}^r$ is $${{\mathcal F}}_r = {{\mathcal F}}_r(1) := C_{\infty}(g_1,\ldots,g_r)_0,$$ the subfield of isobaric elements of weight 0 of $\widetilde{{{\mathcal F}}}_r$;
(2.5.4) The function field of $M^r(N):= {{\Gamma}}(N)\setminus {{\Omega}}^r$ is ${{\mathcal F}}_r(N)$, the field of meromorphic functions on $M^r(N)$ algebraic over ${{\mathcal F}}_r$.
[**2.6 Proposition.**]{}
**
- The field $\widetilde{{{\mathcal F}}}_r(N)$ is generated over $C_{\infty}$ by the Eisenstein series $E_{{{\boldsymbol{u}}}} = E_{1,{{\boldsymbol{u}}}}$ (${{\boldsymbol{u}}}\in {{\mathcal T}}(N)$). It is galois over $\widetilde{{{\mathcal F}}}_r$ with Galois group $$\widetilde{G}(N) := \{{{\gamma}}\in {\rm GL}(r,A/M)~|~\det {{\gamma}}\in {{\mathbb F}}^*\}.$$
- The field ${{\mathcal F}}_r(N)$ is generated over $C_{\infty}$ by the functions $E_{{{\boldsymbol{u}}}}/E_{{{\boldsymbol{v}}}}$ (${{\boldsymbol{u}}},{{\boldsymbol{v}}}\in {{\mathcal T}}(N)$). It is galois over ${{\mathcal F}}_r$ with group $G(N):= \widetilde{G}(N)/Z$. Here $Z \cong {{\mathbb F}}^*$ is the subgroup of $\widetilde{G}(N)$ of scalar matrices with entries in ${{\mathbb F}}^*$.
<!-- -->
- As ${{\Gamma}}$ acts without fixed points on $\Psi^r$, $\widetilde{M}^r(N) = {{\Gamma}}(N)\setminus \Psi^r$ is an étale Galois cover of $\widetilde{M}^r = {{\Gamma}}\setminus \Psi^r$ with group ${{\Gamma}}/{{\Gamma}}(N) \stackrel{\cong}{{{\longrightarrow}}} \widetilde{G}(N)$. Now $E_{{{\boldsymbol{u}}}}$ is ${{\Gamma}}(N)$-invariant and, as (2.4.1) and (2.4.3) show, algebraic over $\widetilde{F}_r$, i.e., $E_{{{\boldsymbol{u}}}} \in \widetilde{{{\mathcal F}}}_r(N)$. Furthermore, the relation $E_{{{\boldsymbol{u}}}}({{\gamma}}{{\boldsymbol{\omega}}}) = E_{{{\boldsymbol{u}}}{{\gamma}}}({{\boldsymbol{\omega}}})$ for ${{\boldsymbol{\omega}}}\in \Psi^r$, ${{\gamma}}\in {{\Gamma}}$ implies ${{\gamma}}= 1$ if ${{\gamma}}\in \widetilde{G}(N)$ fixes all the $E_{{{\boldsymbol{u}}}}$. Therefore, $\widetilde{{{\mathcal F}}}_r(N) = \widetilde{{{\mathcal F}}}_r(E_{{{\boldsymbol{u}}}}~|~{{\boldsymbol{u}}}\in {{\mathcal T}}(N))$ by Galois theory. In view of (2.4.6) the coefficient functions $\ell_i(N,.)$ and therefore (by the well-known commutation relations between the $g_i(~.~)$ and the $\ell_i(N,.)$) also the $g_i$ are polynomials in the $E_{{{\boldsymbol{u}}}}$. Thus in fact $\widetilde{{{\mathcal F}}}_r(N) = C_{\infty}(E_{{{\boldsymbol{u}}}}~|~{{\boldsymbol{u}}}\in {{\mathcal T}}(N))$.
- The argument for ${{\mathcal F}}_r(N)$ is similar. The quotient $G(N) = \widetilde{G}(N)/Z$ by $Z$ as a Galois group comes from the fact that $Z$ acts trivially on ${{\Omega}}^r$.
[**Remark.**]{} By (2.4.3) we may also write $\widetilde{F}_r(N) = C_{\infty}(d_{{{\boldsymbol{u}}}}~|~{{\boldsymbol{u}}}\in {{\mathcal T}}(N))$ and ${{\mathcal F}}_r(N) =
C_{\infty}(d_{{{\boldsymbol{u}}}}/d_{{{\boldsymbol{v}}}}~|~{{\boldsymbol{u}}},{{\boldsymbol{v}}}\in {{\mathcal T}}(N))$.
[**2.7 Remark.**]{} As is well known, the smooth analytic space $M^r(N)={{\Gamma}}(N)\setminus {{\Omega}}^r$ is strongly related with the moduli scheme $M^r(N)/K$ of Drinfeld $A$-modules of rank $r$ with a structure of level $N$ ([@11], [@10], [@15]). Let $K(N) \subset C_{\infty}$ be the field extension of $K$ generated by the $N$-division points of the Carlitz module. Then $K(N)/K$ is finite abelian with group $(A/N)^*$ and ramification properties similar to those of cyclotomic extensions of $\mathbb Q$ [@26]. Let $K_+(N) \subset K(N)$ be the fixed field of ${{\mathbb F}}^* \hookrightarrow (A/N)^*$, the “maximal real subextension” of $K(N)|K$. Then $K_+(N)$ is contained in ${{\mathcal K}}_r(N)=K(E_{{{\boldsymbol{u}}}}/E_{{{\boldsymbol{v}}}}~|~{{\boldsymbol{u}}},{{\boldsymbol{v}}}\in {{\mathcal T}}(N))$, and is actually the algebraic closure of $K$ in ${{\mathcal K}}_r(N)$. Now $M^r(N)/K$ is a smooth $K$-scheme with function field ${{\mathcal K}}_r(N)$, whose set of $C_{\infty}$-points (in fact, its analytification over $C_{\infty}$) is given by $$(M^r(N)/K)(C_{\infty}) \stackrel{\cong}{{{\longrightarrow}}} \underset{\sigma}{\stackrel{\bullet}{\bigcup}}M^r(N)_{\sigma} =
\underset{\sigma}{\stackrel{\bullet}{\bigcup}} ({{\Gamma}}(N)\setminus {{\Omega}}^r)_{\sigma},$$ where $\sigma$ runs through the set of $K$-embeddings of $K_+(N)$ into $C_{\infty}$, i.e., the Galois group ${\rm Gal}(K_+(N)|K) = (A/N)^*/{{\mathbb F}}^*$. Correspondingly, the analytification of $M^r(N)/K \underset{K_+(N)}{\times} C_{\infty}$ is $M^r(N)$, which justifies our notation $M^r(N)$ for ${{\Gamma}}(N)\setminus {{\Omega}}^r$. In the language of pre-Grothendieck algebraic geometry, $M^r(N)/K$ is “defined over $K_+(N)$”.
The group ${\rm GL}(r,A/N)$ acts naturally on the set of $N$-level structures of a fixed Drinfeld module of rank $r$, thus on $M^r(N)/K$, which identifies $M^r/K = ({\rm Proj}\,K[g_1,\ldots,g_r])_{(g_r\not=0)}$ with the quotient of $M^r(N)/K$ by this group. Moreover, the action is compatible with that of $G(N) = {{\Gamma}}/{{\Gamma}}(N)Z \hookrightarrow {\rm GL}(r,A/N)/Z$ on the components $M^r(N)_{\sigma}$. All of this may be transferred to the spaces $\widetilde{M}^r(N) = {{\Gamma}}(N)\setminus \Psi^r$ and their function fields $\widetilde{{{\mathcal F}}}_r(N)$. As we don’t really need it, we omit the details.
In the sequel of the paper, we will construct a compactification $\overline{M}^r(N)$ of $M^r(N)$ (and, similarly, a horizontal compactification of $\widetilde{M}^r(N)$), i.e., a projective $C_{\infty}$-variety $\overline{M}^r(N)$ with set of $C_{\infty}$-points $\overline{M}^r(N)(C_{\infty}) = {{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$, into which $M^r(N)$ embeds as a dense open subvariety, and compatible with the above-described group actions.
[**3. The boundary components**]{}
From now on, we assume that $r \geq 2$.
The set $\mathfrak U_s$ of $s$-dimensional subspaces $U$ of $V = K^r$ is in canonical bijection with ${\rm GL}(r,K)/P_s(K)$ through $$\begin{array}{rcl}
{\rm GL}(r,K)/P_s(K) & \stackrel{\cong}{{{\longrightarrow}}} & \mathfrak U_s.\\
{{\gamma}}& \longmapsto & V_s{{\gamma}}^{-1}
\end{array}\leqno{(3.1)}$$
As the action of ${{\Gamma}}$ on $\mathfrak U_s$ is transitive, we may replace the left hand side with ${{\Gamma}}/ {{\Gamma}}\cap P_s(K)$. Let $(A/N)^r_{\rm prim}$ be the set of primitive elements of $(A/N)^r$, that is, of elements that belong to a basis of the free $(A/N)$-module $(A/N)^r$. Then, as is easily verified, the map $${{\Gamma}}(N) \setminus {{\Gamma}}/{{\Gamma}}\cap P_{r-1}(K) {{\longrightarrow}}(A/N)^r_{\rm prim}/{{\mathbb F}}^* =: {{\mathcal C}}_r(N)\leqno{(3.2)}$$ that associates with the double class of ${{\gamma}}\in {{\Gamma}}$ the first column of ${{\Gamma}}$ (evaluated modulo $N$, and modulo the scalar action of ${{\mathbb F}}^*$) is well-defined and bijective. Together with (3.1) we find that the space of orbits on $\mathfrak U_{r-1}$ of ${{\Gamma}}(N)$ is $${{\Gamma}}(N)\setminus \mathfrak U_{r-1} \stackrel{\cong}{{{\longrightarrow}}} {{\mathcal C}}_r(N).\leqno{(3.3)}$$ This allows us to describe the components of codimension 1 of $${{\Gamma}}(N)\setminus \overline{\Psi}^r = \underset{1 \leq s \leq r}{\stackrel{\bullet}{\bigcup}} \:\:
\underset{U\in {{\Gamma}}(N)\setminus\mathfrak U_s}{\stackrel{\bullet}{\bigcup}} {{\Gamma}}_U(N)\setminus \Psi_U \leqno{(3.4)}$$ and, analogously, of ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$. Here ${{\Gamma}}_U = \{{{\gamma}}\in {{\Gamma}}~|~U {{\gamma}}= U\}$, which acts from the left on $\Psi_U$ (see (1.5)), and ${{\Gamma}}_U(N) := {{\Gamma}}_U \cap {{\Gamma}}(N)$. We put $$\widetilde{M}_U(N) := {{\Gamma}}_U(N)\setminus \Psi_U \mbox{ and } M_U(N) := {{\Gamma}}_U(N)\setminus {{\Omega}}_U \leqno{(3.5)}$$ and call the components $\widetilde{M}_U(N)$ resp. $M_U(N)$ with $\dim(U) = r-1$ the [*cuspidal divisors*]{} or simply the [*cusps*]{} of ${{\Gamma}}(N)\setminus \overline{\Psi}^r$ (or ${{\Gamma}}(N) \setminus \overline{{{\Omega}}}^r$, or of ${{\Gamma}}(N)$). Each of these sets is in canonical bijection with ${{\mathcal C}}_r(N)$. For later use we specify a system of representatives, namely the set $S$ of monic elements of $(A/N)^r_{\rm prim}$. Here, some ${{\boldsymbol{n}}}= (n_1,\ldots,n_r) \in (A/N)^r_{\rm prim}$ is [*monic*]{} if the first non-vanishing $n_i$ has a monic representative $n'_i \in A$ of degree less than $d = \deg\,N$.
The cardinality $c_r(N)$ of ${{\mathcal C}}_r(N)$ is an easy arithmetic function of $N$ and $r$, given by the following formula.
[**3.6 Lemma.**]{} [*Let $N = \underset{1\leq i \leq t}{\prod} {{\frak p}}_i^{s_i}$ be the decomposition of $N$ into powers of different primes ${{\frak p}}_i$ of $A$. Write $q_i=q^{\deg\,{{\frak p}}_i}$. Then $$c_r(N) = (q-1)^{-1} \prod_{1 \leq i \leq t} (q_i^r-1) q_i^{(s_i-1)r}.$$*]{}
We must determine $\#(A/N)^r_{\rm prim}$, which by the Chinese Remainder Theorem is multiplicative. So we may assume that $t=1$, $N = {{\frak p}}_1^{s_1}$ with some prime ${{\frak p}}_1$, $q_1 = q^{\deg\,{{\frak p}}_1}$. Then $$\#(A/N)^r_{\rm prim} = \#(A/{{\frak p}}_1)_{\rm prim}^r \#({{\frak p}}_1/{{\frak p}}_1^{s_1})^r,$$ as some element of $(A/N)^r$ is primitive if and only if its reduction $\bmod\,{{\frak p}}_1$ is. Now $\#(A/{{\frak p}}_1)^r_{\rm prim} =
q_1^r-1$ and $\#({{\frak p}}_1/{{\frak p}}_1^{s_1})^r = q_1^{(s_1-1)r}$, and we are done.
In the case of smaller dimension $s < r-1$ we get a similar description of ${{\Gamma}}(N)\setminus \mathfrak U_s$, which is in $1-1$-correspondence with the set of $(r-s)$-subsets of $(A/N)^r$ that are part of an $(A/N)$-basis of $(A/N)^r$, modulo the action of the group $\{{{\gamma}}\in {\rm GL}(r-s,A/N)~|~\det\,{{\gamma}}\in {{\mathbb F}}^*\}$. We leave the details to the reader, as we will only need the case $s=1$. Here, likewise, $${{\Gamma}}(N)\setminus \mathfrak U_1 \stackrel{\cong}{{{\longrightarrow}}} {{\Gamma}}(N)\setminus {{\Gamma}}/{{\Gamma}}\cap P_1(K) \stackrel{\cong}{{{\longrightarrow}}} {{\mathcal C}}_r(N),
\leqno{(3.7)}$$ where the double class of ${{\gamma}}\in {{\Gamma}}$ is mapped to the last row vector of ${{\gamma}}$ (reduced modulo $N$, and modulo the action of ${{\mathbb F}}^*$). In particular, $$\#({{\Gamma}}(N)\setminus \mathfrak U_1) = \#{{\mathcal C}}_r(N) = c_r(N),$$ which of course could also be seen via the duality of projective spaces over the finite ring $A/N$.
[**4. Behavior of Eisenstein series at the boundary.**]{}
In the whole section, $N$ is a fixed monic element of $A$ of degree $d \geq 1$.
(4.1) Let ${{\boldsymbol{u}}}$ be an element of ${{\mathcal T}}(N) = (N^{-1}A/A)^r \setminus \{0\}$. We start with the relation from (1.12) $$E_{k,{{\boldsymbol{u}}}}({{\gamma}}(U,i)) = E_{k,{{\boldsymbol{u}}}{{\gamma}}}(U,i)\leqno{(4.1.1)}$$ for ${{\gamma}}\in {{\Gamma}}$, $(U,i) \in \overline{\Psi}^r$. Suppose that $U = V_s \cdot {{\gamma}}^{-1}$ with some $1 \leq s < r$ and ${{\gamma}}\in {{\Gamma}}$. Now we read off from (4.1.1):
(4.1.2) The vanishing behavior of $E_{k,{{\boldsymbol{u}}}}$ around the boundary component $\Psi_U$ is the same as the behavior of $E_{k,{{\boldsymbol{u}}}{{\gamma}}}$ around the standard component $\Psi_{V_s}$.
We say that
(4.1.3) ${{\boldsymbol{u}}}$ [*belongs to*]{} $U$ if ${{\boldsymbol{u}}}\in {{\mathcal T}}(N) \subset K^r/A^r = V/L_V$ is represented by an element of $U \subset V$.
In view of ${{\boldsymbol{u}}}{{\gamma}}= {{\boldsymbol{u}}}$ for ${{\gamma}}\in {{\Gamma}}(N)$, this property depends only on the ${{\Gamma}}(N)$-orbit of $U$.
[**4.2 Proposition.**]{}
**
- Suppose that ${{\boldsymbol{u}}}$ does not belong to $U$. Then $E_{k,{{\boldsymbol{u}}}}$ vanishes identically on $\Psi_{U}$.
- If ${{\boldsymbol{u}}}$ belongs to $U$, then $E_{k,{{\boldsymbol{u}}}}$ restricts to $\Psi_U \cong \Psi^s$ like an Eisenstein series $E_{k,{{\boldsymbol{u}}}'}$ of rank $s$, ${{\boldsymbol{u}}}' \in {{\mathcal T}}_s(N) = (N^{-1}A/A)^s \setminus \{0\}$. In particular, it doesn’t vanish identically on $\Psi_U$.
In view of (4.1.2), it suffices to verify the assertions for $U=V_s$. Suppose that ${{\boldsymbol{u}}}= (u_1,\ldots,u_r)$ does not belong to $V_s$, that is, $u_i \not= 0$ for some $i$ with $1 \leq i \leq r-s$. Let ${{\boldsymbol{\omega}}}= ({{\omega}}_1,\ldots,{{\omega}}_r)$ be an element of the fundamental domain $\widetilde{{{\boldsymbol{F}}}}$ described in (1.15). We have $$E_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) = N^{-k} \sum_{{{\boldsymbol{a}}}\in A^r \atop {{\boldsymbol{a}}}\equiv N{{\boldsymbol{u}}}(\bmod N)} (a_1 {{\omega}}_1+ \cdots + a_r{{\omega}}_r)^{-k} =
N^{-k} \sum_{{{\boldsymbol{a}}}} ({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k}.$$ In each term, $a_i \not= 0$, which by (1.8.1) forces that $({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k}$ tends to zero, uniformly in the ${{\boldsymbol{a}}}$, if ${{\boldsymbol{\omega}}}$ approaches $\Psi_{V_s}$. That is, $E_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) {{\longrightarrow}}0$, and $E_{k,{{\boldsymbol{u}}}} \equiv 0$ on $\Psi_{V_s}$. Suppose that ${{\boldsymbol{u}}}$ belongs to $V_s$. As before, each term $({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k}$ tends to zero uniformly in ${{\boldsymbol{a}}}$, as long as at least one of $a_1,\ldots,a_{r-s} \not=0$. Therefore, $\lim\,E_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) = E_{k,{{\boldsymbol{u}}}'}({{\boldsymbol{\omega}}}')$ with ${{\boldsymbol{u}}}' = (u_{r-s+1},\ldots ,u_r)$ if ${{\boldsymbol{\omega}}}$ tends to $(0,\ldots,0,{{\omega}}'_{r-s+1},\ldots,{{\omega}}'_r) = (0,\ldots,0,{{\boldsymbol{\omega}}}')$.
We define the space $${\rm Eis}_k(N) := \sum_{{{\boldsymbol{u}}}\in (N^{-1}A/A)^r} C_{\infty} E_{k,{{\boldsymbol{u}}}}\leqno{(4.3)}$$ of Eisenstein series of weight $k$ and level $N$.
[**4.4 Lemma.**]{} [ *The vector space ${\rm Eis}_k(N)$ is generated by the $E_{k,{{\boldsymbol{u}}}}$ with ${{\boldsymbol{u}}}$ primitive of level $N$ (i.e., $N'{{\boldsymbol{u}}}\not= 0$ for each proper divisor $N'$ of $N$) and even by $E_{k,{{\boldsymbol{u}}}}$ (${{\boldsymbol{u}}}\in N^{-1}S$), where $S$ is the set of representatives for ${{\mathcal C}}_r(N) = (A/N)^r_{\rm prim}/{{\mathbb F}}^*$ given in $(3.5)$.*]{}
Let $N'$ be a monic divisor of $N$, where $N' = 1$ is allowed. Then the distribution relation $$(N/N')^k \sum_{(N/N'){{\boldsymbol{u}}}= {{\boldsymbol{v}}}} E_{k,{{\boldsymbol{u}}}} = E_{k,{{\boldsymbol{v}}}}\leqno{(4.4.1)}$$ holds, where ${{\boldsymbol{v}}}\in (N'^{-1}A/A)^r$ and $E_{k,0} = E_k$ is the Eisenstein series without level. It shows that Eisenstein series of lower level $N'$ may be expressed as linear combinations of those with level $N$. The result now follows from $$E_{k,c{{\boldsymbol{u}}}} = c^{-k}E_{k,{{\boldsymbol{u}}}} \quad (c \in {{\mathbb F}}^*), \leqno{(4.4.2)}$$ which is immediate from the definition of $E_{k,{{\boldsymbol{u}}}}$.
(4.5) We will show that these are all the relations between Eisenstein series of weight $k$, following the strategy of Hecke in [@27], which has been introduced to the function field situation in the case of $r=2$ by Cornelissen [@9]. Let
$$F_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) := N^{-k} \sum_{{{\boldsymbol{a}}}\in A^r\,{\rm primitive} \atop {{\boldsymbol{a}}}\equiv N {{{\boldsymbol{u}}}}\,(\bmod N)} ({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k}
\leqno{(4.5.1)}$$ be the partial sum of $E_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})$ with primitive ${{\boldsymbol{a}}}$, i.e., where ${{\boldsymbol{a}}}= (a_1,\ldots,a_r)$ satisfies $\underset{1 \leq i \leq r}{\sum} A a_i = A$. Then:
(4.5.2) The [*restricted Eisenstein series*]{} $F_{k,{{\boldsymbol{u}}}}$ is well-defined as a function on $\Psi^r$ of weight $k$ and invariant under ${{\Gamma}}(N)$. Like $E_{k,{{\boldsymbol{u}}}}$, it satisfies the functional equation $$F_{k,{{\boldsymbol{u}}}}({{\gamma}}{{\boldsymbol{\omega}}}) = F_{k,{{\boldsymbol{u}}}{{\gamma}}}({{\boldsymbol{\omega}}})$$ under ${{\gamma}}\in {{\Gamma}}$.
Let $\mu:\,A {{\longrightarrow}}\{0,\pm 1\}$ be the Möbius function:\
$\mu(a) = (-1)^n \mbox{ if } a = \epsilon \underset{1 \leq j \leq n}{\prod} {{\frak p}}_j$ with $n$ different monic primes ${{\frak p}}_j$ of $A$ and $\epsilon \in {{\mathbb F}}^*$, and zero otherwise. (As the empty product evaluates to 1, $\mu(a) = 1$ if $a \in {{\mathbb F}}^*$.) Then $\underset{b\,{\rm monic},\, b|a}{\sum} \mu(b) = 1$ if $a \in {{\mathbb F}}^*$ and 0 otherwise, and the usual formalism holds. Möbius inversion yields $$F_{k,{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) = \sum_{t\in(A/N)^*} \sum_{a\in A\,{\rm monic}\atop
at \equiv 1(\bmod N)} \mu(a)a^{-k} E_{k,t{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}). \leqno{(4.5.3)}$$ In particular, $F_{k,{{\boldsymbol{u}}}}$ lies in ${\rm Eis}_k(N)$, and so has a strongly continuous extension to $\overline{\Psi}^r$. We deduce that $$\dim({\rm Eis}_k(N)) \geq \dim(\sum_{{{\boldsymbol{n}}}\in {{\mathcal T}}(N)} C_{\infty} F_{k,{{\boldsymbol{u}}}}). \leqno{(4.5.4)}$$ Recall that by (3.7) the set of 1-dimensional boundary components of ${{\Gamma}}(N)\setminus \overline{\Psi}^r$ is in $1-1$-correspondence with ${{\mathcal C}}_r(N)$, or with its set $S$ of representatives in (3.5). We let $\Psi_{{{\boldsymbol{n}}}}^1 \cong \Psi^1$ be the component corresponding to ${{\boldsymbol{n}}}\in S$.
[**4.6 Proposition.**]{}
**
- Given ${{\boldsymbol{n}}}\in S$ there exists a unique ${{\boldsymbol{n}}}' \in S$ such that $F_{k,{{\boldsymbol{n}}}'/N}$ doesn’t vanish at $\Psi_{{{\boldsymbol{n}}}}^1$.
- The rule ${{\boldsymbol{n}}}{{\longrightarrow}}{{\boldsymbol{n}}}'$ establishes a permutation of $S$.
First we note that $\Psi_{{{\boldsymbol{n}}}}^1$, where ${{\boldsymbol{n}}}= (0,\ldots,0,1)$, equals $\Psi_{V_1}$. In view of (4.5.2) and the transitivity of ${{\Gamma}}$ on the set $\{\Psi_{{{\boldsymbol{n}}}}^1~|~{{\boldsymbol{n}}}\in S\}$, it is enough to show that there is a unique ${{\boldsymbol{n}}}'$ such that $F_{k,{{\boldsymbol{n}}}'/N}$ doesn’t vanish at $\Psi_{V_1}$.
Consider a term $({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k} = (a_1{{\omega}}_1+ \cdots+ a_r{{\boldsymbol{\omega}}}_r)^{-k}$ of $N^kF_{k,{{\boldsymbol{n}}}'/N}$, where $\gcd(a_1,\ldots,a_r) = 1$ and ${{\boldsymbol{\omega}}}\in \tilde{{{\boldsymbol{F}}}}$ as in the proof of (4.2). If one of $a_1,\ldots,a_{r-1}$ doesn’t vanish then $\lim({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k} = 0$, uniformly in ${{\boldsymbol{a}}}$, if ${{\boldsymbol{\omega}}}$ tends to $(0,\ldots,0,{{\omega}}_r)$. Hence $$\lim F_{k,{{\boldsymbol{n}}}'/N}({{\boldsymbol{\omega}}}) = \lim N^{-k} \underset{{{\boldsymbol{a}}}\equiv {{\boldsymbol{n}}}'(\bmod N)}{\sum_{{{\boldsymbol{a}}}\in A^r {\rm \,primitive}\atop a_1\ldots a_{r-1}=0}}
({{\boldsymbol{a}}}{{\boldsymbol{\omega}}})^{-k},$$ which can be non-zero only if $a_r \in {{\mathbb F}}^*$ (and in fact $a_r = 1$, as ${{\boldsymbol{n}}}'$ is required to be monic). This implies that ${{\boldsymbol{n}}}' = (0,\ldots,0,1)$. Conversely, that choice of ${{\boldsymbol{n}}}'$ gives $F_{k,{{\boldsymbol{n}}}'/N}(0,\ldots,0,1) = N^{-k}{{\omega}}_r^{-k} \not= 0$. That is, ${{\boldsymbol{n}}}' = {{\boldsymbol{n}}}= (0,\ldots,0,1)$ is as wanted.
[**4.7 Corollary.**]{} [*The restricted Eisenstein series $F_{k,{{\boldsymbol{u}}}}$, where ${{\boldsymbol{u}}}\in N^{-1}S$, are linearly independent and form a basis of ${\rm Eis}_k(N)$. The dimension of ${\rm Eis}_k(N)$ equals $\#(S) = c_r(N)$.*]{}
(4.4) + (4.5.4) + (4.6)
[**4.8 Proposition.**]{}
**
- The Eisenstein series $E_{{{\boldsymbol{u}}}} = E_{1,{{\boldsymbol{u}}}}$ ($u \in {{\mathcal T}}(N)$) of weight $1$ and level $N$ separate points of ${{\Gamma}}(N)\setminus \overline{\Psi}^r$. That is, if ${{\boldsymbol{\omega}}}, {{\boldsymbol{\omega}}}' \in \overline{\Psi}^r$ satisfy $E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) = E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}')$ for all ${{\boldsymbol{u}}}\in {{\mathcal T}}(N)$, then there exists ${{\Gamma}}\in {{\Gamma}}(N)$ such that ${{\boldsymbol{\omega}}}' = {{\gamma}}{{\boldsymbol{\omega}}}$.
- The same statement for ${{\Gamma}}(N)\setminus \overline{\Psi}^r$ replaced with ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$.
We start with the identity $$NX \prod_{{{\boldsymbol{u}}}\in {{\mathcal T}}(N)}(1-E_{{{\boldsymbol{u}}}}X) = \phi_N(X) = NX + \sum_{1 \leq i \leq rd} \ell_i(N)X^{q^{i}}, \leqno{(4.8.1)}$$ which comes from (2.4.1) and (2.4.3). Here the right hand side is the $N$-division polynomial of the general Drinfeld module $\phi$ of rank $\leq r$, which lives above $\overline{\Psi}^r$. The coefficients $\ell_i(N)$, where $1 \leq i \leq rd$ ($d = \deg N$), are ${{\Gamma}}$-invariant functions on $\overline{\Psi}^r$ of weights $q^{i}-1$.
Hence the data $\{E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})~|~{{\boldsymbol{u}}}\in {{\mathcal T}}(N)\}$ determines the values on ${{\boldsymbol{\omega}}}$ of the $\ell_i(N,{{\boldsymbol{\omega}}})$ and therefore, taking the known relations between the $\ell_i(N)$ and the $g_i$ ($1 \leq j \leq r$) into account, the coefficients $g_1({{\boldsymbol{\omega}}}),\ldots,g_r({{\boldsymbol{\omega}}})$ of the $T$-division polynomial $$\phi_T^{{{\boldsymbol{\omega}}}}(X) = TX+\sum_{1 \leq j \leq r} g_j({{\boldsymbol{\omega}}})X^{q^{j}}$$ of the Drinfeld module $\phi^{{{\boldsymbol{\omega}}}}$ that corresponds to ${{\boldsymbol{\omega}}}\in \overline{\Psi}^r$. Suppose that $\phi^{{{\boldsymbol{\omega}}}}$ has rank $s$ ($1 \leq s \leq r$), that is, $g_s({{\boldsymbol{\omega}}}) \not= 0$, $g_{s+1}({{\boldsymbol{\omega}}}) = \cdots = g_r({{\boldsymbol{\omega}}}) = 0$.
If then $\phi^{{{\boldsymbol{\omega}}}}$ determines an $A$-lattice in $C_{\infty}$ of rank $r$, hence a point ${{\boldsymbol{\omega}}}\in \Psi^r$ up to the action of ${{\Gamma}}$. That is, if $E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}) = E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}}')$ for all ${{\boldsymbol{u}}}$ then ${{\boldsymbol{\omega}}}' = {{\gamma}}{{\boldsymbol{\omega}}}$ with some ${{\gamma}}\in {{\Gamma}}$. The relation $E_{{{\boldsymbol{u}}}}({{\gamma}}{{\boldsymbol{\omega}}}) = E_{{{\boldsymbol{u}}}{{\gamma}}}({{\boldsymbol{\omega}}})$ moreover shows that ${{\boldsymbol{u}}}{{\gamma}}= {{\boldsymbol{u}}}$ for all ${{\boldsymbol{u}}}$, that is, ${{\gamma}}$ lies in fact in ${{\Gamma}}(N)$.
If , the $A$-lattice corresponding to $\phi^{{{\boldsymbol{\omega}}}}$ has rank $s$ and is given by an embedding $i:\: U \hookrightarrow
C_{\infty}$ of some $s$-dimensional subspace $U$ of $K^r$ (i.e., of $L_U = A^r \cap U \hookrightarrow C_{\infty}$). Now Proposition 4.2 allows to determine $U$ up to ${{\Gamma}}(N)$-equivalence. Choose one such $U$; then ${{\boldsymbol{\omega}}}\in \Psi_U$ is determined through $\phi^{{{\boldsymbol{\omega}}}}$ and the $E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})$ up to an element of ${{\Gamma}}_U$ and, in fact (with the same argument as in the case ), up to an element of ${{\Gamma}}_U(N) = {{\Gamma}}_U \cap {{\Gamma}}(N)$. This shows (i); the proof of (ii) is identical.
[**5. The projective embedding.**]{}
In this section, we show that ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$ is the set of $C_{\infty}$-points of a closed subvariety of some projective space. This allows us to endow it with the structure of projecive variety, which then will be labelled with the symbol $\overline{M}^r(N)$, the [*Eisenstein compactification*]{} of $M^r(N)$.
Throughout, $N\in A$ of degree $d \geq 1$ is fixed.
(5.1) We define the [*Eisenstein ring of level*]{} $N$, ${\bf Eis}(N)$, as the $C_{\infty}$-subalgebra of $\widetilde{{{\mathcal F}}}_r(N)$ generated by the Eisenstein series $E_{{{\boldsymbol{u}}}} = E_{1,{{\boldsymbol{u}}}}$ of level $N$ and weight 1. It is graded with respect to weight: its $k$-th piece ${\bf Eis}_k(N)$ is the $C_{\infty}$-space generated by monomials of degree $k$ in the $E_{{{\boldsymbol{u}}}}$. In particular, ${\bf Eis}(N)$ is generated as an algebra by ${\bf Eis}_1(N) = {\rm Eis}_1(N)$, a vector space of dimension $c_r(N)$ (see (4.7)). We also let ${\bf Mod} = {\bf Mod}(1) = C_{\infty}[g_1,\ldots,g_r]$ be the graded algebra of modular forms of type zero for ${{\Gamma}}$ ([@20] 1.7).
[**5.2 Proposition.**]{}
**
- ${\bf Eis}(N)$ contains the algebra ${\bf Mod}$;
- ${\bf Eis}(N)$ is integral over ${\bf Mod}$;
- ${\bf Eis}(N)$ contains all the Eisenstein series $E_{k,{{\boldsymbol{u}}}}$ of arbitrary weight $k$.
<!-- -->
- The argument in the proof of Proposition 4.8 shows that $g_1,\ldots,g_r \in {\bf Eis}(N)$.\
- The $E_{{{\boldsymbol{u}}}}$ are the zeroes of the monic polynomial $N^{-1}X^{q^{rd}} \phi_N(X^{-1})$ with coefficients $N^{-1} \ell_i(N) \in {\bf Mod}$, where $\phi_N(X)$ is as in (4.8.1).\
- Let ${{\Lambda}}$ be any rank-$r$ $A$-lattice in $C_{\infty}$ and $G_{k,{{\Lambda}}}(X)$ be its $k$-th Goss polynomial ([@23] 2.17, [@16] 3.4). It is of shape $$G_{k,{{\Lambda}}}(X) = \sum_{0 \leq i \leq k} a_i({{\Lambda}})X^{k-i}, \leqno{(5.2.1)}$$ where $a_i$ is a modular form of weight $i$ and type $0$, that is, $a_i \in {\bf Mod}$. (This follows from [@16] 3.4(ii).) The characteristic property of Goss polynomials ([*loc. cit*]{} 3.4(i)) implies $$E_{k,{{\boldsymbol{u}}}} = G_{k,{{\Lambda}}}(E_{1,{{\boldsymbol{u}}}}).\leqno{(5.2.2)}$$ Now (iii) is a consequence of (5.2.1) and (5.2.2).
(5.3) We define $\mathbb P = \mathbb P(N)$ as the projective space $\mathbb P({\rm Eis}_1(N)^{\wedge})$ associated with the dual vector space ${\rm Eis}_1(N)^{\wedge}$ of ${\rm Eis}_1(N)$. As a scheme, $$\mathbb P = {\rm Proj}\, R,\leqno{(5.3.1)}$$ where $R := {\rm Sym}({\rm Eis}_1(N))$ is the symmetric algebra on ${\rm Eis}_1(N)$. Consider the map $$j_N:\: \overline{{{\Omega}}}^r {{\longrightarrow}}\mathbb P \leqno{(5.3.2)}$$ to the $C_{\infty}$-valued points of $\mathbb P$ that with the class of $(U,i)$ associates the class (up to scalars) of the linear form $E_{{{\boldsymbol{u}}}} \longmapsto E_{{{\boldsymbol{u}}}}(U,i)$. Then:
(5.3.3) $j_N$ is well-defined, as the $E_{{{\boldsymbol{u}}}}$ have weight 1 and for each $(U,i)$ there exists ${{\boldsymbol{u}}}$ such that $E_{{{\boldsymbol{u}}}}(U,i) \not= 0$.
(5.3.4) $j_N({{\gamma}}(U,i)) = j_N(U,i)$ for ${{\gamma}}\in {{\Gamma}}(N)$, as the $E_{{{\boldsymbol{u}}}}$ are ${{\Gamma}}(N)$-invariant. Here and in the following, we write $j_N(U,i)$ for $j_N$ (class of $(U,i)$).
(5.3.5) As a map from ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$ to $\mathbb P$, $j_N$ is injective, due to Proposition 4.8.
(5.4) The graded algebra $R$ is supplied with a canonical homomorphism $$\epsilon:\: R {{\longrightarrow}}{\bf Eis}(N),\leqno{(5.4.1)}$$ which is the identity on $R_1 = {\rm Eis}_1(N) = {\bf Eis}_1(N)$ and surjective, since ${\bf Eis}_1(N)$ generates ${\bf Eis}(N)$. Let $J$ be the kernel of $\epsilon$. Since ${\bf Eis}(N)$ is a domain, $J$ is a (homogeneous) prime ideal of $R$, and in particular, saturated ([@25] p. 125). Then $j_N$ maps ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$ to the vanishing variety $V(J) \subset \mathbb P$ of $J$.
(5.5) Let $x \in V(J)$ be given. The proof of Proposition 4.8 shows that there exists an element $(U,i)$ of $\overline{\Psi}^r$, well-defined up to the action of ${{\Gamma}}(N)$, such that $j_N(U,i) = x$. Viz., for simplicity choose a representative $\widetilde{x} \in {\rm Eis}_1(N)^{\wedge} = {\rm Hom}_{C_{\infty}}({\rm Eis}_1(N),C_{\infty})$ and put $\widetilde{x}_{{{\boldsymbol{u}}}} := \widetilde{x}(E_{{{\boldsymbol{u}}}})$. Interpreting $\widetilde{x}_{{{\boldsymbol{u}}}}$ as a value of $E_{{{\boldsymbol{u}}}}$, the $\widetilde{x}_{{{\boldsymbol{u}}}}$ determine the values of the coefficient forms $g_1,\ldots,g_r$ as in (4.8), thus (if $g_r\not=0$) a point $\widetilde{{{\boldsymbol{\omega}}}} \in \Psi^r$ up to the action of ${{\Gamma}}(N)$. The corresponding point ${{\boldsymbol{\omega}}}\in {{\Omega}}^r$ is independent of the choice of $\widetilde{x}$ and serves the purpose. If $g_s \not= 0$, $g_{s+1} = \ldots = g_r = 0$ then, as in (4.8), the $s$-dimensional $K$-space $U$ and its boundary coponent $\Psi_U$ is determined up to ${{\Gamma}}(N)$-equivalence by the (non-) vanishing of the $\widetilde{x}_{{{\boldsymbol{u}}}}$. Choosing one such $U$, there exists an embeddding $i:\: U \hookrightarrow
C_{\infty}$, unique up to ${{\Gamma}}_U(N)$, that fits the given data. Then the class of $(U,i)$ in $\overline{{{\Omega}}}^r$ is as wanted. That is, $$j_N:\: {{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r \stackrel{\cong}{{{\longrightarrow}}} V(J) \leqno{(5.6)}$$ is in fact bijective. Furthermore, the restriction of $j_N$ to a stratum ${{\Gamma}}_U(N)\setminus {{\Omega}}_U$ of ${{\Gamma}}(N)\setminus
\overline{{{\Omega}}}^r$ is analytic with respect to the analytification of $V(J)$, as the $E_{{{\boldsymbol{u}}}}$ are.
(5.7) Next, we consider the canonical morphism $\kappa:\: V(J) {{\longrightarrow}}\overline{M}^r$ defined as follows: Choose elements $G_i\in R = {\rm Sym}({\rm Eis}_1(N))$ such that $\epsilon(G_i) = g_i$ ($1 \leq i \leq r$, see (5.4)). For given $x \in V(J)$, $G_i$ may be evaluated on $\widetilde{x}$ (notation as in (5.5)), and we put $$\kappa(x) = (G_1(\widetilde{x}): \ldots : G_r(\widetilde{x})),$$ which is independent of the choice of $\widetilde{x}$ above $x$, and of the choices of the $G_i$. Furthermore, the diagram $$\begin{array}{ccl}
{{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r & \stackrel{j_N}{{{\longrightarrow}}} & V(J) \vspace{0.2cm}\\
\downarrow \pi & & \hspace*{0.3cm}\downarrow \kappa\vspace{0.2cm}\\
{{\Gamma}}\setminus \overline{{{\Omega}}}^r & \stackrel{j}{{{\longrightarrow}}} & \overline{M}^r = {\rm Proj}\, C_{\infty}[g_1,\ldots,g_r]
\end{array} \leqno{(5.7.1)}$$ commutes, where the left vertical arrow is the canonical projection $\pi$.
[**5.8 Proposition.**]{} [*$j_N$ is a strong homeomorphism.* ]{}
This follows essentially from Theorem 2.3, that is, from the corresponding property of $j$. As in the proof of (2.3), $j_N$ is strongly continuous, so we must show that it is also an open map. Consider diagram (5.7.1), where $\pi$ and therefore $j \circ \pi = \kappa \circ j_N$ are open. As $\kappa:\: V(J) {{\longrightarrow}}\overline{M}^r$ is set-theoretically the quotient map of the finite group $G(N) = {{\Gamma}}/{{\Gamma}}(N)\cdot Z$, which acts through homeomorphisms on $V(J)$, the openness of $\kappa \circ j_N$ implies the openness of $j_N$.
By (5.6) and (5.8), we may use $j_N$ to endow ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$ with the structure of (the set of $C_{\infty}$-points of) the projective subvariety $V(J)$ of $\mathbb P$, compatible with the analytic structures and the strong topologies on both sides. By construction, $V(J)$ equals the projective variety associated with the graded algebra ${\bf Eis}(N)$. We collect what has been shown.
[**5.9 Theorem.**]{}
*Let $N$ be a non-constant monic element of $A$.*
- The set ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$ is the set of $C_{\infty}$-points of an irreducible projective variety $\overline{M}^r(N)$ over $C_{\infty}$, the [*Eisenstein compactification*]{} of $M^r(N)$, which may be described as the variety ${\rm Proj}\,{\bf Eis}(N)$ associated with the Eisenstein ring ${\bf Eis}(N)$. It is a closed subvariety of the projective space $\mathbb P = \mathbb P({\rm Eis}_1(N)^{\wedge})$ attached to the dual of the vector space ${\rm Eis}_1(N)$ of Eisenstein series of level $N$ and weight $1$, which has dimension $c_r(N)$. The open subvariety $M^r(N) = {{\Gamma}}(N)\setminus {{\Omega}}^r$ of $\overline{M}^r(N)$ is characterized as $\{x \in \overline{M}^r(N)~|~E_{{{\boldsymbol{u}}}}(x) \not= 0 \quad \forall \, {{\boldsymbol{u}}}\in {{\mathcal T}}(N)\}$.
- The set ${{\Gamma}}(N)\setminus \overline{\Psi}^r$ is the set of $C_{\infty}$-points of an irreducible variety $\overline{\widetilde{M}}^r(N)$ over $C_{\infty}$, which may be described as the variety ${\rm Spec}\,{\bf Eis}(N) \setminus \{I\}$, where $I$ is the irrelevant ideal of the graded ring ${\bf Eis}(N)$. It is a subvariety of the affine space attached to ${\rm Eis}_1(N)^{\wedge}$, endowed with an action of the multiplicative group $\mathbb G_m$, and such that $$\mathbb G_m \setminus \overline{\widetilde{M}}^r(N) \stackrel{\cong}{{{\longrightarrow}}} \overline{M}(N).$$
\(i) has been shown above (see Proposition 4.2 for the last assertion), and the proof of (ii) is - mutatis mutandis - identical.
[**5.10 Remark.**]{} We point out the following functorial properties of the construction of $\overline{M}(N)$ and $\overline{\widetilde{M}}^r(N)$.
- It is compatible with level changes, to wit: Let $N'$ be a multiple of $N$ and $G(N,N')$ the quotient group ${{\Gamma}}(N)/{{\Gamma}}(N')$. The action of ${{\Gamma}}(N)$ on $\Psi^r$ induces an action of $G(N,N')$ on ${{\Gamma}}(N')\setminus \Psi^r =
\widetilde{M}^r(N')$ such that $G(N,N') \setminus \widetilde{M}^r(N') = \widetilde{M}^r(N)$. Further, the fixed space of ${{\Gamma}}(N)$ in ${\rm Eis}_1(N')$ is ${\rm Eis}_1(N)$; hence $G(N,N')$ acts effectively on ${\rm Eis}_1(N')$ with fixed space ${\rm Eis}_1(N)$. Let $\widetilde{j}_N:\:
{{\Gamma}}(N)\setminus \overline{\Psi}^r \hookrightarrow {\rm Eis}_1(N)^{\wedge}$ be the morphism analogous to $j_N$ and implicitly referred to in Theorem 5.9(ii). Then the diagram $$\begin{array}{ccc}
{{\Gamma}}(N')\setminus \overline{\Psi}^r & \stackrel{\widetilde{j}_{N'}}{{{\longrightarrow}}} & {\rm Eis}_1(N')^{\wedge}\vspace{0.2cm}\\
\downarrow & & \hspace*{0.3cm}\downarrow \vspace{0.2cm}\\
{{\Gamma}}(N)\setminus \overline{\Psi}^r & \stackrel{\widetilde{j}_N}{{{\longrightarrow}}} & {\rm Eis}_1(N)^{\wedge}
\end{array} \leqno{(5.10.1)}$$ is commutative and compatible with the action of $G(N,N')$, where the vertical arrows are the canonical projections. In particular, the action of $G(N,N')$ on $\widetilde{M}^r(N')$ with quotient $\widetilde{M}^r(N)$ extends to $\overline{\widetilde{M}}^r(N')$ with quotient $\overline{\widetilde{M}}^r(N)$. Factoring out the multiplicative group $\mathbb G_m$, we find similarly that $G(N,N')$ acts on $\overline{M}^r(N')$ with quotient $\overline{M}^r(N)$.
- The construction of the Eisenstein compactification $\overline{M}^r(N)$ (and likewise of $\overline{\widetilde{M}}^r(N)$) is hereditary in the following sense. Let ${{\Omega}}_U$ be a boundary component ($U \in \mathfrak U_s$, $s < r$) and\
$$M_U(N):= {{\Gamma}}_U(N)\setminus {{\Omega}}_U \stackrel{\cong}{{{\longrightarrow}}} \{{{\gamma}}\in {\rm GL}(s,A)~|~{{\gamma}}\equiv 1 (\bmod N)\}
\setminus {{\Omega}}^s$$ its image in $\overline{M}^r(N)$. Then the Zariski closure $\overline{M}_U(N)$ of $M_U(N)$ in $\overline{M}^r(N)$ is composed of the $M_{U'}(N)$, where $U' \in \mathfrak U$ and $U' \subset U$, and is isomorphic with the variety $\overline{M}^s(N)$. This is seen by assuming, without restriction, that $U = V_s$, in which case the description of $\overline{M}_{V_s}(N)$ is identical with that of $\overline{M}^s(N)$.
[**5.11 Remark.**]{} The idea of using Eisenstein series for a projective embedding of $M^r(N)$ is taken from [@28]. However, Kapranov’s construction has the drawback that it fails to be canonical (it depends on the choice of a certain bound $m_0$, see [@28] Proposition 1.12). Instead, our Proposition 4.8 assures that it suffices to consider Eisenstein series of weight 1, which culminates in the canonical description $\overline{M}^r(N) = {\rm Proj}\,{\bf Eis}(N) \hookrightarrow
\mathbb P({\rm Eis}_1(N)^{\wedge})$ with its functorial properties.
[**6. Tubular neighborhood of cuspidal divisors.**]{}
In this section we show that each point $x$ on a cuspidal divisor, i.e., on a boundary component $M_U(N)$ of $\overline{M}^r(N)$ of codimension 1, possesses a neighborhood $Z$ isomorphic with $B \times W$, where $W$ is an open admissible affinoid neighborhood of $x$ on $M_U(N)$ and $B$ a ball, and such that the map $\pi:\: Z {{\longrightarrow}}W$ derived from the canonical projection $\pi_U:\: \overline{M}^r(N) {{\longrightarrow}}M_U(N)$ is the projection to the second factor.
(6.1) As usual, it suffices to treat the case where $x$ is represented by ${{\boldsymbol{\omega}}}^{(0)} \in {{\Omega}}_{V_{r-1}} \stackrel{\cong}{{{\longrightarrow}}}
{{\Omega}}^{r-1}$. For simplicity, we use the canonical isomorphism as an identification. We may further assume that ${{\boldsymbol{\omega}}}^{(0)}$ belongs to the fundamental domain ${{\boldsymbol{F}}}'$ of ${{\Gamma}}' = {\rm GL}(r-1,A)$ in ${{\Omega}}^{r-1}$, that is, ${{\boldsymbol{\omega}}}^{(0)} = (0:{{\omega}}_2^{(0)}: \ldots: {{\omega}}_r^{(0)})$, where $\{1={{\omega}}_r^{(0)},\ldots,{{\omega}}_2^{(0)}\}$ is an SMB of its lattice. Let $X \subset {{\Omega}}^{r-1}$ be the subspace $$X=\{{{\boldsymbol{\omega}}}' = ({{\omega}}'_2:\ldots:{{\omega}}'_r = 1)~|~|{{\omega}}'_i| = |{{\omega}}_i^{(0)}|,\: 2 \leq i \leq r\}.\leqno{(6.1.1)}$$ Then, in fact, $X \subset {{\boldsymbol{F}}}'$ and $X$ is an admissible open affinoid subspace, whose structure has been investigated in [@20], Theorem 2.4. (All of this collapses for $r=2$ to $X = {{\boldsymbol{F}}}_1 = {{\Omega}}^1 = \{{\rm point}\}$.)
We next put
(6.1.2) $Y_c =$\
$\{{{\boldsymbol{\omega}}}\in {{\Omega}}^r~|~{{\boldsymbol{\omega}}}=({{\omega}}_1:\ldots:{{\omega}}_r)~|~({{\omega}}_2:\ldots:{{\omega}}_r) \in X~|~d({{\omega}}_1, \langle {{\omega}}_2,\ldots,{{\omega}}_r\rangle_{K_{\infty}})
\geq c\}$ for some large $c$ in the value group $q^{{{\mathbb Q}}}$ of $C_{\infty}$. Here $d({{\omega}},\langle~.~\rangle_{K_{\infty}})$ is the distance function to the $K_{\infty}$-space generated by ${{\omega}}_2,\ldots,{{\omega}}_r=1$. It is an admissible open subspace of ${{\Omega}}^r$. Note that ${{\boldsymbol{\omega}}}\in Y_c$ in particular implies $|{{\omega}}_1| \geq c$.
[**6.2 Lemma.**]{}
*Suppose that $c > |{{\omega}}_2^{(0)}|$. Then:*
- If ${{\gamma}}\in {{\Gamma}}$ satisfies ${{\gamma}}(Y_c) \cap Y_c \not= \emptyset$ then ${{\gamma}}\in {{\Gamma}}\cap P_{r-1}$;
- If ${{\gamma}}\in {{\Gamma}}(N)$ is such that ${{\gamma}}(Y_c) \cap Y_c \not= \emptyset$ then ${{\gamma}}$ has the shape $${{\gamma}}= \begin{array}{|c|c|}\hline
1 & u_2,\ldots,u_r \\ \hline
0 & \\
\vdots & {{\gamma}}' \\
0 & \\ \hline
\end{array}$$ where $u_2,\ldots,u_r \in NA$ and ${{\gamma}}'$ runs through a finite subgroup of ${{\Gamma}}'(N) = {{\Gamma}}'\cap {{\Gamma}}(N)$ consisting of strictly upper triangular matrices (i.e., with ones on the diagonal). On the other hand, each ${{\gamma}}$ of this form with ${{\gamma}}' = 1$ stabilizes $Y_c$.
<!-- -->
- Let ${{\boldsymbol{\omega}}}= ({{\omega}}_1:\ldots:{{\omega}}_r) \in Y_c$ be such that ${{\gamma}}{{\boldsymbol{\omega}}}= ({{\omega}}'_1:\ldots:{{\omega}}'_r) \in Y_c$ with ${{\gamma}}= ({{\gamma}}_{i,j}) \in {{\Gamma}}$ (recall that ${{\omega}}_r = {{\omega}}'_r = 1$). Let further ${{\Lambda}}$ be the lattice ${{\Lambda}}_{{{\boldsymbol{\omega}}}} = \langle {{\omega}}_1,\ldots,{{\omega}}_r \rangle_A$ and $\alpha:= {\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})$. Now $\{{{\omega}}'_r,\ldots,{{\omega}}'_1\}$ is a basis of $\alpha^{-1}{{\Lambda}}$ and, since ${{\gamma}}{{\boldsymbol{\omega}}}\in Y_c$, ${{\omega}}'_r,\ldots,{{\omega}}'_2$ are the first $r-1$ elements of an SMB of $\alpha^{-1}{{\Lambda}}$, so $\alpha {{\omega}}'_r,\ldots,\alpha{{\omega}}'_2$ are the first $r-1$ elements of an SMB of ${{\Lambda}}$. Then $|\underset{1\leq j \leq r}{\sum} {{\gamma}}_{i,j}{{\omega}}_j| = |\alpha{{\omega}}'_i| = |{{\omega}}_i|$ holds for $i = 2,\ldots,r$ in view of (1.8.2). If ${{\gamma}}_{i,1} \not= 0$ then $|\underset{1\leq j \leq r}{\sum}{{\gamma}}_{i,j} {{\omega}}_j| \geq |{{\omega}}_1| > |{{\omega}}_i|$, contradiction.
- The entry ${{\gamma}}_{1,1} = 1$ is obvious, as is the fact that each ${{\gamma}}$ with ${{\gamma}}' = 1$ stabilizes $Y_c$. The possible ${{\gamma}}'$ are those that fix $X$. The stabilizer ${{\Gamma}}'_X$ of $X$ in ${{\Gamma}}'$ equals $$\{{{\gamma}}' = ({{\gamma}}_{i,j})_{2\leq i,j \leq r} \in {{\Gamma}}' ~|~|{{\gamma}}_{i,j}| \leq |{{\omega}}_i^{(0)}/{{\omega}}_j^{(0)}|\},$$ matrices with a block structure
$$\begin{BMAT}(b){|c:cc|}{|c:cc|}
B_1 & & \ast \\
& B_2 & \\
0 & & \ddots\\
\end{BMAT}
\medskip$$
with zeroes below the blocks, and each block $B_k$ an invertible matrix over ${{\mathbb F}}$. The number of such ${{\gamma}}'$ is finite, and the congruence condition ${{\gamma}}' \equiv 1 (\bmod N)$ forces each block to equal 1. Hence ${{\gamma}}'$ is strictly upper triangular.
(6.3) We write $G$ for the group that occurs in (6.2)(ii), i.e., $G:= \{{{\gamma}}\in {{\Gamma}}(N)~|~{{\gamma}}(Y_c) \cap Y_c \not= \emptyset\}$, $G_1 := \{{{\gamma}}\in G~|~{{\gamma}}' = 1\}$ and $G' := G/G_1 ={{\Gamma}}'_X$ for the group of possible ${{\gamma}}'$.
(6.4) Consider the function ${{\boldsymbol{\omega}}}\longmapsto t({{\boldsymbol{\omega}}}) := e^{-1}_{N{{\Lambda}}}({{\omega}}_1)$ on ${{\Omega}}^r$, where now ${{\Lambda}}= \langle {{\omega}}_2,\ldots,{{\omega}}_r\rangle_A$. Its most important properties are:
(6.4.1) $t$ is well-defined, as ${{\omega}}_1$ does not belong to $N{{\Lambda}}$;
(6.4.2) it is holomorphic and has a unique strongly continuous extension to ${{\Omega}}^r \cup {{\Omega}}_{V_{r-1}}$, where $t \equiv 0$ on ${{\Omega}}_{V_{r-1}}$;
(6.4.3) $t({{\gamma}}{{\boldsymbol{\omega}}}) = t({{\boldsymbol{\omega}}})$ for ${{\gamma}}\in G$;
(6.4.4) Fix ${{\boldsymbol{\omega}}}' = ({{\omega}}_2:\ldots,{{\omega}}_r) \in X$. Then the image of the map $$\begin{array}{rll}
t_{{{\boldsymbol{\omega}}}'}:\: \{{{\omega}}\in C_{\infty}~|~({{\omega}}:{{\boldsymbol{\omega}}}') \in Y_c\} & {{\longrightarrow}}& C_{\infty}\\
{{\omega}}& \longmapsto & t({{\omega}}:{{\boldsymbol{\omega}}}')
\end{array}$$ is a pointed ball $B_{\rho}^* := \{z \in C_{\infty}~|~0 < |z| \leq \rho\}$ for some $\rho = \rho(c) \in q^{{{\mathbb Q}}}$, and is independent of the choice of ${{\boldsymbol{\omega}}}' \in X$. The function $c \longmapsto \rho(c)$ is strictly monotonically decreasing with $\underset{c\to\infty}{\lim} \rho(c) = 0$.
As for proofs, (6.4.1) is obvious, and (6.4.2) comes from trivial estimates. [*Proof*]{} of (6.4.3): As ${\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})=1$ for ${{\gamma}}\in G$, $({{\gamma}}{{\boldsymbol{\omega}}})_2,\ldots,({{\gamma}}{{\boldsymbol{\omega}}})_r$ generate the same lattice ${{\Lambda}}= \langle {{\omega}}_2,\ldots,{{\omega}}_r\rangle_A$. Hence $$t({{\gamma}}{{\boldsymbol{\omega}}}) = e_{N{{\Lambda}}}(({{\gamma}}{{\boldsymbol{\omega}}})_1)^{-1} = e_{N{{\Lambda}}}({{\omega}}_1)^{-1} = t({{\boldsymbol{\omega}}}),$$ in view of $({{\gamma}}{{\boldsymbol{\omega}}})_1 \equiv {{\omega}}_1(\bmod N{{\Lambda}})$ and the $N{{\Lambda}}$-invariance of $e_{N{{\Lambda}}}$.
[*Proof*]{} of (6.4.4). The assertion that ${\rm im}(t_{{{\boldsymbol{\omega}}}'})$ is a pointed ball $B_{\rho}^*$ for some $\rho$ is a general fact of rigid analysis (see e.g. [@22] Lemma 10.9.1). Expansion of the product for $|e_{N{{\Lambda}}}({{\omega}})|$ shows that it depends only on the distance $d({{\omega}},K_{\infty}{{\Lambda}})$ and the values $|{{\omega}}_2|,\ldots,|{{\omega}}_r|$, as $\{{{\omega}}_r,\ldots,{{\omega}}_2\}$ is an SMB of ${{\Lambda}}$. Since the $|{{\omega}}_i|$ are constant on $X$, the independence of $\rho(c)$ of the choice of ${{\boldsymbol{\omega}}}'$ follows. The last statement is obvious. $\Box$
[**6.5 Proposition.**]{} [ *Let $\pi$ be the projection ${{\boldsymbol{\omega}}}= ({{\omega}}_1:\cdots:{{\omega}}_r) \longmapsto {{\boldsymbol{\omega}}}' = ({{\omega}}_2:\ldots:{{\omega}}_r)$ from $Y_c$ to $X$ and $\rho$ as in $(6.4.4)$. Then $t \times \pi$ induces an isomorphism $$G\setminus Y_c \stackrel{\cong}{{{\longrightarrow}}} B_{\rho}^* \times (G'\setminus X)$$ of analytic spaces.*]{}
For each ${{\boldsymbol{\omega}}}' \in X$, $t_{{{\boldsymbol{\omega}}}'}$ provides an isomorphism $$t_{{{\boldsymbol{\omega}}}'}:\: (N{{\Lambda}})\setminus \{{{\omega}}\in C_{\infty}~|~({{\omega}}:{{\boldsymbol{\omega}}}') \in Y_c\} \stackrel{\cong}{{{\longrightarrow}}} B_{\rho}^*,$$ as it is bijective and $$\frac{d}{d{{\omega}}} t_{{{\boldsymbol{\omega}}}'}({{\omega}}) = -t_{{{\boldsymbol{\omega}}}'}({{\omega}})^2 \not= 0 \quad \mbox{(since $\frac{d}{d{{\omega}}} e_{N{{\Lambda}}}({{\omega}})=1$)}.
\leqno{(6.5.1)}$$ (Here ${{\Lambda}}$ always denotes the lattice $\langle {{\omega}}_2,\ldots,{{\omega}}_r\rangle_A$ associated with ${{\boldsymbol{\omega}}}$!) Therefore, also $$(t,\pi):\: G_1\setminus Y_c \stackrel{\cong}{{{\longrightarrow}}} B_{\rho}^* \times X$$ is bijective, and is in fact an isomorphism, as, due to (6.5.1), its Jacobian matrix is invertible in each point. The group $G' = G/G_1$ acts on both sides (trivially on $B_{\rho}^*$) and due to (6.4.3), the map $(t,\pi)$ is $G'$-equivariant. Therefore $$G\setminus Y_c = G'\setminus (G_1\setminus Y_c) \stackrel{\cong}{{{\longrightarrow}}} G' \setminus (B_{\rho}^* \times X)
= B_{\rho}^* \times (G'\setminus X).$$
(6.6) As an admissible open affinoid in the smooth space ${{\Gamma}}'(N)\setminus {{\Omega}}^{r-1}$, the space $W:= G'\setminus X$ is itself smooth and affinoid. Let $B = B_{\rho}$ be the unpunctured ball of radius $\rho$ and $Z := B \times W$.
Eisenstein series extend uniquely (as strongly continuous functions and therefore, as $Z$ is smooth, by Bartenwerfer’s criterion [@1] also as analytic functions) from $G\setminus Y_c \stackrel{\cong}{{{\longrightarrow}}} B^* \times W$ to $Z$. The following result is a generalization of [@14] Korollar 2.2.
[**6.7 Proposition.**]{} [ *Let ${{\boldsymbol{u}}}= (u_1,\ldots,u_r) \in {{\mathcal T}}(N)$, $u_1 = a/N$ with $\deg a < d = \deg N$. Then the Eisenstein series $E_{{{\boldsymbol{u}}}} =
E_{1,{{\boldsymbol{u}}}}$, regarded as a function on $Z$, has a zero of order $|a|^{r-1}$ along the divisor ($t=0$) of $Z$.* ]{}
The proof has been given for the case $r=2$ in [@14]. There, equivalently, the pole order of $E_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})^{-1} =
d_{{{\boldsymbol{u}}}}({{\boldsymbol{\omega}}})' = e_{{{\boldsymbol{\omega}}}}({{\boldsymbol{u}}}{{\boldsymbol{\omega}}})$ has been determined, where the product expansion of $e_{{{\boldsymbol{\omega}}}}$ was used. The proof generalizes without difficulty to the case of higher rank $r$. For another approach, see [@28], Lemma 1.23.
(6.8) Again by Bartenwerfer’s criterion, the open embedding $$B^* \times W \stackrel{\cong}{{{\longrightarrow}}} G\setminus Y_c \hookrightarrow {{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$$ extends to an injective map $$i:\: Z = B \times W \hookrightarrow {{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r,$$ since $Z$ is smooth, thus normal. Its image is $${\rm im}(i) = G \setminus Y_c \cup G'\setminus X = G \setminus (Y_c \cup X).$$ We will show that $i$ is in fact an open embedding, i.e., an isomorphism of $Z=B \times W$ with ${\rm im}(i)$. That is, given the class $[{{\boldsymbol{\omega}}}'] \in W = G' \setminus X$ of ${{\boldsymbol{\omega}}}' \in X$ with corresponding point $[{{\boldsymbol{\omega}}}] = (0,[{{\boldsymbol{\omega}}}']) \in Z$, we must show that the canonical map $${{\mathcal O}}_{G\setminus(Y_c \cup X),[{{\boldsymbol{\omega}}}]} {{\longrightarrow}}{{\mathcal O}}_{Z,[{{\boldsymbol{\omega}}}]}$$ of analytic local rings is an isomorphism. In fact, it suffices to show the corresponding isomorphism without dividing out the group $G$. In other words, we must show that the canonical injection $${{\mathcal O}}_{Y_c \cup X,{{\boldsymbol{\omega}}}} \hookrightarrow {{\mathcal O}}_{B\times X,{{\boldsymbol{\omega}}}} = {{\mathcal O}}_{B,0} \widehat{\otimes} {{\mathcal O}}_{X,{{\boldsymbol{\omega}}}'}$$ is bijective. ($\widehat{\otimes}$ is the topological tensor product of the two local rings, and we use “${{\boldsymbol{\omega}}}$” both for the point $({{\omega}}_1 = 0, {{\boldsymbol{\omega}}}')$ of $Y_c \cup X$ and the point $(t=0,{{\boldsymbol{\omega}}}')$ of $B \times X$.) Now the left hand side contains ${{\mathcal O}}_{X,{{\boldsymbol{\omega}}}'}$ via the projection $\pi:\: Y_c \cup X {{\longrightarrow}}X$, and it suffices to show that it also contains a uniformizer $u$ of $Y_c\cup X$ along $X$, i.e., some $u$ that as a germ of a function on $B \times X$ presents a zero of order $1$ along ($t=0$). Then ${{\mathcal O}}_{Y_c\cup X,{{\boldsymbol{\omega}}}}$ encompasses the local ring point ${\rm Sp}(C_{\infty}\langle u\rangle)$ at $u=0$, that is, ${{\mathcal O}}_{B,0}$, and we are done. By Proposition 6.7, we can take as $u$ the Eisenstein series $E_{{{\boldsymbol{u}}}}$ with ${{\boldsymbol{u}}}= (\frac{1}{N},0,\ldots,0)\in {{\mathcal T}}(N)$.
Therefore we have shown the following result.
[**6.9 Theorem.**]{} [ *Let $M_U(N) = {{\Gamma}}(N)\setminus {{\Omega}}_U$ with $U \in \mathfrak U_{r-1}$ be a cuspidal divisor on $\overline{M}^r(N)$, ${{\boldsymbol{\omega}}}^{(0)}$ a point on ${{\Omega}}_U$ with class $[{{\boldsymbol{\omega}}}^{(0)}]$ in $M_U(N)$, and $\pi_U:\: {{\Omega}}^r {{\longrightarrow}}{{\Omega}}_U$ the canonical projection. There exists an admissible open affinoid neighborhood $X$ of ${{\boldsymbol{\omega}}}^{(0)}$ in ${{\Omega}}_U$ and an admissible open subspace $Y$ of ${{\Omega}}^r$ characterized by $$Y = \{{{\boldsymbol{\omega}}}\in {{\Omega}}^r~|~\pi_U({{\boldsymbol{\omega}}}) \in X \mbox{ and ${{\boldsymbol{\omega}}}$ sufficiently close to $\pi_U({{\boldsymbol{\omega}}})$}\}$$ such that $Y \cup X$ is isomorphic with a product $B \times X$ and the subspace $Z := {{\Gamma}}(N)\setminus (Y\cup X)$ of ${{\Gamma}}(N)\setminus \overline{{{\Omega}}}^r$ is isomorphic with $B\times W$, where $B$ is a ball and $W$ is an admissible open affinoid neighborhood of $[{{\boldsymbol{\omega}}}^{(0)}]$ in $M_U(N)$, quotient of $X$ by a finite group $G'$ of automorphisms. The second projection $Z{{\longrightarrow}}W$ comes from $\pi_U$, divided out by the action of ${{\Gamma}}(N)$, and the first projection $Z{{\longrightarrow}}B$ is given by an explicit uniformizer.*]{}
(If $U = V_{r-1}$ then $X,Y$ and $t$ are specified in (6.1) and (6.4), and ’sufficiently close’ means $d({{\omega}}_1,\langle {{\omega}}_2,\ldots,{{\omega}}_r\rangle_{K_{\infty}}) \geq c$ for some constant $c$, or equivalently, $|t({{\boldsymbol{\omega}}})| \leq \rho$ for some $\rho$ depending on $c$.) $\Box$
[**6.10 Corollary.**]{} [*$[{{\boldsymbol{\omega}}}^{(0)}]$ is a smooth point of $\overline{M}^r(N)$.*]{}
$B$ and $W$ are smooth.
[**6.11 Remark.**]{} Theorem 6.9 allows to expand holomorphic functions on ${{\Omega}}^r$ of weight $k$ for ${{\Gamma}}(N)$ (so-called weak modular forms, see (7.1)) as Laurent series in $t$, where the coefficients are holomorphic functions on ${{\Omega}}_U$. This is crucial for the theory of modular forms. For the case $r=2$, see e.g. [@24] or [@15]; for higher rank $r$, Basson and Breuer have started investigations in this direction in [@2] and [@3]. See also [@4], [@5], [@6].
[**6.12 Remark.**]{} Analogous tubular neighborhoods along cuspidal divisors may also be constructed for $\overline{\widetilde{M}}^r(N)$. The proof for $\overline{M}^r(N)$ given in (6.4)–(6.8) may easily be adapted.
[**7. Modular forms.**]{}
In this section, we define the ring of modular forms for ${{\Gamma}}(N)$ and relate it with the ring of sections of the very ample line bundle of $\overline{M}^r(N)$ given by the embedding $j_N$.
Theorem 7.9 gives several different descriptions of modular forms. The assumptions of the preceding sections remain in force. Thus $r \geq 2$ and $N \in A$ is monic of degree $d \geq 1$.
(7.1) Let ${{\mathcal O}}(1)$ be the usual twisting line bundle on $\mathbb P = \mathbb P({\rm Eis}_1(N)^{\wedge})$ and $\mathfrak M := j_N^*({{\mathcal O}}(1))$ its restriction to the subvariety $j_N:\: \overline{M}^r(N) \hookrightarrow \mathbb P$. Tracing back the definitions, one sees that the sections of $\mathfrak M^{\otimes k}$ restricted to the open analytic subspace $M^r(N) = {{\Gamma}}(N)\setminus {{\Omega}}^r$ are just the functions $f$ on ${{\Omega}}^r$ subject to
- $f$ is holomorphic;
- for ${{\boldsymbol{\omega}}}\in {{\Omega}}^r$ and ${{\gamma}}\in {{\Gamma}}(N)$, the rule $$f({{\gamma}}{{\boldsymbol{\omega}}}) = {\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})^k f({{\boldsymbol{\omega}}})$$ holds (${\rm aut}({{\gamma}},{{\boldsymbol{\omega}}}) = \underset{1\leq i \leq r}{\sum} {{\gamma}}_{r,i}{{\omega}}_i$, ${{\boldsymbol{\omega}}}$ normalized such that ${{\omega}}_r=1$).
For further use, we baptize functions $f$ satisfying (i) and (ii) as [*weak modular forms*]{} of weight $k$ for ${{\Gamma}}(N)$. Later we will define modular forms as weak modular forms that additionally satisfy certain boundary conditions discussed below.
(7.2) First, we recall the fundamental domain ${{\boldsymbol{F}}}$ for ${{\Gamma}}$ on ${{\Omega}}^r$: $${{\boldsymbol{F}}}= \{{{\boldsymbol{\omega}}}\in {{\Omega}}^r~|~\{{{\omega}}_r=1,{{\omega}}_{r-1},\ldots,{{\omega}}_1\}
\mbox{ is an SMB of its lattice ${{\Lambda}}_{{{\boldsymbol{\omega}}}}$}\}.\leqno{(7.2.1)}$$ The group ${{\Gamma}}/{{\Gamma}}(N)$ acts on $\overline{M}^r(N)$ and thus on weak modular forms of weight $k$ and level $N$ through $$f\longmapsto f_{[{{\gamma}}]_k}, \mbox{ where } f_{[{{\gamma}}]_k}({{\boldsymbol{\omega}}}) = {\rm aut}({{\gamma}},{{\boldsymbol{\omega}}})^{-k} f({{\gamma}}{{\boldsymbol{\omega}}}).\leqno{(7.2.2)}$$ (I.e., the formula is valid for ${{\gamma}}\in {{\Gamma}}$, and ${{\gamma}}\in {{\Gamma}}(N)$ acts trivially.) For simplicity, we choose and fix a system RS of representatives for ${{\Gamma}}/{{\Gamma}}(N)$.
(7.3) Let now $f$ be a weak modular form of some weight $k\in {{\mathbb N}}$. Consider the conditions on $f$:
- $f$ extends to a holomorphic section of $\mathfrak M^{\otimes k}$ over $\overline{M}^r(N)$;
- $f$ extends to a strongly continuous section of $\mathfrak M^{\otimes k}$ over $\overline{M}^r(N)$ (that is, $f$ regarded as a ${{\Gamma}}(N)$-invariant homogeneous function of weight $k$ on $\Psi^r$ has a strongly continuous extension to $\overline{\Psi}^r$);
- $f$ is integral over the ring ${\bf Mod} = C_{\infty}[g_1,\ldots,g_r]$ of modular forms of type 0 for ${{\Gamma}}$;
- $f$ along with all its conjugates $f_{[{{\gamma}}]_k}$ (${{\gamma}}\in RS$) is bounded on ${{\boldsymbol{F}}}$.
[**7.4 Theorem.**]{} [ *For weak modular forms $f$ of weight $k$ and level $N$, we have the following implications: [(a]{}$^{\rm st})$ $\Rightarrow$ [(a)]{} $\Leftrightarrow$ [(b)]{} $\Leftrightarrow$ [(c)]{}. If the variety $\overline{M}^r(N)$ happens to be normal, then the four conditions are equivalent.*]{}
is trivial.
The elementary symmetric functions in the $f_{[{{\gamma}}]_k}$ (${{\gamma}}\in RS$) are invariant under ${{\Gamma}}/{{\Gamma}}(N)$, that is, under ${{\Gamma}}$ and have a strongly continuous extension to $\overline{\Psi}^r$. As ${{\Gamma}}\setminus \overline{\Psi}^r = \overline{M}^r$ is normal, these extensions are in fact holomorphic by [@1] and therefore modular forms for ${{\Gamma}}$. Hence $f$ satisfies an integral equation $$f^n+a_1f^{n-1}+\cdots + a_n = 0 \mbox{ with } a_i \in {\bf Mod}.\leqno{(7.4.1)}$$ Suppose that $f$ is subject to an equation (7.4.1). Then it also holds for $f$ replaced by $f_{[{{\gamma}}]_k}$, and $f$ along with all its conjugates is bounded on ${{\boldsymbol{F}}}$, as the $a_i$ are ([@20] Proposition 1.8).
As in , $f$ satisfies an equation of type (7.4.1) with the elementary symmetric functions in the $f_{[{{\gamma}}]_k}$ (${{\gamma}}\in RS$) as coefficients $a_i$ up to sign. From the boundedness of the $f_{[{{\gamma}}]_k}$ we conclude the boundedness of the $a_i$ on ${{\boldsymbol{F}}}$, which in turn implies $a_i \in {\bf Mod}$ ([@20] Proposition 1.8).
Suppose that $P(f) = 0$ with $P(X) = X^n+a_1X^{n-1} + \cdots + a_n$ and coefficients $a_1,\ldots,a_n \in {\bf Mod}$. Regarding $f$ as a homogeneous and ${{\Gamma}}(N)$-invariant function of weight $k$ on $\Psi^r$, we must show that it extends to a strongly continuous function on $\overline{\Psi}^r$.
Let ${{\boldsymbol{\omega}}}= (U,i)$ be a boundary point, ${{\boldsymbol{\omega}}}\in \overline{\Psi}^r \setminus \Psi^r$, and let $({{\boldsymbol{\omega}}}_{\ell})_{\ell \in {{\mathbb N}}}$ be a sequence of elements of $\Psi^r$ that tends to ${{\boldsymbol{\omega}}}$. Without restriction, replacing $f$ with some transform $f_{[{{\gamma}}]_k}$ if necessary, we may assume that ${{\boldsymbol{\omega}}}\in \Psi_{V_s}$ for some $1 \leq s < r$, ${{\boldsymbol{\omega}}}= (0,\ldots,0,{{\omega}}_{r-s+1},\ldots,
{{\omega}}_r)$, where ${{\boldsymbol{\omega}}}$ lies in the corresponding fundamental domain ${{\boldsymbol{F}}}_s$, see (1.15.4). We will show:
- $(f({{\boldsymbol{\omega}}}_{\ell}))_{\ell \in {{\mathbb N}}}$ converges in $C_{\infty}$.\
Therefore, $\overline{f}({{\boldsymbol{\omega}}}) := \underset{\ell\to\infty}{\lim} f({{\boldsymbol{\omega}}}_{\ell})= \underset{\underset{{{\boldsymbol{\omega}}}'\to {{\boldsymbol{\omega}}}}{{{\boldsymbol{\omega}}}'\in \Psi^r}}{\lim}
f({{\boldsymbol{\omega}}}')$ exists;
- The so-defined extension $\overline{f}$ of $f$ to $\overline{\Psi}^r$ is strongly continuous, of weight $k$ and ${{\Gamma}}(N)$-invariant.
Put $\overline{a}_i := \underset{\ell\to \infty}{\lim} a_i({{\boldsymbol{\omega}}}_{\ell})$, which exists as $a_i$ is a modular form for ${{\Gamma}}$, and let $\overline{P}(X) = X^n+ \underset{1\leq i \leq n}{\sum} \overline{a}_iX^{n-i}$ be the limit polynomial. Elementary estimates show that $f({{\boldsymbol{\omega}}}_{\ell})$ is close to a zero of $\overline{P}$ if $\ell \gg 0$. Hence the set of limit points of $(f({{\boldsymbol{\omega}}}_{\ell}))_{\ell \in {{\mathbb N}}}$ is contained in the set $Z(\overline{P})$ of zeroes of $\overline{P}$, and each $f({{\boldsymbol{\omega}}}_{\ell})$ is close to one of them for $\ell \gg 0$. Let $Y$ be a small neighborhood (w.r.t. the strong topology) of ${{\boldsymbol{\omega}}}$ in $\overline{\Psi}^r$. Then for $Y$ and $\epsilon >0$ small enough, $$\Psi^r \cap Y = \underset{x\in Z(\overline{P})}{\stackrel{\bullet}{\bigcup}} Y_x,\leqno{(7.4.2)}$$ where $Y_x := \{{{\boldsymbol{\omega}}}' \in \Psi^r \cap Y~|~|f({{\boldsymbol{\omega}}}')-x| < \epsilon\}$. We may further choose $Y$ such that $$\begin{array}{l}
\Psi^r \cap Y = \{{{\boldsymbol{\omega}}}' \in \Psi~|~({{\omega}}'_{r-s+1},\ldots,{{\omega}}'_r) \\
\mbox{lies in a fixed connected open affinoid neighborhood $X$ of} \\
{{\boldsymbol{\omega}}}\mbox{ in } \Psi_{V_s} \mbox{ and }
d({{\omega}}'_i,\langle{{\omega}}'_{r-s+1},\ldots,{{\omega}}'_r\rangle_{K_{\infty}}) \geq c \mbox{ for } 1 \leq i \leq r-s\}
\end{array}\leqno{(7.4.3)}$$ for sufficiently large $c\in q^{{{\mathbb Q}}}$. Such a set is connected as an analytic space.
Now the occurrence of at least two different zeroes $x$ in (7.4.2) would contradict the connectedness of $\Psi^r \cap Y$. Hence there exists only one limit point $x$, which equals $$\overline{f}({{\boldsymbol{\omega}}}) := \lim_{\ell \to \infty} f({{\boldsymbol{\omega}}}_{\ell}) = \underset{\underset{{{\boldsymbol{\omega}}}'\to {{\boldsymbol{\omega}}}}{{{\boldsymbol{\omega}}}'\in \Psi^r}}{\lim} f({{\boldsymbol{\omega}}}'),$$ and (A) is established. The fact (B) that $\overline{f}$ is strongly continuous is seen by a modification of the above argument, working now with approximating sequences $({{\boldsymbol{\omega}}}_{\ell})_{\ell \in {{\mathbb N}}}$ for ${{\boldsymbol{\omega}}}$ with ${{\boldsymbol{\omega}}}_{\ell} \in \overline{\Psi}^r$. Also the properties of weight $k$ and ${{\Gamma}}(N)$-invariance turn over from $f$ to $\overline{f}$.
Finally, suppose $\overline{M}^r(N)$ (and thus $\overline{\widetilde{M}}^r(N)$) is normal. Then the existence of a holomorphic extension $\overline{f}$ of $f$ follows, again by Bartenwerfer’s criterion, from the existence of a strongly continuous extension. Hence in this case, (a) implies in fact (a$^{\rm st}$), and all four conditions are equivalent.
[**7.5 Definition.**]{} We define the Satake compactification $M^r(N)^{\rm Sat}$ of $M^r(N)$ as the normalization of $\overline{M}^r(N)$ in its function field ${{\mathcal F}}_r(N)$. It is a normal projective $C_{\infty}$-variety provided with an embedding $\iota:\: M^r(N) \hookrightarrow M^r(N)^{\rm Sat}$ and a finite birational morphism $\nu:\: M^r(N)^{\rm Sat} {{\longrightarrow}}\overline{M}^r(N)$ such that $\nu \circ \iota$ is the identity on $M^r(N)$. Likewise, we define $\widetilde{M}^r(N)^{\rm Sat}$ as the normalization of $\overline{\widetilde{M}}^r(N)$ in its function field $\widetilde{{{\mathcal F}}}_r(N)$. It has similar properties and is supplied with an action of ${{\mathbb G}}_m$ such that ${{\mathbb G}}_m\setminus \widetilde{M}^r(N)^{\rm Sat} \stackrel{\cong}{{{\longrightarrow}}} M^r(N)^{\rm Sat}$.
[**7.6 Corollary**]{} (to the proof of Theorem 7.4; see also [@28], proof of Proposition 1.23): [*The varieties $\overline{\widetilde{M}}^r(N)$ and $\overline{M}^r(N)$ are unibranched, that is, the canonical maps $$\widetilde{\nu}:\: \widetilde{M}^r(N)^{\rm Sat} {{\longrightarrow}}\overline{\widetilde{M}}^r(N) \mbox{ and }
\nu:\: M^r(N)^{\rm Sat} {{\longrightarrow}}\overline{M}^r(N)$$ are bijective.*]{}
Since the open subspace $\widetilde{M}^r(N)$ of $\overline{\widetilde{M}}(N)$ is smooth, it suffices to consider boundary points $[{{\boldsymbol{\omega}}}]$ of $\overline{\widetilde{M}}(N)$ represented by ${{\boldsymbol{\omega}}}$ as in the proof of . Then we have to show that $[{{\boldsymbol{\omega}}}]$ has at most one pre-image in the normalization $\widetilde{M}^r(N)^{\rm Sat}$. However, this follows from the connectedness of the sets $\Psi^r \cap Y$ in (7.4.3). (If there were several pre-images of $[{{\boldsymbol{\omega}}}]$ then $Y\setminus \{{{\boldsymbol{\omega}}}\}$ and also $\Psi^r\cap Y$ had to split into several components for $Y$ sufficiently small.) The argument for $\overline{M}^r(N)$ follows the same lines.
[**7.7 Remark.**]{} We know from (6.10) that the singular locus of $\overline{M}^r(N)$ is contained in $$\overline{M}^r_{\leq r-2}(N) := \bigcup_{U \in \mathfrak U \atop \dim U \leq r-2} M_U(N)$$ and therefore has codimension $\geq 2$, as expected for a normal variety. Together with unibranchedness this suggests that $\overline{M}^r(N)$ should itself be normal, i.e., $M^r(N)^{\rm Sat} = \overline{M}^r(N)$. However, this is not a formal implication, and the question of normality of $\overline{M}^r(N)$ is still open.
(7.8) At least, $\nu:\: M^r(N)^{\rm Sat} {{\longrightarrow}}\overline{M}^r(N)$ is bijective by (7.6) and therefore (since it is a finite morphism) a strong homeomorphism of the sets of $C_{\infty}$-points. As we don’t know whether $\nu$ is always an isomorphism (see Section 8 for examples), we make the following double definition.
(7.8.1) A [*strong modular form*]{} of weight $k$ and level $N$ is a weak modular form $f$ that satisfies condition (7.3)(a$^{\rm st}$), that is, $f$ extends to a holomorphic section of $\mathfrak M^{\otimes k}$ over $\overline{M}^r(N)$. A [*modular form*]{} of weight $k$ and level $N$ is a weak modular form $f$ that satisfies (7.3)(a), i.e., the boundary condition is relaxed to: $f$ extends to a strongly continuous section of $\mathfrak M^{\otimes k}$, or equivalently (by (7.6) and Bartenwerfer’s criterion), $f$ extends holomorphically to a section of $\nu^*(\mathfrak M^{\otimes k})$ over the Satake compactification $M^r(N)^{\rm Sat}$.
(7.8.2) We let ${\bf Mod}_k´^{\rm st}(N)$ resp. ${\bf Mod}_k(N)$ be the $C_{\infty}$-spaces of (strong) modular forms of weight $k$ and $${\bf Mod}^{\rm st}(N) = \underset{k\geq 0}{\oplus} {\bf Mod}^{\rm st}_k(N),\: {\bf Mod}(N) = \underset{k\geq 0}{\oplus} {\bf Mod}_k(N)$$ the corresponding graded $C_{\infty}$-algebras. Then $${\bf Eis}(N) \subset {\bf Mod}^{\rm st}(N) \subset {\bf Mod}(N).\leqno{(7.8.3)}$$ The common quotient field of the three rings is the field $\widetilde{{{\mathcal F}}}_r(N)$. By (7.4) the following criterion holds.
[**7.9 Theorem.**]{}
*Let $f$ be a weak modular form of weight $k$ and level $N$. Then the following three conditions are equivalent:*
- $f \in {\bf Mod}(N)$;
- $f$ is integral over ${\bf Mod} = C_{\infty}[g_1,\ldots,g_r]$;
- $f$ and all its conjugates $f_{[{{\gamma}}]_k}$ (${{\gamma}}\in RS$) are bounded on the fundamental domain ${{\boldsymbol{F}}}$.
(7.10) Let $J$ be the ideal of Eisenstein relations in $R = {\rm Sym}({\rm Eis}_1(N))$, see (5.4). To the exact sequence $$0 {{\longrightarrow}}J {{\longrightarrow}}R {{\longrightarrow}}{\bf Eis}(N) {{\longrightarrow}}0$$ corresponds an exact sequence of sheaves (in the algebraic sense) on the variety $\mathbb P = {\rm Proj}(R)$ $$0 {{\longrightarrow}}\mathfrak J {{\longrightarrow}}{{\mathcal O}}_{{{\mathbb P}}} {{\longrightarrow}}{{\mathcal O}}_{\overline{M}^r(N)} {{\longrightarrow}}0,\leqno{(7.10.1)}$$ where we regard the structure sheaf ${{\mathcal O}}_{\overline{M}^r(N)} $ of $\overline{M}^r(N)$ as a sheaf on ${{\mathbb P}}$ with support in $\overline{M}^r(N) \hookrightarrow {{\mathbb P}}$. It remains exact upon tensoring with the sheaf ${{\mathcal O}}(k)= {{\mathcal O}}(1)^{\otimes k}$ over ${{\mathbb P}}$, where $k>0$. As ${{\mathcal O}}(1)$ restricted to $\overline{M}^r(N)$ is the sheaf $\mathfrak M$ of strong modular forms, we find the exact sequence $$0 {{\longrightarrow}}\mathfrak J(k) {{\longrightarrow}}{{\mathcal O}}_{{{\mathbb P}}}(k) {{\longrightarrow}}\mathfrak M(k) {{\longrightarrow}}0.\leqno{(7.10.2)}$$ The first part of its exact cohomology sequence reads: $$\begin{array}{c}
0 {{\longrightarrow}}H^0({{\mathbb P}},\mathfrak J(k)) {{\longrightarrow}}H^0({{\mathbb P}},{{\mathcal O}}_{{{\mathbb P}}}(k)) \stackrel{\alpha}{{{\longrightarrow}}} H^0({{\mathbb P}},\mathfrak M(k)) {{\longrightarrow}}\vspace{0.2cm}\\
{{\longrightarrow}}H^1({{\mathbb P}},\mathfrak J(k)) {{\longrightarrow}}H^1({{\mathbb P}},{{\mathcal O}}_{{{\mathbb P}}}(k)) {{\longrightarrow}}\ldots
\end{array}$$ Now,
- $H^1({{\mathbb P}},{{\mathcal O}}_ {{{\mathbb P}}}(k))$ vanishes (see e.g. [@25] III Theorem 5.1);
- $H^0({{\mathbb P}},\mathfrak M(k)) = H^0(\overline{M}^r(N),\mathfrak M^{\otimes k}) = {\bf Mod}^{\rm st}_k(N)$;
- ${\rm im}(\alpha)$ is the subspace ${\bf Eis}_k(N)$ of strong modular forms that belong to the Eisenstein algebra ${\bf Eis}(N)$.
Hence $H^1({{\mathbb P}},\mathfrak J(k))$ measures the difference between ${\bf Eis}_k(N)$ and ${\bf Mod}^{\rm st}(N)$. By standard properties ([@25] III Theorem 5.2), $H^1({{\mathbb P}},\mathfrak J(k))$ vanishes for large $k$. As it is always finite-dimensional, we see:
[**7.11 Corollary.**]{} [ *For $k$ sufficiently large, ${\bf Mod}_k^{\rm st}(N)$ agrees with its subspace ${\bf Eis}_k(N)$. In particular, the Eisenstein algebra ${\bf Eis}(N)$ has finite codimension in the algebra ${\bf Mod}^{\rm st}(N)$.* ]{}
[**7.12 Corollary.**]{} [ *$\overline{M}^r(N)={\rm Proj}({\bf Eis}(N)) = {\rm Proj}({\bf Mod}^{\rm st}(N))$.*]{}
The first equality has been shown in Section 5, the second is a formal consequence of the definition of ${\bf Mod}^{\rm st}(N)$, but follows also from $\dim({\bf Mod}^{\rm st}(N)/{\bf Eis}(N)) < \infty$.
(7.13) Consider the projective variety ${\rm Proj}({\bf Mod}(N))$ attached to the algebra of modular forms. It is normal (as ${\bf Mod}(N)$ is integrally closed), provided with a natural map to ${\rm Proj}({\bf Mod}^{\rm st}(N)) = \overline{M}^r(N)$, and birational with $\overline{M}^r(N)$, and thus agrees with the normalization, i.e., $$M^r(N)^{\rm Sat} = {\rm Proj}({\bf Mod}(N)). \leqno{(7.13.1)}$$
[**7.14 Corollary.**]{}
*The three assertions are equivalent:*
- $\overline{M}^r(N)$ is normal.
- ${\bf Mod}^{\rm st}(N) = {\bf Mod}(N)$;
- ${\bf Mod}^{\rm st}(N)$ has finite codimension in ${\bf Mod}(N)$.
\(i) $\Rightarrow$ (ii) is the last assertion of Theorem 7.4, and (ii) $\Rightarrow $ (iii) is trivial. Suppose that (iii) holds. Then ${\rm Proj}({\bf Mod}(N)) = {\rm Proj}({\bf Mod}^{\rm st}(N)) = \overline{M}^r(N)$, and it follows that the latter is normal.
[**7.15 Corollary.**]{} [ *Suppose that $\overline{M}^r(N)$ fails to be normal. Then there exist arbitrarily large weights $k$ such that ${\bf Mod}_k(N)$ is strictly larger than its subspace ${\rm Eis}_k(N)$.*]{}
As all the ${\bf Mod}_k(N)$ have finite dimension, this follows from the last corollary.
(7.16) We conclude this section with an observation about ${\bf Mod}^{\rm st}(N)$. Given ${{\boldsymbol{u}}}\in {{\mathcal T}}(N)$, we let ${\bf Eis}(N)_{E_{{{\boldsymbol{u}}}}}$ be the localization w.r.t. $E_{{{\boldsymbol{u}}}}$, i.e., ${\rm Spec}({\bf Eis}(N)_{E_{{{\boldsymbol{u}}}}})$ is the open subvariety of $\overline{\widetilde{M}}^r(N)$ where $E_{{{\boldsymbol{u}}}}$ doesn’t vanish. Hence $$\overline{\widetilde{M}}^r(N) = \bigcup_{{{\boldsymbol{u}}}\in {{\mathcal T}}(N)} {\rm Spec}({\bf Eis}(N)_{E_{{{\boldsymbol{u}}}}}) =\bigcup_{{{\boldsymbol{u}}}\in N^{-1}S}
\ldots,$$ where $S$ is the set of representatives of $(A/N)^r_{\rm prim}/{{\boldsymbol{F}}}^*$ given in (3.5). Similarly, $$\overline{M}^r(N) = \bigcup_{{{\boldsymbol{u}}}\in N^{-1}S} \overline{M}^r(N)_{(E_{{{\boldsymbol{u}}}} \not= 0)}.$$ A weak modular form of weight $k$ extends to a section of $\mathfrak M^{\otimes k}$ if and only if its restriction to each $\overline{M}^r(N)_{(E_{{{\boldsymbol{u}}}} \not= 0)}$ has the corresponding property, that is, belongs to ${\bf Eis}(N)_{E_{{{\boldsymbol{u}}}}}$. Therefore we may describe ${\bf Mod}^{\rm st}(N)$ as the intersection $${\bf Mod}^{\rm st}(N) = \bigcap_{{{\boldsymbol{u}}}\in N^{-1}S} {\bf Eis}(N)_{E_{{{\boldsymbol{u}}}}} \leqno{(7.16.1)}$$ in $\widetilde{{{\mathcal F}}}(N)$.
[**8. Examples and concluding remarks.**]{}
The preceding immediately raises a number of important questions and desiderata.
[**8.1 Question.**]{} Do the Eisenstein and Satake compactifications $\overline{M}^r(N)$ and $M^r(N)^{\rm Sat}$ always coincide, i.e., is $\overline{M}^r(N)$ always normal?
(8.2) Describe the singularities of both compactifications and construct natural desingularizations together with a modular interpretation! (See [@31] for some results.)
(8.3) How far do the algebras ${\bf Eis}(N)$, ${\bf Mod}^{\rm st}(N)$, ${\bf Mod}(N)$ differ, if at all? Describe their Hilbert functions, that is, the dimensions of their pieces in dimension $k$, and find presentations for these algebras!
Almost nothing about these questions is known when the rank $r$ is larger than 2. We will briefly present the state-of-the-art in the case where , which we assume until (8.9). Here the $\overline{M}^2(N)$ are smooth curves [@11], so the Eisenstein and Satake compactification agree, and therefore ${\bf Mod}^{\rm st}(N) = {\bf Mod}(N)$. The genera of the $\overline{M}^2(N)$ have been determined by Goss [@23] and, with a different method, by the author [@13].
(8.4) Let $d \geq 1$ be the degree of $N$, and suppose that $$N = \prod_{1 \leq i \leq t} \mathfrak p_i^{s_i}$$ is the prime decomposition, where the $\mathfrak p_{i}$ are different monic prime polynomials. As in (3.6), write $q_i = q^{\deg\,\mathfrak p_i}$. We define $${{\lambda}}(N) := \prod_{1\leq i \leq t}q_i^{2s_i-2} (q_i^2-1),\leqno{(8.4.1)}$$ which appears in the formulas below. (Note that ${{\lambda}}(N) = \varphi(N)\epsilon(N)$ with the arithmetic functions $\varphi$, $\epsilon$ defined in [@17] 1.5.) Then the numbers $g(N) = $ genus of the modular curve $\overline{M}^2(N)$, $c(N)=c_2(N)=$ number of cusps of $M^2(N)$, $\deg(\mathfrak M) =$ degree of the line bundle of modular forms over $\overline{M}^2(N)$ and $\dim\,{\bf Mod}_k(N) = \dim\,{\bf Mod}_k^{\rm st}(N)$ are given by
(8.4.2) $g(N) = 1+{{\lambda}}(N)(q^d-q-1)/(q^2-1)$;
(8.4.3) $c(N) = {{\lambda}}(N)/(q-1)$;
(8.4.4) $\deg(\mathfrak M) = {{\lambda}}(N)q^d/(q^2-1)$;
(8.4.5) $\dim\,{\bf Mod}_k(N) = ((k-1)q^d+q+1){{\lambda}}(N)/(q^2-1)$.
Here (8.4.2) and (8.4.3) may be found in [@23] and [@13] (such data for other Drinfeld modular curves are collected in [@17]) and (8.4.4) is from [@15] VII 6.1. The last formula (8.4.5) is an immediate consequence of the Riemann-Roch theorem provided that $k \geq 2$; for $k=1$, Riemann-Roch and Serre duality yield only $$\dim\,{\bf Mod}_1(N) =c(N)+\dim\,H^1(\overline{M}^2(N),\mathfrak M),\leqno{(8.4.6)}$$ where $c(N) = \dim\,{\bf Eis}_1(N) = \dim\,{\rm Eis}_1(N)$ and $\dim\,H^1(\overline{M}^2(N),\mathfrak M) =
\dim\,{\bf Mod}_1^2(N)$ with the space ${\bf Mod}_1^2(N)$ of double cuspidal (double zeroes at the cusps) modular forms of weight 1 ([@15] p. 92). However by the next result, (8.4.5) is valid for $k=1$, too.
[**8.5 Proposition.**]{} [ *For $r=2$ we have ${\rm Eis}_1(N) = {\bf Mod}_1(N)$, of dimension $c(N)$.*]{}
As a proof of this basic fact so far has not been published, we give a brief sketch here.
By Corollary 4.7, $\dim\,{\rm Eis}_1(N) = c(N)$. Since moreover the space of cusp forms ${\bf Mod}_1^1(N)$ is a complement of ${\rm Eis}_1(N)$ in ${\bf Mod}_1(N)$ (this is a consequence of Proposition 4.6), it suffices to show that there are no non-trivial cusp forms of weight 1.
Assume that $f \in {\bf Mod}_1^1(N)$. Then $f^p$ is a cusp form of weight $p \geq 2$ for ${{\Gamma}}(N)$, where $p = {\rm char}({{\boldsymbol{F}}})$. For $0 \leq i \leq p-2$, the residue ${\rm res}_e {{\omega}}^{i} f^p({{\omega}}) d{{\omega}}$ of the differential form ${{\omega}}^{i}f^p({{\omega}})d{{\omega}}$ on ${{\Omega}}= {{\Omega}}^2$ at the oriented edge $e$ of the Bruhat-Tits tree of ${\rm PGL}(2,K_{\infty})$ vanishes for each $e$, as is immediate from the definition of ${\rm res}_e$, see [@34] Definition 9. Hence the image ${\rm res}(f^p)$ under Teitelbaum’s isomorphism ([*loc.cit.*]{}, Theorem 16) of cusp forms of weight $p$ with the space of cocyles of a certain type vanishes, and so do $f^p$ and $f$ itself.
[**8.6 Remark.**]{} All the formulas and results in (8.4) and (8.5) have generalizations
- to other congruence subgroups of ${{\Gamma}}$, e.g., Hecke congruence subgroups ${{\Gamma}}_0(N)$, ${{\Gamma}}_1(N)$, etc., see [@17];
- to more general Drinfeld rings $A$ than $A={{\mathbb F}}[T]$, e.g., $A$ the affine ring of an elliptic curve over ${{\mathbb F}}$, see [@15] pp. 92–93.
As to the relationship between ${\bf Eis}(N)$ and ${\bf Mod}(N)$, there is the following result of Cornelissen [@9].
[**8.7 Theorem**]{} (Cornelissen). [*Let $r=2$. The algebra ${\bf Mod}(N)$ of modular forms for ${{\Gamma}}(N)$ is generated by ${\bf Mod}_1(N) = {\rm Eis}_1(N)$ and the space ${\bf Mod}_2^1(N)$ of cusp forms of weight 2.*]{}
In fact, it is not difficult (using [@14] Korollar 2.2) to show that ${\bf Mod}_2^1(N)$ above may be replaced with the space ${\bf Mod}_2^2(N)$ of double cuspidal forms of weight 2, which under $f({{\omega}}) \longmapsto f({{\omega}})d{{\omega}}$ corresponds to the $g(N)$-dimensional space of holomorphic differentials on $\overline{M}^2(N)$. Still, this doesn’t completely answer the question (8.3) of whether ${\bf Eis}(N) = {\bf Mod}(N)$ in this case. The only positive results in this direction seem to be the following two examples.
[**8.8 Example.**]{} Suppose that $r=2$ and $d = \deg\,N=1$. Then $g(N)=0$, that is, $\overline{M}^2(N)$ is a projective line, and $\deg(\mathfrak M)=q$. Therefore, ${\bf Eis}(N)={\bf Mod}(N)$ and $\dim\,{\bf Mod}_k(N) = 1+kq$ in this case. From this, a presentation of ${\bf Mod}(N)$ may be derived ([@9], see also [@35], [@36], which also study the spaces ${\bf Mod}_k(N)$ as modules under the action of ${{\Gamma}}/{{\Gamma}}(N) = {\rm GL}(2,{{\mathbb F}})$).
[**8.9 Example**]{} (Cornelissen [@8]). Let again $r=2$, $q=2$, and $d = \deg\,N=2$. Then ${\bf Eis}(N) = {\bf Mod}(N)$.
As the possible genera $g(N)$ here are positive (they may take the values 4, 5 and 6), this case is less trivial than (8.8). The equality of the two rings, i.e., the projective normality of the Eisenstein embedding $\overline{M}^2 \hookrightarrow {{\mathbb P}}$, is based on a numerical criterion of Castelnuovo. Unfortunately, the validity of this argument is strictly limited to the requirements of Example 8.9.
For the next example, we return to the general case, where $r \geq 2$ is arbitrary.
[**8.10 Example.**]{} Suppose that $d = \deg\,N=1$. After a coordinate change, we may assume $N=T$. This case has been extensively studied by Pink and Schieder [@32]. Actually, they consider a certain ${{\mathbb F}}$-variety $Q_V$, but which after base extension with $C_{\infty}$ and some translational work may also be seen as our $\overline{M}^r(T)$. Their results (overlapping with (8.8) if $r=2$) give
- ${\bf Eis}(T) = {\bf Mod}(T)$ (Theorem 1.7 in [@32]), i.e., the normality of $R_V = {\bf Eis}(T)$;
- a presentation through generators and relations (Theorem 1.6);
- the Hilbert function of ${\bf Eis}(T)$ (Theorem 1.10).
Further, they construct and discuss a desingularization $B_V$ of $Q_V$ (Section 10). Hence, concerning our questions (8.1)–(8.3), nothing is left to desire. But note that these satisfactory and complete results refer only to the (isolated?) case where $\deg\,N = 1$. None of our current knowledge excludes the possibility that always ${\bf Eis}(N) = {\bf Mod}(N)$, or the sheer opposite possibility that the two rings differ almost always and (8.8), (8.9), (8.10) are just extreme boundary cases.
[99]{} Bartenwerfer, Wolfgang: Der erste Riemannsche Hebbarkeitssatz im nichtarchimedischen Fall. J. Reine Angew. Math. 286/87 (1976), 144–163. Basson, Dirk: A product formula for the higher rank Drinfeld discriminant function. J. Number Theory 178 (2017), 190–200. Basson, Dirk; Breuer, Florian: On certain Drinfeld modular forms of higher rank. J. Théor. Nombres Bordeaux 29 (2017), 827–843. Basson, Dirk; Breuer, Florian; Pink, Richard: Drinfeld modular forms of arbitrary rank. Part I: Analytic theory. ArXiv 1805.12335. Basson, Dirk; Breuer, Florian; Pink, Richard: Drinfeld modular forms of arbitrary rank. Part II: Comparison with algebraic theory. ArXiv 1865.12337 Basson, Dirk; Breuer, Florian; Pink, Richard: Drinfeld modular forms of arbitrary rank. Part III: Examples. ArXiv 1805.12339. Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold: Non-archimedean analysis. Grundlehren der Math. Wiss. 261, Springer-Verlag 1984. Cornelissen, Gunther: Geometrical properties of modular forms over rational function fields. Doctoral Thesis Ghent 1997. Cornelissen, Gunther: Drinfeld modular forms of weight one. J. Number Theory 67 (1997), 215–228. Deligne, Pierre; Husemöller, Dale: Survey of Drinfel’d modules. Contemp. Math. 67 (1987), 25–91. Drinfeld, V.G.: Elliptic modules (Russian) Mat. Sb. 94 (1974) 594–627. English Translation: Math. USSR-Sbornik 23 (1976), 561–592. Fresnel, Jean; van der Put, Marius: Rigid analytic geometry and its applications. Progr. Math. 218, Birkhäuser 2004. Gekeler, Ernst-Ulrich: Drinfeld-Moduln und modulare Formen über rationalen Funktionenkörpern. Bonner Math. Schriften 119, 1980. Gekeler, Ernst-Ulrich: Modulare Einheiten für Funktionenkörper. J. Reine Angew. Math. 348 (1984), 94–115. Gekeler, Ernst-Ulrich: Drinfeld modular curves. Lecture Notes in Math. 1231, Springer-Verlag 1986. Gekeler, Ernst-Ulrich: On the coefficients of Drinfeld modular forms. Invent. Math. 93 (1988), 667–700. Gekeler, Ernst-Ulrich: Invariants of some algebraic curves related to Drinfeld modular curves. J. Number Theory 90 (2001), 166–183. Gekeler, Ernst-Ulrich: Towers of ${\rm GL}(r)$-type of modular curves. J. Reine Angew. Math. (to appear). DOI 10.1515/crelle-2017-0012. Gekeler, Ernst-Ulrich: On Drinfeld modular forms of higher rank. J. Théor. Nombres Bordeaux 29 (2017), 875–902. Gekeler, Ernst-Ulrich: On Drinfeld modular forms of higher rank II. submitted. ArXiv 1708.04197. Gekeler, Ernst-Ulrich: On Drinfeld modular forms of higher rank III: The analogue of the $k/12$-formula. J. Number Theory 192 (2018), 293–306 Gerritzen, Lothar; van der Put, Marius: Schottky groups and Mumford curves. Lecture Notes in Math. 817, Springer-Verlag 1980. Goss, David: $\pi$-adic Eisenstein series for function fields. Comp. Math. 41 (1980), 3–38. Goss, David: Modular forms for ${{\mathbb F}}_r[T]$. J. Reine Angew. Math. 317 (1980), 16–39. Häberli, Simon: Satake compactification of Analytic Drinfeld modular varieties. Doctoral thesis, ETH Zürich 2018. Hartshorne, Robin: Algebraic Geometry. Graduate Texts in Math. 52, Springer-Verlag 1977. Hayes, David: Explicit class field theory for rational function fields. Trans. AMS 189 (1974), 77–91. Hecke, Erich: Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Sem. Univ. Hamburg 5 (1927), 199–224. Kapranov, Michail: Cuspidal divisors on the modular varieties of elliptic modules (Russian) Izv. Akad. Nauk USSR 51 (1987), 568–583. English Translation: Math. USSR-Izv. 30 (1988), 533-547. Lang, Serge: Algebra. Addison-Wesley 1965. Neukirch, Jürgen: Algebraic Number Theory. Grundlehren der Math. Wiss. 322. Springer-Verlag 1999. Pink, Richard: Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank. Manuscripta Math. 140 (2013), 333–361. Pink, Richard; Schieder, Simon: Compactification of a period domain associated to the general linear group over a finite field. J. Algebraic Geometry 23 (2014), 201–243. Schneider, Peter; Stuhler, Ulrich: The cohomology of $p$-adic symmetric spaces. Invent. Math. 105 (1991), 47–122. Teitelbaum, Jeremy: The Poisson kernel for Drinfeld modular curves. J. AMS 4 (1991), 491–511. Varela Roldan, Enrico: Arithmetic of Eisenstein series of level $T$ for the function field modular group ${\rm GL}(2,{{\mathbb F}}_q[T])$. J. Number Theory 167 (2016), 180–201. Varela Roldan, Enrico: Representation theory of Drinfeld modular forms of level $T$. J. Number Theory 172 (2017), 63–92.
Ernst-Ulrich Gekeler\
Fachrichtung Mathematik der Universität des Saarlandes\
Campus E2 4, 66123 Saarbrücken, Germany, gekeler@math.uni-sb.de
|
---
abstract: 'Most of known RR Lyraes are type ab RR Lyraes (RRLab), and they are the excellent tool to map the Milky Way and its substructures. We find that 1148 RRLab stars determined by Drake et al.(2013) have been observed by spectroscopic surveys of SDSS and LAMOST. We derived radial velocity dispersion, circular velocity and mass profile from 860 halo tracers in our paper I. Here, we present the stellar densities and radial velocity distributions of thick disk and halo of the Milky Way. The 288 RRLab stars located in the thick disk have the mean metallicity of \[Fe/H\]$=-1.02$. Three thick disk tracers have the radial velocity lower than 215 km $\rm s^{-1}$. With 860 halo tracers which have a mean metallicity of \[Fe/H\]$=-1.33$, we find a double power-law of $n(r) \propto r^{-2.8}$ and $n(r) \propto r^{-4.8}$ with a break distance of 21 kpc to express the halo stellar density profile. The radial velocity dispersion at 50 kpc is around 78 km $\rm s^{-1}$.'
author:
- Iminhaji Ablimit and Gang Zhao
title: The Density Profile and Kinematics of the Milky Way with RR Lyrae Stars
---
Introduction
============
The recent photometric and spectroscopic surveys have been providing information of stars to measure the stellar number density profile and kinematics of the Galactic disk and halo (e.g., Bland-Hawthorn & Gerhard 2016), and the stellar density profile is the one of the key issues for knowing the nature of the Milky Way. The Galactic thick disk is different from the Galactic thin disk by its unique chemistry, older age and higher elevation (e.g. Bensby 2014; Hawkins et al. 2015). It has been showed that an edge in the stellar disk could be in the range $R_{\rm GC} = 10-15$ kpc from the surveys data (Habing 1988; Minniti et al. 2011).
The Milky Way’s stellar halo is an important component to understand our Galaxy’s formation history. Many observational studies have claimed the broken power-law slope of space density distributions with RR Lyraes (RRLs), globular clusters and blue horizontal branch stars so on (e.g., Saha 1985; Wetterer & McGraw 1996; Miceli et al. 2008; Bell et al. 2008; Watkins et al. 2009; Sear et al. 2010, 2011; Deason et al. 2011). The theoretical study of Font et al. (2011) predict the broken power-law slope of the mass-density profile. The study of Deason et al. (2013) always found the broken stellar halo profile of the Galaxy, and they theoretically explained the origin of the break radius in the RRL profiles in terms of the apocentric distance of the satellites that were accreted. More recently, Iorio et al. (2017) demonstrated the properties of the Galactic stellar halo with RRL in the Gaia data, and they showed that the inner halo ($R_{\rm GC} < 28$ kpc) stellar density profile is well approximated by a single power-law with exponent $\alpha = -2.96$. Cohen et al. (2017) presented RRLs determined in the Palomar Transient Facility database, and derived the stellar density profile with the power-law index of -4 for the outer halo of the Milky Way.
RRL is a very good standard candle because of the narrow luminosity-metallicity-relation in the visual band and period-luminosity-metallicity relations in the near-infrared wavelengths. Besides, most of RRL stars are type ab RRLs (RRLab), and they are old ($>10$ Gyr) with low metallicity, so that they are mainly distributed in the bulge, thick disk and halo (Smith 1995; Demarque et al. 2000).
In this paper, we investigate density profiles of the Galactic thick disk and halo by using 1148 RRLab variable stars presented in our previous paper (Ablimit & Zhao 2017, hereafter paper I). We briefly describe our RRL sample and methods in §2. The kinematic and density information of the thick disk and halo are discussed in §3. In the last section (§4), the conclusions are given.
The sample and methods {#sec:model}
======================
The RRLab sample of this work were selected from 12227 RRLab stars of Catalina Surveys Data Release 1 (Drake et al. 2013a), and the cross matches were made with Sloan Digital Sky Survey (SDSS) spectroscopic data release 8 (DR8) and Large sky Area Multi-Object fiber Spectroscopic Telescope (LAMOST) DR4 within an angular distance of 3 arcsecond. LAMOST is a Chinese national scientific research facility operated by National Astronomical Observatories, Chinese Academy of Sciences (Zhao et al. 2006, 2012; Cui et al. 2012). We found 797 LAMOST matched RRLab stars and 351 SDSS matched RRLab stars with the corrected radial velocities (uncertainty $<15$ $\rm km\,\rm s^{-1}$) and reliable metallicities (in the region of Galactic centric distance $\leq 50$ kpc, for more details see paper I).
An absolute magnitude-metallicity relation has been adopted for distance determination (Sandage 1981). The method of Chaboyer (1999) and Cacciari & Clementini (2003) is used in this work as, $$M_{\rm V} = (0.23 \pm 0.04)([\rm Fe/H] + 1.5) + (0.59 \pm 0.03),$$ where \[Fe/H\] is the metallicity of an RR Lyrae star. We find our overall uncertainties is around 0.15 mag from the uncertainties from the photometric calibration and the variations in metallicity and uncertainty in RRab absolute magnitudes (also see Dambis et al. (2013) for the uncertainty in absolute magnitude). The uncertainties of $\sim 7\%$ in distances are expected by our overall uncertainties of $M_{\rm V}$. The heliocentric $d$ and Galactocentric distances $R_{\rm GC}$ can be derived from the equations, $$d = 10^{(<V> - M_{\rm V} + 5)/5}\, {\rm kpc},$$ where $<V>$ average magnitudes were corrected for interstellar medium extinction using Schlegel et al. (1998) reddening maps, and $$R_{\rm GC} = (R_\odot - d{\rm cos}\,b\, {\rm cos}\,l)^2 + d^2{\rm cos}^2\,b\, {\rm sin}^2\,l +
d^2{\rm sin}^2\,b \, {\rm kpc},$$ where $R_\odot$, $l$ and $b$ are the distance from the sun to the Galactic center (8.33 kpc in this work, see Gillessen et al. 2009), Galactic longitude and latitude of the stars, respectively.
We obtain the fundamental (Galactic) standard of rest (FSR) of stars by using the heliocentric radial velocities ($V_{\rm h}$, the corrected ones:see our paper one) and the solar peculiar motion of (U, V, W) = (11.1, 12, 7.2) km ${\rm s}^{-1}$ (Binney & Dehnen 2010) which are defined in a right-handed Galactic system with U pointing toward the Galactic center, V in the direction of rotation, and W toward the north Galactic pole. The value of $235\pm7$ km ${\rm s}^{-1}$ is taken for the local standard of rest ($\rm{V}_{\rm lsr}$, Reid at al. 2014) in the equation below, $$V_{\rm FRS} = V_{\rm h} + {\rm U} {\rm cos}\, b\, {\rm cos}\, l + ({\rm V} + {\rm V}_{\rm lsr})
{\rm cos}\, b\, {\rm sin}\, l + {\rm W} {\rm sin}\, b.$$
For deriving the spatial density of our RRLab sample, we followed the density calculation as a function of Galactocentric distance described by Wetterer & McGraw (1996), as $${\rho}({R_{\rm GC}}) = \frac{1}{4\pi {R^2_{\rm GC}} f({R_{\rm GC}})}\frac{dN}{dR}$$ $N$ is the number of RRLab as a function of distance and $f({R_{\rm GC}})$ is the fraction of the total halo volume at $R_{\rm GC}$ sampled by the survey. The efficiency or completeness of sampling (selection process) is a way to achieve each $f({R_{\rm GC}})$ for each individual field. Drake et al. (2013a) discussed the Catalina Surveys efficiency of RRL sampling, and their Figure 13 showed the detection completeness as a function of magnitude. We followed Drake et al. (2013a) for $f({R_{\rm GC}})$ by adopting completeness as 70% for $V < 17.5$ mag and it is gradually reach to 0% from $V = 17.5$ to 20 mag.
Results
========
Our sample of 1148 RRLab stars contains 288 thick disc stars with $1
< |z| < 4$ kpc, and also 860 halo stars with $|z| > 4$ kpc (see Figure 3 of paper I). We use the equations introduced above section to demonstrate the stellar number density and velocity distributions of the thick disk and halo.
The thick disk profile
----------------------
Figure 1 demonstrates the stellar density map in the ${R_{\rm GC}}$–Z plane for the thick disk, the bin size is $2\times0.5$ kpc. We assume the thick disk has a shape of cylinder, and get the volume from the ${R_{\rm GC}}$ and Z (Z as a height). The thick disk has a range of $1< |z| < 4$ kpc in the vertical direction, and there is a gap between -1 and 1 kpc because of thin disk, RRLab stars are old and metal-poor stars. The tomographic map distributed in a Galactocentric distance range of 4.5–14.5 kpc. There are two ring areas in the Figure 1, the high density ring showed at the region of ${R_{\rm GC}}= 8$ – $10.5$ kpc and $Z= -1.5$ – $-2.5$ kpc, while the relatively lower density ring showed up at the region of ${R_{\rm GC}}= 8$ – $ 9.5$ kpc and $Z= 2.2$ – $ 2.8$ kpc (for the similar results see Newberg et al (2003), Juri$\acute{\rm c}$ et al. (2008) and Ivezi$\acute{\rm c}$ et al. (2008)). LAMOST survey covers different areas at different longitudes. Thus, these two density regions might be caused by the selection effect of LAMOST. From $Z= -1.5$ – $- 4$ kpc the density distribution yield a heart shape between ${R_{\rm GC}}= 7$ – $ 13$ kpc, and gradually decreases until ${R_{\rm GC}}= 14.5$ kpc (the outer edge). Combining our results of the thick disk and halo, we agree the conclusion of Liu et al. (2017) which is that the disk smoothly transit to the halo without any truncation (also see Liu et al. 2017).
The metallicity and velocity profiles of the thick disk RRLab stars are given in the upper and lower panels of Figure 2. Most of the RRLab are metal-poor stars which have \[Fe/H\] around or lower than -1 dex while few of them with around 0. The red line in the upper panel of the figure is the mean value (\[Fe/H\]$=-1.02$) of all the thick disk tracers. There are three stars which have the radial velocity $<$ -215 km ${\rm s}^{-1}$ around 7.5 kpc, while other the RRLab have the radial velocities higher than -210 km ${\rm s}^{-1}$ (see $V_{\rm FSR}$ distribution in the lower panel of Figure 2).
The halo profile
----------------
The number density of RRL has been on debate such as break or no break in the density distribution. Ivezi$\acute{\rm c}$ et al. (2000) claimed the existence of a break in the density distribution in the halo at ${R_{\rm GC}}\sim 50$ kpc by using 148 SDSS RRLs. However, Ivezi$\acute{\rm c}$ et al. (2004) and Vivas & Zinn (2006) have found no break until $\sim$ 60 or 70 kpc. A broken-power law has been considered as a better number density profile for the RRLs by a number of works (e.g., Saha 1985; Sesar et al. 2007; Keller et al. 2008; Watkins et al. 2009; Akhter et al. 2012; Faccioli et al. 2014). Drake et al. (2013b) presented 1207 RRLs taken by the Caltalina Survey’s Mount Lemmon telescopes, and found the number density out to 100 kpc with $\sim$70% detection efficiency and a break appeared around 50 kpc (see the Figure 12 of Drake et al. (2013b)). They claimed their density profile is good agreement with the Watkins et al.(2009), and the different break is caused by the density enhanced by RRLs in the Sagittarius stream leading and trailing arms. We further analyze 860 RRLab stars from Drake et al. (2013a) by combining LAMOST DR4 and SDSS DR8 data to show the density profile in the 9–50 kpc range. We consider a spherical averaged number density and fit by following formula,
$$n(R_{\rm GC}) = n_0 \left\{ \begin{array}{ll}
(\frac{R_0}{R_{\rm GC}})^{\alpha} & \textrm{if $R_{\rm min}<{R_{\rm GC}} < R_0$}\\
\\
(\frac{R_0}{R_{\rm GC}})^{\beta} & \textrm{if $R_0<{R_{\rm GC}} < R_{\rm max}$},
\end{array} \right.$$
and derive the following values for the halo parameters : $n_0 = 0.35\pm0.18$ ${\rm kpc}^{-3}$, $R_0=21\pm2$ kpc, $\alpha=2.8\pm0.4$ and $\beta=4.8\pm0.4$ (see Figure 3). Watkins et al. (2009) gave their best results as: $(n_0,\,R_0,\,\alpha,\,\beta)=(0.26\,{\rm kpc}^{-3},\,23\,{\rm kpc},\,2.4\,,4.5)$. Faccioli et al. (2014) demonstrated 318 RRLs observed by Xuyi Schmidt telescope photometric survey, and obtained the density profiles by including and removing the possible Sagittarius RRLs. There is not a significant difference between their results with and without Sagittarius RRLs. Their spherical double-power model results are shown in their table 2, and the result with all RRLs are $n_0 = 0.42\pm0.16$ ${\rm kpc}^{-3}$, $R_0=21.5\pm2.2$ kpc, $\alpha=2.3\pm0.5$ and $\beta=4.8\pm0.5$. It is obvious that our result are strongly supports the results of Watkins et al. (2009) and Faccioli et al. (2014).
Figure 4 shows the velocity $V_{\rm FSR}$ distributions of 860 RRLab stars. In general, the radial velocities of RRLab stars are smoothly distributed with the distance, and a small part has a higher radial velocities at inner Galaxy, this may caused by the stream effect. We find the radial velocity dispersion at 50 kpc is $\sim$78 km ${\rm s}^{-1}$ (for more details also see paper I), which is smaller than $\sim$90 km ${\rm s}^{-1}$ derived by Cohen et al. (2017). We used 860 RRLab stars to measure the radial velocity dispersion for the halo based on the SDSS and LAMOST spectroscopic surveys, and Cohen et al. (2017) determined the radial velocity dispersion by only using 112 RRLs based on the moderate resolution spectra with Deimos on the Keck 2 Telescope.
Conclusions
===========
In this work, we have investigated the density profiles and velocity distributions of the thick disk and halo of the Milky Way, based on 1148 RRLab variables with precise distances (7% uncertainty) and reliable radial velocities (uncertainty $<$ 15 km $\rm s^{-1}$) presented in paper I. The 288 thick disk RRLab stars have the mean metallicity of \[Fe/H\]$=-1.02$. Despite three RRLab with the radial velocity lower than -215 km $\rm s^{-1}$, other disk tracers distributed in a region $>$ -210 km $\rm s^{-1}$. Our result shows that the edge of the thick disk is around 14.5 kpc. The halo of the Milky Way have been studying by using RRLab variables. Comparing to previous works, we present a larger sample (860) of halo RRLab variables with the mean metallicity of \[Fe/H\]$=-1.33$. The stellar density distribution of the halo tracers can be well fitted by a broken power-law, and power law index of -2.8 for $<$ 21 kpc & the index of -4.8 for $\geq$ 21 kpc. This density distribution is agreed by most of other works, especially by the works which used RRLs as the tracer. The radial velocity dispersion at 50 kpc is $\sim$78 km $\rm s^{-1}$, and few halo tracers show high radial velocities while that of others smoothly distributed.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by JSPS International Postdoctoral Fellowship of Japan (P17022, JSPS KAKENHI grant no. 17F17022), and also supported by National Natural Science Foundation of China under grant number 11390371 and 11233004. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
**REFERENCES**
Ablimit, I. & Zhao, G. 2017, ApJ, 846, 10
Akhter, S., Da Costa, G. S. et al. 2012, ApJ, 756, 23
Bell, E. F., Zucker, D. B., Belokurov, V., et al. 2008, ApJ, 680, 295
Bensby, T. 2014, A&A, 562, A71
Binney, J. J., & Dehnen, W. 2010, , 403, 1829
Bland-Hawthorn, J. & Gerhand, O. 2016, ARA&A, 54, 529
Cacciari, C., & Clementini, G. 2003, in Stellar Candles for the Extragalactic Distance Scale, ed. D. Alloin & W. Gieren (Lecture Notes in Physics, Vol. 635; Berlin: Springer), 105
Chaboyer, B. 1999, in Post-Hipparcos Cosmic Candles, ed. A. Heck & F. Caputo (Astrophysics and Space Science Library, Vol. 237; Dordrecht: Kluwer), 111
Cohen, G. J., Sesar, B., Bahnolzer, S., et al. 2017, arXiv:1710.01276v1
Cui, X-Q., Zhao, Y-H., Chu, Y-Q. et al., 2012, RAA, 12, 1197
Dambis, A. K., Berdnikov, L. N., Kniazev, A. Y. et al. 2013, , 435, 3206
Deason, A. K., Belokurov, V. & Evans, N. W. 2011, MNRAS, 416, 2903
Deason, A. J., Belokurov, V., Evans, N. W. & Johnston, K. V. 2013, , 763, 113
Demarque, P., Zinn, R., Lee, Y-W. & Yi, S. 2000, AJ, 119, 1398
Drake, A. J., Catelan, M., Djorgovski, S. G., et al. 2013a, , 763, 32
Drake, A. J., Catelan, M., Djorgovski, S. G., et al. 2013b, , 765, 154
Faccioli, L., Smith, M. C. et al. 2014, ApJ, 788, 105
Gillessen, S., Eisenhuaer, F., Trippe, S., et al. 2009, , 692, 1075
Habing, H. 1988, A&A 200, 40
Hawkins, K., Jofre, P, Masseron, T. & Gilmore, G. 2015, MNRAS, 453, 758
Iorio, G., Belokurov, V., Erkal, D. et al. 2017, arXiv:1707.03833v2
Iverzi$\acute{\rm c}$, $\check{\rm Z}$. et al. 2000, AJ, 120, 963
Iverzi$\acute{\rm c}$, $\check{\rm Z}$. et al. 2004, in Clemens D., Shah R. Y., Brainerd T., eds, APS Conf. Ser. Vol. 317, Milky Way Survey: The Structure and Evolution of Our Galaxy. Astron. Soc. Pac., San Francisco, p. 179
Iverzi$\acute{\rm c}$, $\check{\rm Z}$., Sesar, B., Juri$\acute{\rm c}$, M. et al. 2008, ApJ, 684, 287
Juri$\acute{\rm c}$, M., Iverzi$\acute{\rm c}$, $\check{\rm Z}$., Brooks A. et al. 2008, ApJ, 673, 864
Keller, S. C., Murphy, S. et al. 2008, ApJ, 678, 851
Liu, C., Xu, Y., Wan, J-C. et al. 2017, RAA, 17, 96L
Miceli, A., Rest, A., Stubbs, C. W. et al. 2008, ApJ, 678, 865
Minniti, D., Saito, R., Alonso-Garcia, J., Lucas, P. & Hempel, M. 2011, ApJ, 733, L43
Newberg, H. J., Yanny, B., Rockosi, C. et al. 2002, ApJ, 569, 245
Reid, M. J., Menten, K. M., Brunthaler, A. et al., 2014, , 783, 130
Saha, A. 1985, ApJ 289, 310
Sandage, A. R. 1981, , 248, 161
Sesar, B. et al., 2007, ApJ, 134, 2236
Sesar, B. et al., 2010, ApJ, 708, 717
Sesar, B. et al., 2011, ApJ, 731, 4
Smith, H. A. 1995, IrAJ 22, 228S
Watkins, L. L., Evans, N. W., Belokurov, V. et al. 2009, MNRAS, 398, 1757
Wetterer, C. J. & McGraw, J. T. 1996, AJ, 112, 1046
Zhao, G., Chen, Y.-Q., Shi, J.-R. et al. 2006, ChJAA, 6, 265
Zhao, G., Zhao, Y.-H., Chu, Y.-Q., Jing, Y.-P., Deng, L.-C. 2012, RAA, 12, 723
![The mean spatial density for RRLab stars. The color codes for each $R_{\rm GC}$–Z bin. There is gap between -1 and 1 kpc, because the thick disk is considered as a disk of $1<|Z|<4$ kpc.[]{data-label="fig:1"}](f1){width="4.5in"}
![The upper panel shows the metallicity distribution of the thick disk tracers (the red line represents the mean value), and the velocity distribution of the fundamental standard rests are shown in the lower panel.[]{data-label="fig:1"}](f2a "fig:"){width="4.1in"} ![The upper panel shows the metallicity distribution of the thick disk tracers (the red line represents the mean value), and the velocity distribution of the fundamental standard rests are shown in the lower panel.[]{data-label="fig:1"}](f2b "fig:"){width="4.7in"}
![The number density of the halo RRLab stars as a function of Galactocentric distance in a range of 9–50 kpc. The blue line is the fit result of Watkins et al. (2009), and the red line is the result derived in this work.[]{data-label="fig:1"}](f3){width="4.5in"}
![The velocity distribution of the fundamental standard rests $V_{\rm FSR}$ of the halo RRLab stars.[]{data-label="fig:1"}](f4){width="4.7in"}
|
---
abstract: 'We study separability criteria in multipartite quantum systems of arbitrary dimensions by using the Bloch representation of density matrices. We first derive the norms of the correlation tensors and obtain the necessary conditions for separability under partition of tripartite and four-partite quantum states. Moreover, based on the norms of the correlation tensors, we obtain the separability criteria by matrix method. Using detailed examples, our results are seen to be able to detect more entangled states than previous studies. Finally, necessary conditions of separability for multipartite systems are given under arbitrary partition.'
author:
- Hui Zhao$^1$
- 'Mei-Ming Zhang $^1$'
- 'Naihuan Jing$^{2,3}$'
- 'Zhi-Xi Wang$^4$'
title: Separability criteria based on Bloch representation of density matrices
---
Introduction
============
Quantum entanglement is a significant feature in quantum physics. It plays an important role in many ways such as quantum teleportation [@ea] and cryptography [@hbb; @grz; @shh]. Therefore, determination of a state being entangled or not is a significant issue in quantum information theory. Although many efforts have been devoted to the study of this problem [@wcz; @cw; @gl; @hr; @hj; @fwz; @xzw; @zzf; @zff; @yz; @zgj], it is still open except some special cases. Since norms of the Bloch vectors have a close relationship to separability criteria, research on related problem has attracted attention recently.
The norms of correlation tensors in the Bloch representation of quantum states were used to improve the separability criterion [@bf; @bk]. The norms of correlation tensors for density matrices in lower dimensions were discussed in [@js; @kg]. In [@lww], the norms of the correlation tensors for quantum state with systems less than or equal to four have been investigated. Some sufficient or necessary conditions of separability by using the norms of the correlation tensors of density matrices were presented in [@jid; @jiv; @asm; @lwl]. The relations between the norms of the correlation tensors and the detection of genuinely multipartite entanglement in tripartite quantum systems have also been established in [@ljw]. In [@ys], Sufficient and necessary condition of full separability for 3-qubit systems was derived. The relation among bipartite concurrence, concurrence of assistance and genuine tripartite entanglement for $2\otimes2\otimes n$ dimensional quantum states was presented in [@yss]. In [@vh], non full separability criterion in multipartite quantum systems based on correlation tensors was discussed. Using correlation tensors, the authors in [@syl] have provided full separability criteria for bipartite and multipartite quantum states.
In this paper, we study necessary conditions of separability for multipartite quantum states based on correlation tensors. It is known that any n-partite pure state that can be written as a tensor product $|\varphi\rangle\langle\varphi|=|\varphi_A\rangle\langle\varphi_A|\otimes |\varphi_{\overline{A}}\rangle\langle\varphi_{\overline{A}}|$ with respect to some bipartition $A\overline{A}$ ($A$ denoting some subset of subsystems and $\overline{A}$ its complement) is called biseparable. And any mixed state that can be decomposed into a convex sum of biseparable pure states is called biseparable. Consequently, any non-biseparable mixed state is called genuinely multipartite entangled. However, we mainly study separability under specific partition rather than biseparable case for multipartite quantum states. Therefore, our methods can detect entangled states rather than genuinely multipartite entangled states. In Section 2, we present an upper bound for the norm of correlation tensors and separability criteria under any partition by constructing matrices for tripartite quantum states. By a detailed example, our results are seen to outperform previously published results. In Section 3, we generalize an inequality of the norm of the correlation tensors for four-partite states and derive the necessary conditions of separability under different partition for four-partite quantum states. We also give examples to show that our criteria can detect more entangled states than previous available results. In Section 4, we generalize the norm of correlation tensors to multipartite quantum systems and obtain necessary conditions of separability under k-partitions. Comments and conclusions are given in Section 5.
Separability criteria for tripartite Quantum States {#sec2}
===================================================
We first consider the separability criteria for tripartite quantum states. Let $H_n^{d_n}$ ($n=1,2,3$) be $d_n$-dimensional Hilbert spaces. Let $\lambda_{i_f}^{(f)}$, $i_f=1,\cdots,d_f^2-1$, $f=1,2,3,$ denote the mutually orthogonal generators of the special unitary Lie algebra $\mathfrak{su}{(d_f)}$ under a fixed bilinear form, and $I$ the $d_m\times d_m$ identity matrix ($m=1,2,3$). A tripartite state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}$ can be written as follows: $$\begin{aligned}
&&\rho=\frac{1}{d_1d_2d_3}I\otimes I\otimes I+\sum_{f=1}^3\frac{d_f}{2d_1d_2d_3}\sum_{i_f=1}^{d_f^2-1}t_{i_f}^f\lambda_{i_f}^{(f)}\otimes I\otimes I+\\
&&+\sum_{1\leq f<g\leq 3}\frac{d_fd_g}{4d_1d_2d_3}\sum_{i_f=1}^{d_f^2-1}\sum_{i_g=1}^{d_g^2-1}t_{i_fi_g}^{fg}\lambda_{i_f}^{(f)}\otimes\lambda_{i_g}^{(g)}\otimes I+\frac{1}{8}\sum_{i_1=1}^{d_1^2-1}\sum_{i_2=1}^{d_2^2-1}\sum_{i_3=1}^{d_3^2-1}t_{i_1i_2i_3}^{123}\lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)}\otimes\lambda_{i_3}^{(3)},\nonumber\end{aligned}$$ where $\lambda_{i_f}^{(f)}$ or $\lambda_{i_g}^{(g)}$ ($(f)$ or $(g)$ refers the position of $\lambda_{i_f}$ or $\lambda_{i_g}$ in the tensor product) stands for the operators with $\lambda_{i_f}$ or $\lambda_{i_g}$ on $H_{d_f}$ or $H_{d_g}$, and $I$ on the respective spaces, $t_{i_f}^f=tr(\rho\lambda_{i_f}^{(f)}\otimes I\otimes I),$ $t_{i_fi_g}^{fg}=tr(\rho\lambda_{i_f}^{(f)}\otimes\lambda_{i_g}^{(g)}\otimes I),$ $t_{i_1i_2i_3}^{123}=tr(\rho\lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)}\otimes\lambda_{i_3}^{(3)})$. Let $T^{(f)}, T^{(fg)}, T^{(123)}$ be the vectors (tensors) with entries $t_{i_f}^f, t_{i_fi_g}^{fg}, t_{i_1i_2i_3}^{123}$ respectively. And $\|\cdot\|$ stand for the Hilbert-Schmidt norm or Frobenius norm, then we have $\|T^{(f)}\|^2=\sum_{i_f=1}^{d_f^2-1}(t_{i_f}^f)^2$, $\|T^{(fg)}\|^2=\sum_{i_f=1}^{d_f^2-1}\sum_{i_g=1}^{d_g^2-1}(t_{i_fi_g}^{fg})^2$ and $\|T^{(123)}\|^2=\sum_{i_1=1}^{d_1^2-1}\sum_{i_2=1}^{d_2^2-1}\sum_{i_3=1}^{d_3^2-1}(t_{i_1i_2i_3}^{123})^2$. The trace norm is defined as the sum of the singular values of the matrix $A\in\mathbb{R}^{m\times n}$, i.e., $\|A\|_{tr}=\sum_i\sigma_i=tr\sqrt{A^\dag A}$, where $\sigma_i$, $i=1,\cdots,min(m,n)$, are the singular values of the matrix $A$ arranged in descending order.
In particular, for any pure state $\rho\in H_1^{d_1}\otimes H_2^{d_2}$, $2\leq d_1\leq d_2$, we have $\rho=\frac{1}{d_1d_2}I_{d_1}\otimes I_{d_2}+\frac{1}{2d_2}\sum_{i_1=1}^{d_1^2-1}t^1_{i_1}\lambda_{i_1}^{(1)}\otimes I_{d_2}+\frac{1}{2d_1}\sum_{i_2=1}^{d_2^2-1}t^2_{i_2}I_{d_1}\otimes \lambda_{i_2}^{(2)}+\frac{1}{4}\sum_{i_1=1}^{d_1^2-1}\sum_{i_2=1}^{d_2^2-1}t^{12}_{i_1i_2}\lambda_{i_1}^{(1)}\otimes \lambda_{i_2}^{(2)}$.
\[lemma:1\] Let $\rho\in H_1^{d_1}\otimes H_2^{d_2}$ be a pure state, for $d_1\leq d_2$, $$\begin{aligned}
\|T^{(12)}\|^2\leq \frac{4(d_2^2-1)}{d_2^2}.\end{aligned}$$
Let $\rho_1$ and $\rho_2$ be the density matrices with respect to the subsystem $H_1$ and $H_2$. For a pure state $\rho$, we have $tr(\rho^2)=1$ and $tr(\rho_1^2)=tr(\rho_2^2)$, i.e., &&tr(\^2)=+T\^[(1)]{}\^2+T\^[(2)]{}\^2+T\^[(12)]{}\^2=1,\
&&tr(\_1\^2)=+T\^[(1)]{}\^2=tr(\_2\^2)=+T\^[(2)]{}\^2. Therefore $$\begin{aligned}
\|T^{(12)}\|^2=\frac{4(d_2^2-1)}{d_2^2}-\frac{2(d_1+d_2)}{d_1d_2}\|T^{(2)}\|^2\leq\frac{4(d_2^2-1)}{d_2^2}.\end{aligned}$$
$\square$
\[prop:1\] Let $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}$ be a pure state, for $d_1\leq d_2\leq d_3$, $$\begin{aligned}
\|T^{(123)}\|^2\leq8(1-\frac{d_1d_2+d_1d_3+d_2d_3-d_1-d_2}{d_1d_2d_3^2}).\end{aligned}$$
Let $\rho_{i_1}$ and $\rho_{i_2i_3}$ be the density matrices with respect to the subsystem $H_{d_{i_1}}$, $i_1=1, 2, 3$, and $H_{d_{i_2}d_{i_3}}$, $1\leq i_2<i_3\leq3$. For a pure state $\rho$, we have $tr(\rho^2)=1$ and $tr(\rho_{i_1}^2)=tr(\rho_{i_2i_3}^2)$, i.e., $$\begin{aligned}
tr(\rho^2)&=&\frac{1}{d_1d_2d_3}+\frac{1}{2}(\frac{1}{d_2d_3}\|T^{(1)}\|^2+\frac{1}{d_1d_3}\|T^{(2)}\|^2+\frac{1}{d_1d_2}\|T^{(3)}\|^2)+\frac{1}{4}(\frac{1}{d_3}\|T^{(12)}\|^2\nonumber\\
&+&\frac{1}{d_2}\|T^{(13)}\|^2+\frac{1}{d_1}\|T^{(23)}\|^2)+\frac{1}{8}\|T^{(123)}\|^2=1,\\
tr(\rho_{i_1}^2)&=&\frac{1}{d_{i_1}}+\frac{1}{2}\|T^{(i_1)}\|^2=tr(\rho_{i_2i_3}^2)=\frac{1}{d_{i_2}d_{i_3}}+\frac{1}{2}(\frac{1}{d_{i_3}}\|T^{(i_2)}\|^2+\frac{1}{d_{i_2}}\|T^{(i_3)}\|^2)+\frac{1}{4}\|T^{(i_2i_3)}\|^2\nonumber.\end{aligned}$$ Therefore $$\begin{aligned}
\|T^{(123)}\|^2&=&8(1-\frac{1}{d_1d_2d_3})-4(\frac{1}{d_2d_3}\|T^{(1)}\|^2+\frac{1}{d_1d_3}\|T^{(2)}\|^2+\frac{1}{d_1d_2}\|T^{(3)}\|^2)\nonumber\\
&-&2(\frac{1}{d_3}\|T^{(12)}\|^2+\frac{1}{d_2}\|T^{(13)}\|^2+\frac{1}{d_1}\|T^{(23)}\|^2)\nonumber\\
&\leq&8(1-\frac{d_1d_2+d_1d_3+d_2d_3-d_1-d_2}{d_1d_2d_3^2})-4[\frac{d_2(d_3-1)}{d_2d_3}\|T^{(1)}\|^2\nonumber\\
&+&\frac{d_1(d_3-1)}{d_1d_3}\|T^{(2)}\|^2+\frac{d_3+d_1d_2-(d_1+d_2)}{d_1d_3}\|T^{(3)}\|^2]\nonumber\\
&\leq&8(1-\frac{d_1d_2+d_1d_3+d_2d_3-d_1-d_2}{d_1d_2d_3^2}).\end{aligned}$$
$\square$
[**Remark 1.**]{} When $d_1=d_2=d_3=d$, we can obtain that $\|T^{(123)}\|\leq\frac{1}{d^3}(8d^3-24d+16)$. Proposition \[prop:1\] is a generalization of the Theorem 1 given in [@lww].
For the tripartite quantum state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}$, we denote the bipartitions as follows: $f|gh, fg|h$ and $f\neq g\neq h\in\{1, 2, 3\}$. We first consider the separability of $\rho$ under bipartition $f|gh$. Then we construct the matrix $S(\rho_{f|gh})$ by $$\begin{aligned}
S(\rho_{f|gh})=
\left( \begin{array}{cccc}
1& \displaystyle(T^{(g)})^t&\displaystyle(T^{(h)})^t& \displaystyle(T^{(gh)})^t \\
\displaystyle T^{(f)} &\displaystyle T^{(fg)}&\displaystyle T^{(fh)}&\displaystyle T^{(fgh)}\\
\end{array}
\right ).\end{aligned}$$ Using this matrix and the inequality for 1-body correlation tensors $\|T^{(j)}\|^2\leq \frac{2(d_j-1)}{d_j}(j=f, g, h)$[@vh] and Lemma \[lemma:1\], we get the following separability criterion.
\[thm:1\] If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}$ is separable under the bipartition $f|gh$, then $$\begin{aligned}
\|S(\rho_{f|gh})\|_{tr}\leq\sqrt{\frac{(3d_f-2)(9d_gd_h^2-2d_h^2-2d_gd_h-4d_g)}{d_fd_gd_h^2}}.\end{aligned}$$
A tripartite mixed state $\rho$ is separable under bipartition $f|gh$ whenever it can be expressed as $$\begin{aligned}
\rho_{f|gh}=\sum_lp_l\rho_l^{(f)}\otimes\rho_l^{(gh)},\end{aligned}$$ where the probabilities $p_l>0, \sum_lp_l=1$. Let $\rho_l^{(f)}=\frac{1}{d_f}I_{d_f}+\frac{1}{2}\sum_{i_f=1}^{d_f^2-1}t_{li_f}^f\lambda_{i_f}^{(f)}$, $\rho_l^{(gh)}=\frac{1}{d_gd_h}I_{d_g}\otimes I_{d_h}+\frac{1}{2d_h}\sum_{i_g=1}^{d_g^2-1}t_{li_g}^g \lambda_{i_g}^{(g)}\otimes I_{d_h}+\frac{1}{2d_g}\sum_{i_h=1}^{d_h^2-1}t_{li_h}^hI_{d_g}\otimes \lambda_{i_h}^{(h)}+\frac{1}{4}\sum_{i_g=1}^{d_g^2-1}\sum_{i_h=1}^{d_h^2-1}t_{li_gi_h}^{gh}\lambda_{i_g}^{(g)}\otimes\lambda_{i_h}^{(h)}$. Let $v_l^f, v_l^g, v_l^h$ and $v_l^{gh}$ be the column vectors with entries $t_{li_f}^f, t_{li_g}^{g}, t_{li_h}^{h}$ and $t_{li_gi_h}^{gh}$ respectively. Therefore, $$\begin{aligned}
&&T^{(f)}=\sum_lp_lv_l^f,\ T^{(g)}=\sum_lp_lv_l^g,\ T^{(h)}=\sum_lp_lv_l^h,\ T^{(fg)}=\sum_lp_lv_l^f(v_l^g)^t,\nonumber\\ &&T^{(fh)}=\sum_lp_lv_l^f(v_l^h)^t,\
T^{(gh)}=\sum_lp_lv_l^{gh},\ T^{(fgh)}=\sum_lp_lv_l^f(v_l^{gh})^t,\end{aligned}$$ where $t$ stands for transpose. Then the matrix $S(\rho_{f|gh})$ can be written as $$\begin{aligned}
S(\rho_{f|gh})&=&\sum_lp_l
\left( \begin{array}{ccccccccccccccc}
1& \displaystyle(v_l^g)^t& \displaystyle(v_l^h)^t& \displaystyle(v_l^{gh})^t \\
\displaystyle v_l^f &\displaystyle v_l^f(v_l^g)^t&\displaystyle v_l^f(v_l^h)^t&\displaystyle v_l^f(v_l^{gh})^t\\
\end{array}
\right )
\nonumber\\
&=&\sum_lp_l
\left( \begin{array}{ccccccccccccccc}
1\\
\displaystyle v_l^f\\
\end{array}
\right )
\left( \begin{array}{ccccccccccccccc}
1&\displaystyle(v_l^g)^t&\displaystyle(v_l^h)^t&\displaystyle(v_l^{gh})^t\\
\end{array}
\right ).\end{aligned}$$ Thus, $$\begin{aligned}
\|S(\rho_{f|gh})\|_{tr}&\leq&\sum_lp_l\|
\left( \begin{array}{ccccccccccccccc}
1\\
\displaystyle v_l^f\\
\end{array}
\right )
\left( \begin{array}{ccccccccccccccc}
1&\displaystyle(v_l^g)^t&\displaystyle(v_l^h)^t&\displaystyle(v_l^{gh})^t\\
\end{array}
\right )\|_{tr}\nonumber\\
&=&\sum_lp_l\|
\left( \begin{array}{ccccccccccccccc}
1\\
\displaystyle v_l^f\\
\end{array}
\right )\|
\|\left( \begin{array}{ccccccccccccccc}
1&\displaystyle(v_l^g)^t&\displaystyle(v_l^h)^t&\displaystyle(v_l^{gh})^t\\
\end{array}
\right )^t\|\nonumber\\
&=&\sum_lp_l\sqrt{1+\|v_l^f\|^2}\sqrt{1+\|v_l^g\|^2+\|v_l^h\|^2+\|v_l^{gh}\|^2}\nonumber\\
&\leq&\sqrt{\frac{(3d_f-2)(9d_gd_h^2-2d_h^2-2d_gd_h-4d_g)}{d_fd_gd_h^2}},\end{aligned}$$ where we have used $\|A+B\|_{tr}\leq\|A\|_{tr}+\|B\|_{tr}$ for matrices $A$ and $B$ and $\||a\rangle\langle b|\|_{tr}=\||a\rangle\|\||b\rangle\|$ for vectors $|a\rangle$ and $|b\rangle$.
$\square$
[**Remark 2.**]{} We may analyze the bipartition $fg|h$ by using similar methods above. If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}$ is separable under the bipartition $fg|l$, then S(\_[fg|h]{})\_[tr]{}. We next consider the full separability of $\rho$. Denote the matrix $$\begin{aligned}
S(\rho_{f|g|h})=
\left( \begin{array}{ccccccccccccccc}
\displaystyle(T^{(h)})^t& \displaystyle(T^{(gh)})^t \\
\displaystyle T^{(fh)}&\displaystyle T^{(fgh)}\\
\end{array}
\right ).\end{aligned}$$ Using this matrix and the inequality for 1-body correlation tensors $\|T^{(j)}\|^2\leq \frac{2(d_j-1)}{d_j}(j=f, g, h)$[@vh], we get the following separability criterion.
\[thm:2\] If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}$ is fully separable, then $$\begin{aligned}
\|S(\rho_{f|g|h})\|_{tr}\leq\frac{2(d_h-1)}{d_h}\sqrt{\frac{(3d_f-2)(3d_g-2)}{d_fd_g}}.\end{aligned}$$
A tripartite mixed state $\rho$ is fully separable whenever it can be expressed as $$\begin{aligned}
\rho_{f|g|h}=\sum_lp_l\rho_l^{(f)}\otimes\rho_l^{(g)}\otimes\rho_l^{(h)},\end{aligned}$$ where the probabilities $p_l>0, \sum_lp_l=1$. Let $\rho_l^{(f)}=\frac{1}{d_f}I_{d_f}+\frac{1}{2}\sum_{i_f=1}^{d_f^2-1}t_{li_f}^f\lambda_{i_f}^{(f)}$, $\rho_l^{(g)}=\frac{1}{d_g}I_{d_g}+\frac{1}{2}\sum_{i_g=1}^{d_g^2-1}t_{li_g}^g\lambda_{i_g}^{(g)}$, $\rho_l^{(h)}=\frac{1}{d_h}I_{d_h}+\frac{1}{2}\sum_{i_h=1}^{d_h^2-1}t_{li_h}^h\lambda_{i_h}^{(h)}$. Let $v_l^f, v_l^g$ and $v_l^h$ be the column vectors with entries $t_{li_f}^f, t_{li_g}^{g}$ and $t_{li_h}^{h}$ respectively. Therefore $$\begin{aligned}
T^{(h)}=\sum_lp_lv_l^h, T^{(fh)}=\sum_lp_lv_l^f(v_l^h)^t, T^{(gh)}=\sum_lp_lv_l^g\otimes v_l^h, T^{(fgh)}=\sum_lp_lv_l^f(v_l^g\otimes v_l^h)^t.\end{aligned}$$ It follows that the matrix $S(\rho_{f|g|h})$ can be written as $$\begin{aligned}
S(\rho_{f|g|h})=\sum_lp_l
\left( \begin{array}{ccccccccccccccc}
\displaystyle (v_l^h)^t&\displaystyle(v_l^g\otimes v_l^h)^t\\
\displaystyle v_l^f (v_l^h)^t&\displaystyle v_l^f (v_l^g\otimes v_l^h)^t\\
\end{array}
\right )
=\sum_lp_l
\left( \begin{array}{ccccccccccccccc}
1\\
\displaystyle v_l^f\\
\end{array}
\right )
\left( \begin{array}{ccccccccccccccc}
1&\displaystyle(v_l^g)^t\\
\end{array}
\right )
\otimes\ (v_l^h)^t.\end{aligned}$$ Taking the norm, we get that $$\begin{aligned}
\|S(\rho_{f|g|h})\|_{tr}&\leq&\sum_lp_l\|
\left( \begin{array}{ccccccccccccccc}
1\\
v_l^f\\
\end{array}
\right )
\left( \begin{array}{ccccccccccccccc}
1&(v_l^g)^t\\
\end{array}
\right )
\otimes(v_l^h)^t\|_{tr}
=\sum_lp_l\|
\left( \begin{array}{ccccccccccccccc}
1\\
v_l^f\\
\end{array}
\right )\|
\|\left( \begin{array}{ccccccccccccccc}
1\\
v_l^g\\
\end{array}
\right )
\otimes v_l^h\|\nonumber\\
&=&\sum_lp_l\|
\left( \begin{array}{ccccccccccccccc}
1\\
v_l^f\\
\end{array}
\right )\|
\|\left( \begin{array}{ccccccccccccccc}
1\\
v_l^g\\
\end{array}
\right )\|
\|v_l^h\|
\leq\sum_lp_l\|v_l^h\|\sqrt{1+\|v_l^f\|^2}\sqrt{1+\|v_l^g\|^2}\nonumber\\
&\leq&\frac{2(d_h-1)}{d_h}\sqrt{\frac{(3d_f-2)(3d_g-2)}{d_fd_g}}.\end{aligned}$$
$\square$
[**Remark 3.**]{} By the above Theorem \[thm:1\] and Theorem \[thm:2\], we have obtained upper bounds for 1-2, 2-1 and 1-1-1 separable quantum states. With these bounds a complete classification of tripartite quantum states has been derived.
In Ref. [@lww], the authors analyzed necessary conditions of separability under arbitrary partition for four-patite quantum states by using the norms of correlation tensors. We can derive the necessary condition of separability under arbitrary partition for tripartite state $\rho\in H_1^d\otimes H_2^d\otimes H_3^d$ by using methods as similar as [@lww], i.e., $$\begin{aligned}
\label{spe}
\|T^{(123)}\|^2\leq\left\{
\begin{array}{lr}
\frac{8(d-1)^2(d+1)}{d^3},\ if \ \rho\ is \ 1-2 \ separable;\\
\frac{8(d-1)^3}{d^3}, \ if \ \rho \ is \ 1-1-1 \ separable.
\end{array}
\right.\end{aligned}$$ ***Example 1.*** Consider the quantum state $\rho\in H_1^2\otimes H_2^2\otimes H_3^2$: $$\begin{aligned}
\rho=\frac{x}{8}I_8+(1-x)|\phi\rangle\langle\phi|,\end{aligned}$$ where $|\phi\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$, $I_{8}$ stands for the $8\times8$ identity matrix and $x\in [0,1]$. By our Theorem \[thm:1\] and Theorem \[thm:2\], we have $f_1(x)=\|S(\rho_{f|gh})\|_{tr}-2\sqrt{3}=-3x+4-2\sqrt{3}$ and $f_3(x)=\|S(\rho_{f|gh})\|_{tr}-2=-(3+\sqrt{2})x+\sqrt{2}+1$ respectively. When $f_1(x)>0$ or $f_2(x)>0$, $\rho$ is not separable under the bipartition $f|gh$ or fully separable. When $d=2$, according to inequality (\[spe\]), we have $f_2(x)=\|T^{(123)}\|^2-3=3x^2-6x$ and $f_4(x)=\|T^{(123)}\|^2-1=3x^2-6x+2$. And $\rho$ is not separable under the bipartition $f|gh$ or fully separable for $f_2(x)>0$ or $f_4(x)>0$ respectively. Fig. 1 shows that $\rho$ is not separable under the bipartition $f|gh$ for $0\leq x<0.179$ by using Theorem \[thm:1\], while according to inequality (\[spe\]), it cannot detect whether the $\rho$ is separable under the bipartition $f|gh$ or not. And $\rho$ is not fully separable for $0\leq x<0.547$ by using Theorem \[thm:2\], while according to inequality (\[spe\]), $\rho$ is not fully separable for $0\leq x<0.427$. This shows that Theorem \[thm:1\] and Theorem \[thm:2\] detect more entangled states.
![The $f_1(x)$ from Theorem \[thm:1\] (solid straight line), $f_2(x)$ from inequality (\[spe\]) (dashed curve line), $f_3(x)$ from Theorem \[thm:2\] (dotted straight line) and $f_4(x)$ from inequality (\[spe\])(dash-dot curve line).](THREE.eps "fig:"){width="8cm"}\
Separability criteria for four-partite Quantum Systems
======================================================
We next consider the separability criteria for four-partite quantum states. Let $H_n^{d_n}$ $(n=1, 2, 3, 4)$ be $d_n$-dimensional Hilbert spaces. For $f=1, 2, 3, 4$, $i_f=1,\cdots,d_f^2-1$, let $\lambda_{i_f}^{(f)}$ denote the mutually orthogonal generators of the special unitary Lie algebra $\mathfrak{su}(d_f)$ under a fixed bilinear form, and $I$ the $d_m\times d_m$, $m=1, 2, 3, 4$, identity matrix. A four-partite state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ can be written as follows: $$\begin{aligned}
\rho&=&\frac{1}{d_1d_2d_3d_4}I\otimes I\otimes I\otimes I+\frac{1}{2d_gd_hd_e}\sum_{f=1}^4\sum_{i_f=1}^{d_f^2-1}t_{i_f}^f\lambda_{i_f}^{(f)}\otimes I\otimes I\otimes I+\cdots\nonumber\\
&+&\frac{1}{16}\sum_{i_1=1}^{d_1^2-1}\sum_{i_2=1}^{d_2^2-1}\sum_{i_3=1}^{d_3^2-1}\sum_{i_4=1}^{d_4^2-1}t_{i_1i_2i_3i_4}^{1234}\lambda_{i_1}^{(1)}\otimes \lambda_{i_2}^{(2)}\otimes \lambda_{i_3}^{(3)}\otimes \lambda_{i_4}^{(4)},\end{aligned}$$ where $\lambda_{i_f}^{(f)}$ ($(f)$ indicates the position of $\lambda_{i_f}$ in the tensor product) stands for the operators on $H_{d_f}$, $I$ on the remaining appropriate factors, $t_{i_f}^f=tr(\rho\lambda_{i_f}^{(f)}\otimes I\otimes I\otimes I)$, $t_{i_fi_g}^{fg}=tr(\rho\lambda_{i_f}^{(f)}\otimes\lambda_{i_g}^{(g)}\otimes I\otimes I), t_{i_fi_gi_h}^{fgh}=tr(\rho\lambda_{i_f}^{(f)}\otimes\lambda_{i_g}^{(g)}\otimes \lambda_{i_h}^{(h)}\otimes I)$, and $t_{i_1i_2i_3i_4}^{1234}=tr(\rho\lambda_{i_1}^{(1)}\otimes \lambda_{i_2}^{(2)}\otimes \lambda_{i_3}^{(3)}\otimes \lambda_{i_4}^{(4)})$. Let $T^{(f)}, T^{(fg)}, T^{(fgh)}, T^{(1234)}$ be the vectors (tensors) with entries $t_{i_f}^{f}, t_{i_fi_g}^{fg}, t_{i_fi_gi_h}^{fgh}, t_{i_1i_2i_3i_4}^{1234}$, respectively, where $1\leq f<g<h\leq4$, then we get $\|T^{(f)}\|^2=\sum_{i_f=1}^{d_f^2-1}(t_{i_f}^{f})^2$, $\|T^{(fg)}\|^2=\sum_{i_f=1}^{d_f^2-1}\sum_{i_g=1}^{d_g^2-1}(t_{i_fi_g}^{fg})^2$, $\|T^{(fgh)}\|^2=\sum_{i_f=1}^{d_f^2-1}\sum_{i_g=1}^{d_g^2-1}\sum_{i_h=1}^{d_h^2-1}(t_{i_fi_gi_h}^{fgh})^2$, and $\|T^{(1234)}\|^2=\sum_{i_1=1}^{d_1^2-1}\sum_{i_2=1}^{d_2^2-1}\\ \sum_{i_3=1}^{d_3^2-1}\sum_{i_4=1}^{d_4^2-1}(t_{i_1i_2i_3i_4}^{1234})^2$.
\[prop:2\] Let $\rho\in H_1^{d_1}\otimes\cdots\otimes H_4^{d_4}$ be a pure state, for $2\leq d_1\leq d_2\leq d_3\leq d_4$, $$\begin{aligned}
\|T^{(1234)}\|^2\leq16(1-\frac{d_2d_3d_4+d_1d_3d_4+d_1d_2d_4+d_1d_2d_3-d_1-d_2-d_3+d_4}{2d_1d_2d_3d_4^2}).\end{aligned}$$
Let $\rho_{i_1}$ and $\rho_{i_2i_3i_4}$ be the density matrices with respect to the subsystem $H_{d_{i_1}}$ $(i_1=1,\cdots,4)$ and $H_{d_{i_2}d_{i_3}d_{i_4}}$ $(1\leq i_2<i_3<i_4\leq4)$ respectively. For a pure state $\rho$, we have $tr(\rho^2)=1$ and $tr(\rho_{i_1}^2)=tr(\rho_{i_2i_3i_4}^2)$. Therefore $$\begin{aligned}
\|T^{(1234)}\|^2&=&16(1-\frac{1}{d_1d_2d_3d_4})-8(\frac{1}{d_2d_3d_4}\|T^{(1)}\|^2+\cdots+\frac{1}{d_1d_2d_3}\|T^{(4)}\|^2)\nonumber\\
&-&4(\frac{1}{d_3d_4}\|T^{(12)}\|^2+\cdots+\frac{1}{d_1d_2}\|T^{(34)}\|^2)-2(\frac{1}{d_4}\|T^{(123)}\|^2+\cdots+\frac{1}{d_1}\|T^{(234)}\|^2)\nonumber\\
&\leq&16(1-\frac{1}{d_1d_2d_3d_4})-8(\frac{1}{d_2d_3d_4}\|T^{(1)}\|^2+\cdots+\frac{1}{d_1d_2d_3}\|T^{(4)}\|^2)\nonumber\\
&-&\frac{4}{d_4}(\frac{1}{d_3}\|T^{(12)}\|^2+\cdots+\frac{1}{d_1}\|T^{(34)}\|^2)-2(\frac{1}{d_4}\|T^{(123)}\|^2+\cdots+\frac{1}{d_1}\|T^{(234)}\|^2)\nonumber\end{aligned}$$ $$\begin{aligned}
&\leq&16(1-\frac{1}{d_1d_2d_3d_4})-8(\frac{1}{d_2d_3d_4}\|T^{(1)}\|^2+\cdots+\frac{1}{d_1d_2d_3}\|T^{(4)}\|^2)\nonumber\\
&-&\frac{8}{d_4}[\frac{d_2d_3d_4+d_1d_3d_4+d_1d_2d_4+d_1d_2d_3-d_1-d_2-d_3-d_4}{d_1d_2d_3d_4}\nonumber\\
&+&\frac{1}{2}(\frac{d_2d_3d_4-d_2-d_3-d_4}{d_2d_3d_4}\|T^{(1)}\|^2+\cdots+\frac{d_1d_2d_3-d_1-d_2-d_3}{d_1d_2d_3}\|T^{(4)}\|^2)\nonumber\\
&-&\frac{1}{2}(\|T^{(123)}\|^2+\cdots+\|T^{(234)}\|^2)]-2(\frac{1}{d_4}\|T^{(123)}\|^2+\cdots+\frac{1}{d_1}\|T^{(234)}\|^2)\nonumber\\
&\leq&16(1-\frac{d_2d_3d_4+d_1d_3d_4+d_1d_2d_4+d_1d_2d_3-d_1-d_2-d_3+d_4}{2d_1d_2d_3d_4^2})\nonumber\\
&-&8(\frac{d_2d_3d_4-d_2-d_3+d_4}{2d_2d_3d_4^2}\|T^{(1)}\|^2
+\frac{d_1d_3d_4-d_1-d_3+d_4}{2d_1d_3d_4^2}\|T^{(2)}\|^2\nonumber\\
&+&\frac{d_1d_2d_4-d_1-d_2+d_4}{2d_1d_2d_4^2}\|T^{(3)}\|^2
+\frac{2d_4+d_1d_2d_3-d_1-d_2-d_3}{2d_1d_2d_3d_4}\|T^{(4)}\|^2)\nonumber\\
&-&2(\frac{1}{2d_4}\|T^{(123)}\|^2+\frac{2d_4-d_3}{2d_3d_4}\|T^{(124)}\|^2+\frac{2d_4-d_2}{2d_2d_4}\|T^{(134)}\|^2
+\frac{2d_4-d_1}{2d_1d_4}\|T^{(234)}\|^2)\nonumber\\
&\leq&16(1-\frac{d_2d_3d_4+d_1d_3d_4+d_1d_2d_4+d_1d_2d_3-d_1-d_2-d_3+d_4}{2d_1d_2d_3d_4^2}).\end{aligned}$$
$\square$
[**Remark 4.**]{} When $d_1=d_2=d_3=d_4=d$, we can obtain $\|T^{(1234)}\|\leq\frac{16(d^2-1)^2}{d^4}$. Thus, Proposition \[prop:2\] generalizes Theorem 2 in [@lww].
For a four-partite quantum state $\rho$ on $H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$, the bipartitions and tripartitions are the following: $f|ghe,fg|he,fgh|e$ and $f|g|he, fg|h|e, f|gh|e$ where $f\neq g\neq h\neq e=1,2,3,4$. We first consider the separability of $\rho$ under the bipartition $f|ghe$. Define the matrix $S(\rho_{f|ghe})$ by S(\_[f|ghe]{})=(
[cc]{} 1&(T\^[(g)]{})\^t\
T\^[(f)]{}&T\^[(fg)]{}\
). And by the inequality $\|T^{(j)}\|^2\leq \frac{2(d_j-1)}{d_j}(j=f, g)$[@vh], we get the following separability criterion.
\[thm:3\] If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ is separable under the bipartition $f|ghe$, then $$\begin{aligned}
\|S(\rho_{f|ghe})\|_{tr}\leq\sqrt{\frac{(3d_f-2)(3d_g-2)}{d_fd_g}}.\end{aligned}$$
A four-partite mixed state $\rho$ is separable under bipartition $f|ghe$ whenever it can be expressed as $$\begin{aligned}
\rho_{f|ghe}=\sum_lp_l\rho_l^{(f)}\otimes\rho_l^{(ghe)},\end{aligned}$$ where the probabilities $p_l>0, \sum_lp_l=1$. Let $\rho_l^{(f)}=\frac{1}{d_f}I_{d_f}+\frac{1}{2}\sum_{i_f=1}^{d_f^2-1} t_{li_f}^f\lambda_{i_f}^{(f)}, \rho_l^{(ghe)}=\frac{1}{d_gd_hd_e}I_{d_g}\otimes I_{d_h}\otimes I_{d_e}+\frac{1}{2}(\frac{1}{d_hd_e}\sum_{i_g=1}^{d_g^2-1} t_{li_g}^g\lambda_{i_g}^{(g)}\otimes I_{d_h}\otimes I_{d_e}+\cdots)+\frac{1}{4}(\frac{1}{d_e}\sum_{i_g=1}^{d_g^2-1}\sum_{i_h=1}^{d_h^2-1} t_{li_gi_h}^{gh}\lambda_{i_g}^{(g)}\otimes\lambda_{i_h}^{(h)}\otimes I_{d_e}+\cdots)+\frac{1}{8}\sum_{i_g=1}^{d_g^2-1}\sum_{i_h=1}^{d_h^2-1}\sum_{i_e=1}^{d_e^2-1} t_{li_gi_hi_e}^{ghe}\lambda_{i_g}^{(g)}\otimes\lambda_{i_h}^{(h)}\otimes \lambda_{i_e}^{(e)}$. Let $v_l^f$ and $v_l^g$ be the column vectors with entries $t_{li_f}^f$ and $t_{li_g}^g$ respectively. Therefore $$\begin{aligned}
&&T^{(f)}=\sum_lp_lv_l^f, T^{(g)}=\sum_lp_lv_l^g, T^{(fg)}=\sum_lp_lv_l^f(v_l^g)^t,\end{aligned}$$ and then it follows that the matrix $S(\rho_{f|ghe})$ can be written as $$\begin{aligned}
S(\rho_{f|ghe})=\sum_lp_l
\left( \begin{array}{cc}
1&(v_l^g)^t\\
v_l^f&v_l^f(v_l^g)^t\\
\end{array}
\right )
=\sum_lp_l
\left( \begin{array}{cc}
1\\
v_l^f\\
\end{array}
\right )
\left( \begin{array}{ccccccccccccccc}
1&(v_l^g)^t\\
\end{array}
\right ).\end{aligned}$$ Thus, $$\begin{aligned}
\|S(\rho_{f|ghe})\|_{tr} &\leq&\sum_lp_l\|
\left( \begin{array}{c}
1\\
v_l^f\\
\end{array}
\right )
\left( \begin{array}{cc}
1&(v_l^g)^t\\
\end{array}
\right )\|_{tr}
=\sum_lp_l\|
\left( \begin{array}{c}
1\\
v_l^f\\
\end{array}
\right )\|
\left( \begin{array}{cc}
1&(v_l^g)^t\\
\end{array}
\right )^t\|\nonumber\\
&=&\sum_lp_l\sqrt{1+\|v_l^f\|^2}\sqrt{1+\|v_l^g\|^2}\leq\sqrt{\frac{(3d_f-2)(3d_g-2)}{d_fd_g}}.\end{aligned}$$
$\square$
We next consider the separability of $\rho$ under bipartition $fg|he$. Denote the matrix $$\begin{aligned}
S(\rho_{fg|he})=
\left( \begin{array}{cc}
1&(T^{(he)})^t \\
T^{(fg)}&T^{(fghe)}\\
\end{array}
\right ).\end{aligned}$$ And by Lemma \[lemma:1\], we have the following separability criterion.
\[thm:4\] If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ is separable under bipartition $fg|he$, then $$\begin{aligned}
\|S(\rho_{fg|he})\|_{tr}\leq\frac{\sqrt{(5d_g^2-4)(5d_e^2-4)}}{d_gd_e}.\end{aligned}$$
A four-partite mixed state $\rho$ is separable under bipartition $fg|he$ whenever it can be expressed as $$\begin{aligned}
\rho_{fg|he}=\sum_lp_l\rho_l^{(fg)}\otimes\rho_l^{(he)},\end{aligned}$$ where the probabilities $p_l>0, \sum_lp_l=1$.
Let $\rho_l^{(fg)}=\frac{1}{d_fd_g}I_{d_f}\otimes I_{d_g}+\cdots+\frac{1}{4}\sum_{i_f=1}^{d_f^2-1}\sum_{i_g=1}^{d_g^2-1} t_{li_fi_g}^{fg}\lambda_{i_f}^{(f)}\otimes\lambda_{i_g}^{(g)}$, and $\rho_l^{(he)}=\frac{1}{d_hd_e}I_{d_h}\otimes I_{d_e}+\cdots+\frac{1}{4}\sum_{i_h=1}^{d_h^2-1}\sum_{i_e=1}^{d_e^2-1}t_{li_hi_e}^{he}\lambda_{i_h}^{(h)}\otimes\lambda_{i_e}^{(e)}$. Let $v_l^{fg}$ and $v_l^{he}$ be the column vectors with entries $t_{li_fi_g}^{fg}$ and $t_{li_hi_e}^{he}$ respectively. Therefore $$\begin{aligned}
&&T^{(fg)}=\sum_lp_lv_l^{fg}, T^{(he)}=\sum_lp_lv_l^{he}, T^{(fghe)}=\sum_lp_lv_l^{fg}(v_l^{he})^t.\end{aligned}$$ Then the matrix matrix $S(\rho_{fg|he})$ can be written as $$\begin{aligned}
S(\rho_{fg|he})&=&\sum_lp_l
\left( \begin{array}{cc}
1&(v_l^{he})^t \\
v_l^{fg}&v_l^{fg}(v_l^{he})^t\\
\end{array}
\right )
=\sum_lp_l
\left( \begin{array}{cc}
1&(v_l^{fg})^t\\
\end{array}
\right )^t
\left( \begin{array}{cc}
1&(v_l^{he})^t\\
\end{array}
\right ).\end{aligned}$$ Thus, $$\begin{aligned}
\|S(\rho_{fg|he})\|_{tr}&\leq&\sum_lp_l\|
\left( \begin{array}{cc}
1&(v_l^{fg})^t\\
\end{array}
\right )^t
\left( \begin{array}{cc}
1&(v_l^{he})^t\\
\end{array}
\right )\|_{tr}\nonumber\\
&=&\sum_lp_l\|
\left( \begin{array}{cc}
1&(v_l^{fg})^t\\
\end{array}
\right )^t\|
\|\left( \begin{array}{cc}
1&(v_l^{he})^t\\
\end{array}
\right )^t\|\nonumber\\
&=&\sum_lp_l\sqrt{1+\|v_l^{fg}\|^2}\sqrt{1+\|v_l^{he}\|^2}\nonumber\\
&\leq&\frac{\sqrt{(5d_g^2-4)(5d_e^2-4)}}{d_gd_e},\end{aligned}$$ where we have used the triangular inequality of the trace norm.
$\square$
We next consider the separability of $\rho$ under tripartition $f|g|he$. We define the matrix $S(\rho_{f|g|he})$ by $$\begin{aligned}
S(\rho_{f|g|he})=
\left( \begin{array}{cc}
(T^{(h)})^t& T^{(gh)^t} \\
T^{(fh)}&T^{(fgh)}\\
\end{array}
\right ),\end{aligned}$$ with the matrix, the inequality $\|T^{(j)}\|^2\leq \frac{2(d_j-1)}{d_j}(j=f, g, h)$[@vh] and similar methods of Theorem \[thm:2\], then we obtain the following separability criterion.
\[thm:5\] If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ is separable under tripartition $f|g|he$, then $$\begin{aligned}
\|S(\rho_{f|g|he})\|_{tr}\leq\frac{2(d_h-1)}{d_h}\sqrt{\frac{(3d_f-2)(3d_g-2)}{d_fd_g}}.\end{aligned}$$
[**Remark 5.**]{} We may analyze the bipartition $fgh|e$ and tripartitions $fg|h|e$, $f|gh|e$ by using similar methods of Theorem \[thm:2\] and Theorem \[thm:4\] above, respectively. If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ is separable under bipartition $fgh|e$, then $\|S_(\rho_{fgh|e})\|_{tr}\leq\sqrt{\frac{(3d_f-2)(3d_e-2)}{d_fd_e}}.$ If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ is separable under tripartition $fg|h|e$ or $f|gh|e$, then $\|S(\rho_{fg|h|e})\|_{tr}\leq\frac{2(d_f-1)}{d_f}\sqrt{\frac{(3d_h-2)(3d_e-2)}{d_hd_e}}$ or $\|S(\rho_{f|gh|e})\|_{tr}\leq\frac{2(d_g-1)}{d_g}\sqrt{\frac{(3d_f-2)(3d_e-2)}{d_fd_e}}$, respectively.
We next consider the full separability of $\rho$. Denote the matrix $$\begin{aligned}
S(\rho_{f|g|h|e})=
\left( \begin{array}{cccc}
(T^{(e)})^t& (T^{(he)})^t& (T^{(ge)})^t&(T^{(ghe)})^t \\
T^{(fe)}&T^{(fhe)}&T^{(fge)}&T^{(fghe)}\\
\end{array}
\right ),\end{aligned}$$ Using this matrix and the inequality for 1-body correlation tensors $\|T^{(j)}\|^2\leq \frac{2(d_j-1)}{d_j}(j=f, g, h, e)$[@vh], we get the following separability criterion.
\[thm:6\] If the state $\rho\in H_1^{d_1}\otimes H_2^{d_2}\otimes H_3^{d_3}\otimes H_4^{d_4}$ is fully separable, then $$\begin{aligned}
\|S(\rho_{f|g|h|e})\|_{tr}\leq\frac{2(d_e-1)}{d_e}\sqrt{\frac{(3d_f-2)(3d_g-2)(3d_h-2)}{d_fd_gd_h}}.\end{aligned}$$
A four partite mixed state $\rho$ is fully separable whenever it can be expressed as $$\begin{aligned}
\rho_{f|g|h|e}=\sum_lp_l\rho_l^{(f)}\otimes\rho_l^{(g)}\otimes\rho_l^{(h)}\otimes\rho_l^{(e)},\end{aligned}$$ where the probabilities $p_l>0, \sum_lp_l=1$. Let $\rho_l^{(f)}=\frac{1}{d_f}I_{d_f}+\frac{1}{2}\sum_{i_f=1}^{d_f^2-1}t_{li_f}^f\lambda_{i_f}^{(f)}$, $\rho_l^{(g)}=\frac{1}{d_g}I_{d_g}+\frac{1}{2}\sum_{i_g=1}^{d_g^2-1}t_{li_g}^g\lambda_{i_g}^{(g)}$, $\rho_l^{(h)}=\frac{1}{d_h}I_{d_h}+\frac{1}{2}\sum_{i_h=1}^{d_h^2-1}t_{li_h}^h\lambda_{i_h}^{(h)}$, $\rho_l^{(e)}=\frac{1}{d_e}I_{d_e}+\frac{1}{2}\sum_{i_e=1}^{d_e^2-1}t_{li_e}^e\lambda_{i_e}^{(e)}$. Let $v_l^f, v_l^g, v_l^h$ and $v_l^e$ be the column vectors with entries $t_{li_f}^f, t_{li_g}^{g}, t_{li_h}^{h}$ and $t_{li_e}^{e}$ respectively. Therefore, $$\begin{aligned}
&& T^{(e)}=\sum_lp_lv_l^e,\ T^{(fe)}=\sum_lp_lv_l^f(v_l^e)^t, \ T^{(ge)}=\sum_lp_lv_l^g\otimes v_l^e, \ T^{(he)}=\sum_lp_lv_l^h\otimes v_l^e, \nonumber\\ &&T^{(fge)}=\sum_lp_lv_l^f(v_l^g\otimes v_l^e)^t, \ T^{(ghe)}=\sum_lp_lv_l^g\otimes v_l^h\otimes v_l^e,\ T^{(fhe)}=\sum_lp_lv_l^f(v_l^h\otimes v_l^e)^t,\nonumber\\
&& T^{(fghe)}=\sum_lp_lv_l^f(v_l^g\otimes v_l^h\otimes v_l^e)^t,\end{aligned}$$ where $t$ stands for transpose. Then the matrix $S(\rho_{f|g|h|e})$ can be written as $$\begin{aligned}
S(\rho_{f|g|h|e})&=&\sum_lp_l
\left( \begin{array}{ccccccccccccccc}
(v_l^e)^t& (v_l^h\otimes v_l^e)^t& (v_l^g\otimes v_l^e)^t& (v_l^g\otimes v_l^h\otimes v_l^e)^t \\
v_l^f(v_l^e)^t & v_l^f(v_l^h\otimes v_l^e)^t& v_l^f(v_l^g\otimes v_l^e)^t&v_l^f(v_l^g\otimes v_l^h\otimes v_l^e)^t\\
\end{array}
\right )
\nonumber\\
&=&\sum_lp_l
\left( \begin{array}{c}
1\\
v_l^f\\
\end{array}
\right )
\left( \begin{array}{cc}
1&(v_l^g)^t\\
\end{array}
\right )\otimes
\left( \begin{array}{cc}
1&(v_l^h)^t\\
\end{array}
\right )\otimes (v_l^e)^t.\end{aligned}$$ Thus, $$\begin{aligned}
\|S(\rho_{f|g|h|e})\|_{tr}&\leq&\sum_lp_l\|
\left( \begin{array}{c}
1\\
v_l^f\\
\end{array}
\right )
\left( \begin{array}{cc}
1&(v_l^g)^t\\
\end{array}
\right )\otimes
\left( \begin{array}{cc}
1&(v_l^h)^t\\
\end{array}
\right )\otimes (v_l^e)^t
\|_{tr}\nonumber\\
&=&\sum_lp_l\|
\left( \begin{array}{c}
1\\
v_l^f\\
\end{array}
\right )\|
\|\left( \begin{array}{c}
1\\
v_l^g\\
\end{array}
\right )\otimes
\left( \begin{array}{c}
1\\
v_l^h\\
\end{array}
\right )\otimes v_l^e\|\nonumber\\
&=&\sum_lp_l\|
\left( \begin{array}{c}
1\\
v_l^f\\
\end{array}
\right )\|
\|\left( \begin{array}{c}
1\\
v_l^g\\
\end{array}
\right )\|
\|\left( \begin{array}{c}
1\\
v_l^h\\
\end{array}
\right )\|
\|v_l^e\|\nonumber\\
&=&\sum_lp_l \|v_l^e\|\sqrt{1+\|v_l^f\|^2}\sqrt{1+\|v_l^g\|^2}\sqrt{1+\|v_l^h\|^2}\nonumber\\
&\leq&\frac{2(d_e-1)}{d_e}\sqrt{\frac{(3d_f-2)(3d_g-2)(3d_h-2)}{d_fd_gd_h}}.\end{aligned}$$
$\square$
[**Remark 6.**]{} From Theorem \[thm:3\] to Theorem \[thm:6\], we have derived the upper bounds for 1-3, 3-1, 2-2, 1-1-2, 1-2-1, 2-1-1 and 1-1-1-1 separable quantum states. Thus, we can obtain a complete classification of four-partite quantum states with these bounds.
***Example 2.*** Consider the quantum state $\rho\in H_1^2\otimes H_2^2\otimes H_3^2\otimes H_4^2$, $$\begin{aligned}
\rho=x|\psi\rangle\langle\psi|+\frac{1-x}{16}I_{16},\end{aligned}$$ where $|\psi\rangle=\frac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle)$ and $I_{16}$ stands for the $16\times 16$ identity matrix. By Theorem \[thm:4\], we have $f_1(x)=\|S(\rho_{fg|he})\|_{tr}-4=\sqrt{1+x^2}+2\sqrt{2}x+\frac{x-x^2}{1+x^2}-4$ and when $f_1(x)>0$, $\rho$ is not separable under bipartition $fg|he$. When $d=2$, from Ref. [@lww], one has that $f_2(x)=\|T^{(1234)}\|^2-\frac{16}{d^4}(d^2-1)^2=9x^2-9$ and $\rho$ is not separable under bipartition $fg|he$ for $f_2(x)>0$. Fig.2 shows that $\rho$ is not separable under bipartition $fg|he$ for $0.915< x\leq 1$ by Theorem \[thm:4\], while using Theorem 3 in Ref. [@lww], it cannot detect whether the $\rho$ is inseparable under bipartition $fg|he$. Thus, our method detects more entangled states than that of Ref. [@lww].
![The function $f_1(x)$ from Theorem \[thm:4\] (solid line) and $f_2(x)$ from Ref. [@lww] (dashed line).](FOUR1.eps "fig:"){width="8cm"}\
***Example 3.*** Consider the quantum state $\rho\in H_1^2\otimes H_2^2\otimes H_3^2\otimes H_4^2$, $$\begin{aligned}
\rho=x|\varphi\rangle\langle\varphi|+\frac{1-x}{16}I_{16},\end{aligned}$$ where $|\varphi\rangle=\frac{1}{2}(|0001\rangle+|0010\rangle+|0100\rangle+|1000\rangle)$ and $I_{16}$ stands for the $16\times16$ identity matrix. By Theorem \[thm:3\] and Theorem \[thm:5\], we have $f_1(x)=\|S_(\rho_{f|ghe})\|_{tr}-2=\frac{4+2x^2}{2\sqrt{4+x^2}}+x-2$ and $f_3(x)=\|S(\rho_{f|g|he})\|_{tr}-2=\frac{6+3\sqrt{2}}{4}x-2$ respectively. When $f_1(x)>0$ or $f_3(x)>0$, $\rho$ is not separable under bipartition $f|ghe$ or tripartition $f|g|he$ respectively. When $d=2$, we have $f_2(x)=\|T^{(1234)}\|^2-4=4x^2-4$ and $f_4(x)=\|T^{(1234)}\|^2-3=4x^2-3$ from Theorem 3 in Ref. [@lww]. And $\rho$ is not separable under bipartition $f|ghe$ or tripartition $f|g|he$ for $f_2(x)>0$ or $f_4(x)>0$ respectively. From Fig. 3, $\rho$ is not separable under bipartition $f|ghe$ for $0.783< x\leq 1$ by Theorem \[thm:3\], while using the Theorem 3 in Ref. [@lww], it can not detect whether the $\rho$ is not separable under bipartition $f|ghe$. And $\rho$ is not separable under tripartition $f|g|he$ for $0.781< x\leq 1$ by Theorem \[thm:5\], while using the method in Ref. [@lww], $\rho$ is not separable under tripartition $f|g|he$ for $0.866< x\leq 1$. Therefore Theorem \[thm:3\] and Theorem \[thm:5\] detect more entangled states than Theorem 3 in Ref. [@lww].
![The function $f_1(x)$ from Theorem \[thm:3\] (solid curve line), $f_2(x)$ from Theorem 3 in Ref. [@lww] (dashed curve line), $f_3(x)$ from Theorem \[thm:5\] (dotted straight line) and $f_4(x)$ from Theorem 3 in Ref. [@lww] (dash-dot curve line).](FOUR2.eps "fig:"){width="8cm"}\
Separability criteria for multipartite Quantum Systems
======================================================
We finally consider the separability criteria for $n$-partite quantum states. Let $H_i^{d}$ ($i=1, \cdots, n$) denote $d$-dimensional Hilbert spaces. Let $\lambda_{i_f}^{(f)}\,,f=1,\cdots, n$, $i_f=1,\cdots,d^2-1$ be the mutually orthogonal generators of the special unitary Lie algebra $\mathfrak{su}(d)$ under a fixed bilinear form, and $I$ the $d\times d$ the identity matrix. A $n$-partite state $\rho\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$ can be written as follows: $$\begin{aligned}
\rho&=&\frac{1}{d^n}I\otimes\cdots\otimes I+\frac{1}{2d^{n-1}}\sum_{f=1}^n\sum_{i_1=1}^{d^2-1}t_{i_1}^{f}\lambda_{i_1}^{(f)}\otimes I\otimes\cdots\otimes I+\cdots\nonumber\\
&+&\frac{1}{2^n}\sum_{i_1,\cdots,i_n=1}^{d^2-1}t_{i_1,\cdots,i_n}^{1\cdots n}\lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)}\otimes\cdots\otimes\lambda_{i_n}^{(n)},\end{aligned}$$ where $\lambda_{i_f}^{(f)}$ ($(f)$ refers the position of $\lambda_{i_f}$ in the tensor product) stands for the operators with $\lambda_{i_f}$ on $H_{d_f}$, and $I$ on the remaining spaces, $t_{i_1}^{f}=tr(\rho\lambda_{i_1}^{(f)}\otimes I\otimes\cdots\otimes I),\cdots,$ $t_{i_1,\cdots,i_n}^{1\cdots n}=tr(\rho\lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)}\otimes\cdots\otimes\lambda_{i_n}^{(n)})$. Let $T^{(f)},\cdots, T^{(1\cdots n)}$ be the vectors (tensors) with entries $t_{i_f}^{f}, \cdots, t_{i_1\cdots i_n}^{1\cdots n}$ respectively where $1\leq f\leq n$, then we get $\|T^{(f)}\|^2=\sum_{i_f=1}^{d^2-1}(t_{i_f}^{f})^2,\cdots, \|T^{(1\cdots n)}\|^2=\sum_{i_1,\cdots,i_n=1}^{d^2-1}(t_{i_1\cdots i_n}^{1\cdots n})^2$. Define further $A_1=\sum_{f=1}^n\|T^{(f)}\|^2, \cdots, A_n=\|T^{(1\cdots n)}\|^2$.
\[lemma:2\] Let $\rho\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$ be a pure state, $$\begin{aligned}
\|T^{(1\cdots n)}\|^2\leq(\frac{2}{d})^n\frac{(n-2)d^n-nd^{n-2}+2}{n-2}.\end{aligned}$$
Let $\rho_{i_1}$ and $\rho_{i_2\cdots i_n}$ be the density matrices with respect to the subsystem $H_{d_{i_1}}, i_1=1,\cdots,n$, and $H_{d_{i_2}\cdots d_{i_n}}, 1\leq i_2<\cdots<i_n\leq n$. As for a pure state $\rho$ we have $tr(\rho^2)=1$ and $tr(\rho_{i_1}^2)=tr(\rho_{i_2\cdots i_n}^2)$. Therefore when $n=2$, we get $\rho_1^2=\rho_2^2$ and $\|T^{(12)}\|\leq\frac{4(d^2-1)}{d^2}$ [@lww]. When $n>2$, we have $$\begin{aligned}
A_n&=&2^n(1-\frac{1}{d^n})-\frac{2^{n-1}}{d^{n-1}}A_1-\frac{2^{n-2}}{d^{n-2}}A_2-\cdots-\frac{2}{d}A_{n-1}\nonumber\\
&=&2^n(1-\frac{1}{d^n})-\frac{2^{n-1}}{d^{n-1}}A_1-\frac{2^{n-2}}{d^{n-2}}[\frac{n(d^{n-2}-1)}{(n-2)d^n}+\frac{d^{n-2}-n+1}{2(n-2)d^{n-2}}A_1-\frac{n-3}{8(n-2)d^{n-3}}A_3-\cdots\nonumber\\
&-&\frac{1}{d(n-2)2^{n-1}}A_{n-1}]-\cdots-\frac{2}{d}A_{n-1}\nonumber\end{aligned}$$ $$\begin{aligned}
&=&(\frac{2}{d})^n\frac{(n-2)d^n-nd^{n-2}+2}{n-2}-\frac{2^n(d^{n-2}-1)}{2(n-2)d^{n-1}}A_1-\frac{2^n}{8(n-2)d^{n-3}}A_3\nonumber\\
&-&\frac{2^{n+1}}{16(n-2)d^{n-4}}A_4-\cdots-\frac{2^n(n-3)}{d(n-2)2^{n-1}}A_{n-1}\nonumber\\
&\leq&(\frac{2}{d})^n\frac{(n-2)d^n-nd^{n-2}+2}{n-2}.\end{aligned}$$
$\square$
[**Remark 7.**]{} When the dimensions of each system for tripartite and four-partite quantum states are the same, Lemma \[lemma:2\] specializes to Proposition \[prop:1\] and Proposition \[prop:2\].
For the $n$-partite quantum state $\rho\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$, we denote the general $k$-partite decompositions of $\rho$ as follows: $\{a^1\},\cdots,\{a^{k_1}\},\{c_1^1, c_2^1\},\cdots,\{c_1^{k_2}, c_2^{k_2}\},\cdots,\{e_1^{k_j},\cdots,e_j^{k_j}\}$, and $\sum_{m=1}^jk_m=k, \sum_{m=1}^jmk_m=n$. Denote the upper bounds of the $j$-body corralation tensors associated to partition $(12\cdots j)$ by $w_{12\cdots j}$, namely, $\|T^{(1)}\|^2\leq\frac{2(d-1)}{d}=w_1$, $\|T^{(12)}\|^2\leq\frac{4(d^2-1)}{d^2}=w_{12}$, and $\|T^{(12\cdots j)}\|^2\leq(\frac{2}{d})^j\frac{(j-2)d^j-jd^{j-2}+2}{j-2}=w_{12\cdots j}$ (cf. Lemma \[lemma:2\]).
\[thm:7\] Let $\rho\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$ be an $n$-partite $k$-separable quantum state. We have $$\begin{aligned}
\|T^{(1\cdots n)}\|\leq(w_1)^{k_1}(w_{12})^{k_2}\cdots(w_{12\cdots j})^{k_j},\end{aligned}$$ where $\sum_{m=1}^jk_m=k$, $\sum_{m=1}^jmk_m=n$.
Assume that $|\varphi\rangle\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$ is a pure state, say $|\varphi\rangle=|a^1\rangle\otimes \cdots\otimes |a^{k_1}\rangle\otimes \cdots\otimes |e_1^{k_j}\cdots e_j^{k_j}\rangle$, then we have $$\begin{aligned}
t_{i_1\cdots i_n}^{1\cdots n}&&=tr(|\varphi\rangle\langle\varphi|\lambda_{i_1}^{(1)}\otimes\lambda_{i_2}^{(2)}\otimes\cdots\otimes\lambda_{i_n}^{(n)})\nonumber\\
&&=tr(|a^1\rangle\langle a^1|\lambda_{i_1}^{(1)})\cdots tr(|a^{k_1}\rangle\langle a^{k_1}|\lambda_{i_{k_1}}^{({k_1})})\cdots tr(|e_1^{k_j}\cdots e_j^{k_j}\rangle\langle e_1^{k_j}\cdots ej^{k_j}|\nonumber\\
&&\lambda_{i_{k_1+2k_2+\cdots+(j-1)k_{j-1}+1}}^{({k_1+2k_2+\cdots+(j-1)k_{j-1}+1})}\otimes \cdots\otimes\lambda_{i_{k_1+2k_2+\cdots+jk_j}}^{({k_1+2k_2+\cdots+jk_j})})\nonumber\\
&&=t_{i_1}^1\cdots t_{i_{k_1}}^{k_1}\cdots t_{i_{k_1+2k_2+\cdots+(j-1)k_{j-1}+1},\cdots, i_{k_1+2k_2+\cdots+jk_j}}^{{k_1+2k_2+\cdots+(j-1)k_{j-1}+1},\cdots,{k_1+2k_2+\cdots+jk_j}}.\end{aligned}$$ Thus, $$\begin{aligned}
\|T^{(1\cdots n)}\|^2&=&\sum_{i_1,\cdots,i_n=1}^{d^2-1}(t_{i_1\cdots i_n}^{1\cdots n})^2\nonumber\\
&=&\|T^{(1)}\|^2\cdots\|T^{(k_1)}\|^2\cdots\|T^{({k_1+2k_2+\cdots+(j-1)k_{j-1}+1},\cdots,{k_1+2k_2+\cdots+jk_j})}\|\nonumber\\
&\leq&(w_1)^{k_1}(w_{12})^{k_2}\cdots(w_{12\cdots j})^{k_j}.\end{aligned}$$ In general for any mixed state $\rho\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$ with ensemble representation $\rho=\sum_ip_i|\varphi_i\rangle\langle\varphi_i|$ where $\sum_ip_i=1$, we derive that $$\begin{aligned}
\|T^{(1\cdots n)}(\rho)\|^2=\|\sum_ip_iT^{(1\cdots n)}(|\varphi_i\rangle)\|^2&\leq& \sum_ip_i\|T^{(1\cdots n)}(|\varphi_i\rangle)\|^2\nonumber\\
&\leq&(w_1)^{k_1}(w_{12})^{k_2}\cdots(w_{12\cdots j})^{k_j}.\end{aligned}$$
$\square$
[**Remark 8.**]{} When $\rho\in H_1^d\otimes H_2^d\otimes H_3^d\otimes H_4^d$ be a four-partite quantum state. Explicitly by Theorem \[thm:7\], we have that T\^[(1234)]{}{
[lr]{} (d-1)(d\^3-3d+2), if is 1-3 separable;\
(d\^2-1)\^2, if is 2-2 separable;\
(d\^2-1)(d-1)\^2, if is 1-1-2 separable;\
(d-1)\^4, if is 1-1-1-1 separable.
. This shows that Theorem 3 in [@lww] is a special case of Theorem \[thm:7\].
[*Example*]{} 4. Consider the quantum state $\rho\in H_1^d\otimes\cdots\otimes H_5^d$, $$\begin{aligned}
\rho=x|\psi\rangle\langle\psi|+\frac{1-x}{32}I_{32},\end{aligned}$$ where $|\psi\rangle=\frac{1}{\sqrt{2}}(|00000\rangle+|11111\rangle)$ and $I_{32}$ stands for the $32\times32$ identity matrix. Because of $\|T^{(12345)}\|^2=\sum_{i_1,i_2,i_3,i_4,i_5=1}^{d^2-1}t_{i_1i_2i_3i_4i_5}^{12345}$, we see that $\|T^{(12345)}\|^2=16x^2$. By Theorem \[thm:7\] T\^[(12345)]{}\^2{
[lr]{} (d-1)(d\^2-1)\^2, is 1-4 or 1-2-2 separable;\
(d\^2-1)(d\^3-3d+2), is 2-3 separable;\
(d-1)\^2(d\^3-3d+2), is 1-1-3 separable;\
(d-1)\^5, is 1-1-1-1-1 separable.
. Thus, for $\frac{3}{4}<x\leq\frac{\sqrt{3}}{2}$, $\rho$ will not be 1-4 or 1-2-2 separable. For $\frac{\sqrt{3}}{2}<x\leq1$ and $\frac{1}{2}<x\leq\frac{3}{4}$, $\rho$ will not be 2-3 separable or 1-1-3 separable respectively. For $\frac{1}{4}<x\leq\frac{1}{2}$, $\rho$ will be not 1-1-1-1-1 separable.
Conclusion
============
We have studied necessary conditions of separability for multipartite quantum states based on correlation tensors, and we have derived the upper bound for the norms of correlation tensors and the separability criterion under any partition by constructing a matrix for tripartite and four-partite quantum state. We also obtain the separability criteria under various partitions by using matrix method. Furthermore, we have studied the norm of correlation tensors for $\rho\in H_1^d\otimes\ H_2^d\otimes\cdots\otimes H_n^d$ to obtain necessary conditions of separability under any k-partition. Several examples are given under different partitions of the quantum state and our criteria are seen to be able to judge more general situations. In particular, explicit examples are given to show for tripartite and four-partite states.
**Acknowledgements** This work is supported by the National Natural Science Foundation of China under grant Nos. 11101017, 11531004, 11726016 and 11675113, and Simons Foundation under grant No. 523868, Key Project of Beijing Municipal Commission of Education (KZ201810028042).
[99]{}
Ekert, A.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. **67**, 661, 1991.
Hillery, M., Buzek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A **59**, 1829, 1999.
Gisin, N., Ribordy, G., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. **74**, 145, 2002.
Schauer, S., Huber, M., Hiesmayr, B. C.: Experimentally feasible security check for n-qubit quantum secret sharing. Phys. Rev. A **82**, 062311, 2010.
Wu, S. J., Chen, X. M., Zhang, Y. D.: A necessary and sufficient criterion for multipartite separable states. Phys. Lett. A **275**, 244, 2000.
Chen, K., Wu, L. A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. **3**, 193, 2003.
Gurvits, L.: Classical complexity and quantum entanglement. J. Comput. **69**, 448, 2004.
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. **81**, 865, 2009.
Hassan, A. S. M., Joag, P. S.: An experimentally accessible geometric measure for entanglement in N-qudit pure states. Phys. Rev. A **80**, 042302, 2009.
Fei, S. M., Wang, Z. X., Zhao, H.: A note on entanglement of formation and generalized concurrence. Phys. Lett. A **329**, 414, 2004.
Xie, C., Zhao, H., Wang, Z. X.: Separability of density matrices of graphs for multipartite systems. Electron. J. Comb. **20**, 2013.
Zhao, H., Zhang, X. H., Fei, S. M., Wang, Z. X.: Characterization of four-qubit states via bell inequalities. Chinese Sci. Bull. **58**, 2334, 2013.
Zhao, H., Fei, S. M., Fan, J., Wang, Z. X.: Inequalities detecting entanglement for arbitrary bipartite systems. Int. J. Quant. Inform. **12**, 1450013, 2014.
Yu, X. Y., Zhao, H.: Separability of tripartite quantum states with strong positive partial transposes. Int. J. Theor. Phys. **54**, 292, 2015.
Zhao, H., Guo, S., Jing, N. H., Fei, S. M.: Construction of bound entangled states based on permutation operators. Quantum Inf. Process. **15**, 1529, 2016.
Bloch, F.: Nuclear induction. Phys. Rev. **70**, 460, 1946.
Byrd, M. S., Khaneja, N.: Characterization of the positivity of the density matrix in terms of the coherence vector representation. Phys. Rev. A, **68**, 062322, 2003.
Jakobczyk, L., Siennicki, M.: Geometry of bloch vectors in two-qubit system. Phys. Lett. A **286**, 383, 2001.
Kimura, G.: The bloch vector for N-Level systems. Phys. Lett. A **314**, 339, 2003.
Li, M., Wang, Z., Wang, J., Shen, S., Fei,S.M.: The norms of Bloch vectors and classification of four qudits quantum states. Europhys. Lett. A **125**, 20006, 2019.
de Vicente, J. I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. **7**, 624, 2007.
de Vicente, J. I.: Further results on entanglement detection and quantification from the correlation matrix criterion. J. Phys. A **41**, 065309, 2008.
Hassan, A. S. M., Joag, P .S.: Separability criterion for multipartite quantum states based on the bloch reprsentation of density matrices. Quantum Inf. Comput. **8**, 0773, 2008.
Li, M., Wang, J., Fei, S. M., Li-Jost, X. Q.: Quantum separability criteria for arbitrary dimensional multipartite states. Phys. Rev. A **89**, 022325, 2014.
Li, M., Jia, L., Wang, J., Shen, S., Fei, S. M.: Measure and detection of genuine multipartite entanglement for tripartite systems. Phys. Rev. A **96**, 052314, 2017.
Yu, C. S., Song, H. S.: Separability criterion of tripartite qubit systems. Phys. Rev. A **72**, 022333, 2005.
Yu, C. S., Song, H. S.: Entanglement monogamy of tripartite quantum states. Phys. Rev. A **77**, 032329, 2008.
de Vicente, J. I., Huber, M.: Multipartite entanglement detection from correlation tensors. Phys. Rev. A **84**, 062306, 2011.
Shen, S. Q., Yu, J., Li, M., Fei, S. M.: Improved separability criteria based on bloch representation of density matrices. Scientific Reports **6**, 28850, 2016.
|
---
abstract: 'Thin films of $\alpha$-MoGe show progressively reduced $T_{c}$’s as the thickness is decreased below 30 nm and the sheet resistance exceeds 100 $\Omega/\Box$. We have performed far-infrared transmission and reflection measurements for a set of $\alpha$-MoGe films to characterize this weakened superconducting state. Our results show the presence of an energy gap with ratio $2\Delta_0/k_BT_{c} = 3.8 \pm 0.1$ in all films studied, slightly higher than the BCS value, even though the transition temperatures decrease significantly as film thickness is reduced. The material properties follow BCS-Eliashberg theory with a large residual scattering rate except that the coherence peak seen in the optical scattering rate is found to be strongly smeared out in the thinner superconducting samples. A peak in the optical mass renormalization at $2\Delta_0$ is predicted and observed for the first time.'
author:
- 'H. Tashiro$^1$'
- 'J. M. Graybeal$^1$'
- 'D. B. Tanner$^1$'
- 'E. J. Nicol$^2$'
- 'J. P. Carbotte$^3$'
- 'G. L. Carr$^4$'
title: 'The unusual thickness dependence of superconductivity in $\alpha$-MoGe thin films'
---
Disorder and reduced dimensionality affect the physical properties of metallic systems in a number of ways. Anomalous diffusion leads to localization of electrons and a related enhancement of the Coulomb interaction via reduced screening[@anderson79; @altshuler80], seen as an increase in $\mu^{*}$, the renormalized Coulomb interaction parameter. In a system of lower dimensions, the coupling to disorder increases, and pronounced effects are expected. Disorder-driven localization and the related enhancement of the Coulomb interaction inherently compete with the attractive interaction in superconducting metals[@maekawa81; @anderson83], described by the electron-phonon spectral density $\alpha^2F(\omega)$[@carbotte]. This competition reduces the transition temperature. Of particular interest are two-dimensional (2D) superconductors in which the degree of disorder can be adjusted by varying the appropriate parameters. In an ideal 2D system, the relevant parameter is normally considered to be the sheet resistance, $R_{\Box}$. The sheet resistance is determined by two factors: the (possibly thickness dependent) conductivity $\sigma$ and the film thickness $d$.
Amorphous MoGe ($\alpha$-MoGe) thin films are thought to be a model system for studying the interplay between superconductivity and disorder. Several transport experiments have revealed a sharp reduction in the superconducting transition temperature $T_{c}$ with increasing $R_{\Box}$, even in the weakly localized regime[@graybeal84; @graybeal85; @strongin; @raffy]. The suppression of $T_c$ has been attributed to localization and an increase in the Coulomb interaction[@maekawa81]. In this Letter, we explore the $T_c$ suppression in $\alpha$-MoGe thin films with different thickness via temperature-dependent far-infrared transmittance and reflectance. A strong suppression of $T_{c}$ with increasing $R_{\Box}$ is observed. The superconducting energy gap is also reduced, but the ratio of gap energy to transition temperature and the normal-state conductivity, both of which could be dependent on the disorder-driven Coulomb interaction, are not affected at all.
Our films were prepared by co-magnetron sputtering from elemental targets onto rapidly rotating (3 rev/sec or 1Å deposited/rev) single-crystal r-cut sapphire substrates (1 mm thick). A 75Å$\alpha$-Ge underlayer was first laid down on the substrates to ensure smoothness of the subsequently deposited MoGe films. For films prepared in similar fashion, no sign of crystalline inclusions were observed by x-ray and transmission electron microscopy. This procedure is known to yield uniform and homogeneous amorphous films of near ideal stoichiometry[@graybeal84; @graybeal85]. A thickness monitor gave the film thickness; the remaining parameters of our films, in Table \[tab:table1\], were all determined from optical measurements, described below.
Film $d$ (nm) $T_{c}$ (K) $R_{\Box}$ $(\Omega)$ $2\Delta_{0}$ (cm$^{-1})$ $2\Delta_{0}/kT_{c}$ $n_{s}\footnotemark[1]$
------ ---------- ------------- ----------------------- --------------------------- ---------------------- -------------------------
A 4.3 $<1.8$ 505 - - -
B 8.3 4.5 260 12 3.7 1.20
C 16.5 6.1 131 16 3.9 1.49
D 33 6.9 69 18 3.8 1.66
: \[tab:table1\] Parameters for MoGe films.
Far-infrared measurements were performed at beamlines U10A and U12IR of the National Synchrotron Light Source at Brookhaven National Laboratory. U12IR, equipped with a Sciencetech SPS200 Martin-Puplett interferometer, was used for frequencies between 5 and 50 cm$^{-1}$. A Bruker IFS-66v/S rapid scan Fourier-transform interferometer at U10A was used over 20–100 cm$^{-1}$. A bolometer operating at 1.7 K provided excellent sensitivity; its window is responsible for the high-frequency cutoff of 100 cm$^{-1}$. The films were in an Oxford Instruments Optistat bath cryostat, which enabled sample temperatures of 1.7–20 K. Transmittance $T(\omega)$ and reflectance $R(\omega)$ of four films were taken at various temperatures below $T_{c}$. The normal-state transmittance and reflectance were taken at 10 K.
For a metal film of thickness $d \ll \lambda$, the wavelength of the far-infrared radiation, and $d \ll \{\delta, \lambda_{L}\}$, the skin depth (normal state) or penetration depth (superconducting state), the transmittance across the film into the substrate and the single-bounce reflectance from the film are both determined by the film’s complex conductivity $\sigma=\sigma_{1}+i\sigma_{2}$ according to[@palmertinkham; @gao96prb] $$\label{Eq:FilmTransmittance}
T_{f} = \frac{4n}
{(Z_{0}\sigma_{1}d +n+1)^{2}+ (Z_{0}\sigma_{2} d)^{2}},$$ $$\label{Eq:FilmReflectance}
R_{f} = \frac{(Z_{0}\sigma_{1}d + n- 1)^{2} + (Z_{0}\sigma_{2}d )^{2}}
{( Z_{0}\sigma_{1}d +n+1)^{2}+ (Z_{0}\sigma_{2} d)^{2}},$$ where $n$ is the refractive index of the substrate and $Z_{0}$ is the impedance of free space ($4\pi/c$ in cgs; 377 $\Omega$ in mks). Although Eqs. \[Eq:FilmTransmittance\] and \[Eq:FilmReflectance\] describe the physics of the thin film on a thick substrate, the transmittance and reflectance are influenced by multiple internal reflections within the substrate and the (weak) substrate absorption. After accounting for these effects[@gao96prb], measurements of $T$ and $R$ at each frequency determine $\sigma_{1}$ and $\sigma_{2}$. Beginning with Palmer and Tinkham[@palmertinkham], this approach has been used a number of times in the past to obtain the optical properties of superconducting thin films.
We used the broadband far-infrared transmittance to determine the transition temperature. The normal-state transmission is temperature independent (on account of the dominant residual scattering). When, as the sample temperature is decreased slowly, superconductivity occurs, the broadband transmission increases. We call $T_c$ that temperature at which a measurable transmission increase first occurs. Finally, the normal-state infrared transmission, via Eq. (1), gives $R_\Box = 1/\sigma d$. (The frequency-independent transmission tells us that the normal-state $\sigma_1(\omega) = \mbox{constant} \gg \sigma_2$.)
Figure \[fig:ratio\] shows $T_{s}/T_{n}$ and $R_{s}/R_{n}$ at several temperatures for three films; the thinnest film did not superconduct at the lowest achievable temperature in our apparatus. The shape of the transmission curve is determined by a competition between $\sigma_{1}$ and $\sigma_{2}$ in Eq. (\[Eq:FilmTransmittance\]). At low frequency the ratio goes to zero as $\sim\omega^{2}$ due to the kinetic inductance of the superfluid, which yields $\sigma_{2s} \sim 1/\omega$ while at the same time $\sigma_{1s} \sim 0$. The frequency of the maximum of $T_{s}/T_{n}$ occurs very close to the superconducting gap frequency $\omega_{g}=2\Delta/\hbar$ because $\sigma_{1s}$ rises toward the normal-state value above the gap. At high frequencies $T_{s}/T_{n} = 1$.
![\[fig:ratio\] (Color online) Measured transmittance and reflectance ratios of three MoGe films at several temperatures.](Fig1.eps){width="0.99\hsize"}
The data in Fig. \[fig:ratio\] clearly show that the gap shrinks as temperature increases toward $T_{c}$. At a given reduced temperature $T/T_{c}$, the gap shifts to lower energy as the film becomes thinner. The suppression of $T_{c}$ with decreasing thickness (increasing $R_{\Box}$) is confirmed as well. Fits to these data using the dirty-limit, finite-temperature Mattis-Bardeen (MB)[@mattis] conductivity expressions were good, consistent with the signal-to-noise ratio in the data, giving $2\Delta_{0}/k_{B}T_{c} = 3.8 \pm 0.1$, slightly higher than the BCS weak coupling limit of 3.5. Changes in $2\Delta_{0}/k_{B}T_{c}$ with thickness are much smaller than the $T_{c}$ reduction and not monotonic. (See Table \[tab:table1\] for the fit results.)
![\[fig:conductivity\] (Color online) Real (filled circles) and imaginary (open circles) parts of the optical conductivity for three $\alpha$-MoGe thin films. The data are taken at 2.2 K. The Mattis-Bardeen conductivity is also shown.](Fig2.eps){width="0.7\hsize"}
Figure \[fig:conductivity\] shows the 2.2 K results for the real and imaginary parts of the optical conductivity, $\sigma_{1}(\omega)$ and $\sigma_{2}(\omega)$, for each film. The Mattis-Bardeen conductivities are also shown. The gap of $2\Delta$ in the absorption spectrum is evident. All three films have approximately the same normal state conductivity, $\sigma_{N} \sim
4000$ $\Omega^{-1}$cm$^{-1}$ obtained from transmittance measurement of film in the normal state; the superconducting-state $\sigma_1(\omega)$ approaches this value at high frequencies. A similar value (4080 $\Omega^{-1}$cm$^{-1}$) is found by transport measurements. Thus, we conclude that the normal-state conductivity (or resistivity) is independent of the thickness of the film.
As the data is clearly in the dirty limit, the fitting with the MB expressions is quite adequate for obtaining the value of $\Delta_0$. However, in order to elucidate further features of the data, discuss changes in $T_c$, and make predictions, we will now move to more sophisticated calculations using BCS-Eliashberg theory. Figure \[fig:conductivity165\] shows the results for the real and imaginary parts of the optical conductivity, $\sigma_{1}(\omega)$ and $\sigma_{2}(\omega)$, for film C. The lines are results of numerical calculations for the conductivity based on the Eliashberg equations and the Kubo formula for the current-current correlation function[@marsiglio]. The electron-phonon spectral function was taken from that obtained through inversion of tunneling data on amorphous Mo[@kimli] and its mass enhancement parameter $\lambda$ is fixed at 0.9. The Coulomb repulsion $\mu^*$ was adjusted to obtain the measured value of $T_c$. Other parameters are the impurity scattering rate $1/\tau^{imp}=3.5$ eV and the plasma energy $\Omega_p=10.7$ eV. We will see later how these were obtained from the conductivity data itself. The agreement with the data for $\sigma_1$ is best at the lowest temperature considered, with small deviations for $T$ near $T_c$. This is true for all three films. The theory for $\sigma_{2}$ agrees with the $\sim
1/\omega$ low-frequency behavior but tends to be below the experiment, especially at higher frequencies. The fit is less good with increasing $T$, although the qualitative behavior is given correctly.
![\[fig:conductivity165\] (Color online) $\sigma_{1}(\omega)$ and $\sigma_{2}(\omega)$ at various temperatures for the 16.5 nm MoGe film. The points are the data and the lines are the results of our Eliashberg calculations.](Fig3.eps){width="0.7\hsize"}
As changing the thickness of the sample could change both $\mu^*$ and the electron-phonon interaction, there is some choice in fitting the data with Eliashberg theory. In Fig. \[fig:tcmustar\], we show results for $T_c$ and the gap ratio as a function of $\mu^*$ for three values of $\lambda$. For fixed $\lambda$, the points on the $T_c$ curve are from the experimental data for the MoGe films, illustrating the $\mu^*$ needed to obtain the $T_c$. With $\mu^*$ and $\lambda$ now fixed, the experimental points for the gap ratio can be compared to the prediction and there is good agreement. It is clear from this figure that keeping the ratio at 3.8 can be achieved through a change in $\mu^*$ as suggested in[@anderson79; @altshuler80; @maekawa81; @anderson83] but one cannot rule out additional small changes in $\lambda$. In fact, Höhn and Mitrović[@hohn] in their Eliashberg analysis of tunneling data on disordered Pb films found evidence for a change in both these parameters with changing $E_F\tau^{imp}$, where $E_F$ is the Fermi energy. Here such differences will not matter as we are in an impurity-dominated regime and the optics is not sensitive to the $\mu^*$ or $\lambda$ value as we will explain. A $\lambda$ of order 1 is needed, however, to get the measured value of the gap to $T_c$. For definiteness, we only change $\mu^*$ leaving $\alpha^2F(\omega)$ fixed.
![\[fig:tcmustar\] (Color online) Dependence of $T_c$ and $2\Delta/k_BT_c$ on Coulomb repulsion $\mu^*$ for three values of electron-phonon mass enhancement $\lambda$.](Fig4.eps){width="0.6\hsize"}
To proceed with the analysis, we introduce the optical self-energy $\Sigma^{op}(T,\omega)$ and use the extended Drude model, where the conductivity is written $\sigma(T,\omega)= (i\Omega_p^2/4\pi)/(\omega-2\Sigma^{op}(T,\omega))$. The real part of $\Sigma^{op}$ gives the optical mass renormalization $\lambda^{op}(T,\omega)$ with $\omega \lambda^{op}(T,\omega)=-2
\Sigma_1^{op}(T,\omega)$ and its imaginary part is related to the optical scattering rate according to $1/\tau^{op}(T,\omega)=-2
\Sigma_2^{op}(T,\omega)$. These quantities are shown in Fig. \[fig:tauop\] for the thickest and thinnest superconducting samples at $T=2.2K$. To obtain $1/\tau^{op}$, we had to use an impurity scattering rate of 3.5 eV. For $v_F\sim 1.5\times 10^8$ cm/sec, this rate corresponds to a mean free path of $\sim 0.3$ nm. This value, while small, is consistent with other estimates and is much less than the thickness of the films[@graybeal84]. Hence surface scattering is not important, $R_{\Box} \propto 1/d$, and the normal-state conductivity does not depend on sheet resistance.
![\[fig:tauop\] (Color online) Optical scattering rate $1/\tau^{op}(\omega)$ and mass renormalization $1+\lambda^{op}(\omega)$ for the thickest and thinnest superconducting films. Points are data and lines are Eliashberg calculations for the extreme dirty limit.](Fig5.eps){width="0.6\hsize"}
It is important to understand that the peaks in $1/
\tau^{op}(\omega)$ are the optical equivalent of density of states coherence peaks. The calculation for the thickest film fits the data well but for the thinner film the peak is very much attenuated, perhaps indicating a new effect outside standard Eliashberg theory. In their tunneling study of the metal-insulator transition in aluminum films, Dynes and coworkers[@dynes] found a similar effect, namely, a broadening of the density of states coherence peak with increased sheet resistance. The lower panel of Fig. \[fig:tauop\] gives the optical effective mass in the superconducting state. For both samples as $\omega\to 0$, this quantity is very large, of the order of 1000, which is comparable to heavy fermion masses, although its origin is quite different. These values reflect directly the large impurity scattering and are related to the decrease in superfluid density with decreasing $\tau^{imp}$. In an Eliashberg superconductor, the superfluid density ($n_s$) at $T=0$ in the clean-limit case is given by $n_s^{\rm clean}(T=0)=n/(1+\lambda)$, where $n$ is the electron density. In the dirty limit where $1/[2\Delta_0\tau^{imp}(1+\lambda)]\gg 1$, it is instead given by $n_s^{\rm dirty}(T=0)=n\pi\Delta_0\tau^{imp}$ with $\lambda$ dropping out[@marsiglio]. For fixed $n$, this gives immediately a relation between the superfluid density $n_s$, $T_c$, and $\sigma_n$ [@uemura; @homes; @argument] The superfluid density so estimated is shown in Table \[tab:table1\]; its variation is due entirely to the change in $T_c$. We note also the large peak at $2\Delta_0$ in $1+\lambda^{op}$, predicted by theory and seen in the data.
In summary, the observed strong suppression of $T_c$ with increasing $R_\Box$ while the ratio $2\Delta_0/kT_c$ remains constant at the intermediate value of 3.8 can be easily accounted for by a small decrease in $\lambda$, an increase in $\mu^*$ or a combination of both. The large residual scattering rate of our MoGe films makes their optical response indistinguishable from BCS; yet, because of Anderson’s theorem, $2\Delta_0/kT_c$ remains bigger than 3.54. In such dirty samples a large value of the optical effective mass is predicted as well as a peak at $\omega=2\Delta_0$, with a rapid decrease as $\omega$ is increased. Both effects are observed, the peak for the first time. Moreover, the optical scattering rate shows the expected coherence peak in the thickest film considered but is strongly suppressed in the thinnest superconducting one. This effect cannot be understood within BCS-Eliashberg theory and may indicate new physics.
Research is supported by the U.S. Department of Energy through Contract DE-FG02-02ER45984 at the University of Florida, DE-AC02-98CH10886 at the Brookhaven National Laboratory, NSERC of Canada and the Canadian Institute for Advanced Research.
[99]{}
P. W. Anderson, E. Abrahams, and T. V. Ramakrishnan, Phys. Rev. Lett. [**43**]{}, 718 (1979).
B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev. Lett. [**44**]{}, 1288 (1980).
S. Maekawa and H. Fukuyama, J. Phys. Soc. Jpn. [**51**]{}, 1380 (1981).
P. W. Anderson, K. A. Muttalib, and T. V. Ramakrishnan, Phys. Rev. B [**28**]{}, 117 (1983).
J. P. Carbotte, Rev. Mod. Phys. [**62**]{}, 1027 (1990).
J. M. Graybeal and M. R. Beasley, Phys. Rev. B [**29**]{}, 4167 (1984).
J. M. Graybeal, Physica, [**135B**]{}, 113 (1985). M. Strongin, R. S. Thompson, O. F. Kammerer, and J. F. Crow, Phys. Rev. B [**1**]{}, 1078 (1970).
H. Raffy, R. B. Laibowitz, P. Chaudhari, and S. Maekawa, Phys. Rev. B [**28**]{}, 6607 (1983).
L. H. Palmer and M. Tinkham, Phys. Rev. [**165**]{}, 588 (1968).
F. Gao, G. L. Carr, C. D. Porter, D. B. Tanner, G. P. Williams, C. J. Hirschmugl, B. Dutta, X. D. Wu, and S. Etemad Phys. Rev. B [**54**]{}, 700 (1996).
D. C. Mattis and J. Bardeen, Phys. Rev. [**111**]{}, 412 (1958).
F. Marsiglio and J. P. Carbotte in [*Physics of Conventional and High $T_c$ Superconductors*]{}, ed. H. H. Bennemann and J. B. Ketterson (Springer-verlag, Berlin, 2002) Vol. 1, p. 233.
D. B. Kimhi and T. H. Geballe, Phys. Rev. Lett. [**45**]{}, 1039 (1980).
T. Höhn and B. Mitrović, Z. Phys. B [**93**]{}, 173 (1994).
R. C. Dynes, J. P. Garno, G. B. Hertel and T. P. Orlando, [Phys. Rev. Lett.]{} [**53**]{}, 2437 (1984).
Y.J. Uemura, L.P. Le, G.M. Luke, B.J. Sternlieb, W.D. Wu, J.H. Brewer, T.M. Riseman, C.L. Seaman, M.B. Maple, M. Ishikawa, D.G. Hinks, J.D. Jorgensen, G. Saito, and H. Yamochi, [Phys. Rev. Lett.]{} [**66**]{}, 2665 (1991).
C. C. Homes, S. V. Dordevic, M. Strongin, D. A. Bonn, Ruixing Liang, W. N. Hardy, Seiki Komiya, Yoichi Ando, G. Yu, N. Kaneko, X. Zhao, M. Greven, D. N. Basov, and T. Timusk, [Nature]{} [**430**]{}, 539 (2004).
The “missing area” in $\sigma_1(\omega)$ gives the weight of the delta function and, consequently, the superfluid density: $
4\pi n_se^2/c^2 = 1/\lambda_L^2(T=0)
=\sigma_n 4\pi^2\Delta_0/c^2
= 75 \sigma_n T_c/c^2
$. If $\sigma_n$ is constant, then $n_s \propto T_c$.
|
---
abstract: |
The “Capelli problem” for the symmetric pairs $(\gl\times \gl,\gl)$ $(\gl,\g{o})$, and $(\gl,\g{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter (see [@KostantSahi1], [@KostantSahi2], [@KnopSahi], [@Okounkov]). In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\g g,\g k):=(\gl(m|2n),\g{osp}(m|2n))$ and $(\gl(m|n)\times\gl(m|n),\gl(m|n))$, acting on $W:=\sS^2(\C^{m|2n})$ and $\C^{m|n}\otimes(\C^{m|n})^*$.
To achieve this goal, we first prove that the center of the universal enveloping algebra of the Lie superalgebra $\g g$ maps surjectively onto the algebra $\sPD(W)^{\g g}_{}$ of $\g g$-invariant differential operators on the superspace $W$, thereby providing an affirmative answer to the “abstract” Capelli problem for $W$. Our proof works more generally for $\gl(m|n)$ acting on $\sS^2(\C^{m|n})$ and is new even for the “ordinary” cases ($m=0$ or $n=0$) considered by Howe and Umeda in [@HoweUmeda].
We next describe a natural basis $\left\{ D_{\lambda }\right\} $ of $\sPD(W)^{\g g}$, that we call the Capelli basis. Using the above result on the abstract Capelli problem, we generalize the work of Kostant and Sahi [@KostantSahi1], [@KostantSahi2], [@Sahi] by showing that the spectrum of $D_{\lambda }$ is given by a polynomial $c_{\lambda }$, which is characterized uniquely by certain vanishing and symmetry properties.
We further show that the top homogeneous parts of the eigenvalue polynomials $c_{\lambda }$ coincide with the spherical polynomials $d_{\lambda }$, which arise as radial parts of $\g k$-spherical vectors of finite dimensional $\g g$-modules, and which are super-analogues of Jack polynomials. This generalizes results of Knop and Sahi [@KnopSahi].
Finally, we make a precise connection between the polynomials $c_\lambda$ and the shifted super Jack polynomials of Sergeev and Veselov [@SerVes] for special values of the parameter. We show that the two families are related by a change of coordinates that we call the Frobenius transform.\
*Keywords:* Lie superalgebras, the Capelli problem, super Jack polynomials, shifted super Jack polynomials.\
*MSC2010:* 17B10, 05E05.
author:
- 'Siddhartha Sahi[^1] , Hadi Salmasian[^2]'
title: 'The Capelli problem for $\gl(m|n)$ and the spectrum of invariant differential operators'
---
Introduction
============
One of the most celebrated results in classical invariant theory is the Capelli identity for $n\times n$ matrices. It plays a fundamental role in Hermann Weyl’s book [@Weyl]. The Capelli identity is profoundly connected to the representation theory of the Lie algebra $\gl(n)$. The well-known article of Roger Howe [@HoweRem] elucidated this connection by giving a conceptual proof of the Capelli identity.
Later on, Howe and Umeda [@HoweUmeda] generalized the Capelli identity to the setting of multiplicity-free spaces by posing and solving two general questions, which they called the *abstract* and *concrete* Capelli problems. Let $\g g$ be a complex reductive Lie algebra, and $W$ be a multiplicity-free $\g g$-space. The abstract Capelli problem asks whether the centre $\bfZ(\g g)$ of $\bfU(\g g)$, the universal enveloping algebra of $\g g$, maps surjectively onto the algebra $\sPD(W)^{\g g}$ of $\g g$-invariant polynomial-coefficient differential operators on $W$. The concrete Capelli problem asks for explicit elements of $\bfZ(\g g)$ whose images generate $\sPD(W)^\g g$.
Around the same time as [@HoweUmeda], Kostant and Sahi [@KostantSahi1], [@KostantSahi2] considered a slightly different question, which we shall refer to here as the Capelli *eigenvalue* problem. It turns out that the algebra $\sPD(W)^{\g g}$ admits a natural basis $D_{\lambda }$ – the *Capelli basis* – which is indexed by the monoid of highest weights of $\g g$-modules $V_{\lambda }$ occurring in the symmetric algebra $\sS\left( W\right) $. The Capelli eignevalue problem asks for determination of the eigenvalue of $D_{\mu }$ on $V_{\lambda }$. It turns out that these eignvalues are of the form $\varphi
_{\mu }\left( \lambda +\rho \right) $, where $\varphi _{\mu }$ is a symmetric polynomial and $\rho $ is a certain “rho-shift". Although [@KostantSahi1], [@KostantSahi2] are written in the context of symmetric spaces, similar ideas work for the multiplicity-free setting, see [@Knop].
In [@Sahi], it is shown that the polynomials $\varphi _{\mu }$ are uniquely characterized by certain vanishing conditions. In fact [@Sahi] considers a general class of polynomials, depending on several parameters, and in [@KnopSahi] it is shown that a one-parameter subfamily of these polynomials is closely related to *Jack polynomials*. More precisely, the Knop–Sahi polynomials are inhomogeneous polynomials, but their top degree terms are the (homogenenous) Jack polynomials. In the special case where the value of the parameter $\theta$ corresponds to a symmetric space, the Knop–Sahi polynomials are the eigenvalue polynomials $\varphi _{\lambda }$, and the Jack polynomials are the spherical polynomials in $V_{\lambda }$. Many properties of the Knop–Sahi polynomials were subsequently proved by Okounkov and Olshanski [@Okounkov], who worked with a slight modification of these polynomials which they call *shifted Jack polynomials*.
The goal of our paper is to extend this circle of ideas to the Lie superalgebra setting. The Jack polynomials for $\theta=1,\frac{1}{2},2$ are spherical polynomials of the symmetric pairs $$(\gl(n)\times \gl(n),\gl(n)),\
(\gl(n),\g{o}(n)),\
(\gl(2n),\g{sp}(2n)).$$ In the Lie superalgebra setting the last two come together and there are only two pairs to consider, namely $$(\gl(m|n)\times \gl(m|n),\gl(m|n))\text{ and }(\gl(m|2n),\g{osp}(m|2n)).$$ The extension of the theory to the first case is easier and, as we show in Appendix \[appxB\], can be achieved by combining the work of Molev [@Molev] with that of Sergeev–Veselov [@SerVes]. Therefore in the body of the paper we concentrate on the second case, which corresponds to the action of $\gl(m|2n)$ on $\sS^2(\C^{m|2n})$.
In the extension of the aforementioned theory to Lie superalgebras, at least two serious difficulties arise. The first issue is that complete reducibility of finite dimensional representations fails in the case of Lie superalgebras. Complete reducibility is crucial in the approach to the abstract Capelli problem in [@HoweUmeda] and [@GoodmanWallach], where it is needed to split off the $\g g$-invariants in $\bfU(\g g)$ and $\sPD(W)^{\g g}$.
The second issue that arises is that the Cartan–Helgason Theorem is not known in full generality for Lie superalgebras. In particular, it is not immediately obvious that every $\gl(m|2n)$-module that appears in $\sP(\sS^2(\C^{m|2n}))$ has an $\g{osp}(m|2n)$-invariant vector. Furthermore, in the purely even case the fact that a spherical vector is determined uniquely by its radial component follows from the $KAK$ decomposition, which does not have an analogue in the context of supergroups. These issues complicate the formulation and proof of Theorem \[prpgw\] and Theorem \[MAINTHM\].
To achieve our goal in this paper, we have to overcome the above difficulties. Our first main result is Theorem \[prpgw\], where we give an affirmative answer to the abstract Capelli problem of Howe and Umeda for the Lie superalgebra $\gl(m|n)$ acting on $\sS^2(\C^{m|n})$. In fact in Theorem \[prpgw\] we prove a slightly more precise statement that a $\gl(m|n)$-invariant differential operator of order $d$ is in the image of an element of $\bfZ(\gl(m|n))$ which has order $d$ with respect to the standard filtration of $\bfU(\gl(m|n))$. The latter refinement is needed in the proofs of Theorem \[MAINTHM\] and Theorem \[thmconnSV\], which we will elaborate on below. For the superpair $(\gl(m|n)\times\gl(m|n),\gl(m|n))$, the corresponding abstract Capelli theorem indeed follows directly from the work of Molev [@Molev], as explained in Appendix \[appxB\] (see Theorem \[thabsscap\]).
Our second main goal concerns the Capelli eigenvalue problem for the Lie superalgebra $\gl(m|2n)$ acting on $\sS^2(\C^{m|2n})$. Extending [@Sahi Theorem 1], we show that the eigenvalues of the Capelli operators are given by polynomials $c_\lambda$ given in Definition \[dfcl\] that are characterized by suitable symmetry and vanishing conditions (see Theorem \[thm-unqclam\]). Furthermore, in Theorem \[MAINTHM\] we show that the top degree homogeneous term of the eigenvalue polynomial $c_\lambda$ is equal to the spherical polynomial $d_\lambda$ given in Definition \[dfbr\]. This extends the result proved by Knop and Sahi in [@KnopSahi] explained above. The corresponding result for the superpair $(\gl(m|n)\times \gl(m|n),\gl(m|n))$ is Theorem \[THMAppB2\]. In this case the spherical and eigenvalue polynomials turn out to be the well known *supersymmetric Schur polynomials* and *shifted supersymmetric Schur* polynomials [@Molev].
Our third main goal is to establish a precise relation between our eigenvalue polynomials $c_\lambda$ and certain polynomials called *shifted super Jack polynomials*. In connection with the eigenstates of the deformed Calogero–Moser–Sutherland operators [@SerVes], Sergeev and Veselov define a family of $(m+n)$-variable polynomials $SP^*_\flat$, parametrized by $(m,n)$-hook partitions $\flat$ (see Definition \[dfnhookprn\]). In the case of the superpair $(\gl(m|2n),\g{osp}(m|2n))$, in Proposition \[prpQlam\] and Theorem \[thmconnSV\], we prove that the $c_\lambda$ and the $SP^*_\flat$ are related by the *Frobenius transform*, see Definition \[FrobT\]. The corresponding statement for the pair $(\gl(m|n)\times\gl(m|n),\gl(m|n))$ is Theorem \[THMAB3\].
It is worth mentioning that in [@SerVes], the shifted super Jack polynomials are obtained as the image of the shifted Jack polynomials under a certain *shifted Kerov map*, and the fact that the image of the shifted Kerov map is a polynomial is indeed a nontrivial statement which is proved in [@SerVes] indirectly. However, our definition of $c_\lambda$ is more conceptual, and the proofs are more straightforward, as they are based on the Harish–Chandra homomorphism and the solution of the abstract Capelli problem. Furthermore, the work of Sergeev and Veselov does not address the relation with spherical representations of Lie superalgebras. Our paper establishes this connection.
We now outline the structure of this article. Section \[section1\] defines the basic notation that is used throughout the rest of the paper. The solution to the abstract Capelli problem for $\gl(m|n)$ acting on $\sS^2(\C^{m|n})$ is given in Section [prfof571]{}. In Section \[Secsuperpr\] we study spherical highest weight modules of $\gl(m|2n)$. We prove in Proposition \[le-dlam\] and Remark \[rmk-v\*\] that every irreducible $\gl(m|2n)$-submodule of $\sS(\sS^2(\C^{m|2n}))$ or $\sP(\sS^2(\C^{m|2n}))$ has a unique (up to scalar) nonzero $\g{osp}(m|2n)$-fixed vector. It is worth mentioning that Proposition \[le-dlam\] does not follow from the work of Alldridge and Schmittner [@AllSch], since they need to assume that the highest weight is “high enough” in some sense. In Section \[Sec-Sec5\], we prove Theorem \[MAINTHM\] and Theorem \[thm-unqclam\] (see the second goal above). Section \[SecRelSer\] is devoted to connecting the eigenvalue polynomials $c_\lambda$ to the shifted super Jack polynomials of Sergeev and Veselov [@SerVes]. Appendix \[sec-pflem\] contains the proof of Proposition \[DGBRVVV\]. Finally, in Appendix B we outline the proofs of our main theorems for the case of $\gl(m|n)\times\gl(m|n)$ acting on $\C^{m|n}\otimes (\C^{m|n})^*$.
We now briefly describe the structure of our proofs. There is no mystery in the formulation of the statement of Theorem \[prpgw\], but its proof is *not* a simple generalization of any of the existing proofs in the purely even case (i.e., when $n=0$) that we are aware of. Our approach is inspired by the proof given by Goodman and Wallach in [@GoodmanWallach Sec. 5.7.1], but it diverges quickly because their argument relies heavily on the complete reducibility of rational representations of a reductive algebraic group. Our proof proceeds by induction, after we show that the symbol of an invariant differential operator is in the image of $\bfZ(\gl(m|n))$. This symbol is in the span of invariant tensors $\bft_\sigma$ defined in where $\sigma$ is a permutation. The next step is to reduce the latter problem to the case where $\sigma$ is a cycle of consecutive letters. Finally, we prove that in this special case, $\bft_\sigma$ is in the image of the *Gelfand elements* $\mathrm{str}(\mathbf E^d)\in\bfZ(\gl(m|n))$ defined in Lemma \[gelfzhel\].
The idea behind the proof of Theorem \[MAINTHM\] is as follows. Let $D_\lambda$ be a Capelli operator, and let $z_\lambda\in\bfZ(\gl(m|2n))$ be an element in the inverse image of $D_\lambda$, whose existence follows from Theorem \[prpgw\]. We show that modulo the natural isomorphism induced by the trace form, the spherical polynomial $d_\lambda$ is the diagonal restriction of the symbol of the polynomial-coefficient differential operator corresponding to the radial part of $z_\lambda$. Furthermore, we show that the eigenvalue polynomial $c_\lambda$ is the image of the radial part of $z_\lambda$ (see Lemma \[cmulZa\]). We combine the latter two statements, as well as the refinement of the abstract Capelli problem obtained in Theorem \[prpgw\], to prove Theorem \[MAINTHM\].
Finally, the proof of Theorem \[thmconnSV\] goes as follows. By considering the action of $z_\lambda$ on the lowest weight vector of an irreducible $\gl(m|2n)$-module $V_\mu$, we prove in Proposition \[prpQlam\] that $c_\lambda(\mu)$ is a polynomial in the highest weight $\mu^*$ of the contragredient representation $V_\mu^*$. Denoting the latter polynomial $c_\lambda^*$, we verify that after a *Frobenius transform* (see Definition \[FrobT\]), the polynomial $c_\lambda^*$ satisfies the supersymmetry and vanishing properties of the shifted super Jack polynomials of [@SerVes]. It then follows that the two polynomials coincide up to a scalar multiple.
We now elaborate on some of the new techniques and ideas introduced in our paper. Our method of proof of Theorem \[prpgw\] yields a recursive procedure for expressing a given invariant differential operator explicitly as the image of an element of $\bfZ(\gl(m|n))$ using Gelfand elements. In the setting of ordinary Lie algebras, Howe and Umeda [@HoweUmeda] obtain such a formula for the *generators* of the algebra of invariant differential operators. By contrast, even in this setting, our construction is more general and gives explicit pre-images for a *basis* of the algebra.
Another new idea is the construction of spherical vectors in tensor representations of $\gl(m|2n)$ using symbols of the Capelli operators $D_\lambda$ (see Lemma \[lemZnZk=0\]).\
**Acknowledgement.** We thank the referee for useful comments and pointing to references for Lemma \[gelfzhel\]. Throughout this project, we benefited from discussions with Alexander Alldridge, Ivan Dimitrov, Roe Goodman, Roger Howe, Friedrich Knop, James Lepowsky, Nolan Wallach, and Tilmann Wurzbacher. We thank them for fruitful and encouraging discussions. The second author is supported by an NSERC Discovery Grant.
Notation {#section1}
========
We briefly review the basic theory of vector superspaces and Lie superalgebras. For more detailed expositions, see for example [ChWabook]{} or [@Musson]. Throughout this article, all vector spaces will be over $\C$. Let $\SVec$ be the symmetric monoidal category of $\Ztwo$-graded vector spaces, where $\Ztwo:=\left\{\eev,\ood\right\}$. Objects of $\SVec$ are of the form $U=U_\eev\oplus U_\ood$. The parity of a homogeneous vector $u\in U$ is denoted by $|u|\in\Ztwo$. For any two $\Ztwo$-graded vector spaces $U$ and $U'$, the vector space $\Hom_\C
(U,U^{\prime})$ is naturally $\Ztwo$-graded, and the morphisms of $\SVec$ are defined by $\Mor_{\SVec}(U,U^{\prime
}):=\Hom_\C
(U,U^{\prime})_\eev$. The symmetry isomorphism of $\SVec$ is defined by $$\label{symisomdf}
\braid_{U,U^{\prime }}:U\otimes U^{\prime }\to U^{\prime }\otimes U \ ,\
u\otimes u^\prime
\mapsto (-1)^{|u|\cdot |u^{\prime }|}u^{\prime }\otimes u.$$ We remark that throughout this article, the defining relations which involve parities of vectors should first be construed as relations for homogeneous vectors, and then be extended by linearity to arbitrary vectors.
The identity element of the associative superalgebra $
\mathrm{End}_\C(U):=%
\Hom_\C(U,U)
$ will be denoted by $1_U$. Note that $\mathrm{End}_\C(U)$ is a Lie superalgebra with the standard commutator $[A,B]:=AB-(-1)^{|A|%
\cdot|B|}BA$. Set $U^*:=\Hom_\C(U,\C^{1|0})$. The map $$\label{U'U*}
U^{\prime}\otimes U^* \to
\Hom_\C(U,U^{\prime }) \ ,\
u^{\prime}\otimes u^*\mapsto T_{u^{\prime}\otimes u^*},$$ where $T_{u^{\prime}\otimes u^*}(u):=\lag u^*,u\rag u^{\prime }$ for all $u\in U$, is an isomorphism in the category $\SVec$.
The natural representation (in the symmetric monoidal category $\SVec$) of the symmetric group $S_d$ on $U^{\otimes d}$ is explicitly given by $$\label{dfTsigm}
\sigma\mapsto T_{U,d}^\sigma\ \,,\,\ T^{\sigma}_{U,d}( v_1\otimes \cdots
\otimes v_{d}) := (-1)^{\seps\left(\sigma^{-1};v_1,\ldots,v_{d}\right)}
v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(d)}\,,$$ where $$\label{sepssign}
\seps(\sigma;v_1,\ldots,v_d):= \sum_{\substack{ 1\leq r<s\leq d \\ %
\sigma(r)>\sigma(s) }} |v_{\sigma(r)}|\cdot|v_{\sigma(s)}|.$$ The supersymmetrization map $\ssym^d_U:U^{\otimes d}\to U^{\otimes d}$ is defined by $$\label{symdUeq}
\ssym^d_U :=\frac{1}{d!} \sum_{\sigma\in S_d} T^\sigma_{U,d}.$$ We set $$\sS^d(U):=\ssym^d_U\left(U^{\otimes d}\right)
,\
\sS(U):=\bigoplus_{d\ge 0}\sS
^d(U),\
\sP(U):=\sS(U^*),
\text{ and }
\sP^d(U):=\sS^d( U^*).$$ Every $\eta\in U^*_\eev$ extends canonically to a homomorphism of superalgebras $$\label{dfheta}
\sfh_\eta:\sS(U)\to \C\cong \C^{1|0}$$ defined by $\sfh_\eta \big(
\ssym^d_U(u_1\otimes \cdots \otimes u_d) \big)
:=\eta(u_1)\cdots \eta(u_d) $ for $d\geq 0$ and $u_1,\ldots,u_d\in U$.
If $\g g$ is a Lie superalgebra, then $\bfU(\g g)$ denotes the universal enveloping algebra of $\g g$, and $$\C=\bfU^0(\g g)\subset
\bfU^1(\g g)\subset\cdots
\subset
\bfU^d(\g g)\subset\cdots$$ denotes the standard filtration of $\bfU(\g g)$. If $(\pi,U)$ is a $\g g$-module, we define $$U^\g g:=\{u\in U\ :\ \pi(x)u=0\text{ for every }x\in\g g\}.$$ If $(\pi^{\prime },U^{\prime })$ is another $\g g$-module, then $\Hom_\C%
(U,U^{\prime })$ is also a $\g g$-module, with the action $$\label{xcdtTT}
(x\cdot T)u:=\pi^{\prime}(x)Tu-(-1)^{|T||x|}T\pi(x)u\,\$$ for $x\in\g g$, $T\in\Hom_\C%
(U,U^{\prime })$, and $u\in U$. The special case $U^*:=\Hom_\C(U,\C^{1|0})$, where $\C^{1|0}$ is the trivial $%
\g g$-module, is the *contragredient* $\g g$-module. Moreover, the map is $\g g$-equivariant. Set
$$\Hom_\g g(U,U^{\prime }):= \Hom_\C(U,U^{\prime})^{\g g} \ \text{ and }\ \mathrm{%
End}_\g g(U):=\mathrm{End}_\C(U)^\g g.$$
The category of $\g g$-modules is a symmetric monoidal category, and $$\Mor_{\g g\text{-mod}}(U,U^{\prime })\cong \Hom_\g g(U,U^{\prime })_\eev
.$$
Fix a $\Ztwo$-graded vector space $W$. For every homogeneous $w\in W$, let $\partial_w$ be the superderivation of $\sP(W)$ with parity $|w|$ that is defined uniquely by $$\label{pww*}
\partial_w(w^*):=(-1)^{|w|}\lag w^*,w\rag\text { for every } w^*\in W^*\cong %
\sP^1(W).$$ Thus, $\partial_w(a_1a_2)=(\partial_w a_1)a_2+(-1)^{|w|\cdot|a_1|}a_1\partial_wa_2$ for every $a_1,a_2\in\sP(W)$. For every $b\in \sS(W)$ we define $\partial_b\in%
\mathrm{End}_\C(\sP(W))$ as follows. First we set $$\partial_{w_1\cdots w_r}:=\partial_{w_1}\cdots\partial_{w_r} \text{ for homogeneous }%
w_1,\ldots,w_r\in W,$$ and then we extend the definition of $\partial_{b}$ to all $b\in \sS(W)$ by linearity.
Let $\sPD(W)$ be the associative superalgebra of polynomial-coefficient differential operators on $W$. More explicitly, $\sPD(W)$ is the subalgebra of $\mathrm{End}_\C(\sP(W))$ spanned by elements of the form $a\partial_b$, where $a\in \sP(W)$ and $b\in\sS(W)$. If $W$ is a $\g g$-module, then $\sPD%
(W)$ is a $\g g$-invariant subspace of $\mathrm{End}_\C(\sP(W))$ with respect to the $\g g$-action on $\mathrm{End}_\C(\sP(W))$ defined in . Furthermore, the map $$\label{eq-sfm:}
\sfm:\sP(W)\otimes \sS(W)\to \sPD(W)\ ,\ a\otimes b\mapsto a\partial_b$$ is an isomorphism of $\g g$-modules (but not of superalgebras). The superalgebra $\sPD(W)$ has a natural filtration given by $$\sPD^d(W):=\sfm\left(\sP(W)\otimes\left(\bigoplus_{r=0}^d \sS%
^r(W)\right)\right)\text{ for }d\geq 0.$$ For $D\in\sPD(W)$, we write $\mathrm{ord}(D)=d$ if $%
D\in\sPD^d(W)$ but $D\not\in\sPD^{d-1}(W)$. For every $d\geq 0$, the $d$-th order symbol map $$\widehat\sfs_d:\sPD^{d}(W)\to \sP(W)\otimes \sS^d(W)$$ is defined by $$\label{eq-shmphh}
\widehat\sfs_d\left(\sPD^{d-1}(W)\right)=0\text{\, and\ \,} \widehat\sfs_d(%
\sfm(a))=a\ \text{ for }\ a\in \sP(W)\otimes \sS^d(W).$$ If $D_r\in\sPD^{d_r}(W)$ for $1\leq r\leq k$, then $$\label{whsD1Dk}
\widehat\sfs_{d_1+\cdots+d_k}(D_1\ldots D_k)= \widehat\sfs_{d_1}(D_1)\cdots
\widehat\sfs_{d_k}(D_k),$$ where the product on the right hand side takes place in the superalgebra $\sP(W)\otimes\sS(W)$.
The abstract Capelli problem for $W:=\sS^2(V)$ {#prfof571}
==============================================
Fix $V:=\C^{m|n}$ and set $\g g:=\gl(V):=\gl(m|n)$. We fix bases $\bfe%
_1,\ldots,\bfe_m$ for $V_\eev=\C^{m|0}$ and $\bfe_{\oline 1},\ldots,\bfe_{%
\oline n} $ for $V_\ood=\C^{0|n}$. Throughout this article we will use the index set $$\mathcal{I}_{m,n}:=\{1,\ldots,m,\oline 1,\ldots,\oline n\}.$$ We define the parities of elements of $\mathcal{I}_{m,n}$ by $$\left|i\right|: =\left|\bfe_i\right |=
\begin{cases}
\eev & \text{ for }i\in\{1,\ldots,m\}, \\
\ood & \text{ for } i\in\{\oline 1,\ldots,\oline n\}.%
\end{cases}$$
For an associative superalgebra $\cA=\cA_\eev\oplus\cA_\ood$, let $\mathrm{Mat}_{m,n}(\cA)$ be the associative superalgebra of $(m+n)\times (m+n)$ matrices with entries in $\cA$, endowed with the $\Z_2$-grading obtained by the $(m,n)$-block form of its elements. More precisely, $\mathrm{Mat}_{m,n}(\cA)_\eev$ consists of matrices in $(m,n)$-block form $$\label{blockformgg}
\begin{bmatrix}
A & B \\
C & D%
\end{bmatrix}%$$ such that the entries of the $m\times m$ matrix $A$ and the $n\times n$ matrix $D$ belong to $\cA_\eev$, whereas the entries of the matrices $B$ and $C$ belong to $\cA_\ood$. Similarly, $\mathrm{Mat}_{m,n}(\cA)_\ood$ consists of matrices of the form such that the entries of $A$ and $D$ belong to $\cA_\ood$, whereas the entries of $B$ and $C$ belong to $\cA_%
\eev$. The supertrace of an element of $\mathrm{Mat}_{m,n}(\cA)$ is defined by $$\str\left(
\begin{bmatrix}
A & B \\
C & D%
\end{bmatrix}
\right)=\mathrm{tr}(A)-\mathrm{tr}(D).$$
Using the basis $\left\{\bfe_i\,:\,i\in\mathcal{I}_{m,n}\right\}$, we can represent elements of $\g g$ by $(m+n)\times (m+n)$ matrices, with rows and columns indexed by elements of $\mathcal{I}_{m,n}$. For every $i,j\in%
\mathcal{I}_{m,n}$, let $E_{i,j}$ denote the element of $\g g$ corresponding to the matrix with a 1 in the $(i,j)$-entry and 0’s elsewhere. The standard Cartan subalgebra of $\g g$ is $$\g h:=\spn_\C\{E_{1,1},\ldots,E_{m,m},E_{\oline 1,\oline 1},\ldots, E_{%
\oline n,\oline n}\},$$ and the standard characters of $\g h$ are $\eps_1,\ldots,\eps_m,\eps_{\oline %
1},\ldots,\eps_{\oline n}\in\g h^*$, where $\eps_i(E_{j,j})=\delta_{i,j} $ for every $i,j\in\mathcal{I}_{m,n} $. The standard root system of $\g g$ is $%
\Phi:=\Phi^+\cup \Phi^-$ where $$\Phi^+:=\Big\{
\eps_k^{}-\eps_l^{} \Big\}_{1\leq k<l\leq m} \cup \left\{ \eps_k^{}-\eps_{%
\oline l}^{} \right\} _{ 1\leq k\leq m,\,1\leq l\leq n} \cup \left\{ \eps_{%
\oline k}^{}-\eps_{\oline l}^{} \right\}_{1\leq k<l\leq n}$$ and $\Phi^-:=-\Phi^+$. For $\alpha\in\Phi$, we set $\g g_\alpha:=\{x\in\g g\,:\, [h,x]=\alpha(h)x\text{ for every }h\in\g h\}$. Then $\g g=\g n^-\oplus\g h\oplus\g n^+$, where $\g n^\pm:=\bigoplus_{\alpha\in\Phi^\pm}\g g_\alpha$. Finally, the supersymmetric bilinear form $$\label{kappa}
\kappa:\g g\times\g g\to\C\ ,\ \kappa(X,Y):=\mathrm{str}(XY)\text{ for }
X,Y\in\g g,$$ is nondegenerate and $\g g$-invariant.
Set $W:=\sS^2(V)$. The standard representation of $\g g$ on $V$ gives rise to canonically defined representations on $V^{\otimes d}$, $W$, $W^*$, $\sS(W)$, and $\sP(W)$. From now on, we denote the latter representations of $\g g$ on $\sS(W)$ and $\sP(W)$ by $\rho$ and $\check\rho$ respectively.
The goal of the rest of this section is to prove Theorem \[prpgw\], which gives an affirmative answer to the abstract Capelli problem for the $\g g$-module $W$. Set $$\bfZ(\g g):=\{x\in\bfU(\g g)\,:\,[x,y]=0\text{ for every }y\in\bfU(\g g)\},$$ where $[\,\cdot,\cdot\,]$ denotes the standard superbracket of $\bfU(\g g)$.
\[gelfzhel\] Let $\mathbf E\in\mathrm{Mat}_{m,n}(\bfU(\g g))_\eev$ be the matrix with entries $$\mathbf E_{i,j}
:=
(-1)^{|i|\cdot|j|}E_{j,i}
\text{ for }i,j\in\mathcal I_{m,n}.$$ Then $\mathrm{str}(\mathbf E^d)\in\bfZ(\g g)$ for every $d\geq 1$.
This lemma can be found for example in [@sergeev82] or [@scheunert]. For the reader’s convenience, we outline a proof (in the case $n=0$ it is due to Žhelobenko [@Zhel]).\
**Step 1.** Let $\mathbf Z=[z_{i,j}]_{i,j\in\mathcal I_{m,n}}^{}
$ be an element of $\mathrm{Mat}_{m,n}(\bfU(\g g))_\eev$ that satisfies $$\label{xijEkl}
[z_{i,j},E_{k,l}]
=
(-1)^{|i|\cdot |j|+|l|\cdot |j|}
\delta_{i,k}z_{l,j}
-
(-1)^{|j|\cdot |k|+|j|}\delta_{l,j}z_{i,k}
\text{ for }
i,j,k,l\in\mathcal I_{m,n}.$$ Set $z:=\mathrm{str}(\mathbf Z)=\sum_{i\in \mathcal I_{m,n}}(-1)^{|i|}
z_{i,i}$. Then $
\left[z,E_{k,l}\right]=0
$ for $k,l\in\mathcal I_{m,n}$, and thus $z\in\bfZ(\g g)$.\
**Step 2.** Let $\mathbf Z=[z_{i,j}]_{i,j\in\mathcal I_{m,n}}^{}$ and $\mathbf Z'=[z'_{i,j}]_{i,j\in\mathcal I_{m,n}}^{}
$ be elements of $\mathrm{Mat}_{m,n}(\bfU(\g g))_\eev$ that satisfy . Set $\mathbf Z'':=\mathbf Z\mathbf Z'$, so that $\mathbf Z''=[z''_{i,j}]_{i,j\in\mathcal I_{m,n}}^{}$ where $z''_{i,j}:=\sum_{r\in \mathcal I_{m,n}}z_{i,r}z'_{r,j}$. Then $$\begin{aligned}
\left[z''_{i,j},E_{k,l}\right]
&=
\sum_{r\in\mathcal I_{m,n}}z_{i,r}[z'_{r,j},E_{k,l}]
+
\sum_{r\in\mathcal I_{m,n}}
(-1)^{(|r|+|j|)(|k|+|l|)}[z_{i,r},E_{k,l}]z'_{r,j}\\
&=(-1)^{|j|\cdot|i|+|j|\cdot |l|}\delta_{i,k}z''_{l,j}
-
(-1)^{|j|\cdot |k|+|j|}\delta_{l,j}z''_{i,k},\end{aligned}$$ so that the entries of $\mathbf Z''$ satisfy as well.\
**Step 3.** Since holds for the entries of $\mathbf E$, induction on $d$ and Step 2 imply that holds for the entries of $\mathbf E^d$ for every $d\geq 1$, and thus $\mathrm{str}(\mathbf E^d)\in \bfZ(\g g)$ by Step 1.
Let $\left\{\bfe^*_i\ :\ i\in\mathcal{I}_{m,n}\right\}$ be the basis of $V^*$ dual to the basis $\left\{\bfe_i\ :\ i\in\mathcal{I}_{m,n}\right\}$. Set $$\label{eq-dfxijyij}
x_{i,j}:=\frac{1}{2}\left(\bfe_i\otimes \bfe_j+(-1)^{|i|\cdot |j|} \bfe%
_j\otimes \bfe_i\right) \,\text{ and }\ y_{i,j}:= \bfe^*_j\otimes \bfe^*_i+
(-1)^{|i|\cdot|j|} \bfe^*_i\otimes\bfe^*_j,$$ for $i,j\in\mathcal{I}_{m,n}$. The $x_{i,j}$ span $W$, and therefore they generate $\sS(W)$. Similarly, the $y_{i,j}$ span $W^*$, and therefore they generate $\sP(W)$. Moreover, $$x_{i,j}=(-1)^{|i|\cdot |j|}x_{j,i},\ y_{i,j}=(-1)^{|i|\cdot |j|}y_{j,i},\
\text{and } \lag y_{i,j},x_{p,q}\rag=\delta_{i,p}\delta_{j,q}
+(-1)^{|i|\cdot |j|}\delta_{i,q}\delta_{j,p}.$$
The action of $\g g$ on $\sS(W)$ can be realized by the polarization operators $$\label{polrhoEi-j}
\rho(E_{i,j}):=\sum_{r\in\mathcal{I}_{m,n}}x_{i,r}\sfD_{{j,r}}
\ \text{ for }i,j\in\mathcal I_{m,n},$$ where $\sfD_{{i,j}}:\sS(W)\to \sS(W)$ is the superderivation of parity $%
|i|+|j|$ uniquely defined by $$\label{pari-j}
\sfD_{i,j}(x_{k,l}):=\delta_{i,k}\delta_{j,l}+(-1)^{|i|\cdot|j|}\delta_{i,l}%
\delta_{j,k} \text{ for }i,j,k,l\in\mathcal{I}_{m,n}.$$ Similarly, the action of $\g g$ on $\sP(W)$ is realized by the polarization operators $$\label{ppooll}
\check\rho(E_{i,j})=-(-1)^{|i|\cdot|j|} \sum_{r\in\mathcal{I}_{m,n}}
(-1)^{|r|}y_{r,j}\partial_{{r,i}}\ \text{ for }
i,j\in\mathcal I_{m,n},$$ where $\partial_{{i,j}}:=\partial_{x_{i,j}}$ is the superderivation of $%
\sP(W)$ corresponding to $x_{i,j}\in W$, as in . Let $U=U_\eev\oplus U_\ood$ be a $\Ztwo$-graded vector space. For every $\cA%
\subseteq\mathrm{End}_\C(U)$, we set $$\cA^{\prime }:= \{B\in\mathrm{End}_\C(U)\ :\ [A,B]=0 \,\text{ for every }A\in%
\cA
\}.$$
\[rmk-dblcomm\] Fix finite dimensional $\Ztwo$-graded vector spaces $U$ and $U'$, and set $$\cA:=\mathrm{End}_\C(U)\otimes 1_{U'}\subset
\mathrm{End}_\C(U\otimes U').$$ Then $\cA'=1_U\otimes \mathrm{End}_\C(U')$.
\[UAcomm\] Let $U=U_\eev\oplus U_\ood$ be a finite dimensional $\Ztwo$-graded vector space, and let $\cA\subseteq\mathrm{End}_\C(U)_\eev$ be a semisimple associative algebra. Then $(\cA')'=\cA$.
Since $\cA$ is semisimple and purely even, both $U_\eev$ and $U_\ood$ can be expressed as direct sums of irreducible $\cA$-modules. It follows that $U\cong\bigoplus_{\tau}
U_\tau\otimes V_\tau,
$ where the $U_\tau$ are mutually non-isomorphic irreducible $\cA$-modules and $V_\tau\cong
\Hom_{\cA}(U_\tau,U)$. Since the decomposition of $U$ into irreducibles is homogeneous, the multiplicity spaces $V_\tau$ are $\Ztwo$-graded, whereas the $U_\tau$ are purely even. By Artin-Wedderburn theory [@Jacobson], $\cA=\bigoplus_{\tau}
\mathrm{End}_\C(U_\tau)\otimes 1_{V_\tau}^{}$. Thus by Remark \[rmk-dblcomm\], $
\cA'=\bigoplus_{\tau}1_{U_\tau}^{}\otimes \mathrm{End}_\C(V_\tau)
$ and $
(\cA')'=
\bigoplus_{\tau}
\mathrm{End}_\C(U_\tau)\otimes 1_{V_\tau}^{}=\cA
$.
\[lem-supcmmt\] For every $d\geq 1$, the superalgebra $\mathrm{End}_\g g(V^{\otimes d})$ is spanned by the operators $T^\sigma_{V,d}$ defined in . In particular, $
\mathrm{End}_\g g(V^{\otimes d})
=
\mathrm{End}_\g g(V^{\otimes d})_\eev
=\Mor_{\g g\text{-\upshape mod}}(V^{\otimes d},
V^{\otimes d}
)
$.
Set $\cA:=\mathrm{Span}_\C\{T_{V,d}^\sigma\ :\ \sigma\in S_d\}\subset\mathrm{End}_\C(V^{\otimes d})_\eev$. Since $\cA$ is a homomorphic image of the group algebra $\C[S_d]$, it is a semisimple associative algebra, and Lemma \[UAcomm\] implies that $(\cA')'=\cA$.
Let $\cB$ be the associative subalgebra of $\mathrm{End}_\C(V^{\otimes d})$ generated by the image of $\bfU(\g g)$. From [@sergeev Theorem 1] or [@BereleRegev Theorem 4.14] it follows that $\cA'=\cB$. Therefore we obtain that $\mathrm{End}_\g g(V^{\otimes d})=\cB'=(\cA')'=\cA$.
The canonical isomorphism $$\label{V2dV*2dE}
V^{\otimes 2d}\otimes V^{*\otimes 2d}\cong \mathrm{End}_\C(V^{\otimes 2d})$$ given in maps $v_1\otimes\cdots\otimes v_{2d}\otimes
v_1^*\otimes \cdots\otimes v_{2d}^*\in V^{\otimes 2d}\otimes V^{*\otimes 2d}
$ to the linear map $$V^{\otimes 2d} \to V^{\otimes 2d}\ ,\ w_1\otimes \cdots\otimes w_{2d}
\mapsto \lag v^*_{2d},w_1\rag
\cdots \lag v_1^*,w_{2d}\rag
v_1\otimes\cdots\otimes v_{2d} .$$ For every $\sigma\in S_{2d}$, let $\widetilde\bft_\sigma\in V^{\otimes
2d}\otimes V^{*\otimes 2d}$ be the element corresponding to $
T^{\sigma^{-1}}_{V,2d}$ via the isomorphism , where $
T^{\sigma^{-1}}_{V,2d}
$ is defined as in . It is easy to verify that $$\widetilde\bft_\sigma=\sum_{i_1,\ldots,i_d\in\mathcal{I}_{m,n}} (-1)_{}^{ \seps%
\left(\sigma;\bfe_{i_1},\ldots,\bfe_{i_{2d}}^{}\right) } \bfe_{i_{\sigma(1)}} \otimes
\cdots \otimes \bfe_{i_{\sigma(2d)}} \otimes \bfe_{i_{2d}}^{*}\otimes
\cdots\otimes \bfe_{i_1}^{*},$$ where $
\seps\left(\sigma;\bfe_{i_1},\ldots,\bfe_{i_{2d}}^{}\right)
$ is defined in . Now set $\sigma_\circ:=(1,2)\cdots (2d-1,2d)\in S_{2d}$ and $$H_{2d}:=\{\sigma\in
S_{2d}\,:\,\sigma\sigma_\circ=
\sigma_\circ\sigma\}.$$ Let $\psf_d^{}:V^{\otimes 2d}\to
V^{\otimes 2d}$ and $\psf_d^{*}:V^{*\otimes 2d}\to
V^{*\otimes 2d}$ be the canonical projections onto $\sS^d(W)$ and $\sS^d(W^*)\cong\sP^d(W)$, so that $$\label{dfpsfdd*}
\psf_d^{}=\frac{1}{2^d d!}\sum_{\sigma\in H_{2d}}T_{V,2d}^{\sigma}
\ \text{ and }\
\psf_d^*=\frac{1}{2^d d!}\sum_{\sigma\in H_{2d}}T_{V^*,2d}^{\sigma}.$$ For every $\sigma\in S_{2d}$, let $\bft_\sigma\in \sP^d(W)\otimes \sS^d(W)$ be defined by $$\bft_\sigma:=
\left(
\braid^{}_{V^{\otimes
2d},V^{*\otimes 2d}}
\circ
(\psf^{}_d\otimes \psf^{*}_d) \right) \big(\widetilde\bft_\sigma\big),$$ where $\braid^{}_{V^{\otimes
2d},V^{*\otimes 2d}}$ is the symmetry isomorphism, defined in . The action of the linear transformation $T_{V^{},2d}^{\sigma_1^{}}\otimes
T_{V^*,2d}^{\sigma_2^{}}$ on $V^{\otimes 2d}\otimes V^{*\otimes 2d}\cong\mathrm{End}_\C(V^{\otimes 2d})$ is given by $$T\mapsto T_{V,2d}^{\sigma_1^{}}T
T_{V,2d}^{\pi\sigma_2^{-1}\pi^{-1}}
\text{ for every }T\in
\mathrm{End}_\C(V^{\otimes 2d}),$$ where $\pi\in S_{2d}$ is defined by $\pi(i)=2d+1-i$ for every $1\leq i\leq 2d$. Thus from it follows that $$\begin{aligned}
\label{tsigmeqqqqn}
\bft_\sigma&= \left(
\frac{1}{d!2^d}\right)^2 \sum_{\sigma^{\prime },\sigma^{\prime \prime
}\in H_{2d}}
\bft_{\sigma^{\prime }\sigma\sigma^{\prime \prime }}
\\
&= \frac{1}{2^d}\sum_{i_1,\ldots,i_{2d}\in\mathcal{I}_{m,n}} \!\!\!\!\!\!(-1)^{|{i_1}|+\cdots+|{i_{2d}
}|
+
\seps
\left(
\sigma;\bfe_{i_1}^{},\ldots,\bfe_{i_{2d}}^{}\right)} y_{i_{2d-1},i_{2d}}^{}
\cdots y_{i_{1},i_{2}}^{} \otimes x_{i_{\sigma(1)},i_{\sigma(2)}}^{} \cdots
x_{i_{\sigma(2d-1)},i_{\sigma(2d)}}^{}. \notag\end{aligned}$$
\[lem2..5\] Let $\sigma\in S_{2d}$, $\sigma'\in S_{2d'}$, and $\sigma''\in S_{2d''}$ be such that $d'+d''=d$ and $$\sigma(r):=\begin{cases}
\sigma'(r)&\text{ if }1\leq r\leq 2d',\\
\sigma''(r-2d')+2d'&\text{ if }2d'+1\leq r\leq 2d.
\end{cases}$$ Then $\bft_\sigma=
\bft_{\sigma'}\bft_{\sigma''}$ as elements of the superalgebra $\sP(W)\otimes \sS(W)$.
Follows immediately from the explicit summation formula for $\bft_\sigma$ given in .
\[taud0dr\] Let $\sigma\in S_{2d}$. Then the following statements hold.
- $\bft_\sigma=\bft_{\sigma'\sigma\sigma''}$ for every $\sigma',\sigma''\in H_{2d}$.
- There exist $\sigma',\sigma''\in H_{2d}$ and integers $0=d_0<d_1< \cdots< d_{r-1}<d_r=d
$ such that $$\label{eqTAUdecc}
\sigma'\sigma\sigma''=\tau_{d_0,d_1}^{}
\tau_{d_1,d_2}^{}\cdots \tau_{d_{r-1},d_r^{}}^{},$$ where $\tau_{a,b}$ for $a<b$ denotes the cycle $
(2a+2,2a+4,\ldots,2b-2,2b)
$.
\(i) Follows from .
\(ii) First we show that there exist integers $a_1,\ldots,a_d,b_1,\ldots,b_d$ and a permutation $\tau\in S_d$ such that $$2s-1\leq a_s,b_s\leq 2s
\text{ and }
\sigma(a_s)=b_{\tau(s)}
\text{ for }
1\leq s\leq d.$$ To this end, we construct an undirected bipartite multigraph $\mathcal G$, with vertex set $$\mathcal V
:=\{\mathbf a_1,\ldots,\mathbf a_d\}\cup\{\mathbf b_1,\ldots,\mathbf b_d\},$$ and $m_{s,s'}:=\left|M_{s,s'}\right|$ edges between $\mathbf a_s$ and $\mathbf b_{s'}$, labelled by elements of the set $M_{s,s'}$, where $$M_{s,s'}:=\{\sigma(2s-1),\sigma(2s)\}\cap \{2s'-1,2s'\}\,
\text{ for all }1\leq s,s'\leq d
.$$ Elements of $M_{s,s'}$ correspond to equalities of the form $\sigma(a_s)=b_{s'}$, where $2s-1\leq a_s\leq 2s$ and $2s'-1\leq b_{s'}\leq 2s'$. Every vertex in $\mathcal G$ has degree two, and therefore $\mathcal G$ is a union of disjoint cycles. It follows that $\mathcal G$ has a perfect matching, i.e., a set of $d$ edges which do not have a vertex in common. Using this matching we define $\tau\in S_d$ so that the edges of the matching are $(\mathbf{a}_s,\mathbf b_{\tau(s)})$. The label on each edge $(\mathbf{a}_s,\mathbf b_{\tau(s)})$ of the matching is the corresponding $b_{\tau(s)}$, and $a_s:=\sigma^{-1}\big(b_{\tau(s)}\big)$.
Now let $\sigma_1,\sigma_2\in H_{2d}$ be defined by $$\sigma_1(t):=\begin{cases}
2s-1& \text{ if }t=2\tau(s)-1\text{ for }1\leq s\leq d,\\
2s& \text{ if }t=2\tau(s)\text{ for }1\leq s\leq d.
\end{cases}
\ \ \text{ and }\ \
\sigma_2:=\prod_
{\substack{1\leq s\leq d\\[.3mm]
a_s-b_{\tau(s)}\not\in 2\Z}}\!\!\!
(2s-1,2s),$$ and set $\sigma_3:=\sigma_2\sigma_1\sigma$. It is easy to check that that $\sigma_3(a_s)=a_s$ for $1\leq s\leq d$. Next set $$\sigma_4:=\prod_{\substack{1\leq s\leq d\\ a_s=2s}}(2s-1,2s)\text{\ \,and\,\ }\sigma_5:=\sigma_4\sigma_3\sigma_4^{-1},$$ so that $\sigma_5(2s-1)=2s-1$ for $1\leq s\leq d$. Now let $\tau\in S_d$ be defined by $\tau(s):=\frac{1}{2}\sigma_5(2s)$ for $1\leq s\leq d$. Choose $0=d_0<d_1<\cdots<d_r=d$ and $\tau_\circ\in S_d$ such that $$\tau_\circ^{}\tau\tau_\circ^{-1}=
\prod_{s=1}^d(d_{s-1}+1,\ldots,d_s).$$ Finally, let $\sigma_\circ\in H_{2d}$ be defined by $$\sigma_\circ(2s):=
2\tau_\circ(s)
\,\text{ and }\,
\sigma_\circ(2s-1):=
2\tau_\circ(s)-1
\,\text{ for }\,1\leq s\leq d
.$$ It is straightforward to check that $\sigma_\circ^{}\sigma_5\sigma_\circ^{-1}
=\tau_{d_0,d_1}^{}\cdots \tau_{d_{r-1},d_r^{}}^{}$, so that holds for $\sigma':=\sigma_\circ^{}\sigma_4^{}\sigma_2^{}\sigma_1^{}$ and $\sigma'':=\sigma_4^{-1}\sigma_\circ^{-1}$.
\[prp-g-inv-sPWW\] For every $d\geq 1$, $$\label{sSdttsPd}
\left(
\sP^d(W)\otimes \sS^d(W)\right)^\g g
=\mathrm{Span}_\C\left\{\bft_\sigma\,:\,\sigma\in S_{2d}\right\}
.$$ Furthermore, $\left(\sP(W)\otimes \sS(W)\right)^\g g
=\bigoplus_{d\geq 0}\left(
\sP^d(W)\otimes \sS^d(W)\right)^\g g$.
For every $k,l\geq 0$, the action of $\g g$ on $\sP(W)\otimes\sS(W)$ leaves $\sP^k(W)\otimes \sS^l(W)$ invariant. It follows that $\left(\sP(W)\otimes \sS(W)\right)^\g g=\bigoplus_{k,l\geq 0}\left(\sP^k(W)\otimes \sS^l(W)\right)^\g g$. By considering the action of the centre of $\g g$ we obtain that $$\label{sSksSlk=l}
\left(\sP^k(W)\otimes \sS^l(W)\right)^\g g=\{0\} \text{ unless }k=l.$$ Next we prove . Recall the definition of $\psf_d^{}$ and $\psf_d^*$ from . It suffices to prove that the canonical projection $$\label{eq-split}
V^{*\otimes 2d}\otimes V^{\otimes 2d}
\xrightarrow{\psf^*_d\otimes\psf_d^{}}
\sS^d(W^*)\otimes \sS^d(W)$$ is surjective on $\g g$-invariants. This follows from $
\sS^d(W^*)\otimes \sS^d(W)=
\left(
V^{*\otimes 2d}\otimes V^{\otimes 2d}
\right)^{H_{2d}\times H_{2d}}
$ and the fact that $\psf_d^*\otimes \psf_d^{}$ restricts to the identity map on $\sS^d(W^*)\otimes \sS^d(W)$.
For every $i,j\in\mathcal{I}_{m,n}$, let $\varphi_{i,j} \in \sP(W)\otimes\sS
(W)$ be defined by $$\varphi_{i,j}:=(-1)^{|i|\cdot|j|} \sum_{r\in\mathcal{I}%
_{m,n}}(-1)^{|r|}y_{r,j}\otimes x_{r,i}.$$ Note that $$\label{sfmPHIEIJ}
\sfm(\varphi_{i,j})=-\check\rho(E_{i,j}) .$$
\[lem2..8\] Fix $d\geq 1$ and let $\sigma:=(2,4,\ldots,2d)\in S_{2d}$, that is, $\sigma(2r)=2r+2$ for $1\leq r\leq d-1$, $\sigma(2d)=2$, and $\sigma(2r-1)=2r-1$ for $1\leq r\leq d$. Then $$\bft_{\sigma}
=(-2)^d\,\widehat\sfs_d\big(
\check\rho\big(\mathrm{str}\big(\mathbf E^d\big)\big)
\big),$$ where $\mathbf E$ is the matrix defined in Lemma \[gelfzhel\].
For $i_1,\ldots,i_{2d}\in\mathcal I_{m,n}$, we set $$\seps'(i_1,\ldots,i_{2d}):={\sum_{s=1}^{2d}|i_s|+\sum_{s=3}^{2d}
|i_2|\cdot|i_s|+\sum_{s=1}^{d-1}|i_{2s+1}|\cdot|i_{2s+2}|}
.$$ By a straightforward but tedious sign calculation, we obtain from that $$\begin{aligned}
\bft_\sigma&=
\frac{1}{2^d}
\sum_{i_1,\ldots,i_{2d}\in\mathcal I_{m,n}}
(-1)^{\seps'(i_1,\ldots,i_{2d})}_{}y_{i_{2d-1},i_{2d}}\cdots y_{i_{1},i_{2}}
\otimes x_{i_{1},i_{\sigma(2)}}\cdots x_{i_{2d-1},i_{\sigma(2d)}}\\
&
=
\frac{1}{2^d}\sum_{t_1,\ldots,t_{d}\in\mathcal I_{m,n}}
\left(
(-1)_{}^{|t_1|+\sum_{s=1}^d|t_{s}|\cdot |t_{s+1}|}
\prod_{s=1}^d\varphi_{t_{s+1},t_s}
\right)\end{aligned}$$ where $t_s:=i_{2s}$ for $1\leq s\leq d$ and $t_{d+1}:=i_2$. Using and we can write $$\begin{aligned}
\widehat\sfs_d\big(
\check\rho\big(\mathrm{str}\big(\mathbf E^d\big)\big)
\big)&=
\sum_{t_1,\ldots,t_d\in\mathcal I_{m,n}}
\!\!\!\!\!\!(-1)^{|t_1|+\sum_{s=1}^d|t_s|\cdot |t_{s+1}|}
\;\widehat\sfs_d\left(
\check\rho(E_{t_2,t_1})\cdots\check\rho(E_{t_1,t_d})
\right)\\
&=
\sum_{t_1,\ldots,t_d\in\mathcal I_{m,n}}
\!\!\!\!\!\!(-1)^{|t_1|+\sum_{s=1}^d|t_s|\cdot |t_{s+1}|}
\;
\widehat\sfs_1\left(
\check\rho(E_{t_2,t_1})\right)\cdots
\widehat\sfs_1\left(
\check\rho(E_{t_1,t_d})
\right)\\
&=(-2)^d\bft_\sigma.
\qedhere\end{aligned}$$
Let $V:=\C^{m|n}$, $W:=\sS^2(V)$, $\g g:=\gl(V)=\gl(m|n)$, and $\check\rho$ be as above. Our first main result is the following theorem.
\[prpgw\] *(Abstract Capelli Theorem for $W:=\sS^2(V)$.)* For every $d\geq 0$, we have $$\sPD^{ d}(W)^{\g g}=
\check\rho\left(\mathbf{Z}(\g g)\cap \bfU^d(\g g)\right).$$
The inclusion $\supseteq $ is obvious by definition. The proof of the inclusion $\subseteq$ is by induction on $d$. The case $d=0$ is obvious. Next assume that the statement holds for all $d'<d$. Fix $D\in\sPD^{d}(W)^{\g g}$ such that $\mathrm{ord}(D)=d$. It suffices to find $z_D\in\bfZ(\g g)\cap \bfU^{d}(\g g)$ such that $
\widehat\sfs_{d}
\left(\check\rho(z_D)-D\right)=0
$.\
**Step 1.** Choose $\varphi^D\in\sP(W)\otimes \sS(W)$ such that $\sfm(\varphi^D)=D$. We can write $$\varphi^D=\varphi^D_0+
\cdots+\varphi^D_{d}
\text{ where }\varphi^D_r\in\sP(W)\otimes \sS^r(W)
\text{ for }0\leq r\leq d,$$ and Lemma \[prp-g-inv-sPWW\] implies that $\varphi^D_r\in\left(\sP^r(W)\otimes \sS^r(W)\right)^{\g g}$ for $0\leq r\leq d$. Furthermore, $$\widehat\sfs_{d}(D)=
\widehat\sfs_{d}\left(\sfm
\left(\varphi^D\right)\right)
=
\widehat\sfs_{d}\left(\sfm
\left(\varphi_{d}^D\right)\right)=
\varphi_{d}^D.$$ By Lemma \[prp-g-inv-sPWW\], $\varphi^D_{d}$ is a linear combination of the tensors $\bft_{\sigma}$ for $\sigma\in S_{2d}$. Thus to complete the proof, it suffices to find $z_{ \sigma}\in\bfZ(\g g)\cap \bfU^{d}(\g g)$ such that $\widehat\sfs_{d}\left(
\check\rho(z_{\sigma})\right)
=\bft_{\sigma}$.\
**Step 2.** Fix $\sigma\in S_{2d}$ and let $0=d_0<d_1\cdots<d_{r-1}<d_r=d$ be as in Proposition \[taud0dr\](ii). Set $\sigma_s:=\left(2,\ldots,2(d_{s+1}-d_{s})\right)\in S_{2(d_{s+1}-d_{s})}^{}$ for $0\leq s\leq r-1$. From Proposition \[taud0dr\](i) and Lemma \[lem2..5\] it follows that $$\bft_\sigma=\bft_{\sigma_0^{}}\cdots
\bft_{\sigma_{r-1}^{}}.$$ Now set $z_s:=(-\frac{1}{2})^{d_{s+1}-d_{s}}\,\mathrm{str}
\left(\mathbf E^{d_{s+1}-d_{s}}\right)$ for $0\leq s\leq r-1$, so that $z_s\in\bfZ(\g g)$ by Lemma \[gelfzhel\]. By Lemma \[lem2..8\], $$\label{ds+1-ds}
\widehat\sfs_{d_{s+1}-d_{s}}\left(
\check\rho(z_s)\right)=\bft_{\sigma_s}
\text{ for }0\leq s\leq r-1.$$ From it follows that $$\begin{aligned}
\widehat\sfs_{d}(\check\rho(z_0\cdots z_{r-1}))
&=
\widehat\sfs_{d}(\check\rho(z_0)\cdots \check\rho(z_{r-1}))\\
&=\widehat\sfs_{d_1-d_0}(\check\rho(z_0))
\cdots
\widehat\sfs_{d_{r}-d_{r-1}}(\check\rho(z_{r-1}))
=
\bft_{\sigma_0^{}}\cdots
\bft_{\sigma_{r-1}^{}}=\bft_{\sigma},\end{aligned}$$ which, as mentioned in Step 1, completes the proof of the theorem.
The symmetric superpair $\big(\gl(m|2n),\g{osp}(m|2n)\big)$ {#Secsuperpr}
===========================================================
From now on, we assume that the odd part of $V$ is even dimensional. In other words, we set $V:=\C^{m|2n}$. We set $W:=\sS^2(V)$ and $\g g:=\gl
(V)\cong\gl(m|2n)$, as before. Let $\beta\in W^*$ be a nondegenerate, symmetric, even, bilinear form on $V$, so that for every homogeneous $v,v'\in V$ we have $\beta(v,v^{\prime })=0 $ if $|v|\neq
|v^{\prime }|$, and $\beta(v^{\prime},v)=(-1)^{|v|\cdot |v^{\prime}|}\beta(v,v^{\prime }) $. Since $\beta\in W^*$ is even, we have $$(\check\rho(x)\beta)(w)=-(-1)^{|x|\cdot |\beta|}\beta(\rho(x)
w)=-\beta(\rho(x)w) \text{ for }x\in\g g,\ w\in W.$$ Set $\g k:=\g{osp}(V,\beta):=\{x\in\gl(V)\, :\, \check\rho(x)\beta=0\}$, so that $$\g k=\left\{x\in\g g\, :\, \beta(x\cdot v,v^{\prime }) +(-1)^{|x|\cdot |v|}
\beta(v,x\cdot v^{\prime })=0\text{ for }v,v^{\prime }\in V \right\}.$$ In this section we prove that every irreducible $\g g$-submodule of $\sP(W)$ contains a non-zero $\g k$-invariant vector. We remark that this statement does not follow from [@AllSch Thm A]. Recall the decomposition $\g g=\g n^-\oplus\g h\oplus\g n^+$ from Section \[prfof571\]. Let $J_{2n}$ be the $2n\times 2n$ block-diagonal matrix defined by $$J_{2n}:=\mathrm{diag}(\underbrace{J,\ldots,J}_{n\text{ times}}) \text{ where
} J:=%
\begin{bmatrix}
0 & 1 \\
-1 & 0%
\end{bmatrix}
.$$ Let $I_m$ denote the $m\times m$ identity matrix. Without loss of generality we can fix a homogeneous basis $\{\bfe_i\, :\, i\in\mathcal{I}_{m,2n}\}$ for $V\cong\C^{m|2n}$ such that $$\label{beiej=c}
\left[ \beta(\bfe_i{},\bfe_j^{}) \right]_{i,j\in\mathcal{I}_{m,2n}} =%
\begin{bmatrix}
I_m & 0 \\
0 & J_{2n}%
\end{bmatrix}%
.$$
Set $\g a:=\left\{h\in\g h\,:\,\eps_{\oline{2k-1}}^{}(h)=\eps_{\oline{2k}%
}^{}(h)\text{ for }1\leq k\leq n\right\}$ and $$\g t:=\left\{h\in\g h\,:\,\eps_k^{}(h)=0\text{ for }1\leq k\leq m\text{ and }%
\eps_{\oline{2l-1}}^{}(h)=-\eps_{\oline{2l}}^{}(h)\text{ for }1\leq l\leq
n\right\}.$$ Then $\g t=\g k\cap \g h$ and $\g h=\g t\oplus\g a$. Set $$\label{gamkldf}
\gamma_k:=\eps_k\big|_{\g a} \text{ for }1\leq k\leq m\text{ and } \gamma_{%
\oline{l}}^{}:=\eps_{\oline{2l}}^{}\big|_{\g a} \text{ for }1\leq l\leq n.$$ The restricted root system of $\g g$ corresponding to $\g a$ will be denoted by $\Sigma:=\Sigma^+\cup\Sigma^-$, where $\Sigma^+:=\Sigma_\eev^+\cup\Sigma_%
\ood^+$ is explicitly given by $$\Sigma_\eev^+:=\Big\{\gamma_k^{}-\gamma_l^{}\Big\}
_{1\leq k<l\leq m}\cup \Big\{\gamma_{\oline k}-\gamma_{\oline l}\Big\}
_{1\leq k<l\leq n} \ \text{ and }\ \Sigma_\ood^+:=\Big\{\gamma_k^{}-\gamma_{%
\oline{l}}^{}\Big\}
_{1\leq k\leq m,\,1\leq l\leq n}.$$ Furthermore, as usual $\Sigma^-=-\Sigma^+$.
For every $\gamma\in\Sigma
$ we set $\g g_{\gamma}:=\{x\in\g g\,:\,[h,x]=\gamma(h)x\text{ for every }%
h\in\g a\}$. Then $$\label{iwasawa}
\g g=\g k\oplus\g a \oplus\g u^+\text{ where } \g u^\pm:=\bigoplus_{\gamma%
\in\Sigma^\pm}\g g_\gamma\subseteq\g n^\pm.$$
\[rmkbasisK\] Recall from Section \[prfof571\] that using the basis $\{\bfe_i\, :\, i\in\mathcal I_{m,2n}\}$ of $V=\C^{m|2n}$, we can represent every element of $\g g$ as an $(m,2n)$-block matrix whose rows and columns are indexed by the elements of $\mathcal I_{m,2n}$. The following matrices form a spanning set of $\g k$.
- $E_{k,l}-E_{l,k}$ for $1\leq k\neq l\leq m$,
- $E_{\oline{2l-1},\oline{2k-1}}-E_{\oline{2k},\oline{2l}}$ for $1\leq k,l\leq n$,
- $E_{\oline{2l-1},\oline{2k}}+E_{\oline{2k-1},\oline{2l}}$ for $1\leq k,l\leq n$,
- $E_{\oline{2l},\oline{2k-1}}+E_{\oline{2k},\oline{2l-1}}$ for $1\leq k,l\leq n$,
- $E_{k,\oline{2l-1}}+E_{\oline{2l},k}$ for $1\leq k\leq m$ and $1\leq l\leq n$,
- $E_{k,\oline{2l}}-E_{\oline{2l-1},k}$ for $1\leq k\leq m$ and $1\leq l\leq n$.
Every irreducible finite dimensional representation of $\g g$ is a highest weight module. Unless stated otherwise, the highest weights that we consider will be with respect to the standard Borel subalgebra $\g b:=\g %
h\oplus\g n^+$. In this case, the highest weight $\lambda\in\g h^*$ can be written as $$\lambda=\sum_{k=1}^m\lambda_k^{}\eps_k^{}+ \sum_{l=1}^{2n}\lambda_{m+l}^{}\eps_{%
\oline l}\in\g h^*,$$ such that $\lambda_k-\lambda_{k+1}\in\Z_{\geq 0}$ for every $k\in\{1,\ldots,m-1\}\cup\{m+1,\ldots,m+2n-1\}$, where $\Z_{\geq 0}:=\{x\in\Z\,:\,x\geq 0\}$.
\[lem-dimleq1\] Let $U$ be an irreducible finite dimensional $\g g$-module. Then $\dim U^\g k\leq 1$.
Let $\lambda\in\g h^*$ be the highest weight of $U$ and set $U':=\bigoplus_{\mu\neq\lambda}U(\mu)$, where $U(\mu)$ is the $\mu$-weight space of $U$. Clearly $U'$ is a $\Ztwo$-graded subspace of $U$ of codimension one, which is invariant under the action of $\g a\oplus\g u^-$. Note also that $U^\g k$ is a $\Ztwo$-graded subspace of $U$. Suppose that $\dim U^\g k\geq 2$, so that $U'\cap U^\g k\neq\{0\}$. Fix a non-zero homogeneous vector $u\in U'\cap U^\g k$. The PBW Theorem and the Iwasawa decomposition imply that $$U=\bfU(\g g)u=
\bfU(\g a\oplus \g u^-)\bfU(\g k)u
=\bfU(\g a\oplus\g u^-)u\subseteq U',$$ which is a contradiction.
A *partition* is a sequence $\flat:=(\flat_1,\flat_2,\flat_3,\ldots)$ of integers satisfying $
\flat_k\geq \flat_{k+1}\geq 0
$ for every $k\geq 1$, and $\flat_k=0$ for all but finitely many $k\geq 1$. The *size* of $\flat$ is defined to be $|\flat|:=\sum_{k=1}^\infty \flat_k$. The *transpose* of $
\flat$ is denoted by $\flat^{\prime }:=(\flat_1^{\prime },\flat_2^{\prime
},\flat_3^{\prime }\ldots)$, where $$\flat_k^{\prime }:=|\{l\geq 1\ :\
\flat_l\geq k\}|.$$
\[dfnhookprn\] A partition $(\flat_1,\flat_2,\flat_3,\ldots)$ that satisfies $\flat_{m+1}\leq n$ is called an *$(m|n)$-hook partition*. The set of $(m|n)$-hook partitions of size $d$ will be denoted by $\mathrm{H}_{m,n,d}$, and we set $\mathrm{H}_{m,n}:=\bigcup_{d=0}^\infty
\mathrm{H}_{m,n,d}$.
We now define a map $$\label{bfGam}
\boldsymbol{\Gamma}:\mathrm{H}_{m,n}\to \g a^*$$ as follows. To every $\flat=(\flat_1,\flat_2,\flat_3,\ldots)\in\mathrm{H}%
_{m,n}$ we associate the element $\lambda=\boldsymbol{\Gamma}(\flat)\in\g %
a^*$ given by $$\label{frmLamBk}
\lambda:=\sum_{k=1}^m2\flat_k\gamma_k+ \sum_{l=1}^n2\max\big\{\flat^{\prime
}_l-m,0\big\}\gamma_{\oline l} \in \g a^*.$$ For integers $m,n,d\geq 0$, set $$\mathrm{E}_{m,n,d}:=\boldsymbol{\Gamma}(\mathrm{H}_{m,n,d}):= \big\{
\lambda\in\g a^*\,:\,\lambda=\boldsymbol{\Gamma}(\flat)\text{ for some }%
\flat\in \mathrm{H}_{m,n,d} \big\}$$ and $$\label{Emndf}
\mathrm{E}_{m,n}:= \bigcup_{d=0}^\infty\mathrm{E}_{m,n,d}.$$
Because of the decomposition $\g h=\g a\oplus\g t$, every $\lambda\in\g a^*$ can be extended trivially on $\g t$ to yield an element of $\g h^*$. It is straightforward to verify that for every $\lambda\in\mathrm{E}_{m,n}$, the above extension of $\lambda$ to $\g h^*$ is the highest weight of an irreducible $\g g
$-module. We will denote the latter highest weight module by $V_\lambda$. From [@ChWa Thm 3.4] or [@brini] it follows that $$\label{prpSWW}
\sS^d(W)\cong\bigoplus_{\lambda\in\mathrm{E}_{m,n,d}} \!V_{\lambda}\ \ \
\text{as \,$\g g$-modules}.$$
\[prpChWaSe\] From it follows that $$\label{prpSWW*}
\sP^d(W)\cong\bigoplus_{\lambda\in\mathrm{E}_{m,n,d}}
V_\lambda^*
\cong
\bigoplus_{\mu\in\mathrm{E}^*_{m,n,d}}
V_\mu^{}$$ where $\mathrm{E}_{m,n,d}^*$ is the set of all $\mu\in\g a^*$ of the form $$\label{eqhwofVl}
\mu=-\sum_{i=1}^m 2\max\{\flat_{m+1-i}-n,0\}\gamma_i-\sum_{j=1}^n 2\flat'_{n+1-j}\gamma_{\oline j}
\text{ for some }\flat\in\mathrm{H}_{m,n,d},$$ and $V_\mu$ denotes the $\g g$-module whose highest weight is the extension of $\mu$ trivially on $\g t$ to $\g h^*$. The decomposition follows from the fact that for every $\lambda\in \mathrm{E}_{m,n,d}$, the contragredient module $V_\lambda^*$ has highest weight equal to the extension to $\g h^*$ of $-\lambda^-$, where $\lambda^-$ is the highest weight of $V_\lambda$ with respect to the opposite Borel subalgebra $\g b^-:=\g h\oplus\g n^-$. From the calculation of $\lambda^-$ given in [@OlPr Thm 6.1] or [@ChWabook Sec. 2.4.1], it follows that the highest weight of $V_\lambda^*$ is of the form .
Set $$\label{Emn*df}
\mathrm{E}_{m,n}^*:=\bigcup_{d=0}^\infty \mathrm{E}^*_{m,n,d}.$$ By the $
\g g$-equivariant isomorphism , $$\begin{aligned}
\label{spdgCB}
\displaystyle\sPD(W)^\g g_{} &\cong \big(\sP(W)\otimes\sS(W)\big)^\g g \cong
\bigoplus_{\lambda,\mu\in\mathrm{E}_{m,n}^{*}} (V_\lambda\otimes V_\mu^*)^\g %
g \cong \bigoplus_{\lambda,\mu\in\mathrm{E}_{m,n}^{*}} \Hom_\g %
g(V_\mu,V_\lambda).\end{aligned}$$ Every non-zero element of $\Hom_\g g(V_\mu,V_\lambda)$ should map $V_\mu^{\g %
n^+}$ to $V_\lambda^{\g n^+}$, and is uniquely determined by the image of a highest weight vector in $V_\mu^{\g n^+}$. But $\dim(V_\lambda^{\g %
n^+})=\dim(V_\mu^{\g n^+})=1$, so that $\dim\left(\Hom_\g g(V_\lambda,V_\mu)\right)\leq 1 $ with equality if and only if $\lambda=\mu$.
\[DefDla\] For $\lambda\in\mathrm{E}_{m,n}^*$, let $D_\lambda\in \sPD(W)^\g g$ be the $\g g$-invariant differential operator that corresponds via the sequence of isomorphisms to $1_{V_\lambda}^{}\in\mathrm{End}_\C(V_\lambda)$.
From it follows that $\{D_\lambda\,:\,\lambda\in \mathrm{E}^*
_{m,n}\}$ is a basis for $\sPD(W)^\g g$. In the purely even case (that is, when $n=0$), the latter basis is sometimes called the *Capelli basis* of $\sPD(W)^\g g$.
Let $\beta\in W^*$ be the symmetric bilinear form defining $\g k$, as at the beginning of Section \[Secsuperpr\]. Let $\sfh_\beta:\sS(W)\to \C$ be the extension of $\beta$, defined as in . Set $$\label{tildhbeta}
\widetilde\sfh_\beta:\sP(W)\otimes \sS(W)\to
\sP(W) \ ,\ a\otimes b\mapsto \sfh_\beta(b)a.$$ Recall the map $\sfm:\sP(W)\otimes\sS(W)\to \sPD(W)$ defined in .
\[le-dlam\] Let $\lambda\in\mathrm{E}^*_{m,n}$. Then $\dim V_{\lambda}^\g k=1$, and the vector $$\mathbf d_\lambda:=(\widetilde \sfh_\beta\circ \sfm^{-1})(D_\lambda).$$ is a non-zero $\g k$-invariant vector of $V_\lambda$.
Since $\beta\in (W^*)^\g k$, the map $\sfh_\beta$ is $\g k$-equivariant. It follows that $\widetilde\sfh_\beta=\yekP\otimes \sfh_\beta$ is $\g k$-equivariant, and therefore $\mathbf d_\lambda\in V_\lambda^\g k$. By Lemma \[lem-dimleq1\], it suffices to prove that $\mathbf d_\lambda\neq 0$. Fix $d\geq 0$ such that $\lambda\in \mathrm{E}_{m,n,d}^*$, so that $V_\lambda\in \sP^d(W)$ and $V_\lambda^*\subseteq \sS^d(W)$.\
**Step 1.** The linear map $\sfh_\beta\big|_{V_\lambda^*}:V_\lambda^*\to\C$ can be represented by evaluation at a vector $v_\circ\in V_\lambda$, that is, $\sfh_\beta(v^*):=\lag v^*,v_\circ\rag$ for every $v^*\in V_\lambda^*$.
Let $\iota_\lambda:V_\lambda\otimes V_\lambda^*\to \Hom_\C(V_\lambda,V_\lambda)$ be the isomorphism . For every $v\otimes v^*\in V_\lambda\otimes V_\lambda^*$, $$\widetilde\sfh_\beta
\circ\iota_\lambda^{-1}(T_{v\otimes v^*}^{})
=
\widetilde\sfh_\beta
(v\otimes v^*)
=
\sfh_\beta(v^*)v
=
\lag v^*,v_\circ\rag v
=
T_{v\otimes v^*}(v_\circ).$$ It follows that $\widetilde\sfh_\beta\circ \iota_\lambda^{-1}(T)=Tv_\circ$ for every $T\in\mathrm{End}_\C(V_\lambda)$. In particular, $$\mathbf d_\lambda=\widetilde\sfh_\beta\circ\iota_\lambda^{-1}(1_\lambda)=v_\circ,$$ so that to complete the proof we need to show that $v_\circ\neq 0$, or equivalently, that $\sfh_\beta\big|_{V_\lambda^*}\neq 0$.\
**Step 2.** Let $\{x_{i,j}\ :\ {i,j\in\mathcal I_{m,2n}}\}$ be the generating set of $\sS(W)$ that is defined in . We define the superderivations $\sfD_{i,j}$, for $i,j\in\mathcal I_{m,2n}$, as in . Assume that $\sfh_\beta\big|_{V_\lambda^*}=0$. We will reach a contradiction in Step 3. Fix $a\in V_\lambda^*\subseteq \sS^d(W)$. We claim that $$\label{j1i1jkik}
\sfh_\beta\left(
\sfD_{j_1,i_1}\cdots\sfD_{j_k,i_k}(a)
\right)=0
\text{ for every }
k\geq 0,\ i_1,j_1,\ldots,i_k,j_k\in\mathcal I_{m,2n},$$ where $\sfD_{i,j}$ is defined as in . The proof of is by induction on $k$. For $k=0$, it follows from the assumption that $\sfh_\beta\big|_{V_\lambda^*}=0$. Next assume $k=1$. Set $$\label{eqdfiprm}
i'_1:=\begin{cases}
i_1&\text{ if }|i_1|=\eev,\\
\oline{2k}&\text{ if }i_1=\oline{2k-1}\text{ where }
1\leq k\leq n,\\
\oline{2k-1}&\text{ if }i_1=\oline{2k}\text{ where }
1\leq k\leq n,
\end{cases}$$ Then $\rho\big(E_{i_1',j_1^{}}\big)a\in V_\lambda$, so that $\sfh_\beta\big(\rho\big(E_{i_1',j_1^{}}\big)a\big)=0$. But from it follows that $$\begin{aligned}
\sfh_\beta
\big(\rho\big(E_{i_1',j_1^{}}\big)a\big)
&=\sum_{r\in\mathcal I_{m,2n}}\sfh_\beta
\big(x_{i_1',r}\big)
\sfh_\beta\big(\sfD_{j_1,r}(a)\big)\\
&=
\sum_{r\in\mathcal I_{m,2n}}
\beta(\bfe_{i_1'},\bfe_r)
\sfh_\beta\big(\sfD_{j_1,r}(a)\big)=
\pm\sfh_\beta\big(\sfD_{j_1,i_1}(a)\big),\end{aligned}$$ so that $\sfh_\beta(\sfD_{j_1,i_1}(a))=0$.
Finally, assume that $k>1$. We define $i_1',\ldots,i_k'\in\mathcal I_{m,2n}$ from $i_1^{},\ldots,i_k^{}$ according to . Then $\rho\big(E_{i_1',j_1^{}}\big)\cdots \rho\big(E_{i_k',j_k^{}}\big)a\in V_\lambda$, so that $
\sfh_\beta\big(
\rho\big(E_{i_1',j_1^{}}\big)\cdots \rho\big(E_{i_k',j_k^{}}\big)a
\big)
=0$. However, we can write $$\label{rhEkkk}
\rho\big(E_{i_1',j_1^{}}\big)\cdots \rho\big(E_{i_k',j_k^{}}\big)a
=
\sum_{r_1,\ldots,r_k\in\mathcal I_{m,2n}}
x_{i_1',r_1^{}}\cdots x_{i_k',r_k^{}}
\sfD_{j_1^{},r_1^{}}\cdots
\sfD_{j_k^{},r_k^{}}(a)
+R_a,$$ where $R_a$ is a sum of elements of $\sS(W)$ of the form $b\sfD_{s_1,t_1}\cdots\sfD_{s_\ell,t_\ell}(a)$, with $b\in\sS(W)$ and $\ell<k$. Since $\sfh_\beta$ is an algebra homomorphism, the induction hypothesis implies that $\sfh_\beta(R_a)=0$. Therefore implies that $$\begin{aligned}
\sfh_\beta
\big(
\rho\big(E_{i_1',j_1^{}}\big)\cdots \rho\big(E_{i_k',j_k^{}}\big)a
\big)
&=
\sum_{r_1,\ldots,r_k\in\mathcal I_{m,2n}}
\sfh_\beta\big(x_{i_1',r_1^{}}\big)\cdots \sfh_\beta(x_{i_k',r_k^{}}\big)
\sfh_\beta\big(\sfD_{j_1^{},r_1^{}}\cdots
\sfD_{j_k^{},r_k^{}}(a)\big)\\
&=\pm \sfh_\beta
\big(\sfD_{j_1^{},i_1^{}}\cdots
\sfD_{j_k^{},i_k^{}}(a)\big),\end{aligned}$$ so that $\sfh_\beta
\big(\sfD_{j_1^{},i_1^{}}\cdots
\sfD_{j_k^{},i_k^{}}(a)\big)=0$.\
**Step 3.** Set $\mathcal J:=\{(k,\oline l)\,:\,1\leq k\leq m\text{ and }1\leq l\leq 2n\}\subset
\mathcal I_{m,2n}\times \mathcal I_{m,2n}$. Every $a\in\sS(W)$ can be written as $$a=\sum_{S\subseteq \mathcal J}a_Sx_S$$ where $a_S\in\sS(W_\eev)$ and $x_S:=\prod_{(i,j)\in S}x_{i,j}$ for each $S\subseteq \mathcal J$. Now fix $S\subseteq\mathcal J$ and set $ \sfD_S:=\prod_{(i,j)\in S}\sfD_{i,j}$, so that $\sfD_S(a)=\pm a_S$. From it follows that $$\label{Hbetya}
\sfh_\beta\big(
\sfD_{j_1,i_1}
\cdots\sfD_{j_k,i_k})(a_S)\big)
=0\text{ for every }k\geq0,\
i_1,\ldots,i_k,j_1,\ldots,j_k\in\{1,\ldots,m\}.$$ But $\sS(W_\eev)\cong \sP(W_\eev^*)$, and therefore means that the multi-variable polynomial $a_S\in\sP(W_\eev^*)$ and all of its partial derivatives vanish at a fixed point $\beta_\eev:=\beta\big|_{W_\eev}\in W_\eev^*$. Consequently, $a_S=0$. As $S\subseteq \mathcal J$ is arbitrary, it follows that $a=0$.
\[rmk-v\*\] A slight modification of the proof of Proposition \[le-dlam\] shows that for every $\lambda\in\mathrm{E}_{m,n}$ we have $\dim V_\lambda^\g k=1$ as well. To this end, instead of $\beta\in W^*$ we use the $\g k$-invariant vector $$\beta^*:=-\frac{1}{4}(x_{1,1}+\cdots+x_{m,m})
+\frac{1}{2}(x_{\oline 1,\oline 2}+\cdots+x_{\oline{2n-1},\oline{2n}})
\in W.$$
The eigenvalue and spherical polynomials $c_\protect\lambda$ and $d_\lambda$ {#Sec-Sec5}
=============================================================================
As in the previous section, we set $V:=\C^{m|2n}$, so that $\g g:=\gl(V)=\gl(m|2n)$. Recall the decompostion of $\sP(W)$ into irreducible $\g g$-modules given in and , that is, $$\sP(W)=\bigoplus_{\lambda\in\mathrm{E}_{m,n}^*}V_\lambda.$$ For $\lambda\in \mathrm{E}_{m,n}^*$, let $D_\lambda\in\sPD(W)^\g g$ be the $\g g$-invariant differential operator as in Definition \[DefDla\]. Since the decomposition is multiplicity-free, for every $%
\mu\in\mathrm{E}_{m,n}^*$ the map $$D_\lambda:V_\mu\to V_\mu$$ is multiplication by a scalar ${c}_\lambda(\mu)\in \C$. Fix $d\geq 0$ such that $\lambda\in\mathrm{E}_{m,n,d}^*$, so that $V_\lambda\sseq \sP^d(W)$. By Theorem \[prpgw\] we can choose $z_\lambda\in \mathbf{Z}(\g g)\cap\bfU^d(\g g)$ such that $$\label{choicezl}
\check\rho(z_\lambda)=D_\lambda.$$ Next choose $z_{\lambda,\g k}\in\g k\bfU(%
\g g)\cap\bfU^d(\g g)$, $z_{\lambda,\g u^+}\in\bfU(\g g)\g u^+\cap\bfU^d(\g %
g)$, and $z_{\lambda,\g a}\in\bfU(\g a)\cap\bfU^d(\g g)$ such that $$\begin{aligned}
\label{bfUgan}
z_\lambda=z_{\lambda,\g k}+z_{\lambda,\g u^+}+z_{\lambda,\g a},\end{aligned}$$ according to the decomposition $\bfU(\g g)=\left( \g k\bfU(\g g) + \bfU(\g g)%
\g u^+\right)\oplus\bfU(\g a) $. For $\mu\in\g a^*$ let $$\sfh_\mu:\bfU(%
\g a)\cong \sS(\g a)\to \C$$ be the extension of $\mu$ defined as in .
\[cmulZa\] $c_\lambda(\mu)=\sfh_\mu(z_{\lambda,\g a})$ for every $\lambda,\mu\in\mathrm{E}_{m,n}^*$. In particular, $c_\lambda\in\sP(\g a^*)$.
Let $v_\mu$ denote a highest weight vector of $V_\mu\subseteq\sP(W)$. By Remark \[rmk-v\*\], the contragredient module $V_\mu^*\subseteq\sS(W)$ contains a non-zero $\g k$-invariant vector $v_\circ^*$. From the decomposition $\g g=\g k\oplus\g a\oplus\g u^+$ and the PBW Theorem it follows that $V_\mu=\bfU(\g k)v_\mu$. This means that if $\lag v_\circ^*,v_\mu\rag=0$, then $\lag v_\circ^*,V_\mu\rag=0$, which is a contradiction. Thus $\lag v_\circ^*,v_\mu\rag \neq 0$, and $$\begin{aligned}
c_\lambda(\mu)\lag v_\circ^*,v_\mu\rag
&
=\lag v_\circ^*,D_\lambda v_\mu\rag
=
\lag v_\circ^*,\check\rho(z_\lambda)v_\mu\rag\\
& =
\lag v_\circ^*,
(\check\rho(z_{\lambda,\g k})+
\check\rho(z_{\lambda,\g u^+})+
\check\rho(z_{\lambda,\g a})
)v_\mu
\rag
=\lag
v_\circ^*,\check\rho(z_{\lambda,\g a})v_\mu
\rag=\sfh_\mu(z_{\lambda,\g a})\lag v_\circ^*,v_\mu\rag.\end{aligned}$$ It follows that $c_\lambda(\mu)=\sfh_\mu(z_{\lambda,\g a})$.
\[dfcl\] For every $\lambda\in\mathrm{E}_{m,n}^*$, the polynomial $c_\lambda\in\sP(\g a^*)$ is called the *eigenvalue polynomial* associated to $\lambda$.
\[rmkdgclam\] Note that by Theorem \[prpgw\], if $\lambda\in\mathrm{E}_{m,n,d}^*$ then $\deg(c_\lambda)\leq d$.
\[lem-clmuvanish\] Let $\lambda\in\mathrm{E}_{m,n,d}^*$. Then $c_\lambda(\lambda)=d!$ and $ c_\lambda(\mu)=0$ for all other $\mu$ in $\bigcup_{d'=0}^d \mathrm{E}_{m,n,d'}^*$.
Note that $D_\lambda\in\sfm(V_\lambda\otimes V_\lambda^*)\subseteq\sfm\left(\sP^d(W)\otimes\sS^d(W)\right)$ where $\sfm$ is the map defined in . Thus if $d'<d$ then $D_\lambda\sP^{d'}(W)=\{0\}$. Next assume $d'=d$. Then the map $$\label{sPDPWPW-}
\sPD(W)\otimes \sP(W)\to\sP(W)
\ ,\ D\otimes p\mapsto Dp$$ is $\g g$-equivariant, and since $D_\lambda\in\sfm(V_\lambda\otimes V_\lambda^*)$, the map restricts to a $\g g$-equivariant map $V_\mu\to V_\lambda$ given by $p\mapsto D_\lambda p$. By Schur’s Lemma, when $\lambda\neq \mu$, the latter map should be zero.
Finally, to prove that $\mathbf c_\lambda(\lambda)=d!$, we consider the bilinear form $$\label{bilformbb}
\boldsymbol\beta:\sP^d(W)\times \sS^d(W)\to\C\
,\
\boldsymbol\beta(a,b):=\partial_ba.$$ By a straightforward calculation one can verify that $\boldsymbol\beta(a,b)=d!\lag a,b\rag$, where $\lag\cdot,\cdot\rag$ denotes the duality pairing between $\sP^d(W)\cong\sS^d(W)^*$ and $\sS^d(W)$. Next we choose a basis $v_1,\ldots,v_t$ for $V_\lambda$. Let $v_1^*,\ldots,v_t^*$ be the corresponding dual basis of $V_\lambda^*$. From it follows that $$D_\lambda v_k=\sfm(\sum_{l=1}^t v_l\otimes v_l^*)v_k=\sum_{i=1}^tv_l\partial_{v_l^*}(v_k)=
d! v_k$$ for every $1\leq k\leq t$.
Let $\beta^*\in W$ be the $\g k$-invariant vector that is given in Remark \[rmk-v\*\], and set $$\iota_\g a:\g a\to W\ ,\ \iota_\g a(h):=\rho(h)\beta^*.$$ By duality, the map $\iota_\g a$ results in a homomorphism of superalgebras $$\label{dfiotaa*}
\iota_\g a^*:\sP(W)\cong\sS(W^*)\to\sS(\g a^*)\cong \sP(\g a),$$ which is defined uniquely by the relation $\lag \iota^*_\g a (w^*),h\rag =%
\lag w^*,\iota_\g a(h)\rag
$ for $w^*\in W^*$, $h\in\g a$.
Let $\kappa$ be the supertrace form on $\g g$ defined as in . The restriction of $\kappa$ to $\g a$ is a non-degenerate symmetric bilinear form and yields a canonical isomorphism $\sfj:\g a^*\to\g a$ which is defined as follows. For every $\xi\in\g a^*$, $$\sfj(\xi):=x_\xi\text{ if and only if }\kappa(\,\cdot\,,x_\xi)=\xi .$$ Let $\sfj^*:\sP(\g a)\to\sP(\g a^*)$ be defined by $\sfj^*(p):=p\circ\sfj$ for every $p\in\sP(\g a)$. For $\lambda\in\mathrm{E}_{m,n}^*$, let $\mathbf{d}_\lambda\in\sP(W)^\g k$ be the $\g k$-invariant vector in $V_\lambda$ as in Proposition \[le-dlam\].
\[dfbr\] For $\lambda\in\mathrm{E}_{m,n,d}^*$, we define the *spherical polynomial* $d_\lambda$ to be $$d_\lambda:=\sfj^*\circ\iota^*_\g a(\mathbf{d}_\lambda)\in\sP^d(\g a^*).$$
\[DGBRVVV\] The restriction of $\iota_\g a^*$ to $\sP(W)^\g k$ is an injection. In particular, $d_\lambda\neq 0$ for every $\lambda$.
The proof of Proposition \[DGBRVVV\] will be given in Section \[sec-pflem\].
Recall the definitions of $\widetilde\sfh_\beta$ and $z_{\lambda,\g a}$ from and .
\[lemZnZk=0\] Let $\lambda\in\mathrm{E}^*_{m,n,d}$, and let $z_{\lambda,\g a}$ be defined as in . Then $$d_\lambda=\big(\sfj^*\circ\iota^*_\g a\circ\widetilde\sfh_\beta\circ\widehat \sfs_d\big)(\check\rho(z_{\lambda,\g a})).$$
Let $D_\lambda$ be the $\g g$-invariant differential operator as in Definition \[DefDla\]. Since we have $D_\lambda\in\sfm(\sP^d(W)\otimes\sS^d(W))$, from we obtain $$\label{formdlda}
\mathbf d_\lambda=
\widetilde\sfh_\beta(\sfm^{-1}(D_\lambda))
=
\widetilde\sfh_\beta(\widehat\sfs_d(D_\lambda))
.$$ From it follows that $$d_\lambda=\big(\sfj^*\circ\iota^*_\g a\circ\widetilde\sfh_\beta\circ\widehat\sfs_d\big)(\check\rho(z_{\lambda})).$$ By the decomposition and the fact that $z_{\lambda,\g k},z_{\lambda,\g u^+},z_{\lambda,\g a}\in\bfU^d(\g g)$, it is enough to prove that if $x\in\g k\bfU^{d-1}(\g g)$ or $x\in\bfU^{d-1}(\g g)\g u^+$, then $$\label{iotac1P}
\big(
\iota_\g a^*
\circ
\widetilde\sfh_\beta\circ \widehat \sfs_d\big)(\check\rho(x))
=0.$$ First we prove for $x\in\g k\bfU^{d-1}(\g g)$. By the PBW Theorem there exist $x^\circ,x^-\in\bfU(\g g)$ such that such that
- $x=x^\circ+x^-$,
- $x^\circ$ is a sum of monomials of the form $x_1\cdots x_d$ where $x_1\in\g k$ and $x_2,\ldots,x_d\in\g g$,
- $x^{-}\in \bfU^{d-1}(\g g)$.
Recall that $\mathrm{ord}(D)$ denotes the order of a differential operator $D\in\sPD(W)$. Since we have $\mathrm{ord}(\check\rho(x))\leq 1$ for every $x\in \g g$, from it follows that $\widehat\sfs_d(\check\rho(x^-))=0$ and $$\label{wsdrhoc}
\widehat\sfs_d
(\check\rho(x_1\cdots x_d))
=
\widehat\sfs_1(\check\rho(x_1))\cdots
\widehat\sfs_1(\check\rho(x_d)).$$ Since $\iota_\g a^*$ and $\widetilde\sfh_\beta$ are homomorphisms of superalgebras, from it follows that in order to prove , it is enough to verify that $$\label{pWWcsig1}
\big(
\iota_\g a^*\circ\widetilde\sfh_\beta\circ\widehat\sfs_1
\big)
(\check\rho(x_1))=0
\text{ for every }x_1\in\g k.$$ To verify , we use the generators $\{x_{i,j}\,:\,i,j\in\mathcal I_{m,2n}\}$ and $\{y_{i,j}\,:\,i,j\in\mathcal I_{m,2n}\}$ defined as in . From it follows that $$\label{S1cRho}
\widehat\sfs_1(\check\rho(E_{i,j}))=
-(-1)^{|i|\cdot|j|}
\sum_{r\in\mathcal I_{m,2n}}
(-1)^{|r|}y_{r,j}x_{r,i}\,
\text{ for }i,j\in\mathcal I_{m,2n},$$ and consequently, $$\big(
\iota_\g a^*\circ\widetilde\sfh_\beta\circ\widehat\sfs_1
\big)
(\check\rho(E_{i,j}))
=
\begin{cases}
-\iota_\g a^*(y_{i,j}) & \text{ if }|i|=|j|=\eev,\\
\iota_\g a^*(y_{\oline{2k},j}) &
\text{ if }
i=\oline{2k-1}\text{ for }1\leq k\leq n,\text{ and }|j|=\ood,\\
-\iota_\g a^*(y_{\oline{2k-1},j}) &
\text{ if }
i=\oline{2k}\text{ for }1\leq k\leq n,\text{ and }|j|=\ood,\\
0& \text{ if }|i|\neq |j|.
\end{cases}$$ If $x_1\in\g k_\ood$, then $x_1$ is a linear combination of the $E_{i,j}$ such that $|i|\neq |j|$, and therefore $\big(\iota_\g a^*\circ\widetilde\sfh_\beta\circ\widehat\sfs_1
\big)
(\check\rho(x_1))=0$. If $x_1\in\g k_\eev$, then $x_1$ is a linear combination of elements of the cases (i)–(iv) in Remark \[rmkbasisK\], and in each case, we can verify that $\big(\iota_\g a^*\circ\widetilde\sfh_\beta\circ\widehat\sfs_1
\big)
(\check\rho(x_1))=0$.
The proof of for $x\in\bfU(\g g)^{d-1}\g u^+$ is similar. As in the case $x\in\g k\bfU(\g g)$, the proof can be reduced to showing that $$\label{pWWcsig111}
\big(
\iota_\g a^*\circ\widetilde\sfh_\beta\circ\widehat\sfs_1
\big)
(\check\rho(x))=0
\text{ for every }x\in\g u^+.$$ It is easy to verify directly that $\g u^+$ is spanned by the $E_{i,j}$ for $i,j$ satisfying at least one of the following conditions.
- $1\leq i<j\leq m$,
- $i=\oline k$ and $j=\oline l$ where $1\leq k< l\leq 2n$ and $
(k,l)\not\in\{(2t-1,2t)\,:\,1\leq t\leq n\}
$.
Therefore follows from the above calculation of $\big(\iota_\g a^*\circ\widetilde\sfh_\beta\circ\widehat\sfs_1
\big)
(\check\rho(E_{i,j}))$, together with the fact that $\iota_\g a^*(y_{i,j})=0$ unless $i=j\in\{1,\ldots,m\}$, or $i=\oline{2k-1}$ and $j=\oline{2k}$ where $1\leq k\leq n$, or $i=\oline{2k}$ and $j=\oline{2k-1}$ where $1\leq k\leq n$.
For any polynomial $p\in \sP(\g a^*)$, we write $\oline p$ for the homogeneous part of highest degree of $p$. Recall that if $\lambda\in\mathrm{E}_{m,n}^*$, then $c_\lambda$ denotes the eigenvalue polynomial, as in Definition \[dfcl\], and $d_\lambda$ denotes the spherical polynomial, as in Definition \[dfbr\]. We have the following theorem.
\[MAINTHM\] For every $\lambda\in\mathrm{E}_{m,n}^*$, we have $\oline c_\lambda=d_{\lambda}$.
Let $\gamma_i$ be the basis of $\g a^*$ defined in , and let $\{h_i\,:\,i\in\mathcal{I}_{m,n}\}$ be the dual basis of $\g a$. Assume that $\lambda\in\mathrm{E}_{m,n,d}^*$, so that $V_\lambda\subset \sP^d(W)$. Recall the definition of $z_{\lambda,\g a}$ from . By the PBW Theorem, we can write $$z_{\lambda,\g a}=\sum_{k_1,\ldots,k_m,l_{\oline 1},\ldots,l_{\oline n}\geq 0}
u^{}_{k_1,\ldots,k_m,l_{\oline 1},\ldots,l_{\oline n}}
h_1^{k_1}\cdots h_m^{k_m}
h_{\oline 1}^{l_{\oline 1}}\cdots
h_{\oline n}^{l_{\oline n}},$$ where only finitely many of the scalar coefficients $u^{}_{k_1,\ldots,k_m,l_{\oline 1},\ldots,l_{\oline n}}
$ are nonzero. Now consider the polynomial $p_\lambda=p_\lambda(t_1,\ldots,t_{m+n})$ in $m+n$ variables $t_1,\ldots,t_{m+n}$, defined by $$p_\lambda(t_1,\ldots,t_{m+n}):=\sum_{k_1,\ldots,k_m,l_{\oline 1},\ldots,l_{\oline n}\geq 0}
u^{}_{k_1,\ldots,k_m,l_{\oline 1},\ldots,l_{\oline n}}
t_1^{k_1}\cdots t_m^{k_m}
t_{m+ 1}^{l_{\oline 1}}\cdots
t_{m+ n}^{l_{\oline n}},$$ so that $
z_{\lambda,\g a}=p_\lambda\left(h_1^{},\ldots,h_m^{},h_{\oline 1},\ldots,h_{\oline n}\right)
$. We first note that $\deg(p_\lambda)=d$. Indeed since $z_{\lambda,\g a}\in\bfU^d(\g g)$, it follows that $\deg(p_\lambda)\leq d$. If $\deg(p_\lambda)<d$, then $\mathrm{ord}(\check\rho(z_{\lambda,\g a}))<d$, and Lemma \[lemZnZk=0\] implies that $d_\lambda=0$, which contradicts Proposition \[DGBRVVV\].
Fix $\xi:=\sum_{i\in\mathcal I_{m,n}}a_i\gamma_i\in\g a^*$. Lemma \[cmulZa\] implies that $$\begin{aligned}
\label{cl=pol}
c_\lambda(\xi)
&=p_\lambda(\xi(h_1^{}),\ldots,\xi(h_m^{}),
\xi(h_{\oline 1}),\ldots,\xi(h_{\oline n}))
=p_\lambda(a_1^{},\ldots,a_m^{},a_{\oline 1},\dots,a_{\oline n}).\end{aligned}$$ To complete the proof, it suffices to show that for all $\xi\in\g a^*$ we have $$\begin{aligned}
\label{dlxxi}
d_\lambda(\xi)
=
%\big(\iota_\g a^*\circ \widetilde\sfh_\beta\circ\widehat\sfs_d\big)
%(\check\rho(z_{\lambda,\g a}))(\sfj^{-1}(\xi))
%=
\oline p_\lambda
\big(a_1^{},\ldots,a_m^{},a_{\oline 1},\ldots,a_{\oline n}\big),\end{aligned}$$ where $\oline p_\lambda$ denotes the homogeneous part of highest degree of $p_\lambda$. Set $\mathbf D_i:=\check\rho(h_i)\in\sPD(W)$ for every $i\in\mathcal I_{m,n}$. Since $\g a$ is commutative, we have $\mathbf D_i\mathbf D_j=\mathbf D_j\mathbf D_i$ for every $i,j\in\mathcal I_{m,n}$, and thus $$\check\rho(z_{\lambda,\g a})
=
p_\lambda(\mathbf D_1,\ldots,\mathbf D_m,\mathbf D_{\oline 1},\ldots,\mathbf D_{\oline n}).$$ The last relation together with the fact that $\deg(p_\lambda)=d$ imply that $$\begin{aligned}
\widehat\sfs_d
(\check\rho(z_{\lambda,\g a}))&=
{\oline p}_\lambda
\big(\widehat\sfs_1(\mathbf D_1),
\ldots,
\widehat\sfs_m(\mathbf D_m),
\widehat\sfs_1(\mathbf D_{\oline{1}}),
\ldots,
\widehat\sfs_1(\mathbf D_{\oline{n}})
\big),\end{aligned}$$ Using we obtain $$\label{bigyehhhh}
\widetilde\sfh_\beta(\widehat\sfs_d
(\check\rho(z_{\lambda,\g a})))
=
\oline p_\lambda\big(-y_{1,1},\ldots,-y_{m,m},
-2y_{\oline 1,\oline 2},\ldots,-2y_{\oline{2n-1},\oline {2n}}\big).$$ It is straightforward to verify that $$\lag \iota_\g a^*(y_{k,k}),\sfj(\xi)\rag =-a_k\,\text{ for }
1\leq k\leq m
\ \text{ and }\
\lag \iota_\g a^*(y_{\oline{2l-1},\oline{2l}}),\sfj(\xi)\rag=-\frac12 a_{\oline l}
\,\text{ for }
1\leq l\leq n.$$ Thus, by composing both sides of with $\sfj^*\circ \iota_\g a^*$ and then evaluating both sides at $\xi$, we obtain $$\begin{aligned}
%\label{dlxxi}
d_\lambda(\xi)
=
\big(\sfj^*\circ \iota_\g a^*\circ \widetilde\sfh_\beta\circ\widehat\sfs_d\big)
(\check\rho(z_{\lambda,\g a}))(\xi)
=
\oline p_\lambda
\big(a_1^{},\ldots,a_m^{},a_{\oline 1},\ldots,a_{\oline n}\big).\end{aligned}$$ This establishes and completes the proof.
Our final result in this section is a characterization of the eigenvalue polynomial $c_\lambda$ by its symmetry and vanishing properties. Let $$\label{HC+dfneq}
\HC^+:\bfU(\g g)\to \bfU(\g a)$$ denote the Harish–Chandra projection defined by the composition $$\bfU(\g g) \xrightarrow{\ \psf^+\ }\bfU(\g h) \xrightarrow{\ \sfq\ }\bfU(\g %
a)$$ where $\psf^+:\bfU(\g g)\to\bfU(\g h)$ is the projection according to the decomposition $$\bfU(\g g)=\left(\bfU(\g g)\g n^+ +\g n^-\bfU(\g g)\right)\oplus\bfU(\g h)$$ and $\sfq:\bfU(\g h)\to\bfU(\g a)$ is the projection corresponding to the decomposition $\g h=\g a\oplus\g t$. For every $\lambda\in\mathrm{E}_{m,n}^*$, let $z_\lambda\in\bfZ(\g g)$ be defined as in . A precise description of the algebra $\psf^+(\bfZ(\g g))\subseteq \bfU(\g h)$ is known (see for example [@sergeev82], [@sergeev3], [@KacZ], [@Gorelik], or [@ChWabook Sec. 2.2.3]), and implies that $\psf^+(\bfZ(\g g))$ is equal to the subalgebra of $\bfU(\g h)$ generated by the elements $$\label{dfnGd}
G_d:=\sum_{k=1}^m\left(
E_{k,k}+\frac{m+1}{2}-n-k
\right)^d
+(-1)^{d-1}
\sum_{l=1}^{2n}
\left(
E_{\oline{l},\oline{l}}+\frac{m+1}{2}+n-l
\right)^d$$ for every $d\geq 1$. Let $\mathbf I(\g a^*)\subseteq\sP(\g a^*)$ be the subalgebra that corresponds to $\sfq(\psf^+(\bfZ(\g g))$ via the canonical isomorphism $\bfU(\g a)\cong\sS(\g a)\cong \sP(\g a^*)$.
\[thm-unqclam\] Let $\lambda\in\mathrm{E}_{m,n,d}^*$. Then ${c}_\lambda$ is the unique element of $
\mathbf I(\g a^*)
$ that satisfies $\deg(c_\lambda)\leq d$, $c_\lambda(\lambda)=d!$, and $c_\lambda(\mu)=0$ for all other $\mu$ in $\bigcup_{d'=0}^d \mathrm{E}_{m,n,d'}^*
$.
By Remark \[rmkdgclam\] and Lemma \[lem-clmuvanish\], only the uniqueness statement requires proof.\
**Step 1.** We prove that $$\label{eqIa*=}
\mathbf I(\g a^*)\cap
\bigoplus_{d'=0}^d\sP^{d'}(\g a^*)=\mathrm{Span}_\C
\left\{ c_\lambda\,:\,
\lambda\in\bigcup_{d'=0}^d\mathrm{E}_{m,n,d'}^*
\right\}
.$$ Fix $z\in\bfZ(\g g)$ and let $\mu\in\mathrm{E}_{m,n}^*$. Let $v_\mu\in V_\mu$ be a highest weight vector and let $v_\circ^*\in V_\mu^*$ be a nonzero $\g k$-invariant vector. As shown in the proof of Lemma \[cmulZa\], we have $\lag v_\circ^*,v_\mu\rag\neq 0$. Since $z\in\bfZ(\g g)$, we have $z-\psf^+(z)\in\bfU(\g g)\g n^+\cap\g n^-\bfU(\g g)$, and therefore $$\begin{aligned}
\lag v_\circ^*,\check\rho(z)v_\mu\rag
&=
\lag v_\circ^*,\check\rho(\psf^+(z))v_\mu\rag\\
&=
\lag v_\circ^*,\check\rho(\sfq(\psf^+(z)))v_\mu\rag=
\mu(\sfq(\psf^+(z)))\lag v_\circ^*,v_\mu\rag
=
\mu(\HC^+(z))\lag v_\circ^*,v_\mu\rag.\end{aligned}$$ Since $\check\rho(z)\in\sPD(W)^\g g$, we can write $\check\rho(z)$ as a linear combination of Capelli operators (see Definition \[DefDla\]), say $\check\rho(z)=\sum_k{a_k}D_{\lambda_k}$. It follows that the map $\mu\mapsto\mu(\HC^+(z))$ agrees with $\sum_k a_k c_{\lambda_k}(\mu)$ for $\mu\in\mathrm{E}_{m,n}^*$. Since $\mathrm{E}_{m,n}^*$ is Zariski dense in $\g a^*$, the latter two maps should agree for all $\mu\in\g a^*$ as well. This implies that $$\mathbf I(\g a^*)\subseteq\mathrm{Span}_\C\{c_\lambda\,:\,\lambda\in\mathrm{E}_{m,n}^*\}.$$ Furthermore, from Theorem \[MAINTHM\] and Proposition \[DGBRVVV\] it follows that the homogeneous part of highest degree of every nonzero element of $$\mathrm{Span}_\C\left\{c_\lambda\,:\,\lambda\in\bigcup_{d'>d}\mathrm{E}_{m,n,d'}^*\right\}$$ has degree strictly bigger than $d$. Consequently, the left hand side of is a subset of its right hand side. The reverse inclusion follows from the above argument by choosing $z:=z_\lambda$ where $z_\lambda$ is defined in .\
**Step 2.** Set $N_{m,n,d}:=\sum_{d'=0}^d\big|\mathrm{E}_{m,n,d'}^*\big|$. From Step 1 it follows that $$\dim\left(
\mathbf I(\g a^*)\cap
\bigoplus_{d'=0}^d\sP^{d'}(\g a^*)
\right)
\leq N_{m,n,d}.$$ Now consider the linear map $$L_{m,n,d}:\mathbf I(\g a^*)\cap
\bigoplus_{d'=0}^d\sP^{d'}(\g a^*)\to\C^{N_{m,n,d}}
\ ,\
p\mapsto \big(p(\mu)\big)_{\mu\in\bigcup_{d'=0}^d
\mathrm{E}_{m,n,d'}^*}.$$ From Lemma \[lem-clmuvanish\] it follows that the vectors $L_{m,n,d}(c_\lambda)$, for $\lambda\in\bigcup_{d'=0}^d\mathrm{E}_{m,n,d'}^*$, form a nonsingular triangular matrix, so that they form a basis of $\C^{N_{m,n,d}}$. Consequently, $L_{m,n,d}$ is an invertible linear transformation. In particular, for every $\lambda\in\mathrm{E}_{m,n,d}^*$, the polynomial $c_\lambda$ is the unique element of $\mathbf I(\g a^*)\cap
\bigoplus_{d'=0}^d\sP^{d'}(\g a^*)$ that satisfies $c_\lambda(\lambda)=d!$ and $c_\lambda(\mu)=0$ for all other $\mu\in\bigcup_{d'=0}^d \mathrm{E}_{m,n,d'}^*$.
Relation with Sergeev–Veselov polynomials for $\theta=\frac{1}{2}$ {#SecRelSer}
==================================================================
The main result of this section is Theorem \[thmconnSV\], which describes the precise relation between the eigenvalue polynomials $
c_\lambda$ of Definition \[dfcl\] and the shifted super Jack polynomials of Sergeev–Veselov [SerVes]{}. As in the previous section, we set $V:=\C^{m|2n}$, so that $\g g:=\gl(V)=\gl(m|2n)$. The map $\omega:\g g\to\g g$ given by $%
\omega(x)=-x$ is an anti-automorphism of $\g g$, that is, $$\omega([x,y])=(-1)^{|x|\cdot|y|}[\omega(y),\omega(x)]$$ for $x,y\in\g g$. Therefore $\omega$ extends canonically to an anti-automorphism $\omega:\bfU(%
\g g)\to\bfU(\g g)$, that is, $$\omega(xy)=(-1)^{|x|\cdot|y|}\omega(y)\omega(x)
\ \text{ for }x,y\in\bfU(\g g).$$
Recall that $\HC^+:
\bfU(\g g)\to \bfU
(\g a)$ is the Harish–Chandra projection as in . Let $$\HC^-:\bfU(\g g)\to \bfU%
(\g a)$$ be the opposite Harish–Chandra projection defined by the composition $$\bfU(\g g) \xrightarrow{\ \psf^-\ }\bfU(\g h) \xrightarrow{\ \sfq\ }\bfU(\g %
a)$$ where where $\psf^-:\bfU(\g g)\to\bfU(\g h)$ is the projection according to the decomposition $$\bfU(\g g)=\left(\bfU(\g g)\g n^- +\g n^+\bfU(\g g)\right)\oplus\bfU(\g h).$$ It is straightforward to verify that $$\label{HC+HC-p}
\HC^-(\omega(z))=\omega(\HC^+(z))\text{ for }z\in\bfU(\g g).$$
\[Dfndualmu\] For every $\mu\in\mathrm{E}_{m,n}^*$, we use $\mu^*$ to denote the unique element of $\mathrm{E}_{m,n}$ that satisfies $V_\mu^*\cong V_{\mu^*}^{}$.
Using formulas and , one can see that the map $\mu\mapsto \mu^*$ is not linear. Nevertheless, the following proposition still holds.
\[prpQlam\] Let $\lambda\in\mathrm{E}_{m,n,d}^*$. Then there exists a unique polynomial $c_\lambda^*\in\sP(\g a^*)$ such that $\deg(c_\lambda^*)\leq d$ and $
c_\lambda(\mu)=c_\lambda^*(\mu^*)
$ for every $\mu\in\mathrm{E}_{m,n}^*$.
First we prove the existence of $c^*_\lambda$. Recall that the action of $D_\lambda:V_\mu\to V_\mu$ is by the scalar $c_\lambda(\mu)$. Let $v_{\mu^*}$ denote the highest weight of $V_{\mu^*}$, and define $v_\mu^-\in (V_{\mu^*})^*\cong V_\mu$ by $\lag v_\mu^-,v_\mu^{*}\rag =1$ and $\lag v_\mu^-,\bfU(\g n^-)v_{\mu^*}\rag=0$. It is straightforward to verify that $v_\mu^-\in (V_\mu)^{\g n^-}$, and thus $v_{\mu}^-$ is the lowest weight vector of $V_\mu^{}$. It follows immediately that the lowest weight of $V_\mu^{}$ is $-\mu^*$. Choose $z_\lambda\in\bfZ(\g g)$ as in . By considering the $\g h$-action on $\bfU(\g g)$ we obtain $
z_\lambda-\psf^-(z_\lambda)\in\bfU(\g g)\g n^-
$, and therefore $$c_\lambda(\mu)v_\mu^-=D_\lambda v_\mu^-=\check\rho(z_\lambda)v_\mu^-
=\check\rho(\psf^-(z_\lambda))v_\mu^-=
\sfh_{-\mu^*}(\HC^-(z_\lambda))v_\mu^-,$$ where $\sfh_{-\mu^*}:\bfU(\g a)\cong\sS(\g a)\to\C$ is defined as in . It is straightforward to check that the map $$\label{eq-dfQlamb}
c_\lambda^*(\nu):=
\sfh_{-\nu}(\HC^-(z_\lambda))$$ is a polynomial in $\nu\in\g a^*$. From and Theorem \[prpgw\] it follows that $\deg(c^*_\lambda)\leq d$. Finally, uniqueness of $c^*_\lambda$ follows from the fact that $\mathrm{E}_{m,n}$ is Zariski dense in $\g a^*$.
We now recall the definition of the algebra of *shifted supersymmetric polynomials* $$\Lambda_{m,n,\frac12}^\natural\subseteq\sP(\C^{m+n}),$$ introduced in [@SerVes Sec. 6]. For $1\leq k\leq r$, let $\mathsf{e}_{k,r}$ be the $k$-th unit vector in $\C^r$. Then the algebra $\Lambda_{m,n,\frac12}^\natural$ consists of polynomials $f(\sfx_1,\ldots,\sfx_m,\sfy_1,\ldots,\sfy_n)$, which are separately symmetric in $\sfx:=(\sfx_1,\ldots,\sfx_m)$ and in $\sfy:=(\sfy_1,\ldots,\sfy_n)$, and which satisfy the relation $$\textstyle f(\sfx+\frac{1}{2}\mathsf{e}_{k,m},\sfy-\frac{1}{2}\mathsf{e}_{l,n})=f(\sfx-\frac{1}{2}\mathsf{e}_{k,m},\sfy+\frac{1}{2}\mathsf{e}_{l,n})$$ on every hyperplane $\sfx_k+\frac12 \sfy_l=0$, where $1\leq k\leq m$ and $1\leq
l\leq n$.
In [@SerVes Sec. 6], Sergeev and Veselov introduce a basis of $\Lambda_{m,n,\frac12}^\natural$, $$\label{SPbinLL}
\left\{
SP_\flat^*\in\Lambda_{m,n,\frac12}^\natural\ :\
\flat\in\mathrm{H}_{m,n}\right\},$$ indexed by the set $\mathrm{H}_{m,n}$ of $(m|n)$-hook partitions (see Definition \[dfnhookprn\]). The polynomials $SP_\flat^*$ are called *shifted super Jack polynomials* and they satisfy certain vanishing conditions, given in [@SerVes Eq. (31)].
Recall the map $\boldsymbol\Gamma:\mathrm{H}_{m,n}\to \g a^*$ from . Given $\flat=(\flat_1,\flat_2,\flat_3,\ldots)\in\mathrm{H}_{m,n}$, we set $%
\flat_k^*:=\max\{\flat_k^{\prime }-m,0\}$ for every $k\geq 1$. Fix $\mu\in\mathrm{E}_{m,n}^*$, and let $\mu^*\in \mathrm{E}_{m,n}$ be as in Definition \[Dfndualmu\]. From the definition of the map $\boldsymbol{\Gamma}$ it follows that there exists a $\flat\in\mathrm{H}_{m,n}$ such that $$\label{fordfGAM}
\mu^*=\boldsymbol{\Gamma}(\flat)=
\sum_{i=1}^m2\flat_i\gamma_i+\sum_{j=1}^n2\flat_j^* \gamma_{\oline j}.$$ Now let $\mathsf{F}:\mathrm{H}_{m,n}\to \C^{m+n}$ be the *Frobenius map* of \[Sec. 6\][SerVes]{}, defined as follows. For every $\flat\in\mathrm{H}_{m,n}$, we have $\mathsf{F}(\flat)=\left(\sfx_1(\flat),\ldots,\sfx_m(\flat),\sfy_1(\flat),\ldots,\sfy_n(\flat)\right)$, where
$$\label{xkfykfl}
\begin{cases}
\sfx_k(\flat):=\flat_k-\frac12(k-\frac12)
-\frac12(2n-\frac m2) & \text{ for }1\leq k\leq m, \\
\sfy_l(\flat):=\flat_l^*-2(l-\frac12)+\frac12(4n+m) & \text{ for }1\leq l\leq n.%
\end{cases}$$
\[lem6..3\] The map $\boldsymbol\Gamma\circ\mathsf{F}^{-1}$ defined initially on $\mathsf{F}(\mathrm{H}_{m,n})$ extends uniquely to an affine linear map $\Psi:\C^{m+n}\to \g a^*$.
This is immediate from the formulas and , and the Zariski density of $\mathsf{F}(\mathrm{H}_{m,n})$ in $\C^{m+n}$.
\[FrobT\] For $p\in\sP(\g a^*)$, we define its *Frobenius transform* to be $$\mathscr{F}(p):=p\circ\Psi.$$
Note that $$\label{degpdegFpeq}
\deg(p)=\deg(\mathscr{F}(p))
\text{ for every }p\in\sP(\g a^*).$$ We now relate the shifted super Jack polynomials $SP_\flat^*$ to the dualized eigenvalue polynomials $c_\lambda^*$ from Proposition \[prpQlam\].
\[thmconnSV\] Let $\lambda\in\mathrm{E}_{m,n,d}^*$ and let $\flat\in\mathrm{H}_{m,n,d}$ be such that $\lambda^*=\boldsymbol\Gamma(\flat)$. Then we have $$\label{Ql=GGSP*}
\mathscr{F}(c_\lambda^*)=
\frac{d!}{H(\flat)}SP^*_\flat,$$ where $
H(\flat)
:=\prod_{k\geq 1}\prod_{l=1}^{\flat_k}
\left(
\flat_k-l+1+\frac12(\flat_l'-k)
\right)
$.
By , Proposition \[prpQlam\], and the results of [@SerVes Sec. 6], both sides of are in $\bigoplus_{d'=0}^d\sP^{d'}(\C^{m+n})$. Next we prove that $$\label{FQinL}
\mathscr{F}(c^*_\lambda)
\in
\Lambda^\natural_{m,n,\frac12}\,.$$ Note that $\bfZ(\g g)$ is invariant under the anti-automorphism $\omega$. Let $z_\lambda\in\bfZ(\g g)\cap\bfU^d(\g g)$ be as in . From we obtain $\HC^-(z_\lambda)=\omega(\HC^+(\omega(z_\lambda)))$, and thus $\HC^-(z_\lambda)$ is in the subalgebra of $\bfU(\g a)\cong\sS(\g a)$ that is generated by $\omega(\sfq(G_d))$ for $d\geq 1$, where $G_d$ is given in . Observe that $\sfq(G_d)$ is obtained from $G_d$ by the substitutions $$E_{k,k}\mapsto h_k\text{ for }1\leq k\leq m
\ \text{ and }\
E_{\oline{2l-1},\oline{2l-1}},E_{\oline{2l},\oline{2l}}\mapsto \frac{1}{2}h_{\oline l}
\text{ for }
1\leq l\leq n,$$ where $\{h_i\,:\,i\in\mathcal{I}_{m,n}\}$ is the basis for $\g a$ that is dual to the $\g a^*$-basis $\{\gamma_i\,:\,i\in\mathcal{I}_{m,n}\}$, defined in . By a straightforward calculation we can verify that $\mathscr{F}(c^*_\lambda)$ is in the algebra generated by the polynomials $$\label{2x+ny-n}
\sum_{k=1}^m(2\sfx_k+n)^d+
(-1)^{d-1}\sum_{l=1}^n\left(\sfy_l-n+\frac12\right)^d+\left(\sfy_l-n-\frac12\right)^d$$ for every $d\geq 1$. Furthermore, the polynomials belong to $\Lambda_{m,n,\frac12}^\natural$. This completes the proof of .
Next we fix $\mu\in\mathrm{E}_{m,n}^*$ and choose $\flat^{(\mu)}\in \mathrm{H}_{m,n}$ such that $\mu^*=\boldsymbol\Gamma(\flat^{(\mu)})$, where $\mu^*\in\mathrm{E}_{m,n}$ is as in Definition \[Dfndualmu\]. Then $$\textstyle
\mathscr{F}(c^*_\lambda)
\left(\mathsf{F}\left(\flat^{(\mu)}\right)\right)
=
c^*_\lambda(\mu^*)=c_\lambda(\mu)
.$$ From [@SerVes Eq. (31)] we have $SP^*_\flat\left(\mathsf{F}\left(\flat^{(\mu)}\right)
\right)=H\left(\flat^{(\mu)}\right)$ for $\mu=\lambda$, and $SP_\flat^*\left(\mathsf{F}\left(\flat^{(\mu)}\right)\right)=0$ for all other $\mu\in\bigcup_{d'=0}^d\mathrm{E}_{m,n,d'}^*$. The equality now follows from the latter vanishing property and Theorem \[thm-unqclam\], or alternatively from the discussion immediately above [@SerVes Eq. (31)].
**Appendix**
Proof of Proposition \[DGBRVVV\] {#sec-pflem}
================================
In this appendix we prove Proposition \[DGBRVVV\]. The proof of this proposition is similar to the proof of Proposition \[le-dlam\], although somewhat more elaborate. Recall the generators $\{x_{i,j}\}_{i,j\in\mathcal I_{m,2n}}$ for $\sS(W)$ and $\{y_{i,j}\}_{i,j\in\mathcal I_{m,2n}}$ for $\sP(W)$, defined in . We consider the total ordering $\prec$ on $\mathcal I_{m,2n}$ given by $$1\prec\cdots\prec m\prec \oline 1\prec\cdots\prec \oline{2n}.$$ Set $
\mathcal I':=\left\{\big(\oline{2k-1},
\oline{2k}\big)\,:\,1\leq k\leq n\right\}
$ and $\mathcal I'':=
\left\{
(i,j)\in\mathcal I_{m,2n}\times \mathcal I_{m,2n}\ ,\
i\prec j
\right\}
\setminus
\mathcal I'
$. For every $a_1,\ldots,a_m,a_{\oline{1}},\ldots,a_{\oline{n}}\in\C$, we set $$\label{dfbfx}
\bfx:=-\sum_{k=1}^m a_k h_k+\sum_{l=1}^n a_{\oline{l}}h_{\oline{l}}
\in\g a,$$ where $\{h_i\,:\,i\in\mathcal I_{m,n}\}$ is the basis of $\g a$ that is dual to $\{\gamma_i\,:\,i\in\mathcal I_{m,n}\}$ defined in . Let $
\sfh_{\bfx}:\sP(\g a)\cong\sS(\g a^*)\to \C
$ be as defined in . For any $\bfx\in\g a$ as above, set $$\xi_{\bfx}:=
\frac12\sum_{k=1}^ma_kx_{k,k}+\sum_{l=1}^n a_{\oline l}x_{\oline{2l-1},\oline{2l}}\in W,$$ and let $
\sfh_{\xi_{\bfx}}:\sP(W)\cong\sS(W^*)\to \C
$ be as defined in . Observe that $\sfh_{\bfx}\circ\iota_\g a^*=\sfh_{\xi_\bfx}$ where $\iota_\g a^*$ is as defined in . In particular, for every $a\in\sP(W)$, the set $$S_a:=\left\{\bfx\in\g a\ :\ \sfh_{\xi_{\bfx}}(a)=0\right\}$$ is Zariski closed in $\g a$.
Let $\partial_{i,j}:=\partial_{x_{i,j}}$ denote the superderivation of $\sP(W)$ that is defined according to .
Let $\mathbf d\in\sP(W)^\g k$ such that $\sfh_{\xi_{\bfx}}(\mathbf d)= 0$ for every $\bfx\in\g a$. Then $$\label{hvcthht}
\sfh_{\xi_{\bfx}}
\big(
\partial_{{i_1,j_1}}\cdots\partial_{{i_s,j_s}}
\mathbf d\big)=0\
\text{ for all }
\bfx\in\g a,\ s\geq 1,\
\text{and }(i_1,j_1),\ldots,(i_s,j_s)\in\mathcal I''.$$
We use induction on $s$. First assume that $s=1$. Then $$\label{chrRsph}
\check\rho(x)\mathbf d\text{ for all }x\in\g k.$$ Set $x:=E_{k,l}-E_{l,k}$ for $1\leq k<l\leq m$. Then and imply that $$\begin{aligned}
0&=\sfh_{\xi_{\bfx}}(\check\rho(x)\mathbf d)\\
&=-\sum_{r\in\mathcal I_{m,2n}}
(-1)^{|r|}\sfh_{\xi_{\bfx}}(y_{r,l})
\sfh_{\xi_{\bfx}}(\partial_{{r,k}}\mathbf d)
+
\sum_{r\in\mathcal I_{m,2n}}
(-1)^{|r|}\sfh_{\xi_{\bfx}}(y_{r,k})
\sfh_{\xi_{\bfx}}(\partial_{{r,l}}\mathbf d)\\
&=(-a_l+a_k)\sfh_{\xi_{\bfx}}(\partial_{{k,l}}\mathbf d),\end{aligned}$$ from which it follows that $\sfh_{\xi_{\bfx}}(\partial_{{k,l}}d_\lambda)=0$ for all $\bfx\in\g a$ as in which satisfy $a_k\neq a_l$. But the set of all $\bfx\in\g a$ given as in which satisfy $a_k\neq a_l$ for all $1\leq k<l\leq m$ is a Zariski dense subset of $\g a$, and it follows that $\sfh_{\xi_{\bfx}}(\partial_{x_{k,l}}\mathbf d)=0$ for every $\bfx\in\g a$. A similar argument for each of the cases (ii)–(vi) of Remark \[rmkbasisK\] (where in cases (ii)–(iv) we assume $k\neq l$) proves for $s=1$.
Next we define an involution $i\mapsto i^\dagger$ on $\mathcal I_{m,2n}$ by $$k^\dagger:=k\text{ for }1\leq k\leq m,\
(\oline{2l-1})^\dagger:=\oline{2l}\text{ for }1\leq l\leq n,\text{ and }
(\oline{2l})^\dagger:=\oline{2l-1}\text{ for }1\leq l\leq n.$$ Let $\mathsf f:\mathcal I_{m,2n}\to\mathcal I_{m,n}$ be defined by $\mathsf f(k)=k$ for $1\leq k\leq m$ and $\mathsf f\big(\oline{2l-1}\big)
=\mathsf f\big(\oline{2l}\big)=\oline{l}$ for $1\leq l\leq n$. Fix an element $x\in\g k$ that belongs to the spanning set of $\g k$ given in Remark \[rmkbasisK\]. (When $x$ is chosen from one of the cases (ii)–(iv) in Remark \[rmkbasisK\], we assume that $k\neq l$.) Then there exist $p,q\in\mathcal I_{m,2n}$ such that $$\label{pprecqqpre}
p\prec q,\ p^\dagger\prec q^\dagger,\ \text{and }
\check\rho(x)=\sum_{r\in\mathcal I_{m,2n}}
\left(\pm y_{r,p}^{}\partial_{{r,q}^{}}
\pm y_{r,q^\dagger_{}}
\partial_{{r,p^\dagger}}\right).$$ Now fix $x_1,\ldots,x_{s+1}\in\g k$ such that every $x_k$, for $1\leq k\leq s+1$, is an element of the spanning set of $\g k$ given in Remark \[rmkbasisK\]. For every $1\leq k\leq s+1$, if $x_k$ is chosen from the cases (ii)–(iv) in Remark \[rmkbasisK\], then we assume that $k\neq l$. Choose $(p_1,q_1),\ldots,(p_{s+1},q_{s+1})\in\mathcal I_{m,2n}\times\mathcal I_{m,2n}$ corresponding to $x_1,\ldots,x_{s+1}$ which satisfy . For $1\leq u\leq s+1$, we define $$p_{u,S}:=\begin{cases}
p_u& \text{ if }u\in S,\\
(q_u)^\dagger&\text{ if }u\not\in S,
\end{cases}
\ \ \text{ and }\ \
q_{u,S}:=\begin{cases}
q_u& \text{ if }u\in S,\\
(p_u)^\dagger&\text{ if }u\not\in S.
\end{cases}$$ Then $$\begin{aligned}
\label{x1xs+1s}
\check\rho&(x_1)\cdots\check\rho(x_{s+1})\mathbf d
\\
&=\sum_{S\subseteq \{1,\ldots,s+1\}}\
\sum_{r_1,\ldots,r_{s+1}\in\mathcal I_{m,2n}}
\left(\pm y_{r_1,p_{1,S}}^{}\cdots y_{r_{s+1},p_{s+1,S}}^{}
\partial_{{r_1,q_{1,S}}^{}}^{}\cdots
\partial_{{r_{s+1},q_{s+1,S}}^{}}(\mathbf d)\right)+R_{\mathbf d},
\notag\end{aligned}$$ where $R_{\mathbf d}$ is a sum of terms of the form $b\partial_{p_1',q_1'}\cdots \partial_{p_{t}', q_{t}'}(\mathbf d)$, with $b\in\sP(W)$ and $t\leq s$. By the induction hypothesis, $\sfh_{\xi_\bfx}(R_{\mathbf d})=0$ for every $\bfx\in\g a$. Thus implies that $$\begin{aligned}
\label{Ssub1s++1}
0&=\sfh_{\xi_\bfx}
(\check\rho(x_1)\cdots\check\rho(x_{s+1})\mathbf d)\\
&=
\sum_{S\subseteq \{1,\ldots,s+1\}}
\pm
a_{\mathsf f(p_{1,S}^{})}^{}\cdots
a_{\mathsf f(p_{s+1,S}^{})}^{}
\sfh_{\xi_\bfx}\left(
\partial_{{(p_1)^\dagger,q_1^{}}}
\cdots
\partial_{{(p_{s+1})^\dagger,q_{s+1}^{}}}
\mathbf d\right).
\notag\end{aligned}$$ From it follows that $p_u\prec (q_u)^\dagger$ for every $1\leq u\leq s+1$. the monomial $$a_{\mathsf f(p_{1}^{})}^{}\cdots
a_{\mathsf f(p_{s+1}^{})}^{}$$ appears in exactly once. Therefore the right hand side of can be expressed as $$\psi(\bfx)
\sfh_{\xi_\bfx}\left(
\partial_{{(p_1)^\dagger,q_1^{}}}
\cdots
\partial_{{(p_{s+1})^\dagger,q_{s+1}^{}}}
\mathbf d\right),$$ where $\psi\in\sP(\g a)$ is a nonzero polynomial. It follows that the set consisting of all $\bfx\in\g a$ which satisfy $\psi(\bfx)\neq 0$ is a Zariski dense subset of $\g a$. Consequently, the set $$\left\{
\bfx\in\C^{m+n}\ :\
\sfh_{\xi_\bfx}\left(
\partial_{{(p_1)^\dagger,q_1^{}}}
\cdots
\partial_{{(p_{s+1})^\dagger,q_{s+1}^{}}}
\mathbf d\right)=0
\right\}$$ is both Zariski dense and Zariski closed. This completes the proof of .
We are now ready to prove Proposition \[DGBRVVV\].
By the equality $\sfh_{\bfx}\circ\iota_\g a^*=\sfh_{\xi_\bfx}$, it suffices to show that for every $\mathbf d\in\sP(W)^\g k$, if $\sfh_{\xi_{\bfx}}(\mathbf d)= 0$ for every $\bfx\in\g a$, then $\mathbf d=0$. Let $\mathscr D\sseq\sP(W)$ be the subalgebra generated by $$\Big\{y_{k,k}\,:\,1\leq k\leq m
\Big
\}
\cup
\Big\{y_{\oline{2l-1},\oline{2l}}\,:\,1\leq l\leq n
\Big\}.$$ Then $
\mathbf d=\sum_{S\subseteq\mathcal I''}a_S^{}y_S^{}
$, where $y_S^{}:=\prod_{(i,j)\in S}y_{i,j}$ and $a_S^{}\in\mathscr D$ for every $S\subseteq \mathcal I''$. Now we fix $S\subseteq \mathcal I''$ and set $\widetilde\partial:=\prod_{(i,j)\in S}\partial_{{i,j}}
$, so that $\widetilde\partial(\mathbf d)=za_S^{}$ for some scalar $z\neq 0$. By , $$\label{hthetAS}
\sfh_{\xi_\bfx}(a_S^{})=
\frac{1}{z}\sfh_{\xi_\bfx}
\left(\widetilde\partial(\mathbf d)\right)=0\text{ for every }
\bfx\in\g a.$$ From it follows that $a_S^{}=0$. Since $S\subseteq \mathcal I''$ is arbitrary, we obtain $\mathbf d=0$.
The Capelli problem for $\g{gl}(V)\times\g{gl}(V)$ acting on $V\otimes V^*$ {#appxB}
===========================================================================
In this appendix, we show that the main results of Sections \[prfof571\]–\[SecRelSer\], including the abstract Capelli theorem and the relation between the eigenvalue polynomials $c_\lambda$ and the shifted super Jack polynomials of [@SerVes Eq. (31)], extend to the case of $\gl(V)\times \gl(V)$ acting on $W:=V\otimes V^*$, where $V:=\C^{m|n}$. These extensions can be proved along the same lines. However, they can also be deduced from the results of [@Molev], and we sketch the necessary arguments below.
Recall the triangular decomposition $\gl(m|n)=\g n^-\oplus\g h\oplus\g n^+$ from Section \[prfof571\]. Set $$\label{canoglgl}
\g g:=\gl(V)\times\gl(V)\cong \gl(m|n)\times \gl(m|n).$$ Recall that $\mathrm{H}_{m,n}$ is the set of $(m,n)$-hook partitions, as in Definition \[dfnhookprn\]. We define a map $\boldsymbol\Gamma:\mathrm{H}_{m,n}\to\g h^*$ by $\boldsymbol\Gamma(\flat):=
\sum_{k=1}^m\flat_k\eps_k+\sum_{l=1}^n
\flat_{ l}^*\eps_{\oline l}
$, where $\flat_l^*:=\max\{\flat'_l-m,0\}$. Set $\breve{\mathrm{E}}_{m,n,d}:=\boldsymbol\Gamma(\mathrm{H}_{m,n,d})$, and let $
\breve{\mathrm{E}}_{m,n}
:=\bigcup_{d=0}^\infty
\breve{\mathrm{E}}_{m,n,d}$. For every $\flat\in{\mathrm{H}}_{m,n}$, let $s_\flat^{}$ denote the *supersymmetric Schur polynomial* defined in [@Molev Eq. (0.2)], and let $s_\flat^*$ denote the *shifted supersymmetric Schur polynomial* defined in [@Molev Sec. 7].
We can decompose $\sP(W)$ and $\sS(W)$ into irreducible $\g g$-modules, that is, $$\label{adv-PW}
\sP^d(W)\cong
\bigoplus_{\mu\in\breve{\mathrm{E}}_{m,n,d}}
V_\mu^*\otimes V_\mu^{}
\,\text{ and }\,
\sS^d(W)\cong
\bigoplus_{
\mu\in\breve{\mathrm{E}}_{m,n,d}}
V_\mu^{}\otimes V_\mu^{*},$$ where $V_\mu$ is the irreducible $\gl(V)$-module of highest weight $\mu$, and $V_\mu^*$ is the contragredient of $V_\mu$. An argument similar to the proof of Lemma \[prp-g-inv-sPWW\], and based on a decomposition similar to , implies that $$\sPD^d(W)^\g g
\cong
\bigoplus_{k=0}^d
\left(\sP^k(W)\otimes \sS^k(W)\right)^\g g
\cong
\bigoplus_{k=0}^d\,\bigoplus_{\lambda\in \breve{\mathrm{E}}_{m,n,k}}\C D_\lambda,$$ where $D_\lambda$ is the $\g g$-invariant differential operator that corresponds to the identity map in $\Hom_{\gl(V)\times\gl(V)}^{}(V_\lambda^*\otimes V_\lambda^{},V_\lambda^{*}\otimes V_\lambda^{})$. As in , the $\g g$-action on $\sP(W)$ can be realized by polarization operators, and therefore we obtain a Lie superalgebra homomorphism $\g g\to\sPD(W)$. The latter map extends to a homomorphism of associative superalgebras $$\label{rhocheckII}
\check\rho:\bfU(\g g)\to \sPD(W).$$ There exists a canonical tensor product decomposition $
\bfU(\g g)\cong\bfU(\gl(V))\otimes \bfU(\gl(V))
$ corresponding to . Set $
\bfZ^d(\gl(V)):=\bfZ(\gl(V))\cap\bfU^d(\gl(V))
$ for every integer $d\geq 0$. The next theorem extends Theorem \[prpgw\].
\[thabsscap\] *(Abstract Capelli Theorem for $W:=V\otimes V^*$.)* The restrictions $$\bfZ^d(\gl(V))\otimes 1\to\sPD^d(W)^{\g g}
\text
{ and }
1\otimes\bfZ^d(\gl(V))\to\sPD^d(W)^{\g g}$$ of the map $\check\rho$ given in are surjective.
The statement is a consequence of the results of [@Molev]. In [@Molev Thm 7.5], a family of elements $\mathbb S_\flat\in\bfZ^{|\flat|}(\gl(V))$ is constructed which is parametrized by partitions $\flat\in\mathrm{H}_{m,n}$. Furthermore, in [@Molev Thm 8.1] it is proved that the map $\bfZ^{|\flat|}(\gl(V))\otimes 1\to\sPD^{|\flat|}(W)^\g g$ takes $\mathbb S_\flat$ to a differential operator $\Delta_\flat\in\sPD^{|\flat|}(W)^{\g g}$, which is defined in [@Molev Sec. 8]. To complete the proof of surjectivity of the map $\bfZ^d(\gl(V))\otimes 1\to\sPD^d(W)^{\g g}$, it is enough to show that for every $d\geq 0$, the sets $$\big\{\Delta_\flat\,:\,\flat\in\mathrm{H}_{m,n,k}, 0\leq k\leq d\big\}\text{ and }
\big\{D_\lambda\,:\,\lambda\in\breve{\mathrm{E}}_{m,n,k},0\leq k\leq d\big\}$$ span the same subspace of $\sPD^d(W)^\g g$. Since the above two sets have an equal number of elements, it is enough to show that the elements of $\big\{\Delta_\flat\,:\,\flat\in\mathrm{H}_{m,n,k}, 0\leq k\leq d\big\}$ are linearly independent. To prove the latter statement, we note that the spectrum of $\Delta_\flat$ can be expressed in terms of the Harish-Chandra image of $\mathbb S_\flat$, which by [@Molev Thm 7.5] is equal to the shifted supersymmetric Schur polynomial $s_\flat^*$. Since the $s_\flat^*$ are linearly independent (see [@Molev Cor. 7.2] and subsequent remarks therein), the operators $\Delta_\flat$ are also linearly independent.
For $\mu\in\breve{\mathrm{E}}_{m,n}$, set $W_\mu:=V_\mu^*\otimes V_\mu$. The operator $D_\lambda$ acts on $W_\mu$ by a scalar $c_\lambda(\mu)\in\C$. For $\flat\in\mathrm{H}_{m,n}$, let $\breve{H}(\flat)$ denote the product of the hook lengths of all of the boxes in the Young diagram representation of $\flat$. We define a map $$\breve{\boldsymbol\Gamma}_\circ:
\breve{\mathrm{E}}_{m,n}\to
\C^{m+n}
\ \,,\ \,
\sum_{k=1}^m\flat_k\eps_k+\sum_{l=1}^n\flat_l^*\eps_{\oline l}\mapsto
(\flat_1,\ldots,\flat_m,\flat_1^*,\ldots,\flat_n^*).$$ As in Section \[Sec-Sec5\], we denote the homogeneous part of highest degree of any $p\in\sP(\g h^*)$ by $\oline p$.
\[THMAppB2\] Let $\lambda\in\breve{\mathrm{E}}_{m,n,d}$ and let $\flat\in\mathrm{H}_{m,n,d}$ be such that $\boldsymbol\Gamma(\flat)=\lambda$. Then $$c_\lambda=\frac{d!}{\breve{H}(\flat)}s_\flat^*
\circ\breve{\boldsymbol\Gamma}_\circ
.$$ In particular, $\oline c_\lambda=
\frac{d!}{\breve{H}(\flat)}
s_\flat^{}\circ \breve{\boldsymbol\Gamma}_\circ$.
Follows from [@Molev Thm 7.3] and [@Molev Thm 7.5].
The Frobenius map $$\mathsf{F}:\mathrm{H}_{m,n}\to \C^{m+n}$$ of [@SerVes Sec. 6] is given by $\mathsf{F}(\flat):=(\sfx(\flat),\ldots,\sfx_m(\flat),\sfy_1(\flat),\cdots,\sfy_n(\flat))$, where $$\begin{cases}
\sfx_k(\flat):=\flat_k-k+\frac12(1-n+m)
& \text{ for }1\leq k\leq m, \\
\sfy_l(\flat):=\flat_l^*-l+\frac12(m+n+1) & \text{ for }1\leq l\leq n.%
\end{cases}$$ As in Lemma \[lem6..3\], the map $\boldsymbol\Gamma\circ\mathsf{F}^{-1}$ extends to an affine linear map $\Psi:\C^{m+n}\to \g h^*$. Thus we can define the Frobenius transform $$\mathscr{F}:\sP(\g h^*)\to\sP(\C^{m+n})$$ as in Definition \[FrobT\], namely by $\mathscr{F}(p):=p\circ\Psi$.
For $\lambda\in\breve{\mathrm{E}}_{m,n,d}$, by Theorem \[thabsscap\] there exists an element $z_\lambda\in \bfZ(\gl(V))\cap\bfU^d(\gl(V))$ such that $D_\lambda=\check\rho(1\otimes z_\lambda)$. Then for a highest weight vector $v_\mu^*\otimes v_\mu^{}\in V_\mu^*\otimes V_\mu^{}\cong W_\mu$, we have $$c_\lambda(\mu)v_\mu^*\otimes v_\mu^{}
=
D_\lambda
v_\mu^*\otimes v_\mu^{}
=
v_\mu^*\otimes\check\rho(z_\lambda)v_\mu^{}
=\mu(\HC^+(z_\lambda))v_\mu^*\otimes v_\mu^{},$$ where $\HC^+:\bfU(\gl(V))\to \bfU(\g h)$ is the Harish-Chandra projection corresponding to the decomposition $
\bfU(\gl(V))=
\left(\bfU(\gl(V))
\g n^+
+
\g n^-\bfU(\gl(V))
\right)
\oplus\bfU(\g h)
$. From the description of the image of the Harish-Chandra projection (see for example [@sergeev82], [@sergeev3], [@KacZ], [@Gorelik], or [@ChWabook Sec. 2.2.3]), we obtain explicit generators for $\HC^+(\bfZ(\gl(V))$, similar to the $G_d$ defined in . Furthermore, by the canonical isomorphism $\bfU(\g h)\cong\sS(\g h)\cong \sP(\g h^*)$, we can consider $\HC^+(\bfZ(\gl(V))$ as a subalgebra of $\sP(\g h^*)$, which we henceforth denote by $\mathbf{I}(\g h^*)$. A direct calculation using the explicit generators of $\mathbf{I}(\g h^*)$ proves that $\mathscr{F}(\mathbf I(\g h^*))\subseteq\Lambda^\natural_{m,n,1}$, where $\Lambda^\natural_{m,n,1}$ consists of polynomials $f(\sfx_1,\ldots,\sfx_m,\sfy_1,\ldots,\sfy_n)$, which are separately symmetric in $\sfx:=(\sfx_1,\ldots,\sfx_m)$ and in $\sfy:=(\sfy_1,\ldots,\sfy_n)$, and which satisfy the relation $$\textstyle f(\sfx+\frac{1}{2}\mathsf{e}_{k,m},\sfy-\frac{1}{2}\mathsf{e}_{l,n})=f(\sfx-\frac{1}{2}\mathsf{e}_{k,m},\sfy+\frac{1}{2}\mathsf{e}_{l,n})$$ on every hyperplane $\sfx_k+ \sfy_l=0$, where $1\leq k\leq m$ and $1\leq
l\leq n$. Now let $SP_\flat^*$ be the basis of $\Lambda^\natural_{m,n,1}$ introduced in [@SerVes Sec. 6] for $\theta=1$. An argument similar to the proof of Theorem \[thmconnSV\] yields the following statement.
\[THMAB3\] Let $\lambda\in\breve{\mathrm{E}}_{m,n,d}$, and let $\flat\in\mathrm{H}_{m,n,d}$ be chosen such that $\lambda=\boldsymbol\Gamma(\flat)$, then $$\mathscr{F}(c_\lambda)=\frac{d!}{\breve{H}(\flat)}SP_\flat^*.$$
[99]{} Alldridge, A.; Schmittner, S., *Spherical representations of Lie supergroups*. J. Funct. Anal. 268 (2015), no. 6, 1403–1453.
Berele, A.; Regev, A., * Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras*. Adv. Math. 64 (1987), no. 2, 118–175.
Brini, A.; Huang, R. Q.; Teolis, A. G. B.,*The umbral symbolic method for supersymmetric tensors*. Adv. Math. 96 (1992), 123–193.
Cheng, S.-J.; Wang, W., *Howe duality for Lie superalgebras*. Compositio Math. 128 (2001), 55–94.
Cheng, S.-J.; Wang, W., *Dualities and representations of Lie superalgebras*, Graduate Studies in Mathematics, 144. American Mathematical Society, Providence, RI, 2012. xviii+302 pp.
Goodman, R.; Wallach, N. R., *Symmetry, representations, and invariants*. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009. xx+716 pp.
Gorelik, M., *The Kac construction of the centre of U(g) for Lie superalgebras*. J. Nonlinear Math. Phys. 11 (2004), no. 3, 325–349.
Howe, R., *Remarks on classical invariant theory*. Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570.
Howe, R.; Umeda, T., *The Capelli identity, the double commutant theorem, and multiplicity-free actions*. Math. Ann. 290 (1991), no. 3, 565–619.
Jacobson, N., *Basic algebra II.* Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp.
Kac, V. G. *Laplace operators of infinite-dimensional Lie algebras and theta functions*. Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 2, Phys. Sci., 645–647.
Kostant, B.; Sahi, S., *Jordan algebras and Capelli identities*. Inv. Math. 112 (1993), no. 3, 657–664.
Knop, F., *Some remarks on multiplicity free spaces*. Representation theories and algebraic geometry (Montreal, PQ, 1997), 301–317, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Kluwer Acad. Publ., Dordrecht, 1998.
Knop, F.; Sahi, S., *Difference equations and symmetric polynomials defined by their zeros*. Internat. Math. Res. Notices (1996), no. 10, 473–486.
Kostant, B.; Sahi, S., *The Capelli identity, tube domains, and the generalized Laplace transform*. Adv. Math. 87 (1991), no. 1, 71–92.
Molev, A., *Factorial supersymmetric Schur functions and super Capelli identities*. Kirillov’s seminar on representation theory, 109–137, Amer. Math. Soc. Transl. Ser. 2, 181, Amer. Math. Soc., Providence, RI, 1998.
Musson, I., *Lie superalgebras and enveloping algebras*. Graduate Studies in Mathematics, 131. American Mathematical Society, Providence, RI, 2012. xx+488 pp.
Okounkov, A.; Olshanski, G., *Shifted Jack polynomials, binomial formula, and applications*. Math. Res. Lett. 4 (1997), no. 1, 69–78.
Olshanskii, G. I.; Prati, M. C., *Extremal weights of finite-dimensional representations of the Lie superalgebra $\gl_{n|m}$*. Nuovo Cimento A (11) 85 (1985), no. 1, 1–18.
Sahi, S., *The spectrum of certain invariant differential operators associated to a Hermitian symmetric space*. Lie theory and geometry, 569–576, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994.
Scheunert, M., *Eigenvalues of the Casimir operators for the general linear, the special linear, and the orthosymplectic Lie superalgebras*. J. Math. Phys. 24 (1983), no. 11, 2681–2688.
Sergeev, A. N., *Invariant polynomial functions on Lie superalgebras*. C. R. Acad. Bulgare Sci. (1982), no. 5, 573–576.
Sergeev, A. N., *Tensor algebra of the identity representation as a module over the Lie superalgebras $GL(n,m)$ and $Q(n)$*. (Russian) Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430.
Sergeev, A. N., *The invariant polynomials on simple Lie superalgebras*. Represent. Theory 3 (1999), 250–280.
Sergeev, A. N.; Veselov, A. P., * Generalised discriminants, deformed Calogero–Moser–Sutherland operators and super-Jack polynomials*. Adv. Math. 192 (2005) 341–375.
Weyl, H., *The Classical Groups. Their Invariants and Representations*. Princeton University Press, Princeton, N.J., 1939. xii+302 pp.
Želobenko, D. P., *Compact Lie groups and their representations*. Translated from the Russian by Israel Program for Scientific Translations. Translations of Mathematical Monographs, Vol. 40. American Mathematical Society, Providence, R.I., 1973. viii+448 pp.
[^1]: Department of Mathematics, Rutgers University, `sahi@math.rutgers.edu`.
[^2]: Department of Mathematics and Statistics, University of Ottawa, `hsalmasi@uottawa.ca`.
|
---
abstract: |
We obtain a presentation of the quantum Schur algebras ${S_v(2,d)}$ by generators and relations. This presentation is compatible with the usual presentation of the quantized enveloping algebra ${\mathbf{U}}=
\mathbf{U}_v(\mathfrak{gl}_2)$. In the process we find new bases for ${S_v(2,d)}$. We also locate the ${{\mathbb Z}}[v,v^{-1}]$-form of the quantum Schur algebra within the presented algebra and show that it has a basis which is closely related to Lusztig’s basis of the ${{\mathbb Z}}[v,v^{-1}]$-form of ${\mathbf{U}}$.
author:
- Stephen Doty and Anthony Giaquinto
date: 'December 1, 2000'
title: 'Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of ${\mathfrak {gl}}_2$'
---
Introduction {#AAA}
============
In [@DG] we gave a description of the rational Schur algebra $S_{{{\mathbb Q}}}(2,d)$ in terms of generators and relations. This description is compatible with the usual presentation of the universal enveloping algebra $U(\mathfrak{gl}_2)$. We also described the integral Schur algebra $S_{{{\mathbb Z}}}(2,d)$ as a certain subalgebra of the rational version and an integral basis was exhibited. In this paper, we formulate and prove quantum versions of those results.
Consider the Drinfeld-Jimbo quantized enveloping algebra ${\mathbf{U}}$ corresponding to the Lie algebra ${{\mathfrak {gl}}}_2$. It has a natural two-dimensional module $E$, and thus the tensor product $E^{{\otimes}d}$ is also a module for ${\mathbf{U}}$. Let $$\rho_d:{\mathbf{U}}\rightarrow {\operatorname{End}}(E^{{\otimes}d})$$ be the corresponding representation. Amongst the various equivalent definitions of the quantum Schur algebra ${S_v(2,d)}$ is that it is precisely the image of the homomorphism $\rho_d$. (We shall not need to consider the other definitions in this work.) Since ${S_v(2,d)}$ is a homomorphic image of ${\mathbf{U}}$, it is natural to ask for an efficient generating set of ${\operatorname{Ker}}(\rho_d)$, thereby giving a presentation of ${S_v(2,d)}$.
In what follows, we obtain a precise answer to this question. Recall that ${\mathbf{U}}$ is generated by elements $e$, $f$, ${{K_1}}^{\pm 1}$, ${{K_2}}^{\pm
1}$ subject to various well-known relations (see section 3). Now in the representation $\rho_d$, it is easy to see that ${{K_1}}{{K_2}}=v^d$, so we may use this relation to eliminate ${{K_2}}$ (or ${{K_1}}$) from the generating set for ${S_v(2,d)}$. Having done this, then our main result is that the only additional relation needed to give the desired presentation of ${S_v(2,d)}$ is the minimal polynomial of ${{K_1}}$ in ${\operatorname{End}}(E^{{\otimes}d})$.
Set ${{\mathcal A}}={{\mathbb Z}}[v,v^{-1}]$. The algebra ${\mathbf{U}}$ contains a certain ${{\mathcal A}}$-subalgebra ${{\mathbf{U}}_{\mathcal{A}}}$ which may be viewed as a quantum version of the integral form $U_{{{\mathbb Z}}}(\mathfrak{gl}_2)$ of the classical enveloping algebra. The algebra ${{\mathbf{U}}_{\mathcal{A}}}$, originally constructed by Lusztig, is generated by the $v$-divided powers of $e$ and $f$ along with ${{K_1}}^{\pm 1}$ and ${{K_2}}^{\pm 1}$. It has an ${{\mathcal A}}$-basis which is a quantum analog of Kostant’s basis for the integral form of the classical enveloping algebra $U(\mathfrak{gl}_2)$. The image of the homomorphism $\rho_d$ upon restriction to ${{\mathbf{U}}_{\mathcal{A}}}$ gives us an “integral” Schur algebra ${S_{\mathcal{A}}(2,d)}$, which can be used to define a version over any ${{\mathcal A}}$-algebra. We show that the integral Schur algebra has a basis which is closely related to Lusztig’s basis of ${{\mathbf{U}}_{\mathcal{A}}}$.
Although the results of this paper are quantum versions of those appearing in [@DG], the techniques are somewhat different, since some of the arguments given in that paper did not quantize directly. In particular, the treatment here of the degree zero part (generated by the images of $K_1^{\pm 1}$ and $K_2^{\pm 1}$) of ${S_v(2,d)}$ is totally different. Specifically, we exhibit an idempotent basis of this subalgebra, whereas in [@DG] a PBW-type basis of the degree zero part was used. The idempotent basis is more amenable to computations and is precisely the kind of basis needed to handle the general Schur algebras $S(n,d)$ and their quantizations. Another difference between the results of this paper and the results of [@DG] is that here we do not obtain analogues of the “restricted PBW-basis”, although we believe such results should be true in the quantum case.
Finally, it is clear that analogous results hold in general, for any $n$ and $d$, although one cannot expect to obtain such precise reduction formulas in the general case as those given here and in [@DG]. The authors expect to treat the general case in a later paper.
Statement of results {#AA}
====================
The main results of this paper are contained in the following theorems. Let ${{\mathcal A}}={{\mathbb Z}}[v,v^{-1}]$ with fraction field ${{\mathbb Q}}(v)$. Each of the first three theorems gives a presentation of the quantum Schur algebra ${S_v(2,d)}$ in terms of generators and relations. The first result gives a presentation which is similar to that of ${{\mathbf{U}}}_v(\mathfrak{sl}_2)$.
{#AAa0}
Over ${{\mathbb Q}}(v)$, the quantum Schur algebra ${S_v(2,d)}$ is isomorphic to the algebra generated by $e$, $f$, $K^{\pm 1}$ subject to the relations: $$\begin{aligned}
& KK^{-1}=K^{-1}K=1,\label{AAa0:a}\\
&K eK^{- 1} =v^{2}e,\qquad
K fK^{- 1} =v^{- 2}f,\label{AAa0:b}\\
& ef-fe=\frac{K - K^{-1}}{v-{v^{-1}}},\label{AAa0:c}\\
& (K-v^d)(K-v^{d-2})\cdots (K-v^{-d+2})(K-v^{-d})=0.\label{AAa0:d}\end{aligned}$$
The next two results give presentations of ${S_v(2,d)}$ which are similar to that of ${{\mathbf{U}}}_v(\mathfrak{gl}_2)$.
{#AAa}
Over ${{\mathbb Q}}(v)$, the quantum Schur algebra ${S_v(2,d)}$ is isomorphic to the algebra generated by $e$, $f$, ${{K_1}}^{\pm 1}$ subject to the relations: $$\begin{aligned}
& {{K_1}}{{K_1}}^{-1}={{K_1}}^{-1}{{K_1}}=1,\label{AAa:a}\\
&{{K_1}}e{{K_1}}^{- 1} =ve,\qquad
{{K_1}}f{{K_1}}^{- 1} =v^{- 1}f\label{AAa:b}\\
& ef-fe=\frac{v^{-d}{{K_1}}^2 - v^{d}{{K_1}}^{-2}}{v-{v^{-1}}},\label{AAa:c}\\
& ({{K_1}}-1)({{K_1}}-v)({{K_1}}-v^2)\cdots ({{K_1}}-v^d)=0.\label{AAa:d}\end{aligned}$$
By a change of variable ($K_2=v^{d}{K_{1}^{-1}}$) we obtain another equivalent presentation of ${S_v(2,d)}$.
{#AAb}
Over ${{\mathbb Q}}(v)$, the quantum Schur algebra ${S_v(2,d)}$ is isomorphic to the algebra generated by $e$, $f$, ${{K_2}}^{\pm 1}$ subject to the relations: $$\begin{aligned}
& {{K_2}}{{K_2}}^{-1}={{K_2}}^{-1}{{K_2}}=1,\label{AAb:a}\\
&{{K_2}}e{{K_2}}^{- 1} =v^{- 1}e,\qquad
{{K_2}}f{{K_2}}^{- 1} =v f\label{AAb:b}\\
& ef-fe=\frac{v^d{{K_2}}^{-2} - v^{-d}{{K_2}}^{2}}{v-{v^{-1}}},\label{AAb:c}\\
& ({{K_2}}-1)({{K_2}}-v)({{K_2}}-v^2)\cdots ({{K_2}}-v^d)=0.\label{AAb:d}\end{aligned}$$
For indeterminates $X,X^{-1}$ satisfying $X X^{-1} = X^{-1} X = 1$ and any $t\in {{\mathbb N}}$ we formally set $${\begin{bmatrix}X\\#2\end{bmatrix}}=\prod_{s=1}^{t}
\frac{Xv^{-s+1}-X^{-1}v^{s-1}}{v^s-v^{-s}}.$$ This expression will make sense if $X$ is replaced by any invertible element of a ${{\mathbb Q}}(v)$-algebra. The next result describes the ${{\mathcal A}}$-form ${S_{\mathcal{A}}(2,d)}$ in terms of the above generators, and gives an ${{\mathcal A}}$-basis for the algebra. In this we can take ${S_v(2,d)}$ to be given by either presentation \[AAa\] or \[AAb\], but we always assume that ${{K_1}}$ and ${{K_2}}$ are related by the condition ${{K_1}}{{K_2}}=v^d$.
{#section}
\[AAc\] The integral Schur algebra ${S_{\mathcal{A}}(2,d)}$ is isomorphic to the ${{\mathcal A}}$-subalgebra of ${S_v(2,d)}$ generated by $$e^{(m)}:=\frac{e^m}{[m]!},\quad f^{(m)}:=\frac{f^m}{[m]!}\quad
(m\in {{\mathbb N}}),\qquad {{K_1}}^{\pm 1}.$$ The preceding statement is true when ${{K_1}}$ is replaced by ${{K_2}}$. Moreover, an ${{\mathcal A}}$-basis for ${S_{\mathcal{A}}(2,d)}$ is the set consisting of all $${e^{(a)}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}{f^{(c)}}$$ such that the natural numbers $a,b_1,b_2,c$ are constrained by the conditions $a+b_1+c\leq d,\ b_1+b_2=d$. Another such basis consists of all $$f^{(a)} {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}e^{(c)}$$ such that the natural numbers $a,b_1,b_2,c$ satisfy the constraints $a+b_2+c\leq d,\ b_1+b_2=d$.
The following reduction formulas make it possible, in principle, to express the product of two basis elements as an ${{\mathcal A}}$-linear combination of basis elements.
{#section-1}
\[AAd\] In ${S_{\mathcal{A}}(2,d)}$ we have the following reduction formulas for all $s\ge1$ and all $a,b_1,b_2,c\in {{\mathbb N}}$ with $b_1+b_2=d$: $$\begin{gathered}
\begin{aligned}{e^{(a)}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}{f^{(c)}}=& \\
\sum_{k=s}^{\min(a,c)}(-1)^{k-s}&{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}
{e^{(a-k)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}}{f^{(c-k)}}
\end{aligned}\label{AAd:a}\\
\begin{aligned}
f^{(a)}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}e^{(c)}=& \\
\sum_{k=s}^{\min(a,c)}(-1)^{k-s}&{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_2+k\\#2\end{bmatrix}}
{f^{(a-k)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}}{e^{(c-k)}}
\end{aligned}\label{AAd:b}\end{gathered}$$ where $s=a+b_1+c-d$ in (a) and $s=a+b_2+c-d$ in (b).
The next theorem, the quantum analogue of [@DG Thm. 2.4], provides yet another kind of basis for ${S_{\mathcal{A}}(2,d)}$. In this case we will deduce it as a direct consequence of Theorem \[AAc\], while in [@DG] the analogue of the idempotent basis in Theorem \[AAc\] was not needed. We remark that the analogues of the integral idempotent bases in Theorem \[AAc\] above do hold in the classical situation.
{#section-2}
\[AAe\] The set $$\left\{{e^{(a)}}{\begin{bmatrix}K_1\\#2\end{bmatrix}} {f^{(c)}}\quad\mid\quad a,b,c\in {{\mathbb N}},\
a+b+c\leq d\right\}$$ is an ${{\mathcal A}}$-basis for ${S_{\mathcal{A}}(2,d)}$. Another such basis is given by the set $$\left\{{f^{(a)}} {\begin{bmatrix}K_2\\#2\end{bmatrix}}
{e^{(c)}} \quad\mid\quad a,b,c\in {{\mathbb N}},\ a+b+c\leq d\right\}.$$
Quantized enveloping algebras {#A}
=============================
{#Aa}
The Drinfeld-Jimbo quantized enveloping algebra ${\mathbf{U}}={\mathbf{U}}_v(\mathfrak{gl}_2)$ is defined to be the ${{\mathbb Q}}(v)$-algebra with generators $e$, $f$, ${{K_1}}$, ${K_{1}^{-1}}$, ${{K_2}}$, ${K_{2}^{-1}}$ and relations $$\begin{aligned}
& {{K_1}}{{K_2}}={{K_2}}{{K_1}},\label{Aa:a}\\
& {{K_i}}{{K_i}}^{-1}={{K_i}}^{-1}{{K_i}}=1 \quad (i=1,2),\label{Aa:b}\\
& {{K_1}}e{K_{1}^{-1}}=ve, \quad {{K_1}}f{K_{1}^{-1}}=v^{-1}f,\label{Aa:c}\\
& {{K_2}}e{K_{2}^{-1}}={v^{-1}}e, \quad {{K_2}}f{K_{1}^{-1}}=vf,\label{Aa:d}\\
& ef-fe=\frac{{{K_1}}{K_{2}^{-1}}- {K_{1}^{-1}}{{K_2}}}{v-{v^{-1}}}\label{Aa:e}.\end{aligned}$$ Let ${\mathbf{U}}^+$ (respectively ${\mathbf{U}}^-$) be the subalgebra generated by $e$ (respectively $f$) and let ${\mathbf{U}}^0$ be the subalgebra generated by $K_1^{\pm 1}$, $K_2^{\pm 1}.$ There are ${{\mathbb Q}}(v)$-vector space isomorphisms $${\mathbf{U}}\cong {\mathbf{U}}^+{\otimes}{\mathbf{U}}^0 {\otimes}{\mathbf{U}}^- \cong {\mathbf{U}}^-{\otimes}{\mathbf{U}}^0 {\otimes}{\mathbf{U}}^+$$ and moreover it is well-known that the algebra ${\mathbf{U}}$ has PBW-type bases $\left\{ e^a {{K_1}}^{b_{1}} {{K_2}}^{b_{2}}f^c\right\}$ and $\left\{ f^a
{{K_1}}^{b_{1}} {{K_2}}^{b_{2}}e^c\right\}$ where $a,c\in {{\mathbb N}}$ and $b_1,b_2\in
{{\mathbb Z}}.$
In the algebra ${\mathbf{U}}$, set $K=K_1K_2^{-1}$, and define ${\mathbf{U}}_v(\mathfrak{sl}_2)$ to be the subalgebra of ${\mathbf{U}}$ generated by $e$, $f$, and $K^{\pm 1}$. The familiar relations that these elements satisfy are easily deducible from - .
{#Ab}
For $r,s \in {{\mathbb Z}}$ define $$\begin{aligned}
& [r]=\frac{v^r-v^{-r}}{v-{v^{-1}}}\label{Ab:a}\\
& {\begin{bmatrix}r\\#2\end{bmatrix}} = \frac{[r][r-1]\cdots [r-s+1]}{[1][2]\cdots
[s]}.\label{Ab:b}\end{aligned}$$ These satisfy the well-known identities $$\begin{aligned}
& [r+s]=v^{-s}[r]+v^r [s]\label{Ab:c}\\
& {\begin{bmatrix}r+1\\#2\end{bmatrix}}=v^{-s}{\begin{bmatrix}r\\#2\end{bmatrix}}+v^{r-s+1}{\begin{bmatrix}r\\#2\end{bmatrix}}.
\end{aligned}$$ For every $m,t\in {{\mathbb N}},$ $c\in {{\mathbb Z}}$ and any element $X$ an ${{\mathbb Q}}(v)$-algebra define $$\begin{aligned}
& [m]!=[m][m-1]\cdots [1]\label{Ab:f}\\
& X^{(m)}=\frac{X^m}{[m]!}\label{Ab:g}\\
& {\begin{bmatrix}X; c\\#3\end{bmatrix}} = \begin{cases} \displaystyle{ \prod_{i=1}^t
\frac{Xv^{c-i+1}-X^{-1}v^{-c+i-1}}{v^i-v^{-i}} }
\quad \text{if}\quad t\neq 0 \\
1 \quad \text{if} \quad t=0.\end{cases}\quad
(X\quad \text{invertible}) \label{Ab:h}\end{aligned}$$ We note that ${\begin{bmatrix}X\\#2\end{bmatrix}}$, as defined in section 2, coincides with the element ${\begin{bmatrix}X; 0\\#3\end{bmatrix}}$.
In [@Lu], Lusztig investigates many relations which hold in quantized enveloping algebras. Those which we will need are contained in the following Lemma.
{#section-3}
\[Ad\] For any $c,n\in {{\mathbb Z}}$ and $m,t,t'\in {{\mathbb N}}$ the following identities hold in ${\mathbf{U}}$: $$\begin{gathered}
{{K_1}}^n\,e\, K_1^{-n}=v^n\,e, \quad {{K_2}}^n\,e\, K_2^{-n}=v^{-n}\,e, \label{Ad:a}\\
{{K_1}}^n\,f\, K_1^{-n}=v^{-n}\,f, \quad {{K_2}}^n\,f\, K_2^{-n}=v^{n}\,f,\label{Ad:b}\\
{\begin{bmatrix}{{K_1}}; c\\#3\end{bmatrix}} \,e=e\, {\begin{bmatrix}{{K_1}}; c+1\\#3\end{bmatrix}},\quad
{\begin{bmatrix}{{K_1}}; c\\#3\end{bmatrix}} \,f=f\, {\begin{bmatrix}{{K_1}}; c-1\\#3\end{bmatrix}},\label{Ad:c}\\
{\begin{bmatrix}{{K_2}}; c\\#3\end{bmatrix}} \,e=e\, {\begin{bmatrix}{{K_2}}; c-1\\#3\end{bmatrix}},\quad
{\begin{bmatrix}{{K_2}}; c\\#3\end{bmatrix}} \,f=f\, {\begin{bmatrix}{{K_2}}; c+1\\#3\end{bmatrix}},\label{Ad:d}\\
{f^{(m)}}\, e= e\, {f^{(m)}} - {\begin{bmatrix}{{K_1}}{K_{2}^{-1}}; m-1\\#3\end{bmatrix}}{f^{(m-1)}},\label{Ad:e}\\
f\, {e^{(m)}}= {e^{(m)}}\, f -{e^{(m-1)}}{\begin{bmatrix}{{K_1}}{K_{2}^{-1}}; m-1\\#3\end{bmatrix}},\label{Ad:f}\\
{\begin{bmatrix}{{K_i}}; c+1\\#3\end{bmatrix}}=v^{t+1}{\begin{bmatrix}{{K_i}}; c\\#3\end{bmatrix}}
+v^{t-c}{{K_i}}^{-1}{\begin{bmatrix}{{K_i}}; c\\#3\end{bmatrix}},\label{Ad:g}\\
{\begin{bmatrix}{{K_i}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_i}}; -t\\#3\end{bmatrix}}={\begin{bmatrix}t+t'\\#2\end{bmatrix}}{\begin{bmatrix}{{K_i}}\\#2\end{bmatrix}},\label{Ad:h}\\
{\begin{bmatrix}{{K_i}}; c\\#3\end{bmatrix}}=\sum_{j=0}^t v^{c(t-j)}{\begin{bmatrix}c\\#2\end{bmatrix}}{{K_i}}^{-j}{\begin{bmatrix}{{K_i}}\\#2\end{bmatrix}}\quad (c\geq 0).\label{Ad:i}\end{gathered}$$
All of these identities are special cases of those appearing on pages 269-270 of [@Lu].
The algebra ${{\mathcal B}}_d$
==============================
In this section we define a homomorphic image of ${\mathbf{U}}$ which will turn out to be isomorphic to the quantum Schur algebra ${S_v(2,d)}$.
{#Ba}
Let $d$ be a fixed nonnegative integer and define ${{\mathcal B}}_d$ to be the ${{\mathbb Q}}(v)$-algebra generated by $e$, $f$, ${{K_1}}^{\pm 1}$, ${{K_2}}^{\pm 1}$ subject to relations \[Aa\]-, along with the additional relations $$\begin{aligned}
&{{K_1}}{{K_2}}= v^d \label{Ba:a}\\
& ({{K_1}}-1)({{K_1}}-v)\cdots ({{K_1}}-v^d)=0\label{Ba:b}\end{aligned}$$ Note that relation (b) can be replaced by the equivalent relation $$\label{Ba:c}
({K_{1}^{-1}}-1)({K_{1}^{-1}}-{v^{-1}})\cdots ({K_{1}^{-1}}-v^{-d})=0$$ and in the presence of relation (a) it can also be replaced by either $$\begin{aligned}
& ({{K_2}}-1)({{K_2}}-v)\cdots ({{K_2}}-v^{d})=0,\quad {\text {or}}\label{Ba:d}\\
& ({K_{2}^{-1}}-1)({K_{2}^{-1}}-{v^{-1}})\cdots ({K_{2}^{-1}}-v^{-d})=0.\label{Ba:e}\end{aligned}$$
The defining relations of ${{\mathcal B}}_d$ are invariant if $e$ and $f$ are interchanged along with $K_1$ and $K_2$. These interchanges therefore induce an automorphism of ${{\mathcal B}}_d$. We shall often make use of this property, which we will call [*symmetry*]{}, in the sequel.
Let ${{\mathcal B}}_d^0$ be the subalgebra of ${{\mathcal B}}_d$ generated by ${{K_1}}^{\pm 1}$, ${{K_2}}^{\pm 1}.$ It follows from relations \[Ba\]- that $\dim({{\mathcal B}}_d^0)=d+1$.
{#section-4}
\[Bb\] In the algebra ${{\mathcal B}}_d^0$ we have $${\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}=0={\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}},\qquad
{\begin{bmatrix}{K_{1}^{-1}}\\#2\end{bmatrix}}=0={\begin{bmatrix}{K_{2}^{-1}}\\#2\end{bmatrix}}.$$
Using definition \[Ab\] we have that $$\begin{aligned}
{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}&=\prod_{i=1}^{d+1}
\frac{{{K_1}}v^{1-i}-{K_{1}^{-1}}v^{i-1}}{v^i-v^{-i}}\\
&=\prod_{i=1}^{d+1} \frac{({K_{1}^{-1}}v^{1-i})(K_1^2-v^{2(i-1)})}{v^i-v^{-i}}\\
&=\prod_{i=1}^{d+1} \frac{({K_{1}^{-1}}v^{1-i})(K_1-v^{(i-1)})(K_1+v^{(i-1)})}{v^i-v^{-i}}\\
&=0\quad {\text{by relation \ref{Ba}\eqref{Ba:b}.}}\end{aligned}$$ The other equalities follow in a similar manner using \[Ba\]-.
More generally we have
{#section-5}
\[Bc\] In the algebra ${{\mathcal B}}_d^0$ we have $${\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}=0$$ whenever $b_1+b_2=d+1.$
$$\begin{aligned}
{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}&=\prod_{i=1}^{b_1}
\frac{{{K_1}}v^{1-i}-{K_{1}^{-1}}v^{i-1}}{v^i-v^{-i}}
\prod_{i=1}^{b_2}
\frac{{{K_2}}v^{1-i}-{K_{2}^{-1}}v^{i-1}}{v^i-v^{-i}}\\
&=\prod_{i=1}^{b_1}
\frac{{{K_1}}v^{1-i}-{K_{1}^{-1}}v^{i-1}}{v^i-v^{-i}}
\prod_{i=1}^{b_2}
\frac{{K_{1}^{-1}}v^{d+1-i}-{{K_2}}v^{-d+i-1}}{v^i-v^{-i}}\\
&=(-1)^{b_2}{\begin{bmatrix}d+1\\#2\end{bmatrix}}{\begin{bmatrix}K_1\\#2\end{bmatrix}}\\
&=0 \quad {\text{by
\ref{Ba}\eqref{Ba:b}}}.\end{aligned}$$
An immediate consequence of the preceding lemma is that $${\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}=0$$ whenever $b_1+b_2\ge d+1.$ When $b_1+b_2\leq d$, these elements are non-zero, but we shall see that the most important case is $b_1+b_2=d.$
{#section-6}
\[Bd\] Suppose that $b_1+b_2=d$ and $t\in {{\mathbb N}}.$ Then for $i=1,2$ the following identities hold in ${{\mathcal B}}_d^0$: $$\begin{gathered}
K_i {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}=v^{b_i}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\label{Bd:a}\\
{\begin{bmatrix}K_i; c\\#3\end{bmatrix}} {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}={\begin{bmatrix}b_i+c\\#2\end{bmatrix}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\label{Bd:b}\end{gathered}$$
The second equality of the lemma follows immediately from the first using definition \[Ab\], and so we only need to prove (a). Consider the case $i=1$. We have $$\begin{aligned}
({{K_1}}&- v^{b_1}){\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\\
& = ({{K_1}}-v^{b_1}) \prod_{i=1}^{b_1}
\frac{{{K_1}}v^{1-i}-{K_{1}^{-1}}v^{i-1}}{v^i-v^{-i}}
\prod_{i=1}^{b_2}
\frac{{{K_2}}v^{1-i}-{K_{2}^{-1}}v^{i-1}}{v^i-v^{-i}}\\
& = (-1)^{b_2}({{K_1}}-v^{b_1}) \prod_{\substack{ i=1\\i\neq
b_1+1}}^{d+1}
\frac{{{K_1}}v^{1-i}-{K_{1}^{-1}}v^{i-1}}{v^i-v^{-i}}\\
&= (-1)^{b_2}\,({{K_1}}-v^{b_1})\prod_{\substack{ i=1\\i\neq
b_1+1}}^{d+1}
\frac{v^{1-i}{K_{1}^{-1}}({{K_1}}^2-v^{2(i-1)})}{v^i-v^{-i}}\\
&=0\quad {\text {by \ref{Ba}\eqref{Ba:b}.}}\end{aligned}$$ This proves identity (a) for $i=1.$ The case $i=2$ follows from symmetry.
The following result establishes the structure of the algebra ${{\mathcal B}}_d^0$; it will later be crucial in determining the structure of the entire algebra ${{\mathcal B}}_d$.
{#section-7}
\[Be\] In the algebra ${{\mathcal B}}_d^0$, the set $$\left\{ {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\quad | \quad b_1+b_2=d\right\}$$ is a basis of mutually orthogonal idempotents whose sum is the identity.
By Lemma \[Bd\] we have that $$\left({\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\right)^2={\begin{bmatrix}b_1\\#2\end{bmatrix}}{\begin{bmatrix}b_2\\#2\end{bmatrix}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}={\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}$$ and so each ${\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}$ is an idempotent. Now suppose that $b_1+b_2=b_1'+b_2'=d$ and $b_1\neq b_1'.$ Then either $b_1+b_2'\geq d+1$ or $b_1'+b_2\geq
d+1$ and so orthogonality follows by Lemma \[Bc\]. Thus in the algebra ${{\mathcal B}}_d^0$ (which has dimension $d+1$), we have a set of $d+1$ distinct mutually orthogonal idempotents. It follows that these must form a basis and that their sum is the identity.
These idempotents have pleasant commutation relations with the elements $e$ and $f$, given in the next lemma.
{#section-8}
\[Bf\] Suppose that $b_1+b_2=d$ and $a\in {{\mathbb N}}.$ Then in the algebra ${{\mathcal B}}_d$ $$\begin{aligned}
& {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}e^a =\begin{cases}
e^a\,{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}\quad \text{if}\quad
b_1\geq a\\ \\
0\quad \text{if}\quad b_1<a \end{cases}\\
& e^a {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}=\begin{cases}
{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}} e^a \quad \text{if}\quad
b_2\leq a\\ \\
0\quad \text{if}\quad b_2>a \end{cases}\\
& f^a {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}=\begin{cases}
{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}f^a\quad \text{if}\quad
b_1\geq a\\ \\
0\quad \text{if}\quad b_1<a. \end{cases}\\
& {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}f^a =\begin{cases}
f^a {\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}\quad \text{if}\quad
b_2\leq a\\ \\
0\quad \text{if}\quad b_2>a. \end{cases}\end{aligned}$$
Each of the relations is similar to prove, so we will only verify the first one. By \[Ad\] and \[Ad\] we have $${\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\,
e^a=e^a\, {\begin{bmatrix}{{K_1}}; a\\#3\end{bmatrix}} {\begin{bmatrix}{{K_2}}; -a\\#3\end{bmatrix}}$$ and thus $${\begin{bmatrix}b_2+a\\#2\end{bmatrix}}^{-1}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\,e^a\,{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}} =
{\begin{bmatrix}b_2+a\\#2\end{bmatrix}}^{-1} e^a\, {\begin{bmatrix}{{K_1}}; a\\#3\end{bmatrix}}
{\begin{bmatrix}{{K_2}}; -a\\#3\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}.$$ Now using \[Ad\] and \[Ad\], this equality becomes $${\begin{bmatrix}b_2+a\\#2\end{bmatrix}}^{-1}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\,{\begin{bmatrix}{{K_2}}; a\\#3\end{bmatrix}}\, e^a=
{\begin{bmatrix}b_2+a\\#2\end{bmatrix}}^{-1}\,e^a\, {\begin{bmatrix}{{K_1}}; a\\#3\end{bmatrix}}\,
{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}.$$ Transforming this further using \[Ad\] and \[Bd\] yields $${\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, e^a ={\begin{bmatrix}b_2+a\\#2\end{bmatrix}}^{-1}\, e^a \left( \sum_{j=0}^{b_1}
v^{a(b_1-j)}{\begin{bmatrix}a\\#2\end{bmatrix}}{{K_1}}^{-j}{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}\right)
{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}.$$ Since ${\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}=0$ whenever $j<a$ and ${\begin{bmatrix}a\\#2\end{bmatrix}}=0$ whenever $j>a$, the only possible non-zero term on the right side of the equality corresponds to the case $j=a.$ Consequently, ${\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, e^a = 0$ whenever $b_1<a$ and if $b_1\geq a$ then $${\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, e^a=v^{a(b_1-a)}{{K_1}}^{-a}{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}.$$ The remaining part of claim (a) of the lemma now follows using \[Bd\].
Remark
------
An immediate consequence of the preceding lemma is that $$e^a{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}=0={\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}f^a\quad \text{whenever}\quad a+b_1\geq d+1$$ and $$f^a{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}=0={\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}e^a\quad \text{whenever}\quad a+b_2\geq d+1.$$ In particular, this implies that both $e$ and $f$ are nilpotent of index $d+1$.
The results obtained thus far show that the sets $$\left\{ {e^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {f^{(c)}}\mid a,c\leq d, b_1+b_2=d \right\}$$ and $$\left\{ f^{(a)}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {f^{(c)}}\mid a,c\leq d, b_1+b_2=d \right\}$$ are spanning sets for the algebra ${{\mathcal B}}_d.$ We will henceforth refer to elements of these spanning sets simply as [*monomials*]{}. These sets, however, are not bases. To establish this fact, we need some terminology. Define the [*fake degree*]{} of a monomial ${e^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\,
{f^{(c)}}$ (respectively ${f^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {e^{(c)}}$) to be $a+b_1+c$ (respectively $a+b_2+c$) and, in both cases, define its [*height*]{} to be $a+c$.
{#section-9}
\[Bg\] Suppose that $b_1+b_2=d$. Then in the algebra ${{\mathcal B}}_d$ all monomials of the form ${e^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {f^{(c)}}$ (respectively ${f^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {e^{(c)}}$) of fake degree $d+1$ are expressible as ${{\mathbb Q}}(v)$-linear combinations of monomials of the same form of strictly smaller fake degree and height.
By symmetry it is enough to prove the claim for monomials of the form ${e^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {f^{(c)}}$. We use induction on height. The base case here is height one since there are no height zero monomials of fake degree $d+1$. Consider a monomial of fake degree $d+1$ and height $s\geq 1$. Suppose that $a\geq c$. Then $a\geq 1$ and by Lemma \[Bf\] we have $${e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}} \,e\,{f^{(c)}}=[a]{e^{(a)}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}{f^{(c)}}$$ and so the claim will follow once we show that ${e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}} \,e\,{f^{(c)}}$ can be expressed in the desired form. But, by \[Ad\] and \[Bd\] we have $$\begin{aligned}
{e^{(a-1)}}&{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}} \,e\,{f^{(c)}}\\
&={e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}}\left( {f^{(c)}}e +{\begin{bmatrix}{{K_1}}{K_{2}^{-1}}; c-1\\#3\end{bmatrix}} {f^{(c-1)}}\right)\\
&= {e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}} {f^{(c)}}e \\
& \qquad + [b_1-b_2+c-1]{e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}}{f^{(c-1)}}\end{aligned}$$ where the second term on the right hand side is zero if $c=0.$ However, in general, the second term is a monomial of fake degree $d$ and height $s-2$, which is of the desired form. To analyze the first term, we need to use the fact that if $M$ is a monomial of fake degree $d'$ and height $s'$, then $Me$ is an ${{\mathbb Q}}(v)$-linear combination of monomials of fake degree at most $d'$ and height at most $s'+1$ (this claim can be verified using \[Ad\] and \[Bd\]). Now by induction, ${e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}}{f^{(c)}}$ is expressible as an ${{\mathbb Q}}(v)$-linear combination of monomials of fake degree at most $d$ and height at most $s-2$. Thus, by our claim, ${e^{(a-1)}}{\begin{bmatrix}K_1\\#1 \end{bmatrix}\begin{bmatrix}K_2\\#2\end{bmatrix}}{f^{(c)}}e$ is expressible as a linear combination of terms of fake degree at most $d$ and height at most $s-1$. This completes the proof in the case $a\geq c$; the case $a\leq
c$ is similar and is omitted.
Identifications
===============
In this section we show that ${{\mathcal B}}_d$ is isomorphic to ${S_v(2,d)}$.
{#Ae}
Let $E$ be an ${{\mathbb Q}}(v)$-module with basis $\{e_1,e_2\}$. There is a canonical representation $\rho: {\mathbf{U}}\rightarrow {\operatorname{End}}_{{{\mathbb Q}}(v)}(E)$ defined by: $$e \mapsto \begin{pmatrix} 0&1\\ 0&0\end{pmatrix},\quad
f \mapsto \begin{pmatrix} 0&0\\ 1&0\end{pmatrix},\quad
K_1 \mapsto \begin{pmatrix} v&0\\ 0&1\end{pmatrix},\quad
K_2 \mapsto \begin{pmatrix} 1&0\\ 0&v\end{pmatrix}.$$ Since ${\mathbf{U}}$ is a bialgebra, we obtain a representation $$\rho_d: {\mathbf{U}}\rightarrow {\operatorname{End}}_{{{\mathbb Q}}(v)}(E^{{\otimes}d})$$ in the $d$th tensor power of $E$, for every $d\in {{\mathbb N}}$. Specifically, $\rho_d =\rho^d
\circ \Delta ^{d-1}$, where $\Delta^{d-1}:{\mathbf{U}}\rightarrow {\mathbf{U}}^{{\otimes}d}$ is the iterated comultiplication map. As stated earlier, ${S_v(2,d)}$ is the image of the homomorphism $\rho_d.$
{#section-10}
\[Ca\] Write $\overline{X}$ for the image of $X\in {\mathbf{U}}$ under the representation $\rho_d: {\mathbf{U}}\rightarrow {\operatorname{End}}_{{{\mathbb Q}}(v)}(E^{{\otimes}d})$. Then we have the identities $$\begin{gathered}
{{\overline{K_1}}}{{\overline{K_2}}}= v^d \\
({{\overline{K_1}}}-1)({{\overline{K_1}}}-v)\cdots ({{\overline{K_1}}}-v^d)=0\end{gathered}$$
We have ${{\overline{K_1}}}{{\overline{K_2}}}= \rho_d({{K_1}}{{K_2}})=\rho_1({{K_1}}{{K_2}})^{{\otimes}d}$ since $\Delta ({{K_i}})={{K_i}}{\otimes}{{K_i}}$. But by \[Ae\], $$\rho_1({{K_1}}{{K_2}})= \begin{pmatrix} v&0\\ 0&1\end{pmatrix}
\begin{pmatrix} 1&0\\ 0&v\end{pmatrix}=v\begin{pmatrix} 1&0\\ 0&1\end{pmatrix}
=v1_V$$ and so $\rho_1({{K_1}}{{K_2}})^{{\otimes}d}=v^d1_{V^{{\otimes}d}}$, which proves claim (a). The case $d=1$ for claim (b) follows immediately since $\rho_1({{K_1}})=\begin{pmatrix} v&0\\ 0&1\end{pmatrix}$. Now $\rho_d
({{K_1}})={{K_1}}^{{\otimes}d}$ is a diagonal matrix whose entries are $1,v,\ldots, v^d$ (not counting multiplicities). The claim follows.
Henceforth we shall omit the bar, writing the image of $X$ simply as $X$ instead of $\overline{X}$.
{#section-11}
\[Cb\] The Schur algebra ${S_v(2,d)}$ is isomorphic as a ${{\mathbb Q}}(v)$-algebra to ${{\mathcal B}}_d$. Moreover, the set of all monomials ${e^{(a)}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}{f^{(c)}}$ ($a+b_1+c\leq
d$) is a basis over ${{\mathbb Q}}(v)$. Similarly, another such basis consists of all the monomials ${f^{(a)}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}{e^{(c)}}$ ($a+b_2+c\leq d$).
By the preceding lemma we see that the surjection $\rho_d:{\mathbf{U}}\to {S_v(2,d)}$ factors through ${{\mathcal B}}_d$, giving the commutative diagram $$\begin{CD}
{\mathbf{U}}@>>> {S_v(2,d)}\\
@VVV @AAA\\
{{\mathcal B}}_d @= {{\mathcal B}}_d
\end{CD}$$ in which all arrows are surjections. By Theorem \[Bg\], the set of all monomials of the form ${e^{(a)}}{\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}{f^{(c)}}$ satisfying $a+b_1+c\leq
d$ spans the algebra ${{\mathcal B}}_d$. This spanning set is in one-to-one correspondence with the set of all monomials in 4 commuting variables of total degree $d$ (set one of the variables equal to 1 to get the correspondence). It is well known that this number is the dimension of ${S_v(2,d)}$. Hence the map ${{\mathcal B}}_d \rightarrow {S_v(2,d)}$ must be an isomorphism and the spanning set is a basis. A similar argument establishes that the set of all ${f^{(a)}}\, {\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}\, {e^{(c)}}$ satisfying $a+b_2+c\leq d$ is also a basis of ${{\mathcal B}}_d$. This completes the proof.
{#section-12}
Theorems \[AAa\] and \[AAb\] follow immediately from the previous result by using the relation ${{K_1}}{{K_2}}=v^d$ to remove either ${{K_1}}$ or ${{K_2}}$ from the generating set. However, the results of the preceding theorem are not sufficient to establish Theorem \[AAc\] since we only know that the coefficients in the reduction formulas are elements of ${{\mathbb Q}}(v)$, not necessarily of ${{\mathcal A}}$. In the next section we indeed show that these coefficients all lie in ${{\mathcal A}}$. The process by which we prove this does not rely on Theorem \[Cb\] and so the forthcoming results give alternative proofs of Theorems \[AAa\] and \[AAb\] as well as a proof of \[AAc\].
We now prove Theorem \[AAa0\]. In the algebra ${S_v(2,d)}$, set $K=v^{-d}K_1^2$. Using Lemma \[Bd\] and Theorem \[Be\] we obtain the equality $$K_1^2=v^dK=\sum
_{b_1+b_2=d}v^{2b_1}{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}} =
\left(\sum
_{b_1+b_2=d}v^{b_1}{\begin{bmatrix}{{K_1}}\\#2\end{bmatrix}}{\begin{bmatrix}{{K_2}}\\#2\end{bmatrix}}\right)^2.$$ Thus, $e$, $f$, and $K^{\pm 1}$ comprise another generating set for ${S_v(2,d)}$. Relations (a)-(c) of Theorem \[AAa0\] are easily seen to be equivalent to the corresponding parts of Theorem \[AAa\]. Relation (d) of these theorems are also equivalent. To see this, note that $$\begin{aligned}
\prod _{i=0}^d(K-v^{d-2i})&=v^{-d^2-d}\prod _{i=0}^d(K_1^2-v^{2(d-i)})\\
&=v^{-d^2-d}\prod _{i=0}^d(K_1-v^{(d-i)})(K_1+v^{(d-i)}).\end{aligned}$$ But by Theorem \[AAa\] this expression is equal to zero. This completes the proof of Theorem \[AAa0\].
The integral reduction
======================
In this section we prove Theorems \[AAc\], \[AAd\], and \[AAe\]. For typeographic convenience, we will throughout this section use the abbreviation ${K_{b_1,b_2}}$ in place of the more cumbersome notation ${\begin{bmatrix}K_1\\b_1\end{bmatrix}\begin{bmatrix}K_2\\b_2\end{bmatrix}}$.
{#section-13}
\[Da\] In the algebra ${{\mathcal B}}_d$ we have the equality $${e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}=\sum_{k=1}^{\min(a,c)}(-1)^{k-1}
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}$$ for all $a,b_1,b_2,c\in {{\mathbb N}}$ satisfying $a+b_1+c=d+1$ and $b_1+b_2=d$. Similarly we have the equality $$f^{(a)}{K_{b_1,b_2}}e^{(c)}=
\sum_{k=1}^{\min(a,c)}(-1)^{k-1}{\begin{bmatrix}b_2+k\\#2\end{bmatrix}}
{f^{(a-k)}}{K_{b_1-k,b_2+k}}{e^{(c-k)}}$$ for all $a,b_1,b_2,c\in {{\mathbb N}}$ satisfying $a+b_2+c=d+1$ and $b_1+b_2=d$.
By symmetry it suffices to prove the first reduction formula only. Suppose first that $c=0$. Then the claim follows from the remark following Lemma \[Bf\]. If $c>0$ we will prove the desired result by induction on $a$. The base case here is $a=0$, which also follows from the remark mentioned above. The identity to be proved can be rewritten in the form $$\label{Da:a}
0=\sum_{k=0}^{\min(a,c)}(-1)^k
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}.$$ Suppose this equation is satisfied for some fixed quadruple $a,b_1,b_2,c$ satisfying the conditions $a+b_1+c=d+1$, $b_1+b_2=d$ and $c\geq 0$. From this we will derive the result for the case $a+1, b_1-1,b_2+1,c$. The idea is to multiply on the right by $e$ and commute it all the way to the left. Using Lemma \[Ad\] we obtain $$\begin{aligned}
0&=\sum_{k=0}^{\min(a,c)}(-1)^k
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}e\\
&=\sum_{k=0}^{\min(a,c)}(-1)^k
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}{e^{(a-k)}}{K_{b_1+k,b_2-k}}\\
&\hspace{2in} \times \left(
e{f^{(c-k)}}-{\begin{bmatrix}{{K_1}}{K_{2}^{-1}}; c-k-1\\#3\end{bmatrix}}{f^{(c-k-1)}}\right).\end{aligned}$$ Then by Lemmas \[Bd\] and \[Bf\] this equation becomes $$\begin{aligned}
0&=\sum_{k=0}^{\min(a,c)}(-1)^k
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}{e^{(a-k)}}e{K_{b_1+k-1,b_2-k+1}}{f^{(c-k)}}\\
&-\sum_{k=0}^{\min(a,c)}(-1)^k
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}[b_1-b_2+k+c-1]{e^{(a-k)}}{K_{b_1+k,b_2-k}}
{f^{(c-k-1)}}\end{aligned}$$ which is equivalent to the identity $$\begin{aligned}
0&=\sum_{k=0}^{\min(a,c)}(-1)^k
{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}[a-k+1]{e^{(a-k+1)}}{K_{b_1+k-1,b_2-k+1}}
{f^{(c-k)}}\\
&+\sum_{k=1}^{1+\min(a,c)}(-1)^{k-1}
{\begin{bmatrix}b_1+k-1\\#2\end{bmatrix}}[a-b_1-k+1]\\
&\hspace{2in}\times {e^{(a-k+1)}}{K_{b_1+k-1,b_2-k+1}}
{f^{(c-k)}}.\end{aligned}$$ (The equality $b_1-b_2+k+c-1=-(a-b_1-k)$ follows from the hypotheses $a+b_1+c=d+1$ and $b_1+b_2=d$.) Writing $m$ for $\min(a,c)$ we obtain $$\begin{aligned}\label{Da:b}
0&=\sum_{k=1}^{m} R
{e^{(a-k+1)}}{K_{b_1+k-1,b_2-k+1}}{f^{(c-k)}}\\
+[a+1]&{e^{(a+1)}}{K_{b_1,b_2}}{f^{(c)}}+(-1)^m[a-b_1-m]{e^{(a-m)}}{K_{b_1+m,b_2-m}}{f^{(c-m-1)}}
\end{aligned}$$ where $$\begin{aligned}
R&={\begin{bmatrix}b_1+k\\#2\end{bmatrix}}[a-k+1]-{\begin{bmatrix}b_1+k-1\\#2\end{bmatrix}}[a-b_1-k+1]\\ &
=[a+1]{\begin{bmatrix}b_1+k-1\\#2\end{bmatrix}}.\end{aligned}$$ Now the last term in is zero if $m=c$. Otherwise $m=a<c$ and the term takes the form $$\begin{aligned}
(-1)^a{\begin{bmatrix}b_1+a\\#2\end{bmatrix}}&[-b_1]{K_{b_1+a,b_2-a}}
{f^{(c-a-1)}}\\
&=
(-1)^{a+1}[a+1]{\begin{bmatrix}b_1+a\\#2\end{bmatrix}}{K_{b_1+a,b_2-a}}{f^{(c-a-1)}}.\end{aligned}$$ This shows that all the terms in equation have a common factor of $[a+1]$. Putting these terms together and dividing by $[a+1]$ we obtain the equality $$0=\sum_{k=0}^M (-1)^k{\begin{bmatrix}b_1+k-1\\#2\end{bmatrix}}{e^{(a-k+1)}}
{K_{b_1+k-1,b_2-k+1}}{f^{(c-k)}}$$ where $M=m=\min(a+1,c)$ in the case $m=c$ and $M=m+1=a+1=\min(a+1,c)$ otherwise. In either case $M=\min(a+1,c)$ and the induction is complete.
{#section-14}
\[Db\] Suppose $a,b_1,b_2,c\in {{\mathbb N}}$ with $b_1+b_2=d$. Then if $s=a+b_1+c-d>0$ we have the equality $${e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}=\sum_{k=s}^{\min(a,c)}(-1)^{k-s}{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}
{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}$$ and if $s=a+b_2+c-d$ we have the equality $$f^{(a)}{K_{b_1,b_2}}e^{(c)}
= \sum_{k=s}^{\min(a,c)}(-1)^{k-s}{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_2+k\\#2\end{bmatrix}}
{f^{(a-k)}}{K_{b_1-k,b_2+k}}{e^{(c-k)}}.$$
By symmetry we need only verify the first equality. We proceed by induction on $s$. The case $s=1$ is the content of Theorem \[Da\]. Let $a,b_1,b_2,c$ be given such that $a+b_1+c-d=s+1$ and $b_1+b_2=d$. If $c<s$ then $a+b_1\geq d+1$ and so by Lemma \[Bf\] we have $${e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}=0.$$ By a similar argument one sees that this also holds if $a<c$. Hence we may assume that both $a$ and $c$ are $\geq s$. It is enough to prove the result for the case $a\geq c\geq s$ since the other case is similar.
Thus $a\geq 1$ and we have by induction $$\begin{aligned}
&{e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}= \frac{e}{[a]}{e^{(a-1)}}{K_{b_1,b_2}}{f^{(c)}}\\
&= \frac{e}{[a]}\sum_{k=s}^{\min(a-1,c)}(-1)^{k-s}
{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}
{e^{(a-1-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}\\
&=\frac{1}{[a]}\sum_{k=s}^{\min(a-1,c)}(-1)^{k-s}
{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}
[a-k]{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}\\
&=\frac{[a-s]}{[a]}{\begin{bmatrix}b_1+s\\#2\end{bmatrix}}{e^{(a-s)}}{K_{b_1+s,b_2-s}}
{f^{(c-s)}}\\
&\ + \frac{1}{[a]}\sum_{k=s+1}^{\min(a-1,c)}(-1)^{k-s}
{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}[a-k]
{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}.\end{aligned}$$ Now the first term of the last equality can be expanded by Theorem \[Da\] since $(a-s)+(b_1+s)+(c-s)=a+b_1+c-s=d+1$. Putting this in and shifting the index of summation we obtain the equalities $$\begin{aligned}
&e^{(a)} {K_{b_1,b_2}}{f^{(c)}}\\
&=
\frac{[a-s]}{[a]}{\begin{bmatrix}b_1+s\\#2\end{bmatrix}}\sum_{k=1}^{c-s}(-1)^{k-1}{\begin{bmatrix}b_1+s+k\\#2\end{bmatrix}}
{e^{(a-s-k)}}{K_{b_1+s+k,b_2-s+k}}{f^{(c-s-k)}}\\
&\qquad -\frac{1}{[a]}\sum_{k=s+1}^{\min(a-1,c)}(-1)^{k-(s+1)}
{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}[a-k]
{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}\\
&=
\frac{[a-s]}{[a]}{\begin{bmatrix}b_1+s\\#2\end{bmatrix}}\sum_{k=s+1}^{c}(-1)^{k-(s+1)}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}
{e^{(a-k)}}{K_{b_1+k,b_2+k}}{f^{(c-k)}}\\
&\qquad -\frac{1}{[a]}\sum_{k=s+1}^{\min(a-1,c)}(-1)^{k-(s+1)}
{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}[a-k]
{e^{(a-k)}}{K_{b_1+k,b_2-k}}{f^{(c-k)}}\end{aligned}$$ Now the second term in the last equality above can be taken from $s+1$ to $c$ since $\min(a-1,c)$ is different from $c$ only if $a=c$, in which case the additional term in the sum will be zero (the factor $[a-k]$ is zero when $k=c=a$). Putting the two sums together and using the identity $$\frac{[a-s]}{[a]}{\begin{bmatrix}b_1+s\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}-\frac{[a-k]}{[a]}
{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}={\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}.$$ we obtain $${e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}=\sum_{k=s+1}^c
(-1)^{k-(s+1)}{\begin{bmatrix}k-1\\#2\end{bmatrix}}{\begin{bmatrix}b_1+k\\#2\end{bmatrix}}{e^{(a-k)}}
{K_{b_1+k,b_2-k}}{f^{(c-k)}}$$ and this completes the induction.
{#Dc}
Theorem \[Db\] combines with the isomorphism ${{\mathcal B}}_d\cong {S_v(2,d)}$ to prove Theorem \[AAd\].
We now prove Theorem \[AAc\]. First recall the representation $$\label{Dc:a}
\rho_d: {\mathbf{U}}\rightarrow {\operatorname{End}}_{{{\mathbb Q}}(v)}(E^{{\otimes}d})$$ of section \[Ae\]. Let ${{\mathbf{U}}_{\mathcal{A}}}$ be the subalgebra of ${\mathbf{U}}$ generated by all ${e^{(m)}}$, ${f^{(m)}}$ ($m\in {{\mathbb N}}$) and ${{K_i}}^{\pm 1}$. The map in restricts to give a map ${{\mathbf{U}}_{\mathcal{A}}}\rightarrow {\operatorname{End}}_{{{\mathbb Q}}(v)}(E^{{\otimes}d})$. Let $E_{{{\mathcal A}}}$ be the ${{\mathcal A}}$-submodule of $E$ spanned by the canonical basis elements $e_1$ and $e_2$. It is clear that $E_{{\mathcal A}}$ is stable under the action of ${{\mathbf{U}}_{\mathcal{A}}}$ and hence so is $E_{{{\mathcal A}}}^{{\otimes}d}$. Thus the image of the above map is contain in ${\operatorname{End}}_{{{\mathcal A}}}(E_{{{\mathcal A}}}^{{\otimes}d})$. Consequently we have a representation $$\label{Dc:b}
\rho_d^{{{\mathcal A}}}: {{\mathbf{U}}_{\mathcal{A}}}\rightarrow {\operatorname{End}}_{{{\mathcal A}}}(E_{{{\mathcal A}}}^{{\otimes}d})$$ and by a result of Du [@Du] (see also [@Gr]) the algebra ${S_{\mathcal{A}}(2,d)}$ is precisely the image of this representation.
A fundamental result in [@Lu] (applied to the ${\mathfrak
{gl}}_2$ case) is that ${{\mathbf{U}}_{\mathcal{A}}}$ is a free ${{\mathcal A}}$-module with basis $$\label{Dc:c}
\left\{ e^{(a)} {{K_1}}^{\delta_1} {{K_2}}^{\delta_2}{K_{b_1,b_2}}f^{(c)}\quad | \quad
a,b_1,b_2,c\in {{\mathbb N}},\quad \delta_i\in \{0,1\}.\right\}$$ Hence ${S_{\mathcal{A}}(2,d)}$ is spanned over ${{\mathcal A}}$ by the images of the basis elements of (\[Dc:c\]). By Lemma \[Bd\] and Theorem \[Be\] the elements ${{K_1}}^{\delta_1} {{K_2}}^{\delta_2}{K_{b_1,b_2}}$ with $\delta_i\in
\{0,1\}$ and $b_1,b_2$ arbitrary are ${{\mathcal A}}$-linear combinations of those ${K_{b_1,b_2}}$ with $b_1+b_2=d$. Therefore ${S_{\mathcal{A}}(2,d)}$ is spanned by all ${e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}$ with $b_1+b_2=d$. But by Theorem \[Db\] we see that the set of all such terms satisfying the condition $a+b_1+c\leq d$ is a spanning set for ${S_{\mathcal{A}}(2,d)}$ over ${{\mathcal A}}$. Being linearly independent over ${{\mathbb Q}}(v)$, this set is also linearly independent over ${{\mathcal A}}$. By symmetry, the set ${f^{(a)}}{K_{b_1,b_2}}{e^{(c)}}$ with $b_1+b_2=d$ and $a+b_2+c\leq d$ is linearly independent over ${{\mathcal A}}$. This completes the proof of Theorem \[AAc\].
It remains to prove Theorem \[AAe\]. By Lemma \[Bd\] and Theorem \[Be\] it follows that $$\begin{aligned}
{\begin{bmatrix}K_1\\#2\end{bmatrix}} = {\begin{bmatrix}K_1\\#2\end{bmatrix}}\cdot 1
&= {\begin{bmatrix}K_1\\#2\end{bmatrix}}\sum_{b_1+b_2=d} {K_{b_1,b_2}}\\
&= \sum_{b_1+b_2=d}{\begin{bmatrix}b_1\\#2\end{bmatrix}} {K_{b_1,b_2}}\end{aligned}$$ and similarly we have $${\begin{bmatrix}K_2\\#2\end{bmatrix}} = \sum_{b_1+b_2=d}{\begin{bmatrix}b_2\\#2\end{bmatrix}} {K_{b_1,b_2}}.$$ Moreover, the matrix of coefficients in these equations is, with respect to an appropriate ordering of its rows and columns, triangular with $1$’s on the main diagonal. So these equations can be inverted over ${{\mathcal A}}$ to obtain formulas expressing each ${K_{b_1,b_2}}$ ($b_1+b_2=d$) as an ${{\mathcal A}}$-linear combination of the ${\begin{bmatrix}K_1\\#2\end{bmatrix}}$ or the ${\begin{bmatrix}K_2\\#2\end{bmatrix}}$ as $b$ ranges from $0$ to $d$. Thus it follows that the sets $\{1, {\begin{bmatrix}K_1\\#2\end{bmatrix}}, \dots,
{\begin{bmatrix}K_1\\#2\end{bmatrix}}\}$ and $\{1, {\begin{bmatrix}K_2\\#2\end{bmatrix}}, \dots,
{\begin{bmatrix}K_2\\#2\end{bmatrix}}\}$ are spanning sets (over ${{\mathcal A}}$) for the algebra $S_{{\mathcal A}}^0(n,d)$. Since it is already known that the rank of this free ${{\mathcal A}}$-module is $d+1$ and since the ring ${{\mathcal A}}$ is commutative, it follows that the two sets are in fact ${{\mathcal A}}$-bases for $S_{{\mathcal A}}^0(n,d)$.
{#section-15}
\[Dd\] Let $a,b,c$ be nonnegative integers satisfying $a+b+c>d$. Then the elements ${e^{(a)}}{\begin{bmatrix}K_1\\#2\end{bmatrix}} {f^{(c)}}$ and ${f^{(a)}}{\begin{bmatrix}K_2\\#2\end{bmatrix}} {e^{(c)}}$ of degree $a+b+c$ are each expressible as ${{\mathcal A}}$-linear combinations of elements of the same form but of degree not exceeding $d$.
From the remarks preceding the lemma we know that ${\begin{bmatrix}K_1\\#2\end{bmatrix}}$ and ${\begin{bmatrix}K_2\\#2\end{bmatrix}}$ are expressible as ${{\mathcal A}}$-linear combinations of the idempotents ${K_{b_1,b_2}}$ ($b_1+b_2=d$). Thus the elements ${e^{(a)}}{\begin{bmatrix}K_1\\#2\end{bmatrix}} {f^{(c)}}$ (resp., ${f^{(a)}}{\begin{bmatrix}K_2\\#2\end{bmatrix}} {e^{(c)}}$) are expressible as ${{\mathcal A}}$-linear combinations of elements of the form $$\label{Dd:a}
{e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}\quad (\text{resp., } {f^{(a)}}{K_{b_1,b_2}}{e^{(c)}})$$ where $b_1+b_2=d$ and where the fake degree $a+b_1+c$ (resp., $a+b_2+c$) is strictly greater than $d$. By Theorem \[AAd\] it follows that each element of the form above is expressible as an ${{\mathcal A}}$-linear combination of elements of the same form $$\label{Dd:b}
{e^{(a')}} K_{u_1,u_2} {f^{(c')}} \quad (\text{resp., }
{f^{(a')}} K_{u_1,u_2} {e^{(c')}})$$ where $a'<a$, $c'<c$, $u_1+u_2=d$, and $a'+u_1+c'\le d$ (resp., $a'+u_2+c'\le d$). Now by expressing $K_{u_1,u_2}$ as an ${{\mathcal A}}$-linear combination of elements of the form ${\begin{bmatrix}K_1\\#2\end{bmatrix}}$, (resp., ${\begin{bmatrix}K_2\\#2\end{bmatrix}}$) where $0\le b' \le d$ we obtain (via left and right multiplication by appropriate elements) formulas expressing each of the elements of the form in terms of ${{\mathcal A}}$-linear combinations of elements of the form $$\label{Dd:c}
{e^{(a')}} {\begin{bmatrix}K_1\\#2\end{bmatrix}} {f^{(c')}} \quad (\text{resp., }
{f^{(a')}} {\begin{bmatrix}K_2\\#2\end{bmatrix}} {e^{(c')}}).$$ If an element of this form satisfies the constraint $a'+b'+c' \le d$ then we leave it be, but for those elements which do not satisfy this constraint we repeat the entire process given above, replacing the element by an ${{\mathcal A}}$-linear combination of elements of the same form, in which for each element the degree of $e$ is strictly smaller, as is the degree of $f$. After repeating the process finitely many times we obtain the desired result.
Now we can prove Theorem \[AAe\]. By Theorem \[AAc\] we know that the set $$B=\left\{{e^{(a)}}{K_{b_1,b_2}}{f^{(c)}}\mid a+b_1+c\leq d,\ b_1+b_2=d\right\}$$ is an ${{\mathcal A}}$-basis for ${S_{\mathcal{A}}(2,d)}$. Now consider the set $$B'=\left\{{e^{(a)}}{\begin{bmatrix}K_1\\#2\end{bmatrix}} {f^{(c)}}\mid a+b+c\leq d\right\}.$$ The sets $B$ and $B'$ have the same cardinality and the ring ${{\mathcal A}}$ is commutative, thus it will follow that $B'$ is a basis once we can show that it spans.
We know that ${S_{\mathcal{A}}(2,d)}$ is spanned by elements of the form ${e^{(a)}}{\begin{bmatrix}K_1\\#2\end{bmatrix}} {f^{(c)}}$ since $S_{{\mathcal A}}= S_{{\mathcal A}}^+ S_{{\mathcal A}}^0 S_{{\mathcal A}}^-$. By the above lemma we know that each such element not satisfying the constraint $a+b+c\le d$ is expressible as an ${{\mathcal A}}$-linear combination of elements which do satisfy that constraint. It follows that the set $B'$ is a spanning set for ${S_{\mathcal{A}}(2,d)}$. This proves the first part of Theorem \[AAe\]. The second part of the theorem follows by symmetry.
[9999]{}
S. Doty and A. Giaquinto, Presenting Schur algebras as quotients of the universal enveloping algebra of ${{\mathfrak {gl}}}_2$, [*preprint*]{}, Loyola Univ. Chicago, Sept. 2000.
J. Du, A note on quantized Weyl reciprocity at roots of unity, [*Algebra Colloq.*]{} [**2**]{} (1995), no. 4, 363-372.
R. Green, $q$-Schur algebras as quotients of quantized enveloping algebras, [*J. Algebra*]{} [**185**]{} (1996), no. 3, 660-687.
G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, [*J. Amer. Math. Soc.*]{} [**3**]{} (1990), no. 1, 257-296.
Mathematical and Computer Sciences
Loyola University Chicago
Chicago, Illinois 60626 U.S.A.
E-mail: doty@math.luc.edu
tonyg@math.luc.edu
|
---
author:
- 'Qixiang Yang and Tao Qian [^1]'
date:
title: '**The duality about function set and Fefferman-Stein Decomposition** '
---
Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, the authors establish the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ on basis of the dual on function set which has special topological structure. The function in Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$ can be written as the certain combination of $D+1$ functions in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D) \bigcap L^{\infty}(\mathbb{R}^D)$.
To get such decomposition, [**(i),**]{} The authors introduce some auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ defined via wavelet expansions. The authors proved $\tls\subsetneqq L^{1}(\rr^D) \bigcup \tls \subset {\rm WE}^{1,\,q}(\rr^D)\subset L^{1}(\rr^D) + \tls$ and $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ is strictly contained in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. [**(ii),**]{} The authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by function set $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. [**(iii),**]{} We also consider the dual of $\mathrm{WE}^{1,\,q}(\mathbb R^D)$. As a consequence of the above results, the authors get also Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ by Banach space $L^{1}(\rr^D) + \tls$.
Although Fefferman-Stein type decomposition when $D=1$ was obtained by C.-C. Lin et al. \[Michigan Math. J. 62 (2013), 691-703\], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the cases $D\ge2$, which needs some new methodology.
Introduction and main results {#s1}
=============================
The Riesz transforms on $\rr^D$ ($D\ge2$), which are natural generalizations of the Hilbert transform on $\rr$, may be the most typical examples of Calderón-Zygmund operators (see, for example, [@g08; @s70; @s93] and references therein). It is well known that the Riesz transforms have many interesting properties, for example, they are the simplest, nontrivial, ¡°invariant¡± operators under the action of the group of rotations in the Euclidean space $\rr^D$, and they also constitute typical and important examples of Fourier multipliers. Moreover, they can be used to mediate between various combinations of partial derivatives of functions. All these properties make the Riesz transforms ubiquitous in mathematics and useful in various fields of analysis such as partial differential equations and harmonic analysis (see [@s70; @s93] for more details on their applications).
The Riesz transform characterization of Hardy spaces plays important roles in the real variable theory of Hardy spaces (see, for example, [@fs; @s70]). Via this Riesz transform characterization of the Hardy space $\hon$ and the duality between $\hon$ and the space of functions with bounded mean oscillation, $\bmo$, Fefferman and Stein [@fs] further obtained the nowadays so-called Fefferman-Stein decomposition of $\bmo$. Later, Uchiyama [@u82] gave a constructive proof of the Fefferman-Stein decomposition of $\bmo$. Since then, many articles focus on the classical Riesz transform characterization and the Fefferman-Stein decomposition of different variants of Hardy spaces and BMO spaces; see, for example, [@ccyy; @cg; @g79; @ll; @yzn] and references therein. Recently, Lin et al. [@lly] established the Hilbert transform characterization of Triebel-Lizorkin spaces ${\dot{F}^0_{1,\,q}(\rr)}$ and the Fefferman-Stein decomposition of Triebel-Lizorkin spaces ${\dot{F}^0_{\fz,\,q'}(\rr)}$ for each $q\in[2,\fz)$. Yang et al. [@yql] obtained the Fefferman-Stein decomposition for $Q$-spaces $Q_\az(\rr^D)$ and the Riesz transform characterization of $P^\az(\rr^D)$, the predual of $Q_\az(\rr^D)$, for any $\az\in[0,\fz)$.
As was pointed out by Lin et al. in [@lly Remark 1.4], the approach used in [@lly] for the Hilbert transform characterization of Triebel-Lizorkin spaces ${\dot{F}^0_{1,\,q}(\rr)}$ can not be applied to $\tls$ when $D\ge2$, which needs to develop some new skills. In this article, motivated by some ideas from [@lly; @yql], we establish the Riesz transform characterization of Triebel-Lizorkin spaces ${\dot{F}^0_{1,\,q}(\rr^D)}$ and the Fefferman-Stein decomposition of Triebel-Lizorkin spaces ${\dot{F}^0_{\fz,\,q'}(\rr^D)}$ for all $D\in\nn:=\{1,\,2,\,\ldots\}$ and $q\in[2,\fz)$.
In order to state the main results of this article, we now recall the definition of the Triebel-Lizorkin space $\tls$ from [@tr1]; see also [@tr2; @tr3; @tr4; @fjw]. Let $\sch$ and $\schd$ be the *Schwartz space* and its *dual* respectively, and $\pd$ the *class of all polynomials on $\rr^D$*. Following [@tr1], we also let $$\schi:=\lf\{\vz\in\sch:\ \int_{\rr^D}\vz(x)x^{\az}\,dx=0
\ {\rm for\ all}\ \az\in\zz^D_+\r\}$$ and $\schid$ be its dual. Here and hereafter, $\zz_+:=\nn\cup\{0\}$, $\zz^D_+:=(\zz_+)^D$ and, for any $\az:=(\az_1,\ldots,\az_D)\in\zz^D_+$ and $x:=(x_1,\ldots,x_D)\in\rr^D$, $x^\az:=x_1^{\az_1}\cdots x_D^{\az_D}$.
\[da.a\] Let $\vz\in\sch$ satisfy $\supp(\widehat{\vz})\st\{\xi\in\rr^D:\
\frac12 \le|\xi|\le 2\}$, $|\widehat{\vz}(\xi)|\ge c>0$ if $\frac35 \le|\xi|\le \frac53$, and $\sum_{j\in\zz}|\widehat{\vz}(2^j\xi)|=1$ if $\xi\neq0$, where $c$ is a positive constant. Write $\vz_j(\cdot):=2^{Dj}\vz(2^{j}\cdot)$ for any $j\in\zz$. Let $q\in(1,\fz)$. Then the *homogeneous Triebel-Lizorkin space* $\tls$ is defined to be the set of all $f\in\schid$ such that $$\|f\|_{\tls}:=\lf\|\lf\{\sum_{j\in\zz}
\lf|\vz_j\ast f\r|^q\r\}^{1/q}\r\|_{\lon}<\fz.$$
\[ra.n\] (i) It is well known that $\schid=\schd/\pd$ with equivalent topologies; see, for example, [@ywy Proposition 8.1] and [@s16 Theorem 6.28] for an exact proof.
\(ii) From [@fjw p.42], it follows that $\dot{F}^0_{1,\,2}(\rr^D)=\hon$ with equivalent norms. Obviously, for any $q\in[2,\fz)$, $\hon\st\tls$.
Now we recall the definition of the dual space of $\tls$, $\dtl$, from [@fj p.70], where $1/q+1/q'=1$.
\[da.b\] Let $q\in(1,\fz)$. Then the *homogeneous Triebel-Lizorkin space* $\dot{F}^{0,\,q}_\fz(\rr^D)$ is defined to be the set of all $f\in\schid$ such that $$\|f\|_{\dot{F}^0_{\fz,\,q}(\rr^D)}:=\sup_{\{Q:\ \rm dyadic\ cube\}}
\lf\{\frac1{|Q|}\int_{Q}\sum_{j=-\log_2\ell(Q)}^\fz\lf|\vz_j\ast f(x)\r|^q
\,dx\r\}^{1/q}<\fz,$$ where the supremum is taken over all dyadic cubes $Q$ in $\rr^D$ and $\ell(Q)$ denotes the *side length* of $Q$.
\[ra.c\] (i) From [@fjw p.42], it follows that $\dot{F}^0_{\fz,\,2}(\rr^D)=\bmo$ with equivalent norms.
\(ii) It was shown in [@fj (5.2)] that, for each $q\in(1,\fz)$, $\dtl$ is the dual space of $\tls$. In particular, $\bmo$ is the dual space of $\hon$, which was proved before in [@fs].
Next we recall the definition of the $1$-dimensional Meyer wavelets from [@w97]; see also [@lly; @m92; @QY] for a different version. Let $\Phi\in C^\fz(\rr)$, the *space of all infinitely differentiable functions on $\rr$*, satisfy $$\label{ph1}
0\le\Phi(\xi)\le\frac1{\sqrt{2\pi}}\quad {\rm for\ any\ }\xi\in\rr,$$ $$\label{ph2}
\Phi(\xi)=\Phi(-\xi)\quad {\rm for\ any\ }\xi\in\rr,$$ $$\label{ph3}
\Phi(\xi)=\frac1{\sqrt{2\pi}}\quad {\rm for\ any\ }\xi\in[-2\pi/3,2\pi/3],$$ $$\label{ph4}
\Phi(\xi)=0\quad {\rm for\ any\ }\xi\in(-\fz,4\pi/3]\cup[4\pi/3,\fz),$$ and $$\label{ph5}
\lf[\Phi(\xi)\r]^2+\lf[\Phi(\xi-2\pi)\r]^2=\frac1{2\pi}
\quad {\rm for\ any\ }\xi\in[0,2\pi].$$ In what follows, the *Fourier transform* and the *reverse Fourier transform* of a suitable function $f$ on $\rr^D$ are defined by $$\widehat{f}(\xi):=(2\pi)^{-D/2}\int_{\rr^D}e^{-i\xi x}f(x)\,dx \quad
{\rm for\ any\ }\xi\in\rr^D,$$ respectively, $$\check{f}(x):=(2\pi)^{-D/2}\int_{\rr^D}e^{ix\xi}f(\xi)\,d\xi \quad
{\rm for\ any\ }x\in\rr^D.$$
From [@w97 Proposition 3.2], it follows that $\phi:=\check{\Phi}$ (the *“farther" wavelet*) is a scaling function of a *multiresolution analysis* defined as in [@w97 Definition 2.2]. The *corresponding function* $m_{\phi}$ of $\phi$, satisfying $\widehat{\phi}(2\cdot)=m_{\phi}(\cdot)\widehat{\phi}(\cdot)$, is a $2\pi$-periodic function which equals $\sqrt{2\pi}\Phi(2\cdot)$ on the interval $[-\pi,\pi)$.
Furthermore, by [@w97 Theorem 2.20], we construct a $1$-dimensional wavelet $\psi$ (the *“mother" wavelet*) by setting $\widehat{\psi}(\xi):=e^{i\xi/2}m_{\phi}(\xi/2+\pi)\Phi(\xi/2)$ for any $\xi\in\rr$. It was shown in [@w97 Proposition 3.3] that $\psi$ is a real-valued $C^{\fz}(\rr)$ function, $\psi(-1/2-x)=\psi(-1/2+x)$ for all $x\in\rr$, and $$\label{ps1}
\supp\lf(\widehat{\psi}\r)\st[-8\pi/3,-2\pi/3]\cup[2\pi/3,8\pi/3].$$ Such a wavelet $\psi$ is called a *$1$-dimensional Meyer wavelet*.
Let $D\in\nn\cap[2,\fz)$ and $\vec{0}:=(\overbrace{0,\ldots,0}^{D\ {\rm times}})$. The $D$-dimensional Meyer wavelets are constructed by tensor products as follows. Let $x:=(x_1,\ldots,x_D)\in\rr^D$, $E_D:=\{0,1\}^D\bh\{\vec{0}\}$ and, for any $\lz:=(\lz_1,\ldots,\lz_D)\in E_D$, define $$\psi^{\lz}(x):=\phi^{\lz_1}(x_1)\cdots\phi^{\lz_D}(x_D),$$ with $\phi^{\lz_j}(x_j):=\phi(x_j)$ if $\lz_j=0$ and $\phi^{\lz_j}(x_j):=\psi(x_j)$ if $\lz_j=1$. As in [@w97], for any $(\lz,j,k)\in\blz_D:=\{(\lz,j,k):\ \lz\in E_D,\ j\in\zz,\ k\in\zz^D\}$ and $x\in\rr^D$, we let $\psi^\lz_{j,\,k}(x):=2^{Dj}\psi^{\lz}(2^jx-k)$ and, for $\lz=\vec{0}$ and any $k:=(k_1,\ldots,k_D)$, let $\psi^{\vec{0}}_{j,\,k}(x)
:=2^{Dj}\phi(2^jx_1-k_1)\cdots\phi(2^jx_D-k_D)$ and $\psi^{\vec{0}}(x):=\phi(x_1)\cdots\phi(x_D)$.
By [@w97 Proposition 3.1] and arguments of tensor products, we know that, for any $(\lz,j,k)\in\blz_D$, $\psi^{\lz}_{j,\,k}\in\schi$. Thus, for any $(\lz,j,k)\in\blz_D$ and any $f\in\schid$, let $a^{\lz}_{j,\,k}(f):=\langle f,\psi^{\lz}_{j,\,k}\rangle$, where $\langle \cdot,\cdot\rangle$ represents the duality between $\schid$ and $\schi$. From the proof of [@fjw Theorem (7.20)], it follows that, for any $f\in\schid$, $$\label{x.x}
f=\sum_{\lz\in E_D}\sum_{j\in\zz}\sum_{k\in\zz^D}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}\quad {\rm in}\quad
\schid.$$ Moreover, by [@w97 Proposition 5.2], we know that $\{\psi^{\lz}_{j,\,k}\}_{(\lz,j,k)\in\blz_D}$ is an orthonormal basis of $\ltw$.
For any $\ell\in\{1,\ldots,D\}$ and any $f\in\sch$, denote by $R_{\ell}(f)$ the *Riesz transform* of $f$, which is defined by setting $$\widehat{R_\ell(f)}(\xi):=-i\frac{\xi_\ell}{|\xi|}\widehat{f}(\xi)
\quad {\rm for\ any\ }\xi\in\rr^D.$$ Since and hold true, by [@yql (5.2)], we know that, for any $\ell\in\{1,\ldots,D\}$, $(\lz,j,k),\,(\wz{\lz},\wz{j},\wz{k})\in\blz_D$ and $|j-\wz{j}|\ge2$, we have $$\label{b.d}
\lf( R_{\ell}\lf(\psi^{\lz}_{j,\,k}\r),\psi^{\wz{\lz}}_{\wz{j},\,\wz{k}}
\r)=0,$$ where $(\cdot,\cdot)$ denotes the inner product in $\ltw$.
Now we recall the wavelet characterization of $\tls$ and $\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$ (see, for example, [@fjw Theorem (7.20)]). For $j\in \mathbb{Z}$ and $k=(k_1,\cdots, k_{D})\in \mathbb{Z}^{D}$, denote $Q_{j,k}=\prod\limits^{D}_{l=1} [2^{-j}k_l, 2^{-j}(1+k_{l})[$.
\[ta.d\] Let $q\in(1,\fz)$. Then
\(i) $f\in\tls$ if and only if $f\in\schid$ and $$\cj_f:=\lf\|\lf\{\sum_{(\lz,\,j,\,k)\in\blz_D}
\lf[2^{Dj}\lf|a^{\lz}_{j,\,k}(f)\r|\chi\lf(2^jx-k\r)\r]^q
\r\}^{1/q}\r\|_{\lon}<\fz,$$ where $\chi$ denotes the characteristic function of the cube $[0,1)^D$. Moreover, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\frac1C\|f\|_{\tls}\le\cj_f\le C\|f\|_{\tls}.$$
\(ii) $f\in \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$ if and only if $f\in\schid$ and there exists $C>0$ such that for all dyadic cube $Q$, $$\sum_{(\lz,\,j,\,k)\in\blz_D, Q_{j,k} \subset Q} 2^{(q-1) j D} |a^{\lambda}_{j,k}|^{q} \leq C|Q|.$$
\[ra.o\] By Remark \[ra.n\](ii) and Theorem \[ta.d\], we also obtain the wavelet characterization of $\hon$ as in [@m92 p.143].
To consider Fefferman-Stein type decomposition for $\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$, we need to study some properties relative to frequency. Hence we use Meyer wavelets to introduce the auxiliary function spaces $\loq$. We consider the linear functional on these function sets and consider some exchangeability of Riesz transform and some sums of orthogonal projector operator defined by Meyer wavelets.
Let $q\in(1,\fz)$ and $f\in\schid$. For any $s\in\zz$, $N\in\nn$ and $t\in\{0,\ldots,N+1\}$, let $$\label{b.a}
P_{s,\,N}f:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N\le j\le s\}}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}
\quad {\rm in}\quad \schid.$$ For each $t\in\{0,\ldots,N+1\}$, let, in $\schid$, $$\label{b.x}
T^{(1)}_{s,\,t,\,N}(f):=\begin{cases}
0, \ \ \ \ &t=0, \\
\displaystyle\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-t+1\le j\le s\}}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}, \ \ \ \
&t\in\{1,\ldots,N+1\}
\end{cases}$$ and $$\label{b.y}
T^{(2)}_{s,\,t,\,N}(f):=\begin{cases}
\displaystyle\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N\le j\le s-t\}}
a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}, \ \ \ \ &t\in\{0,\ldots,N\}, \\
0, \ \ \ \ &t=N+1.
\end{cases}$$
\[da.e\] Then the *space $\liq$* is defined to be the space of all $f\in\schid$ such that $$\|f\|_{\liq}:=\sup_{\{s\in\nn,\,N\in\nn\}}
\sup_{t\in\{0,\ldots,N+1\}}
\lf[\lf\|T^{(1)}_{s,\,t,\,N}(f)\r\|_{{\dot{F}^0_{\fz,\,q}(\rr^D)}}
+\lf\|T^{(2)}_{s,\,t,\,N}(f)\r\|_{\li}\r]<\fz.$$
It is easy to see that
For $1<q\leq \infty$, $\liq= L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$ are Banach spaces.
\[da.e111\] The relative *space $\loq$* is defined to be the space of all $f\in\schid$ such that $$\|f\|_{\loq}:=\sup_{\{s\in\nn,\,N\in\nn\}}
\min_{t\in\{0,\ldots,N+1\}}\lf[\lf\|T^{(1)}_{s,\,t,\,N}(f)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t,\,N}(f)\r\|_{\lon}\r]<\fz.$$ Further, for $f\in L^{1}(\mathbb{R}^{D}) \bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$, we define $$\|f\|_{\{1, q\}}:= \min (\|f\|_{L^{1}}, \|f\|_{\dot{F}^{0}_{1,q}}).$$ For $f\in L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{ 1,q}(\mathbb{R}^{D})$, we define $$\|f\|_{1,q}:= \inf\limits_{f+g\in L^{1} + \dot{F}^{0}_{1,q}} \{\|f\|_{L^{1}} + \|g\|_{\dot{F}^{0}_{1,q}}\}.$$
\[ra.k\] The spaces $\loq$ and $\liq$, with $q\in(1,\fz)$, when $D=1$ were introduced by Lin et al. [@lly p.693], respectively, [@lly p.694], which were denoted by $L^{1,\,q}(\rr)$, respectively, $L^{\fz,\,q}(\rr)$. To distinguish these spaces with the well-known Lorentz spaces, we use the notation $\loq$ and $\liq$ which indicate that these spaces are defined via wavelet expansions. Recall also that the space $\loq$ was also called the relative $L^1$ space in [@lly p.693].
We know, $\forall N\geq 1$, the function $P_{N}f(x)= \sum\limits_{(\epsilon,j,k)\in \Lambda_{D}, |j|+|k|\leq 2^{N}} a^{\lambda}_{j,k} \psi^{\lambda}_{j,k}(x)\in \schi$. Set $ A= \loq$ or $ L^{1}(\mathbb{R}^{D}) \bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ or $L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$. If $f\in A$, then $P_{N}f\in A$. It is easy to see that
\[pr.1.9\] For $1\leq q<\infty$,
\(i) $\loq$ is complete with the above induced norm.
\(ii) The functions in $\schi$ are dense in $A$.
\[re:Banach,function\] Let $q\in[2,\fz)$. It was shown in [@tr3 p.239] that the dual space of $\tls$ is $\dtl$. Further $\dot{F}^{0}_{1,2}(\mathbb{R}^{D})= H^{1}(\mathbb{R}^{D}) \subset L^{1}(\mathbb{R}^{D}).$ Hence $$\label{eq:Hardy}
L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,2}(\mathbb{R}^{D})={\rm WE}^{1,2} (\mathbb{R}^{D})= L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,2}(\mathbb{R}^{D})=L^{1}(\mathbb{R}^{D}).$$ Let $2<q<\infty$. $L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ are Banach spaces. $WE^{1,q} (\mathbb{R}^{D})$ are function sets, not Banach spaces. Moreover, the following equalities are [**not**]{} true $$L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})= {\rm WE}^{1,q} (\mathbb{R}^{D})= L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D}).$$ In fact, the above two equal signs both have to be changed to the inclusion sign “$\subset$".
For $A$, we can use distributions to define their dual elements.
For $1\leq q<\infty$ and the function set $A\subset L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$, we call $l$ to be a dual element of $A$, if $l\in \schid$ and $$\sup\limits_{f\in \schi, \|f\|_{A}\leq 1} |\langle l, f\rangle|<\infty.$$ We write $l\in A'$.
$A'$ is a linear space. In fact, for $\alpha, \beta\in \mathcal{C}$ and $l_1, l_2\in A'$, we know that $\alpha l_1 +\beta l_2\in A'$. Further, $L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ is the linearization function space of the set $L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ or the set $ {\rm WE}^{1,q} (\mathbb{R}^{D}).$ The dual elements on the set $L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ or on the set ${\rm WE}^{1,q} (\mathbb{R}^{D})$ are the same as which on the linear space $L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$. Now we are ready to state the first main auxiliary result of this paper.
\[th:111\] For $q\in [2,\infty)$, we have $$\big(L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'= \big({\rm WE}^{1,q} (\mathbb{R}^{D})\big)'=\big( L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'= L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
For $q=2$, due to the equation (\[eq:Hardy\]), the above Theorem \[th:111\] is evident. For general $q$, the proof of this theorem will be given in the final section.
Next we state the second main auxiliary result which will be needed in the proof of our Fefferman-Stein type decomposition. We will use certain exchangeability of Meyer wavelets and Riesz transform to prove Theorem \[ta.h\] in section 2.
\[ta.h\] Let $D\in\nn$ and $q\in[2,\fz)$. Then $f\in\schid$ belongs to $\tls$ if and only if $f\in\loq$ and $\{R_{\ell}(f)\}_{\ell=1}^D\st\loq$. Moreover, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\frac1C\|f\|_{\tls}\le\sum_{\ell=0}^D\|R_{\ell}(f)\|_{\loq}
\le C\|f\|_{\tls},$$ where $R_0:={\rm Id}$ denotes the *identity operator*.
\[ra.l\] If $D=1$, Theorem \[ta.h\] is just [@lly Theorem 1.3].
Fefferman-Stein decomposition says, for some function space $A$, there exists some space $B$ satisfying $B\nsubseteq A$ such that, for $f\in A$, there exist $f_{l}\in B$ such that $$f=\sum\limits^{D}_{l=0}R_{l} f_{l}.$$ The functions in B have better properties than those in A. But a function in $A$ has been written as a linear combination of a function in $B$ and the $n$ images of functions in $B$ under correspondingly the $n$ Riesz transformations. Such a result brings certain conveniences in PDE and in harmonic analysis. The following theorems \[ta.x\] and \[ta.i\] tell us that we have also Fefferman-Stein decomposition for ${\dot{F}^0_{\fz,\,q}(\rr^D)}$.
By Remark \[ra.k\](iv), we know that, for any $q\in[2,\fz)$, $\tls\st\loq$ and ${\rm WE}^{\fz,\,q'}(\rr^D)\st\dtl$. The following conclusions indicate that the above inclusions of sets are proper, which are extensions of [@lly Remark 1.8]. The proof of theorem \[ta.x\] will be given at section 3.
\[ta.x\] Let $D\in\nn$ and $q\in[2,\fz)$. Then
\(i) $\tls\subsetneqq L^{1}(\rr^D) \bigcup \tls \subset {\rm WE}^{1,\,q}(\rr^D)\subset L^{1}(\rr^D) + \tls$;
\(ii) $ {\rm WE}^{\fz,\,q'}(\rr^D)\subsetneqq\dtl.$
Combining Theorem \[ta.h\], Remark \[ra.k\](iv) and some arguments analogous to those used in the proof of [@lly Theorem 1.7], we obtain the following Fefferman-Stein decomposition of ${\dot{F}^0_{\fz,\,q}(\rr^D)}$, the proof will be given in the final section.
\[ta.i\] Let $D\in\nn$ and $q\in(1,2]$. Then $f\in{\dot{F}^0_{\fz,\,q}(\rr^D)}$ if and only if there exist $\{f_\ell\}_{\ell=0}^D\in \liq
%\st {\dot{F}^0_{\fz,\,q}(\rr^D)}
$ such that $f=f_0+\sum_{\ell=1}^D R_{\ell}\lf(f_\ell\r).$
By Theorems \[th:111\] and \[ta.i\], we have
\[ta.cor\] Let $D\in\nn$ and $q\in[2,\fz)$. Then $f\in\schid$ belongs to $\tls$ if and only if $f\in L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$ and $\{R_{\ell}(f)\}_{\ell=1}^D\st L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})$. Moreover, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\frac1C\|f\|_{\tls}\le\sum_{\ell=0}^D\|R_{\ell}(f)\|_{L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})}
\le C\|f\|_{\tls},$$ where $R_0:={\rm Id}$ denotes the *identity operator*.
\[ra.m\] Theorem \[ta.i\] when $D=1$ is just [@lly Theorem 1.7].
The organization of this article is as follows.
In Section \[s2\], via the definition of the space ${\rm WE}^{1,\,q}(\rr^D)$, the boundedness of Riesz transforms on $\tls$, the Riesz transform characterization of $\hon$ and some ideas from [@lly; @yql], we prove Theorem \[ta.h\], namely, establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$. Comparing with the corresponding proof of [@yql Subsection 6.2], the main innovation of this proof is that we regard the corresponding parts of the norms of Riesz transforms $\{R_\ell(f_{s_1,\, N_1})\}_{\ell=1}^D$ in ${\rm WE}^{1,\,q}(\rr^D)$ as a whole to choose $t^1_{s,\, N}\in\{0,\ldots, N+1\}$ such that below holds true, while not to choose $t^\ell_{s,\, N}\in\{0,\ldots, N+1\}$ such that below holds true for each $\ell\in\{1,\ldots,D\}$ separately as in [@yql (6.6)]. Using this technique, we successfully overcome those difficulties described in [@lly Remark 1.4].
In Section \[s3\], we prove Theorem \[ta.x\]. To this end, we first give a $1$-dimensional Meyer wavelet satisfying $\psi(0)\neq0$ (see Example \[ec.a\] below), which is taken from [@w97 Exercise 3.2]. By using such a $1$-dimensional Meyer wavelet satisfying $\psi(0)\neq0$, we then finish the proof of Theorem \[ta.x\] via tensor products and some arguments from the proof of [@lly Remark 1.8]. Comparing with that proof of [@lly Remark 1.8], we make an additional assumption that $\psi(0)\neq0$ here, which is needed in the estimate below.
In Section 4, we give the proof of Theorems \[th:111\], \[ta.i\] and \[ta.cor\].
Finally, we make some conventions on notation. Throughout the whole paper, $C$ stands for a [*positive constant*]{} which is independent of the main parameters, but it may vary from line to line. If, for two real functions $f$ and $g$, $f\le Cg$, we then write $f\ls g$; if $f\ls g\ls f$, we then write $f\sim g$. For $q\in(1,\fz)$, let $q'$ be the *conjugate number* of $q$ defined by $1/q+1/q'=1$. Let $\mathcal{C}$ be the set of complex numbers and $\nn:=\{1,2,\ldots\}$. Furthermore, $\langle\cdot,\cdot\rangle$ and $(\cdot,\cdot)$ represent the duality relation, respectively, the $\ltw$ inner product.
Proof of Theorem \[ta.h\] {#s2}
=========================
In this section, we prove Theorem \[ta.h\]. To this end, we need to recall some well known results.
The following conclusion is taken from [@fjw Corollary (8.21)].
\[ta.f\] Let $D\in\nn$ and $q\in(1,\fz)$. Then the Riesz transform $R_{\ell}$ for each $\ell\in\{1,\ldots,D\}$ is bounded on $\tls$.
\[ra.p\] From Remark \[ra.n\](ii) and Theorem \[ta.f\], it follows that the Riesz transform $R_{\ell}$ for each $\ell\in\{1,\ldots,D\}$ is bounded on $\hon$.
The Riesz transform characterization of $\hon$ can be found in [@s70 p.221].
\[ta.g\] Let $D\in\nn$. The space $\hon$ is isomorphic to the space of all functions $f\in\lon$ such that $\{R_{\ell}(f)\}_{\ell=1}^D\st\lon$. Moreover, there exists a positive constant $C$ such that, for all $f\in\hon$, $$\frac1C\|f\|_{\hon}\le\|f\|_{\lon}+\sum_{\ell=1}^D\|R_{\ell}(f)\|_{\lon}
\le C\|f\|_{\hon}.$$
The following lemma is completely analogous to [@lly Lemma 2.2], the details being omitted.
\[lb.j\] Let $D\in\nn$ and $q\in[2,\fz)$. If $f\in\loq$, then, for any $j\in\zz$, $Q_{j}(f)\in\hon$, where $Q_{j}(f):=\sum_{(\lz,\,k)\in E_D\times\zz^D}a^{\lz}_{j,\,k}(f)\psi^{\lz}_{j,\,k}$. Moreover, there exists a positive constant $C$ such that, for all $j\in\zz$ and $f\in\loq$, $$\lf\|Q_j(f)\r\|_{\hon}\le C\|f\|_{\loq}.$$
We first show the necessity of Theorem \[ta.h\]. By Remark \[ra.k\](iv) and Theorem \[ta.f\], we have $$\sum_{\ell=0}^D\|R_\ell(f)\|_{\loq}\ls\sum_{\ell=0}^D\|R_\ell(f)\|_{\tls}
\ls\|f\|_{\tls},$$ which completes the proof of the necessity of Theorem \[ta.h\].
Now we show the sufficiency of Theorem \[ta.h\]. To this end, for any $f\in\loq$ such that $\{R_{\ell}(f)\}_{\ell=1}^D\st\loq$, it suffices to show that, for any $s_1\in\zz$, $N_1\in\nn$ and $f_{s_1,\,N_1}:=P_{s_1,\,N_1}f$ defined as in , we have $$\label{b.b}
\lf\|f_{s_1,\,N_1}\r\|_{\tls}\ls\sum_{\ell=0}^D
\lf\|R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\loq},$$ where the implicit constant is independent of $s_1$, $N_1$ and $f$.
Indeed, assume that holds true for the time being. Owing to , for any $\ell\in\{1,\ldots,D\}$, there exists a sequence $\{f^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}
\st\cc$ such that $$R_{\ell}\lf(f_{s_1,\,N_1}\r)
:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s_1-N_1-1\le j\le s_1+1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}\quad{\rm in}\quad \schid.$$ By this and the orthogonality of $\{\psi^{\lz}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}$, we know that, for each $\ell\in\{1,\ldots,D\}$, $$\begin{aligned}
\label{b.x1}
&\lf\|R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\loq}\\
&\noz\hs=\sup_{\{\wz{s}\in\zz,\,\wz{N}\in\nn\}}\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}\r.\\
&\noz\hs\hs\lf.+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1+1,\,
\wz{s}-\wz{N}\ge s_1-N_1-1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}\r.\\
&\noz\hs\hs\lf.+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1+1,\,
\wz{s}-\wz{N}\ge s_1-N_1-1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}R_{\ell}(f)\r\|_{\tls}
+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
R_{\ell}(f)\r\|_{\lon}\r]\\
&\noz\hs\le\lf\|R_{\ell}(f)\r\|_{\loq}<\fz\end{aligned}$$ and, similarly, $$\begin{aligned}
\label{b.x2}
&\lf\|f_{s_1,\,N_1}\r\|_{\loq}\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1+1,\,
\wz{s}-\wz{N}\ge s_1-N_1-1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}\r.\\
&\noz\hs\hs\lf.+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}
\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1,\,
\wz{s}-\wz{N}\ge s_1-N_1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}(f_{s_1,\,N_1})\r\|_{\tls}
+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}(f_{s_1,\,N_1})\r\|_{\lon}\r]\\
&\noz\hs=\sup_{\gfz{\wz{s}\in\zz,\,\wz{N}\in\nn}{\wz{s}\le s_1,\,
\wz{s}-\wz{N}\ge s_1-N_1}}
\min_{t\in\{0,\ldots,\wz{N}+1\}}
\lf[\lf\|T^{(1)}_{\wz{s},\,t,\,\wz{N}}(f)\r\|_{\tls}
+\lf\|T^{(2)}_{\wz{s},\,t,\,\wz{N}}(f)\r\|_{\lon}\r]\\
&\noz\hs\le\|f\|_{\loq}<\fz.\end{aligned}$$ From , and , we deduce that $$\lf\|f_{s_1,\,N_1}\r\|_{\tls}\ls\sum_{\ell=0}^D
\lf\|R_{\ell}(f)\r\|_{\loq}.$$ This, together with Theorem \[ta.d\] and the Levi lemma, implies that $f\in\tls$ and $$\begin{aligned}
\|f\|_{\tls}
&\ls\lf\|\lf\{\sum_{(\lz,\,j,\,k)\in\blz_D}
\lf[2^{Dj}\lf|a^{\lz}_{j,\,k}(f)\r|\chi\lf(2^jx-k\r)\r]^q
\r\}^{1/q}\r\|_{\lon}\\
&\sim\lim_{N_1,s_1\to\fz}\lf\|\lf\{\sum_{\{(\lz,\,j,\,k)\in\blz_D:\
s_1-N_1\le j\le s_1\}}
\lf[2^{Dj}\lf|a^{\lz}_{j,\,k}(f)\r|\chi\lf(2^jx-k\r)\r]^q
\r\}^{1/q}\r\|_{\lon}\\
&\sim\lim_{N_1,\,s_1\to\fz}\lf\|f_{s_1,\,N_1}\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}(f)\r\|_{\loq},\end{aligned}$$ which are the desired conclusions.
Thus, to finish the proof of the sufficiency of Theorem \[ta.h\], we still need to prove . To this end, fix $s_1\in\zz$ and $N_1\in\nn$. In order to obtain the $\loq$-norms of $\{R_{\ell}(f_{s_1,\,N_1})\}_{\ell=0}^D$, by and , it suffices to consider $s:=s_1+1$ and $N:=N_1+2$ in and . For such $s$ and $N$, there exist $t^{(0)}_{s,\,N},\,t^{(1)}_{s,\,N}
\in\{0,\ldots,N+1\}$ such that $$\begin{aligned}
\label{b.g}
&\lf\|T^{(1)}_{s,\,t^{(0)}_{s,\,N},\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t^{(0)}_{s,\,N},\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\\
&\noz\hs=\min_{t\in\{0,\ldots,N+1\}}
\lf[\lf\|T^{(1)}_{s,\,t,\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t,\,N}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\end{aligned}$$ and $$\begin{aligned}
\label{b.f}
&\sum_{\ell=1}^D\lf[\lf\|T^{(1)}_{s,\,t^{(1)}_{s,\,N},\,N}R_{\ell}
\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t^{(1)}_{s,\,N},\,N}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r]\\
&\noz\hs=\min_{t\in\{0,\ldots,N+1\}}\sum_{\ell=1}^D
\lf[\lf\|T^{(1)}_{s,\,t,\,N}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\tls}
+\lf\|T^{(2)}_{s,\,t,\,N}R_{\ell}\lf(f_{s_1,\,N_1}\r)\r\|_{\lon}\r].\end{aligned}$$
In the remainder of this proof, to simplify the notation, *we let $g_1:=f_{s_1,\,N_1}$ for any fixed $s_1$ and $N_1$, $t_j:=t^{(j)}_{s,\,N}$ and $T_{i,\,j}:=T^{(i)}_{s,\,t^{(j)}_{s,\,N},\,N}$ for any $i\in\{1,2\}$ and $j\in\{0,1\}$*.
We then consider the following three cases.
**Case I**. $t_0=t_1$. In this case, we write $g_1=a_1+a_2$, where $$a_1:=\sum_{j=s-t_0+1}^s Q_j\lf(g_1\r)
\quad {\rm and}\quad a_2:=\sum_{j=s-N}^{s-t_0}
Q_j\lf(g_1\r).$$ By , we have $a_2=T_{2,\,0}(g_1)\in\lon$ and $$\label{x.o}
\lf\|a_2\r\|_{\lon}
=\lf\|T_{2,\,0}\lf(g_1\r)\r\|_{\lon}
\le\|g_1\|_{\loq},$$ which, together with Lemma \[lb.j\] and $\hon\st\lon$, further implies that $$Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\in\hon$$ and $$\begin{aligned}
\label{b.h}
&\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\hs\le \lf\|Q_{s-t_0}\lf(g_1\r)\r\|_{\hon}
+\lf\|Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, by this, $\hon\st\lon$ and , we obtain $$a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\in\lon$$ and $$\begin{aligned}
\label{b.i}
&\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\noz\hs\le\lf\|a_2\r\|_{\lon}
+\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Moreover, for each $\ell\in\{1,\ldots,D\}$, we have $$\begin{aligned}
\label{x.y}
T_{2,\,1}R_{\ell}\lf(g_1\r)
&=T_{2,\,1}R_{\ell}\lf(a_2
+Q_{s-t_0+1}\lf(g_1\r)\r)\\
&\noz=T_{2,\,1}R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)+Q_{s-t_0-1}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r)+T_{2,\,1}R_{\ell}Q_{s-t_0+1}\lf(g_1\r)\\
&\noz=R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)+Q_{s-t_0-1}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r)+T_{2,\,1}R_{\ell}Q_{s-t_0+1}\lf(g_1\r).\end{aligned}$$ Hence, by , , $\hon\st\lon$, Remarks \[ra.o\] and \[ra.p\], and Lemma \[lb.j\], we conclude that, for any $\ell\in\{1,\ldots,D\}$, $$\begin{aligned}
{\rm II}^{(\ell)}:&=R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)+Q_{s-t_0-1}\lf(g_1\r)\r]\r)
+T_{2,\,1}R_{\ell}Q_{s-t_0+1}\lf(g_1\r)\in\lon\end{aligned}$$ and $$\begin{aligned}
\label{b.j}
\lf\|{\rm II}^{(\ell)}\r\|_{\lon}
&\le\lf\|T_{2,\,1}R_{\ell}\lf(g_1\r)\r\|_{\lon}\\
&\noz\hs+\lf\|T_{2,\,1}R_{\ell}
\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)+Q_{s-t_0+1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}\\
&\noz\hs+\lf\|R_{\ell}\lf(Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)+Q_{s-t_0+1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}\\
&\noz\hs+\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)+Q_{s-t_0+1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}+
\lf\|g_1\r\|_{\loq}.\end{aligned}$$
From , it follows that, for each $\ell\in\{1,\ldots,D\}$, there exist $\{\tau^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}\st\cc$ such that $$\begin{aligned}
{\rm I}^{(\ell)}:&=R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N-1\le j\le s-t_0-1\}}
\tau^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}\end{aligned}$$ and $$R_{\ell}Q_{s-t_0+1}\lf(g_1\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-t_0\le j\le s-t_0+2\}}
\tau^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}.$$
For any $h\in\li$ and $j_0\in\zz$, let $$P_{j_0}(h):=\sum_{k\in\zz^D}\lf\langle h,\psi^{\vec{0}}_{j_0,\,k}
\r\rangle\psi^{\vec{0}}_{j_0,\,k},$$ where $\langle\cdot,\cdot\rangle$ represents the duality between $\li$ and $\lon$. We claim that $P_{j_0}(h)\in\li$. Indeed, by $
|\langle h,\psi^{\vec{0}}_{j_0,\,k}\rangle|\ls2^{-Dj_0/2}
$ and $\psi^{\vec{0}}\in\sch$, we know that, for all $x\in\rr^D$, $$\lf|P_{j_0}(h)(x)\r|\ls\sum_{k\in\zz^D}2^{-Dj_0/2}
\lf|\psi^{\vec{0}}_{j_0,\,k}(x)\r|
\ls\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}\lf(2^{j_0}x-k\r)\r|\ls1.$$ Let $$h_0:=P_{s-t_0}(h)=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\
j\le s-t_0-1\}}a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.$$ Thus, $h_0\in\li$ and $\|h_0\|_{\li}\ls1$ by the above claim.
Moreover, from , we observe that, for any $\ell\in\{1,\ldots,D\}$, $$\lf|\lf\langle {\rm I}^{(\ell)},h\r\rangle\r|
=\lf|\lf\langle {\rm I}^{(\ell)},h_0\r\rangle\r|
=\lf|\lf\langle {\rm II}^{(\ell)},h_0\r\rangle\r|
\le\lf\|{\rm II}^{(\ell)}\r\|_{\lon}\lf\|h_0\r\|_{\li},$$ which, combined with $\|h_0\|_{\li}\ls1$ and , further implies that $$\lf\|{\rm I}^{(\ell)}\r\|_{\lon}\ls\lf\|{\rm II}^{(\ell)}\r\|_{\lon}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ From this, Theorem \[ta.g\] and , it follows that $$a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\in\hon$$ and $$\begin{aligned}
&\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\hon}\\
&\hs\sim\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\hs\hs+\sum_{\ell=0}^D\lf\|R_{\ell}\lf(a_2
-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\lf\|g_1\r\|_{\loq}+
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq},\end{aligned}$$ which, together with Remark \[ra.n\](ii), and Lemma \[lb.j\], further implies that $$\begin{aligned}
\label{b.l}
\lf\|a_2\r\|_{\tls}&\ls\lf\|a_2\r\|_{\hon}\\
&\noz\ls\lf\|a_2-\lf[Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r]\r\|_{\hon}\\
&\noz\hs+\lf\|Q_{s-t_0}\lf(g_1\r)
+Q_{s-t_0-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\lf\|g_1\r\|_{\loq}+
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Furthermore, by , we find that $$\lf\|a_1\r\|_{\tls}=\lf\|T_{1,\,0}
\lf(g_1\r)\r\|_{\tls}\le\lf\|g_1\r\|_{\tls},$$ which, combined with , implies that $g_1=a_1+a_2\in\tls$ and $$\lf\|g_1\r\|_{\tls}\le\lf\|a_1\r\|_{\tls}
+\lf\|a_2\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ This finishes the proof of **Case I**.
**Case II**. $t_0>t_1$. In this case, we write $g_1=b_1+b_2+b_3$, where $$b_1:=\sum_{j=s-t_1+1}^s Q_j\lf(g_1\r),
\quad b_2:=\sum_{j=s-t_0+1}^{s-t_1}
Q_j\lf(g_1\r) \quad {\rm and}\quad b_3:=\sum_{j=s-N}^{s-t_0}
Q_j\lf(g_1\r).$$ Similar to , for any $\ell\in\{1,\ldots,D\}$, we know that $$\begin{aligned}
T_{2,\,1}R_{\ell}\lf(g_1\r)
&=T_{2,\,1}R_{\ell}\lf(b_3+b_2
+Q_{s-t_1+1}\lf(g_1\r)\r)\\
&=R_{\ell}\lf(b_3+b_2
-\lf[Q_{s-t_1}\lf(g_1\r)+Q_{s-t_1-1}\lf(g_1\r)\r]\r)\\
&\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)
+T_{2,\,1}R_{\ell}Q_{s-t_1+1}\lf(g_1\r)\\
&=:{\rm I}^{(\ell)}_1+ {\rm I}^{(\ell)}_2+{\rm I}^{(\ell)}_3.\end{aligned}$$
For any $h\in\li$, let $
h_1:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j\le s-t_1-1\}}
a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.
$ Similar to the proof of $h_0\in\li$, we have $h_1\in\li$ and $$\label{x.z}
\lf\|h_1\r\|_{\li}\ls1.$$ By , we know that, for any $\ell\in\{1,\ldots,D\}$, there exists a sequence $\{f^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}
\st\cc$ such that $${\rm I}^{(\ell)}_1=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-N-1\le j\le s-t_1-1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}\quad \mathrm{and}\quad
{\rm I}^{(\ell)}_3=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j=s-t_1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k},$$ which imply that $$\lf|\lf\langle {\rm I}^{(\ell)}_1,h\r\rangle\r|
=\lf|\lf\langle {\rm I}^{(\ell)}_1,h_1\r\rangle\r|
=\lf|\lf\langle {\rm I}^{(\ell)}_1+{\rm I}^{(\ell)}_3,h_1\r\rangle\r|
=\lf|\lf\langle T_{2,\,1}R_{\ell}\lf(g_1\r)
-{\rm I}^{(\ell)}_2,h_1\r\rangle\r|.$$ Hence, by this, , , $\hon\st\lon$, Remarks \[ra.o\] and \[ra.p\], and Lemma \[lb.j\], we conclude that $$\begin{aligned}
\label{b.u}
\lf\|{\rm I}^{(\ell)}_1\r\|_{\lon}
&\ls\lf[\lf\|T_{2,\,1}R_{\ell}\lf(g_1\r)\r\|_{\lon}
+\lf\|{\rm I}^{(\ell)}_2\r\|_{\lon}\r]\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|{\rm I}^{(\ell)}_2\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}+
\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, ${\rm I}^{(\ell)}_1\in\lon$. Moreover, by Remark \[ra.p\] and Lemma \[lb.j\], we have $$\begin{aligned}
\label{x.u}
&\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\hs\ls\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ From this, $\hon\st\lon$ and , we deduce that $$\begin{aligned}
\label{x.v}
\lf\|R_{\ell}\lf(b_2+b_3\r)\r\|_{\lon}
&\le\lf\|{\rm I}^{(\ell)}_1\r\|_{\lon}
+\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
On the other hand, for any $h\in\li$, let $
\wz{h}_0:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j\ge s-t_0+2\}}
a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.
$ Similar to the proof of $h_0\in\li$, we have $h-\wz{h}_0\in\li$ and $\|h-\wz{h}_0\|_{\li}\ls1$, which further implies that $$\label{x.w}
\lf\|\wz{h}_0\r\|_{\li}\le\lf\|h\r\|_{\li}
+\lf\|h-\wz{h}_0\r\|_{\li}\ls1.$$ By , we know that $$\begin{aligned}
&R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ s-t_0+2\le j\le s-t_1+1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k},\end{aligned}$$ which implies that $$\begin{aligned}
\label{x.n}
&\lf\langle R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r),h\r\rangle\\
&\noz\hs=\lf\langle R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r),\wz{h}_0\r\rangle\\
&\noz\hs=\lf\langle R_{\ell}\lf(b_3+b_2
-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r),\wz{h}_0\r\rangle.\end{aligned}$$ From an argument similar to that used in , it follows that $$\begin{aligned}
\label{x.t}
&\lf\|R_{\ell}\lf(Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\hs\ls\lf\|Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, by , , , $\hon\st\lon$ and , we conclude that $$\begin{aligned}
&\lf\|R_{\ell}\lf(b_2-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\lf\|R_{\ell}\lf(b_3+b_2
-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\lf\|R_{\ell}\lf(b_3+b_2\r)\r\|_{\lon}
+\lf\|R_{\ell}\lf(Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r)\r\|_{\hon}\\
&\hs\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$ Therefore, by this, $\hon\st\lon$ and , we obtain $$\begin{aligned}
\lf\|R_{\ell}\lf(b_2\r)\r\|_{\lon}
&\le\lf\|R_{\ell}\lf(b_2
-\lf[Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs+\lf\|R_{\ell}\lf(Q_{s-t_0+1}\lf(g_1\r)
+Q_{s-t_0+2}\lf(g_1\r)\r)\r\|_{\hon}\\
&\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq},\end{aligned}$$ which, together with , implies that $$\begin{aligned}
\label{x.s}
\lf\|R_{\ell}\lf(b_3\r)\r\|_{\lon}
&\le\lf\|R_{\ell}\lf(b_3+b_2\r)\r\|_{\lon}
+\lf\|R_{\ell}\lf(b_2\r)\r\|_{\lon}\\
&\noz\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Furthermore, by , we know that $$\lf\|b_3\r\|_{\lon}=
\lf\|T_{2,\,0}\lf(g_1\r)\r\|_{\lon}
\ls\lf\|g_1\r\|_{\loq},$$ which, combined with and Theorem \[ta.g\], implies that $b_3\in\hon$ and $$\begin{aligned}
\label{b.o}
\lf\|b_3\r\|_{\hon}
&\sim\lf\|b_3\r\|_{\lon}
+\sum_{\ell=1}^D\lf\|R_{\ell}\lf(b_3\r)\r\|_{\lon}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$ By and Remark \[ra.n\](ii), we know that $b_3\in\tls$ and $$\label{b.q}
\lf\|b_3\r\|_{\tls}\ls\lf\|b_3\r\|_{\hon}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$
Moreover, by , we obtain $$\lf\|b_1+b_2\r\|_{\tls}=
\lf\|T_{1,\,0}\lf(g_1\r)\r\|_{\tls}
\ls\lf\|g_1\r\|_{\loq},$$ which, together with and Remark \[ra.n\](ii), further implies that $$\lf\|g_1\r\|_{\tls}\le\lf\|b_1+b_2\r\|_{\tls}
+\lf\|b_3\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ This finishes the proof of **Case II**.
**Case III**. $t_0<t_1$. In this case, we write $g_1=e_1+e_2+e_3$, where $$e_1:=\sum_{j=s-t_0+1}^s Q_j\lf(g_1\r),
\quad e_2:=\sum_{j=s-t_1+1}^{s-t_0}
Q_j\lf(g_1\r) \quad{\rm and}\quad e_3:=\sum_{j=s-N}^{s-t_1}
Q_j\lf(g_1\r).$$ Similar to , for any $\ell\in\{1,\ldots,D\}$, we have $$\begin{aligned}
\label{x.m}
T_{2,\,1}R_{\ell}\lf(g_1\r)
&=T_{2,\,1}R_{\ell}\lf(e_3
+Q_{s-t_1+1}\lf(g_1\r)\r)\\
&\noz=R_{\ell}\lf(e_3
-\lf[Q_{s-t_1}\lf(g_1\r)+Q_{s-t_1-1}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{2,\,1}R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)+T_{2,\,1}R_{\ell}Q_{s-t_1+1}\lf(g_1\r)\\
&\noz=:{\rm II}^{(\ell)}_1+ {\rm II}^{(\ell)}_2+{\rm II}^{(\ell)}_3.\end{aligned}$$ For any $h\in\li$, let $$h_2:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j\le s-t_1-1\}}
a^{\lz}_{j,\,k}(h)\psi^{\lz}_{j,\,k}.$$ By an argument similar to that used in the proof of $h_0\in\li$, we conclude that $h_2\in\li$ and $\|h_2\|_{\li}\ls1$. By , we know that, for any $\ell\in\{1,\ldots,D\}$, there exists a sequence $\{f^{\lz,\,\ell}_{j,\,k}\}_{(\lz,\,j,\,k)\in\blz_D}
\st\cc$ such that $${\rm II}^{(\ell)}_1=\sum_{\{(\lz,\,j,\,k)
\in\blz_D:\ s-N-1\le j\le s-t_1-1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k}$$ and $$T_{2,\,1}R_{\ell}Q_{s-t_1+1}\lf(g_1\r)
=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\ j=s-t_1\}}
f^{\lz,\,\ell}_{j,\,k}\psi^{\lz}_{j,\,k},$$ which, together with , imply that $$\lf|\lf\langle {\rm II}^{(\ell)}_1,h\r\rangle\r|
=\lf|\lf\langle {\rm II}^{(\ell)}_1,h_2\r\rangle\r|
=\lf|\lf\langle {\rm II}^{(\ell)}_1+{\rm II}^{(\ell)}_3,h_2\r\rangle\r|
=\lf|\lf\langle T_{2,\,1}R_{\ell}\lf(g_1\r)
-{\rm II}^{(\ell)}_2,h_2\r\rangle\r|.$$ Hence, by this, $\|h_2\|_{\li}\ls1$, , $\hon\st\lon$, Remarks \[ra.o\] and \[ra.p\], and Lemma \[lb.j\], we conclude that $$\begin{aligned}
\label{b.m}
\lf\|{\rm II}^{(\ell)}_1\r\|_{\lon}
&\ls\lf\|T_{2,\,1}R_{\ell}\lf(g_1\r)\r\|_{\lon}
+\lf\|{\rm II}^{(\ell)}_2\r\|_{\lon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|{\rm II}^{(\ell)}_2\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|R_{\ell}\lf(Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}\\
&\noz\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}+
\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Thus, ${\rm II}^{(\ell)}_1\in\lon$.
By , we have $e_2+e_3\in\lon$ and $$\label{b.s}
\lf\|e_2+e_3\r\|_{\lon}
=\lf\|T_{2,\,0}\lf(g_1\r)\r\|_{\lon}
\ls\lf\|g_1\r\|_{\loq}.$$ For any $h\in\li$, let $$\wz{h}_1:=\sum_{\{(\lz,\,j,\,k)\in\blz_D:\
j\le s-t_1-2\}}h^{\lz}_{j,\,k}\psi^{\lz}_{j,\,k}.$$ Similar to the proof of $h_0\in\li$, we have $\|\wz{h}_1\|_{\li}\ls1$. We notice that $$\begin{aligned}
&\lf\langle e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r],h\r\rangle\\
&\hs=\lf\langle e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r],\wz{h}_1\r\rangle
=\lf\langle e_3,\wz{h}_1\r\rangle
=\lf\langle e_3+e_2,\wz{h}_1\r\rangle.\end{aligned}$$ Therefore, by this, and $\|\wz{h}_1\|_{\li}\ls1$, we obtain $$\begin{aligned}
\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\lon}&\ls\lf\|e_3+e_2\r\|_{\lon}
\ls\lf\|g_1\r\|_{\loq}.\end{aligned}$$ Hence $e_3-[Q_{s-t_1}(g_1)
+Q_{s-t_1-1}(g_1)]\in\lon$. From this, and Theorem \[ta.g\], we deduce that $e_3-[Q_{s-t_1}(g_1)
+Q_{s-t_1-1}(g_1)]\in\hon$ and $$\begin{aligned}
&\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\hon}\\
&\hs\sim\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\hs\hs+\sum_{\ell=1}^D\lf\|R_{\ell}
\lf(e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r)\r\|_{\lon}\\
&\hs\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$ Then, by this, $\hon\st\lon$ and Lemma \[lb.j\], we know that $e_3\in\lon$ and $$\begin{aligned}
\label{b.z}
\lf\|e_3\r\|_{\lon}
&\le\lf\|e_3-\lf[Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r]\r\|_{\lon}\\
&\noz\hs+\lf\|Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_1-1}\lf(g_1\r)\r\|_{\hon}\\
&\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\noz\end{aligned}$$ For each $\ell\in\{1,\ldots, D\}$, we observe that $$\begin{aligned}
T_{1,\,1}R_{\ell}\lf(g_1\r)
=T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_1}\lf(g_1\r)\r).\end{aligned}$$ By this and , we know that, for any $\ell\in\{1,\ldots, D\}$, $$T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_1}\lf(g_1\r)\r)\in\tls$$ and $$\lf\|T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_1}\lf(g_1\r)\r)\r\|_{\tls}\ls
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.$$ This, together with $$\label{b.tx}
\lf\|e_1\r\|_{\tls}
=\lf\|T_{1,\,0}\lf(g_1\r)\r\|_{\tls}
\le\lf\|g_1\r\|_{\loq}\quad({\rm see}\ (\ref{b.g})),$$ Theorems \[ta.d\] and \[ta.f\], and , further implies that, for each $\ell\in\{1,\ldots,D\}$, $$\begin{aligned}
\label{b.v}
&\lf\|T_{1,\,1}R_{\ell}\lf(e_2
+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\noz\hs\le\lf\|T_{1,\,1}R_{\ell}\lf(e_1+e_2
+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}
+\lf\|T_{1,\,1}R_{\ell}\lf(e_1\r)\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|R_{\ell}\lf(e_1\r)\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|e_1\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}
+\lf\|g_1\r\|_{\loq}.\end{aligned}$$
Furthermore, for any $\ell\in\{1,\ldots,D\}$, we notice that $$\begin{aligned}
\label{x.l}
T_{1,\,1}R_{\ell}\lf(e_2
+Q_{s-t_1}\lf(g_1\r)\r)&=R_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r]\r)\\
&\noz\hs+T_{1,\,1}R_{\ell}
\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r).\end{aligned}$$ By Theorems \[ta.d\] and \[ta.f\], Remark \[ra.n\](ii) and Lemma \[lb.j\], we conclude that $$\begin{aligned}
&\lf\|T_{1,\,1}R_{\ell}
\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\hs\ls\lf\|R_{\ell}\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\hs\ls\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r\|_{\tls}\\
&\hs\ls\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\tls},\end{aligned}$$ which, together with and , implies that $$\begin{aligned}
\label{y.u}
&\lf\|R_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r)\r\|_{\tls}\\
&\noz\hs\le\lf\|T_{1,\,1}R_{\ell}\lf(e_2
+Q_{s-t_1}\lf(g_1\r)\r)\r\|_{\tls}\\
&\noz\hs\hs+\lf\|T_{1,\,1}R_{\ell}
\lf(Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r)\r\|_{\tls}\\
&\noz\hs\ls\lf\|g_1\r\|_{\loq}+
\sum_{\ell=1}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Now we need a useful identity from [@sw p.224, (2.9)] that, for all $f\in\ltw$, $$\label{y.x}
\sum_{\ell=1}^D R^2_\ell(f)=-f.$$
From $e_2-[Q_{s-t_1+1}(g_1)
+Q_{s-t_0}(g_1)]\in\ltw$ and , we deduce that $$\begin{aligned}
e_2-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]=\sum_{\ell=1}^D R^2_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)+Q_{s-t_0}\lf(g_1\r)\r]\r)
\in\tls,\end{aligned}$$ which, combined with Theorem \[ta.f\] and , implies that $$\begin{aligned}
\label{y.y}
&\lf\|e_2-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r\|_{\tls}\\
&\noz\hs\le\sum_{\ell=1}^D \lf\|R^2_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r)\r\|_{\tls}\\
&\noz\hs\ls\sum_{\ell=1}^D \lf\|R_{\ell}\lf(e_2
-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r)\r\|_{\tls}\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Again, by Remark \[ra.n\](ii) and Lemma \[lb.j\], we obtain $$\begin{aligned}
\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r\|_{\tls}\!\ls\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r\|_{\hon}
\ls\lf\|g_1\r\|_{\loq},\end{aligned}$$ which, together with , implies that $$\begin{aligned}
\label{b.w}
\lf\|e_2\r\|_{\tls}
&\le\lf\|e_2-\lf[Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r]\r\|_{\tls}\\
&\noz\hs+\lf\|Q_{s-t_1+1}\lf(g_1\r)
+Q_{s-t_0}\lf(g_1\r)\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq}.\end{aligned}$$
Combining with , and , we obtain $$\begin{aligned}
\lf\|g_1\r\|_{\tls}&\le\sum_{j=1}^3\lf\|e_j\r\|_{\tls}
\ls\sum_{\ell=0}^D\lf\|R_{\ell}\lf(g_1\r)\r\|_{\loq},\end{aligned}$$ which completes the proof of **Case III** and hence Theorem \[ta.h\].
Proof of Theorem \[ta.x\] {#s3}
=========================
In this section, we prove Theorem \[ta.x\]. To this end, we first provide a $1$-dimensional Meyer wavelet satisfying $\psi(0)\neq0$, which is taken from [@w97 Exercise 3.2].
\[ec.a\] Let $$f(x):=\begin{cases}
e^{-1/x^2}, \ \ \ \ &x\in(0,\fz),\\
0, \ \ \ \ &x\in(-\fz,0],
\end{cases}$$ $f_1(x):=f(x)f(1-x)$ and $g(x):=\lf[\int_{-\fz}^\fz f_1(t)\,dt\r]^{-1}
\int_{-\fz}^x f_1(t)\,dt$ for all $x\in\rr$. Let $\xi\in\rr$ and $\Phi(\xi):=\frac1{\sqrt{2\pi}}\cos
(\frac{\pi}2 g(\frac{3}{2\pi}|\xi|-1))$. Then, by [@w97 Exercise 3.2], we know that $\Phi\in C^\fz(\rr)$ satisfies through . Following the construction of the $1$-dimensional Meyer wavelet, we obtain the “father" wavelet $\phi$, the corresponding function $m_{\phi}$ of $\phi$, and the “mother" wavelet $\psi$.
By the proof of [@w97 Proposition 3.3(ii)], we know that, for any $x\in\rr$, $$\psi(x)=\frac1{\sqrt{2\pi}}\int_{-\fz}^\fz
\cos\lf([x+1/2]\xi\r)\az(\xi)\,d\xi,$$ where, for each $\xi\in\rr$, $\az(\xi):=m_{\phi}(\xi/2+\pi)\Phi(\xi/2)$ is an even function supported in $[-8\pi/3,-2\pi/3]\cup[2\pi/3,8\pi/3]$.
Now we show that $\psi(0)\neq0$. Indeed, by the facts that $\az$ is even, $\supp(\az)\st[-8\pi/3,-2\pi/3]\cup[2\pi/3,8\pi/3]$ and $m_{\phi}$ is $2\pi$-periodic, we have $$\begin{aligned}
\psi(0)&=\frac2{\sqrt{2\pi}}\int_0^\fz \cos(\xi/2)\az(\xi)\,d\xi
=\frac2{\sqrt{2\pi}}\int_{2\pi/3}^{8\pi/3}\cos(\xi/2)\az(\xi)\,d\xi\\
&=\frac2{\sqrt{2\pi}}\int_{2\pi/3}^{8\pi/3}\cos(\xi/2)
m_{\phi}(\xi/2+\pi-2\pi)\Phi(\xi/2)\,d\xi\\
&=2\int_{2\pi/3}^{8\pi/3}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi\\
&=2\int_{2\pi/3}^{\pi}\cos(\xi/2)\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi
+2\int_{\pi}^{4\pi/3}\cdots+2\int_{4\pi/3}^{8\pi/3}\cdots
=:{\rm J}_1+{\rm J}_2+{\rm J}_3.\end{aligned}$$
We first estimate ${\rm J}_1$. Observe that, for any $\xi\in[2\pi/3,\pi]$, by , we know that $\Phi(\xi/2)=1/\sqrt{2\pi}$. Moreover, from $\frac12\le \frac{3}{2\pi}|\xi-2\pi|-1\le1$ and the construction of $g$, we deduce that $g(\frac12)\le g(\frac{3}{2\pi}|\xi-2\pi|-1)\le g(1)=1$. Hence, by the construction of $\Phi$, we obtain $$0\le\Phi(\xi-2\pi)=\frac1{2\pi}\cos\lf(\frac{\pi}2 g\lf(
\frac{3}{2\pi}|\xi-2\pi|-1\r)\r)\le\frac1{\sqrt{2\pi}}\cos\lf(\frac{\pi}2
g\lf(\frac12\r)\r),$$ which further implies that $$\begin{aligned}
\lf|{\rm J}_1\r|&\le2\int_{2\pi/3}^{\pi}\cos(\xi/2)\frac1{\sqrt{2\pi}}
\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\frac1{\sqrt{2\pi}}\,d\xi\\
&=\frac1{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
\int_{2\pi/3}^{\pi}\cos(\xi/2)\,d\xi
=\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r).\end{aligned}$$
Now we deal with ${\rm J}_2$. Observe that, for any $\xi\in[\pi,4\pi/3]$, by , we know that $\Phi(\xi/2)=1/\sqrt{2\pi}$. Moreover, from $0\le \frac{3}{2\pi}|\xi-2\pi|-1\le\frac12$ and the construction of $g$, it follows that $0=g(0)\le g(\frac{3}{2\pi}|\xi-2\pi|-1)\le g(\frac12)$. Hence, by the construction of $\Phi$ again, we have $$1\ge\Phi(\xi-2\pi)\ge\frac1{\sqrt{2\pi}}\cos\lf(\frac{\pi}2
g\lf(\frac12\r)\r),$$ which further implies that $$\begin{aligned}
\lf|{\rm J}_2\r|&=-2\int^{4\pi/3}_{\pi}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi\\
&\ge-2\int^{4\pi/3}_{\pi}\cos(\xi/2)\frac1{\sqrt{2\pi}}
\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\frac1{\sqrt{2\pi}}\,d\xi\\
&=-\frac1{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
\int^{4\pi/3}_{\pi}\cos(\xi/2)\,d\xi
=\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r).\end{aligned}$$
Finally, we estimate ${\rm J}_3$. We first write $$\label{x.r}
\lf|{\rm J}_3\r|=-2\int^{8\pi/3}_{4\pi/3}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi
\ge-2\int^{2\pi}_{4\pi/3}\cos(\xi/2)
\Phi(\xi-2\pi)\Phi(\xi/2)\,d\xi.$$ Observe that, for any $\xi\in[4\pi/3,2\pi]$, by , we know that $\Phi(\xi-2\pi)=1/\sqrt{2\pi}$. Moreover, by $0\le \frac{\xi}2\frac{3}{2\pi}-1\le\frac12$ and the construction of $g$, we conclude that $0=g(0)\le g(\frac{\xi}2\frac{3}{2\pi}-1)\le g(\frac12)$. Hence, from the construction of $\Phi$, we deduce that $$1\ge\Phi(\xi/2)\ge\frac1{\sqrt{2\pi}}\cos\lf(\frac{\pi}2
g\lf(\frac12\r)\r),$$ which, together with , further implies that $$\begin{aligned}
\lf|{\rm J}_3\r|
&\ge-2\int_{4\pi/3}^{2\pi}\cos(\xi/2)\frac1{\sqrt{2\pi}}
\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\frac1{\sqrt{2\pi}}\,d\xi\\
&=-\frac1{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
\int_{4\pi/3}^{2\pi}\cos(\xi/2)\,d\xi
=\frac{\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r).\end{aligned}$$
Combining the estimates of ${\rm J}_1$, ${\rm J}_2$ and ${\rm J}_3$ and the construction of $g$, we obtain $$\begin{aligned}
|\psi(0)|&\ge\lf|{\rm J}_3+{\rm J}_2\r|-\lf|{\rm J}_1\r|
=\lf|{\rm J}_3\r|+\lf|{\rm J}_2\r|-\lf|{\rm J}_1\r|\\
&\ge\frac{\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
+\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)
-\frac{2-\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)\\
&=\frac{\sqrt{3}}{\pi}\cos\lf(\frac{\pi}2g\lf(\frac12\r)\r)>0,\end{aligned}$$ which completes the proof of Example \[ec.a\].
Now we are ready to prove Theorem \[ta.x\].
\(I) Suppose that $q\in[2,\fz)$, $\phi$ and $\Phi$ are defined as in the construction of the $1$-dimensional Meyer wavelets. Moreover, we assume that the $1$-dimensional Meyer wavelet $\psi$ satisfies $\psi(0)\neq0$.
For any $x:=(x_1,\ldots,x_D)\in\rr^D$, let $\psi^{\vec{0}}(x):=\phi(x_1)\cdots\phi(x_D)$. From [@lly (5.1)], we deduce that, for all $x\in\rr^D$, $$\label{c.a}
\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}(x-k)\r|
=\prod_{\ell=1}^D\sum_{k_\ell\in\zz}\lf|\phi\lf(x_\ell-k_\ell\r)\r|\ls1.$$
For any $j\in\zz$, $k\in\zz^D$ and $x\in\rr^D$, we write $\psi^{\vec{0}}_{j,\,k}(x):=2^{Dj/2}\psi^{\vec{0}}\lf(2^jx-k\r)$. Let $f\in\lon$. For any $j\in\zz$, define $P_j(f):=\sum_{k\in\zz^D}\langle f,\psi^{\vec{0}}_{j,\,k}\rangle
\psi^{\vec{0}}_{j,\,k}$. Then, for any $j\in\zz$, by , $$\begin{aligned}
\label{c.b}
\lf\|P_j(f)\r\|_{\lon}&\le\int_{\rr^D}\int_{\rr^D}
|f(y)|\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}(2^jy-k)\r|
\lf|2^{Dj}\psi^{\vec{0}}(2^jx-k)\r|\,dx\,dy\\
&\noz\ls\int_{\rr^D}\int_{\rr^D}
|f(y)|\sum_{k\in\zz^D}\lf|\psi^{\vec{0}}(x-k)\r|\,dy\ls\|f\|_{\lon}.\end{aligned}$$
The inclusion relation in (i) of Theorem \[ta.x\] is easy to see. In order to prove (i) of Theorem \[ta.x\], it suffices to show that $\tls\subsetneqq L^{1}( \rr^D )\bigcup \tls$.We first observe that $\psi^{\vec{0}}\in\lon$. Indeed, $$\lf\|\psi^{\vec{0}}\r\|_{\lon}=\prod_{\ell=1}^D\lf\|\phi\r\|_{\lon}
=\lf\|\phi\r\|_{\lon}^D<\fz.$$ To show $\psi^{\vec{0}}\not\in\tls$, let $a_{j,\,k}(\psi^{\vec{0}}):=(\psi^{\vec{0}},\psi_{j,\,k})$ for any $j\in\zz$ and $k\in\zz^D$, where $(\cdot,\cdot)$ represents the $\ltw$ inner product. Let $\xi:=(\xi_1,\ldots,\xi_D),\,\eta:=(\eta_1,\ldots,\eta_D)\in\rr^D$. Then, by the multiplication formula (see [@sw p.8, Theorem 1.15]), , , and the assumption $\psi(0)\neq0$, we obtain $$\begin{aligned}
\label{3.4x}
\lf|a_{j,\,0}\lf(\psi^{\vec{0}}\r)\r|&=\prod_{\ell=1}^D\lf|\int_{\rr}
\widehat{\phi}\lf(-\xi_\ell\r)
2^{-j/2}\widehat{\psi}\lf(2^{-j}\xi_\ell\r)\,d\xi_\ell\r|\\
&=\prod_{\ell=1}^D\lf|\int_{-4\pi/3}^{4\pi/3}
\Phi\lf(\xi_\ell\r)
2^{-j/2}\widehat{\psi}\lf(2^{-j}\xi_\ell\r)\,d\xi_\ell\r|\noz\\
&=\prod_{\ell=1}^D\lf|\int_{-2^{2-j}\pi/3}^{2^{2-j}\pi/3}
\Phi\lf(2^j\eta_\ell\r)2^{j/2}\widehat{\psi}\lf(\eta_\ell\r)\,d\eta_\ell\r|\noz\\
&=\prod_{\ell=1}^D\lf|\int_{-8\pi/3}^{8\pi/3}
\Phi\lf(2^j\eta_\ell\r)2^{j/2}\widehat{\psi}\lf(\eta_\ell\r)\,d\eta_\ell\r|\noz\\
&\sim2^{Dj/2}\prod_{\ell=1}^D\lf|\int_{-8\pi/3}^{8\pi/3}
\widehat{\psi}\lf(\eta_\ell\r)
\,d\eta_\ell\r|\sim2^{Dj/2}|\psi(0)|^D\gtrsim2^{Dj/2}\noz,\end{aligned}$$ provided that $j<-M$ for some positive integer $M$ large enough. Therefore, we have $$\begin{aligned}
&\int_{\rr^D}\int_{\rr^D}
\lf\{\sum_{j\in\zz,\,k\in\zz^D}\lf[2^{Dj/2}\lf|a_{j,\,k}\lf(\psi^{\vec{0}}\r)\r|
\chi\lf(2^jx-k\r)\r]^q\r\}^{1/q}\,dx\\
&\hs\ge\int_{\rr^D}
\lf\{\sum_{j=-\fz}^{-M-1}2^{Djq/2}\lf|a_{j,\,0}\lf(\psi^{\vec{0}}\r)\r|^q
\chi\lf(2^jx-k\r)\r\}^{1/q}\,dx\\
&\hs\gtrsim\int_{\rr^D}
\lf\{\sum_ {j=-\fz}^{-M-1}2^{Djq}\chi\lf(2^jx-k\r)\r\}^{1/q}\,dx\\
&\hs\gtrsim\sum_{m=M}^\fz\int_{B(0,\,2^{m+1})\bh B(0,\,2^m)}
\lf\{\sum_{j=-\fz}^{-m-1}2^{Djq}\r\}^{1/q}\,dx=\fz,\end{aligned}$$ which, combined with Theorem \[ta.d\], implies that $\psi^{\vec{0}}\not\in\tls$. This finishes the proof of (i) of Theorem \[ta.x\].
\(II) Then we use Daubechies wavelets to proof (ii) of Theorem \[ta.x\]. We know that there exist some integer $M$ and a Daubechies scale function $\Phi^{0}(x)\in C^{D+2}_{0} ([-2^{M}, 2^{M}]^{D})$ satisfying $$\label{5.1}
C_{D}=\int \frac{-y_{1}}{|y|^{n+1}}
\Phi^{0}(y-2^{M+1}e) dy<0,$$ where $e= (1,1,\cdots, 1)$. Let $\Phi(x)= \Phi^{0}(x-2^{M+1}e)$ and let $f$ be defined as $$\label{5.2}
f(x)=\sum\limits_{j\in 2\mathbb{N}} \Phi(2^{j}x).$$
For $j,j'\in 2\mathbb{N}, j\neq j'$, the supports of $\Phi(2^{j}x)$ and $\Phi(2^{j'}x)$ are disjoint. Hence the above $f(x)$ in (\[5.2\]) belongs to $L^{\infty}(\mathbb{R}^{D})$. The same reasoning gives, for any $j'\in \mathbb{N}$, $$\sum\limits_{j\in \mathbb{N}, 2j> j'} \Phi(2^{2j}x)\in
L^{\infty}(\mathbb{R}^{D}).$$ Now we compute the wavelet coefficients of $f(x)$ in (\[5.2\]). For $(\lambda',j',k')\in \Lambda_{D}$, let $f^{\lambda'}_{j',k'}=
\langle f,\ \Phi^{\lambda'}_{j',k'}\rangle$. We divide two cases: $j'<0$ and $j'\geq 0$.
For $j'<0$, since the support of $f$ is contained in $[-3 \cdot
2^{M}, 3 \cdot 2^{M}]^{D}$, we know that if $|k'|> 2^{2M+5}$, then $f^{\lambda'}_{j',k'}=0$. If $|k'|\leq 2^{2M+5}$, we have $$|f^{\lambda'}_{j',k'}| \leq C 2^{Dj'} \int |f(x)| dx
\leq C 2^{Dj'}.$$ For $j'\geq 0$, by orthogonality of the wavelets, we have $$f^{\lambda'}_{j',k'}= \Big\langle f,\ \Phi^{\lambda'}_{j',k'}\Big\rangle
= \Big\langle \sum\limits_{j\in \mathbb{N}, 2j> j'} \Phi(2^{2j}\cdot),\
\Phi^{\lambda'}_{j',k'}\Big\rangle.$$ By the same reasoning, for the case $j'\geq0$, we know that if $|k'|> 2^{2M+5}$, then $f^{\lambda'}_{j',k'}=0$. Since $\sum\limits_{j\in \mathbb{N}, 2j>
j'} \Phi(2^{2j}x)\in L^{\infty}$, if $|k'|\leq 2^{2M+5}$, we have $$|f^{\lambda'}_{j',k'}| \leq C\int
|\Phi^{\lambda'}_{j',k'}(x)| dx \leq C .$$ By the above estimation of wavelet coefficients of $f(x)$ and by the wavelet characterization of $\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$ in (ii) of Theorem \[ta.d\], we conclude that $f\in \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$. That is to say, $$\label{eq:1111}f\in L^{\infty}(\mathbb{R}^{D})\bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
Since $\Phi^{0}\in C^{D+2}_{0}([-2^{M}, 2^{M}]^{D})$, we know that $$\Phi(x)= \Phi^{0}(x-2^{M+1}e)\in C^{D+2}_{0}([2^{M}, 3 \cdot 2^{M}]^{D}).$$ Further, if $|x|\leq 2^{M-1}$ and $y\in [2^{M},
3 \cdot 2^{M}]^{D} $, then $|x-y|> 2^{M-1}$. Hence $R_{1}\Phi(x)$ is smooth in the ball $\{x:\ |x|\leq 2^{M-1}\}$.
Applying (\[5.1\]), there exists a positive $\delta>0$ such that for $|x|<\delta$, there holds $R_{1}\Phi(x)<\frac{C_{D}}{2}<0.$ That is to say, if $2^{2j}|x|<\delta$, then $R_{1}\Phi(2^{2j}x)<\frac{C_{D}}{2}<0$. Hence $$\label{eq:1112} R_{1}f(x)\notin L^{\infty}(\mathbb{R}^{D}).$$
The equations (\[eq:1111\]), (\[eq:1112\]) and the continuity of Riesz operators on $\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$ implies the conclusion (ii).
The proof of Theorems \[th:111\], \[ta.i\] and \[ta.cor\]
=========================================================
We prove first Theorem \[th:111\]
If $l \in (L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D}))'$, then $$\sup\limits_{f\in \schi, \|f\|_{L^{1} \bigcup \dot{F}^{0}_{1,q} }\leq 1} |\langle l, f\rangle|<\infty.$$ That is to say, $$\label{e1e}\sup\limits_{f\in \schi, \|f\|_{L^{1} }\leq 1} |\langle l, f\rangle|<\infty \mbox { and }$$ $$\label{e2e}\sup\limits_{f\in \schi, \|f\|_{ \dot{F}^{0}_{1,q} }\leq 1} |\langle l, f\rangle|<\infty.$$ The condition (\[e1e\]) means $l\in L^{\infty}(\mathbb{R}^{D})$, the condition (\[e2e\]) means $l\in \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$. Hence we have the following inclusion relation: $$\label{eq:inc.1}
\big(L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)' \subset L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
Further, it is known that $$L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D}) \subset {\rm WE}^{1,q} (\mathbb{R}^{D}) \subset L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D}).$$ Hence we have $$\label{eq:inc.2}
\big(L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'\subset \big( {\rm WE}^{1,q} (\mathbb{R}^{D})\big)' \subset \big(L^{1}(\mathbb{R}^{D})\bigcup \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'.$$
Moreover, we know that $$\label{eq:inc.3}
\big(L^{1}(\mathbb{R}^{D})+ \dot{F}^{0}_{1,q}(\mathbb{R}^{D})\big)'= L^{\infty}(\mathbb{R}^{D}) \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$
The equations (\[eq:inc.1\]) , (\[eq:inc.2\]) and (\[eq:inc.3\]) implies the Theorem \[th:111\].
Then we prove Theorem \[ta.i\].
By the continuity of the Riesz operators on the $\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$, we know that if $f_{l}\in \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})\bigcap
L^{\infty}(\mathbb{R}^{D})$, then $$\sum\limits_{0\leq l\leq D}R_{l}f_{l}(x) \in
\dot{F}^{0}_{\infty,q}(\mathbb{R}^{D}).$$
Now we prove the converse result. Let $$\begin{array}{c} B=\Big\{(g_{0},g_{1},\cdots,g_{D}): g_{l}\in {\rm WE}^{1,q'}(\mathbb{R}^{D}),
l=0,\cdots,D\Big\},\\
\tilde{B}=\Big\{(g_{0},g_{1},\cdots,g_{D}): g_{l}\in L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q'}(\mathbb{R}^{D}),
l=0,\cdots,D\Big\},\end{array}$$ where $B\subset \tilde{B}$. The norm of $B$ and $\tilde{B}$ is defined as follows respectively $$\begin{array}{c}\|(g_{0},g_{1},\cdots,g_{D})\|_{B}=
\sum\limits^{D}_{l=0}\|g_{l}\|_{{\rm WE}^{1,q'}},\\
\|(g_{0},g_{1},\cdots,g_{D})\|_{\tilde{B}}=
\sum\limits^{D}_{l=0}\|g_{l}\|_{L^{1} + \dot{F}^{0}_{1,q'}}.\end{array}$$ We define $$\begin{array}{c}S=\Big\{(g_{0},g_{1},\cdots,g_{D})\in B: g_{l}=R_{l}g_{0}, l=0, 1,\cdots, D\Big\},\\
\tilde{S}=\Big\{(g_{0},g_{1},\cdots,g_{D})\in \tilde{B}: g_{l}=R_{l}g_{0}, l=0, 1,\cdots, D\Big\},\end{array}$$ where $S\subset \tilde{S}.$
By Theorem \[ta.h\], $g_{0}\rightarrow
(g_{0},R_{1}g_{0},\cdots,R_{D}g_{0})$ define a norm preserving map from $\dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$ to $S$. Hence the set of continuous linear functionals $f$ on $\dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$ is equivalent to the set of bounded linear map on the set $S$. According to Theorem \[th:111\], which is also the set of continuous linear functionals on Banach space $\tilde{S}$.
According to Theorem \[th:111\], the continuous linear functionals on $B$ belong to $${\rm WE}^{\infty,q}(\mathbb{R}^{D}) + \cdots + {\rm WE}^{\infty,q}(\mathbb{R}^{D}).$$ $\forall f\in \dot{F}^{0}_{\infty,q}(\mathbb{R}^{D})$, $f$ defines a continuous linear functional $l$ on $\dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$ and also on $\tilde{S}$. Hence there exist $\tilde{f}_{l}\in {\rm WE}^{\infty,q}(\mathbb{R}^{D}), l=0,1,\cdots,D,$ such that for any $g_{0}\in \dot{F}^{0}_{1,q'}(\mathbb{R}^{D})$, $$\begin{aligned}
& &\int_{\mathbb{R}^{D}}f(x) g_{0}(x) dx \\
&=&\int_{\mathbb{R}^{D}}\tilde{f}_{0}(x) g_{0}(x) dx
+\sum\limits^{D}_{l=1} \int_{\mathbb{R}^{D}}
\tilde{f}_{l}(x) R_{l}g_{0}(x) dx\\
&=&\int_{\mathbb{R}^{D}}\tilde{f}_{0}(x) g_{0}(x) dx
-\sum\limits^{D}_{l=1} \int_{\mathbb{R}^{D}} R_{l}(\tilde{f}_{l})(x)
g_{0}(x) dx.\end{aligned}$$
Hence $f(x)= \tilde{f}_{0}(x) -\sum\limits^{D}_{l=1}
R_{l}(\tilde{f}_{l})(x)$.
Finally, we give the proof of Theorem \[ta.cor\].
By the continuity of Riesz operators on $ \dot{F}^{0}_{1,q}(\mathbb{R}^{D}$, there exists a positive constant $C$ such that, for all $f\in\tls$, $$\sum_{\ell=0}^D\|R_{\ell}(f)\|_{L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})}
\le C\|f\|_{\tls}.$$
To prove $$\frac1C\|f\|_{\tls}\le\sum_{\ell=0}^D\|R_{\ell}(f)\|_{L^{1}(\mathbb{R}^{D}) + \dot{F}^{0}_{1,q}(\mathbb{R}^{D})},$$ it is sufficient to prove $$|\langle f,g\rangle|\leq C\big\{\sum\limits^{D}_{l=0} \|R_l f\|_{{\rm WE}^{1,q}(\mathbb{R}^{D})}\big\}\|g\|_{\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})},
\forall g\in \schi \bigcap \dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D}).$$ But $\forall g\in \schi \bigcap\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})$, by Theorem \[ta.i\], there exists $g_{l}$ such that $\|g_l\|_{{\rm WE}^{\infty,q'}(\mathbb{R}^{D})}\leq C\|g\|_{\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})}$ and $g=\sum\limits^{D}_{l=0} R_l g_l.$ Hence, we have $$\begin{array}{rcl}
|\langle f, g\rangle | &= |\langle f, \sum\limits^{D}_{l=0} R_l g_l\rangle |\,\,\, \leq \sum\limits^{D}_{l=0} |\langle f, R_l g_l\rangle |
&= \sum\limits^{D}_{l=0} |\langle R_lf, g_l\rangle | \\ &\leq C \sum\limits^{D}_{l=0} \|R_l f\|_{{\rm WE}^{1,q}(\mathbb{R}^{D})} \|g_{l}\|_{{\rm WE}^{\infty,q'}(\mathbb{R}^{D})}
& \leq C \sum\limits^{D}_{l=0} \|R_l f\|_{{\rm WE}^{1,q}(\mathbb{R}^{D})} \|g\|_{\dot{F}^{0}_{\infty,q'}(\mathbb{R}^{D})}.
\end{array}$$
[**Acknowledgement:**]{} The authors would like to thank Dachun Yang and Xing Fu, who contributed beneficial discussions and useful suggestions to this study.
[99]{}
J. Cao, D.-C. Chang, D. Yang and S. Yang, Riesz transform characterizations of Musielak-Orlicz-Hardy spaces, Trans. Amer. Math. Soc. (to appear) or arXiv: 1401.7373.
M. Christ and D. Geller, Singular integral characterizations of Hardy spaces on homogeneous groups, Duke Math. J. 51 (1984), 547-598.
C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-193.
M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991.
L. Grafakos, Classical Fourier Analysis, Second edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008.
D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27-42.
C.-C. Lin, Y.-C. Lin and Q. Yang, Hilbert transform characterization and Fefferman-Stein decomposition of Triebel-Lizorkin spaces, Michigan Math. J. 62 (2013), 691-703.
C.-C. Lin and H. Liu, ${\rm BMO}_L(\mathbb{H}^n)$ spaces and Carleson measures for Schrödinger operators, Adv. Math. 228 (2011), 1631-1688.
Y. Meyer, Wavelets and Operators, Translated from the 1990 French original by D. H. Salinger, Cambridge Studies in Advanced Mathematics 37, Cambridge University Press, Cambridge, 1992.
S. Nakamura, T. Noi and Y. Sawano, Generalized Morrey spaces and trace operator, Sci. China Math. 59 (2016), 281-336.
T. Qian, Q. Yang, Wavelets and holomorphic functions, Complex Analysis and Operator Theory, DOI: 10.1007/s11785-016-0597-5.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton N.J., 1970.
E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971.
H. Triebel, Spaces of distributions of Besov type on Euclidean $n$-space. Duality, interpolation, Ark. Mat. 11 (1973), 13-64.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition, Johann Ambrosius Barth, Heidelberg, 1995.
H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.
H. Triebel, Theory of Function Spaces. II, Birkhäuser Verlag, Basel, 1992.
Akihito Uchiyama, A constructive proof of the Fefferman-Stein decomposition of $\mathop\mathrm{BMO} (\rr^n)$, Acta Math. 148 (1982), 215-241.
P. Wojtaszczyk, A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts 37, Cambridge University Press, Cambridge, 1997.
D. Yang, C. Zhuo and E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. (2016), DOI: 10.1007/s13163-016-0188-z.
Q. Yang, T. Qian and P. Li, Fefferman-Stein decomposition for $Q$-spaces and micro-local quantities, Nonlinear Analysis, 2016, 145:24-48.
W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, 2005, Springer-Verlag, Berlin, 2010.
Qixiang Yang
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China.
[*E-mail:*]{} `qxyang@whu.edu.cn`
Tao Qian (Corresponding author)
Department of Mathematics, University of Macau, Macao, China
[*E-mails*]{}: ` fsttq@umac.mo`
[^1]: Corresponding author
|
---
abstract: |
We use UV/optical and X-ray observations of 272 radio-quiet Type 1 AGNs and quasars to investigate the [C [iv]{}]{} Baldwin Effect (BEff). The UV/optical spectra are drawn from the *Hubble Space Telescope*, *International Ultraviolet Explorer* and Sloan Digital Sky Survey archives. The X-ray spectra are from the *Chandra* and *XMM-Newton* archives. We apply correlation and partial-correlation analyses to the equivalent widths, continuum monochromatic luminosities, and [$\alpha_{\rm ox}$]{}, which characterizes the relative X-ray to UV brightness. The equivalent width of the [C [iv]{} $\lambda$1549]{} emission line is correlated with both [$\alpha_{\rm ox}$]{} and luminosity. We find that by regressing [$l_\nu$(2500 Å)]{} with EW([C [iv]{}]{}) and [$\alpha_{\rm ox}$]{}, we can obtain tigher correlations than by regressing [$l_\nu$(2500 Å)]{} with only EW([C [iv]{}]{}). Both correlation and regression analyses imply that [$l_\nu$(2500 Å)]{} is not the only factor controlling the changes of EW([C [iv]{}]{}); [$\alpha_{\rm ox}$]{} (or, equivalently, the soft emission) plays a fundamental role in the formation and variation of [C [iv]{}]{}. Variability contributes at least 60% of the scatter of the EW([C [iv]{}]{})-[$l_\nu$(2500 Å)]{} relation and at least 75% of the scatter of the of the EW([C [iv]{}]{})-[$\alpha_{\rm ox}$]{} relation.
In our sample, narrow [Fe K$\alpha$]{} 6.4 keV emission lines are detected in 50 objects. Although narrow [Fe K$\alpha$]{} exhibits a BEff similar to that of [C [iv]{}]{}, its equivalent width has almost no dependence on either [$\alpha_{\rm ox}$]{} or EW([C [iv]{}]{}). This suggests that the majority of narrow [Fe K$\alpha$]{} emission is unlikely to be produced in the broad emission-line region. We do find suggestive correlations between the emission-line luminosities of [C [iv]{}]{} and [Fe K$\alpha$]{}, which could be potentially used to estimate the detectability of the [Fe K$\alpha$]{} line of quasars from rest-frame UV spectroscopic observations.
author:
- |
Jian Wu, Daniel E. Vanden Berk, W. N. Brandt, Donald P. Schneider, Robert R. Gibson,\
and\
Jianfeng Wu
title: 'Probing the Origins of the [C [iv]{}]{} and [Fe K$\alpha$]{} Baldwin Effects'
---
We thank Jane Charlton for providing a number of *HST*/FOS spectra of the core sample AGNs, Eric Feigelson for useful suggestions and advice on statistics, Ohad Shemmer and Dennis Just for discussions on linear regression, and Lanyu Mi for help with the statistical computations.
This work was partially supported by NSF grant AST-0607634 and NASA LTSA grant NAG5-13035.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS website is <http://www.sdss.org/>.
|
---
abstract: 'We obtain exact results for fractional equations of Fokker-Planck type using evolution operator method. We employ exact forms of one-sided Lévy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.'
author:
- 'K. Górska'
- 'K. A. Penson'
- 'D. Babusci'
- 'G. Dattoli'
- 'G. H. E. Duchamp'
title: 'Operator solutions for fractional Fokker-Planck equations'
---
Introduction
============
Ordinary derivatives account for the variation of a given function with respect to a given variable. Fractional derivatives have a more subtle meaning. We use throughout the Euler’s definition of the fractional derivative according to which the derivative of order $\alpha$ ($0<\alpha<1$) of a constant is indeed not zero, but $\partial_{x}^{\alpha}\, 1 \,=\, \frac{x^{-\alpha}}{\Gamma(1 - \alpha)}$ [@AAKilbas06]. Their role in modelling physical phenomena is not intuitive and the treatment of the associated fractional differential equations (FDE) requires extreme care, not only from the mathematical point of view. The generalization of a relaxation equation, with a constant force term, to a fractional form reads [@TFNonnenmacher95; @BJWest10] $$\label{eq1}
\partial_{t}^{\alpha}\, P_{\alpha}(t) \,=\, -\kappa\, P_{\alpha}(t) + \frac{t^{-\alpha}}{\Gamma(1 - \alpha)}\, P + f,$$ where $P = P_{\alpha}(0)$ is the initial condition and $\kappa$ is a constant. Eq. (\[eq1\]) is an $\alpha$th order FDE with $f$ being the non-homogeneous part. The term with $P$ is not a genuine non-homogeneous contribution, but it accounts for the nonvanishing of a constant under fractional derivative. We use in Eq. (\[eq1\]) the Euler definition of fractional derivative because it appears most suitable to treat the dynamical behavior governed by the FFP equation we will discuss later. The problem (\[eq1\]) is mathematically well defined. The apparent singularity at $t=0$ can be removed by multiplying both sides of the equation by $\partial_{t}^{1-\alpha}$, thus getting $$\label{eq2}
\partial_{t}\, P_{\alpha}(t) \,=\, \partial_{t}^{1-\alpha} \left(- \kappa\, P_{\alpha}(t) + f\right).$$ The notion of stationary solution is well defined for an ordinary relaxation differential equation, but not for its fractional counterpart. In common terms *stationary* means that the solution is no more sensitive to time variations and, hence, its (ordinary) time derivative is zero. The notion should be revised for FDE, in accordance with the order of the derivative. The solution of Eq. (\[eq1\]) reads [@BJWest10; @IPodlubny99]: $$\begin{aligned}
\label{eq3}
P_{\alpha}(t) = E_{\alpha}\left(-\kappa\, t^{\alpha}\right) + f\, t^{\alpha} E_{\alpha, \,\alpha + 1}\left(- \kappa\, t^{\alpha}\right),\end{aligned}$$ where $E_{\alpha, \beta}(z) \,=\, \sum_{r = 0}^{\infty} z^{r}/\Gamma(\alpha\, r + \beta)$ is the modified Mittag-Leffler function and reduces to its ordinary case for $\beta = 1$, $E_{\alpha}(z) = E_{\alpha, 1}(z)$ [@AAKilbas06].
The solutions, plotted in Fig. \[fig1\] for different values of $\alpha$, do not display any long time stationary behavior. Quasi-stationary behavior is reached for $\alpha$ approaching the unity. For large $t$ we find $P_{\alpha}(t) \propto t^{\alpha - 1}$, for which the $\alpha$th derivative is vanishing. We can therefore conclude that the solution reaches $\alpha$-*derivative stationary* form.
![\[fig1\] Logarithmic plot of $P_{\alpha}(t)$, (see Eq. (\[eq3\])), for $\kappa = 1$, $P = 1$, $f = 2$ and $\alpha =1/4$ (I), $1/2$ (II), and $5/6$ (III).](fig1.eps)
Fractional Fokker-Planck equations and evolution operators
==========================================================
The Eq. (2) can be generalized to the following partial differential equation $$\label{eq4}
\partial_{t}^{\,\alpha}\, F_{\alpha}(x, t) = \hat{L}_{FP}\, F_{\alpha}(x, t) + \frac{t^{-\alpha}}{\Gamma(1-\alpha)}\, \gamma(x),$$ which has been shown to be tailor suited for study of problems of anomalous diffusion [@RMetzler99]. The initial condition is $F_{\alpha}(x, 0) = \gamma(x)$. From mathematical point of view Eq. (\[eq4\]) is a well-posed Cauchy problem and it is the two-variables generalization of Eq. (\[eq1\]). In Ref. [@RMetzler99] Eq. (\[eq4\]) has been used to model the continuous time random walk with the inclusion of effects of space dependent jump probabilities and $\hat{L}_{FP}$ denotes the Fokker-Planck (FP) operator involving the spatial derivative $\partial_{x}$. The presence of the term with $\gamma(x)$ ensures that Eq. (\[eq4\]) is well defined and describes a process preserving the norm of the distribution $F_{\alpha}(x, t)$, when the time evolves. For any function $h(x)$ its average value over $F_{\alpha}(x, t)$ is defined as $$\label{eqhav}
\langle h(t)\rangle_{\alpha} = \int_{-\infty}^{\infty} h(x)\, F_{\alpha}(x, t)\,\de x\,.$$
The formal solution of Eq. (\[eq4\]) is obtained by using an extension of the evolution operator formalism, introduced by Schrödinger, therefore getting $$\label{eq5}
F_{\alpha}(x, t) \,=\, \hat{U}_{\alpha}(t)\, \gamma(x), \qquad \hat{U}_{\alpha}(t) \,=\, E_{\alpha}(t^{\alpha}\, \hat{L}_{FP}).$$
Below we shall apply Eqs. (\[eq5\]) to three different versions of Fokker-Planck operators $\hat{L}_{FP}$. Limiting for the moment the discussion to $\hat{L}_{FP} \,=\, k\, \partial_{x}^{2}$, where $k$ is the generalized diffusion constant, Eq. (\[eq4\]) can be interpreted as the fractional version of the heat equation [@DVWidder75] and its solution reads $$\label{eq6}
F_{\alpha}(x, t) \,=\, \sum_{r = 0}^{\infty} \frac{(k\, t^{\alpha})^{r}}{\Gamma(1 + \alpha\, r)}\, \left[\partial_{x}^{2\, r}\, \gamma(x)\right],$$ which for $\gamma(x) = x^{n}$ ($n \in \mathbb{Z}$), gives $$\label{eq7}
F_{\alpha}(x, t) \,=\, _{\alpha}H_{n}^{(2)}(x, k\,t^{\alpha}) \,=\, n!\, \sum_{r = 0}^{[n/2]} \frac{x^{n-2\, r}\, (k\, t^{\alpha})^{r}}{(n - 2\, r)!\, \Gamma(1 + \alpha\, r)},$$ which are polynomials in $x$. For $\alpha = 1$ they are known as heat polynomials [@DVWidder75]. Any initial function $\gamma(x)~=~\sum_{n=0}^{\infty} c_{n}\, x^{n}$ allows therefore a solution of the fractional heat equation as the following expansion $$\label{eq8}
F_{\alpha}(x, t) \,=\, \sum_{n=0}^{\infty} c_{n}\, _{\alpha}H_{n}^{(2)}(x, k\,t^{\alpha}).$$
As in the case of conventional heat equation, the series in terms of *fractional* heat polynomials $_{\alpha}H^{(2)}_{n}$ are of limited usefulness since it converges for short times only. As an example, for $\alpha=1$ and $$\label{eqfx}
\gamma(x) = \frac{1}{\sqrt{2\pi}\, \sigma_{x}}\, e^{-x^{2}/(2\, \sigma_{x}^{2})}\,,$$ the convergence is limited to $t < \sigma_{x}^{2}/(4\, k)$. The use of the Gauss-Weierstrass transform [@APPrudnikov92] provides solutions with a well behaved long time behavior and therefore we look for an analogous transform relevant for the fractional case.
We make therefore the assumption that such a transform exists and that we can write $$\label{eq9}
E_{\alpha}(a\, t^{\alpha}) \,=\, \int_{0}^{\infty} n_{\alpha}(s, t)\, e^{a\, s}\, \de s$$ with $n_{\alpha}(s, t)$ being not yet specified functions. The evolution operator in Eq. (\[eq5\]) can therefore be written as $$\label{eq12}
\hat{U}_{\alpha}(t) \,=\, \int_{0}^{\infty} n_{\alpha}(s, t)\, \hat{U}_{1}(s)\, \de s, \qquad \hat{U}_{1}(t) \,=\, e^{t\, \hat{L}_{FP}},$$ and, equivalently, $$\label{eq12a}
F_{\alpha}(x, t) \,=\, \int_{0}^{\infty} n_{\alpha}(s, t)\, F_{1}(x, s)\, \de s$$ holds. Eq. ([\[eq12a\]]{}) provides the link between fractional and ordinary Fokker-Planck equations through the knowledge of $n_{\alpha}(s, t)$. This equation, specified to the case of Eq. (\[eq7\]), leads to the following relation $$\label{eq10a}
_{\alpha}H_{n}^{(2)}(x, t^{\alpha}) \,=\, \int_{0}^{\infty} n_{\alpha}(s, t)\, _{1}H_{n}^{(2)}(x, s)\, \de s,$$ which yields, as a direct consequence of Eq. (\[eq9\]) $$\label{eq10}
\int_{0}^{\infty} n_{\alpha}(s, t)\, \frac{s^{m}}{m!}\, \de s \,=\, \frac{t^{\alpha\, m}}{\Gamma(1 + \alpha\, m)}.$$ According to Eq. (16) in [@EBarkai01], the functions $n_{\alpha}(s, t)$ can be identified as $$\label{eq11}
n_{\alpha}(s, t) \,=\, \frac{1}{\alpha}\, \frac{t}{s^{1 + 1/\alpha}}\, g_{\alpha}\left(\frac{t}{s^{1/\alpha}}\right).$$ The functions $g_{\alpha}(z)$ are the one-sided Lévy stable distributions, recently obtained for $\alpha$ rational in [@KAPenson10; @KGorska11]. For related considerations compare [@ASaa11]. The Eq. (\[eq12\]) is similar to the one reported in Refs. [@RMetzler99] and [@EBarkai01]. The meaning of the fractional evolution operator is that the solution of the fractional Fokker-Planck (FFP) equation of order $\alpha$ is known whenever that the ordinary case, $\alpha = 1$, is available. By simple manipulation of the previous equations (see Eqs. (\[eq9\]) and (\[eq12\])) we can also conclude that $$\label{eq15}
\hat{U}_{\beta}(t) \,=\, \int_{0}^{\infty}\, n_{\beta/\alpha}(s, t)\, \hat{U}_{\alpha}(s)\, \de s, \qquad \beta < \alpha\, .$$ Therefore, the solution of the FFP equation of order $\beta$ can be derived from its $\alpha$ counterpart by a self-reproducing procedure. It should also be noted that, for $\alpha \neq 1$, $\hat{U}_{\alpha}(t_{2} + t_{1}) \neq \hat{U}_{\alpha}(t_{2})\, \hat{U}(t_{1})$. The evolution at different times $t_{2} > t_{1}$ is therefore given by $$\begin{aligned}
\label{eq12b}
\hat{U}_{\alpha}(t_{2}) \hat{U}_{\alpha}(t_{1}) &=& \int_{0}^{\infty} n_{\alpha}(s_{1}, t_{1})\, \de s_{1} \\ [0.7\baselineskip] \nonumber
&& \,\times\, \int_{0}^{\infty} n_{\alpha}(s_{2}, t_{2}) \hat{U}_{1}(s_{1} + s_{2}) \de s_{2}\,.\end{aligned}$$ This formula turns out extremely useful to deal with successive approximations, when the nature of the FP operator does not provide any close form for the operator $\hat{U}_{1}(t)$. The functions $n_{\alpha}(x, t)$ defined in Eq. (\[eq11\]) turn out to be, for $\alpha = 1/k$, $k=2, 3, \ldots$, solutions of general heat equations $\partial_{t}\, u_{1/k}(x, t) = (-1)^{k}\, \partial^{2}_{x}\, u_{1/k}(x, t)$ with the initial condition $u_{1/k}(x, 0) = \delta(x)$. These heat equations have been also obtained in [@EOrsingher09] from purely probabilistic arguments. The case of $u_{\alpha}(x, t)$ for $\alpha = l/k$ and $l > 1$ will be the subject of a forthcoming study.
Specific examples
=================
We can now apply the wealth of the operator techniques known for the conventional FP equation, to solve its fractional version. For instance, starting with Gaussian initial condition of Eq. (\[eqfx\]), we evaluate $\hat{U}_1 (s)\, \gamma(x)$ with the Glaisher formula [@GDattoli97; @GDattoli00] and obtain $$\begin{aligned}
\label{eq12c}
\hat{U}_{1}(s) \gamma(x) &=& \frac{1}{\sqrt{2\pi}\, \sigma_{x}}\, \left(1 + \frac{2 \kappa_1 s}{\sigma_{x}^2}\right)^{-1/2} \nonumber \\
& & \,\times \, \exp\left[-\frac{x^2}{2 \sigma_{x}^2} \left(1 + \frac{2 \kappa_1 s}{\sigma_{x}^2}\right)^{-1}\right],\end{aligned}$$ which, according to formula (\[eqhav\]), gives $\langle x^2(s)\rangle_{1} = \sigma_{x}^{2}\left(1 + \frac{2 k s}{\sigma_{x}^{2}}\right)$, and, by using Eq. (\[eq11\]), allows us to conclude that the $\alpha$- and $t$-dependent variance of $x$ is given by $$\begin{aligned}
\label{eq12d}
\sigma_{x, \alpha}^{2}(t) &=& \int_{0}^{\infty} n_{\alpha}(s, t) \langle x^{2}(s) \rangle_{1}\, \de s \\[0.7\baselineskip] \nonumber
&=& \sigma_{x}^{2} + \frac{2 k t^{\alpha}}{\Gamma(1 + \alpha)}.\end{aligned}$$ Note that for $\gamma(x) = \delta(x)$, we have formally $\sigma_{x}^{2} = 0$ and Eq. (\[eq12d\]) reproduces the defining equation of subdiffusive behavior.
We now move on to more general Fokker-Planck operators. We start by considering the operator $\hat{L}_{FP}~=~k \partial_{x}^{2}~+~[\mathcal{F}/(m_{0}\, \eta_{\alpha})]\, \partial_{x}$, where the second term is due to the action of a constant force $\mathcal{F}$, $\eta_{\alpha}$ is fractional friction coefficient, and $m_{0}$ is the particle mass. The solution of our problem can be written by adding to the Glaisher form a shift term in the Gaussian provided by $\mathcal{F}\, t/(m_{0}\, \eta_{\alpha})$. The solution for different values of $\alpha$ and $t = 2$ are given in Fig. \[fig2\] and the first moment of the distribution is, see Refs. [@RMetzler99] and [@EBarkai01]: $$\label{eq16}
\langle x(t) \rangle_{\alpha} \,=\, - \frac{\mathcal{F}\, t^{\alpha}}{m_{0}\, \eta_{\alpha}\, \Gamma(1 + \alpha)}.$$
![\[fig2\] Plot of the solution of Eq. (\[eq4\]), $F_{\alpha}(x, t)$, with FP operator $\hat{L}_{FP} = \partial_{x}^{2} + 2\partial_{x}$ and initial condition $\gamma(x) = \frac{1}{\sqrt{2 \pi}}\, e^{-x^2/2}$, for $t=2$ and $\alpha =1/4$ (I), $1/2$ (II), and $3/4$ (III).](fig2.eps)
The operator $\hat{L}_{FP} = \frac{2}{\tau}\left(\sigma^2_\epsilon \,\partial_{x}^{2} + \partial_{x} x\right)$ is used in storage ring physics to model the effect of diffusion and damping ($\tau$ is the damping time) of the electron beam due to the synchrotron radiation emitted by the electron in the bending magnets of the ring [@FCiocci00]. $\sigma_{\epsilon}$ is the variance of so-called *equilibrium* distribution (see below). The two processes, (damping and diffusion), yield eventually a stationary solution in the conventional case. Such a condition does not exist for the fractional version.
The evolution operator $\hat{U}_{1}(t)$ associated with the last FP operator can be written in a simple form. By setting indeed $\hat{A} = 2\frac{t}{\tau} \sigma^2_\epsilon \partial_{x}^{2}$ and $\hat{B} = 2\frac{t}{\tau}\partial_{x} x$ we obtain $\left[\hat{A}, \hat{B}\right] = \frac{4 t}{\tau} \sigma^2_\epsilon \hat{A}$, so that the use of conventional operator ordering methods yields [@GDattoli97] $$\label{eq17}
\hat{U}_{1}(t) \,=\, e^{\hat{A} + \hat{B}} = \exp\left(\frac{1-e^{-4 t/\tau}}{4 t/\tau}\, \, \hat{A}\right)\, e^{\hat{B}}\,.$$ In the case in which the initial distribution is the Gaussian the fractional evolution will be characterized by the following variance $$\label{eq18}
\langle x^{2}(t)\rangle_{\alpha} \,=\, \int_{0}^{\infty} n_{\alpha}(s, t)\, \langle x^{2}(s)\rangle_{1}\, \de s$$ with $$\label{eqseps}
\langle x^{2}(t)\rangle_{1} \,=\, \left(\sigma^{2} - \sigma_{\epsilon}^{2}\right)\, e^{-4 t/\tau} + \sigma_{\epsilon}^{2},$$ where $\sigma$ is the variance of the initial distribution. In Eq. (\[eq18\]) $\langle x^{2}(t) \rangle_{\alpha}$ is obtained by replacing, according to Eq. (\[eq9\]) in the second term $e^{-4 t/\tau}$ with $E_{\alpha}\left(- 4 t^{\alpha}/\tau\right)$. (Note that the physical dimension of the damping time $\tau$ is $\left[t^{\alpha}\right]$). In Fig. \[eq3\] we reported the $\langle x^{2}(t)\rangle_{\alpha}$ and, as expected, equilibrium conditions in the conventional sense is not reached. The plot shows however the onset of analogous regimes after the knee-shaped decrease. This is a consequence of the fact that for increasing time the second term in Eq. (\[eqseps\]) becomes dominating with respect to the first, and all solutions approach the $\alpha$-derivative stationary form.
![\[fig3\] Double logarithmic plot of $\langle x^{2}(t)\rangle_{\alpha}$, (see Eq. (\[eq18\])), for $\sigma = 2$, $\sigma_{\epsilon} = 1$, and $\alpha = 3/5$ (I), $4/5$ (II), and $9/10$ (III).](fig3.eps)
Discussion and conclusion
=========================
We can also consider the case of partial fractional differential equations in which the fractional derivatives is acting on the *spatial* coordinates. From the mathematical point of view we have the following Cauchy problem $$\begin{aligned}
\label{eq19}
\partial_{t}\,G_{\alpha}(x, t) &=& - \lambda\, \partial_{x}^{\alpha}\, G_{\alpha}(x, t) \\ [0.7\baselineskip] \nonumber
G_{\alpha}(x, 0) &=& h(x),\end{aligned}$$ In such a context the Levy stable distribution function is going to play a role in the theory of FFP of type Eq. (\[eq19\]). The use of the evolution operator yields a formal solution of the type $G_{\alpha}(x, t) = e^{- \lambda\, t\, \partial_{x}^{\alpha}}\, h(x)$. The series expansion of the exponential may have a limited use only, we look therefore for a more useful representation of the evolution operator. The use of the identity [@KAPenson10] $$e^{- a p^{\alpha}} = \int_{0}^{\infty} g_{\alpha}(\xi)\, \exp(- a^{1/\alpha}\, p\, \xi)\, \de \xi$$ is the naturally suited choice, so that we find $$\label{eq20}
G_{\alpha}(x, t) = \left(\lambda\, t\right)^{-1/\alpha}\, \int_{-\infty}^{x}\, g_{\alpha}\left[\frac{x-\sigma}{(\lambda\, t)^{1/\alpha}}\right]\, h(\sigma)\, \de \sigma\,.$$ This technique (albeit limited to the case $\alpha = 1/2$) has been applied to the study of the relativistic heat equation ($\partial_{t}\, G_{1/2}(x, t) = - \sqrt{1 - \partial_{x}^{2}}\, G_{1/2}(x, t)$) in [@DBabusci] and appears a very promising tool in further applications, possibly involving relativistic quantum mechanics.
Finally, we emphasize that the solutions of Eq. (\[eq4\]) for $1 < \alpha \leq 2$ can also be obtained with the help of the evolution operator and of *two-sided* Lévy stable distributions obtained in [@KGorska11]. The form of Eq. (\[eq9\]) has to be however modified as then the $n_{\alpha}(s, t)$ function has to be replaced by its two-variable counterpart discussed in [@KGorska11]. In this context we refer to Eqs. (5.21) and (5.22) of [@EOrsingher09] where the two-sided case is studied for the heat-type FP equation.
The different topics touched on in this paper have shown that the combined use of techniques from various fields (including statistical mechanics, theory of fractional calculus, ordinary quantum mechanics, etc) offers the appropriate tool to study new phenomena in the theory of anomalous diffusion.
Acknowledgment
==============
The authors acknowledge support from Agence Nationale de la Recherche (Paris, France) under Program PHYSCOMB No. ANR-08-BLAN-0243-2. G. Dattoli thanks the University Paris XIII for financial support and kind hospitality.
[12]{}
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, *Theory and Applications of Fractional Differential Equations* (Elsevier, Amsterdam, 2006).
T. F. Nonnenmacher and R. Metzler, Fractals **3**, 557 (1995)
B. J. West, Front. Physio. **1**:12 (2010).
I. Podlubny, *Franctional Differential Equations* (Academic Press, San Diego, 1999)
R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. **82**, 3563 (1999).
D. V. Widder, *The heat equation* (Academic Press, New York, 1975).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, *Integrals and Series*, vol. 5 (Gordon and Breach, Amsterdam, 1992).
K. A. Penson and K. Górska, Phys. Rev. Lett. **105**, 210604 (2010).
K. Górska and K. A. Penson, Phys. Rev. E **83**, 061125 (2011).
A. Saa and R. Venegeroles, Phys. Rev. E **84**, 026702 (2011).
E. Barkai, Phys. Rev. E **63**, 046118 (2001).
E. Orsingher and L. Beghin, Annals Prob. **37**(1), 206 (2009).
G. Dattoli, P. L. Ottaviani, A. Torre, and L. Vásquez, Riv. Nuovo Cimento **20** (2), 1 (1997).
G. Dattoli, J. Comp. Appl. Math. **118**, 111 (2000).
F. Ciocci, A. Torre, and A. Renieri, *Insertion Devises for Synchrotron Radiation and Free Electron Laser* (World Scientific, Singapore, 2000).
D. Babusci, G. Dattoli, and M. Quatromini, Phys. Rev. A **83**, 062109 (2011) .
|
---
bibliography:
- 'refs.bib'
---
[@Noyel2017] have shown that the map of Asplund’s distances, using the multiplicative LIP law, between an image, $f \in \overline{{\mathcal{I}}}=[0,M]^{D}$, $D \subset {\mathbb R}^N$, $M \in {\mathbb R}$, $M > 0$, and a structuring function $B \in [O,M]^{D_B}$, $D_B \subset {\mathbb R}^N$, namely the probe, is the logarithm of the ratio between a general morphological dilation ${\lambda}_{B} f$ and an erosion $\mu_{B} f$, in the lattice $(\overline{{\mathcal{I}}} , \leq)$: $$As_{B}^{{\mathbin{ \ooalign{$\bigtriangleup$\crcr\hidewidth
\raise.14em\hbox{${\vstretch{0.7}{\hstretch{0.7}{\scriptscriptstyle\times}}}$}\hidewidth}}}}f = \ln \left( \frac{ {\lambda}_{B} f }{ \mu_{B} f }\right) =
\ln{ \left( \frac{ \vee_{h \in D_B} \{ \widetilde{f}(x+h) / \widetilde{B}(h) \} }{ \wedge_{h \in D_B} \{ \widetilde{f}(x+h) / \widetilde{B}(h) \} }\right) }, \> \text{ with } f > 0 \text{ and } \widetilde{f}= \ln{\left( 1 - f/M \right)}.
\label{eq:map_As_la_mu_general_se}$$
The morphological dilation and erosion with a translation invariance (in space and in grey-level) are defined, for additive structuring functions as [@Serra1988; @Heijmans1990]: $$\begin{array}{ccc}
(\delta_B(f))(x) &=& \vee_{h \in D_B} \left\{ f(x - h) + B(h) \right\} = (f \oplus B) (x) \> \text{, (dilation)}\\
(\varepsilon_B(f))(x) &=& \wedge_{h \in D_B} \left\{ f(x + h) - B(h) \right\} = (f \ominus B) (x) \> \text{, (erosion)}\\
\end{array}
\label{eq:erode_dilate_funct_add}$$ and for multiplicative structuring functions as [@Heijmans1990]: $$\begin{array}{ccl}
\vee_{h \in D_B} \left\{ f(x - h) . B(h) \right\} &=& (f \dot{\oplus} B) (x) \> \text{, (dilation)}\\
\wedge_{h \in D_B} \left\{ f(x + h) / B(h) \right\} &=& (f \dot{\ominus} B) (x) \> \text{, (erosion)}.\\
\end{array}
\label{eq:erode_dilate_funct_mult}$$ If $f>0$ and $B>0$, we have $(f \dot{\oplus} B) = \exp{( \vee_{h \in D_B} \left\{ \ln{f(x-h)} + \ln{B(h)} ) \right\})} = \exp{ (\ln f \oplus \ln B )}$. Let us define, the reflected structuring function by $\overline{B}(x)=B(-x)$ and $\widehat{f} = \ln{(- \widetilde{f})}$. We obtain:$$\begin{array}{cll}
{\lambda}_{B} f &=& \vee_{h \in D_B} \left\{ \widetilde{f}(x+h) / \widetilde{B}(h) \right\} = \vee_{-h \in D_B} \left\{ \widetilde{f}(x-h) / \widetilde{B}(-h) \right\}= (-\widetilde{f}) \dot{\oplus} (-1/\widetilde{\overline{B}})\\
&=& \exp{ \left( \vee_{-h \in D_B} \left\{ \ln{(-\widetilde{f}(x-h))} + \ln{(-1/\widetilde{\overline{B}}(h)) } \right\} \right) } \> \text{, with } \widetilde{f}<0 \text{ and } \widetilde{\overline{B}} <0\\
&=& \exp{ \left( \ln{(-\widetilde{f})} \oplus (- \ln{(-\widetilde{\overline{B}})) } \right) } = \exp{ \left( \widehat{f} \oplus (- \widehat{\overline{B}}) \right)}.
\end{array}
\label{eq:dem:la}$$ Similarly, we have $\mu_{B} f = (-\widetilde{f}) \dot{\ominus} (-\widetilde{B}) = \exp{ \left( \widehat{f} \ominus \widehat{B} \right) } $. Using the previous expressions of ${\lambda}_{B} f$ and $\mu_{B} f$ into equation \[eq:map\_As\_la\_mu\_general\_se\], we obtain: $$As_{B}^{{\mathbin{ \ooalign{$\bigtriangleup$\crcr\hidewidth
\raise.14em\hbox{${\vstretch{0.7}{\hstretch{0.7}{\scriptscriptstyle\times}}}$}\hidewidth}}}}f = \ln \left( \frac{ {\lambda}_{B} f }{ \mu_{B} f }\right)
= \ln \left( \frac{ \exp{ \left( \widehat{f} \oplus (- \widehat{\overline{B}}) \right) } } { \exp{ \left( \widehat{f} \ominus \widehat{B} \right)} }\right)
= \left[ \widehat{f} \oplus (- \widehat{\overline{B}}) \right] - \left[ \widehat{f} \ominus \widehat{B} \right]
= \delta_{-\widehat{\overline{B}}} \widehat{f} - \varepsilon_{\widehat{B}} \widehat{f}.
\label{eq:map_As_la_mu_general_se_simple}$$ Therefore, the map of Asplund’s distances with the LIP multiplication is the difference between a dilation and an erosion with an additive structuring function (i.e. a morphological gradient).
|
---
author:
- Kazumi Okuyama
- and Kazuhiro Sakai
bibliography:
- 'paper.bib'
title: 'Multi-boundary correlators in JT gravity'
---
Introduction {#sec:intro}
============
Jackiw-Teitelboim (JT) gravity [@Jackiw:1984je; @Teitelboim:1983ux] is a very useful toy model to study various issues in quantum gravity and holography. As discussed in [@Almheiri:2014cka; @Maldacena:2016upp; @Jensen:2016pah; @Engelsoy:2016xyb], JT gravity is holographically dual to the low energy Schwarzian sector of the Sachdev-Ye-Kitaev (SYK) model [@Sachdev; @kitaev2015simple]. In a recent paper [@Saad:2019lba] Saad, Shenker and Stanford showed that the partition function of JT gravity on asymptotically Euclidean AdS spacetime is equal to the partition function of a certain double-scaled random matrix model and the contributions of higher genus spacetimes originated from the splitting and joining of baby universes is captured by the $1/N$ expansion of the matrix model. See also [@Stanford:2019vob; @Blommaert:2019wfy; @Okuyama:2019xbv; @Johnson:2019eik; @Kapec:2019ecr; @Betzios:2020nry] for related works in this direction. This opens up an interesting avenue to study the effect of topology change in holography using the powerful techniques of the random matrix theory. This connection between JT gravity and the random matrix model comes from the fact that the density of states in Schwarzian theory is exactly equal to the planar (genus-zero) eigenvalue density of the random matrix model which arises in the topological recursion of the Weil-Petersson volume [@Eynard:2007kz]. This connection is very interesting from the viewpoint of holography. It clearly shows that JT gravity is dual to an ensemble of boundary theories and the partition function on asymptotic AdS spacetime with renormalized boundary length ${\beta}$ is interpreted as the ensemble average ${\langle}Z({\beta}){\rangle}={\langle}\operatorname{Tr}e^{-{\beta}H}{\rangle}$ over the random Hamiltonian $H$.
In our previous paper [@Okuyama:2019xbv], we showed that JT gravity is nothing but a special case of the Witten-Kontsevich topological gravity [@Witten:1990hr; @Kontsevich:1992ti] and we studied the partition function ${\langle}Z({\beta}){\rangle}$ of JT gravity on spacetime with a single asymptotic boundary in detail. In particular, we found that ${\langle}Z({\beta}){\rangle}$ is written as the expectation value of the macroscopic loop operator in 2d gravity [@Banks:1989df]. The important difference of JT gravity from the known example of 2d gravity is that infinitely many couplings $t_k$ are turned on with a specific value $t_k={\gamma}_k$ with $$\begin{aligned}
{\gamma}_0={\gamma}_1=0,\quad {\gamma}_k=\frac{(-1)^k}{(k-1)!}~~~(k\geq2).
\end{aligned}
\label{eq:gammavalue}$$ By generalizing the approach of Zograf [@Zograf:2008wbe], we found that the contributions of the higher genus topologies can be systematically computed by making use of the KdV constraint obeyed by the partition function. As emphasized in [@Zograf:2008wbe], this method serves as a very fast algorithm for the higher genus computation compared to the Mirzakhani’s recursion relation for the Weil-Petersson volume [@mirzakhani2007simple]. We also found that in the low temperature regime the genus expansion can be reorganized in the following scaling limit, which we call the ’t Hooft limit $$\begin{aligned}
\hbar\to0,~{\beta}\to\infty\quad\text{with}\quad{\lambda}=\hbar{\beta}~~~\text{fixed},
\end{aligned}
\label{eq:tHooft}$$ where $\hbar\sim e^{-S_0}$ is the genus-counting parameter. In this limit the free energy $\log {\langle}Z({\beta}){\rangle}$ admits the ’t Hooft expansion $$\begin{aligned}
\log {\langle}Z({\beta}){\rangle}=\sum_{k=0}^\infty \hbar^{k-1}{\mathcal{F}}_k({\lambda})
\end{aligned}$$ and we found the analytic form of the first few terms of ${\mathcal{F}}_k({\lambda})$. It turned out that leading term ${\mathcal{F}}_0({\lambda})$ is given by the Legendre transform of the effective potential for a probe eigenvalue in the matrix model.
In the present paper, we will study the partition function of JT gravity on spacetimes with multiple boundaries by generalizing the method of KdV equation in [@Okuyama:2019xbv]. We find that the KdV constraints for the connected part of the multi-boundary correlator $ {\langle}\prod_i Z({\beta}_i){\rangle}_{{\mbox{\scriptsize c}}}$ is obtained by acting the “boundary creation operators” $B({\beta}_i)$ to the original KdV equation for the potential $u(x)$. Our boundary creation operator $B({\beta})$ is the same as the one discussed in the old 2d gravity literature [@Moore:1991ir; @Ginsparg:1993is] which is based on the idea that the macroscopic loop operator is expanded in terms of the microscopic loop operators in the limit ${\beta}\to0$, up to the so-called non-universal terms which scale with negative powers of ${\beta}$. We can systematically compute the genus expansion of the correlator ${\langle}\prod_i Z({\beta}_i){\rangle}_{{\mbox{\scriptsize c}}}$ by solving this KdV constraints recursively. Most of the computation can be done away from the “on-shell” value of the couplings . In particular, we define the off-shell generalization of the effective potential and its Legendre transform, the off-shell free energy. We find that the multi-boundary correlators can be written in terms of a certain combination of the off-shell free energy in the ’t Hooft limit. We also study the WKB expansion of the Baker-Akhiezer (BA) functions.[^1]
In this paper we will focus on the two-point function and compute its genus expansion using the above formalism. We also compute its low temperature expansion and study its behavior in the ’t Hooft limit as well. It turns out that the two-point function in JT gravity is expressed in terms of the error function, which is a natural generalization of the known result of pure topological gravity [@okounkov2002generating]. From the bulk gravity viewpoint, the connected part of the two-point function ${\langle}Z({\beta}_1) Z({\beta}_2){\rangle}_{{\mbox{\scriptsize c}}}$ corresponds to a Euclidean wormhole (also known as the “double trumpet” [@Saad:2018bqo]) connecting the two asymptotically AdS boundaries with renormalized lengths ${\beta}_1,{\beta}_2$. The analytic continuation of the two-point function ${\langle}Z({\beta}+{\mathrm{i}}t) Z({\beta}-{\mathrm{i}}t){\rangle}_{{\mbox{\scriptsize c}}}$, known as the spectral form factor (SFF), is of particular interest in the context of quantum chaos and the SFF is widely studied in the SYK model and JT gravity [@Garcia-Garcia:2016mno; @Cotler:2016fpe; @Saad:2018bqo; @Saad:2019pqd]. We find the analytic form of the SFF in the ’t Hooft limit and show that the SFF in JT gravity exhibits the characteristic feature of the so-called ramp and plateau, as expected for a chaotic system with random matrix statistics of eigenvalues.
In a recent interesting paper [@Marolf:2020xie], Marolf and Maxfield considered the boundary creation operators in the context of the AdS/CFT correspondence and made some interesting argument on the baby universe Hilbert space building upon the earlier works by Coleman [@Coleman:1988cy] and by Giddings and Strominger [@Giddings:1988cx; @Giddings:1988wv]. The argument in [@Marolf:2020xie] is mostly based on the intuition coming from a simple toy model, which is not a full-fledged JT gravity. It is interesting to ask how our boundary creation operator $B({\beta})$ fits into the story in [@Marolf:2020xie], but we do not have a clear understanding of it. We make some preliminary remarks on this problem in section \[sec:boundary\] and leave the details for a future work.
This paper is organized as follows. In section \[sec:genus\], we compute the genus expansion of multi-boundary correlators using the KdV constraint obeyed by these correlators. In section \[sec:wkb\], we consider the WKB expansion of the Baker-Akhiezer functions and the ’t Hooft expansion of the multi-boundary correlators. Along the way, we define the off-shell extension of the effective potential and the free energy. In section \[sec:lowT\], we compute the low temperature expansion of the two-boundary correlator. In section \[sec:SFF\], we study the spectral form factor in JT gravity and show that it exhibits the ramp and plateau behavior as expected for chaotic system. In section \[sec:boundary\], we consider the free boson/fermion representation of the $\tau$-function and discuss the boundary creation operator in this formalism. Finally we conclude in section \[sec:conclusion\]. In appendix \[sec:micro\], we consider the wavefunction of microscopic loop operators in the ’t Hooft limit.
Genus expansion {#sec:genus}
===============
Basics and conventions
----------------------
In this paper we will generalize our method [@Okuyama:2019xbv] developed for one-boundary partition function to the case of multi-boundary correlators. To begin with, let us summarize basics, notations and conventions.
As we showed in [@Okuyama:2019xbv], JT gravity can be regarded as a special case of the general Witten-Kontsevich topological gravity [@Witten:1990hr; @Kontsevich:1992ti]. In this model the intersection numbers $$\begin{aligned}
\label{eq:intersec}
\langle\kappa^\ell\tau_{d_1}\cdots\tau_{d_n}\rangle_{g,n}
=\int_{\overline{\cal M}_{g,n}}
\kappa^\ell\psi_1^{d_1}\cdots\psi_n^{d_n},\qquad
\ell,d_1,\ldots,d_n\in{{\mathbb Z}}_{\ge 0}\end{aligned}$$ play the role of correlation functions. They are defined on a closed Riemann surface $\Sigma$ of genus $g$ with $n$ marked points $p_1,\ldots,p_n$. We let ${\cal M}_{g,n}$ denote the moduli space of $\Sigma$ and $\overline{\cal M}_{g,n}$ the Deligne-Mumford compactification of ${\cal M}_{g,n}$. $\kappa$ (often denoted as $\kappa_1$ in the literature) is the first Miller-Morita-Mumford class, which is proportional to the Weil-Petersson symplectic form $\omega=2\pi^2\kappa$. $\psi_i$ is the first Chern class of the complex line bundle whose fiber is the cotangent space to $p_i$ and $\tau_{d_i}=\psi_i^{d_i}$. The intersection number in obeys the selection rule $$\begin{aligned}
\int_{\overline{\cal M}_{g,n}}
\kappa^\ell\psi_1^{d_1}\cdots\psi_n^{d_n}=0
\quad \mbox{unless}\quad \ell+d_1+\cdots+d_n=3g-3+n,
\label{eq:selection}\end{aligned}$$ which we will use frequently.
For the above correlation functions one can introduce the generating functions[^2] $$\begin{aligned}
\label{eq:genGF}
\begin{aligned}
G(s,\{t_k\})&:=\sum_{g=0}^\infty {g_{\rm s}}^{2g-2}G_g(s,\{t_k\}),&
F(\{t_k\})&:=\sum_{g=0}^\infty {g_{\rm s}}^{2g-2}F_g(\{t_k\}),\\
G_g(s,\{t_k\})
&:=\left\langle e^{s\kappa+\sum_{d=0}^\infty t_d\tau_d}\right\rangle_g,&
F_g(\{t_k\})
&:=\left\langle e^{\sum_{d=0}^\infty t_d\tau_d}\right\rangle_g.
\end{aligned}\end{aligned}$$ $G$ is actually expressed in terms of $F$ as [@Mulase:2006baa; @Dijkgraaf:2018vnm] $$\begin{aligned}
\label{eq:GFrel}
G(s,\{t_k\})=F(\{t_k+\gamma_k s^{k-1}\}),\end{aligned}$$ where ${\gamma}_k$ is defined in . Using this property we showed [@Okuyama:2019xbv] that JT gravity is nothing but the special case of the general Witten-Kontsevich gravity with $t_k=\gamma_k$. Conversely, we can define a natural deformation of JT gravity by (partially) releasing $t_k$ from the constraint $t_k={\gamma}_k$ and regard them as deformation parameters. This is one of our main ideas in [@Okuyama:2019xbv] and enables us to investigate JT gravity using the techniques of the traditional 2d gravity. In what follows we will apply this prescription to multi-boundary correlators.
In this paper we study the $n$-boundary connected correlator of JT gravity $$\begin{aligned}
\label{eq:onshellcorr}
\langle Z(\beta_1)\cdots Z(\beta_n)\rangle_{{\mbox{\scriptsize c}}}.\end{aligned}$$ We consider two kinds of its generalizations, $Z_n(\beta_1,\ldots,\beta_n;t_0,t_1)$ and $Z_n(\beta_1,\ldots,\beta_n;\{t_k\})$. They are obtained respectively by releasing only $t_0,t_1$ or all $\{t_k\}$ from the constraint $t_k=\gamma_k$. We often call them “off-shell” correlators. They are related to the “on-shell” correlators as $$\begin{aligned}
\begin{aligned}
Z_n(\beta_1,\ldots,\beta_n,t_0,t_1,\{t_k=\gamma_k\}_{k\ge 2})
&=Z_n(\beta_1,\ldots,\beta_n,t_0,t_1),\\
Z_n(\beta_1,\ldots,\beta_n,t_0=0,t_1=0)
&=\langle Z(\beta_1)\cdots Z(\beta_n)\rangle_{{\mbox{\scriptsize c}}}.
\end{aligned}\end{aligned}$$ As in [@Okuyama:2019xbv] we introduce the notations $$\begin{aligned}
\label{eq:rescaledvar}
\hbar:=\frac{1}{\sqrt{2}}{g_{\rm s}},\quad
x :=\hbar^{-1}t_0,\quad
\tau :=\hbar^{-1}t_1\end{aligned}$$ and $$\begin{aligned}
\label{eq:rescaleddiff}
\partial_k :=\frac{\partial}{\partial t_k},\quad
{}^{'}:=\partial_x=\hbar\partial_0,\quad
\dot{~}:=\partial_\tau =\hbar\partial_1.\end{aligned}$$ As discussed in [@Okuyama:2019xbv], ${g_{\rm s}}$ is the genus-counting parameter in the high temperature regime while $\hbar$ is the natural coupling constant in the low temperature ’t Hooft limit . It is also convenient to introduce $$\begin{aligned}
\label{eq:defIn}
I_n(v,\{t_k\})=\sum_{\ell=0}^\infty t_{n+\ell}\frac{v^\ell}{\ell!}
\quad (n\ge 0),\end{aligned}$$ because it is known [@Itzykson:1992ya] that $F_g(\{t_k\})\ (g\ge 2)$ are polynomials in $I_n(u_0,\{t_k\})\ (n\ge 2)$ and in $[1-I_1(u_0,\{t_k\})]^{-1}$, with $$\begin{aligned}
\label{eq:defu0}
u_0&:=\partial_0^2 F_0.\end{aligned}$$ We will work mostly on the partially constrained case $t_k=\gamma_k\ (k\ge 2)$, leaving $t_0$ and $t_1$ as parameters. In this case we further introduce the new variables $$\begin{aligned}
\label{eq:yt-def}
y:=u_0,\quad
t:=1-I_1\end{aligned}$$ and the functions $$\begin{aligned}
\label{eq:Bndef}
B_n(y)
:=\frac{J_n(2\sqrt{y})}{y^{n/2}}
=\sum_{k=\max(0,-n)}^\infty\frac{(-1)^ky^k}{k!(k+n)!}\qquad(n\in{{\mathbb Z}}).\end{aligned}$$ Here, $J_n(z)$ is the Bessel function of the first kind. $B_n$ are related to $I_n$ as $$\begin{aligned}
B_{n-1}&=(-1)^n I_n(y,\{t_0,t_1,t_k=\gamma_k\ (k\ge 2)\})
\quad (n\ge 2)\end{aligned}$$ and satisfy $$\begin{aligned}
\partial_yB_n&=-B_{n+1},\qquad
yB_{n+1}=nB_n-B_{n-1}.\end{aligned}$$ The old variables $(t_0,t_1)$ and the new ones $(y,t)$ are related as $$\begin{aligned}
t_1=B_0-t,\quad
t_0=y(B_1-t_1).
\label{eq:t1t0}\end{aligned}$$ The differentials $\partial_{0,1}$ are then written in the new variables as[^3] $$\begin{aligned}
\partial_0=\frac{1}{t}(\partial_y-B_1\partial_t),\quad
\partial_1=y\partial_0-\partial_t.\end{aligned}$$ The on-shell value $(t_0,t_1)=(0,0)$ corresponds to $(y,t)=(0,1)$. At this value $B_n$ becomes $$\begin{aligned}
\label{eq:Bnonshell}
B_n(0)=\frac{1}{n!},\quad n\ge 0.\end{aligned}$$
Multi-boundary correlators of general topological gravity \[sec:gexpMulti\]
---------------------------------------------------------------------------
In our previous paper [@Okuyama:2019xbv] we formulated how to compute the genus expansion of the partition function $Z_1=Z_{{\mbox{\scriptsize JT}}}$ of JT gravity on a surface with one boundary. In this section we generalize the method to the case of multi-boundary correlators of general topological gravity.
Let us start with the fact that the $n$-boundary correlator of general topological gravity is given by [@Moore:1991ir] $$\begin{aligned}
\label{eq:ZninF}
\begin{aligned}
Z_n(\{\beta_i\},\{t_k\})
&\simeq B(\beta_1)\cdots B(\beta_n)F(\{t_k\}).
\end{aligned}\end{aligned}$$ Here $F$ is defined in and the operator $B(\beta)$ is given by $$\begin{aligned}
\label{eq:Bdef}
B(\beta)
={g_{\rm s}}\sqrt{\frac{\beta}{2\pi}}\sum_{d=0}^\infty\beta^d\partial_d.\end{aligned}$$ As mentioned in section \[sec:intro\], is based on the idea that the macroscopic loop operator $Z({\beta})$ is expanded in terms of the microscopic loop operator $\tau_d$ in the limit ${\beta}\to0$ $$\begin{aligned}
Z({\beta})\simeq {g_{\rm s}}\sqrt{\frac{\beta}{2\pi}}\sum_{d=0}^\infty\beta^d\tau_d
\end{aligned}$$ and the insertion of $\tau_d$ is represented by the derivative ${\partial}_d$ when acting on the free energy $F$. $B(\beta)$ can be thought of as the “boundary creation operator.” We put the symbol “$\simeq$” in , meaning that the equality holds up to an additional non-universal part [@Moore:1991ir] when $3g-3+n<0$. Such a deviation appears, however, only in the genus-zero part of $n=1,2$-boundary correlators, which we will discuss separately in section \[sec:genus0\]. Note that the complex dimension of the moduli space ${\overline{\mathcal{M}}}_{g,n}$ is $3g-3+n$ which becomes negative for $(g,n)=(0,1)$ and $(0,2)$.
As in the case of single boundary the genus expansion can be computed by solving a differential equation which follows from the KdV equation. To see this, let us first introduce $$\begin{aligned}
\begin{aligned}
W_n(\{\beta_i\},\{t_k\})
:=&\ \partial_x Z_n = \hbar\partial_0 Z_n\\
\simeq&\ \hbar B(\beta_1)\cdots B(\beta_n)\partial_0 F,\\
W_0(\{t_k\}):=&\ \partial_x F = \hbar\partial_0 F,\\
u:=&\ {g_{\rm s}}^2\partial_0^2 F=2{\partial}_x^2F.
\end{aligned}
\label{eq:Wudef}\end{aligned}$$ Recall that $u$ satisfies the KdV equation [@Witten:1990hr; @Kontsevich:1992ti] $$\begin{aligned}
\label{eq:uKdV}
\dot{u}=uu'+\frac{1}{6}u'''.\end{aligned}$$ Integrating this equation once in $x=\hbar^{-1}t_0$ we have $$\begin{aligned}
\label{eq:W0KdV}
\dot{W}_0
&=(W_0')^2+\frac{1}{6}W_0'''.\end{aligned}$$ Since $B(\beta_i)$ commutes with $\dot{~}=\partial_\tau$ and ${}'=\partial_x$, we immediately obtain a differential equation for $W_n$ by simply acting $B(\beta_1)\cdots B(\beta_n)$ on both sides of the above equation. For instance, by acting $B(\beta_1)$ on both sides of we obtain $$\begin{aligned}
\label{eq:W1KdV}
\begin{aligned}
\dot{W}_1
&=2W_0'W_1'+\frac{1}{6}W_1'''\\
&=uW_1'+\frac{1}{6}W_1'''
\end{aligned}\end{aligned}$$ for $W_1(\beta_1)$. This is nothing but the differential equation for $W$ in [@Okuyama:2019xbv] with the identification $W_1=W$. By further acting $B(\beta_2)$ on both sides of we obtain $$\begin{aligned}
\label{eq:W2KdV}
\begin{aligned}
\dot{W}_2(\beta_1,\beta_2)
&=2W_1'(\beta_1)W_1'(\beta_2)+2W_0'W_2'(\beta_1,\beta_2)
+\frac{1}{6}W_2'''(\beta_1,\beta_2)\\
&=2W_1'(\beta_1)W_1'(\beta_2)+uW_2'(\beta_1,\beta_2)
+\frac{1}{6}W_2'''(\beta_1,\beta_2).
\end{aligned}\end{aligned}$$ In general the differential equation for $W_n$ may be written as $$\begin{aligned}
\label{eq:WnKdV}
\begin{aligned}
\dot{W}_n(\beta_1,\ldots,\beta_n)
&=\sum_{I\subset N}W_{|I|}'W_{|N-I|}'
+\frac{1}{6}W_n'''(\beta_1,\ldots,\beta_n),
\end{aligned}\end{aligned}$$ where $N=\{1,2,\ldots,n\}$, $W_{|I|}'=W_{|I|}'(\beta_{i_1},\ldots,\beta_{i_{|I|}})$ with $I=\{i_1,i_2,\ldots,i_{|I|}\}$ and the sum is taken for all possible subset $I$ of $N$ including the empty set. The equation uniquely determines $W_n$ in the genus expansion given the genus expansion of $W_k\ (k<n)$ and the genus zero part $W_n^{g=0}$. It is important to note that the non-universal parts are entirely absent in because all the elements other than $W_0'=\frac{u}{2}$ appearing in are equal to or higher than the third derivative of $F_0$ (see the discussion in the next subsection).
Finally $Z_n$ is obtained by merely integrating $W_n$ once in $x$. This can be done order by order in the genus expansion. As a demonstration we will study in detail the case of two-boundary correlator of JT gravity in subsection \[sec:gexp2pt\].
Genus zero part\[sec:genus0\]
-----------------------------
In this section let us consider the genus zero part of the multi-boundary correlator $Z_n^{g=0}$ and calculate $W_n^{g=0}=\partial_x Z_n^{g=0}$. In fact, $Z_n^{g=0}$ has been already calculated in the literature [@Ambjorn:1990ji; @Moore:1991ir; @Ginsparg:1993is]. In what follows we will reproduce the results in our notation.
Restricting to the genus zero part, we have $$\begin{aligned}
\label{eq:Zng0inF}
Z_n^{g=0}(\{\beta_i\})\simeq{g_{\rm s}}^{-2}B(\beta_1)\cdots B(\beta_n)F_0.\end{aligned}$$ Recall that $F_0$ is expressed as [@Itzykson:1992ya] $$\begin{aligned}
\label{eq:F0IZ}
F_0
&=\frac{1}{2}\int_0^{u_0}dv\left(I_0(v,\{t_k\})-v\right)^2,\end{aligned}$$ where $I_0$ and $u_0$ are defined in –. Note that for $v=u_0$ we have $$\begin{aligned}
u_0=I_0(u_0,\{t_k\}).\end{aligned}$$ Using these relations we obtain $$\begin{aligned}
\label{eq:Z1g0}
\begin{aligned}
Z_1^{g=0}(\beta,\{t_k\})
&\simeq{g_{\rm s}}^{-2}B(\beta)F_0\\
&=\frac{1}{{g_{\rm s}}}\sqrt{\frac{\beta}{2\pi}}
\sum_{d=0}^\infty\beta^d
\int_0^{u_0}dv\left(I_0(v,\{t_k\})-v\right)\partial_d I_0(v,\{t_k\})\\
&=\frac{1}{{g_{\rm s}}}\sqrt{\frac{\beta}{2\pi}}
\int_0^{u_0}dv\left(I_0(v,\{t_k\})-v\right)e^{\beta v}.
\end{aligned}\end{aligned}$$ As mentioned above, the last expression is only reliable up to the non-universal part. The non-universal part arises because the correlator is not fully constrained by the intersection number of quantum gravity which is defined only for $3g-3+n\ge 0$. However, by taking derivative with respect to $t_k$ we can insert the microscopic loop operator $\tau_k$ into the bracket of the intersection number and increase $n$ by one. By repeating this procedure we can map the computation of the partition function precisely to the integral over the well-defined moduli space of punctured Riemann surfaces. For the one-point function we can remove the non-universal part by differentiating twice in $t_0$ $$\begin{aligned}
\label{eq:ddZ1g0}
\begin{aligned}
\partial_0^2Z_1^{g=0}(\beta,\{t_k\})
&=\frac{1}{{g_{\rm s}}}\sqrt{\frac{\beta}{2\pi}}
\partial_0^2\int_0^{u_0}dv\left(I_0(v,\{t_k\})-v\right)e^{\beta v}\\
&=\frac{1}{{g_{\rm s}}}\sqrt{\frac{\beta}{2\pi}}
\partial_0\int_0^{u_0}dv e^{\beta v}\\
&=\frac{1}{{g_{\rm s}}}\sqrt{\frac{\beta}{2\pi}}
\partial_0\frac{e^{\beta u_0}}{\beta}.
\end{aligned}\end{aligned}$$ $Z_1^{g=0}$ is obtained by integrating the above relation twice in $t_0$. We impose the boundary condition that $Z_1^{g=0}$ identically vanishes for $u_0\to -\infty$ $$\begin{aligned}
\label{eq:Z1g0bc}
Z_1^{g=0}\big|_{u_0=-\infty}=\partial_0 Z_1^{g=0}\big|_{u_0=-\infty}=0.\end{aligned}$$ This is naturally understood from our viewpoint [@Okuyama:2019xbv] that $Z_1=Z_{{\mbox{\scriptsize JT}}}=\operatorname{Tr}(e^{\beta Q}\Pi)$ is the macroscopic loop operator, in which $Q=\partial_x^2+u$ is approximated as $Q\sim u_0$ at genus zero. Hence we have $$\begin{aligned}
\label{eq:Z1g0true}
\begin{aligned}
Z_1^{g=0}(\beta,\{t_k\})
&=\frac{1}{{g_{\rm s}}}\sqrt{\frac{\beta}{2\pi}}
\int_{-\infty}^{u_0}dv\left(I_0(v,\{t_k\})-v\right)e^{\beta v}.
\end{aligned}\end{aligned}$$ In other words the true genus-zero part of the one-point function including the non-universal term is obtained from by replacing the region of integration from $[0,u_0]$ to $(-\infty,u_0]$. Note that if we set $t_k=\gamma_k\ (k\ge 2)$ the above expression gives the result for JT gravity $$\begin{aligned}
Z_1^{g=0}(\beta,t_0,t_1)
=\frac{1}{2\sqrt{\pi\beta}\hbar}
\int_{-\infty}^{u_0}dv\left(J_0(2\sqrt{v})-t_1\right)e^{\beta v}.\end{aligned}$$ By further setting $t_1=0$ this reproduces our previous result obtained in [@Okuyama:2019xbv].
In a similar manner, we can compute the genus zero part of the two-boundary correlator $$\begin{aligned}
\label{eq:Z2g0}
\begin{aligned}
Z_2^{g=0}(\beta_1,\beta_2,\{t_k\})
&\simeq{g_{\rm s}}^{-2}B(\beta_1)B(\beta_2)F_0\\
&=\sqrt{\frac{\beta_1\beta_2}{(2\pi)^2}}
\sum_{d=0}^\infty\beta_1^d\partial_d
\int_0^{u_0}dv\left(I_0(v,\{t_k\})-v\right)e^{\beta_2 v}\\
&=\sqrt{\frac{\beta_1\beta_2}{(2\pi)^2}}
\int_0^{u_0}dv e^{(\beta_1+\beta_2)v}\\
&=\sqrt{\frac{\beta_1\beta_2}{(2\pi)^2}}
\frac{e^{(\beta_1+\beta_2)u_0}-1}{\beta_1+\beta_2}.
\end{aligned}\end{aligned}$$ We can remove the non-universal part by differentiating once in $t_0$ $$\begin{aligned}
\label{eq:dZ2g0}
\begin{aligned}
\partial_0 Z_2^{g=0}(\beta_1,\beta_2,\{t_k\})
&=\sqrt{\frac{\beta_1\beta_2}{(2\pi)^2}}
\partial_0\frac{e^{(\beta_1+\beta_2)u_0}}{\beta_1+\beta_2}.
\end{aligned}\end{aligned}$$ By imposing the boundary condition $$\begin{aligned}
Z_2^{g=0}\big|_{u_0=-\infty}=0\end{aligned}$$ we obtain $$\begin{aligned}
\label{eq:Z2g0true}
\begin{aligned}
Z_2^{g=0}(\beta_1,\beta_2,\{t_k\})
&=\sqrt{\frac{\beta_1\beta_2}{(2\pi)^2}}
\frac{e^{(\beta_1+\beta_2)u_0}}{\beta_1+\beta_2}.
\end{aligned}\end{aligned}$$ Again the true two-point function is obtained from by extending the integration region to $(-\infty,u_0]$. Given this expression we can easily determine the genus-zero part of the $n$-point function by induction in $n$ $$\begin{aligned}
\label{eq:Zng0main}
\begin{aligned}
Z_n^{g=0}(\{\beta_i\},\{t_k\})
&=\sqrt{\frac{\prod_{i=1}^n\beta_i}{(2\pi)^n}}
\frac{({g_{\rm s}}\partial_0)^{n-2}e^{\sum_{i=1}^n\beta_i u_0}}
{\sum_{i=1}^n\beta_i}\quad (n\ge 2),
\end{aligned}\end{aligned}$$ where we have used the genus-zero version of the KdV flow[^4] $$\begin{aligned}
\partial_k u_0=\partial_0{\mathcal{R}}_{k+1}=\partial_0\frac{u_0^{k+1}}{(k+1)!}.
\label{eq:zero-flow}\end{aligned}$$ Finally, the genus zero part $W_n^{g=0}=\partial_x Z_n^{g=0}$ is obtained from , and as $$\begin{aligned}
\label{eq:Wng0}
\begin{aligned}
W_n^{g=0}(\{\beta_i\},\{t_k\})
&=\sqrt{\frac{\prod_{i=1}^n\beta_i}{2(2\pi)^n}}
\frac{({g_{\rm s}}\partial_0)^{n-1}e^{\sum_{i=1}^n\beta_i u_0}}
{\sum_{i=1}^n\beta_i}\quad (n\ge 1).
\end{aligned}\end{aligned}$$
Two-boundary correlator of JT gravity\[sec:gexp2pt\]
----------------------------------------------------
In this section we focus on the two-boundary correlator of JT gravity and demonstrate in detail how to compute the genus expansion by the method described in section \[sec:gexpMulti\].
Before explaining our method, let us first briefly recall how to compute the correlator by using the method of [@Saad:2019lba]. The correlator is evaluated by the path-integral of JT gravity on two-dimensional surfaces of arbitrary genus with two boundaries. As shown in [@Saad:2019lba], the $n$-boundary correlator is written as a combination of simple building blocks: the partition function of Schwarzian mode on the “trumpet geometry” $Z_{\text{trumpet}}({\beta}_i,b_i)$ and the Weil-Petersson volume $V_{g,n}(b_1,\cdots,b_n)$ of the moduli space of Riemann surface with geodesic boundaries with lengths $b_i\ (i=1,\cdots,n)$ $$\begin{aligned}
Z_{\text{trumpet}}({\beta},b)
&=\frac{e^{-\frac{\gamma b^2}{2\beta}}}{\sqrt{2\pi\beta\gamma^{-1}}},\\
V_{g,n}(b_1,\cdots,b_n)
&=\left\langle
\exp\biggl(2\pi^2\kappa+\sum_{i=1}^n\frac{b_i^2}{2}\psi_i\biggr)
\right\rangle_{g,n},
\end{aligned}
\label{eq:trumpet}$$ where ${\gamma}$ is the asymptotic value of the dilaton field at the boundary of spacetime. Then the genus sum of two-boundary correlator is written as $$\begin{aligned}
\label{eq:Z2decomp}
\begin{aligned}
\langle Z(\beta_1)Z(\beta_2)\rangle_{{\mbox{\scriptsize c}}}&=\langle Z(\beta_1)Z(\beta_2)\rangle_{{\mbox{\scriptsize c}}}^{g=0}
+\langle Z(\beta_1)Z(\beta_2)\rangle_{{\mbox{\scriptsize c}}}^{g\ge 1},
\end{aligned}\end{aligned}$$ where $g=0$ and $g\ge 1$ parts are evaluated respectively as $$\begin{aligned}
\label{eq:gexpZ2formal}
\begin{aligned}
\langle Z(\beta_1)Z(\beta_2)\rangle_{{\mbox{\scriptsize c}}}^{g=0}
&=\int_0^\infty bdb
Z_{\text{trumpet}}({\beta}_1,b) Z_{\text{trumpet}}({\beta}_2,b)
=\frac{\sqrt{\beta_1\beta_2}}{2\pi(\beta_1+\beta_2)},\\
\langle Z(\beta_1)Z(\beta_2)\rangle_{{\mbox{\scriptsize c}}}^{g\ge 1}
&=\sum_{g=1}^\infty e^{-2gS_0}
\int_0^\infty
\prod_{i=1,2}b_idb_i Z_{\text{trumpet}}({\beta}_i,b_i)
V_{g,2}(b_1,b_2)\\
&=\sum_{g=1}^\infty e^{-2gS_0}
\frac{\sqrt{\beta_1\beta_2\gamma^{-2}}}{2\pi}
\left\langle
\frac{e^{2\pi^2\kappa}}{\prod_{i=1,2}(1-\beta_i\gamma^{-1}\psi_i)}
\right\rangle_{g,2}\\
&=\frac{\sqrt{\beta_1\beta_2}}{2\pi}
\sum_{g=1}^\infty {g_{\rm s}}^{2g}
\left\langle
\frac{e^{\kappa}}{\prod_{i=1,2}(1-\beta_i\psi_i)}
\right\rangle_{g,2}.
\end{aligned}\end{aligned}$$ Here we have set $$\begin{aligned}
\gamma=\frac{1}{2\pi^2},\qquad
{g_{\rm s}}=(2\pi^2)^{\frac{3}{2}}e^{-S_0}
\end{aligned}
\label{eq:ga-gs}$$ as in [@Okuyama:2019xbv] and we have used the selection rule . From the above expressions one obtains $$\begin{aligned}
\label{eq:gexpZ2result}
\langle Z(\beta_1)Z(\beta_2)\rangle_{{\mbox{\scriptsize c}}}=\frac{\sqrt{\beta_1\beta_2}}{2\pi}
\left[\frac{1}{\beta_1+\beta_2}
+\left(\frac{1}{16}+\frac{\beta_1+\beta_2}{12}
+\frac{\beta_1^2+\beta_1\beta_2+\beta_2^2}{24}\right){g_{\rm s}}^2
+{\cal O}({g_{\rm s}}^4)\right].\end{aligned}$$ This expansion can be computed up to arbitrary genus in principle given the data of $\langle\kappa^\ell\psi_1^{d_1}\psi_2^{d_2}\rangle_{g,2}$.
Let us now move on to explaining our method described in section \[sec:gexpMulti\]. Using this method the genus expansion can be computed very efficiently, as in the case of one-boundary partition function [@Okuyama:2019xbv]. Regarding the genus zero result we first expand $u,W_1,W_2$ as[^5] $$\begin{aligned}
\label{eq:uW1W2exp}
\begin{aligned}
u&=\sum_{g=0}^\infty {g_{\rm s}}^{2g}u_g,\\
W_1(\beta)
&=\sqrt{\frac{\beta}{4\pi}}e^{\beta y}
\sum_{g=0}^\infty{g_{\rm s}}^{2g}W_{g,1}(\beta),\\
W_2(\beta_1,\beta_2)
&={g_{\rm s}}\sqrt{\frac{\beta_1\beta_2}{8\pi^2}}e^{(\beta_1+\beta_2)y}
\sum_{g=0}^\infty{g_{\rm s}}^{2g}W_{g,2}(\beta_1,\beta_2).
\end{aligned}\end{aligned}$$ The genus zero coefficients are given respectively as $$\begin{aligned}
\label{eq:uW1W2init}
u_0=y,\qquad
W_{0,1}(\beta)=\frac{1}{\beta},\qquad
W_{0,2}(\beta_1,\beta_2)=\frac{1}{t}.\end{aligned}$$ By plugging the expansions into the differential equations , and we obtain the recursion relations $$\begin{aligned}
\label{eq:uW1W2recrel}
\begin{aligned}
-\frac{1}{t}\partial_t(tu_g)
&=\sum_{h=1}^g u_{g-h}\partial_0 u_h
+\frac{1}{12}\partial_0^3 u_{g-1},\\
-\partial_t W_{g,1}(\beta)
&=\sum_{h=1}^g u_h\partial_{0,\beta}W_{g-h,1}(\beta)
+\frac{1}{12}\partial_{0,\beta}^3 W_{g-1,1}(\beta),\\
-\partial_t W_{g,2}(\beta_1,\beta_2)
&=\sum_{h=0}^g\partial_{0,\beta_1}W_{h,1}(\beta_1)
\partial_{0,\beta_2}W_{g-h,1}(\beta_2)\\
&\hspace{1em}
+\sum_{h=1}^g u_h\partial_{0,\beta_1+\beta_2}W_{g-h,2}(\beta_1,\beta_2)
+\frac{1}{12}\partial_{0,\beta_1+\beta_2}^3W_{g-1,2}(\beta_1,\beta_2),
\end{aligned}\end{aligned}$$ where we have introduced the notation $$\begin{aligned}
\partial_{0,\beta}
:=e^{-\beta y}\partial_0 e^{\beta y}
=\partial_0+\beta t^{-1}.\end{aligned}$$ The higher genus coefficients $u_g,W_{g,1},W_{g,2}$ are computed by solving these recursion relations with the initial data .
We next expand $Z_2$ as $$\begin{aligned}
\label{eq:Z2gexp}
\begin{aligned}
Z_2(\beta_1,\beta_2)
&=\frac{\sqrt{\beta_1\beta_2}}{2\pi}
e^{(\beta_1+\beta_2)y}
\sum_{g=0}^\infty{g_{\rm s}}^{2g}Z_{g,2}(\beta_1,\beta_2).
\end{aligned}\end{aligned}$$ The coefficient $Z_{g,2}$ is then obtained from the relation $$\begin{aligned}
\partial_{0,\beta_1+\beta_2}Z_{g,2}(\beta_1,\beta_2)
=W_{g,2}(\beta_1,\beta_2).\end{aligned}$$ As in [@Okuyama:2019xbv], the integration in $t_0$ can be done unambiguously assuming that $Z_{g,2}\ (g\ge 1)$ is a polynomial in $t^{-1}$ without $t$-independent term. We find $$\begin{aligned}
\label{eq:Zg2results}
Z_{0,2}=\frac{1}{\beta_1+\beta_2},\quad
Z_{1,2}=
\frac{\beta_1^2+\beta_1\beta_2+\beta_2^2}{24t^2}
+\frac{2(\beta_1+\beta_2)B_1-B_2}{24t^3}
+\frac{B_1^2}{12t^4},\quad\cdots.\end{aligned}$$ Setting $(y,t)=(0,1)$ with the on-shell values of $B_n$ one can check that with reproduces the expansion .
On multi-boundary correlator of JT gravity
------------------------------------------
Using the method of [@Saad:2019lba] the $n$-boundary connected correlator for $n\ge 3$ is obtained by combining the contribution of $n$ trumpets and the Weil-Petersson volume in $$\begin{aligned}
\begin{aligned}
&\hspace{-1em}
\langle Z(\beta_1)\cdots Z(\beta_n)\rangle_{{\mbox{\scriptsize c}}}\\
&=\sum_{g=0}^\infty e^{-(2g-2+n)S_0}
\int_0^\infty\prod_{i=1}^nb_idb_i Z_{\text{trumpet}}({\beta}_i,b_i)
V_{g,n}(b_1,\cdots,b_n)\\
&=\sum_{g=0}^\infty e^{-(2g-2+n)S_0}
\sqrt{\frac{\prod_{i=1}^n\beta_i}{(2\pi\gamma)^{n}}}
\left\langle
\frac{e^{2\pi^2\kappa}}{\prod_{i=1}^n(1-\beta_i\gamma^{-1}\psi_i)}
\right\rangle_{g,n}\\
&=\sqrt{\frac{\prod_{i=1}^n\beta_i}{(2\pi)^{n}}}
\sum_{g=0}^\infty {g_{\rm s}}^{2g-2+n}
\left\langle
\frac{e^{\kappa}}{\prod_{i=1}^n(1-\beta_i\psi_i)}
\right\rangle_{g,n},
\end{aligned}\end{aligned}$$ where we have set ${\gamma}$ and ${g_{\rm s}}$ as in and have used the selection rule . This expression is reproduced from as follows. For $n\ge 3$ we have $$\begin{aligned}
\begin{aligned}
\langle Z(\beta_1)\cdots Z(\beta_n)\rangle_{{\mbox{\scriptsize c}}}&=Z_n(\{\beta_i\},\{t_k=\gamma_k\})\\
&=B_1(\beta_1)\cdots B_n(\beta_n)F\Big|_{t_k=\gamma_k}.
\end{aligned}\end{aligned}$$ Note that the non-universal part is absent for $n\geq3$ since the dimension of the moduli space $3g-3+n$ is non-negative in this case. Using the relation between $F$ and $G$ we have $$\begin{aligned}
\begin{aligned}
\langle Z(\beta_1)\cdots Z(\beta_n)\rangle_{{\mbox{\scriptsize c}}}&=B_1(\beta_1)\cdots B_n(\beta_n)G\big|_{s=1,t_k=0}\\
&=\sqrt{\frac{\prod_{i=1}^n\beta_i}{(2\pi)^{n}}}
\sum_{g=0}^\infty{g_{\rm s}}^{2g-2+n}
\left\langle\frac{e^{\kappa}}{\prod_{i=1}^n(1-\beta_i\psi_i)}
\right\rangle_{g,n},
\end{aligned}\end{aligned}$$ where we have used $\tau_{d_i}=\psi_i^{d_i}$ as in [@Okuyama:2019xbv].
WKB and ’t Hooft expansions {#sec:wkb}
===========================
In this section we study the WKB expansion of the Baker-Akhiezer wave function and the ’t Hooft expansion of the multi-boundary correlators. Our new formalism is much more efficient than our previous method based on the topological recursion [@Okuyama:2019xbv].
Baker-Akhiezer function\[sec:BAgexp\]
-------------------------------------
As we saw in [@Okuyama:2019xbv] the Baker-Akhiezer functions $\psi_\pm(\xi;\{t_k\})$ are certain two independent solutions of the Schrödinger equation $$\begin{aligned}
\label{eq:Schrxi}
Q\psi&=\xi\psi,\end{aligned}$$ where $$\begin{aligned}
Q:=\partial_x^2+u\end{aligned}$$ and $u$ is defined in . More specifically, in terms of the resolvent $$\begin{aligned}
R(\xi)
=\Bigl\langle x\,\Big|\,\frac{1}{\xi-Q}\,\Big|\,x\Bigr\rangle
=\int_0^\infty d\beta e^{-\beta\xi}W_1(\beta),\end{aligned}$$ $\psi_\pm$ are expressed as $$\begin{aligned}
\psi_\pm = \sqrt{R}e^{\pm S},\quad
S'=\frac{1}{2R}.\end{aligned}$$ Let us introduce $$\begin{aligned}
\begin{aligned}
A_\pm
:=&\,\log\psi_\pm
=\pm S +\frac{1}{2}\log R,\\
v_\pm
:=&\,A_\pm'
=\frac{\pm 1+R'}{2R}.
\end{aligned}\end{aligned}$$ From the differential equation [@Gelfand:1975rn; @BBT] $$\begin{aligned}
2RR''-{R'}^2+4(u-\xi)R^2=-1\end{aligned}$$ we see that $v_\pm$ are solutions to the equation $$\begin{aligned}
\label{eq:Burel}
v^2+v'=\xi-u.\end{aligned}$$ Using this equation we can compute the WKB expansion of $v_\pm$ as follows. Let us assume that $v$ admits the expansion $$\begin{aligned}
v=\sum_{n=0}^\infty\hbar^nv_n.\end{aligned}$$ By plugging this form as well as the genus expansion of $u$ $$\begin{aligned}
u=\sum_{g=0}^\infty(\sqrt{2}\hbar)^{2g}u_g\end{aligned}$$ into , we find $$\begin{aligned}
v_0^2=\xi-u_0,\quad 2v_0v_1+\partial_0v_0=0\end{aligned}$$ at the leading and the next to the leading orders. From these we find[^6] $$\begin{aligned}
\label{eq:Binit}
v_0=\pm z,\quad
v_1
=-\frac{1}{2}\partial_0\log v_0
=\frac{\partial_0 u_0}{4(\xi-u_0)}
=\frac{1}{4tz^2},\end{aligned}$$ where we have introduced the notation $$\begin{aligned}
z := \sqrt{\xi-u_0}.\end{aligned}$$ At the order of $\hbar^n\ (n\ge 2)$ is written as the recursion relation $$\begin{aligned}
v_n
=-\frac{1}{2v_0}
\left(\partial_0 v_{n-1}+\sum_{k=1}^{n-1}v_k v_{n-k}
+\left\{\begin{array}{ll}
2^{\frac{n}{2}}u_{\frac{n}{2}}&\mbox{($n$ even)}\\
0&\mbox{($n$ odd)}
\end{array}\right.
\right).\end{aligned}$$ Solving this recursion relation with the initial condition $v_0=+z$ one can easily calculate the WKB expansion of $v_+$. (Recall that the genus expansion of $u$ is calculated by solving .) In the same way one can calculate $v_-$ starting from the initial condition $v_0=-z$, but instead $v_-$ is obtained from $v_+$ by merely replacing $z$ with $-z$. The same arguments hold for $A_\pm$ and $\psi_\pm$. Therefore, in what follows we omit the subscript “$+$” and write $$\begin{aligned}
v_+ =v,\quad
A_+&=A,\quad
\psi_+=\psi,\end{aligned}$$ with the understanding that $$\begin{aligned}
v_- =v|_{z\to -z},\quad
A_- =A|_{z\to -z},\quad
\psi_- =\psi|_{z\to -z}.\end{aligned}$$ As a side remark, note that is viewed as a Miura transformation. From this viewpoint $v$ can be viewed as a solution to the modified KdV equation $$\begin{aligned}
\label{eq:mKdV}
{\tilde{\partial}}_1 v=\frac{\hbar^2}{6}\partial_0^3 v-v^2\partial_0 v\end{aligned}$$ with $$\begin{aligned}
{\tilde{\partial}}_1:=\partial_1-\xi\partial_0.\end{aligned}$$ It is also possible to compute the WKB expansion of $v$ by directly solving this equation with the initial condition .
Using the above method we obtain $$\begin{aligned}
v_0=z,\quad
v_1=\frac{1}{4tz^2},\quad
v_2=-\frac{5}{32t^2z^5}
+\frac{1}{t^3}\left(-\frac{B_1}{8z^3}+\frac{B_2}{24z}\right)
-\frac{B_1^2}{12t^4z},\quad\cdots.\end{aligned}$$ Next, let us consider the WKB expansion of $A=\log\psi$. We expand $A$ as $$\begin{aligned}
A=\sum_{n=0}^\infty\hbar^{n-1}A_n,\end{aligned}$$ so that we have $$\begin{aligned}
\label{eq:diffAn}
\partial_0A_n=v_n.\end{aligned}$$ By solving this we find $$\begin{aligned}
\label{eq:An}
\begin{aligned}
A_0
&=-\frac{2}{3}tz^3
+\sum_{n=1}^\infty\frac{(-2)^{n+1}B_n}{(2n+3)!!}z^{2n+3},\\
A_1&=-\frac{1}{2}\log|z|-{\frac{1}{2}}\log(4\pi),\\
A_2&=-\frac{5}{48tz^3}-\frac{B_1}{24t^2z},\\
A_3&=\frac{5}{64t^2z^6}
+\frac{1}{t^3}\left(\frac{B_1}{16z^4}-\frac{B_2}{48z^2}\right)
+\frac{B_1^2}{24t^4z^2}.
\end{aligned}\end{aligned}$$ Here $A_1$ is immediately obtained from up to the integration constant $-{\frac{1}{2}}\log(4\pi)$. This constant is universal in the sense that it does not depend on the background. Thus it can be determined by the asymptotic expansion of the Airy function which is the BA function for the pure topological gravity corresponding to the trivial background $t_n=0~(n\geq1)$.[^7] $A_n\ (n\ge 2)$ is also easily obtained assuming that $A_n$ is a polynomial in $t^{-1}$ without $t$-independent term. Getting $A_0$ is less trivial, but one can explicitly check that $A_0$ given in satisfies . One can also check that $$\begin{aligned}
V_{{\mbox{\scriptsize eff}}}(\xi)
\equiv-2A_0
=\frac{4}{3}tz^3
+\sum_{n=1}^\infty\frac{(-1)^n(n+1)!B_n}{(2n+3)!}(2z)^{2n+3}
\label{eq:off-Veff}\end{aligned}$$ is regarded as the off-shell generalization of the effective potential $V_{{\mbox{\scriptsize eff}}}(\xi)$ discussed in [@Saad:2019lba; @Okuyama:2019xbv]: $$\begin{aligned}
V_{{\mbox{\scriptsize eff}}}(\xi)\Big|_{y=0,t=1}
=\frac{1}{2}\sin\bigl(2\sqrt{\xi}\bigr)
-\sqrt{\xi}\cos\bigl(2\sqrt{\xi}\bigr).\end{aligned}$$ In this way one can in principle compute the WKB expansion of $\psi=\exp\left[\sum_{n=0}^\infty\hbar^{n-1}A_n\right]$ up to any order.
We note in passing that the on-shell BA function and its derivative are expanded as $$\begin{aligned}
\psi&=\sum_{n=0}^\infty \hbar^{\frac{2n}{3}-\frac{1}{6}}\Psi_n({\partial}_\eta)\text{Ai}(\eta),\\
{\partial}_x\psi&=\sum_{n=0}^\infty \hbar^{\frac{2n}{3}+\frac{1}{6}}{\widetilde{\Psi}}_n({\partial}_\eta)\text{Ai}(\eta),
\end{aligned}$$ where $\eta=\hbar^{-\frac{2}{3}}z^2$. By matching the WKB expansion of $\psi,{\partial}_x\psi$ and the asymptotic expansion of the Airy function, we find the first few terms of $\Psi_n,{\widetilde{\Psi}}_n$ $$\begin{aligned}
\Psi_0&=1,\quad &{\widetilde{\Psi}}_0=&-{\partial}_\eta,\\
\Psi_1&={\partial}_\eta^2-\frac{4}{15}{\partial}_\eta^5,\quad &{\widetilde{\Psi}}_1=&
\frac{1}{2}-\frac{5{\partial}_\eta^3}{3}+\frac{4 {\partial}_\eta^6}{15},\\
\Psi_2&=-\frac{9 {\partial}_\eta}{8}+\frac{5{\partial}_\eta^4}{2}
-\frac{212{\partial}_\eta^7}{315}+\frac{8 {\partial}_\eta^{10}}{225},\quad &{\widetilde{\Psi}}_2=&
\frac{33 {\partial}_\eta^2}{8}-\frac{9{\partial}_\eta^5}{2}+\frac{268{\partial}_\eta^8}{315}-\frac{8{\partial}_\eta^{11}}{225}.
\end{aligned}$$ Above $\Psi_n$ agrees with the result in our previous paper [@Okuyama:2019xbv] obtained by a different method.
Trace formula
-------------
In [@Okuyama:2019xbv] we showed that the one-boundary partition function is expressed as $$\begin{aligned}
\label{eq:Z1Tr}
\begin{aligned}
Z_1(\beta)
&=\int_{-\infty}^x dx'\langle x'|e^{\beta Q}|x'\rangle
=\operatorname{Tr}\left[e^{\beta Q}\Pi\right],
\end{aligned}\end{aligned}$$ where we have introduced the projector $$\begin{aligned}
\Pi =\int_{-\infty}^x dx'|x'\rangle\langle x'|.\end{aligned}$$ As shown in [@Okuyama:2018aij] the general connected correlator is given by $$\begin{aligned}
\begin{aligned}
Z_n(\{\beta_i\},\{t_k\})
&=\operatorname{Tr}\log
\left(1+\left[-1+\prod_{i=1}^n(1+z_i e^{\beta_iQ})\right]\Pi\right)
\Bigg|_{{\cal O}(z_1\cdots z_n)}\\
&=\operatorname{Tr}\log
\left(1+\sum_{k=1}^n\sum_{i_1<\cdots<i_k}z_{i_1}\cdots z_{i_k}
e^{(\beta_{i_1}+\cdots\beta_{i_k})Q}\Pi\right)
\Bigg|_{{\cal O}(z_1\cdots z_n)}.
\end{aligned}\end{aligned}$$ For instance, two- and three-boundary correlators are written explicitly as $$\begin{aligned}
Z_2(\beta_1,\beta_2)
&=\operatorname{Tr}\left[e^{(\beta_1+\beta_2)Q}\Pi
-e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi\right],
\label{eq:corr-Pi}
\\
Z_3(\beta_1,\beta_2,\beta_3)
&=\operatorname{Tr}\left[e^{(\beta_1+\beta_2+\beta_3)Q}\Pi
+e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi e^{\beta_3 Q}\Pi
+e^{\beta_1 Q}\Pi e^{\beta_3 Q}\Pi e^{\beta_2 Q}\Pi\right.\nonumber\\
&\hspace{2.7em}\left.
-e^{\beta_1 Q}\Pi e^{(\beta_2+\beta_3) Q}\Pi
-e^{\beta_2 Q}\Pi e^{(\beta_3+\beta_1) Q}\Pi
-e^{\beta_3 Q}\Pi e^{(\beta_1+\beta_2) Q}\Pi\right].\end{aligned}$$ In general, $Z_n$ is a sum of the multi-boundary correlator $$\begin{aligned}
\label{eq:cKdef}
\operatorname{Tr}(e^{\beta_1 Q}\Pi \cdots e^{\beta_n Q}\Pi)
=:\exp\left[{{\mathcal{K}}^{(n)}(\beta_1,\ldots,\beta_n)}\right].\end{aligned}$$ In [@Okuyama:2019xbv] we saw that ${\mathcal{K}}^{(1)}={\mathcal{F}}$ admits the ’t Hooft expansion ${\mathcal{F}}=\sum_{k=0}^\infty\hbar^{k-1}{\mathcal{F}}_k$. Similarly, in what follows we explicitly show that ${\mathcal{K}}^{(n)}$ admits the ’t Hooft expansion $$\begin{aligned}
{\mathcal{K}}^{(n)}=\sum_{k=0}^\infty\hbar^{k-1}{\mathcal{K}}^{(n)}_k.\end{aligned}$$
Darboux-Christoffel kernel
--------------------------
Let $|\xi\rangle$ be the energy eigenstate corresponding to $\psi(\xi)$ in section \[sec:BAgexp\], namely $$\begin{aligned}
\begin{aligned}
Q|\xi\rangle &= \xi|\xi\rangle,\\
\langle x|\xi\rangle&=\langle\xi|x\rangle=\psi(\xi,t_0=\hbar x).
\end{aligned}\end{aligned}$$ $|\xi\rangle$ is normalized so that $$\begin{aligned}
\label{eq:Eid}
1=\int_{-\infty}^\infty d\xi |\xi\rangle \langle \xi|.\end{aligned}$$ By inserting $n$ copies of with variables $\xi_i\ (i=1,\ldots,n)$, the multi-boundary correlator is expressed as $$\begin{aligned}
\label{eq:nptint}
\begin{aligned}
e^{{\mathcal{K}}^{(n)}}
&=\operatorname{Tr}(e^{\beta_1 Q}\Pi \cdots e^{\beta_n Q}\Pi)\\
&=
\int_{-\infty}^\infty d\xi_1\cdots\int_{-\infty}^\infty d\xi_n\,
e^{\sum_{i=1}^n\beta_i\xi_i}K_{12}K_{23}\cdots K_{n1},
\end{aligned}\end{aligned}$$ where $$\begin{aligned}
\begin{aligned}
K_{ij}\equiv K(\xi_i,\xi_j)
&=\langle \xi_i|\Pi|\xi_j\rangle\\
&=\int_{-\infty}^x dx'\langle \xi_i|x'\rangle\langle x'|\xi_j\rangle
=\int_{-\infty}^x dx'\psi(\xi_i)\psi(\xi_j)
\end{aligned}\end{aligned}$$ is the Darboux-Christoffel kernel. Since $\psi(\xi)$ satisfies the Schrödinger equation , we see that $$\begin{aligned}
\begin{aligned}
(\xi_i-\xi_j)K_{ij}
&=\int_{-\infty}^x dx'
\left[\left[\left(\partial_{x'}^2+u\right)\psi(\xi_i)\right]\psi(\xi_j)
-\psi(\xi_i)\left(\partial_{x'}^2+u\right)\psi(\xi_j)\right]\\
&=\int_{-\infty}^x dx'
\partial_{x'}
\left[\partial_{x'}\psi(\xi_i)\psi(\xi_j)
-\psi(\xi_i)\partial_{x'}\psi(\xi_j)\right]\\
&=\partial_x\psi(\xi_i)\psi(\xi_j)
-\psi(\xi_i)\partial_x\psi(\xi_j).
\end{aligned}\end{aligned}$$ The Darboux-Christoffel kernel then becomes $$\begin{aligned}
\begin{aligned}
K_{ij}
&=\frac{\partial_x\psi(\xi_i)\psi(\xi_j)
-\psi(\xi_i)\partial_x\psi(\xi_j)}
{\xi_i-\xi_j}\\
&=e^{A(\xi_i)+A(\xi_j)}
\frac{\partial_x A(\xi_i)-\partial_x A(\xi_j)}{\xi_i-\xi_j}
=e^{A(\xi_i)+A(\xi_j)}\frac{v(\xi_i)-v(\xi_j)}{\xi_i-\xi_j}.
\end{aligned}\end{aligned}$$ Plugging this expression into and using the genus expansion of $A(\xi)$ calculated in section \[sec:BAgexp\], one can compute the ’t Hooft expansion of ${\mathcal{K}}^{(n)}$, as we see below.
Saddle point calculation {#sec:saddle}
------------------------
In [@Okuyama:2019xbv] we considered the ’t Hooft expansion of ${\mathcal{F}}=\log Z_{{\mbox{\scriptsize JT}}}$ $$\begin{aligned}
{\mathcal{K}}^{(1)}={\mathcal{F}}=\sum_{k=0}^\infty\hbar^{k-1}{\mathcal{F}}_k\end{aligned}$$ and calculated the first three coefficients ${\mathcal{F}}_k\ (k=0,1,2)$ at the on-shell value $(y,t)=(0,1)$. In what follows let us generalize the calculation to the off-shell as well as the multi-boundary cases.
Let us first consider the off-shell generalization of the above free energy. Using the technique developed in the previous sections we have $$\begin{aligned}
\begin{aligned}
e^{{\mathcal{F}}}
&=
\int_{-\infty}^\infty d\xi e^{\frac{\lambda\xi}{\hbar}}
K(\xi,\xi)\\
&=
\int_{-\infty}^\infty d\xi e^{\frac{\lambda\xi}{\hbar}+2A(\xi)}
\partial_\xi v(\xi)
=
\int_{-\infty}^\infty d\xi e^{\frac{\lambda\xi}{\hbar}+2A(\xi)}
\frac{1}{2z}\partial_z v(\xi)\\
&=
\int_{-\infty}^\infty d\xi
e^{\left[\lambda\xi+2A_0(\xi)\right]\hbar^{-1}
+2A_1(\xi)+ 2A_2(\xi)\hbar+{\cal O}(\hbar^2)}
\frac{1}{2z}\left(1+\partial_z v_1(\xi)\hbar+{\cal O}(\hbar^2)\right).
\end{aligned}
\label{eq:cFpathint}\end{aligned}$$ The above integral is evaluated by the saddle point approximation. The saddle point $\xi_*$ is given by the condition $$\begin{aligned}
\label{eq:saddlecond}
\partial_\xi\left[\lambda\xi+2A_0(\xi)\right]\Big|_{\xi=\xi_*}=0.\end{aligned}$$ This is equivalent to $$\begin{aligned}
\label{eq:laexp}
\begin{aligned}
\lambda
&=\partial_\xi V_{{\mbox{\scriptsize eff}}}(\xi)\Big|_{\xi=\xi_*}\\
&=2tz_*+\sum_{n=1}^\infty\frac{(-1)^n n!B_n}{(2n+1)!}(2z_*)^{2n+1},
\end{aligned}\end{aligned}$$ where $z_\ast := \sqrt{\xi_\ast-y}$ and $V_{{\mbox{\scriptsize eff}}}(\xi)$ is the off-shell effective potential defined in . Inverting this relation we obtain $$\begin{aligned}
\label{eq:zxiexp}
\begin{aligned}
z_*(\lambda)
&=\frac{1}{2t}\lambda+\frac{B_1}{12t^4}\lambda^3
+\left(\frac{B_1^2}{24t^7}-\frac{B_2}{120t^6}\right)\lambda^5
+{\cal O}(\lambda^7),\\
\xi_*(\lambda)
&=y+\frac{1}{4t^2}\lambda^2
+\frac{B_1}{12t^5}\lambda^4
+\left(\frac{7B_1^2}{144t^8}-\frac{B_2}{120t^7}\right)\lambda^6
+{\cal O}(\lambda^8).
\end{aligned}\end{aligned}$$ As in [@Okuyama:2019xbv] let us introduce a new variable $\phi$ as $$\begin{aligned}
\xi=\xi_*+\sqrt{\hbar}\phi.\end{aligned}$$ The integral is then written as $$\begin{aligned}
\begin{aligned}
e^{{\mathcal{F}}}
&=
e^{\left[\lambda\xi_*+2A_0(\xi_*)\right]\hbar^{-1}+2A_1(\xi_*)}
\frac{1}{2z_*}
\int_{-\infty}^\infty \sqrt{\hbar}d\phi
e^{\partial_{\xi_*}^2A_0(\xi_*)\phi^2}
\left(1+{\cal O}(\hbar)\right).
\end{aligned}\end{aligned}$$ By expanding the integrand in $\hbar$, the integral in $\phi$ can be performed as Gaussian integrals. One can in principle calculate ${\mathcal{F}}_k$ up to any order. Evaluating the integral up to ${\cal O}(\hbar)$ for instance, we obtain $$\begin{aligned}
\label{eq:offFn}
\begin{aligned}
{\mathcal{F}}_0
&=\lambda\xi_*+2A_0(\xi_*)
=\lambda\xi_*-\int_y^{\xi_*}d\xi_*'\lambda(\xi_*')\\
&=\int_0^\lambda d\lambda'\xi_\ast(\lambda'),\\
{\mathcal{F}}_1
&=
2A_1(\xi_*)
-\log(2z_*)
+\frac{1}{2}\log\frac{\pi\hbar}{-\partial_{\xi_*}^2A_0(\xi_*)}\\
&=\frac{1}{2}\log(\partial_\lambda\xi_*)-2\log z_*
+\frac{1}{2}\log\frac{\hbar}{32\pi},\\[1ex]
{\mathcal{F}}_2
&=\frac{5(\partial_{\xi_*}^2\lambda)^2}{24(\partial_{\xi_*}\lambda)^3}
-\frac{\partial_{\xi_*}^3\lambda}{8(\partial_{\xi_*}\lambda)^2}
+\frac{\partial_{\xi_*}^2\lambda}{2z_*^2(\partial_{\xi_*}\lambda)^2}
+\frac{1}{z_*^4\partial_{\xi_*}\lambda}
-\frac{17}{24tz_*^3}-\frac{B_1}{12t^2z_*}.
\end{aligned}\end{aligned}$$ In particular, ${\mathcal{F}}_0({\lambda})$ is given by the Legendre transform of the effective potential $V_{{\mbox{\scriptsize eff}}}(\xi)=-2A_0(\xi)$. By using and we see that they are expanded as $$\begin{aligned}
\label{eq:offFnexp}
\begin{aligned}
{\mathcal{F}}_0(\lambda)
&=y\lambda+\frac{\lambda^3}{12t^2}+\frac{B_1\lambda^5}{60t^5}
+\left(\frac{B_1^2}{144t^8}-\frac{B_2}{840t^7}\right)\lambda^7\\
&\hspace{1em}
+\left(\frac{5B_1^3}{1296t^{11}}-\frac{B_1B_2}{720t^{10}}
+\frac{B_3}{15120t^9}\right)\lambda^9
+{\cal O}(\lambda^{11}),\\
{\mathcal{F}}_1(\lambda)
&=\frac{1}{2}\log\frac{\hbar}{4\pi}+\log t-\frac{3}{2}\log\lambda\\
&\hspace{1em}
+\left(\frac{B_1^2}{24t^6}-\frac{B_2}{60t^5}\right)\lambda^4
+\left(\frac{5B_1^3}{108t^9}
-\frac{B_1B_2}{36t^8}+\frac{B_3}{420t^7}\right)\lambda^6
+{\cal O}(\lambda^8),\\
{\mathcal{F}}_2(\lambda)
&=\frac{B_1}{t\lambda}-\frac{B_2}{12t^3}\lambda
+\left(\frac{B_1^3}{18t^7}-\frac{29B_1B_2}{360t^6}
+\frac{B_3}{72t^5}\right)\lambda^3+{\cal O}(\lambda^5).
\end{aligned}\end{aligned}$$ At the on-shell value $(y,t)=(0,1)$ and reduce to $$\begin{aligned}
\lambda=\sin(2z_*)\quad
\Leftrightarrow\quad
z_*=\sqrt{\xi_*}=\frac{1}{2}\arcsin{\lambda}\end{aligned}$$ and the results reproduce those obtained in [@Okuyama:2019xbv]: $$\begin{aligned}
\label{eq:onFn}
\begin{aligned}
{\mathcal{F}}_0(\lambda)
&=\frac{1}{4}\lambda\arcsin(\lambda)^2
+\frac{1}{2}\left(\sqrt{1-\lambda^2}\arcsin\lambda-\lambda\right),\\
{\mathcal{F}}_1(\lambda)
&=-\frac{3}{2}\log\arcsin\lambda
-\frac{1}{4}\log(1-\lambda^2)
+\frac{1}{2}\log\frac{\hbar}{4\pi},\\
{\mathcal{F}}_2(\lambda)
&=\frac{17}{3\arcsin(\lambda)^3}
\left[-1+\frac{1}{\sqrt{1-\lambda^2}}\right]
-\frac{23\lambda}{12(1-\lambda^2)\arcsin(\lambda)^2}\\
&+\frac{1}{12\arcsin(\lambda)}
\left[-2-\frac{2}{\sqrt{1-\lambda^2}}
+\frac{5}{(1-\lambda^2)^{3/2}}\right].
\end{aligned}\end{aligned}$$ In the same manner as above one can calculate the ’t Hooft expansion of the two-boundary correlator. We start from $$\begin{aligned}
\begin{aligned}
e^{{\mathcal{K}}^{(2)}}
&=
\int_{-\infty}^\infty d\xi_1\int_{-\infty}^\infty d\xi_2\,
e^{\frac{\lambda_1\xi_1+\lambda_2\xi_2}{\hbar}}
e^{2A(\xi_1)+2A(\xi_2)}
\left(\frac{v(\xi_1)-v(\xi_2)}{\xi_1-\xi_2}\right)^2.
\end{aligned}\end{aligned}$$ It is clear that the saddle point $\xi_{i*}\ (i=1,2)$ is given by $$\begin{aligned}
\label{eq:saddlecond2}
\partial_{\xi_i}\left[\lambda_i\xi_i+2A_0(\xi_i)\right]
\Big|_{\xi_i=\xi_{i*}}=0.\end{aligned}$$ This is the same relation as in the one-boundary case and thus $\lambda_i$ and $\xi_i$ are related as in –. Evaluating the integral by the saddle point approximation we find $$\begin{aligned}
\label{eq:offKn}
\begin{aligned}
{\mathcal{K}}_0^{(2)}
&=\sum_{i=1}^2\left[\lambda\xi_{i*}+2A_0(\xi_{i*})\right]
=\sum_{i=1}^2{\mathcal{F}}_0(\lambda_i),\\
{\mathcal{K}}_1^{(2)}
&=\sum_{i=1}^2\left[
2A_1(\xi_{i*})
+\frac{1}{2}
\log\frac{\pi\hbar}{-\partial_{\xi_{i*}}^2 A_0(\xi_{i*})}
\right]
-2\log(z_{1*}+z_{2*})\\
&=\sum_{i=1}^2\left[
\frac{1}{2}\log(\partial_{\lambda_i}\xi_{i*})-\log z_{i*}
+\frac{1}{2}\log\frac{\hbar}{8\pi}
\right]
-2\log(z_{1*}+z_{2*}),\\
{\mathcal{K}}_2^{(2)}
&=\Biggl[
\frac{5(\partial_{\xi_{1*}}^2\lambda_1)^2}
{24(\partial_{\xi_{1*}}\lambda_1)^3}
-\frac{\partial_{\xi_{1*}}^3\lambda_1}
{8(\partial_{\xi_{1*}}\lambda_1)^2}
+\frac{(3z_{1*}+z_{2*})\partial_{\xi_{1*}}^2\lambda_1}
{4z_{1*}^2(z_{1*}+z_{2*})(\partial_{\xi_{1*}}\lambda_1)^2}\\
&\hspace{2em}
+\frac{3(5z_{1*}^2+4z_{1*}z_{2*}+z_{2*}^2)}
{8z_{1*}^4(z_{1*}+z_{2*})^2\partial_{\xi_{1*}}\lambda_1}
-\frac{12z_{1*}+5z_{2*}}{24tz_{1*}^3z_{2*}}
-\frac{B_1}{12t^2z_{1*}}
\Biggr]+(\lambda_1\leftrightarrow \lambda_2),
\end{aligned}\end{aligned}$$ where ${\mathcal{F}}_0$ is given in .
At the on-shell value $(y,t)=(0,1)$ the above results reduce to $$\begin{aligned}
\label{eq:onKn}
\begin{aligned}
{\mathcal{K}}_0^{(2)}
=&\,\sum_{i=1}^2\left[\frac{1}{4}\lambda_i\arcsin(\lambda_i)^2
+\frac{1}{2}
\left(\sqrt{1-\lambda_i^2}\arcsin\lambda_i-\lambda_i\right)
\right],\\
{\mathcal{K}}_1^{(2)}
=&\,
-\frac{1}{4}\log[(1-\lambda_1^2)(1-\lambda_2^2)]
-\frac{1}{2}\log(\arcsin\lambda_1\arcsin\lambda_2)\\
&\,-2\log(\arcsin\lambda_1+\arcsin\lambda_2)
+\log\frac{\hbar}{\pi},\\
{\mathcal{K}}_2^{(2)}
=&\,\sum_{k=0}^2\frac{f_k(\lambda_1)+f_k(\lambda_2)}
{(\arcsin\lambda_1+\arcsin\lambda_2)^k}
\quad\mbox{with}\\
&f_0(\lambda)
=\frac{5}{3\arcsin(\lambda)^3}
\left[-1+\frac{1}{\sqrt{1-\lambda^2}}\right]
-\frac{11\lambda}{12(1-\lambda^2)\arcsin(\lambda)^2}\\
&\hspace{4em}
+\frac{1}{12\arcsin(\lambda)}
\left[-2-\frac{2}{\sqrt{1-\lambda^2}}
+\frac{5}{(1-\lambda^2)^{3/2}}\right],\\
&f_1(\lambda)
=
-\frac{2\lambda}{(1-\lambda^2)\arcsin\lambda}
-\left(1-\frac{1}{\sqrt{1-\lambda^2}}\right)
\frac{4}{\arcsin(\lambda)^2},\\
&f_2(\lambda)
=\left(-8+\frac{6}{\sqrt{1-\lambda^2}}\right)\frac{1}{\arcsin\lambda}.
\end{aligned}\end{aligned}$$ In the same way one can calculate ${\mathcal{K}}_k^{(n)}$ for $n\ge 3$. We find that ${\mathcal{K}}_{0,1}^{(n)}(\lambda_1,\ldots,\lambda_n)$ for $n\in{{\mathbb Z}}_{>0}$ take the universal form $$\begin{aligned}
\begin{aligned}
{\mathcal{K}}_0^{(n)}(\lambda_1,\ldots,\lambda_n)
&=\sum_{i=1}^n{\mathcal{F}}_0(\lambda_i),\\
{\mathcal{K}}_1^{(n)}(\lambda_1,\ldots,\lambda_n)
&=\sum_{i=1}^n\left[
\frac{1}{2}\log(\partial_{\lambda_i}\xi_{i*})-\log z_{i*}
+\frac{1}{2}\log\frac{\hbar}{8\pi}
\right]
-\sum_{i=1}^n\log(z_{i*}+z_{i+1,*}),
\end{aligned}\end{aligned}$$ where the subscript $i$ should be identified mod $n$.
We also find that ${\mathcal{K}}_2^{(3)}$ at the on-shell value is given by $$\begin{aligned}
\begin{aligned}
&{\mathcal{K}}^{(3)}_2(\lambda_1,\lambda_2,\lambda_3)\Big|_{y=0,t=1}\\
&=\sum_{i=1}^3\left[{\mathcal{F}}_2(\lambda_i)+\frac{1}{2(z_{i*})^3}\right]\\
&\hspace{.5em}
+\left[\left(
\frac{3(z_{1*})^4+2(z_{2*}+z_{3*})(z_{1*})^3-3z_{2*}z_{3*}(z_{1*})^2
-3z_{2*}z_{3*}(z_{2*}+z_{3*})z_{1*}-2(z_{2*})^2(z_{3*})^2}
{4(z_{1*})^3(z_{1*}+z_{2*})^2(z_{1*}+z_{3*})^2\cos(2z_{1*})}
\right.\right.\\
&\hspace{1.5em}
\left.\left.\phantom{\frac{1}{1}}
-\frac{z_{1*}+z_{2*}}{4(z_{1*}z_{2*})^2}
+\frac{z_{2*}z_{3*}-(z_{1*})^2}
{4(z_{1*})^2(z_{1*}+z_{2*})(z_{1*}+z_{3*})}
\frac{\sin(2z_{1*})}{\cos^2(2z_{1*})}\right)
+\mbox{cyclic perm.}\right],
\end{aligned}\end{aligned}$$ where ${\mathcal{F}}_2$ at the on-shell value is given in .
Low temperature expansion of two-boundary correlator {#sec:lowT}
====================================================
In this section let us consider the low temperature expansion of the two-boundary correlator. More specifically, we consider the situation where $$\begin{aligned}
\label{eq:Tdef}
T=\frac{1}{\beta}=\frac{1}{\beta_1+\beta_2}\end{aligned}$$ is small and calculate the expansion of $Z_2(\beta_1,\beta_2)$ in $T$. To begin with, one can observe that the leading order term in each coefficient $Z_{g,2}$ of the genus expansion is independent of $y$ and has the form ${\cal O}(\beta^{3g-1}t^{-2g})$. We find that they can be summed over genus $$\begin{aligned}
\label{eq:TexpZ2leading}
\begin{aligned}
&\hspace{-1em}
\frac{\sqrt{\beta_1\beta_2}}{2\pi}e^{(\beta_1+\beta_2)y}
\left[\frac{1}{\beta_1+\beta_2}
+\frac{\beta_1^2+\beta_1\beta_2+\beta_2^2}{24t^2}{g_{\rm s}}^2+\cdots\right]\\
&=\frac{t}{2\sqrt{\pi}\hbar(\beta_1+\beta_2)^{3/2}}
e^{(\beta_1+\beta_2)y+\frac{\hbar^2(\beta_1+\beta_2)^3}{12t^2}}
\operatorname{Erf}\left(\frac{\hbar}{2t}\sqrt{\beta_1\beta_2(\beta_1+\beta_2)}
\right)\\
&=\frac{t}{2\sqrt{\pi}h}
e^{\frac{y}{T}+\frac{h^2}{12t^2}}
\operatorname{Erf}\left(\frac{h}{2t}\sqrt{r(1-r)}\right).
\end{aligned}\end{aligned}$$ Here $\operatorname{Erf}(z)$ is the error function $$\begin{aligned}
\operatorname{Erf}(z)=\frac{2}{\sqrt{\pi}}\int_0^z dt e^{-t^2}\end{aligned}$$ and we have introduced the notation $$\begin{aligned}
h=\hbar \beta^{3/2},\quad
r=\frac{\beta_1}{\beta}.\end{aligned}$$ At the on-shell value $(y,t)=(0,1)$ precisely reproduces the result of the Airy case [@okounkov2002generating] (discussed also in our previous paper [@Okuyama:2019xbv]). More generally, including the subleading corrections the two-point function at the on-shell value – is written as $$\begin{aligned}
{\langle}Z({\beta}_1)Z({\beta}_2){\rangle}_{{\mbox{\scriptsize c}}}=\frac{{\sqrt{r(1-r)}}}{2\pi}\left[1+
\sum_{g=1}^\infty ({\sqrt{2}}h)^{2g}\int_{{\overline{\mathcal{M}}}_{g,2}}\frac{e^{T{\kappa}}}
{(1-r\psi_1)(1-(1-r)\psi_2)}\right].
\end{aligned}
\label{eq:two-kappa}$$
Regarding the above result, as in the case of $Z_1$ [@Okuyama:2019xbv] it is natural to make an ansatz $$\begin{aligned}
\label{eq:TexpZ2anz1}
Z_2(\beta_1,\beta_2)
=\frac{e^{\frac{y}{T}+\frac{h^2}{12t^2}}}{2\sqrt{\pi}h}
\sum_{\ell=0}^\infty\frac{T^\ell}{\ell!}z^{(2)}_\ell\end{aligned}$$ with $$\begin{aligned}
z^{(2)}_0=t\operatorname{Erf}\left(\frac{h}{2t}\sqrt{r(1-r)}\right).\end{aligned}$$ The subleading parts $z^{(2)}_\ell\ (\ell\ge 1)$ can also be estimated from the data of the genus expansion , . We find that $z^{(2)}_\ell$ has the structure $$\begin{aligned}
\label{eq:TexpZ2anz2}
z^{(2)}_\ell(\beta_1,\beta_2)
=\operatorname{Erf}\left(\frac{h}{2t}\sqrt{r(1-r)}\right)z_\ell(\beta)
+h\sqrt{\frac{r(1-r)}{\pi}}e^{-\frac{h^2r(1-r)}{4t^2}}
g_\ell(\beta_1,\beta_2)\end{aligned}$$ with $$\begin{aligned}
\label{eq:zlgl}
\begin{aligned}
z_0&=t,\quad z_1=\left(1+\frac{h^4}{60t^4}\right)B_1,\quad\ldots,\\
g_0&=0,\quad
g_1=\left(-\frac{1}{t}+\frac{h^2(1-r+r^2)}{6t^3}\right)B_1,\quad,\ldots.
\end{aligned}\end{aligned}$$ Interestingly, the above $z_\ell$ coincides with the coefficient $z_\ell$ of the low temperature expansion of $Z_1$ studied in [@Okuyama:2019xbv]. In [@Okuyama:2019xbv] we saw that $z_\ell$ can be calculated by solving a set of recursion relations following from the KdV constraint. Similarly, one can derive recursion relations for $g_\ell$, as we will see below. We should emphasize that from $z_\ell$ and $g_\ell$ contain the all-genus information of the intersection numbers at the fixed power of ${\kappa}$, i.e. ${\kappa}^\ell$.
Let us first express the small $T$ expansion of $Z_2$ as $$\begin{aligned}
\label{eq:TexpZ2}
Z_2(\beta_1,\beta_2)
&={\mathcal{A}}\sum_{\ell=0}^\infty\frac{T^{\ell+1}}{\ell!}z_\ell
+{\mathcal{B}}\sum_{\ell=0}^\infty\frac{T^{\ell+1}}{\ell!}g_\ell,\end{aligned}$$ where $$\begin{aligned}
{\mathcal{A}}=\frac{1}{2\sqrt{\pi} h T}
e^{\frac{y}{T}+\frac{h^2}{12t^2}}
\operatorname{Erf}\left(\frac{h}{2t}\sqrt{r(1-r)}\right),\quad
{\mathcal{B}}=\frac{\sqrt{r(1-r)}}{2\pi T}
e^{\frac{y}{T}+\frac{h^2r^3}{12t^2}+\frac{h^2(1-r)^3}{12t^2}}.\end{aligned}$$ Using the properties $$\begin{aligned}
\label{eq:ABdiffrel}
\partial_t {\mathcal{A}}=-\frac{h^2}{6t^3}{\mathcal{A}}-\frac{1}{t^2}{\mathcal{B}},\qquad
\partial_t {\mathcal{B}}=-\frac{h^2[r^3+(1-r)^3]}{6t^3}{\mathcal{B}}\end{aligned}$$ it is not difficult to see that the small $T$ expansion of $W_2=\partial_x Z_2$ takes the form $$\begin{aligned}
\label{eq:TexpW2}
W_2(\beta_1,\beta_2)
&={\mathcal{A}}\sum_{\ell=0}^\infty T^\ell w_\ell(h)
+{\mathcal{B}}\sum_{\ell=0}^\infty T^\ell b_\ell,\end{aligned}$$ where $w_\ell(h)$ is the expansion coefficient for $W_1(\beta)$ introduced in [@Okuyama:2019xbv] and $b_\ell$ is some polynomial in $h,r,t^{-1},B_n\ (n\ge 1)$. The small $T$ expansions of $W_1(\beta_1)$ and $W_1(\beta_2)$ are explicitly written as $$\begin{aligned}
\label{eq:TexpW1W1}
\begin{aligned}
W_1(\beta_1)
&=e^{\frac{ry}{T}+\frac{h^2r^3}{12t^2}}
\sum_{\ell=0}^\infty\left(\frac{T}{r}\right)^{\ell+1}
w_\ell(h r^{3/2}),\\
W_1(\beta_2)
&=e^{\frac{(1-r)y}{T}+\frac{h^2(1-r)^3}{12t^2}}
\sum_{\ell=0}^\infty\left(\frac{T}{1-r}\right)^{\ell+1}
w_\ell(h(1-r)^{3/2}).
\end{aligned}\end{aligned}$$ As we saw in [@Okuyama:2019xbv], $w_\ell(h)$ is determined by the KdV constraint for the one-point function $$\begin{aligned}
\label{eq:W1KdVconst}
\begin{aligned}
-\partial_t W_1
&=\hat{u}\partial_0 W_1
+\frac{h^2T^3}{6}\partial_0^3 W_1,\\
\hat{u}
&=u-y=\sum_{g=1}^\infty 2^g h^{2g} T^{3g} u_g.
\end{aligned}\end{aligned}$$ Similarly, $b_\ell$ can be computed from the KdV constraint for the two-point function $$\begin{aligned}
\label{eq:W2KdVconst}
-\partial_t W_2
=\hat{u}\partial_0 W_2
+\frac{h^2T^3}{6}\partial_0^3 W_2+\partial_0 W_1\partial_0 W_1\end{aligned}$$ by equating the terms proportional to ${\mathcal{B}}$. Note that the last term of is proportional to ${\mathcal{B}}$. If we formally set ${\mathcal{B}}=0$, we obtain the homogeneous equation for $W_2$ which is equivalent to the KdV constraint for the one-point function . This justifies the ansatz for $W_2$ (and thus our original conjectures and for $Z_2$) where the expansion coefficient of the first term is given by that of the one-point function $w_\ell$.
Plugging – into and using the relations one finds that the recursion relation for $b_\ell$ is given by $$\begin{aligned}
\begin{aligned}
&-\partial_t b_\ell
+\frac{h^2}{6t^3}\left[r^3+(1-r)^3-1\right]b_\ell
+\frac{1}{t^2}w_\ell(h)\\
&=\left(\hat{u}\partial_0+\frac{h^2T^3}{6}\partial_0^3\right)
\left({\mathcal{A}}\sum_{j=0}^\ell T^jw_j(h)+{\mathcal{B}}\sum_{j=1}^{\ell-1}T^jb_j\right)
\Bigg|_{{\mathcal{B}},T^\ell}\\
&\hspace{1em}
+\left(\sum_{j=0}^\ell\left(\frac{T}{r}\right)^{j+1}
\left(\partial_0+\frac{r}{tT}+\frac{h^2r^3B_1}{6t^4}\right)
w_j(hr^{3/2})\right)
\cdot\left(r\to 1-r\right)\Bigg|_{T^\ell}.
\end{aligned}\end{aligned}$$ Starting from $b_0=0$ one can compute $b_\ell$. For instance, the first term is $$\begin{aligned}
b_1=\frac{h^2B_1}{6t^4}(1-r+r^2).\end{aligned}$$ From $\partial_x Z_2=W_2$, one can show that $$\begin{aligned}
g_\ell
=t\ell!b_\ell
-t\ell\left[\partial_0 g_{\ell-1}
+\frac{h^2B_1\left(r^3+(1-r)^3\right)}{6t^4}g_{\ell-1}
+\frac{B_1}{t^3}z_{\ell-1}\right].\end{aligned}$$ Starting from $g_0=0$ one can calculate $g_\ell\ (\ell\ge 1)$ up to arbitrary high order. We have verified that this indeed reproduces our conjectured results estimated from the genus expansion.
A few remarks are in order. First, it is worth noting that the two-point function admits a low-temperature expansion of the form $$\begin{aligned}
\label{eq:Z2ptinD}
Z_2(\beta_1,\beta_2)=\operatorname{Erf}(\sqrt{D})Z_1(\beta)\end{aligned}$$ with $$\begin{aligned}
D=\sum_{\ell=0}^\infty T^\ell D_\ell,\quad D_0=\frac{h^2}{4t^2}r(1-r).
\label{eq:Dell}\end{aligned}$$ The structure of $Z_2({\beta}_1,{\beta}_2)$ in is naturally understood from by expanding the error function in $T$. Note that and imply that the two terms in correspond to $$\begin{aligned}
\begin{aligned}
\operatorname{Tr}(e^{({\beta}_1+{\beta}_2)Q}\Pi)&=Z_1(\beta),\\
\operatorname{Tr}(e^{{\beta}_1Q}\Pi e^{{\beta}_2Q}\Pi)&=\text{Erfc}(\sqrt{D})Z_1(\beta),
\end{aligned}
\label{eq:tr-vs-Erfc}\end{aligned}$$ where $$\begin{aligned}
\operatorname{Erfc}(z) = 1-\operatorname{Erf}(z)\end{aligned}$$ is the complementary error function. To calculate $D_\ell$ in , it is convenient to introduce the normalized coefficients $c_\ell:=D_\ell/D_0$ and expand $D$ as $$\begin{aligned}
\label{eq:Dexpansion}
D=D_0\sum_{\ell=0}^\infty T^\ell c_\ell,
\quad c_0=1.\end{aligned}$$ One can rewrite as $$\begin{aligned}
\operatorname{Erf}(\sqrt{D_0})\sum_{\ell=0}^\infty\frac{T^\ell}{\ell!}z_\ell
+2t\sqrt{\frac{D_0}{\pi}}e^{-D_0}
\sum_{\ell=1}^\infty\frac{T^\ell}{\ell!}g_\ell
=\operatorname{Erf}(\sqrt{D})\sum_{\ell=0}^\infty\frac{T^\ell}{\ell!}z_\ell\end{aligned}$$ or $$\begin{aligned}
\begin{aligned}
\frac{\sum_{\ell=1}^\infty\frac{T^{\ell}}{\ell!}g_\ell}
{\sum_{\ell=0}^\infty\frac{T^{\ell}}{\ell!}{\hat{z}}_\ell}
&=\frac{1}{2}\sqrt{\frac{\pi}{D_0}}e^{D_0}
\left(\operatorname{Erf}(\sqrt{D})-\operatorname{Erf}(\sqrt{D_0})\right)\\
&=\frac{1}{2}c_1T
+\left(\frac{c_2}{2}-\frac{c_1^2}{8}-\frac{D_0c_1^2}{4}\right)T^2
+{\cal O}(T^3),
\end{aligned}\end{aligned}$$ where ${\hat{z}}_\ell:=z_\ell/t$. (Note that ${\hat{z}}_0=1$). By comparing both sides of the equation one can express $c_\ell$ in terms of ${\hat{z}}_\ell$ and $g_\ell$. First few of the results read $$\begin{aligned}
\label{eq:cn}
\begin{aligned}
c_1&= 2g_1,\\
c_2&= 2D_0g_1^2+g_1^2-2g_1{\hat{z}}_1+g_2,\\
c_3&=\frac{8}{3}D_0(D_0+1)g_1^3-2{\hat{z}}_1(2D_0+1)g_1^2
+(2D_0g_2+2{\hat{z}}_1^2+g_2-{\hat{z}}_2)g_1-{\hat{z}}_1g_2+\frac{1}{3}g_3.
\end{aligned}\end{aligned}$$ From these expressions one immediately obtains $D_\ell=D_0c_\ell$.
Second, as discussed in [@Okuyama:2019xbv], given the result of the low temperature expansion it is straightforward to take the ’t Hooft limit and one can rearrange the low temperature expansion as the ’t Hooft expansion. From the above results one can compute the ’t Hooft expansion of $D$ $$\begin{aligned}
D=\sum_{n=0}^\infty\hbar^{n-1}{\mathcal{D}}_n,
\end{aligned}
\label{eq:Dexp}$$ where ${\mathcal{D}}_n$ is obtained as a double series expansion in $(\lambda_1,\lambda_2)=(\hbar{\beta}_1,\hbar{\beta}_2)$. Alternatively, from the relation $$\label{eq:DKrel}
\begin{aligned}
\operatorname{Erfc}({\sqrt{D}})Z_1({\beta})
=\operatorname{Tr}(e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi)=e^{\mathcal{K}^{(2)}}
\end{aligned}$$ and the result of $\mathcal{K}^{(2)}$ in , one can calculate ${\mathcal{D}}_n$ as exact functions. Here, the complementary error function can be expanded in $\hbar$ with the help of the asymptotic formula $$\begin{aligned}
\label{eq:erfcexp}
\operatorname{Erfc}(z)
=\frac{e^{-z^2}}{\sqrt{\pi}z}
\sum_{n=0}^\infty\frac{(2n-1)!!}{(-2z^2)^n}.\end{aligned}$$ For instance, the leading term is given by $$\begin{aligned}
{\mathcal{D}}_0={\mathcal{F}}_0({\lambda}_1+{\lambda}_2)-{\mathcal{F}}_0({\lambda}_1)-{\mathcal{F}}_0({\lambda}_2).
\end{aligned}$$ The higher order corrections ${\mathcal{D}}_{n\geq1}$ can also be easily obtained from the result of $\mathcal{K}^{(2)}_{n\geq1}$ in . We verified at the on-shell value $(y,t)=(0,1)$ that the series expansions of ${\mathcal{D}}_n$ obtained from are in perfect agreement with the exact expressions of ${\mathcal{D}}_n\ (n=1,2)$ obtained through .
Third, one might think that $g_\ell$ would be interpreted as the expansion coefficients for $\operatorname{Tr}(e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi)$. This intuition, however, is not precise. Rather, by using , and $\operatorname{Tr}(e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi)$ is rewritten as $$\begin{aligned}
\begin{aligned}
\operatorname{Tr}(e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi)
&=Z_1(\beta)-Z_2(\beta_1,\beta_2)\\
&=\operatorname{Erfc}(\sqrt{D_0})Z_1(\beta)
-{\mathcal{B}}\sum_{\ell=0}^\infty\frac{T^{\ell+1}}{\ell!}g_\ell\\
&=\operatorname{Erfc}(\sqrt{D_0})
\frac{e^{\frac{h^2}{12t^2}+\frac{y}{T}}}{2\sqrt{\pi}h}
\sum_{\ell=0}^\infty\frac{T^\ell}{\ell!}z_\ell
-{\mathcal{B}}\sum_{\ell=0}^\infty\frac{T^{\ell+1}}{\ell!}g_\ell.
\end{aligned}
\label{eq:TexpK2}\end{aligned}$$ This clearly shows that not only $g_\ell$ but also $z_\ell$ are involved in the low temperature expansion of $\operatorname{Tr}(e^{\beta_1 Q}\Pi e^{\beta_2 Q}\Pi)$. By rearranging the low temperature expansion as the ’t Hooft expansion and using the asymptotic expansion formula , we explicitly verified at the on-shell value $(y,t)=(0,1)$ that is indeed in agreement with $e^{{\mathcal{K}}^{(2)}}=e^{\sum_{k=0}^\infty \hbar^{k-1}{\mathcal{K}}_k^{(2)}}$ with ${\mathcal{K}}_k^{(2)}\ (k=0,1,2)$ given in .
Spectral form factor in JT gravity {#sec:SFF}
==================================
In this section we will study the spectral form factor (SFF) in JT gravity using our result of two-point function. The SFF is extensively studied in the SYK model as a useful diagnostic of the late-time chaos [@Garcia-Garcia:2016mno; @Cotler:2016fpe; @Saad:2018bqo; @Saad:2019pqd]. The SFF of chaotic system exhibits a characteristic behavior called the ramp and the plateau. From the bulk gravity perspective, the ramp comes from the Euclidean wormhole connecting the two boundaries. The plateau behavior, on the other hand, is a doubly non-perturbative effect with respect to the Newton’s constant whose bulk gravity interpretation is still missing. From the random matrix model picture, the origin of the plateau can be traced back to the universal eigenvalue correlation given by the so-called sine-kernel formula. However, this argument is based on the matrix model before taking the double-scaling limit and the analytic form of the SFF in the JT gravity case has not been obtained yet as far as we know. Using our result in the previous section, we can explicitly write down the analytic form of SFF in JT gravity and see how the ramp and the plateau come about.
The SFF is defined by analytically continuing the two-boundary correlator ${\langle}Z({\beta}_1)Z({\beta}_2){\rangle}_{{{\mbox{\scriptsize c}}}}$ to a complex value of the boundary length ${\beta}_{1,2}={\beta}\pm{\mathrm{i}}t$. It is convenient to define the normalized SFF by $$\begin{aligned}
g({\beta},t,\hbar):=\frac{{\langle}Z({\beta}+{\mathrm{i}}t)Z({\beta}-{\mathrm{i}}t){\rangle}_{{{\mbox{\scriptsize c}}}}}{{\langle}Z(2{\beta}){\rangle}}.
\end{aligned}$$ Using our result in section \[sec:lowT\], this is given by the error function $$\begin{aligned}
g({\beta},t,\hbar)=\text{Erf}\bigl({\sqrt{D}}\bigr).
\end{aligned}$$ We are interested in the late-time behavior of the SFF at the timescale of order $t\sim\hbar^{-1}$. To study this regime, it is natural to take the ’t Hooft limit[^8] $$\begin{aligned}
\hbar\to0,~ {\beta}\to\infty,~ t\to\infty,\quad\text{with}\quad {\lambda}=\hbar{\beta},~\tau=\hbar t
~~\text{fixed}.
\end{aligned}$$ As we have seen in section \[sec:lowT\], $D$ is expanded as in this ’t Hooft limit. To see the behavior of the ramp and the plateau, it is sufficient to take the first term of the ’t Hooft expansion $$\begin{aligned}
D\approx \hbar^{-1}{\mathcal{D}}_0=\hbar^{-1}\bigl[{\mathcal{F}}_0(2{\lambda})-{\mathcal{F}}_0({\lambda}+{\mathrm{i}}\tau)-{\mathcal{F}}_0({\lambda}-{\mathrm{i}}\tau)\bigr],
\end{aligned}
\label{eq:lead-D}$$ where ${\mathcal{F}}_0({\lambda})$ is given by . In Fig. \[fig:SFF\], we show the plot of SFF in the approximation for $\hbar=1/30$ with several different values of ${\lambda}$. One can see that the SFF exhibits the characteristic feature of the ramp and the plateau. We observe that the timescale of the transition from ramp to plateau depends on ${\lambda}$ as in the pure topological gravity case [@Okuyama:2019xbv].
![Plot of the spectral form factor $g({\beta},t,\hbar)$ as a function of $\tau=\hbar t$ for $\hbar=1/30$.[]{data-label="fig:SFF"}](sff-JT.pdf){width="8cm"}
In [@Saad:2019lba] it is argued that the ramp is reproduced from the genus-zero part of the connected correlator $$\begin{aligned}
{\langle}Z({\beta}+{\mathrm{i}}t)Z({\beta}-{\mathrm{i}}t){\rangle}^{g=0}_{{{\mbox{\scriptsize c}}}}=\frac{{\sqrt{{\beta}^2+t^2}}}{4\pi{\beta}}=\frac{{\sqrt{{\lambda}^2+\tau^2}}}{4\pi{\lambda}}.
\end{aligned}$$ In Fig. \[fig:SFF-genus0\] we show the plot of the genus-zero part (orange dashed curve) and the full result (blue solid curve) for the SFF with $\hbar={\lambda}=1/30$ as an example. One can see that the genus-zero part captures the growing ramp behavior of SFF at early times. This agreement at early times can be shown analytically using the Taylor expansion of the error function and the small ${\lambda}$ behavior of ${\mathcal{F}}_0({\lambda})\sim\frac{{\lambda}^3}{12}$.
![Plot of the spectral form factor for $\hbar={\lambda}=1/30$. The orange dashed curve is the genus-zero result while the blue solid curve is the plot of error function.[]{data-label="fig:SFF-genus0"}](sff-JT-genus0.pdf){width="8cm"}
The appearance of the plateau behavior is almost guaranteed by the functional form of the error function. However, one can pin down the origin of plateau by looking closely at the late-time behavior of the second term $\operatorname{Tr}(e^{{\beta}_1Q}\Pi e^{{\beta}_2Q}\Pi)$ in the connected correlator $Z_2({\beta}_1,{\beta}_2)$ in . As we have seen in section \[sec:saddle\], this term can be evaluated by the saddle point approximation. For ${\beta}_{1,2}={\beta}\pm{\mathrm{i}}t$, the saddle points are given by $$\begin{aligned}
\xi_{1,2}^*={\frac{1}{4}}\arcsin({\lambda}\pm{\mathrm{i}}\tau)^2,
\end{aligned}
\label{eq:saddle-SFF}$$ and the saddle point value is given by $$\begin{aligned}
\operatorname{Tr}(e^{{\beta}_1Q}\Pi e^{{\beta}_2Q}\Pi)\sim \exp\left[\frac{{\mathcal{F}}_0({\lambda}+{\mathrm{i}}\tau)+{\mathcal{F}}_0({\lambda}-{\mathrm{i}}\tau)}{\hbar}\right].
\end{aligned}$$ This contribution decays exponentially at late-times and the SFF approaches the plateau value given by the first term $\operatorname{Tr}(e^{({\beta}_1+{\beta}_2)Q}\Pi)$ in . These saddle points can be thought of as the eigenvalue instantons sitting at the complex conjugate pair of points $E=-\xi^*_{1,2}$ and the transition from ramp to plateau is induced by the pair creation of eigenvalue instantons as advocated in [@Okuyama:2018gfr].
Another interesting phenomenon is that the connected and the disconnected contributions exchange dominance as we lower the temperature. This transition is observed in a coupled SYK model [@Maldacena:2018lmt] and it is expected to occur in JT gravity as well. To see this, let us compare the disconnected part ${\langle}Z({\beta}){\rangle}^2$ and the connected part ${\langle}Z({\beta})^2{\rangle}_{{{\mbox{\scriptsize c}}}}$ and study their behavior as a function of ${\beta}$. Here we set ${\beta}_1={\beta}_2={\beta}$ for simplicity. Since JT gravity becomes a good approximation of the SYK model in the low energy limit, it is useful to study the behavior of two-boundary correlator in the ’t Hooft limit. At the leading order in the ’t Hooft expansion we find $$\begin{aligned}
{\langle}Z({\beta}){\rangle}^2
&\approx \frac{\hbar}{4\pi (\arcsin{\lambda})^3\sqrt{1-{\lambda}^2}}
e^{\frac{2{\mathcal{F}}_0({\lambda})}{\hbar}},\\
{\langle}Z({\beta})^2{\rangle}_{{{\mbox{\scriptsize c}}}}
&\approx \sqrt{\frac{\hbar}{4\pi(\arcsin 2{\lambda})^3\sqrt{1-4{\lambda}^2}}}
e^{\frac{{\mathcal{F}}_0(2{\lambda})}{\hbar}}\text{Erf}
\Biggl({\sqrt{\frac{{\mathcal{F}}_0(2{\lambda})-2{\mathcal{F}}_0({\lambda})}{\hbar}}}\Biggr).
\end{aligned}
\label{eq:con-dis}$$ In Fig. \[fig:exchange\], we show the plot of for $\hbar=1/30$. One can see that at high temperature the disconnected part is dominant, but as we lower the temperature the connected part becomes dominant below some critical temperature. Thus we succeeded to reproduce the transition observed in [@Maldacena:2018lmt] directly from the JT gravity computation. In the bulk gravity picture, this is an analogue of the Hawking-Page transition between two different topologies of spacetime. At high temperature the two disconnected Euclidean black holes are dominant while at low temperature the Euclidean wormhole connecting the two boundaries becomes dominant.
![Plot of the disconnected part ${\langle}Z({\beta}){\rangle}^2$ and the connected part ${\langle}Z({\beta})^2{\rangle}_{{{\mbox{\scriptsize c}}}}$ as a function of ${\lambda}=\hbar{\beta}$ at $\hbar=1/30$.[]{data-label="fig:exchange"}](exchange.pdf){width="8cm"}
Boundary creation operator and Hartle-Hawking state {#sec:boundary}
===================================================
As we have seen in section \[sec:genus\], we can write the connected $n$-point amplitude as $$\begin{aligned}
{\langle}Z({\beta}_1)\cdots Z({\beta}_n){\rangle}_{{{\mbox{\scriptsize c}}}}\simeq B({\beta}_1)\cdots B({\beta}_n)F,
\end{aligned}
\label{eq:B-corr}$$ where $F$ denotes the free energy and the operator $B({\beta})$ is given by . $B({\beta})$ can be thought of as the “boundary creation operator.” The same operator has been considered in the context of 2d gravity in [@Moore:1991ir]. should be understood as the equality up to the non-universal terms at genus-zero for the one- and two-point functions, which should be treated separately.
In a recent paper by Marolf and Maxfield [@Marolf:2020xie], the idea of boundary creation operators is also discussed. The important property of the boundary creation operators is that they all commute and hence can be diagonalized simultaneously. It is argued in [@Marolf:2020xie] that the simultaneous eigenstate of the boundary creation operators, the so-called ${\alpha}$-state, can be thought of as a member of an ensemble and the correlator ${\langle}\prod_i Z({\beta}_i){\rangle}$ is interpreted as the ensemble average. Moreover, by reinterpreting the earlier discussion of baby universes [@Coleman:1988cy; @Giddings:1988cx; @Giddings:1988wv] from the viewpoint of AdS/CFT duality, it is argued that one can define the baby universe Hilbert space from the data of correlators ${\langle}\prod_i Z({\beta}_i){\rangle}$ and this Hilbert space includes many null states due to the bulk diffeomorphism invariance.[^9] To demonstrate these properties, a simple toy model is studied in [@Marolf:2020xie] where the action $S$ of the model has only the topological term given by the Euler characteristic $\chi$ of the 2d spacetime.
In this section we will consider whether the proposal in [@Marolf:2020xie] can be generalized to the JT gravity case. Firstly, the boundary creation operator $B({\beta})$ defined in clearly commutes $$\begin{aligned}
{[}B({\beta}), B({\beta}'){]}=0,
\end{aligned}$$ and hence one can try to diagonalize $B({\beta})$’s simultaneously. One immediate problem is that $B({\beta})$ in does not look like a hermitian operator, thus its eigenvalue is not necessarily a real number. According to the proposal in [@Marolf:2020xie], this problem might be resolved on the physical Hilbert space, which is obtained by taking the quotient of the original Hilbert space by the space of null states. We do not have a clear understanding of how this happens. In the rest of this section, we will examine how the proposal of [@Marolf:2020xie] is generalized or modified in the case of JT gravity.
To study the proposal of [@Marolf:2020xie] in JT gravity, it is convenient to use the free boson or free fermion representation of the Witten-Kontsevich $\tau$-function (see e.g. [@BBT; @Aganagic:2003qj; @Kostov:2009nj; @Kostov:2010nw] and references therein) $$\begin{aligned}
\tau= e^F={\langle}t|V{\rangle},
\end{aligned}
\label{eq:tau-WK}$$ where the state ${\langle}t|$ is given by the coherent state of free boson $$\begin{aligned}
{\langle}t|={\langle}0|\exp\left(\sum_{k=0}^\infty \frac{{\widetilde{t}}_k{\alpha}_{2k+1}}{{g_{\rm s}}(2k+1)!!}\right)
\end{aligned}
\label{eq:bra-t}$$ with ${\alpha}_n$ obeying the usual commutation relation of the free boson $$\begin{aligned}
{[}{\alpha}_n,{\alpha}_m{]}=n{\delta}_{n+m,0},\qquad {\langle}0|{\alpha}_n=0\quad (n<0).
\end{aligned}$$ ${\widetilde{t}}_k$ in is defined by $$\begin{aligned}
{\widetilde{t}}_k=t_k-{\delta}_{k,1}.
\end{aligned}$$ Note that only the odd-modes ${\alpha}_{2n+1}$ appear in the state ${\langle}t|$ in since the KdV hierarchy associated with the Witten-Kontsevich $\tau$-function is a mod 2 reduction of the KP hierarchy. Note that the derivative ${\partial}_k$ with respect to the coupling $t_k$ is mapped to the operator ${\alpha}_{2k+1}$ when acting on the state ${\langle}t|$ $$\begin{aligned}
{\partial}_k{\langle}t|={\langle}t|\frac{{\alpha}_{2k+1}}{{g_{\rm s}}(2k+1)!!}.
\end{aligned}
\label{eq:t-map}$$ In other words, the microscopic loop operator $\tau_k$ corresponds to ${\alpha}_{2k+1}$ up to a normalization constant.
To write down the state $|V{\rangle}$ in , it is useful to introduce the free fermions $\psi_r,\psi_r^*~(r\in\mathbb{Z}+{\frac{1}{2}})$ obeying the anti-commutation relation $$\begin{aligned}
\{\psi_r,\psi_s^*\}={\delta}_{r+s,0},\qquad
\psi_r|0{\rangle}=\psi^*_r|0{\rangle}=0\quad (r>0).
\end{aligned}$$ They are related to ${\alpha}_n$ by the bosonization $$\begin{aligned}
{\alpha}_n=\sum_{r\in\mathbb{Z}+{\frac{1}{2}}}:\psi_r\psi_{n-r}^*:.
\end{aligned}$$ Then $|V{\rangle}$ is written as $$\begin{aligned}
|V{\rangle}=\exp\left(\sum_{m,n=0}^\infty A_{m,n}\psi_{-m-{\frac{1}{2}}}\psi^*_{-n-{\frac{1}{2}}}\right)|0{\rangle}.
\end{aligned}
\label{eq:V-ket}$$ In general any choice of $A_{m,n}$ defines a $\tau$-function of KdV hierarchy, but the Witten-Kontsevich $\tau$-function for the topological gravity corresponds to a specific choice of $A_{m,n}$. The generating function of $A_{m,n}$ for the Witten-Kontsevich $\tau$-function is obtained in [@zhou2013explicit; @zhou2015emergent; @balogh2017geometric]: $$\begin{aligned}
\sum_{m,n=0}^\infty A_{m,n}z^{-m-1}w^{-n-1}=\frac{1}{z-w}+\frac{a(w)b(-z)-a(-z)b(w)}{z^2-w^2},
\end{aligned}
\label{eq:A-gen}$$ where $a(z)$ and $b(z)$ are given by $$\begin{aligned}
a(z)&=\sum_{m=0}^\infty
\left(\frac{-{g_{\rm s}}}{288}\right)^{m}\frac{(6m)!}{(2m)!(3m)!}z^{-3m},\\
b(z)&=-\sum_{m=0}^\infty \left(\frac{-{g_{\rm s}}}{288}\right)^{m}\frac{(6m)!}{(2m)!(3m)!}
\frac{6m+1}{6m-1}z^{-3m+1}.
\end{aligned}
\label{eq:azbz-def}$$ The series $a(z)$ and $b(z)$ appear in the asymptotic expansion of the Airy function and its first derivative, respectively.
The important property of the state $|V{\rangle}$ is that it satisfies the Virasoro constraint [@Fukuma:1990jw; @Dijkgraaf:1990rs] $$\begin{aligned}
{\mathcal{L}}_n|V{\rangle}=0\quad(n\geq-1),
\end{aligned}
\label{eq:Vir-con}$$ where the Virasoro generator ${\mathcal{L}}_n$ is given by $$\begin{aligned}
{\mathcal{L}}_n&=L_n
-\frac{1}{2{g_{\rm s}}}{\alpha}_{2n+3},\\
L_n&={\frac{1}{4}}\sum_{k\in\mathbb{Z}}:{\alpha}_{2k+1}{\alpha}_{2n-2k-1}:
+\frac{1}{16}{\delta}_{n,0}.
\end{aligned}
\label{eq:Ln-def}$$ One can show that $[{\mathcal{L}}_n,{\mathcal{L}}_m]=(n-m){\mathcal{L}}_{n+m}$ for $n,m\geq-1$. The constant term $\frac{1}{16}$ in ${\mathcal{L}}_0$ can be thought of as the conformal weight of the $\mathbb{Z}_2$ twist field. Note that ${\mathcal{L}}_n$ is written as $$\begin{aligned}
{\mathcal{L}}_n= e^{-\frac{1}{3{g_{\rm s}}}{\alpha}_3}L_ne^{\frac{1}{3{g_{\rm s}}}{\alpha}_3}.
\end{aligned}$$ In other words, the linear term $-\frac{1}{2{g_{\rm s}}}{\alpha}_{2n+3}$ in arises from the shift of ${\widetilde{t}}_1=t_1-1$ [@Kac:1991nv; @Itzykson:1992ya].
One can show that $|V{\rangle}$ in satisfies the Virasoro constraint order by order in the ${g_{\rm s}}$-expansion. To see this, let us expand the state $|V{\rangle}$ as $$\begin{aligned}
|V{\rangle}=\sum_{n=0}^\infty {g_{\rm s}}^{n}|V_n{\rangle}.
\end{aligned}$$ From the result of $A_{m,n}$ in , the first two terms are given by $$\begin{aligned}
|V_0{\rangle}&=|0{\rangle},\\
|V_1{\rangle}&=\frac{1}{24}\Bigl(5\psi_{-{\frac{1}{2}}}\psi^*_{-\frac{5}{2}}
+5\psi_{-\frac{5}{2}}\psi^*_{-{\frac{1}{2}}}-7\psi_{-\frac{3}{2}}\psi^*_{-\frac{3}{2}}\Bigr)|0{\rangle}.
\end{aligned}
\label{eq:V01}$$ As an example, let us check the constraint ${\mathcal{L}}_{n}|V{\rangle}=0$ at the order ${\mathcal{O}}({g_{\rm s}}^0)$. At this order the constraint ${\mathcal{L}}_{n}|V{\rangle}=0$ reads $$\begin{aligned}
L_{n}|V_0{\rangle}-\frac{1}{2}{\alpha}_{2n+3}|V_1{\rangle}=0.
\end{aligned}$$ One can easily show that this is indeed satisfied for $|V_{0,1}{\rangle}$ in . The higher order constraint can be shown in a similar manner. In [@Sen:1990rz; @Imbimbo:1990ua], the Virasoro constraint of matrix model is interpreted as the gauge symmetry of closed string field theory in a minimal model background. This suggests that the Virasoro constraint is the analogue of the bulk diffeomorphism invariance discussed in [@Marolf:2020xie].
Now let us consider the Hartle-Hawking state $|\text{HH}{\rangle}$ [@Hartle:1983ai]. As discussed in [@Polchinski:1989fn], it is natural to identify the Hartle-Hawking state $|\text{HH}{\rangle}$ as “the most symmetric state.” In the present case, $|V{\rangle}$ is such a state since $|V{\rangle}$ is invariant under the Virasoro generators . $|V{\rangle}$ can be thought of as the $SL(2,\mathbb{C})$ invariant vacuum corresponding to the identity operator and it is a natural candidate for the no-boundary state. Thus we propose to identify the Hartle-Hawking state $|\text{HH}{\rangle}$ with the state $|V{\rangle}$ in $$\begin{aligned}
|\text{HH}{\rangle}=|V{\rangle}.
\end{aligned}
\label{eq:our-HH}$$ In particular, this state satisfies the equation ${\mathcal{L}}_0|\text{HH}{\rangle}=0$ which corresponds to the Wheeler-DeWitt equation.
Next we consider the interpretation of the correlator ${\langle}\prod_i Z({\beta}_i){\rangle}$ in JT gravity. The correlator here refers to the full correlator including both the connected and the disconnected parts. One can generalize to the full correlator by acting $B({\beta})$’s on the $\tau$-function instead of the free energy $$\begin{aligned}
{\langle}Z({\beta}_1)\cdots Z({\beta}_n){\rangle}&\simeq\frac{B({\beta}_1)\cdots B({\beta}_n){\langle}t|V{\rangle}}
{{\langle}t|V{\rangle}}\\
&=:
\frac{{\langle}t|{\widehat{B}}({\beta}_1)\cdots {\widehat{B}}({\beta}_n)|V{\rangle}}{{\langle}t|V{\rangle}}.
\end{aligned}$$ By using the dictionary the operator ${\widehat{B}}({\beta})$ is written as $$\begin{aligned}
{\widehat{B}}({\beta})=\frac{1}{{\sqrt{2\pi}}}\sum_{n=0}^\infty \frac{{\beta}^{n+{\frac{1}{2}}}}{(2n+1)!!}{\alpha}_{2n+1}.
\end{aligned}$$ It turns out that the non-universal terms at genus-zero are correctly incorporated by extending the summation to all $n\in\mathbb{Z}$. Namely we define the operator ${\widehat{Z}}({\beta})$ by $$\begin{aligned}
{\widehat{Z}}({\beta})=\frac{1}{{\sqrt{2\pi}}}\sum_{n=-\infty}^\infty \frac{{\beta}^{n+{\frac{1}{2}}}}{(2n+1)!!}{\alpha}_{2n+1}.
\end{aligned}
\label{eq:hat-Z}$$ Then the full correlator is given by $$\begin{aligned}
{\langle}Z({\beta}_1)\cdots Z({\beta}_n){\rangle}=
\frac{{\langle}t|{\widehat{Z}}({\beta}_1)\cdots {\widehat{Z}}({\beta}_n)|\text{HH}{\rangle}}{{\langle}t|\text{HH}{\rangle}},
\end{aligned}
\label{eq:HH-corr}$$ where we used our identification $|\text{HH}{\rangle}=|V{\rangle}$. To see that this is the correct prescription, let us consider the genus-zero part of the one-point function $$\begin{aligned}
{\langle}Z({\beta}){\rangle}^{g=0} &=\frac{ {\langle}t| {\widehat{Z}}({\beta})|\text{HH}{\rangle}}{{\langle}t|\text{HH}{\rangle}} \Bigg|_{g=0}
={\langle}0|\exp\left(\sum_{k=1}^\infty \frac{{\widetilde{t}}_k}{{g_{\rm s}}(2k+1)!!}{\alpha}_{2k+1}\right)
{\widehat{Z}}({\beta})|0{\rangle}\\
&=\frac{1}{{\sqrt{2\pi}}{g_{\rm s}}}\sum_{k=1}^\infty {\beta}^{-k-{\frac{1}{2}}}\frac{{\widetilde{t}}_k}{(2k-1)!!(-2k-1)!!}\\
&=\frac{e^{1/{\beta}}}{{\sqrt{2\pi}}{g_{\rm s}}{\beta}^{3/2}}.
\end{aligned}
\label{eq:g0-Z}$$ Here we have used ${\widetilde{t}}_k=\frac{(-1)^k}{(k-1)!}~(k\geq1)$ and $$\begin{aligned}
(2k-1)!!(-2k-1)!!=(-1)^k.
\end{aligned}
\label{eq:fac-rel}$$ Similarly, the genus-zero part of the two-point function becomes $$\begin{aligned}
{\langle}Z({\beta}_1)Z({\beta}_2){\rangle}^{g=0}_{{{\mbox{\scriptsize c}}}}&=\frac{1}{2\pi}\Biggl{\langle}0\Biggr|\sum_{k=0}^\infty
\frac{{\beta}_1^{k+{\frac{1}{2}}}}{(2k+1)!!}{\alpha}_{2k+1}
\sum_{n=0}^\infty
\frac{{\beta}_2^{-n-{\frac{1}{2}}}}{(-2n-1)!!}{\alpha}_{-2n-1}\Biggl|0\Biggr{\rangle}\\
&=\frac{1}{2\pi}\sum_{n=0}^\infty \frac{{\beta}_1^{n+{\frac{1}{2}}}{\beta}_2^{-n-{\frac{1}{2}}}}{(2n-1)!!(-2n-1)!!}
\\
&=\frac{{\sqrt{{\beta}_1{\beta}_2}}}{2\pi({\beta}_1+{\beta}_2)}.
\end{aligned}
\label{eq:g0-ZZ}$$ and agree with the known result of the genus-zero part in JT gravity. Using the relation one can show that ${\widehat{Z}}({\beta})$’s commute at least formally $$\begin{aligned}
{[}{\widehat{Z}}({\beta}_1),{\widehat{Z}}({\beta}_2){]}
&=\frac{1}{2\pi}\sum_{n,k\geq0}\left[\frac{{\beta}_1^{k+{\frac{1}{2}}}}{(2k+1)!!}{\alpha}_{2k+1},
\frac{{\beta}_2^{-n-{\frac{1}{2}}}}{(-2n-1)!!}{\alpha}_{-2n-1}\right]\\
&+\frac{1}{2\pi}\sum_{n,k\geq0}
\left[\frac{{\beta}_1^{-k-{\frac{1}{2}}}}{(-2k-1)!!}{\alpha}_{-2k-1},
\frac{{\beta}_2^{n+{\frac{1}{2}}}}{(2n+1)!!}{\alpha}_{2n+1}\right]\\
&=\frac{1}{2\pi}\sum_{n\geq0}(-1)^n\Bigl({\beta}_1^{n+{\frac{1}{2}}}{\beta}_2^{-n-{\frac{1}{2}}}-{\beta}_1^{-n-{\frac{1}{2}}}{\beta}_2^{n+{\frac{1}{2}}}\Bigr)\\
&=\frac{{\sqrt{{\beta}_1{\beta}_2}}}{2\pi}\left(\frac{1}{{\beta}_1+{\beta}_2}-\frac{1}{{\beta}_1+{\beta}_2}\right)\\
&=0.
\end{aligned}$$
Our proposal is consistent with the identification of the one-point function ${\langle}Z({\beta}){\rangle}$ as the wavefunction of the Hartle-Hawking state, which is usually assumed in 2d gravity literature (see e.g. [@Ginsparg:1993is] and references therein) $$\begin{aligned}
{\langle}Z({\beta}){\rangle}=\Psi_{\text{HH}}({\beta})={\langle}{\beta}|\text{HH}{\rangle},
\end{aligned}$$ where ${\langle}{\beta}|$ is given by $$\begin{aligned}
{\langle}{\beta}|=\frac{{\langle}t|{\widehat{Z}}({\beta})}{{\langle}t|\text{HH}{\rangle}}.
\end{aligned}$$ More generally, the multi-point correlator is written as $$\begin{aligned}
{\langle}Z({\beta}_1)\cdots Z({\beta}_n){\rangle}&={\langle}{\beta}_1,\cdots,{\beta}_n|\text{HH}{\rangle},\\
{\langle}{\beta}_1,\cdots,{\beta}_n|&=\frac{{\langle}t|{\widehat{Z}}({\beta}_1)\cdots {\widehat{Z}}({\beta}_n)}{{\langle}t|\text{HH}{\rangle}}.
\end{aligned}$$
Our expression is different from the proposal in [@Marolf:2020xie] $$\begin{aligned}
{\langle}Z({\beta}_1)\cdots Z({\beta}_n){\rangle}=
\frac{{\langle}\text{HH}|{\widehat{Z}}({\beta}_1)\cdots {\widehat{Z}}({\beta}_n)|\text{HH}{\rangle}}{{\langle}\text{HH}|\text{HH}{\rangle}}.
\end{aligned}
\label{eq:MM-prop}$$ This difference comes from the fact that the bra and the ket are treated asymmetrically in the free boson/fermion representation of the $\tau$-function . In other words, our expression corresponds to a special (Euclidean) time-slicing of the spacetime where the initial state has no boundary and all the boundaries are on the final state. At present, it is not clear to us how to reconcile our and the proposal in [@Marolf:2020xie].
Conclusions and outlook {#sec:conclusion}
=======================
We have studied the multi-boundary correlators in JT gravity using the KdV constraints obeyed by these correlators. Along the way, we have defined the off-shell generalization of the effective potential and have studied the WKB expansion of the Baker-Akhiezer functions as well. In particular, we have computed the genus expansion of the connected two-boundary correlator ${\langle}Z({\beta}_1)Z({\beta}_2){\rangle}_{{\mbox{\scriptsize c}}}$ as well as its low temperature expansion. We have found that the two-point function is written in terms of the error function and the ramp and plateau behavior of the SFF in JT gravity is explained by the functional form of this error function. We have also confirmed the picture put forward in [@Okuyama:2018gfr] that the transition from ramp to plateau is induced by the pair creation of eigenvalue instantons.
There are many interesting open questions. In section \[sec:boundary\] we briefly discussed a possible connection to the recent work by Marolf and Maxfield [@Marolf:2020xie] which clearly deserves further investigation. It would be interesting to construct the ${\alpha}$-state which simultaneously diagonalizes the operator ${\widehat{Z}}({\beta})$ in and see how the argument in [@Marolf:2020xie] is generalized to the JT gravity case. In particular, it is interesting to see what the non-factorized contribution ${\langle}Z({\beta}_1)Z({\beta}_2){\rangle}_{{\mbox{\scriptsize c}}}$ coming from the Euclidean wormhole [@Maldacena:2004rf; @ArkaniHamed:2007js] looks like in the ${\alpha}$-state. The pure topological gravity would be a good starting point to study this problem since the explicit form of the $n$-point correlator is known in the literature [@okounkov2002generating; @buryak; @Alexandrov:2019eah].
It is emphasized in [@Marolf:2020xie] that non-perturbative effects are important to realize the massive truncation of the Hilbert space by the diffeomorphism invariance. The free fermion representation of the state $|V{\rangle}$ in is defined by the asymptotic expansion in ${g_{\rm s}}$ and hence it only makes sense as a perturbative expansion (see ). However, it is possible to include the effect of D-instanton corrections systematically within this framework [@Fukuma:1996hj; @Fukuma:1996bq; @Fukuma:1999tj]. It would be interesting to study such D-instanton effects in JT gravity and see how they affect the argument of diffeomorphism invariance in JT gravity.
In [@Penington:2019kki; @Almheiri:2019qdq] it is argued that the Page curve for the black hole evaporation is correctly reproduced if we include the contribution of replica wormholes in the computation of entropy of Hawking radiation using the replica method in the gravity path integral. One can immediately apply our formalism to compute the contribution of the replica wormholes in pure JT gravity sector. To model the black hole microstates one can add the end of the world (EOW) branes to JT gravity [@Penington:2019kki; @Marolf:2020xie]. It would be interesting to construct a generalization of the JT gravity matrix model which incorporates the degrees of freedom of the EOW branes.
As discussed in [@Maldacena:2019cbz; @Cotler:2019nbi], the matrix model description of JT gravity can be generalized to the 2d de Sitter space by analytically continuing the boundary length ${\beta}$ to imaginary value ${\beta}\to\pm{\mathrm{i}}\ell$. In [@Cotler:2019dcj] the boundary creation/annihilation operators are considered in this de Sitter setting. It would be interesting to see how they are related to our discussion in section \[sec:boundary\].
Finally, it would be interesting to generalize our computation in this paper to JT supergravity [@Stanford:2019vob]. In particular, the genus expansion of JT supergravity on orientable surfaces without time-reversal symmetry can be computed from the Brezin-Gross-Witten $\tau$-function [@norbury]. We will report on the computation of JT supergravity case elsewhere.
This work was supported in part by JSPS KAKENHI Grant Nos. 19K03845 and 19K03856, and JSPS Japan-Russia Research Cooperative Program.
Wavefunction of microscopic loop operators {#sec:micro}
==========================================
In this appendix we will consider the correlator of microscopic loop operators in the presence of one macroscopic loop operator. It is easily obtained by differentiating ${\langle}Z({\beta}){\rangle}$ $$\begin{aligned}
{\partial}_{n_1}{\partial}_{n_2}\cdots {\langle}Z({\beta}){\rangle}={\langle}\tau_{n_1}\tau_{n_2}\cdots Z({\beta}){\rangle}.
\end{aligned}$$ It is convenient to define the normalized correlator $$\begin{aligned}
\big{\langle}\!\big{\langle}\prod_i \tau_{n_i}\big{\rangle}\!\big{\rangle}:
=\frac{{\langle}\prod_i \tau_{n_i} Z({\beta}){\rangle}}{{\langle}Z({\beta}){\rangle}},
\end{aligned}$$ which can be thought of as the wavefunction of microscopic loop operators [@Moore:1991ir; @Ginsparg:1993is].
For instance, the one-point function $\big{\langle}\!\big{\langle}\tau_{n}\big{\rangle}\!\big{\rangle}$ at the leading order is given by $$\begin{aligned}
\big{\langle}\!\big{\langle}\tau_{n}\big{\rangle}\!\big{\rangle}={\partial}_n\log{\langle}Z({\beta}){\rangle}\approx
\frac{1}{\hbar}{\partial}_n {\mathcal{F}}_0({\lambda}).
\end{aligned}
\label{eq:tau-dF}$$ The derivative of ${\mathcal{F}}_0({\lambda})$ with respect to the coupling $t_n$ can be computed by using the fact that $V_{\text{eff}}(\xi)$ and ${\mathcal{F}}_0({\lambda})$ are related by the Legendre transformation. Thus we find $$\begin{aligned}
\frac{{\partial}{\mathcal{F}}_0({\lambda})}{{\partial}t_n}\Bigg|_{{\lambda}~\text{fixed}}&=\frac{{\partial}}{{\partial}t_n}\Bigg|_{{\lambda}~\text{fixed}}
\Bigl({\lambda}\xi_*-V_{\text{eff}}(\xi_*)\Bigr)\\
&={\lambda}\frac{{\partial}\xi_*}{{\partial}t_n}
-V_{\text{eff}}'(\xi_*)\frac{{\partial}\xi_*}{{\partial}t_n}-\frac{{\partial}V_{\text{eff}}(\xi_*)}{{\partial}t_n}\Bigg|_{\xi_*~\text{fixed}}\\
&=-\frac{{\partial}V_{\text{eff}}(\xi_*)}{{\partial}t_n}\Bigg|_{\xi_*~\text{fixed}}.
\end{aligned}
\label{eq:FV-Legendre}$$ In the last step we have used the saddle point equation ${\lambda}=V_{\text{eff}}'(\xi_*)$. From the explicit form of the off-shell effective potential in , one can easily compute the derivative $-{\partial}_n V_{\text{eff}}(\xi_*)$. From and , for the on-shell JT gravity case $t_n={\gamma}_n$ we find the wavefunction of the microscopic loop operator $\tau_n$ at the leading order in the ’t Hooft expansion $$\begin{aligned}
\big{\langle}\!\big{\langle}\tau_{n}\big{\rangle}\!\big{\rangle}=
\frac{\arcsin({\lambda})^{2n+1}}{ 2^{n}(2n+1)!!\hbar}+{\mathcal{O}}(\hbar^0).
\end{aligned}$$ It turns out that the wavefunction is factorized at the leading order in the ’t Hooft expansion $$\begin{aligned}
\big{\langle}\!\big{\langle}\prod_i \tau_{n_i}\big{\rangle}\!\big{\rangle}&\approx e^{-\frac{{\mathcal{F}}_0({\lambda})}{\hbar}}\prod_i{\partial}_{n_i}e^{\frac{{\mathcal{F}}_0({\lambda})}{\hbar}}\\
&\approx\prod_i \frac{{\partial}_{n_i}{\mathcal{F}}_0({\lambda})}{\hbar}\\
&\approx \prod_i \big{\langle}\!\big{\langle}\tau_{n_i}\big{\rangle}\!\big{\rangle}.
\end{aligned}$$
One can go beyond the leading order and compute the higher order correction to the wavefunction of microscopic loop operators by using the off-shell generalization of the free energy ${\mathcal{F}}$ in . After some algebra, we find the first order correction to the ’t Hooft expansion $$\begin{aligned}
\big{\langle}\!\big{\langle}\tau_0^k\prod_{i=1}^m \tau_{n_i}\big{\rangle}\!\big{\rangle}=&\frac{\arcsin({\lambda})^k}{\hbar^{m+k}}
\prod_{i=1}^m\frac{\arcsin({\lambda})^{2n_i+1}}{ 2^{n_i}(2n_i+1)!!}\\
\times &\Biggl[1+\hbar\biggl(2\sum_{i=1}^mn_i+m+k\biggr)\left(
\frac{2\sum_{i=1}^mn_i+m+k-5}{{\sqrt{1-{\lambda}^2}}\arcsin({\lambda})^3}+\frac{{\lambda}}{(1-{\lambda}^2)\arcsin({\lambda})^2}\right)\\
&\qquad+\hbar\frac{5k-k^2}{\arcsin({\lambda})^3}+{\mathcal{O}}(\hbar^2)\Biggr],
\end{aligned}
\label{eq:macro-micro}$$ where $n_i>0~(i=1,\cdots,m)$.
[^1]: We will refer to the $\hbar$-expansion of a function of energy eigenvalue $\xi$ as “the WKB expansion” while the $\hbar$-expansion of a function of the ’t Hooft parameter ${\lambda}$ as “the ’t Hooft expansion.” They are related by the saddle point approximation of the integral such as .
[^2]: $G$ and $F$ in this paper are related to those in our previous paper [@Okuyama:2019xbv] by $G_{\rm here}={g_{\rm s}}^{-2}G_{\rm there}$, $F_{\rm here}={g_{\rm s}}^{-2}F_{\rm there}$.
[^3]: This change of variables was originally introduced by Zograf (see e.g. [@Zograf:2008wbe]).
[^4]: can also be shown by using $\partial_k u_0={\partial}_k{\partial}_0^2F_0$ with $F_0$ in .
[^5]: $W_{g,1}$ is related to $W_g$ in [@Okuyama:2019xbv] by $\beta W_{g,1}=W_g$.
[^6]: For the sake of simplicity we restrict ourselves hereafter to the JT gravity case $t_k=\gamma_k\ (k\ge 2)$, i.e. we set $1-I_1=t$, but the discussion here would be easily generalized to the case of general topological gravity.
[^7]: As reviewed in appendix A of [@Okuyama:2019xbv], the BA function for the pure topological gravity is given by $\psi=\hbar^{-\frac{1}{6}}\text{Ai}\bigl(\hbar^{-\frac{2}{3}}z^2\bigr)$ which has the large $z$ asymptotic expansion $
\psi\approx \frac{1}{{\sqrt{4\pi z}}}e^{-\frac{2z^3}{3\hbar}}
$.
[^8]: $t$ and $\tau$ in this section should not be confused with those used in the previous sections.
[^9]: In a recent paper [@vafa], it is argued that the baby universe Hilbert space must be one-dimensional in a consistent quantum gravity on a spacetime with dimension $d>3$.
|
---
abstract: 'Many microfluidic devices use macroscopic pressure differentials to overcome viscous friction and generate flows in microchannels. In this work, we investigate how the chemical and geometric properties of the channel walls can drive a net flow by exploiting the autophoretic slip flows induced along active walls by local concentration gradients of a solute species. We show that chemical patterning of the wall is not required to generate and control a net flux within the channel, [[ rather]{}]{} channel geometry alone is sufficient. Using numerical simulations, we determine how geometric characteristics of the wall influence channel flow rate, and confirm our results analytically in the asymptotic limit of lubrication theory.'
author:
- Sébastien Michelin
- 'Thomas D. Montenegro-Johnson'
- Gabriele De Canio
- 'Nicolas Lobato-Dauzier'
- Eric Lauga
title: Geometric pumping in autophoretic channels
---
v
Introduction
============
Controlled flow manipulation at the micro- or nano-scale is at the heart of recent developments in microfluidics, including many applications in the field of biological analysis and screening [@whitesides2006]. Generating and controlling a flow within the confined environment of a microfluidic channel requires an external forcing to overcome the viscous stress on the walls. In synthetic micro and nanofluidic systems, this is usually achieved [[ either]{}]{} mechanically, by applying a pressure difference between the inlet and outlet of the domain, or through electrokinetics[[ /electroosmosis, where the flow results from an externally-imposed electric field within the channel [@squires2005; @sia2003microfluidic; @ajdari1995; @ajdari2000]]{}]{}.
However, many biological systems rely on stresses localized at boundaries in order to drive flow, rather than on a global macroscopic forcing. For example, microscopic cilia on the lung epithelium induce a directional flow of mucus through their coordinated beating, acting as a pump [@sleigh1988]. Similar microscale flow forcing at the wall also plays an essential role in the early stages of embryo development [@hirokawa2009] or in the reproduction of mammals, where cilia-driven flow is responsible for the migration of the ovum down the female reproductive tract [@halbert1976].
In a dual [process]{}, cilia-driven flows play an essential role in the self-propulsion of micro-organisms such as *Paramecium* [@brennen1977]; the flow generated by the beating of cilia anchored on the wall of a moving cell is responsible for its locomotion. [For both swimming and pumping]{}, the coordination of neighboring cilia into metachronal wave patterns [has been proven]{} essential to achieving maximum flow rate/swimming speed with a minimum energetic cost [@gueron1999; @michelin2010c; @osterman2011; @hussong2011; @elgeti2013].
Several attempts have been made to reproduce ciliary pumping in the lab through the fabrication of artificial actuated cilia [@dentoonder2008; @fahrni2009; @babataheri2011; @coq2011]. All of them rely, however, on the application of a global electromagnetic forcing field, and generating efficient pumping would require the application of phase-shifted forcing on neighboring cilia [@khaderi2011; @khaderi2012]. This [constraint]{}, as well as the manufacturing process of microscopic cilia, poses important challenges to miniaturization.
Phoretic mechanisms, namely the ability to generate fluid motion near a boundary under the effect of an external field gradient, represent an alternative route for both pumping and swimming systems that require no moving parts. [These mechanisms arise from]{} the interaction of rigid boundaries with neutral or ionic solute species in their immediate environment, and are known to generate the migration of passive particles in external gradients [@anderson1989]. Phoretic motion has recently received renewed attention in the context of artificial self-propelled systems. [Such artificial swimmers]{} generate the [field gradients required for propulsion]{} themselves, for example through chemical reactions catalyzed at their surface, and [thus]{} do not rely on any [external]{} forcing to achieve propulsion [@paxton2004; @howse2007; @golestanian2007; @palacci2013]. [These]{} systems combine two properties: (i) an *activity*, i.e. the ability to release/consume solute species or thermal energy at their surface, and (ii) a phoretic *mobility*, i.e. the ability to create a slip velocity at the boundary from a local tangential gradient of solute concentration (diffusiophoresis), temperature (thermophoresis) or electric potential (electrophoresis). Recently, autophoretic systems have also been considered for generating micro-rotors that rotate without the application of external torques [@yang2014; @yang2015].
In order to generate the surface gradients necessary to their self-propulsion, autophoretic particles must break spatial symmetry. A similar requirement exists for autophoretic pumps. This symmetry-breaking may be achieved for self-propelled particles following three main routes: (i) chemical asymmetry, i.e. patterning the particle surface with active and passive sites (e.g. Janus particles) [@golestanian2007; @howse2007; @theurkauff2012], (ii) spontaneous symmetry-breaking resulting from the advection of the field responsible for the phoretic response by the flow itself [@michelin2013c; @izri2014] and (iii) geometric asymmetry [@shklyaev2014; @michelin2015a].
In this work, we use a combination of theoretical analysis and numerical simulations to investigate whether, and how, autophoretic mechanisms and geometric asymmetry can generate a net flow in a microfluidic channel without imposing any external mechanical forcing, electromagnetic forcing [or chemical patterning]{}. For simplicity we focus on the specific case of diffusiophoresis, where slip velocities are generated at the wall from tangential gradients in the concentration of a solute released from one of the channel walls into the fluid. Because of the similarity between the different phoretic mechanisms, it is expected that the results of the present contribution may easily be generalized to thermo- or electrophoretic systems. Specifically, we follow the classical continuum framework of self-diffusiophoresis [@golestanian2007; @julicher2009; @sabass2012; @michelin2014], and consider how a left-right asymmetry in the wall shape can generate a net flow within the channel which hence acts as a microfluidic pump.
The paper is organized as follows. Section \[sec:equations\] summarizes this continuum framework for the case of the flow within an asymmetric channel, and presents the numerical methods used in this work. Section \[sec:results\] shows how the wall geometric characteristics determine the net flow within the channel. The results are then confirmed analytically in Sec. \[sec:lubrication\] using lubrication theory in the long-wavelength limit, and conclusions and perspectives are presented in Sec. \[sec:conclusions\].
Problem formulation {#sec:equations}
===================
Diffusiophoretic channel
------------------------
We consider a two-dimensional channel of mean height $H$, bounded by a flat bottom boundary ($y=0$) and a top wall with a periodic non-flat profile, $y=h(x)$ [(see illustration in Figure \[fig:schema\])]{}. In the channel gap, filled with a Newtonian fluid of dynamic viscosity $\mu$ and density $\rho$, a solute species of local concentration $C({\mathbf{x}})$ with molecular diffusivity $D$ is present and interacts with the channel walls through a short range potential. When the typical thickness of this interaction region is much smaller than the other length scales of the problem (namely the channel gap and the wavelength), the interaction of the wall with a local solute gradient generates an effective slip velocity at the wall [@anderson1989; @michelin2014] $$\label{eq:mobility}
{\mathbf{u}}_\textrm{slip}=M\boldsymbol{\grad}_\parallel C,$$ where $\boldsymbol{\grad}_\parallel=(\mathbf{1}-{\mathbf{n}}{\mathbf{n}})\cdot\boldsymbol{\grad}$ is the tangential component of the gradient to the surface of local normal ${\mathbf{n}}$ and $M$, the phoretic mobility, is a property of the solute-wall interaction [which]{} may be positive or negative depending on the repulsive or attractive nature of that interaction [@anderson1989]. The chemical properties of the channel walls are also characterized by a chemical activity, i.e. the ability to create or consume the solute species. Here we consider a simple fixed-flux model, for which the activity of the wall is given by a fixed flux of solute per unit area $A$, counted positively (resp. negatively) when solute is released (resp. absorbed) $$\label{eq:activity}
D\,{\mathbf{n}}\cdot\boldsymbol{\grad} C=-A.$$
In the case of self-diffusiophoretic propulsion, locomotion [is often achieved through]{} inhomogeneity in the chemical treatment of the particle [@paxton2004; @howse2007; @golestanian2007]. Recent work has shown that geometric asymmetry of chemically-homogeneous particle alone [is in fact]{} sufficient to ensure locomotion [@shklyaev2014; @michelin2015a]. Here we investigate a similar question, namely the possibility of obtaining a net flow from chemically-homogeneous channel walls using shape asymmetry. We thus assume that the top corrugated wall has homogeneous mobility $M$ and activity $A$. To ensure the existence of a steady state solution, the concentration of the solute [on the bottom wall]{} is assumed to be fixed ($C=C_0$). [Consequently, the fluid velocity on that wall satisfies the no-slip boundary condition.]{} By studying the relative concentration of solute to [that on the bottom wall, we can assume without loss of generality that $C_0=0$.]{}
The phoretic slip velocity generated at the top wall by the wall-solute interaction drives a flow within the channel. When viscous effects dominate inertia (namely, when the Reynolds number ${\mbox{Re}}=\rho \mathcal{U}H/\mu$ is small, with $\mathcal{U}={{\color{black} |AM|}}/D$ the typical phoretic velocity), the flow satisfies [the incompressible Stokes ]{} equations $$\label{eq:stokes}
\mu\grad^2{\mathbf{u}}=\boldsymbol{\grad} p,\qquad \boldsymbol{\nabla}\cdot {\mathbf{u}}=0,$$ for the velocity and pressure fields, ${\mathbf{u}}$ and $p$ respectively. Solute molecules diffuse within the channel, and in general can also be advected by the phoretic flows. However, when diffusive effects dominate (i.e. when the Péclet number, ${\mbox{Pe}}=\mathcal{U}H/D$, is small), the solute dynamics is completely decoupled from the flow, and the solute concentration satisfies Laplace’s equation $$\label{eq:laplace}
\nabla^2 C=0.$$
Equations –, together with the boundary conditions Eqs. – applied at the top wall and the inert boundary conditions $C=0$ and ${\mathbf{u}}=\mathbf{0}$ at the bottom wall, form a closed set of equations that can be solved successively for the solute concentration $C$ and velocity field ${{\color{black} {\mathbf{u}}= (u,v)}}$. From these results, the net flow rate within the channel, $Q$, can be computed as $$\label{eq:flux}
Q=\int_0^{h(x)}u(x,y){\mathrm{d}}y,$$ [which]{} is independent of $x$ because the flow is incompressible.
Asymmetric channel
------------------
The shape of the channel is a periodic function of $x$ characterized by a wavenumber $k=2\pi/L$. A sinusoidal wall will generate a perfectly left-right-symmetric concentration distribution and flow pattern, leading to no net flow along the channel. In the following, we focus on a subset of asymmetric wall shapes, essentially smoothed ratchets, that are formally obtained by mathematically shearing the symmetric sinusoidal profile. The top wall is described in parametric form by $$\label{eq:shape}
x(s)=s-\frac{\gamma}{k} \sin ks,\qquad y(s)=H+a\sin ks,$$ the non-dimensional [[ asymmetry]{}]{} parameter, $\gamma$, determines the asymmetry of the profile, and $H$ and $a$ are the mean channel width and the amplitude of the width fluctuations, respectively (see Figure \[fig:schema\]).
![Asymmetric phoretic channel. The top wall is characterized by constant chemical activity $A$ and mobility $M$. The [bottom wall]{} maintains a fixed concentration and thus flow satisfies no-slip there. The example shown here corresponds to $\gamma=\pi/4$, $a/H=1/2$ and $L/H=2\pi$ with the asymmetric shape of the wall given in Eq. .[]{data-label="fig:schema"}](domain_plot.pdf)
[Hereafter]{}, the problem is non-dimensionalized using $1/k$ as characteristic length, $\mathcal{U}$ as characteristic velocity, and ${{\color{black} |A|}}/Dk$ as characteristic concentration fluctuation. [[ While $A$ and $M$ are both signed quantities, after nondimensionalisation we focus on the case $A=M=1$; changing the sign of either $A$ or $M$ simply reverses the slip velocity forcing and flow rate without changing its magnitude.]{}]{} The problem is now completely [specified]{} by three geometrical quantities, namely the non-dimensional mean gap, $H^*=kH$, the corrugation amplitude, $a^*=ka$, and the [[ asymmetry]{}]{} parameter, $\gamma$.
Numerical method
----------------
Equation for the solute concentration and Eq. for the flow and pressure fields are solved numerically using a boundary integral approach with periodic Green’s functions.
We denote $\Omega$ the fluid domain in a period of the channel gap, $\partial\Omega$ its lower and upper boundaries (the inert and active walls), and ${\mathbf{n}}$ the unit normal vector pointing into the fluid domain. The two-dimensional periodic Green’s function for Laplace’s equation, Eq. , is given by $$\begin{aligned}
\Phi(x,y;\xi,\eta) =&\ \frac{1}{4\pi}\sum_{{{\color{black} n=}}-\infty}^{\infty} \ln\left[(x -
\xi + 2n\pi)^2 + (y - \eta)^2 \right] \nonumber \\
=&\ \frac{1}{4\pi}\ln[2(\cosh(y-\eta) - \cos(x-\xi))].
\label{eq:BIE_diffusion}\end{aligned}$$ Assuming that the channel walls are smooth, the concentration at a point $(x,y)$ on one of the walls can then be computed [using the]{} boundary integral formulation [@banerjee1981boundary] $$\begin{aligned}
\frac{1}{2}C(x,y) = &\int_{\partial\Omega}\left[C(\xi,\eta)\frac{\partial}{\partial
n}(\Phi(x,y;\xi,\eta)) \right.\nonumber\\
&-\left.\Phi(x,y;\xi,\eta)\frac{\partial}{\partial
n}(C(\xi,\eta))\right]\mathrm{d}s(\xi,\eta).\end{aligned}$$ The upper and lower boundaries of the domain are discretized into $200$ straight-line segments, and $C$ and $\mathrm{d}C/\mathrm{d}n$ are assumed constant over each element. For elements on the bottom (resp. top) boundary, $C = 0$ ([resp. $\mathrm{d}C/\mathrm{d}n=-1$)]{} is enforced at the midpoint of each segment. This reduces the boundary integral equation to a dense matrix system for the solution vector containing the unknown $\mathrm{d}C/\mathrm{d}n$ on the lower boundary and $C$ on the upper boundary. The free-space (i.e. singular) component of the Green’s function is isolated and integrated analytically, and all non-singular element integrals are computed with a $16$-point Gaussian quadrature. In order to compute the fluid flow and flow rate in the channel, only the boundary concentration of solute (and not its bulk distribution) is needed, which makes the boundary element method particularly suitable for this problem. [The numerical code was validated against analytical solutions for diffusion in a channel with nontrivial boundary conditions and domain geometry, achieving a relative error of at worst $0.004\%$.]{}
For Stokes flow, the [dimensionless]{} boundary integral equation for boundary force density, $\mathbf{f}$, is given by $$\begin{aligned}
u_j(\mathbf{x}) = \frac{1}{2\pi} &\int_{{{\color{black} \partial\Omega}}}
[S_{ij}(\mathbf{x}-\boldsymbol{\xi})f_i(\boldsymbol{\xi})
\nonumber \\
&- T_{ijk}(\mathbf{x}-\boldsymbol{\xi})u_i(\boldsymbol{\xi})
n_k(\boldsymbol{\xi})]\mathrm{d}s(\boldsymbol{\xi}).
\label{eq:BIE_stokes}\end{aligned}$$ For $\hat{\mathbf{x}} = \mathbf{x} - \boldsymbol{\xi}$ and $r =
|\hat{\mathbf{x}}|$, the two-dimensional, $2\pi$-periodic Green’s functions for Stokes flow are $$\mathbf{S} = \sum_{n = -\infty}^{\infty} \mathbf{I}\ln r_n -
\frac{\hat{\mathbf{x}}_n\hat{\mathbf{x}}_n}{r_n^2}, \quad
\mathbf{T} = \sum_{n = -\infty}^{\infty}
4\frac{\hat{\mathbf{x}}_n\hat{\mathbf{x}}_n\hat{\mathbf{x}}_n}{r_n^4},$$ where $\hat{\mathbf{x}}_n = \left(x - \xi + 2n\pi,y - \eta\right)$. These functions may be expressed in the closed form $$\begin{aligned}
S_{xx} &= K + \hat{y}\partial_{\hat{y}}K - 1, \\
S_{yy} &= K - \hat{y}\partial_{\hat{y}}K, \\
S_{xy} &= -\hat{y}\partial_{\hat{x}}K = S_{yx},\\
T_{xxx} &= 2\partial_{\hat{x}}(2K + \hat{y}\partial_{\hat{y}}K),\\
\end{aligned} \qquad
\begin{aligned}
T_{xxy} &= 2\partial_{\hat{y}}(\hat{y}\partial_{\hat{y}}K),\\
T_{xyy} &= -2\hat{y}\partial_{\hat{x}\hat{y}}K,\\
T_{yyy} &= 2(\partial_{\hat{y}}K - \hat{y}\partial_{\hat{y}\hat{y}}K),\\
T_{ijk} &= T_{kij} = T_{jki},
\end{aligned}$$ for $K = \frac{1}{2}\ln\left[2\cosh(\hat{y}) - 2\cos(\hat{x}) \right]$. The computational procedure for discretizing the domain boundary is identical to that used for the diffusion equation . [[ Constant force elements are assumed, singular integrals have the singularity removed and computed analytically, and non-singular integrals are computed with 16- point Gaussian quadrature.]{}]{}The implementation is based upon the authors’ previously published work [on the optimal swimming of a sheet]{} [@PhysRevE.89.060701], with the addition of Tikhonov regularisation to improve matrix conditioning.
Results {#sec:results}
=======
The role of asymmetry
---------------------
When inertia and solute advection are negligible, the Laplace problem for the solute concentration is linear and decouples from the Stokes flow problem, which is also linear. Breaking the left-right symmetry is thus required in order to create a net flow within the channel, in the same way that symmetry breaking is required [to achieve]{} self-propulsion of autophoretic particles. If the chemical properties of the walls are homogeneous, this asymmetry can only arise from geometry and therefore, purely symmetric profiles such as sinusoidal upper-wall shapes will [yield]{} zero net flux.
In order to analyze the effect of asymmetry, we [first]{} investigate the effect of the [[ asymmetry]{}]{} parameter, $\gamma$, on the flow rate. The numerical results are shown in Figure \[fig:Q\_vs\_shear\]. At $\gamma=0$, we recover zero-net flux, as expected. For asymmetric shapes, we obtain that the flow rate within the channel increases monotonically with $\gamma$.
![Dependence of the net flow rate through the channel, $Q$, with the left-right asymmetry of the top wall, $\gamma$, in the cases $a/H=1/2$, $L/H=2\pi$ and [$0 \leq \gamma \leq \pi$]{}. The four red dots correspond to the different panels illustrated in Figure \[fig:streamlines\].[]{data-label="fig:Q_vs_shear"}](flux_vs_shear.pdf)
![Solute concentration (colors) and streamlines (lines) within the channel with increasing [[ asymmetry]{}]{} in the case $a/H=1/2$ and $L/H=2\pi$. The four panels above correspond to the four red dots in Figure \[fig:Q\_vs\_shear\]. Red (resp. white) streamlines correspond to traversing (resp. recirculating) flow regions.[]{data-label="fig:streamlines"}](streamlines.pdf)
To [gain]{} insight into the origin of this flow rate, we illustrate in Figure \[fig:streamlines\] the dependence of the solute concentration distribution and streamlines with $\gamma$. For a strictly symmetric profile, $\gamma=0$, a flow is induced in the channel but one with no net flux. Indeed, a vertical solute concentration gradient is created between the two walls due to the fixed-flux emission of solute at the upper wall and the constant concentration imposed at the lower wall, where the solute is consumed. The upper wall is not horizontal, and regions of the upper wall located the furthest from the bottom wall are exposed to higher concentration than regions corresponding to the narrowest channel width. This [implies]{} the existence of tangential solute gradients along the wall and, hence, of a slip velocity that drives a flow within the channel. Due to the symmetry of the channel, the flow organizes into two counter-rotating flow cells leading to zero fluid transport across the channel. For positive activity and mobility of the upper wall, the flow is directed away from the active wall (i.e. downward) in the regions where the channel width is greatest, while it is directed [[ toward the active wall (i.e. upward)]{}]{} in the narrowest regions (Figure \[fig:streamlines\]a).
When $\gamma\neq 0$, asymmetry is introduced [in two ways]{}. The asymmetric upper wall can now be decomposed into a longer backward facing section and a shorter forward-facing section. The solute gradient on the former is weaker than on the latter, leading to a stronger left-to-right slip flow along the forward facing section. [Additionally,]{} wall asymmetry increases confinement in the trough along the upper wall leading to higher solute concentration (the rate of production of solute per unit surface is fixed). This asymmetry between the wall sections driving the flow within the channel generates a shape and intensity asymmetry between the two recirculation regions, and a traversing streamtube appears ([marked by dark red streamlines in]{} Figure \[fig:streamlines\]). [This streamtube]{} corresponds to flow regions that do not recirculate, but are transported along the channel, being “pumped” by the phoretic [mechanism]{}. This tube follows a pattern along the channel similar to that of a conveyor belt driven between the two recirculating regions forced by the slip flow on the wall. Along the shorter forward-facing section of the upper wall, it is driven by the stronger slip flow that dictates the direction of the net flow in this case. The tube then separates from the wall where the slip velocity changes sign, and circumnavigates around the counter-rotating flow cell driven by the longer wall section.
As $\gamma$ is increased beyond $\gamma\geq 1$, the slope of the shorter flow-driving wall changes sign, leading to a “folded” geometry that promotes large confinement effects on the solute concentration distribution (see the difference in color scales in Figure \[fig:streamlines\]). This, in turn, enhances the phoretic slip and net flow rate. For strong asymmetry, the traversing streamtube is mostly rectilinear and away from the active wall, except in a narrowing region where it circles around the smaller recirculation region and is driven by the phoretic slip within the trough on the boundary.
This process does not appear to saturate when $\gamma\gg 1$ for fixed amplitude $a$. In this limit, the flow domain can be decomposed into two regions: a [complex]{}, thin region corresponding to long and thin folds in the wall shape where very large concentration gradients are established by confinement, and an outer region where a net unidirectional flow is forced within the channel. Beyond obvious practical considerations regarding the manufacturing of such geometries, [[ the]{}]{} assumptions of the current model would potentially break down when the [[ asymmetry parameter $\gamma$]{}]{} becomes too large, as the phoretic flow become sufficiently intense for advection to be non-negligible (${\mbox{Pe}}\neq 0$). [[ Furthermore, when local concentrations become too large, it is likely that the model of fixed-flux release would be impacted, and more detailed reaction kinetics may be required.]{}]{}
Effect of the pattern amplitude on the flow rate
------------------------------------------------
The flow within the channel is effectively driven by the upper wall, while the no-slip condition on the lower inert wall tends to limit the fluid motion. As a consequence, it is expected that when the channel gap in the narrowest region becomes small ($a\approx H$), the net flow rate should be small, as the flow viscosity will offer maximum resistance there. However, the corrugation amplitude is an essential element to the pumping performance of the device as it determines the gradient along the upper wall between the peak and troughs, and therefore the intensity of the two recirculating regions driving the flow. When $a\ll H$, it is therefore also expected that the flow rate will become negligible. This [intuition]{} is confirmed by our numerical results in the case of weak [[ asymmetry]{}]{} ($\gamma\leq 1$, unfolded geometry, see Figure \[fig:amplitude\]a) for which the net flow rate within the channel displays a maximum at intermediate amplitude and decreases to zero in both limits $a\ll H$ and $a\approx H$. In this case, the limit $a\ll H$ corresponds to a flat upper wall.
![The net flow rate through the channel, $Q$, as a function of its relative amplitude $a/H$ in the case $L/H=2\pi$. [[ (a) Flux for $\gamma=\pi/4$ (as Figure \[fig:streamlines\]b), showing behaviour representative of weak asymmetry (unfolded) channels. (b) Flux for $\gamma=\pi/2$ (as Figure \[fig:streamlines\]d), showing distinct behaviour for strong asymmetry (folded) channels.]{}]{}[]{data-label="fig:amplitude"}](flux_vs_amplitude.pdf)
The behavior of the system is however quite different when the upper wall is folded ($\gamma\geq 1$, [[ strong asymmetry]{}]{}, see Figure \[fig:amplitude\]b). In this case, the limit $a\ll H$ [is not limiting]{} to a flat wall, but to a surface with infinitely thin and almost horizontal folds. [Within these folds,]{} confinement creates very large solute concentrations and concentration gradients. As [noted]{} in the previous section, this limit is [the singular case]{} of a flat wall forced periodically by infinitely large slip velocities in infinitely thin regions. As a consequence, the net flow rate does not decrease to zero for small amplitude, marking a stark difference between the folded and unfolded geometries.
![[Net flow rate through the channel, $Q$, as a function of the channel height $H/L$ for [[ unfolded $\gamma=\pi/4$ and folded $\gamma=\pi/2$ channels with]{}]{} fixed relative amplitude, $a/L = 0.08$.]{}[]{data-label="fig:height"}](flux_vs_height.pdf)
Role of the channel width
-------------------------
We finally turn to the influence of the third geometric characteristic of the channel, namely its mean width-to-length ratio. The limit of small minimum width ($a\approx H$) was already discussed in the previous section and the flow rate within the channel vanishes in that limit due to the diverging hydrodynamic resistance of the channel. When $H$ is large compared to both $a$ and $L$, the relative concentration distribution along the top wall is not influence by the location of the passive wall so the tangential concentration gradients and slip velocity become independent of $H$. As a result the net flow rate through the channel varies linearly with the channel height as for a classical Couette (shear) flow. This is confirmed by our numerical results shown in Figure \[fig:height\].
Long-wavelength prediction {#sec:lubrication}
==========================
When the local height of the channel is small in comparison to the typical longitudinal length of the topography, i.e. $|h'(x)|\ll 1$, the problem can be solved within the [framework of lubrication (long-wavelength) theory]{}. Defining $h(x)=\varepsilon f(x)$ with $f(x)=O(1)$ and $\varepsilon{{\color{black} =H/L}} \ll 1$, [the method consists in]{} solving for the concentration and velocity fields as regular series expansions in $\varepsilon$. On the upper wall, the normal unit vector pointing into the fluid is now written $$\begin{aligned}
{\mathbf{n}}&=\frac{-{\mathbf{e}}_y+h'{\mathbf{e}}_x}{\sqrt{1+h^{'2}}}
\nonumber \\
&=-{\mathbf{e}}_y+\varepsilon f'{\mathbf{e}}_x+\frac{\varepsilon^2f^{'2}}{2}{\mathbf{e}}_y-\frac{\varepsilon^3f^{'3}}{2}{\mathbf{e}}_x+O(\varepsilon^4).\end{aligned}$$ Defining a rescaled vertical coordinate $y=\varepsilon Y$, the Laplace and Stokes flow problems are now given by
$$\begin{aligned}
\frac{1}{\varepsilon^2}{\frac{\partial ^2C}{\partial Y^2}}+{\frac{\partial ^2C}{\partial x^2}}&=0,\label{eq:lub1}\\
\frac{1}{\varepsilon^2}{\frac{\partial ^2u}{\partial Y^2}}+{\frac{\partial ^2u}{\partial x^2}}&={\frac{\partial p}{\partial x}},\label{eq:lub2}\\
\frac{1}{\varepsilon^2}{\frac{\partial ^2v}{\partial Y^2}}+{\frac{\partial ^2v}{\partial x^2}}&=\frac{1}{\varepsilon}{\frac{\partial p}{\partial Y}},\label{eq:lub3}\\
\frac{1}{\varepsilon}{\frac{\partial v}{\partial Y}}+{\frac{\partial u}{\partial x}}&=0,\label{eq:lub4}\end{aligned}$$
and the boundary conditions at $y=\varepsilon f(x)$ become
$$\begin{aligned}
-1&=-\frac{1}{\varepsilon}{\frac{\partial C}{\partial Y}}+\varepsilon\left(f'{\frac{\partial C}{\partial x}}+\frac{f^{'2}}{2}{\frac{\partial C}{\partial Y}}\right)+O(\varepsilon^3 C),\label{eq:lubbc1}\\
u&=\left({\frac{\partial C}{\partial x}}+f'{\frac{\partial C}{\partial Y}}\right)\left(1-\varepsilon^2f^{'2}\right)+O(\varepsilon^4 C),\label{eq:lubbc2}\\
v&=\varepsilon f'\left({\frac{\partial C}{\partial x}}+f'{\frac{\partial C}{\partial Y}}\right)+O(\varepsilon^3 C).\label{eq:lubbc3}\end{aligned}$$
These, together with the conditions ${{\color{black} C}}=0$ and $u=v=0$ at ${{\color{black} Y}}=0$, suggest searching for solutions of the form
$$\begin{aligned}
C(x,{{\color{black} Y}})&=\varepsilon C_1(x,{{\color{black} Y}})+\varepsilon^3 C_3(x,{{\color{black} Y}})+O(\varepsilon^5),\label{eq:expc}\\
u(x,{{\color{black} Y}})&=\varepsilon u_1(x,{{\color{black} Y}})+\varepsilon^3 u_3(x,{{\color{black} Y}}) +O(\varepsilon^5),\label{eq:expu}\\
v(x,{{\color{black} Y}}) & =\varepsilon^2 v_2(x,{{\color{black} Y}})+O(\varepsilon^4),\label{eq:expv}\\
p(x,{{\color{black} Y}}) & = \varepsilon^{-1}p_{-1}(x,{{\color{black} Y}})+\varepsilon p_1(x,{{\color{black} Y}})+O(\varepsilon^3).\label{eq:expp}\end{aligned}$$
The flow rate $Q$ is then [given by]{} $$Q=\varepsilon \int_0^{f(x)}u(x,Y){\mathrm{d}}Y=\varepsilon^2 Q_2+\varepsilon^4 Q_4+O(\varepsilon^6),$$ with $$Q_j=\int_0^fu_{j-1}(x,Y){\mathrm{d}}Y.$$ The flow is incompressible and steady, therefore $Q$ and $Q_j$ do not depend on $x$.
Inserting Eq. into Eqs. and gives at leading order $$C_1(x,Y)=Y.\label{eq:c0}$$ Eqs. , and then provide at $O(\varepsilon)$: $$u_1(x,Y)=\frac{p_{-1}'}{2}(Y^2-Yf)+\frac{Yf'}{f},$$ with $p_{-1}$ the leading-order pressure distribution which is vertically invariant. The function $p_{-1}(x)$ is periodic, therefore $$Q_2=0 \quad \textrm{and}\quad p_{-1}(x)=-6/f.$$ We see that in the lubrication limit, a velocity field is present at $O(\varepsilon)$, which takes the form of two recirculating regions, but does not give rise to any net flow through the channel at this order. After substitution and [application of]{} the continuity equation, we obtain
$$\begin{aligned}
u_1(x,Y)&=3\left(\frac{f'}{f^2}\right)Y^2-2\left(\frac{f'}{f}\right)Y,\label{eq:u1}\\
v_2(x,Y)&=\left(\frac{f'}{f}\right)'Y^2-\left(\frac{f'}{f^2}\right)'Y^3.\label{eq:v2}\end{aligned}$$
At next order, the Laplace problem, Eqs. and , together with Eq. , provide $$C_3(x,Y)=\frac{Y f^{'2}}{2}\cdot$$ The horizontal Stokes flow problem now yields, $$\begin{aligned}
u_3(x,Y)&=2\left[\left(\frac{f'}{f}\right)''\left(\frac{Y^3-Yf^2}{3}\right)-\left(\frac{f'}{f^2}\right)''\left(\frac{Y^4-Yf^3}{4}\right)\right]\nonumber\\
&+\frac{\tilde{p}_1'}{2}(Y^2-{{\color{black} Y}}f)+\left(ff'f''-\frac{f^{'3}}{2}\right)\frac{Y}{f},\end{aligned}$$ with $\tilde{p}_1(x)$ a function of $x$ only. [[ Integrating the previous equation in $Y$ finally provides the flow rate at $O(\varepsilon^4)$]{}]{} $$\begin{aligned}
Q_4&=2\left[ \frac{3f^5}{40}\left( \frac{f'}{f^2}\right)'' - \frac{f^4}{12}\left( \frac{f'}{f}\right)''\right] -\frac{\tilde p'_1}{12}f^3\nonumber\\
&+ \left(\frac{f^2f'f''}{2}-\frac{f f'^3}{4} \right).\end{aligned}$$ Using the periodicity of $\tilde{p}_1$, $Q_4$ can be computed by dividing the previous equation by $f^3$, taking the spatial average in $x$, and integration by parts. The result can be rewritten in terms of the original channel height $h(x)$. [At leading order]{} we obtain that the flow [through]{} the channel [is given by]{} $$\label{eq:flowratelubric}
Q=\frac{11}{30}\frac{{\left\langle h^{'3}/h^2\right\rangle}}{{\left\langle 1/h^3\right\rangle}},$$ [where ${\left\langle \cdot\right\rangle}$ is the spatial average over a period.]{}
We see two important results: (i) the flow rate is intimately linked to the distribution of local slope along the wall $h'(x)$, and (ii) shape asymmetry is essential for the existence of a net flow. Indeed, slip flow along the active wall arises from the orientation of the wall with a component along the leading order solute concentration gradient. A non zero $h'(x)$ is therefore sufficient to guarantee the existence of a local flow but not necessarily of a net flow through the channel. This separation of scales is clearly visible in the lubrication expansion: the leading order flow arises from the local channel geometry (i.e. the fact that the wall is not flat and orthogonal to the leading order solute gradient). However, at this order, the net flow is zero because the flows driven by the forward- and backward-facing walls exactly cancel out. [A net flux]{} results from an imbalance between these local flows which can only be induced by geometric asymmetry. Left-right symmetric profiles are characterized by an even channel height, $h(x)=h(-x)$. Consequently the function $ h^{'3}/h^2$ is odd and thus exactly averages to zero, so that $Q_4$ is identically zero (all higher orders are expected to be zero as well).
The result in Eq. is a weighted algebraic spatial average of the slope of the active wall. More precisely, the leading order flow rate [through]{} the channel, Eq. , is the ratio of two integrals; the numerator is the mean flow forcing due to the asymmetry of the channel, while the denominator is the average hydrodynamic resistance of the channel over a wavelength.
![Ratio between the numerical results and the lubrication (long-wavelength) theory predictions for $\gamma=\pi/10$ and $a=H/3$ as a function of an increasing slenderness, $1/\varepsilon$.[]{data-label="fig:lubrication"}](lubrication_comparison.pdf)
This leading-order prediction is compared to the full numerical simulations in Figure \[fig:lubrication\]. For a fixed [[ asymmetry parameter]{}]{} $\gamma$ and relative amplitude $a/H$, several simulations are performed for increasing $L/H$ (note that as $H$ is reduced, $a$ is reduced in the same amount) and the flow rate through the channel is shown to converge for large slenderness to the prediction of the lubrication theory.
Note that the lubrication result, Eq. , is valid for any ratio [$H/L\ll1$]{}, regardless of the relative magnitude of the mean channel height $H$ and the perturbation amplitude $a$. In the limit of small wall roughness ($a\ll H$), the hydrodynamic resistance (the denominator in Eq. \[eq:flowratelubric\]) is independent of $a$ at leading order and simply scales as $1/H^3$, while the phoretic forcing (the numerator in Eq. \[eq:flowratelubric\]) scales as $a^3/H^2$. As a consequence, $Q$ scales as $a^3 H$ when $a\ll H$.
In the opposite limit $a\sim H$, using classical asymptotic expansions to compute the leading order contribution to the integrals in Eq. [@bender1978], one can show that the flow forcing due to the channel’s asymmetry (i.e. the numerator in Eq. \[eq:flowratelubric\]) remains finite and $O(H)$, but that in contrast the hydrodynamic resistance diverges. More specifically, [[ a standard lubrication calculation]{}]{} leads at leading order to $${\left\langle \frac{1}{h^3}\right\rangle}=\frac{3\sqrt{2}}{16H^{1/2}(H-a)^{5/2}}\cdot$$ As a consequence, $Q\sim H^{3/2}(H-a)^{5/2}$ when $(H-a)\ll H$.
Note that since the limiting factor is [then the channel]{} width at the narrowest point, and its impact on the hydrodynamic resistance, it is expected that this scaling in $(H-a)^{5/2}$ should hold true even when $H$ is not small and could be recovered through a new lubrication expansion limited to the narrow-gap region of the channel [@leal2007], provided the curvature of the wall in that region remains finite.
Conclusions {#sec:conclusions}
===========
The active research in recent years on autophoretic particles has demonstrated that fuel-based mechanisms represent a promising route to designing self-propelled systems that rely only on chemical reactions and the interaction with the immediate environment to create locomotion. The results presented in this work show that this is [also true for]{} the dual problem of pumping flow within a micro-channel, and that geometric asymmetry, rather than chemical patterning of the channel walls, is sufficient to create a net flow. Our results provide insight into the flow dynamics within the channel, and the mechanism leading to the net fluid transport: the breaking of symmetry between two recirculating flow regions driven by wall slip velocity, and the emergence of a conveyor-belt-like flow within the channel.
For simplicity, we focused in this paper on a reduced set of wall shapes with one active wall and the other one passive. [The numerical methodology and the long-wavelength theory, Eq. , are however valid for any periodic channel geometry in two dimensions.]{} Furthermore, these results were obtained within the simplified framework of a fixed-rate release of solute by the active wall. Previous studies on autophoretic self-propelled particles have shown that the exact reaction kinetics, in particular the dependence of the reaction rate on the local solute concentration, may significantly impact the system dynamics [@ebbens2012; @michelin2014]. We expect for example the direction of pumping to be impacted by reaction kinetics, although the basic result showing the emergence of a net flow due to geometric asymmetry of the phoretic wall should remain true. [[ Finally, our study focused on the particular case of self-diffusiophoresis. Because of the formal similarities in the equations of the problem, these results can be generalized easily to other phoretic mechanisms such as self-thermophoresis or self-electrophoresis.]{}]{}
![Concentration distribution and flow streamlines within an annular closed-loop channel with a geometrically-asymmetric inner active wall releasing solute with a fixed flux, and a passive circular outer wall with uniform concentration. The recirculating streamlines are shown in white while the traversing streamlines are plotted in red, and correspond to a clockwise-rotating flow.[]{data-label="fig:ninja"}](shuriken_comp.pdf)
Looking forward, the results of this work could be generalized to a larger range of geometries, including closed-loop channels for which pressure-driven flows can not easily be achieved. This is shown in Figure \[fig:ninja\] where we have adapted our numerical approach to compute the net clockwise flow through a two-dimensional annular channel driven by the geometric asymmetry of the inner active wall.
Acknowledgements {#acknowledgements .unnumbered}
================
SM acknowledges the support of the French Ministry of Defense (DGA). TDMJ is supported by a Royal Commission for the Exhibition of 1851 Research Fellowship. GDC acknowledges support from Associazione Residenti Torrescalla. This work was funded in part by a European Union Marie Curie CIG Grant to EL.
|
---
abstract: 'Mid-late M stars are opportunistic targets for the study of low-mass exoplanets in transit because of the high planet-to-star radius ratios of their planets. Recent studies of such stars have shown that, like their early-M counterparts, they often host multi-resonant networks of small planets. Here, we reanalyze radial velocity measurements of YZ Ceti, an active M4 dwarf for which the HARPS exoplanet survey recently discovered three exoplanets on short-period (P = 4.66, 3.06, 1.97 days) orbits. Our analysis finds that the orbital periods of the inner two planets cannot be uniquely determined using the published HARPS velocities. In particular, it appears likely that the 3.06-day period of YZ Ceti c is an alias, and that its true period is 0.75 days. If so, the revised minimum mass of this planet is less than 0.6 Earth masses, and its geometric transit probability increases to 10%. We encourage additional observations to determine the true periods of YZ Ceti b and c, and suggest a search for transits at the 0.75-day period in TESS lightcurves.'
author:
- Paul Robertson
bibliography:
- 'yzceti\_alias.bib'
title: 'Aliasing in the Radial Velocities of YZ Ceti: An Ultra-Short Period for YZ Ceti c?'
---
Introduction
============
Mid-late M stars are increasingly common targets of exoplanet surveys. [@borucki10] included relatively few such stars in its target list, but its extended mission K2 has revealed several systems of small planets orbiting very low-mass stars [e.g. @mann16; @hirano16]. TESS [@ricker15] has started science operations, and will add many more systems to the catalog of exoplanets around late-type stars. At the same time, a collection of near-infrared Doppler spectrographs is going into operation, beginning with CARMENES [@quirrenbach16] and HPF [@mahadevan14], which will enable ground-based follow-up to determine the masses of planets transiting these cool, faint stars.
Already, several of the most high-profile recent exoplanet discoveries have been around mid-late M stars. TRAPPIST-1, a nearby M8 dwarf, was shown to host a multi-resonant network of seven low-mass exoplanets [@gillon17], three of which lie within the liquid-water habitable zone [HZ; @kopparapu13]. Radial velocity (RV) surveys have also discovered low-mass exoplanets around nearby mid-late M stars. @anglada16 found evidence for a terrestrial-mass planet in the HZ of Proxima Centauri. More recently, @astudillo-defru17 announced the discovery of three Earth-mass exoplanets orbiting the M4.5 dwarf YZ Ceti based on observations from the HARPS spectrograph. At candidate periods of 1.97, 3.06, and 4.66 days, the YZ Ceti system potentially represents another compact multi-harmonic system like TRAPPIST-1. TESS will observe YZ Ceti in late 2018, and all of the reported planets have relatively high ($P \sim 5$%) geometric transit probabilities.
Ground-based exoplanet surveys are plagued by difficulties associated with temporal sampling. Uneven time sampling caused by shared telescope resources, seasonal target observability, weather, and the day/night cycle limit sensitivity in certain regions of frequency space, and can create ambiguities in others. Aliasing occurs when a continuous signal is observed at a cadence such that the observations cannot distinguish between the true signal frequency and a combination of the signal and observing frequencies. RV surveys are commonly hampered by the “1-day alias", and periodograms of RV data will often show peaks at the frequency of a planet ($f_p$) and its alias at $f_a = f_p \pm 1$ day$^{-1}$. This effect was demonstrated most powerfully by @dawson10, who revised the period of 55 Cnc e from 2.8 days [@mcarthur04] to its true value of 0.75 days, where it was later found to transit [@winn11].
In this Letter, we argue that the periods of two of the three planets orbiting YZ Ceti are not well determined due to aliasing in the HARPS RV time series. For planet b, which has a period of either 1.97 or 2.02 days, the difference is primarily important for the efficiency of identifying potential transits. On the other hand, the period of planet c may be 0.75 days rather than 3.06 days, which significantly alters its derived physical properties and geometric transit probability. Given that the available RVs are unable to clearly distinguish between these candidate periods, it will be especially important to examine all potential transit windows in TESS lightcurves of YZ Ceti.
Data and Analysis
=================
Data
----
In this Letter, we have analyzed the HARPS observations of YZ Ceti as presented in Table B.4 of @astudillo-defru17. The quantities derived from time-series spectroscopic observations of YZ Ceti include RVs, as well as spectral properties sensitive to stellar magnetic activity. The activity tracers include the full width at half maximum (FWHM) of the cross-correlation function, the line bisector slopes (BIS), and strengths of the calcium H&K () and absorption lines.
Stellar Activity
----------------
@astudillo-defru17 analyzed photometry of YZ Ceti from the All-Sky Automated Survey [ASAS; @pojmanski97] and the HARPS FWHM values, finding evidence of the stellar rotation period at $P_{rot} \sim 83$ days. They find no evidence for this period in the RV data or the absorption-line activity indicators, an assessment with which we agree. However, we note that the time series (and , at lower S/N) does include several interesting periodicities, including at periods near 500 and 53 days, and possibly also a long-period trend. These periods are difficult to interpret in light of the candidate rotation period at 83 days. The 500-day period is shorter than typical stellar magnetic cycles, but could be a “sub-cycle" of the longer-term magnetic evolution indicated by the trend. Similar behavior has been observed for the Sun [e.g. @wauters16]. It is possible that the rotation period is either 53 or 83 days, and the other period is a typical active region lifetime.
None of the periods identified in the stellar activity indicators appear at significant power in RV. Depending on the model adopted for the planets, we sometimes observe residual power near the $\sim 25$-day harmonic of the 53-day period, but at levels far too low to be statistically significant.
Thus, it appears that the timescales for the primary stellar activity signals, as well as their dominant harmonics, lie far from the periods of the candidate exoplanets. @astudillo-defru17 included a Gaussian process (GP) correlated noise component [@rasmussen05] in their 3-planet model, but we see no evidence for correlated noise from astrophysical variability near the planets under consideration. We find that the derived properties of the planets do not change significantly when including a GP noise model, and that we cannot meaningfully constrain the hyperparameters of the quasi-periodic GP kernel. Thus, for our analysis, we have modeled the HARPS time series as a sum of three Keplerian functions.
RV Period Search
----------------
We sought to identify periodicity in the RV time series using three periodograms, each with different advantages. We first used the traditional Lomb-Scargle periodogram (GLS), as fully generalized by @zk09. We also considered the Bayes factor periodogram (BFP), provided in the `Agatha` software suite by @feng17. The BFP computes the power spectrum by comparing the Bayesian information criterion (BIC) for a periodic signal to that of the noise model at each period, and includes options for correlated noise models and correlations with activity proxies. Finally, we computed the compressed sensing periodogram (CSP) as described by @hara17. Whereas the GLS and BFP evaluate one period at a time, and identify multiple signals iteratively by removing the strongest signal and recomputing, the CSP models the entire frequency parameter space simultaneously by fitting amplitudes to a large library of periodic signals (here, sinusoids). By evaluating all periods simultaneously, the CSP excels at minimizing the impact of aliasing.
We note that the model selection routine in `Agatha` prefers a white noise model with no correlations to activity proxies for the RV series. Thus, the GLS and BFP power spectra are largely similar. However, the BFP offers the advantage of a more robust threshold for statistical significance. Namely, as discussed in @feng17, peaks with power $\ln(BF) > 5$ are generally considered significant.
![Periodograms of the HARPS RVs of YZ Ceti. For the GLS and BFP periodograms, we show the power spectra after successively modeling and subtracting planets d (*blue*) and b (*red*). The pink arrows indicate the periods of the three planets as published by @astudillo-defru17. All three periodograms prefer the 0.75-day period for planet c over its 1-day alias at 3.06 days.[]{data-label="fig:periodograms"}](yzceti_pscomp.pdf){width="\columnwidth"}
All three periodograms identify the 4.66-day signal of YZ Ceti d as the strongest periodicity in the RV time series. However, subsequent analysis of the power spectra reveals ambiguities for the periods of each of the other two proposed planets in the system. In Figure \[fig:periodograms\], we show our periodograms. For the GLS and BFP periodograms, we have successively modeled and subtracted the orbits of planet d and b[^1] in order to study the residual periodicities. The three periodograms again agree on the second-strongest signal, this time at a period of 0.75 days. This period is a 1-day alias of the 3.06-day period attributed to planet c by @astudillo-defru17. The GLS and BFP power spectra also show significant power at the 3.06-day period, while the CSP converges on a single model that prefers the 0.75-day period.
Furthermore, the GLS and BFP periodograms indicate two possible periods for planet b, one at the published value of 1.97 days, and one at a slightly longer period of 2.02 days. Again, the 2.02-day period is a 1-day alias of the 1.97-day period. While the physical properties of planet b are minimally dependent on such a small difference in period, it will be important to identify the correct period for future transit searches.
The CSP does not recover the 2-day planet at any significant amplitude. In general, we find that the detections of planets b and c are marginal, and strongly dependent on the RVs from the first season of high-cadence observations in 2013. The stellar activity indicators suggest YZ Ceti is relatively quiet during this season, and we do not observe periodicity near the planet periods in the activity tracers when isolating the 2013 observations. Thus, there is no particular reason to exclude or disfavor these data. However, it will be important to confirm these planets with additional observations.
Attempts to Break the Period Degeneracies
-----------------------------------------
### Simulated Signals
We attempted to break the degeneracies between the periods of planets b and c, using two techniques. First, we considered the method suggested by @dawson10–which relies on the periodogram peaks and phases of the signals in comparison to simulated time series–to examine the period of planet c. Our application of this technique involves first computing a pair of 3-planet Keplerian fits, one for each candidate period for planet c. Then, using the parameters derived for planet c, we generated a simulated time series for a single planet at the candidate period, using the time stamps and error bars of the original HARPS RVs. At each time $t_i$ in the simulated series, we added a random perturbation drawn from a Gaussian distribution with $\sigma = \sqrt{\sigma_{RV,i}^2 + (1.8~\textrm{m s}^{-1})^2}$, since our 3-planet models yielded a typical “jitter" value of 1.8 . We computed the periodogram of the simulated series, and compared to the GLS of the original RVs after removing planets b and d. Finally, we fit a Keplerian to the simulated data using the “wrong" period (i.e. we fit a 3.06-day Keplerian to the 0.75-day simulated signal) and compared the phase to the modeled phase of the real data. Here, we define “phase" as the mean anomaly ($M_0$) at the first HARPS epoch.
![The GLS periodogram of the YZ Ceti RVs (*black/gray*) after subtracting planets b and d, compared to simulated time series for planets at 3.06 days (*red, middle row*) and 0.75 days (*blue, bottom row*). The periodogram of the original data is shown in all three rows for visual comparison. The dials above each peak show the phase ($M_0$) derived by modeling a Keplerian at that period. We do not show dials at peaks for which we fixed the planet’s phase.[]{data-label="fig:dawson"}](yzceti_cperiod_b2p02.pdf){width="\columnwidth"}
We performed this exercise for models using periods of 1.97 and 2.02 days for planet b. In Figure \[fig:dawson\], we show the results for the test using $P_b = 2.02$ days, although the choice of $P_b$ makes little difference. We show the version using $P_b = 2.02$ primarily because it exhibits the greatest discrepancy between the two hypotheses. For the simulated signals, we find that each candidate period creates significant periodogram power at the 1-day alias, and that the phases of models to the “wrong" period are similar to those derived from the original RVs. The one significant difference we observe between the periodograms of the simulated data and the original is that the periodogram power at the true period of the simulated 3-day planet is significantly stronger than that of the original data. We note, however, that this could be caused by random fluctuation due to the jitter added to the simulated data. Alternatively, it could indicate incompatibility of the 3.06-day period with the 2.02-day period. In general, we find that this test does not unambiguously distinguish between the candidate periods, since a planet at either period can create power at both periodogram peaks, and result in Keplerian models that match the phases derived from fitting to incorrect periods. Furthermore, we elected not to repeat this experiment to distinguish between the candidate periods of planet b because of the greater ambiguity of planet c’s period.
### Model Comparison
Another way to determine which periods are preferred by the current data is to check goodness-of-fit statistics for models to the RVs. We fit Keplerian models to each of the candidate period combinations using the Markov Chain Monte Carlo (MCMC) RV modeling package `RadVel` [@fulton18]. For each model, we set the Keplerian orbital elements of each planet, zero-point offsets and additional white-noise terms (“jitters") for the HARPS velocities before and after the fiber upgrade, as free parameters. The MCMC chains use 150 random walkers, and up to 100000 iterations, although the calculation stops if the chains are found to have converged as determined by yielding a value of the Gelman-Rubin statistic less than 1.003 [@ford06]. Where possible, we have retained the model priors used by @astudillo-defru17. When evaluating models with $P_c = 0.75$ days, we shifted the period prior to uniform between 0.5 and 1 day.
Prior to computing the MCMC models, we excluded two RVs (BJD = 2457258.798094, 2457606.952332), each of which is more than $6\sigma$ from the data mean. We removed these values primarily to prevent a biased estimate of the velocity offset between the pre- and post-upgrade RVs.
We evaluated models for solutions of 0, 1, 2, and 3 planets to compare the relative importance of changing the planet periods to that of adding or removing planets. However, we emphasize that our model comparison is a simple exercise intended primarily to test whether we could distinguish between the aliases for the periods of planets b and c. A more thorough exploration of the parameter space, including, for example, the impact of various noise models or additional planets, is beyond the scope of this work.
For each model configuration, we compared the BIC and the log of the likelihood ($\ln {\mathcal{L}}$). The results of this comparison are summarized in Table \[tab:comparison\]. We have assigned values for the false alarm probability (FAP) by comparing the $\ln {\mathcal{L}}$ values between models with different numbers of planets by using the “unbiased" likelihood improvement $\tilde{Z}$ from @baluev09 [Equation 18]. The FAP then scales approximately as FAP $\propto e^{-\tilde{Z}} \sqrt{\tilde{Z}}$ [e.g. @baluev08]. The FAPs listed in Table \[tab:comparison\] are relative to the highest-likelihood model with 1 fewer planet than the model under consideration.
Unfortunately, we cannot identify any single model that is clearly preferable to the others. The FAP values in Table \[tab:comparison\] show a clear preference for the 2-planet model with $P = 4.66, 0.75$ over the single-planet model, but the case for adding a third planet is marginal compared to the improvement yielded by changing the period of planet c to 18 hours.
Because we used uniform priors in our MCMC models, we may evaluate the relative probabilities of models with the same number of free parameters as $\frac{P_1}{P_2} = e^{\ln {\mathcal{L}}_1 - \ln {\mathcal{L}}_2}$. Thus, for the 3-planet models, we find that the specific configuration proposed by @astudillo-defru17 is approximately 3200 times less likely than our highest-likelihood model, which uses $P_c = 0.75$d. However, the rest of the models are more similar, with no model above the $\frac{P_1}{P_2} = 150$ threshold typically used as a minimum for unambiguously preferring one model over another [e.g. @feroz11]. Thus, rather than choosing a single “best" model, we summarize some qualitative results revealed by this analysis:
- Models with $P_c = 0.75$d are consistently preferred over those with the original 3.06-day period. The 2-planet solution with $P = 4.66, 0.75$d is clearly the best such model, and both of the highest-likelihood 3-planet models have $P_c = 0.75$d.
- In light of the 2-planet solution with $P = 4.66, 0.75$d, the addition of a third planet with a period near 2 days is only marginally supported by our analysis. The FAP for the best 3-planet solution relative to the best 2-planet model is $0.2\%$, suggesting planet b is probably real, but requires additional observations to be confirmed. Interestingly, the best 2-planet solution was identified in the CSP, suggesting that–as argued by @hara17–the compressed sensing technique is especially useful for avoiding ambiguities caused by aliasing.
- Distinguishing between periods of 1.97 and 2.02 days for planet b is particularly difficult. The 1.97-day period is especially disfavored when adopting the 3.06-day period for planet c or excluding it altogether. On the other hand, the best model in Table \[tab:comparison\] uses the 1.97-day period.
----------- ------------------ --------- --------------------- ----------------------
Number of Planet Periods BIC $\ln {\mathcal{L}}$ FAP
Planets (days)
0 N/A 1085.00 -531.80 N/A
1 4.66 1069.90 -510.87 $3.7 \times 10^{-6}$
2 4.66, 1.97 1082.13 -503.60 $>1$
2 4.66, 2.02 1077.21 -501.14 $0.15$
2 4.66, 3.06 1074.17 -499.62 $0.04$
2 4.66, 0.75 1056.91 -490.99 $1.6 \times 10^{-5}$
3 4.66, 1.97, 3.06 1069.94 -484.13 $>1$
3 4.66, 2.02, 3.06 1058.94 -478.63 $0.02$
3 4.66, 2.02, 0.75 1057.59 -477.95 $0.01$
3 4.66, 1.97, 0.75 1053.83 -476.07 $0.002$
----------- ------------------ --------- --------------------- ----------------------
: Comparison of goodness-of-fit statistics for our MCMC models to the RV data. False alarm probability (FAP) is given relative to the highest-likelihood model with 1 fewer planet than the model considered.[]{data-label="tab:comparison"}
Discussion
==========
In Table \[tab:parameters\], we list the modeled and derived parameters for our best fit to the system with planet periods of 4.66, 1.97, and 0.75 days. The values in Table \[tab:parameters\] assume a stellar mass $M_* = 0.13 \pm 0.01 M_\odot$, derived from the @delfosse00 $K$-band mass-luminosity relationship [$K = 6.42 \pm 0.02$; @cutri03] and the *Gaia* DR2 parallax $\pi = 269.36 \pm 0.08$ mas [@gaia_dr2]. We present this solution not as a replacement for the model presented in @astudillo-defru17, but rather to serve as a comparison elucidating the consequence of adopting the shorter value of $P_c$.
If the true period of planet c is in fact 0.75 days, it becomes somewhat unique among the known exoplanets. The minimum masses of the YZ Ceti planets are already the smallest ever discovered with RV, but the revised minimum mass $m_c \sin i = 0.58 M_\oplus$ would establish it as firmly sub-terrestrial in mass. It would be beneficial to acquire additional RV observations of YZ Ceti during TESS observations in order to better evaluate potential activity contributions to the RVs, and more precisely determine the orbital properties of the low-mass planets in this system.
We used the probabilistic mass-radius prediction routine `Forecaster` [@chen17] to estimate the expected radius of planet c under the assumptions that $P_c = 0.75$ days, and that we are viewing the YZ Ceti system edge-on. `Forecaster` predicts a radius $R_c = 0.86 \pm 0.1 R_\oplus$. The $K$-band radius-luminosity relationship of @mann15 yields a radius $R_* = 0.169 \pm 0.001 R_\odot$ for YZ Ceti, which results in a geometric transit probability of 10%, more than double the probability derived from the 3.06-day period. The small stellar radius also yields a relatively high expected transit depth of 0.22%, which could even potentially be observed from the ground [@stefansson17]. Thus, if the true period of YZ Ceti c is 0.75 days, it offers the potential opportunity for high signal-to-noise study of a transiting sub-terrestrial exoplanet orbiting a relatively bright nearby star. TESS is currently scheduled to observe YZ Ceti in Sector 3 (September-October 2018) of its survey of the southern hemisphere. TESS should easily recover the transit signatures of all three planets if they are inclined so as to transit.
If $P_c = 18$ hours, the small orbital separation (high temperature) and small mass (low escape velocity) of the planet could result in significant levels of mass loss. The planet’s atmosphere [e.g. GJ 436b, @ehrenreich15] or surface [e.g. KIC 1255b, @rappaport12] may be escaping, creating an extended tail of material extending from its surface and causing variable transit depths and durations. The expected surface temperature of YZ Ceti c at the 18-hour period [$\sim 1000$ K, according to eq. 5 of @rappaport12] is too low to vaporize silicates, but tidal forces could result in enhanced volcanic activity that would launch dust from the surface. Thus, if the planet (or just its exosphere/tail) is transiting, it may provide a unique opportunity to study its atmospheric and interior composition via transit spectroscopy with JWST.
Conclusion
==========
Our analysis suggests the available HARPS RVs of YZ Ceti are incapable of distinguishing unambiguously between 1-day aliases for the periods of planets b and c. Our periodograms and model comparisons show a slight preference for revising the period of planet c to 0.75 days, but determining an exact period for planet b is more difficult. If the period of planet c is in fact 0.75 days, its minimum mass drops to just above half the Earth’s mass, and its transit probability increases to 10%.
This work has made use of observations collected at the European Southern Observatory under ESO program IDs 180.C-0886(A), 183.C-0437(A), and 191.C-0873(A). The author is grateful to the anonymous referee for an expeditious and helpful review. The author also thanks Michael Endl, Gudmundur Stefansson, and Jason Wright for valuable input on this analysis.
Parameter Planet b Planet c Planet d
--------------------------------------------------------- ------------------------------- -------------------------------- -------------------------------
Period $P$ (d) $1.9689 \pm 4 \times 10^{-4}$ $0.75215 \pm 1 \times 10^{-5}$ $4.6568 \pm 4 \times 10^{-4}$
Time of inferior conjunction $T_C$ (BJD - 2450000) $7662.1 \pm 0.2$ $7661.56 \pm 0.05$ $7657.9 \pm 0.2$
$\sqrt{e} \cos \omega$ $-0.2 \pm 0.4$ $-0.2 \pm 0.3$ $-0.1 \pm 0.3$
$\sqrt{e} \sin \omega$ $0.02 \pm 0.3$ $-0.1 \pm 0.3$ $0.1 \pm 0.3$
RV amplitude $K$ () $1.3 \pm 0.3$ $1.6 \pm 0.3$ $1.8 \pm 0.3$
HARPS pre-upgrade zero-point offset ()
HARPS pre-upgrade white-noise jitter $\sigma_{pre}$ ()
HARPS post-upgrade zero-point offset ()
HARPS post-upgrade white-noise jitter $\sigma_{pre}$ ()
Minimum mass $m \sin i$ ($M_\oplus$) $0.65 \pm 0.15$ $0.58 \pm 0.11$ $1.21 \pm 0.2$
Semi-major axis $a$ (AU) $0.0156 \pm 4 \times 10^{-4}$ $0.0082 \pm 2 \times 10^{-4}$ $0.0276 \pm 7 \times 10^{-4}$
Eccentricity $e$ $0.03^{+0.4}_{-0.03}$ $0.05^{+0.35}_{-0.05}$ $0.02^{+0.25}_{-0.02}$
Longitude of periastron $\omega$ ($^\circ$) $354 \pm 90$ $30 \pm 80$ $315 \pm 120$
[^1]: Ordinarily, the BFP automatically models and subtracts the strongest periodogram peak at each step. For the purpose of determining the period of planet c, we have manually removed planets d and b even though the peaks associated with planet c are stronger.
|
---
abstract: 'We present the results of a survey of nearby, quiescent, non-peculiar, extremely isolated galaxies to search for the gaseous remnants of galaxy formation. Such remnants are predicted to persist around galaxies into the present day by galaxy formation models. We find low-mass companions around 7 of 34 galaxies surveyed. In addition we find 5 galaxies with lopsided distributions. The implications for galaxy formation and the nature of high velocity clouds are discussed.'
author:
- 'D.J. Pisano'
- 'Eric M. Wilcots'
title: 'Assessing the state of galaxy formation.'
---
Introduction
============
What is the current state of galaxy formation in the local universe? There have been many recent detections of clouds near larger spiral galaxies in the local universe. Such detections include clouds around NGC 925 (Pisano, Wilcots, & Elmegreen 1998), IC 10 (Wilcots & Miller 1998), four of five barred Magellanic spirals (Wilcots, Lehman, & Miller 1996), four of 16 low surface brightness dwarf galaxies and four of nine galaxies (Taylor [*[et al.]{}*]{} 1993, 1996), and high-velocity clouds (HVCs) around M101 (Kamphuis 1993), NGC 628 (Kamphuis & Briggs 1992, and in our own Local Group (see Wakker & van Woerden 1997; Blitz [*[et al.]{}*]{} 1999). Typical clouds have 10$^7$-10$^8$ of amounting to 1%-50% of the mass of the primary galaxy.
In numerous cases these clouds have been suggested to be remnant material from the galaxy formation process (e.g. NGC 925, IC 10, NGC 628, etc...). Current models of cold dark matter galaxy formation in which disk galaxies were built up via the accretion of smaller bodies in a hierarchical merging process predict such remnant material to exist (e.g. Navarro, Frenk, & White 1995). Unfortunately the serendipitous nature of these detections inhibit our ability to divine the true origin of the clouds. These clouds could be primordial material, but could also be material ejected via a galactic fountain or superwind, tidal debris from a recent interaction, or simply a small dwarf galaxy companion. Therefore, few, if any, of the clouds represent unambiguous detections of the remnant reservoir of gas from which galaxies formed.
To determine what the current state of galaxy formation is, we have conducted a systematic search for the remnant gas around a sample of extremely isolated and quiescent galaxies. The results from the pilot survey of six galaxies were reported in Pisano & Wilcots 1999, here we report of the current status of the expanded survey.
Sample
======
In order to determine the origin of clouds around other galaxies it is important to have a well-defined sample. We chose galaxies from the Nearby Galaxies Catalog (Tully 1988). The galaxies were classified as isolated such that they had no known companions with $M_B\le$-16 mag within 1 Mpc of them. In addition, galaxies were chosen that were classified as non-peculiar. These two conditions minimize the chance of the galaxy having had a recent interaction or merger so there should be no tidal debris around our sample galaxies. Our galaxies were also chosen to be quiescent (i.e. not Seyferts or starbursts) so that any gas around these galaxies is unlikely to be galactic ejecta from a galactic fountain or superwind.
Finally, galaxies were chosen such that they were large enough and close enough to resolve with the VLA in D configuration and the ATCA .750 configuration (D$_{25}\ge$1, R$\le$45 Mpc), yet far enough away to probe out at least 90 kpc (R$\ge$21 Mpc). This left us with 60 galaxies in the entire sky; we observed 34 of those galaxies
Observations
============
Between November 1997 and February 2000 we observed a total of 34 galaxies with the VLA and ATCA. A total of 600 km s$^{-1}$ was covered at a resolution of 5.2 km s$^{-1}$ for each observation. The sample galaxies had distances between 21 Mpc and 45 Mpc, allowing us to survey out to a radius of 92 - 192 kpc at a resolution of $\sim$1(6.1-12.8 kpc). The resulting observations have $\sigma\simeq$0.5-1$\times$10$^{19}$cm$^{-1}$ per channel for the column density. The mass detection limits are 8.3$\pm$3.5$\times$10$^6$ for a 5$\sigma$ detection over 2 channels (10.4 km s$^{-1}$). The range of mass detection limits comes from varying sensitivity and distance for each galaxy.
Results
=======
Of the 34 galaxies surveyed we detected gas-rich companions in around 7 of them (see figures 1 & 2). Another 5 galaxies have “disturbed” morphologies (figures 3 & 4); either severe warps or lopsided distributions possibly indicative of a recent minor merger. The remaining 22 galaxies are relatively normal with all of the idiosyncrasies we typically see in galaxies such as small warps and asymmetries.
The detected companions have M$_{\HI}$ between 10$^8$ and 10$^9$, which corresponds to 3%-30% of the primary galaxy’s mass in . The companions appear show signatures of rotation, so based on the rotation widths and sizes of the companions we determined their dynamical masses to be between 10$^9$ and 10$^{10}$, which is 0.5%-10% of the main galaxy’s dynamical masses. The ratio of mass to dynamical mass for the companions range from 7%-85%. All clouds detected have spatially coincident optical emission, with the possible exception of UGC 11152. All of these properties are consistent with the gas-rich companions being typical dwarf galaxies.
This does not, however, mean that we have not detected the gaseous remnants of galaxy formation, but simply that these gas clouds formed stars before being accreted by the primary galaxy. It is important to confirm, however, that these gas-rich companions will eventually be accreted. Our companions have projected separations of 20-100 kpc (1-6 R$_{gal}$) in radius and 20-100 km s$^{-1}$ in velocity. These numbers imply an orbit time, which is roughly equal to the dynamical friction timescale, of 5-10 Gyr. These companions turn out to be in relatively stable orbits.
Implications for the nature of High-Velocity Clouds
===================================================
The Blitz [*[et al.]{}*]{} (1999) model for the origin of HVCs in the Local Group suggests that they are primordial material left over from the formation of the Local Group. In this model HVCs are at large distances from the Milky Way (R$\sim$750kpc-1Mpc) and, therefore, have large masses (M$_{\HI}\sim$10$^7$).
While our survey was not optimized to examine the origin of HVCs, there are some intriguing implications from its results. We detected no objects that resembled HVCs (i.e. no gas-rich companions without stars). Furthermore, we detected no companions smaller than 10$^8$ in down to our detection limit at $\sim$10$^7$. This implies that if HVCs are associated with galaxy formation, they must either have masses lower than 10$^7$and/or be at projected separations greater than 140 kpc. The former suggestion is somewhat unlikely, because we do detect larger companions and while one might expect more clouds at lower masses, we do not detect any down to our detection limit. Another possible explanation for our non-detection of HVCs is that they are associated with group formation, and not with the formation of individual, isolated galaxies. Either way, any explanation for a primordial origin of HVCs in the Local Group must account for our non-detection of them around isolated galaxies.
Implications for Galaxy Formation
=================================
The main goal of this work was to assess the state of galaxy formation in the local universe. At this point in time, this work has yielded three main implications for galaxy formation:
First, galaxy formation appears to be an efficient process. As discussed above, we found no companions with masses below 10$^8$ implying that there is not a population of low mass clouds within $\sim$100 kpc of these galaxies.
Second, galaxy formation has basically concluded. Only $\sim$20% of isolated galaxies have low-mass ($\sim$10%M$_{gal}$), gas-rich companions, (which are in stable orbits). Therefore most of the gas must already be in the main galaxy. The mass outside of the main galaxy will take a long time ($\sim$5 Gyr) to be accreted.
Third, galaxy formation may have recently ended. Another 15% of isolated galaxies have disturbed morphologies suggesting that they may have recently (in the last 1 Gyr) undergone a minor merger ($\le$10% M$_{gal}$).
The accretion rates implied by this work (10% of M$_{gal}$ over 6 Gyr for $\sim$40% of galaxies) are consistent with those derived by Toth & Ostriker (1992) from the scale height of the Milky Way, Zaritsky & Rix (1997) from the frequency of asymmetries, and Navarro, Frenk, & White (1995) from galaxy formation simulations. Future work on this project will involve further increasing the number of galaxies surveyed, comparing the stellar properties of the main galaxies and companions with their properties and more detailed comparisons of our detection rates with theoretical predictions.
D.J.P. wishes to thank the Wisconsin Space Grant Consortium for support for this work and the conference organizers for providing a student travel grant to help attend the conference. D.J.P. and E.M.W. were supported by NSF grants AST-9616907 and AST-9875008.
[Blitz, L., Spergel, D.N., Teuben, P.J., Hartmann, D., & Burton, W.B., 1999, , 514, 818 Kamphuis, J., 1993, Ph.D. thesis Kamphuis, J., & Briggs, F.H., 1992, , 253, 335 Navarro, J.F., Frenk, C.S., & White, S.D.M., 1995, , 275, 56 Pisano, D.J., Wilcots, E.M., & Elmegreen, B.G., 1998, , 115, 975 Pisano, D.J., & Wilcots, E.M., 1999, , 117, 2168 Taylor, C.L., Brinks, E., & Skillman, E.D., 1993, , 105, 128 Taylor, C.L., Thomas, D.L., Brinks, E., & Skillman, E.D., 1996, , 107, 143 Toth, G., & Ostriker, J.P., 1992, 389, 5 Tully, R.B., 1988, Nearby Galaxies Catalog (Cambridge: Cambridge Univ. Press) Wakker, B.P., & van Woerden H., 1997, , 35, 217 Wilcots, E.M., Lehman, C., & Miller, B., 1996, , 111, 1575 Wilcots, E.M., & Miller, B., 1998, , 116, 2363 Zaritsky, D., & Rix, H.-W., 1997, , 477, 118 ]{}
|
---
abstract: 'The ability to unconditionally verify the location of a communication receiver would lead to a wide range of new security paradigms. However, it is known that unconditional location verification in classical communication systems is impossible. In this work we show how unconditional location verification can be achieved with the use of quantum communication channels. Our verification remains unconditional irrespective of the number of receivers, computational capacity, or any other physical resource held by an adversary. Quantum location verification represents a new application of quantum entanglement that delivers a feat not possible in the classical-only channel. For the first time, we possess the ability to deliver real-time communications viable only at specified geographical coordinates.'
author:
- 'Robert A. Malaney, School of Electrical Engineering and Telecommunications, University of New South Wales, NSW 2052, Australia. r.malaney@unsw.edu.au'
title: 'Location-Dependent Communications using Quantum Entanglement'
---
The ability to offer a real-time communication channel whose viability is unconditionally a function of the receiver location would offer a range of new information security paradigms and applications. In particular, there are a range of industries and organizations that clearly would be interested in delivering information content in the sure knowledge a recipient receiver is at an *a-priori* agreed upon location (*e.g.* see discussions in [@denning; @classic1; @malaney; @classic]). The ability to guarantee location-sensitive communications requires unconditional (independent of the physical resources held by an adversary) location verification. However, in the classical-only channel such unconditional location verification is impossible. The finite speed of light can only be used to bound the minimum (but not the maximum) range a receiver is from some reference station. Add to the mix that classical information can be copied, and that an adversary can possess unlimited receivers (each of which can be presumed to possess unlimited computational capacity), it is straightforward to see why classical-only unconditional location verification is impossible. It is the purpose of this work to show how the introduction of quantum entanglement into the communication channel overcomes the above concerns, providing for the first time an unconditional location verification protocol.
Quantum teleportation [@tele1], the transfer of unknown quantum state information, is now experimentally verified through a host of experiments, *e.g.* [@tele2; @tele5]. In addition, the key resource underpinning teleportation, quantum entanglement, has been experimentally verified over very large ranges. An entanglement measurement over $144$km, achieved recently using optical free-space communications between two telescopes [@tenrife], proves the validity of ground-station to satellite quantum communications, and is widely seen as a major step in the path towards a global quantum communications network. In such a network it is envisaged that a combination of satellite and fiber optic links will interconnect a multitude of quantum nodes, quantum devices and quantum computers. In optical fiber, transmission of entangled photons is limited to about 100km by losses and de-coherence effects, *e.g.* [@fibre4]. Communications over fiber beyond this range will make use of either quantum repeaters [@repeaters], or the trusted relay paradigm used in a recent deployment of an eight-node quantum network [@trusted].
Experimental verification of quantum superdense coding [@dense1] has also been achieved through a series of experiments, *e.g.* [@dense2; @dense6]. In superdense coding, two bits of classical information can be transferred at the cost of only one qubit.
Teleportation and superdense coding are strongly related, and indeed they are often considered as protocols which are the inverse of each other, differing only in how and when they utilize quantum entanglement. Quantum location verification can be considered a new protocol that differs again in how and when it uses quantum entanglement.
The principal condition for unconditional location verification is that; only a device at *one* unique location (the authorized location) is able to, *immediately* and *correctly*, respond to signals received from multiple reference stations. In the classical-only channel this condition can never be unconditionally guaranteed. However, as we now show, with the introduction of quantum communication channels the condition necessary for unconditional location verification can in fact be guaranteed.
Consider some reference stations at publicly known locations, and a device which is not a reference station (Cliff) that is to be verified at a publicly known location $(x_v,y_v)$. Let us assume that processing times, such as those due to local quantum measurements, are negligible (we discuss later the minor impact of this). We also assume that the reference stations are authenticated and share secure communication channels between each other via quantum key distribution (QKD) [@qcd1; @qcd2], and that all classical communication between Cliff and the reference stations occurs via wireless channels. The use of wireless communications is important since we will require the time delay of all classical communications to be set by the line-of sight-distance between transceivers divided by $c$ (light speed in vacuum).
For two dimensional location verification we require a minimum of three reference stations. Consider $N$ maximally entangled multipartite systems available to a network possessing $k$ reference stations. Consider also that each of the multipartite systems comprises $k$ qubits, with each reference station initially holding one qubit from each of the $N$ systems. The $2^k$ orthogonal basis states of each multipartite state can be written $$\left| S_b \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left| a
\right\rangle _1 \otimes \left| a \right\rangle _2 \ldots \left|
a \right\rangle _k \pm \left| a \right\rangle _1 \otimes \left|
a \right\rangle _2 \ldots \left| a \right\rangle _k } \right)
\label{eq:1a}$$ where ${b = 1,...2^k }$, and the states $ \left| a \right\rangle$ represent $\left| 0 \right\rangle$ or $\left| 1
\right\rangle$ with the index on the state labelling the location (ignoring any *null* state).
Transformation between the basis states can be achieved by a set of $2^k$ unitary transformations induced on the locally held qubits. By this means a $k$-bit message, per entangled state, can be transferred from the stations to Cliff. This is achieved using superdense coding in which the stations encode each message to a specific basis state $ \left|
S_b \right\rangle $, with Cliff decoding the message via a quantum measurement that deterministically discriminates all possible basis states (the state $ \left|
S_b \right\rangle $ is sent directly to Cliff via quantum channels connected to the reference stations).
Quantum location verification builds on this concept of state encoding with one key addition. It must be the case that deterministic discrimination amongst the encoded states is possible, *within a pre-described time bound at only one location*. This can be achieved if the $2^k$ states which encode the $k$-bit messages are made *non-orthogonal* by the introduction of an additional local unitary transformation at each reference station. Let these additional transformations be labelled $U_i^{r}$, where ${r = 1,...k }$ indexes the reference station, and $i=1...N$ references the specific multipartite state to which the local transformation is applied.
Consider the $ith$ encoded multipartite state in which a $k$-bit message is encoded as $\left| {S_b
}\right\rangle$. Then on application of the additional transformations a new state $ \left| {\Upsilon _i } \right\rangle
= U_i^1 \otimes U_i^2 \otimes ...U_i^k \left| {S_b }
\right\rangle$ is produced. Our requirement is that $ \left\langle
{{\Upsilon _i }}
\mathrel{\left | {\vphantom {{\Upsilon _i } {\Upsilon _j }}}
\right. \kern-\nulldelimiterspace}
{{\Upsilon _j }} \right\rangle \ne 0$ when $
\left| {\Upsilon _i } \right\rangle \ne \left| {\Upsilon _j }
\right\rangle$. Ideally, the unitary matrices $U_i^r$ are chosen so that upon measurement of $ \left| {\Upsilon _i } \right\rangle
$ in a measurement basis $ \left| {S_1 } \right\rangle ,\left|
{S_2 } \right\rangle \ldots \left| {S_{2^k } } \right\rangle $, the probability of collapse to each basis state is approximately equal ($1/2^k$).
For quantum location verification to be unconditional it must be impossible for an adversary to map the values of $U_i^{r}$ to specific $k$-bit messages (in our protocol all matrices $U_i^{r}$ and all $k$-bit messages are ultimately sent over a classical channel). This means that there must be some form of randomness applied to the selection of each $U_i^{r}$. One strategy that provides for both a random selection mechanism, and the required non-orthogonal behavior between the states $ \left| {\Upsilon _i }
\right\rangle $, is to allow the $U_i^{r}$ to be constructed from four random real parameters ($\alpha,\beta,\gamma, \phi)$. The unitary matrix at each reference station can then be implemented as $$U = e^{i\phi } R_z \left( \alpha \right)R_y \left( \beta
\right)R_z \left( \gamma \right) , \label{eq:1r}$$ where the rotations $R$ are given by $$R_y (\theta ) = e^{ - i\theta \sigma _y /2} {\rm{ \ and \
}}R_z (\theta ) = e^{ - i\theta \sigma _z /2} ,$$ and with the $\sigma$’s representing the Pauli operators. Classical communication of the additional matrices can be achieved in many ways, such as passing of experimental instructions (e.g. duration of laser pulses), indexing of a large number of matrices, or as a transfer of matrix element information. The latter, which we adopt here, involves the transmission of the values ($\alpha,\beta,\gamma, \phi)$ adopted for each $U_i^r$. Although finite bandwidth of the classical channel limits the precision of this information transfer - required precision is available at the cost of additional bandwidth. In actual deployment, any global phase can be ignored. We discuss later a pragmatic implementation strategy leading to an outcome effectively the same as the outcome derived from Eq. (\[eq:1r\]).
The location verification proceeds by the encoding of a secret sequence onto a set of $N$ entangled systems $ \left| {\Upsilon
_{i=1...N} } \right\rangle $, transmission of each $ \left|
{\Upsilon _{i} } \right\rangle $ to Cliff via quantum channels, followed by transmission of the unitary matrices $U_i^{r}$ (*i.e.* the set ($\alpha,\beta,\gamma, \phi)$) to Cliff by classical channels. Upon receiving this quantum and classical information Cliff can decode and broadcast the decoded sequence via the classical channel. Given that information transfer over the classical channel proceeds at a velocity $c$, location information becomes unconditionally verifiable (as explained later). Ultimately, the verification is based on the inability to clone deterministically the set $ \left| {\Upsilon _i }
\right\rangle $ with fidelity one. Although cloning with lower fidelities is possible, confidence levels on the location verification can be increased to any arbitrary level by increasing $N$.
We now outline the protocol in more detail using well known maximally entangled states. For clarity, we proceed with a one dimensional location verification using just two reference stations, which we henceforth refer to as Alice and Bob. A geometrical constraint for one-dimensional location verification is that the device to be located must lie between Alice and Bob. That is, $ \tau _{AC} + \tau _{BC} = \tau _{AB} $, where $ \tau
_{AC}$ ($ \tau _{BC}$) is the light travel time between Alice (Bob) and Cliff, and where $\tau_{AB}$ is the light travel time between Alice and Bob.
Let Alice share with Bob a set of *N* maximally entangled qubit pairs $ \left|
{\Omega _i^{AB} } \right\rangle$, where the subscript $i = 1
\ldots N$ labels the entangled pairs. Let each of the pairs be described by one of the Bell states $ \left| {\Phi ^ \pm } \right\rangle =
\frac{1}{{\sqrt 2 }}\left( {\left| {00} \right\rangle \pm \left|
{11} \right\rangle } \right)$, $ \left| {\Psi ^ \pm }
\right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left| {01}
\right\rangle \pm \left| {10} \right\rangle } \right)$, with the first qubit being held by Alice and the second by Bob. We will assume an encoding ($ 00 \to \Phi ^ +$ *etc.*) that is public.
Without loss of generality we can assume all pairs are initially in the state $ \left| {\Phi ^ + }
\right\rangle$. After the encoding of a sequence onto a series of entangled pairs, Alice and Bob apply an additional random unitary transformation $U^A_i$ and $U^B_i$, respectively, to their local qubit from each pair. As a consequence, the entangled pairs held by Alice and Bob now form a non-orthogonal set, $$\left| {\Upsilon _i^{AB} } \right\rangle = U_i^A \otimes U_i^B
\left| {\Omega _i^{AB} } \right\rangle \ \ . \label{eq:1}$$ For example, for $\left| {\Phi ^ + } \right\rangle $ Eq. (\[eq:1\]) leads to a state $ \frac{1}{{\sqrt 2 }}\left(
{U_i^A \left| 0 \right\rangle _A \otimes U_i^B \left| 0
\right\rangle _B + U_i^A \left| 1 \right\rangle _A \otimes U_i^B
\left| 1 \right\rangle _B } \right)$.
![Quantum Circuit for Location Verification[]{data-label="fig1"}](fig1 "fig:"){width="3.5in" height="3.4in"}\
A step-by-step exposition of the protocol follows.
$\bullet$ Step 1: Via a secure channel Alice and Bob agree on a mutual random bit sequence $S_{ab}$ that is to be encoded. The encoding is achieved via superdense coding in which two classical bits are encoded using local unitary operators as described by $ I\left| {\Phi ^ + } \right\rangle = \left| {\Phi ^
+ } \right\rangle $, $ \sigma _x \left| {\Phi ^ + } \right\rangle
= \left| {\Psi ^ + } \right\rangle$, $ i\sigma _y \left| {\Phi ^ +
} \right\rangle = \left| {\Psi ^ - } \right\rangle$, and $ \sigma
_z \left| {\Phi ^ + } \right\rangle = \left| {\Phi ^ - }
\right\rangle$. For each pair of entangled qubits, Alice and Bob also agree who will induce the necessary unitary operation on their local qubit in order to encode sequential two-bit segments of $S_{ab}$.
$\bullet$ Step 2: Prior to the transmission of any qubit the transformation $ \left| {\Omega _i^{AB} } \right\rangle
\to \left| {\Upsilon _i^{AB} } \right\rangle$ as described by Eq. (\[eq:1\]) is induced. This set is then transmitted by Alice and Bob to Cliff via two separate quantum channels.
$\bullet$ Step 3: Alice and Bob communicate to Cliff, via separate classical channels, the random matrices $U^A_i$ and $U^B_i$ used to form the set $ \left| {\Upsilon _i^{AB} }
\right\rangle $. This classical information is transmitted in a synchronized manner to Cliff such that for each value of $i$ the $U^A_i$ sent by Alice and the $U^B_i$ sent by Bob arrive simultaneously at Cliff’s publicly announced location $(x_v,y_v)$. It is also ensured that this classical information is received at Cliff *after* the arrival of the corresponding qubit pair of $ \left| {\Upsilon
_i^{AB} } \right\rangle$.
$\bullet$ Step 4: Upon receipt of each matrix pair $U^A_i$, $U^B_i$, Cliff undertakes the transform $\left( {U_i^A
\otimes U_i^B } \right)^\dag \left| {\Upsilon _i^{AB} }
\right\rangle \to \left| {\Omega _i^{AB} } \right\rangle $ before taking a Bell State Measurement (BSM) in order to determine the two-bit segment encoded in the entangled pair. Cliff then immediately broadcasts (classically) the decoded two-bit segment back to Alice and Bob.
$\bullet$ Step 5: Alice checks that the sequence returned to her by Cliff is correctly decoded and notes the round-trip time for the process. Likewise Bob. Alice and Bob can then compare their round-trip times to Cliff (2$\tau_{AC}$ and 2$\tau_{BC}$) in order to verify consistency with Cliff’s publicly reported location $(x_v,y_v)$.
The quantum circuit for the one dimensional quantum location verification just described is given in fig. 1.
Quantum location verification is independent of the physical resources an adversary may possess. In the classical-only channel an adversary can place co-operating devices closer to reference stations and then delay responses in order to defeat any location verification (*e.g.* [@classic1; @classic]). Attempts to remedy this problem by making devices unique and tamper-proof are clearly limited (see discussion in [@classic1]), and cannot provide for unconditional security.
However, in quantum verification multiple devices are of no value. In order to decode immediately, Cliff’s receiver must possess *all* the qubits that comprise each entangled state. Cliff cannot distribute copies of his local qubits to other devices due to the no-cloning theorem [@noclon]. The key point is that for any given location $(x_v,y_v)$ that is to undergo a verification process, one can always find placements for the reference stations such that no other location can be *simultaneously* closer to all of the reference stations than $(x_v,y_v)$ (*e.g.* recall the geometrical constraint in our one dimensional verification). This being the case, an adversary with no device at the location being verified cannot pass the verification test. Even if the adversary possess multiple receivers, an additional round-trip communication time between his devices will be required for decoding. This will result in a round-trip time between at least one reference station and the location $(x_v,y_v)$ being larger than expected. In classical verification the *round-trip* communication between the adversary’s devices is not required.
Extension of the one-dimensional location verification protocol to two-dimensional verification could be a straightforward application of additional bipartite entanglement between Alice and some third reference station, say Dan. This can be achieved by introduction of a new set of Bell states shared between Alice and Dan, with the protocol following a similar exposition to that given. However, perhaps a more elegant solution is the use of multipartite entangled states. For example, consider a Green-Horne-Zeilinger (GHZ) [@ghz1; @ghz2] state in which three qubits are maximally entangled, such as $ \left| S \right\rangle ^ + =
\frac{1}{{\sqrt 2 }}\left( {\left| {000} \right\rangle + \left|
{111} \right\rangle } \right)$. Transformation from this GHZ basis state to one of the eight basis states is achieved by the set of transforms $U_{GHZ}=(\sigma _z \otimes \sigma _z ,I_2
\otimes \sigma _z ,i\sigma _y \otimes \sigma _z ,\sigma _x \otimes
\sigma _z ,I_2 \otimes \sigma _x ,\sigma _z \otimes \sigma _x ,
\sigma _x \otimes \sigma _x ,i\sigma _y \otimes \sigma _x $), where the first (second) operator acts on the first (second) qubit [@ghz6]. A step-by-step quantum location verification using such tripartite states proceeds in similar manner to the bipartite protocol.
Clearly, a security threat to the protocol is the potential ability of an adversary who is in possession of an optimal cloning machine, redistributing the set $\left| {\Upsilon _i }
\right\rangle$ to other devices. If cloning were exact, the verification test would fail because the *round-trip* communication between the devices (needed to decode) would not be required. However, optimal cloning of the set $\left| {\Upsilon
_i } \right\rangle$ can be described by the fidelity, $F_c$, between this set and a cloned set. This is known to be upper bounded by $F_c
\approx 0.7$ for bipartite entanglement and $F_c
\approx 0.6$ for tripartite entanglement [@clone2; @clone3]. As such, for a series of two-bit messages encoded in $N=100$ bipartite states, an optimal cloning machine would have a probability of 1 in $10^{16}$ of passing the verification system even though not at the authorized location. For 100 three-bit messages encoded in tripartite states this decreases to a probability of 1 in $10^{22}$. Arbitrary smaller probabilities are achieved exponentially in $N$.
A key aspect of our protocol is rapid implementation of the random unitary matrices, $U_i^{r}$, at the reference stations. One pragmatic strategy that provides for both a random selection mechanism, and the required non-orthogonal behavior between the states $\left| {\Upsilon _i }
\right\rangle$, is to allow the $U_i^{r}$ to be constructed from random permutations of the Hadamard gate $H$, and the $\pi /8$ gate $T$. It is known that any single-qubit unitary operation can be approximated to arbitrary accuracy from $H$ and $T$ gates (*e.g.* [@chang]), and that standard optical devices can be deployed to induce such gates on polarized photons. In simulations we have explored permutations of the $T$ and $H$ gates as a means of producing the random transforms needed to remove the orthogonality of the original basis. A series of random permutations leading to gates of the form $TTHTHHTTH....$ were performed, and the average orthogonality of the set $\left|
{\Upsilon _i } \right\rangle$ measured. It was found that even with gates using only 5 random combinations (e.g. $THHTH$) the required non-orthogonal properties between the states $\left|
{\Upsilon _i } \right\rangle$ was achieved - with the average fidelity between any two states being $F\sim 0.3$. Similar fidelities were found using the random matrix formulation of Eq. (\[eq:1r\]).
The new quantum protocol we have outlined is aimed at networks in which the quantum channel utilizes fiber and the classical channel utilizes wireless communications. The protocol requires the use of random transformations at the reference stations, and the presence of efficient millisecond quantum memory at the receiver (see [@laur] for state-of-the-art implementation of quantum memory at telecommunication wavelengths). However, implementation of our protocol is simpler when it is assumed that qubits in the quantum channel move with velocity $c$, as no additional transformations are required, and the need for quantum memory is negated (in many set-ups). In such a circumstance the one-dimensional verification protocol would follow a set-up similar to that utilized in recent experiments on entanglement swapping [@swap]. In [@swap], a BSM via linear optics is conducted on a series of entangled photons arriving from different synchronized pulsed sources. Coincidence counting is achieved within the nanosecond range. Using similar techniques, an implementation of location verification over tens of km, to an accuracy of meters is currently possible. Any relaxation of our initial assumptions such as zero processing time, will manifest itself in a (determinable) reduction in the accuracy of the location being verified. Note that even though we have described our protocol under the assumption that all four Bell states can be discriminated in the BSM - this is not a requirement. When using linear optics for BSM only two Bells states can be discriminated (deterministically). In this case our encoding scheme would need to be adjusted to a three message encoding. This has the minor effect of a drop in the channel capacity.
Clearly there are many variants on our protocol, such as the use of teleportation, the use of other entanglement degrees of freedom, and the use of entanglement swapping between the reference stations and the device. For example, a modified verification protocol that uses entanglement swapping can be constructed that entirely negates the requirement for direct transfer of qubits between the reference stations and the device. Location verification would then be possible in a satellite-to-device communications system, provided the satellite and the device shared *a-priori* an entangled resource stored in quantum memory.
Quantum location verification could greatly assist in the authentication of devices within large-scale multihop quantum networks [@longline]. Current quantum authentication techniques require the distribution of secret keys distributed *a priori* amongst potential users [@whyqcd]. However, such keys, whether classical bits or entangled qubits, are subject to unauthorized re-distribution. We also note that quantum location verification can be used within other data-delivery protocols in which real-time data transfer can be communicated to a device successfully *only if* that device is at a specific location. The location verification can be monitored continuously in real time, halting any real-time data transfer upon violation of the verification procedures. An adversary could not continue to receive real-time data without one of his devices being at the specified location.
Quantum location verification represents a new application in the emerging field of quantum communications. It delivers an outcome not possible in the classical-only channel. For the first time, we possess the ability to unconditionally authenticate a communication channel based on the geographical coordinates of a receiver.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work has been supported by the University of New South Wales.
[1]{}
Scott, L. and Denning, D. E., GPS World, April 2003, 40–49 (2003).
Capkun, S. and Hubaux, J-P., IEEE Journal on Selected Areas in Communications, Vol. 24, 221–232 (2006).
Malaney, R. A., IEEE International Conference on Communications (ICC ’07), Glasgow, 1558–1563 (2007).
Chiang, J. T., Haas, J. J. and Hu, Y.-C., WiSec’09, Zurich, Switzerland (2009).
Bennett, C. H. *et al*., Phys. Rev. Lett. 70, 1895–-1899 (1993).
Bouwmeester, D. *et al*., Nature 390, 575–-579 (1997).
Ursin, R. *et al*., Nature 430, 849 (2004).
Ursin, R. *et al*., Nature Physics 3, 481–486 (2007).
Takesue, H. *et al*., New J. Phys. 7, 232 (2005).
Briegel, H. J., D¨ur, W., Cirac, J. I. and Zoller, P., Phys. Rev. Lett. 81, 5932–-5935 (1998).
Poppe, A., Peev, M. and Maurhart, O., International Journal of Quantum Information Vol. 6, No. 2 (2008).
Bennett, C. H. and Wiesner, S. J., Phys. Rev. Lett. 69, 2881 - 2884 (1992).
Mattle, K., *et al*., Phys. Rev. Lett. 76, 4656–-4659 (1996).
Barreiro, J. T., Wei, T. C. and Kwiat, P. G., Nature Physics, Vol.4, 282–286 (2008).
Bennett, C. H. and Brassard, G., Conf. on Computers, Systems and Signal Processing, Bangalore, India, 175–-179 (1984).
Ekert, A. K., Phys. Rev. Lett. 67, 661–-663 (1991).
Wootters, W. K. and Zurek, W. H., Nature 299, 802–803 (1982).
Greenberger, D. M., Horne, M. A. and Zeilinger, A., In: Bell$'$s Theorem, Quantum Theory, and Conceptions of the Universe, M. Kafatos (ed.), (Kluwer, Dordrecht) 73-–76 (1989).
Greenberger, D. M., Horne, M. A., Shimony, A. and Zeilinger, A., Am. J. Phys. 58, 1131–-1143 (1990).
Lamoureux, L. P., Navez, P., Fiurášek, J., and Cerf, N. J. , Phys. Rev. A 69, 040301R (2004).
Karpov, E., Navez, P., and Cerf, N. J., Phys. Rev. A 72, 042314 (2005).
Lee, H. J., Ahn, D. and Hwang, S. W., Phys. Rev. A, 66, 024304 (2002).
Nielsen, M. A. and Chuang, I. L., Quantum Computation and Quantum Information, Cambridge Univ. Press, (2000).
Lauritzen, B., *et al.*, arXiv:quant-ph/0908.2348v1 (2009).
Kaltenbaek, R., Prevedel, R., Aspelmeyer, M. and Zeilinger, A., Phys. Rev. A 79, 040302(R) (2009).
Van Meter, R. Ladd, T. D. Munro, W. J. and Nemoto, K., IEEE/ACM Transactions on Networking, 17(3), 1002-1013 (2009).
Paterson, K. G., Piper. F. and Schack, R., arXiv:quant-ph/0406147v1 (2004).
|
---
abstract: 'The effects of polymer additives on Rayleigh–Taylor (RT) instability of immiscible fluids is investigated using the Oldroyd-B viscoelastic model. Analytic results obtained exploiting the phase-field approach show that in polymer solution the growth rate of the instability speeds up with elasticity (but remains slower than in the pure solvent case). Numerical simulations of the viscoelastic binary fluid model confirm this picture.'
author:
- 'GUIDOBOFFETTA$^{1,2}$, ANDREAMAZZINO$^3$, STEFANOMUSACCHI O$^1$ and LARAVOZELLA$^3$'
title: 'Rayleigh–Taylor instability in a viscoelastic binary fluid'
---
Introduction
============
Mixing of species (e.g. contaminants, tracers and particles) and thermodynamical quantities (e.g. temperature) are dramatically influenced by fluid flows [@D05]. Controlling the rate of mixing in a flow is an objective of paramount importance in many fields of science and technologies with wide-ranging consequences in industrial applications [@WMD01].\
The difficulties of the problem come from the intricate nature of the underlying fluid flow, which involves many active nonlinearly coupled degrees of freedom [@F95], and on the poor comprehension of the way through which the fluid is coupled to the transported quantities. The problem is even more difficult when the transported quantity reacts back to the flow field thus affecting its dynamics. An instance is provided by the heat transport in convection [@S94].\
Mixing emerges as a final stage of successive hydrodynamic instabilities [@DR81] eventually leading to a fully developed turbulent stage. The possibility of controlling such instability mechanisms thus allows one to have a direct control on the mixing process. In some cases the challenge is to enhance the mixing process by stimulating the turbulence transition, in yet other cases the goal is to suppress deleterious instabilities and the ensuing turbulence. Inertial confinement fusion [@CZ02] is an example whose success relies on the control of the famous Rayleigh–Taylor (RT) instability occurring when a heavy, denser, fluid is accelerated into a lighter one. For a fluid in a gravitational field, such instability was first described Lord Rayleigh in the 1880s [@R883] and later generalized to all accelerated fluids by Sir Geoffrey Taylor in 1950 [@T50].
Our attention here is focused on RT instability with the aim of enhancing the perturbation growth-rate in its early stage of evolution. The idea is to inject polymers into the fluid and to study on both analytical and numerical ground how the stability of the resulting viscoelastic fluid is modified. Similar problems were already investigated in more specific context, including RT instability of viscoelastic fluids with suspended particles in porous medium with a magnetic field [@SR92] and RT linear stability analysis of viscoelastic drops in high-speed airstream [@JBF02]. We also mention that the viscoelasticity is known to affect also other kind of instabilities, including Saffman–Taylor instability [@W90; @C99], Faraday waves [@WMK99; @MZ99], the stability of Kolmogorov flow [@BCMPV05], Taylor–Couette flow [@LSM90; @GS96], and Rayleigh–Bénard problem [@VA69; @ST72].
The paper is organized as follows. In Sec. 2 the basic equations ruling the viscoelastic immiscible RT system are introduced together with the phase-field approach. In Sec. 3 the linear analysis is presented and the analytical results shown and discussed in Sec. 4. The resulting scenario is corroborated in Sec. 5 by means of direct numerical simulations of the original field equations.
Governing equations
===================
The system we consider is composed of two incompressible fluids (labeled by 1 and 2) having different densities, $\rho_1$ and $\rho_2 > \rho_1$, and different dynamical viscosities, $\mu_1$ and $\mu_2$, with the denser fluid placed above the less dense one. For more generality, the two fluids are supposed to be immiscible so that the surface tension on the interface separating the two fluids will be explicitly taken into account.
The effects of polymer additives is here studied within the framework of the Oldroyd-B model [@O50; @H77; @BHAC87]. In this model polymers are treated as elastic dumbbells, i.e. identical pairs of microscopic beads connected by harmonic springs. Their concentration is supposed to be low enough to neglect polymer-polymer interactions. The polymer solution is then regarded as a continuous medium, in which the reaction of polymers on the flow is described as an elastic contribution to the total stress tensor of the fluid [see [e.g.]{} @BHAC87].
In order to describe the mixing process of the resulting viscoelastic immiscible fluids we follow the phase-field approach (for a general description of the method see, [e.g.]{}, [@B02; @CH58], and for application to multiphase flows see, [e.g.]{}, [@BCB03; @DSS07; @M07; @CMMV09]). Here, we only recall that the basic idea of the method is to treat the interface between two immiscible fluids as a thin mixing layer across which physical properties vary steeply but continuously. The evolution of the mixing layer is ruled by an order parameter (the phase field) that obeys a Cahn–Hilliard equation [@CH58]. One of the advantage of the method is that the boundary conditions at the fluids interface need not to be specified being encoded in the governing equations. From a numerical point of view, the method permits to avoid a direct tracking of the interface and easily produces the correct interfacial tension from the mixing-layer free energy.
To be more specific, the evolution of the viscoelastic binary fluid is described by the system of differential equations $$\rho_0
\left(\partial_t {\bm v} + {\bm v} \cdot{\bm \partial} {\bm v}\right) =
-{\bm \partial} p + {\bm \partial} \cdot (2 \mu {\bm e} )
+ A \rho_0 {\bm g} \phi
-\phi {\bm \partial} {\cal M}
+{2 \mu \eta \over \tau}
{\bm\partial}\cdot ({\bm \sigma}-\mathbb{I})
\label{eq1}$$ $$\partial_t \phi + {\bm v} \cdot {\bm \partial} \phi = \gamma
\partial^2 {\cal M}
\label{eq2}$$ $$\partial_t {\bm \sigma}+{\bm v} \cdot{\bm \partial} {\bm \sigma} =
({\bm \partial}{\bm v})^T \cdot{\bm \sigma}+
{\bm \sigma}\cdot{\bm \partial}{\bm v}-{2 \over \tau}({\bm \sigma}-
\mathbb{I})
\,\,\, .
\label{eq3}$$ Eq. (\[eq1\]) is the usual Boussinesq Navier–Stokes equation [@KC01] with two additional stress contributions. The first one arises at the interface where the effect of surface tension enters into play [@B02; @YFLS04; @BBCV05], the last term represents the polymer back-reaction to the flow field [@BHAC87].\
In (\[eq1\]), we have defined $\rho_0=(\rho_1 + \rho_2)/2$, $\bf{g}$ is the gravitational acceleration pointing along the $y$-axis, $\mathcal{A}\equiv(\rho_2-\rho_1)/(\rho_2+\rho_1)$ is the Atwood number, $e_{ij}\equiv\left(
\partial_i v_j + \partial_j v_i \right) / 2$ is the rate of strain tensor and $\mu=\mu(\phi)$ is the dynamical viscosity field parametrically defined as [@LS03] $$\frac{1}{\mu} = \frac{1+\phi}{2 \mu_1}+\frac{1-\phi}{2 \mu_2}
\label{eq4}$$ $\phi$ being the phase field governed by (\[eq2\]). The phase field $\phi$ is representative of density fluctuations and we take $\phi=1$ in the regions of density $\rho_1$ and $\phi=-1$ in those of density $\rho_2 \ge \rho_1$. ${\bm \sigma}\equiv \frac{\langle{\bm R}{\bm R} \rangle}{R_0^2}$ is the polymer conformation tensor, ${\bm R}$ being the end-to-end polymer vector ($R_0$ is the polymer length at equilibrium), the parameter $\eta$ is proportional to polymer concentration and $\tau=\tau(\phi)$ is the (slowest) polymer relaxation time which, according to the Zimm model [@DE86], is assumed to be proportional to the viscosity $\mu$ (therefore we have $\tau=\tau_1$ for $\phi=1$ and $\tau=\tau_2$ for $\phi=-1$ with $\mu(\phi)/\tau(\phi)$ constant). Finally, $\gamma$ is the mobility and ${\cal M}$ is the chemical potential defined in terms of the Ginzburg–Landau free energy ${\cal F}$ as [@CH58; @B02; @YFLS04] $${\cal M} \equiv \frac{\delta {\cal F} }{\delta \phi}\qquad\mbox{and}\qquad
{\cal F}[\phi] \equiv \lambda \int_{\Omega} \mathrm{d}\bm{x} \;
\left( \frac{1}{2} |{\bm \partial} \phi|^2+ V(\phi) \right) \, .
\label{eq5}$$ where $\Omega$ is the region of space occupied by the system, $\lambda$ is the magnitude of the free-energy and the potential $V(\phi)$ is $$V(\phi)\equiv \frac{1}{4 \epsilon^2} (\phi^2 -1 )^2
\label{eq6}$$ where $\epsilon$ is the capillary width, representative of the interface thickness.\
The unstable equilibrium state with heavy fluid placed on the top of light fluid is given by $$\bm{v}=\bm{0}\, , \quad \phi(y)=-\tanh\left (\frac{y}
{\epsilon\sqrt{2}}\right )\qquad \mbox{and}\qquad {\bm \sigma}=\mathbb{I}
\label{eq7}$$ corresponding to a planar interface of width $\epsilon$ with polymers having their equilibrium length $R_0$. In this case, the surface tension, ${\cal S}$, is given by [see, for example, @LL00]: $${\cal S} \equiv \lambda \int_{-\infty}^{+\infty} dy \;\left( \frac{1}{2}
|{\bm \partial} \phi|^2+ V(\phi)
\right) = \frac{2\lambda \sqrt{2}}{3\epsilon} \, .
\label{eq8}$$ The sharp-interface limit is obtained by taking the $\lambda$ and $\epsilon$ to zero, keeping $\cal{S}$ fixed to the value prescribed by surface tension [@LS03].
Linear stability analysis
=========================
Let us now suppose to impose a small perturbation on the interface separating the two fluids. Such perturbation will displace the phase field from the previous equilibrium configuration, which minimizes the free energy (\[eq5\]) to a new configuration for which, in general, $\mathcal{M} \neq 0$. We want to determine how the perturbation evolves in time.
Focusing on the two-dimensional case (corresponding to translational invariant perturbations along the $z$ direction), let us denote by $h(x,t)$ the perturbation imposed to the planar interface $y=0$ in a way that we can rewrite the phase-field $\phi$ as: $$\phi = f\left(\frac{y-h(x,t)}{\epsilon \sqrt{2}}\right)\, ,
\label{eq9}$$ where $h$ can be larger than $\epsilon$, yet it has to be smaller than the scale of variation of $h$ (small amplitudes). In this limit we assume the interface to be locally in equilibrium, i.e. $\partial^2 f/\partial y^2 = V'(f)$, and thus $f(y)=-\tanh(y)$ and therefore ${\cal M} = - \lambda \frac{\partial^2 f}{\partial x^2}$ ($'$ denotes derivative with respect to the argument).
Linearizing the momentum equation for small interface velocity we have $$\rho_0 \partial_t v_y = - \partial_y p - \phi \partial_y {\cal M} -
A g \rho_0 \phi + {2 \mu \eta \over \tau}\partial_i\sigma_{i2}
+\mu \left(\partial_x^2 + \partial_y^2 \right)v_y
+2(\partial_y v_y)\partial _y\mu \,.
\label{eq10}$$
Integrating on the vertical direction and using derivations by parts one gets $$\rho_0 \partial_t q = {\cal S} \frac{\partial^2 h}{\partial x^2}+
2 A g \rho_0 h + {2 \mu \eta \over \tau} \Sigma +Q
\label{eq11}$$ where we have defined $$Q\equiv \int_{-\infty}^{+\infty}\mu
\left (\frac{\partial^2 }{\partial x^2} -
\frac{\partial^2 }{\partial y^2}\right )v_y\,dy\qquad
q \equiv \int_{-\infty}^{\infty} v_y\,dy \qquad
\Sigma\equiv \int_{-\infty}^{\infty}\partial_x \sigma_{12}\,dy \, ,
\label{eq12}$$ and we have used the relations $\int (f')^2 dy = 2 \sqrt{2}/(3 \epsilon)$, $\int f f''' dy = 0$, $\int f dy = 2 h$.
Note that, unlike what happens in the inviscid case, Eq. (\[eq11\]) does not involve solely the field $q_y$ but also second-order derivatives of $v_y$. In order to close the equation, let us resort to a potential-flow description. The idea is to evaluate $Q$ for a potential flow $v_y$ and then to plug $Q=Q^{pot}$ into (\[eq11\]) [@M93]. The approximation is justified when viscosity is sufficiently small and its effects are confined in a narrow region around the interface. Because for a potential flow $\partial^2 {\bm v}=0$ we have $$Q^{pot}=
2 \int_{-\infty}^{+\infty}\mu {\partial^2 u_y \over \partial x^2}\, dy=
2 \int_{-\infty}^{0}\mu {\partial^2 u_y \over \partial x^2}\, dy +
2 \int_{0}^{\infty}\mu {\partial^2 u_y \over \partial x^2}\, dy =
(\mu_1+\mu_2){\partial^2 q \over \partial x^2} \, .
\label{eq14}$$ Substituting in (\[eq11\]) and defining $\nu=(\mu_1+\mu_2)/(2 \rho_0)$ one finally obtains $$\partial_t q =
{{\cal S} \over \rho_0} {\partial^2 h \over \partial x^2}+ 2 A g h +
{2 \mu \eta \over \tau \rho_0} \Sigma +
2 \nu {\partial^2 q \over \partial x^2} \, .
\label{eq15}$$ Let us now exploit the equation (\[eq2\]) for the phase field to relate $q_y$ to $h$. For small amplitudes, we have: $$\partial^2{\cal M} = {\lambda \over \epsilon \sqrt{2}}
\left [f'\frac{\partial^4h}{\partial x^4} + {1 \over 2\epsilon^2}
f'''\frac{\partial^2 h}{\partial x^2} \right ]
\label{eq16}$$ and therefore, from (\[eq2\]) $$- {1 \over \epsilon} f' \partial_t h + v_y {1 \over \epsilon} f' =
{\gamma \lambda \over \epsilon} \left [f' \partial_x ^4 h +
{1 \over 2\epsilon^2} f''' \partial_x^2 h \right ] \, .
\label{eq17}$$
Integrating over $y$, observing that $1/(2\sqrt{2}\epsilon) f'$ approaches $\delta(y-h)$ as $\epsilon \to 0$ and using the limit of sharp interface ($\gamma \lambda \to 0$) one obtains $$\partial_t h = v_y(x,h(t,x),t) \equiv v_y^{(int)}(x,t) \, .
\label{eq18}$$
The equation for the perturbation $\sigma_{12}$ of the conformation tensor is obtained by linearizing (\[eq3\]) around $\sigma_{\alpha \beta}=\delta_{\alpha \beta}$ $$\partial_t \sigma_{12}= \partial_{x} v_{y}+ \partial_{y} v_{x} -
\frac{2}{\tau}\sigma_{12}
\label{eq19}$$ from which, exploiting incompressibility, we obtain $$\partial_t \partial_x \sigma_{12}=
(\partial_{x}^2 - \partial_{y}^2) v_{y} -
{2 \over \tau} \partial_x \sigma_{12} -
2 \sigma_{12} \partial_x {1 \over \tau}\, .
\label{eq20}$$ For small amplitude perturbations the last term, which is proportional to $\sigma_{12} \partial_x \phi$, can be neglected at the leading order. Integrating over $y$ and using again the potential flow approximation one ends up with $$\partial_t \Sigma=2 \partial_x^2 q - {2 \over \bar{\tau}} \Sigma
- ({1 \over \tau_1}-{1 \over \tau_2}) \int dy \phi \partial_x \sigma_{12}
\, .
\label{eq21}$$ where we have introduced $\bar{\tau}=2 \tau_1 \tau_2/(\tau_1 + \tau_2)$.
In conclusion, we have the following set of equations (in the $(x,t)$ variables) for the linear evolution of the Rayleigh–Taylor instability in a viscoelastic flow $$\left \{
\begin{array}{lll}
\partial_t h & = & v_y^{(int)} \\
\partial_t q & = &
{{\cal S} \over \rho_0} \partial_x^2 h + 2 A g h +
{2 \nu \eta c \over \bar{\tau}} \Sigma + 2 \nu \partial_x^2 q \\
\partial_t \Sigma & = & 2 \partial_x^2 q - {2 \over \bar{\tau}} \Sigma
- ({1 \over \tau_1}-{1 \over \tau_2}) \int dy \phi \partial_x \sigma_{12}
\, .
\end{array}
\right .
\label{eq22}$$ where $c=4 \mu_1 \mu_2/(\mu_1+\mu_2)^2 \le 1$.
Potential flow closure for the interface velocity
=================================================
The set of equations (\[eq22\]) is not closed because of the presence of the interface velocity $v_y^{(int)}$ and of the integral term in the equation for $\Sigma$. In order to close the system we exploit again the potential flow approximation for which $v_y=\partial_y \psi$.
Taking into account the boundary condition for $y\to \infty$, the potential can be written (e.g. for $y\ge 0$) as $$\psi(x,y,t)=\int_0^{\infty} e^{-k y+i k x} \hat{\psi}(k,t) dk + c.c.
\label{eq23}$$ where “ ” denotes the Fourier transform, and therefore $$v_y(x,y,t)= - \int_0^{\infty} k e^{-k y+i k x} \hat{\psi}(k,t) dk + c.c.
\label{eq24}$$ $$q(x,t)= - 2 \int_0^{\infty} e^{i k x} \hat{\psi}(k,t) dk + c.c.
\label{eq25}$$ and taking a flat interface, $y=0$, at the leading order $$v^{(int)}(x,t)= - \int_0^{\infty} k e^{i k x} \hat{\psi}(k,t) dk + c.c.
\label{eq26}$$ Assuming consistently that also $$\sigma_{12}(x,y,t)= \int_0^{\infty} e^{-k y+i k x} \hat{\sigma}_{12}(k,t) dk + c.c. \, ,
\label{eq26b}$$ in the limit of small amplitudes one has $\int dy \phi \partial_x \sigma_{12}=0$ and the set of equation (\[eq22\]) for the Fourier coefficients becomes $$\left \{
\begin{array}{lll}
\partial_t \hat{h} & = & {k \over 2} \hat{q} \\
\partial_t \hat{q} & = &
- {{\cal S} \over \rho_0} k^2 \hat{h} + 2 A g \hat{h} +
{2 \nu c \eta \over \bar{\tau}} \hat{\Sigma} - 2 \nu k^2 \hat{q} \\
\partial_t \hat{\Sigma} & = & - 2 k^2 q - {2 \over \bar{\tau}} \hat{\Sigma} \, .
\end{array}
\right .
\label{eq27}$$ Restricting first to the case without polymers ($\eta=0$), the growth rate $\alpha_N$ of the perturbation is obtained by looking for a solution of the form $\hat{h} \sim e^{\alpha_N t}$ which gives $$\alpha_N = - \nu k^2 + \sqrt{\omega^2 + (\nu k^2)^2}
\label{eq28}$$ where it has been defined $$\omega=\sqrt{A g k - {{\cal S} \over 2 \rho_0} k^3} \, .
\label{eq29}$$ The expression (\[eq29\]) is the well-known growth rate for a Newtonian fluid in the limit of zero viscosity [@C61], while (\[eq28\]) is a known upper bound to the growth rate for the case with finite viscosity [@MMSZ77].
Let us now consider the case with polymers, i.e. $\eta>0$. The growth rate $\alpha$ is given by the solution of $$(\alpha \bar{\tau})^3 + 2 (\alpha \bar{\tau})^2 (1+\nu k^2 \bar{\tau})+\alpha
\bar{\tau}
\left[4 \nu (1+c \eta) k^2 \bar{\tau} -\omega^2 \bar{\tau}^2 \right]-2 \omega^2 \bar{\tau}^2=0 \, .
\label{eq30}$$ The general solution is rather complicated and not very enlightening. In the limit of stiff polymers, $\bar{\tau} \to 0$, one gets $$\alpha_0 \equiv \lim_{\bar{\tau} \to 0} \alpha = - \nu(1+c \eta) k^2 + \sqrt{\omega^2 + [\nu(1+c \eta) k^2]^2} \, .
\label{eq31}$$ Comparing with (\[eq28\]) one sees that in this limit polymers simply renormalize solvent viscosity. This result is in agreement with the phenomenological definition of $c \eta$ as the zero-shear polymer contribution to the total viscosity of the mixture [@V75]. Therefore, in order to quantify the effects of elasticity on RT instability, the growth rate for viscoelastic cases at finite $\bar{\tau}$ has to be compared with the Newtonian case with renormalized viscosity $\nu(1+c \eta)$.
Another interesting limit is $\bar{\tau} \to \infty$. In this case from (\[eq30\]) one easily obtains that the growth rate coincides with that of the pure solvent (\[eq28\]), i.e. $\alpha_{\infty} = \alpha_N$. The physical interpretation is that in the limit $\bar{\tau} \to \infty$ and at finite time for which polymer elongation is finite, the last term in (\[eq1\]) vanishes and one recovers the Newtonian case without polymers (i.e. $\eta=0$). Of course, this does not mean that in general polymer effects for high elasticity disappear. Indeed in the long-time limit polymer elongation is able to compensate the $1/\tau$ coefficient and in the late, non-linear stages, one expects to observe strong polymer effects at high elasticity.
From equation (\[eq30\]) one can easily show (using implicit differentiation) that $\alpha(\bar{\tau})$ is a monotonic function and, because $\alpha_\infty \ge \alpha_0$, we have that instability rate grows with the elasticity, or the Deborah number, here defined as $De \equiv \omega \bar{\tau}$.
The case of stable stratification, $g \to -g$, is obtained by $\omega^2 \to -\omega^2$ neglecting surface tension. In this case (\[eq30\]) has no solution for positive $\alpha$, therefore polymers alone cannot induce instabilities in a stably stratified fluid.
Numerical results
=================
The analytical results obtained in the previous Sections are not exact as they are based on a closure obtained from the potential flow approximation. While this approximation is consistent for the inviscid limit $\nu=0$ (where it gives the correct result (\[eq29\]) for a Newtonian fluid) for finite viscosity we have shown that it gives a known upper bound to the actual growth rate of the perturbation [@MMSZ77] (this is because the potential flow approximation underestimates the role of viscosity which reduces the instability). Nonetheless, in the case of Newtonian fluid this upper bound is known to be a good approximation of the actual value of the growth rate measured in numerical simulations [@MMSZ77]. Because both $\bar{\tau} \to 0$ and $\bar{\tau} \to \infty$ limits correspond to Newtonian fluids, we expect that also in the viscoelastic case the potential flow description is a good approximation.
To investigate this important point, we have performed a set of numerical simulations of the full model (\[eq1\]-\[eq3\]) in the limit of constant viscosity and relaxation time (i.e. $\mu_1=\mu_2$, $c=1$ and $\tau_1=\tau_2=\bar{\tau}$) in two dimensions by means of a standard, fully dealiased, pseudospectral method on a square doubly periodic domain. The resolution of the simulations is $1024\times 1024$ collocation points (a comparative run at double resolution did not show substantial modifications on the results). More details on the numerical simulation method can be found in [@CMV06] and [@CMMV09].
The basic state corresponds to a zero velocity field, a hyperbolic-tangent profile for the phase field and an uniform distribution of polymers in equilibrium, according to (\[eq7\]). The interface of the basic state is perturbed with a sinusoidal wave at wavenumber $k$ (corresponding to maximal instability for the linear analysis) of amplitude $h_0$ much smaller than the wavelength ($k h_0 =0.05$).
The growth rate $\alpha$ of the perturbation is measured directly by fitting the height of the perturbed interface at different times with an exponential law. For given values of $A\,g$, $\mathcal{S}/ \rho_0$, $\nu$ and $\eta$, this procedure is repeated for different values of $\bar{\tau}$ at the maximal instability wavenumber $k$ (which, for the range of parameters considered here, is always $k=1$, i.e. it is not affected by elasticity). Figure \[fig1\] shows the results for two sets of runs at different values of $\eta$ and $\nu$. As discussed above, we find that the theoretical prediction given by (\[eq30\]) is indeed an upper bound for the actual growth rate of the perturbation. Nevertheless, the bound gives grow rates which are quite close to the numerical estimated values (the error is of the order of $10\%$). The error is smaller for the runs having a larger value of $\eta$ and $\nu$, as was already discussed by [@CMMV09].
![The perturbation growth-rate $\alpha$ normalized with the inviscid growth rate $\omega$ (\[eq29\]) as a function of the Deborah number $De=\omega \bar{\tau}$. Points are the results of numerical simulations of the full set of equations (\[eq1\]-\[eq3\]), lines represent the theoretical predictions obtained from (\[eq30\]). The values of parameters are: $c=1$, $k=1$, $A g=0.31$, $\mathcal{S} / \rho_0=0.019$ and $\eta=0.3$, $\nu=0.3$ (upper points and line) and $\eta=0.5$, $\nu=0.6$ (lower points and line).[]{data-label="fig1"}](fig.eps)
Both theoretical and numerical results show that the effect of polymers is to increase the perturbation growth-rate. $\alpha$ grows with the elasticity and saturates for sufficiently large value of $De$.
Conclusions and perspectives
============================
We investigated the role of polymers on the linear phase of the Rayleigh–Taylor instability in an Oldroyd-B viscoelastic model. In the limit of vanishing Deborah number (i.e. vanishing polymer relaxation time) we recover a known upper bound for the growth rate of the perturbation in a viscous Newtonian fluid with modified viscosity. For finite elasticity, the growth rate is found to increase monotonically with the Deborah number reaching the solvent limit for high Deborah numbers. Our findings are corroborated by a set of direct numerical simulations on the viscoelastic Boussinesq Oldroyd-B model.
Our analysis has been confined to the linear phase of the perturbation evolution. When the perturbation amplitude becomes sufficiently large, nonlinear effects enter into play and a fully developed turbulent regime rapidly sets in [@CC06; @VC09; @BMMV09]. In the turbulent stage we expect more dramatic effects of polymers. In turbulent flows, a spectacular consequence of viscoelasticity induced by polymers is the drag reduction effect: addition of minute amounts (a few tenths of p.p.m. in weight) of long-chain soluble polymers to water leads to a strong reduction (up to $80\%$) of the power necessary to maintain a given throughput in a channel [see [e.g.]{} @T49; @V75]. We conjecture that a similar phenomenon might arise also in the present context. Heuristically, the RT system can indeed be assimilated to a channel inside which vertical motion of thermal plumes is maintained by the available potential energy. This analogy suggests the possibility to observe in the viscoelastic RT system a “drag” reduction (or mixing enhancement) phenomenon, i.e. an increase of the velocity of thermal plumes with respect to the Newtonian case. Whether or not this picture does apply to the fully developed turbulence regime is left for future research.
We thank anonymous Referees for useful remarks.
[99]{}
2003 Computation of multiphase systems with phase field models. [*J. Comput. Phys.*]{} [**190**]{}, 371–397.
2005 Turbulence and coarsening in active and passive binary mixtures. [*Phys. Rev. Lett.*]{} [**95**]{}, 224501-1–224501-4.
1987 [*Dynamics of Polymeric Liquids.*]{} Wiley-Interscience.
2007 The viscoelastic Kolmogorov flow: eddy viscosity and linear stability. [*J. Fluid Mech.*]{} [**523**]{}, 161–170.
2009 Kolmogorov scaling and intermittency in Rayleigh–Taylor turbulence. [*Phys. Rev. E*]{} [**79**]{}, 065301-1–065301-4.
2002 Theory of phase-ordering kinetics. [*Advances in Physics*]{} [**51**]{}, 481–587.
2006 Reynolds number effects on Rayleigh–Taylor instability with possible implications for type Ia supernovae. [*Nature Physics*]{} [**2**]{}, 562–568.
1958 Free energy of a non uniform system. [*J. Chem. Phys.*]{} [**28**]{}, 258–267.
2006 Rayleigh–Taylor turbulence in two-dimensions. [*Phys. Rev. Lett.*]{} [**96**]{}, 134504-1–134504-4
2009 Phase-field model for the Rayleigh–Taylor instability of immiscible fluids. [*J. Fluid Mech.*]{} [**622**]{}, 115–134.
1961 Hydrodynamic and Hydromagnetic Stability. [*New York: Dover*]{}.
2002 Energy transfer in Rayleigh–Taylor instability. [*Phys. Rev. E*]{} [**66**]{}, 026312-1–026312-12.
1999 Saffman–Taylor instability in yield-stress fluids. [*J. Fluid Mech.*]{} [**380**]{}, 363–376.
2005 [Turbulent mixing.]{} [*Ann. Rev. Fluid. Mech.*]{} **37**, 329–356.
2007 [Diffuse interface model for incompressible two-phase flows with large density ratios.]{} [*J. Comput. Phys.*]{} [**226**]{}, 2078–2095.
1986 The Theory of Polymer Dynamics. [*Oxford University Press*]{}.
1981 Hydrodynamic Stability. [*Cambridge University Press*]{}.
1995 Turbulence. The legacy of A. N. Kolmogorov. [*Cambridge University Press*]{}.
1996 Couette-Taylor flow in dilute polymer solutions. [*Phys. Rev. Lett.*]{} [**77**]{}, 1480–1483.
1977 Mechanical models of dilute polymer solutions in strong flows. [*Phys. Fluids*]{} [**20**]{}, S22–S30.
2002 [Rayleigh–Taylor instability of viscoelastic drops at high Weber numbers]{}. [*J. Fluid Mech.*]{} [**453**]{}, 109–132.
2001 Fluids Mechanics, Second Edition. [*Academic Press*]{}.
2000 Fluid Mechanics Volume 6 of Course of Theoretical Physics, Second Edition Revised. [*Butterworth Heinemann*]{}.
1990 A purely elastic instability in Taylor–Couette flow. [*J. Fluid Mech.*]{} [**218**]{}, 537–600.
2003 A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. [*Physica*]{} D [**179**]{}, 211–228.
1977 Unstable normal mode for Rayleigh–Taylor instability in viscous fluids. [*Phys. Fluids*]{} [**20(12)**]{}, 2000–2004.
1993 Effect of viscosity on Rayleigh–Taylor and Richtmyer–Meshkov instabilities. [*Phys. Rev. E*]{} [**47**]{}, 375–383.
2007 Phase-field models for fluid mixtures. [*Math. Comput. Model.*]{} [**45**]{}, 1042–1052.
Faraday instability in a linear viscoelastic fluid. [*Europhys. Lett.*]{} [**45**]{}, 169–174.
1950 On the formulation of rheological equations of state. [*Proc. Roy. Soc. London Ser. A*]{} [**200**]{}, 523–541.
1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. [*Proc. London Math. Soc.*]{} [**14**]{}, 170–177.
1992 Rayleigh-Taylor instability of viscoelastic fluids with suspended particles in porous medium in hydromagnetics [*Czech. J. Phys.*]{} [**42**]{}, 919–926.
1994 High Rayleigh number convection. [Ann. Rev. Fluid. Mech.]{} **26**, 137–168.
1972 Convective stability of a general viscoelastic fluid heated from below. [*Phys. Fluids*]{} [**15**]{}, 534–539.
1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. [*Proc. R. Soc. A*]{} [**201**]{}, 192–197.
1949 Observation on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. [*Proc. 1st International Congress on Rheology*]{} [**2**]{}, 135–141.
1969 Overstability of a viscoelastic fluid layer heated from below. [*J. Fluid Mech.*]{} [**36**]{}, 613–623.
1975 Drag reduction fundamentals. [*AIChE Journal*]{} [**21**]{}, 625–656.
2009 Self-similarity and universality in Rayleigh–Taylor, Boussinesq turbulence. [*Phys. Fluids*]{} [**21**]{}, 015102-1–015102-9.
1999 Faraday waves on a viscoelastic liquid. [*Phys. Rev. Lett.*]{} [**83**]{}, 308–311.
2001 Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation. Springer, New York.
1990 The Taylor-Saffman problem for a non-Newtonian liquid. [*J. Fluid Mech.*]{} [**220**]{}, 413–425 .
2004 A diffuse-interface method for simulating two-phase flows of complex fluids. [*J. Fluid Mech.*]{} [**515**]{}, 293–317.
|
---
abstract: 'By paying attention to the hole-doped two-dimensional systems of antiferromagnetically (strongly) correlated electrons, we discuss the cause of hole-rich phase formation in association with phase separation. We show that the phase diagram obtained from the Maxwell’s construction in the plane of temperature vs. hole density is consistent with one derived from the evaluation of hole-rich and electron-rich phases in real space. We observe that the formation of a hole-rich phase is attributed to the aggregation of hole pairs induced by spin singlet pairs present in the pseudogap phase and that a direct involvement of correlations between hole pairs are not essential for phase separation.'
address: |
$^{1}$ Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea\
$^{2}$ Korea Institute of Advanced Study, Seoul 130-012, Korea
author:
- 'Sung-Sik Lee, Sul-Ah Ahn, and Sung-Ho Suck Salk$^{1,2}$'
title: 'Phase Separation in the Two-Dimensional Systems of Strongly Correlated Electrons; the Role of Spin Singlet Pairs on Hole Pairing Contribution to Hole-rich Phase '
---
Currently there has been an activated interest in the study of stripe modulations owing to strong experimental evidences observed [@Zimmermann; @Tranquada; @Shen; @Jorgensen; @Hundley; @Hammel; @Weidinger; @Harshman; @Cho; @Borsa; @Dong; @Hunt; @Nikolaev] involving hole-doped high $T_c$ cuprates. Phase separation, a phenomenon of broken symmetry at low energies can be manifested as a stripe phase or a phase-separated inhomogeneity. At a critical temperature, the phase separation appears owing to the divergence of compressibility or the zero inverse compressibility. The onset of phase separation can be obtained from the evaluation of the zero inverse compressibility using the well-known Maxwell’s construction. Thus far, various numerical approaches [@Zaanen; @EmeryKivelsonLin; @Seibold; @Luchini; @White; @Putikka; @Dagotto; @Prelovsek; @Castellani; @Poliblanc; @Kohno; @Hellberg; @Shraiman; @Calendra; @Pryadko; @Cosentini; @Gimm; @WhiteScalapino; @HellbergManousakis] to the Hubbard and $t-J$ Hamiltonians have successfully revealed the existence of phase separation in the hole-doped two dimensional systems of strongly correlated electrons. They include the Hatree-Fock (HF), the Lanczos exact diagonalization (LED), the density matrix renormalization group (DMRG), a fixed node quantum Monte Carlo (FNQMC) and Green’s function Monte Carlo (GFMC) methods. Most recently a slave-boson functional integral (SBFI) method of Gimm and Salk[@Gimm] has been proposed to reveal a good agreement with earlier exact numerical studies[@EmeryKivelsonLin; @Hellberg] of phase separation boundary. However it remains to be seen whether the phase separation obtained from the Maxwell’s construction can, indeed, show phase-separated distributions of charge and spin in real space. Currently there exists a great controversy [@WhiteScalapino; @HellbergManousakis; @Prelovsek; @Hellberg] over the issue of whether the phase separation accompanied by the hole-rich phase arises as a consequence of correlations between pairs of holes or not. Here we address this issue, based on our slave-boson functional integral approach[@Gimm; @SSLee], and take the advantage of its usefulness to study the role of charge and spin degrees of freedom on phase separation.
The $t-J$ Hamiltonian to deal with the hole doped systems of antiferromagnetically correlated electrons is written, $$\begin{aligned}
H&=&-t\sum_{\langle i,j\rangle\sigma}\left(c_{i\sigma}^{\dagger}c_{j\sigma}
+\mbox{H.c.}\right)
+J\sum_{\langle i,j\rangle}\left({\bf S}_i\cdot{\bf S}_j
- \frac{n_in_j}{4}\right)\\
\label{tj_ham_1}
&=& -t\sum_{\langle i,j\rangle\sigma} \left(c_{i\sigma}^{\dagger}c_{j\sigma}
+\mbox{H.c.}\right) \nonumber\\
&&-\frac{J}{2} \sum_{\langle i,j\rangle}
\left(c_{i\uparrow}^{\dagger}c_{j\downarrow}^{\dagger}
-c_{i\downarrow}^{\dagger}c_{j\uparrow}^{\dagger}\right)
\left(c_{j\downarrow}c_{i\uparrow}-c_{j\uparrow}c_{i\downarrow}\right).
\label{tj_ham_2}\end{aligned}$$ In order to explicitly see physics involved with the spin and charge degrees of freedom, we rewrite the above equation in the $U(1)$ slave-boson representation with the use of $c_{i\sigma}=f_{i\sigma}b_i^{\dagger}$ as a composite of spinon $f_{i\sigma}$ and holon $b_i$ operators subject to the single occupancy constraint $b_i^{\dagger}b_i + \sum_{\sigma}f_{i\sigma}^{\dagger}f_{i\sigma}=1$ $$\begin{aligned}
H_{t-J}&=&-t\sum_{\langle i,j\rangle\sigma}f_i^{\dagger}f_j b_j^{\dagger}b_i
-\frac{J}{2} \sum_{\langle i,j\rangle} b_ib_j b_j^{\dagger}b_i^{\dagger} \nonumber\\
&&\times\left(f_{i\uparrow}^{\dagger}f_{j\downarrow}^{\dagger}
-f_{i\downarrow}^{\dagger}f_{j\uparrow}^{\dagger}\right)
\left(f_{j\downarrow}f_{i\uparrow}-f_{j\uparrow}f_{i\downarrow}\right).
\label{tj_slave_boson}\end{aligned}$$ The above $U(1)$ slave-boson expression can also be deduced from our earlier $SU(2)$ slave-boson representation[@SSLee] of the $t-J$ Hamiltonian. The second term shows spinon pairing (spin singlet) interactions between adjacent sites. Both terms explicitly manifest coupling between the charge (holon) and spin (spinon) degrees of freedom.
We attempt a multi-faceted study of phase separation by employing various levels of approaches; the Hatree-Fock (HF)[@SSLee] of Hubbard Hamiltonian for a qualitative, comparative study of phase separations, the Monte Carlo diagonalization (MCD)[@RaedtLinden] for an accurate study of correlated electron systems and the slave-boson functional integral (SBFI) method for an investigation of the role of charge (hole) and spin degrees of freedom on phase separation. Extending our earlier HF approach[@SSLee] of Hubbard Hamiltonian to a study of temperature dependence of charge and spin distributions in real space[@SSLeePreprint], we first examine the domain of phase separation in the plane of temperature vs. hole density and compare the phase separation domain derived from the Maxwell’s construction with the one obtained from the direct (real) space calculations of electron-rich and hole-rich phases. Based on HF calculations with $U=4t$, in Fig. \[phase\_U4\] we display the predicted phase separation diagram in the plane of the temperature $T$ vs. hole density $x$ with the two different approaches; one from the use of the Maxwell’s construction (denoted by shaded area in Fig. \[phase\_U4\]) and the other from the direct evaluations of hole (charge) and electron (spin) distributions (denoted as solid diamonds) in the real space of $14\times 14$ lattice. Temperature dependent HF total energies for Maxwell’s construction are obtained from the solution of the following self-consistent equations (Eq. (\[self\_consistent\_m\_mu\])) involving the quasi-particle energy, $$E_{\bf k}^{\pm}= \frac{U}{2}(1-x) - \mu \pm \epsilon_{\bf k}
\label{self_consistency}$$ with $\epsilon_{\bf k} = \sqrt{[-2t(\cos k_x + \cos
k_y)]^2 + \left( \frac{m U}{2} \right)^2 }$ and $m=|\langle c_{i \uparrow}^{\dagger}c_{i \uparrow}
-c_{i \downarrow}^{\dagger}c_{i \downarrow}\rangle|$, by determining the uniform staggered magnetization $m$ and the chemical potential $\mu$ $$\begin{aligned}
1 & = & \frac{1}{N} \sum^{'}_{\bf k} \frac{U}{2 \epsilon_{\bf k}}
\left[ \tanh \left( \frac{E_{\bf k}^+}{2T} \right) - \tanh \left(
\frac{E_{\bf k}^-}{2T} \right) \right] \nonumber \\
x & = & \frac{1}{N} \sum^{'}_{\bf k} \left[
\tanh \left( \frac{E_{\bf k}^+}{2T} \right) +
\tanh \left( \frac{E_{\bf k}^-}{2T} \right) \right].
\label{self_consistent_m_mu}\end{aligned}$$ Encouragingly the two results showed a large overlapping domain of phase separation in the $T-x$ plane. The present mean field results are at best qualitative.
In order to show the doping dependence of phase separation in real space, we now examine the electron and hole distributions interior and exterior to the phase separation boundary. At a chosen hole density of $x = 0.25$ near but below a critical value $x_c\simeq 0.3$ at $T=0.01t$, a stripe phase no longer appears and, instead, an inhomogeneous phase separation is revealed, as is shown in Fig. \[gp\_T10\_d50\]. Although not shown here, such inhomogeneous structures are prevalent near but below the critical hole density $x_c$, but they eventually disappear to yield uniform phase structures beyond $x_c$. It should be noted that despite the fact that no correlations exist between the Hatree-Fock quasiparticles, the phase separation is seen to occur as well known. They are not accurate for strongly correlated (large $U$) electron systems. However, it is to be noted that with currently available, numerically exact methods, the present level of real space calculations of temperature dependent phase separation for a large square lattice are not easily feasible. It is of great interest to see whether there exists a possibility of ’metastable’ phase-separation in a certain region of temperature and doping where density fluctuations are excessively large as is shown by the results from the Maxwell’s construction. This case happened near $T=0.01t$ and $x=0.25$, showing a metastable region of phase separation. In this small domain, the density fluctuations were predicted to be large. A detailed study for verification is necessary in the future.
For an accurate account of strongly correlated electrons for the study of phase separation as a function of antiferromagnetic coupling strength $J$, we now take the MCD method[@RaedtLinden] of the $t-J$ Hamiltonian, Eq. (1) with $J=1.0t$ for a $4\times 4$ square lattice. Fig. \[maxwell\_j1\] shows the computed ground state energy (in units of $t$) shifted by a linear factor $e_H n_e$ as a function of the electron density $n_e$ ($n_e = 1-x$). The solid triangle represents the GFMC calculation by Hellberg and Manousakis[@Hellberg]$^{c}$ and the open circles, our present results. From the Maxwell’s construction the critical electron density (hole density) is predicted to be $n_c=0.684$ ($x_c=0.316$) compared to the value of $n_c = 0.730$ ($x_c=0.270$) obtained from the GFMC. The overall variation of curvature as a function of electron (hole) density is grossly similar between the two approaches, although the predicted ground state energies are not the same. In Fig. \[various\_ps\], we display our MCD predicted phase separation boundary in the plane of electron (hole) density vs. antiferromagnetic interaction strength (Heisenberg coupling constant) in units of $t$ and make comparison with other methods. The MCD agreed very well with the GFMC of Hellberg and Manousakis [@Hellberg]$^{c}$, the LED of Emery [*et al.*]{} [@EmeryKivelsonLin] and the SBFI ($U(1)$ slave-boson functional integral) result of Gimm and Salk[@Gimm]. A salient feature is that all of these methods yielded a similar phase separation boundary, by showing a smoothly decreasing (increasing) trend of the critical electron (hole) density with an intercept near $J/t \sim 3.5$ as the antiferromagnetic interaction strength $J/t$ increases.
However, the accurate MCD calculations can not readily resolve the current controversy over the issue of whether the effect of correlations between the hole pairs is the primary cause of forming the hole-rich phase. For this cause we now explore the $U(1)$ slave-boson functional integral approach of the $t-J$ Hamiltonian (Eq. (\[tj\_slave\_boson\])). This method is advantageous to examine the role of the charge and spin degrees of freedom or the role of the holon and spinon degrees of freedom on phase separation. Earlier we reported that this method[@Gimm] also showed a satisfactory phase separation boundary for all $J$ values ($J/t \leq 1$ and $J/t > 1$) in general agreement with other numerical studies [@EmeryKivelsonLin; @Hellberg]. The spin (spinon) degrees of freedom shown in the second term in Eq. (\[tj\_slave\_boson\]) allows spinon pairing (spin singlet formation) interactions between adjacent sites. Indeed it has been shown from the $U(1)$ slave-boson theories that the spinon pairing, that is, the spin singlet pair order appears below the pseudogap (spin gap) temperature $T^*$ [@KotliarLiu; @SSLee]. This indicates that the motion of paired holes rather than the independent motion of separated holes is energetically preferred in the presence of surrounding electron spin singlet pairs (spinon pairs) below the pseudogap (spin gap) temperature $T^*$. This is because the hole pairs (holon pairs) can readily migrate to the occupied sites of spin pairs (spinon pairs) involving no spin-bond breaking. In summary, we showed that the phase separation diagram obtained from Maxwell’s construction in the plane of temperature vs. hole density is consistent with one derived from the real space (direct lattice) calculations of hole-rich and electron-rich phases; inhomogeneous phase separation appears near $x_c$ at finite temperatures; the formation of the hole-rich phase for phase separation is attributed to the aggregation of hole pairs induced by spin singlet pairs which exist in the spin-gap phase.
One of us (S.H.S.S.) greatly acknowledges generous supports of POSRIB (2001) project at Pohang University of Science and Technology and Korean Ministry of Education (Hakjin Excellence Leadership Program 2001). We thank Chan-Ho Yang for computational assistances.
[99]{} M. v. Zimmermann, A. Vigliante, T. Niemöller, N. Ichikawa, T. Frello, J. Madsen, P. Wochner, S. Uchida, N. H. Andersen, J. M. Tranquada, D. Gibbs, and J. R. Schneider, Europhys. Lett. [**41**]{}, 629 (1998). J. M. Tranquada, J. D. Axe, N. Ichikawa, A. R. Moodenbaugh, Y. Nakamura, and S. Uchida, Phys. Rev. Lett. [**78**]{}, 338 (1997); J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, Phys. Rev. B. [**54**]{}, 7489 (1996); J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature [**375**]{}, 561 (1995); P. Wochner, J. M. Tranquada, D. J. Buttrey, and V. Sachan, Phys. Rev. B, [**57**]{}, 1066 (1998); J. M. Tranquada, D. J. Buttrey, V. Sachan, and J. E. Lorenzo, Phys. Rev. Lett. [**73**]{}, 1003 (1994); V. Sachan, D. J. Buttrey, J. M. Tranquada, J. E. Lorenzo, and G. Shirane, Phys. Rev. B. [**51**]{}, 12 742 (1995); J. M. Tranquada, J. E. Lorenzo, D. J. Buttrey, and V. Sachan, [*ibid.*]{} [**52**]{}, 3581 (1995). A. Lanzara, P. V. Bogdanov, X. J. Zhou, S. A. Kellar, D. L. Feng, E. D. Lu, T. Yoshida, H. Elsaki, A. Fusimori, K. Kishio, J.-I. Shhimoyama, T. Noda, S. Uchida, Z. Hussain, and Z. X. Shen, Nature [**412**]{}, 510 (2001); references therein. J. D. Jorgensen, B. Dabrowski, S. Pei, D. G. Hinks, L. Soderholm, B. morosin, J. E. Schriber, E. L. Venturini, and D. S. Ginley, Phys. Rev. B [**38**]{}, 11 337 (1988). M. F. Hundley, J. D. Thompson, S-W. Cheong, Z. Fisk, and J. E. Schirber, Phys. Rev. B [**41**]{}, 4062 (1990). S.-H. Lee and S-W. Cheong, Phys. Rev. lett. [**79**]{}, 2514 (1997); P. C. Hammel, A. P. Reyes Z. Fisk, M. Takigawa, J. D. Thompson, R. H. Heffner, S-W. Cheong, and J. E. Schirber, Phys. Rev. B, [**42**]{}, 6781 (1990); P. C. Hammel [*et al.*]{}, Physica C [**185-189**]{}, 1095 (1991). A. Weidinger, Ch. Niedermayer, A. Golnik, R. Simon, and E. Recknagel, Phys. Rev. Lett. [**62**]{}, 102 (1989). D. R. Harshman, G. Aeppli, B. Batlogg, G. P. Espinosa, R. J. Cava, A. S. Cooper, L. W. Rupp, E. J. Ansaldo, and D. Ll. Williams, Phy. Rev. Lett. [**63**]{}, 1187 (1989). J. H. Cho, F. Borsa, D. C. Johnston, and D. R. Torgeson, Phys. Rev. B [**46**]{}, 3179 (1992); J. H. Cho, F. C. Chou, and D. C. Johnston, Phys. Rev. Lett. [**70**]{}, 222 (1993). F. Borsa, P. Carretta, A. Lascialfari, D. R. Torgeson, F. C. Chou, and D. C. Johnston, Physica C [**235-240**]{}, 1713 (1994); F. Borsa, P. Carretta, J. H. Cho, F. C. Chou, Q. Hu, D. C. Johnston, A. Lascialfari, D. R. Torgeson, R. J. Gooding, N. M. Salem, and K. J. E. Vos, Phys. Rev. B [**52**]{}, 7334 (1995); F. C. Chou and D. C. Johnston, Phys. Rev. B [**54**]{}, 572 (1996). X. L. Dong, Z. F. Dong, B. R. Zhao, Z. X. Zhao, X. F. Duan, L.-M. Peng, W. W. Huang, B. Xu, Y. Z. Zhang, S. Q. Guo, L. H. Zhao, and L. Li, Phys. Rev. Lett. [**80**]{}, 2701 (1998). A. W. Hunt, P. M. Singer, K. R. Thurber, and T. Imai, Phys. Rev. Lett. [**82**]{}, 4300 (1999); references therein.; E. G. Nikolaev, H. B. Brom, and A. A. Zakharov, Phys. Rev. B [**62**]{}, 3050 (2000). J. Zaanen and O. Gunnarsson, Phys. Rev. B [**40**]{}, 7391 (1989); J. Zaanen and P. B. Littlewood, Phys. Rev. B. [**50**]{}, 7222 (1994). V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. [**64**]{}, 475 (1990); S. A. Kivelson, V. J. Emery, and H. Q. Lin, Phys. Rev. B [**42**]{}, 65223 (1990); S. A. Kivelson and V. J. Emery, in [*Strongly Correlated Electronic Matrials: the Los Alamos Symposium, 1993*]{}, edited by K. S. Bedel [*et al.*]{} (Addison-Wesley, Reading, MA, 1994); V. J. Emery and S. A. Kivelson, Physica [**209C**]{}, 597 (1993). G. Seibold, C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. B [**58**]{}, 13 506 (1998). M. U. Luchini, M. U. Ogata, W. O. Puttika, and T. M. Rice, Physica [**185-189C**]{}, 141 (1991); W. O. Putikka, M. U. Luchini, and T. M. Rice, Phys. Rev. Lett. [**68**]{}, 538 (1992). S. R. White, Phys. Rev. Lett. [**69**]{}, 2863 (1992); Phys. Rev. B [**48**]{}, 10 345 (1993). W. O. Putikka, M. U. Luchini, and T. M. Rice, Phys. Rev. Lett. [**68**]{}, 538 (1992). E. Dagotto, Rev. Mod. Phys. [**66**]{}, 763 (1994); H. Fehske, V. Waas, H. Röder, and H. Bütter, Phys. Rev. B [**44**]{}, 8473 (1991); D. poilblanc, [*ibid.*]{} [**52**]{}, 9201 (1995). P. Prelovšek and X. Zotos, Phys. Rev. B [**47**]{}, 5984 (1993). C. Castellani, C. Di Castro, and M. Grilli, Phys. Rev. B [**40**]{}, 7391 (1989). D. Poliblanc, Phys. Rev. B [**52**]{}, 9201 (1995). M. Kohno, Phys. Rev. B [**55**]{}, 1435 (1997). a) C. S. Hellberg and E. Manousakis, Phys. Rev. Lett. [**78**]{}, 4609 (1997); b) J. Phys. Chem. Solids [**59**]{}, 1818 (1998); c) Phys. Rev. B [**61**]{}, 11 787 (2000). B. Shraiman and E. Siggia, Phys. Rev. Lett. [**60**]{}, 740 (1988); [**61**]{}, 467 (1988); M. Boninsegni and E. Manousakis, Phys. Rev. B [**45**]{}, 4877 (1992); [**46**]{}, 560 (1992). M. Calendra, F. Becca, and S. Sorella, Phys. Rev. Lett. [**81**]{}, 5185 (1998). L. P. Pryadko, S. Kivelson, and D. W. Hone, Phys. Rev. Lett. [**80**]{}, 5651 (1998). A. C. Cosentini, M. Capone, L. Guidoni and G. B. Bachelet, Phys. Rev. B [**58**]{}, R 14685 (1998).
T. H. Gimm and Sung-Ho Suck Salk, Phys. Rev. B [**62**]{}, 13 930 (2000). S. R. White and D. J. Scalapino, Phys. Rev. Lett. [**84**]{}, 3021 (2001); [**80**]{}, 1272 (1998);[**81**]{}, 3227 (1998); Phys. Rev. B [**60**]{}, 753 (1999). S. Hellberg and E. Manousakis, Phys. Rev. lett. [**84**]{}, 3022 (2001); [**83**]{}, 132 (1999). S.-S. Lee and Sung-Ho Suck Salk, Phys. Rev. B [**64**]{}, 052501 (2001); T.-H. Gimm, S.-S. Lee, S.-P. Hong and Sung-Ho Suck Salk, Phys. Rev. B [**60**]{}, 6324 (1999). H. De Raedt and W. von der Linden, Int. J. Mod. Phys. C [**3**]{}, 97 (1992); Phys. Rev. B [**45**]{}, 8787 (1992); W. von der Linden, Phys. Rep. [**220**]{}, 53; H. De Raedt and M. Frick, Phys. Rep. [**231**]{}, 107 (1993); S.-P. Hong, S.-S. Lee, and Sung-Ho Suck Salk, Phys. Rev. B [**62**]{}, 14880 (2000). G. Kotliar and J. Liu, Phys. Rev. B [**38**]{}, 5142 (1988); M. U. Ubbens and P. A. Lee, Phys. Rev. B [**46**]{}, 8434 (1992); [**49**]{}, 6853 (1994); X. G. Wen and P. A. Lee, Phys. Rev. Lett. [**76**]{}, 503 (1996); [**80**]{}, 2193 (1998); N. Nagaosa and P. A. Lee, Phys. Rev. B [**61**]{}, 9166 (2000).
FIGURE CAPTIONS
- The temperature (in the unit of hopping energy $t$) dependence of phase separation as a function of hole doping density $x$ based on the Hatree-Fock approximation with $U=4t$ for a $14\times 14$ square lattice. The shaded region represents phase separation obtained from the Maxwell’s construction; the solid diamond, the phase separation obtained from the numerical evaluation of charge and spin distributions in real space and the open diamond, the uniform phase.
- Spin (electron) and charge (hole) distribution at $x = 50/196 (\simeq 0.25)$ $T=0.01t$. The large circles represent holes (charges) and smaller ones, electrons (spins). The size of circles represents the degree of hole-richness or electron-richness.
- Maxwell’s construction (dotted line) from predicted ground state energies denoted by open circles; the energies (in units of $t$) are shifted by a linear factor $-e_H n_e$, $\left(e_h(x)-e_H n_e \right)/t$ as a function of electron density for $J=1.0t$ with a $4\times 4$ lattice. For comparison the Green’s function Monte Carlo result[@Hellberg]$^{d}$ (denoted by HM) with various lattice sizes up to $11\times 11$ is displayed.
- Phase separation for the hole-doped systems of antiferromagnetically correlated electrons in the plane of Heisenberg coupling strength, $J/t$ and the electron density, $n_e=1-x$. The triangles denoted by HM represent the Green’s function Monte Carlo prediction of Hellberg and Manousakis [@Hellberg]$^{c}$ and the stars denoted by EKL, the exact diagonalization result of Emery [*et al.*]{}[@EmeryKivelsonLin]. The solid circles represent the results from the $U(1)$ slave-boson functional integral approach of Gimm and Salk[@Gimm] and diamonds, our present MCD results.
= 7cm = 6cm
= 6cm = 6cm
|
---
abstract: 'A new model to describe biological invasion influenced by a line with fast diffusion has been introduced by H. Berestycki, J.-M. Roquejoffre and L. Rossi in 2012.The purpose of this article is to present a related model where the line of fast diffusion has a nontrivial range of influence, i.e. the exchanges between the line and the surrounding space has a nontrivial support. We show the existence of a spreading velocity depending on the diffusion on the line. Two intermediate model are also discussed. Then, we try to understand the influence of different exchange terms on this spreading speed. We show that various behaviour may happen, depending on the considered exchange distributions.'
author:
- 'Antoine Pauthier [^1]'
bibliography:
- 'biblio.bib'
title: 'The influence of nonlocal exchange terms on Fisher-KPP propagation driven by a line of fast diffusion'
---
Introduction
============
Models
------
The purpose of this study is a continuation of [@BRR1] in which was introduced, by H. Berestycki, J.-M. Roquejoffre and L. Rossi, a new model to describe biological invasions in the plane when a strong diffusion takes place on a line, given by (\[BRReq2\]).
$$\label{BRReq2}
\begin{cases}
\partial_t u-D \partial_{xx} u= \nub v(t,x,0)-\mub u & x\in\R,\ t>0\\
\partial_t v-d\Delta v=f(v) & (x,y)\in\R\times\R^*,\ t>0\\
v(t,x,0^+)=v(t,x,0^-), & x\in\R,\ t>0 \\
-d\left\{ \partial_y v(t,x,0^+)-\partial_y v(t,x,0^-) \right\}=\mub u(t,x)-\nub v(t,x,0) & x\in\R,\ t>0.
\end{cases}$$
A two-dimensional environment (the plane $\R^2$) includes a line (the line $\{(x,0),\quad x\in \R \}$) in which fast diffusion takes place while reproduction and usual diffusion only occur outside the line. For the sake of simplicity, we will refer to the plane as “the field“ and the line as “the road“, as a reference to the biological situations. The density of the population is designated by $v=v(t,x,y)$ in the field, and $u=u(t,x)$ on the road. Exchanges of densities take place between the field and the road: a fraction $\nu$ of individuals from the field at the road (i.e. $v(x,0,t)$) joins the road, while a fraction $\mu$ of the population on the road joins the field. The diffusion coefficient in the field is $d$, on the road $D$. Of course, the aim is to study the case $D>d$. The nonlinearity $f$ is of Fisher-KPP type, i.e. strictly concave with $f(0)=f(1)=0$. Considering a nonnegative, compactly supported initial datum $(u_0,v_0)\neq(0,0)$, the main result of [@BRR1] was the existence of an asymptotic speed of spreading $c^*$ in the direction of the road. They also explained the dependence of $c^*$ on $D,$ the coefficient of diffusion on the road. In their model, the line separates the plane in two half-planes which do not interact with each other, but only with the line. Moreover, interactions between a half-plane and the line occur only with the limit-condition in (\[BRReq2\]). That is why, in [@BRR1], the authors consider only a half-plane as the field.
New results on (\[BRReq2\]) have been recently proved. Further effects like a drift or a killing term on the road have been investigated in [@BRR2]. The case of a fractional diffusion on the road was studied and explained by the three authors and A.-C. Coulon in [@BRRC] and [@these_AC]. Models with an ignition-type nonlinearity are also studied by L. Dietrich in [@Dietrich1] and [@Dietrich2].
Our aim is to understand what happens when local interactions are replaced by integral-type interactions: exchanges of populations may happen between the road and a point of the field, not necessarily at the road. The density of individuals who jump from a point of the field to the road is represented by $y\mapsto \nu(y)$, from the road to a point of the field by $y \mapsto \mu(y)$. This is a more general model than the previous one, but interactions still only occur in one dimension, the y-axis. We are led to the following system: $$\label{RPeq}
\begin{cases}
\partial_t u-D \partial_{xx} u = -\mub u+\int \nu(y)v(t,x,y)dy & x \in \R,\ t>0 \\
\partial_t v-d\Delta v = f(v) +\mu(y)u(t,x)-\nu(y)v(t,x,y) & (x,y)\in \R^2,\ t>0,
\end{cases}$$ where $\mub = \int \mu(y)dy$, the parameters $d$ and $D$ are supposed constant positive, $\mu$ and $\nu$ are supposed nonnegative, and $f$ is a reaction term of KPP type. Using the notation $\nub=\int\nu,$ we can generalise this to exchanges given by boundary conditions, with $\mu=\mub\d_0$ and $\nu=\nub\d_0.$ Hence, in the same vein as (\[RPeq\]), it is natural to consider the following semi-limit model $$\label{RPSL}
\begin{cases}
\partial_t u-D \partial_{xx} u=-\mub u +\int\nu(y)v(t,x,y)dy & x\in\R,\ t>0\\
\partial_t v-d\Delta v=f(v) -\nu(y)v(t,x,y) & (x,y)\in\R\times\R^*,\ t>0\\
v(t,x,0^+)=v(t,x,0^-), & x\in\R,\ t>0 \\
-d\left\{ \partial_y v(t,x,0^+)-\partial_y v(t,x,0^-) \right\}=\mub u(t,x) & x\in\R,\ t>0
\end{cases}$$ where interactions from the road to the field are local whereas interactions from the field to the road are still nonlocal. We also introduce te symmetrised semi-limit model, where nonlocal interactions are only from the road to the field. $$\label{RPSL2}
\begin{cases}
\partial_t u-D \partial_{xx} u=-\mub u +\nub v(t,x,0) x\in\R,\ t>0\\
\partial_t v-d\Delta v=f(v) +\mu(y)u(t,x) & (x,y)\in\R\times\R^*,\ t>0\\
v(t,x,0^+)=v(t,x,0^-), & x\in\R,\ t>0 \\
-d\left\{ \partial_y v(t,x,0^+)-\partial_y v(t,x,0^-) \right\}=-\nub v(t,x,0) & x\in\R,\ t>0.
\end{cases}$$ All these models are connected with each other, setting the scaling $$\nu_\e(y)= \frac{1}{\e}\nu\lp\frac{y}{\e}\rp,\ \mu_\e(y)= \frac{1}{\e}\mu\lp\frac{y}{\e}\rp.$$ With this scaling, exchanges functions tends to Dirac functions, and integral exchanges tends formally to boundary conditions. For example, the limit $\e\to0$ in (\[RPeq\]) leads to the dynamics of (\[BRReq2\]). This result will be investigating in [@Pauthier2]. A similar study would yield to the same kind of convergence of systems (\[RPSL\]) or (\[RPSL2\]) to (\[BRReq2\]).
Reaction-diffusion equations of the type $$\partial_t u-d\Delta u=f(u)$$ have been introduced in the celebrated articles of Fisher [@fisher] and Kolmogorov, Petrovsky and Piskounov [@KPP] in 1937. The initial motivation came from population genetics. The reaction term are that of a logistic law, whose archetype is $f(u)=u(1-u)$ for the simplest example. In their works in one dimension, Kolmogorov, Petrovsky and Piskounov revealed the existence of propagation waves, together with an asymptotic speed of spreading of the dominating gene, given by $2\sqrt{df'(0)}$. The existence of an asymptotic speed of spreading was generalised in $\R^n$ by D. G. Aronson and H. F. Weinberger in [@AW] (1978). Since these pioneering works, front propagation in reaction-diffusion equations have been widely studied. Let us cite, for instance, the works of Freidlin and Gärtner [@FG] for an extension to periodic media, or [@W2002], [@BHN1] and [@BHN2] for more general domains.
Assumptions
-----------
We always assume that $u_0$ and $v_0$ are nonnegative, bounded and uniformly continuous, with $(u_0,v_0)\not\equiv(0,0)$. Our assumptions on the reaction term are of KPP-type: $$f\in C^1([0,1]), \ f(0)=f(1)=0, \ \forall s\in (0,1),\ 0<f(s)\leq f'(0)s.$$ We extend it to quadratic negative function outside $[0,1].$ Our assumptions on the exchange terms will differ depending on the sections. For the parts concerning the robustness of the results of [@BRR1], that is Proposition \[liouville\] and Theorem \[spreadingthm\], they are the following:
- $\mu$ is supposed to be nonnegative, continuous, and decreasing faster than an exponential function: $\exists M>0,\ a>0$ such that $\forall y\in \R,\ \mu(y)\leq M\exp(-a|y|).$
- $\nu$ is supposed to be nonnegative, continuous and twice integrable, both in $+\infty$ and $-\infty$, id est $$\label{nucond}
\int_0^{+\infty}\int_x^{+\infty}\nu(y)dydx<+\infty, \ \int_{-\infty}^{0}\int_{-\infty}^x\nu(y)dydx<+\infty$$
- We suppose $\mu,\nu \not\equiv 0$, $\nu(0)>0,$ and that both $\nu$ and $\mu$ tend to $0$ as $|y|$ tends to $+\infty.$
For the parts dealing with variations on the spreading speed, we suppose that $\nu$ and $\mu$ are either nonnegative, continuous, compactly supported even functions, either given by a Dirac measure, either the sum of a Dirac measure and a nonnegative, continuous, compactly supported even function.
Results of the paper
--------------------
### Persistence of the results of [@BRR1]
We start with the results that are similar in flavour to those of [@BRR1] concerning the system (\[BRReq2\]) and showing the robustness of the threshold $D=2d$ which was brought out in the paper. The first one concerns the stationary solutions of (\[RPeq\]) and the convergence of the solutions to this equilibrium.
\[liouville\] under the assumptions on $f$, $\nu$, and $\mu$, then:
1. problem (\[RPeq\]) admits a unique positive bounded stationary solution $(U_s,V_s)$, which is x-independent ;
2. for all nonnegative and uniformly continuous initial condition $(u_0,v_0)$, the solution $(u,v)$ of (\[RPeq\]) starting from $(u_0,v_0)$satisfies $\displaystyle
\left(u(t,x),v(t,x,y)\right)\underset{t\to\infty}{\longrightarrow}(U_s,V_s)
$ locally uniformly in $(x,y)\in\R^2$.
The second and main result deals with the spreading in the $x$-direction: we show the existence of an asymptotic speed of spreading $c^*$ such that the following Theorem holds
\[spreadingthm\] Let $(u,v)$ be a solution of (\[RPeq\]) with a nonnegative, compactly supported initial datum $(u_0,v_0)$. Then, pointwise in $y$, we have:
- for all $c>c^*$, $\displaystyle\lim_{t\to\infty}\sup_{|x|\geq ct}(u(x,t),v(x,y,t)) = (0,0)$ ;
- for all $c<c^*$, $\displaystyle\lim_{t\to\infty}\sup_{|x|\leq ct}\lp(u(x,t),v(x,y,t))-\lp U_s,V_s(y)\rp \rp =(0,0) $.
Because $f$ is a KPP-type reaction term, it is natural to look for positive solutions of the linearised system $$\label{RPli}
\begin{cases}
\partial_t u-D \partial_{xx} u = -\mub u+\int \nu(y)v(t,x,y)dy & x \in \R,\ t>0 \\
\partial_t v-d\Delta v = f'(0)v +\mu(y)u(t,x)-\nu(y)v(t,x,y) & (x,y)\in \R^2,\ t>0.
\end{cases}$$ We will construct exponential travelling waves and use them to compute the asymptotic speed of spreading in the $x$-direction. Theorem \[spreadingthm\] relies on the following Proposition:
\[spreading\_speed\]
1. There exists a limiting velocity $c_*$, depending on $D$ and $d$, such that $\forall c>c^*,\ \exists \la>0,\
\exists \phi\in H^1(\R)$ positive such that $\displaystyle
(t,x,y)\mapsto e^{-\la(x-ct)} \begin{pmatrix}
1 \\ \phi(y)
\end{pmatrix}
$ is a solution of (\[RPli\]). No such solution exists if $c<c^*.$
2. If $D\leq 2d$, then $c_*=c_{KPP}=2\sqrt{df'(0)}$. If $D>2d$, then $c_*>c_{KPP}.$
Thesee three results easily extend to the two semi-limit models (\[RPSL\]) and (\[RPSL2\]). We will develop some proofs only for the system (\[RPSL\]), the other being easier.
### Effect of the nonlocal exchanges on the spreading speed
Given all these connected models, a natural question is to understand how different exchange terms influence the propagation. One possible way to see it is to ask if, with similar parameters, some exchange functions give slower or faster spreading speed than other. Our results deal with maximal or locally maximal spreading speed. Throughout the end of the paper, we consider the set of admissible exchange functions from the road to the field for fixed $\mub$ $$\Lambda_{\mub} = \{\mu\in C_0(\R),\mu\geq 0, \int\mu=\mub,\mu \textrm{ is even} \}.$$ Of course, we define $\Lambda_{\nub}$ in a similar fashion. The first result is devoted to the semi-limit case (\[RPSL2\]), where the exchange $\nu$ is a Dirac measure at $y=0$, and $\mu$ is nonlocal. For fixed constants $d,D,\nub,\fp,$ for any function $\mu\in\Lambda_{\mub},$ let $c^*(\mu)$ be the spreading speed associated to the semi-limit system (\[RPSL2\]) with exchange function from the road to the field $\mu.$ Then we have the following property.
\[vitessemaxRPSL2\] Let $c^*_0$ the spreading speed associated with the limit system (\[BRReq2\]) with the same parameters and exchange rate from the road to the field $\mub.$ Then: $$c^*_0=\sup\{c^*(\mu),\ \mu\in\Lambda_{\mub}\}.$$
The second main result is concerned with the other semi-limit case (\[RPSL\]), where the exchange $\mu$ is a Dirac measure, and $\nu$ is nonlocal ; in our study, we consider $\nu$ close to a Dirac measure. Let the exchange term $\nu$ be of the form $$\label{nuperturbe}
\nu(y)=(1-\e)\d_0+\e\up(y)$$ where $$\up\in\Lambda_1:=\{\up\in C_0(\R),\up\geq 0, \int\up=1,\up \textrm{ is even} \}.$$
\[thmdenonacceleration\] For some $\up\in\Lambda_1,$ $\e>0,$ let us consider an exchange function of the form (\[nuperturbe\]). Let $c^*(\nu)$ be the spreading speed associated to (\[RPSL\]) with exchange function $\nu,$ and $c^*_0$ the one associated to (\[BRReq2\]) with same parameters. There exist $m_1>2$ depending on $\fp$, $M_1$ depending on $\mub$ such that:
1. if $D<m_1$ there exist $\e_0$ and $\up\in\Lambda_1$ such that $\forall \e<\e_0,$ $c^*_0<c^*(\nu);$
2. if $\mub>4$ and $D,\fp>M_1$ there exists $\e_0$ such that $\forall \up\in\Lambda_1,$ $\forall \e<\e_0,$ $c^*_0>c^*(\nu).$
Outline and discussion
----------------------
The following section is concerned with the Cauchy problem, stationary solutions and the long time behaviour. Its conclusion is the proof of Proposition \[liouville\]. The third section is devoted to the proof of Proposition \[spreading\_speed\], and we prove Theorem \[spreadingthm\] in the fourth. Our results and methods in these two sections shed a new light on those of [@BRR1] and [@BRR2]. It is striking to find the same condition on $D$ and $d$ for the enhancement of the spreading in one direction. The stationary solutions are nontrivial and more complicated to bring out. The computation of the spreading speed $c^*$ comes from a nonlinear spectral problem, and not from an algebraic system which could be solved explicitly. It also involves some tricky arguments of differential equations.
In the fifth section, we investigate the semi-limit model (\[RPSL\]). This underlines the robustness of the method for this kind of system.
We study in the sixth section the asymptotics $D\to+\infty$ in all cases, which has already been done for the initial model in [@BRR1]. Such an asymptotics for a nonlinearity has also been studied by L. Dietrich in [@Dietrich2].
We prove Proposition \[vitessemaxRPSL2\] in the seventh section. We show that in the semi-limit case (\[RPSL2\]), the spreading speed is maximal for a concentrate exchange term, that is for the initial limit system (\[BRReq2\]). Such a result may be linked to the case of a periodic framework found in [@matano].
It could be expected a similar result in the other semi-limit case (\[RPSL\]). We prove by two different ways that it is not true. We first investigate the case of a self-similar approximation of a Dirac measure for the nonlocal exchange $\nu.$ For these kind of exchange functions, the Dirac measure is a local minimizer for the spreading speed. This is the purpose of the eighth section.
Considering that, a natural guess would be that in the semi-limit case (\[RPSL\]) the Dirac measure is a local minimizer anyway. Once again, this is not true. This is the purpose of the last section: we prove that any behaviour may happen in a neighbourhood of concentrate exchange term. More precisely, we prove in Theorem \[thmdenonacceleration\] that if $c^*_0$ is the spreading speed associated to the limit system (\[BRReq2\]), considering a perturbated exchange function of the form $\nu=(1-\e)\d_0+\e\up,$ that is mainly boundary conditions with a small integral contribution, then
- for some ranges of parameters $D,\mub,\fp,$ in the neighbourhood of $\e=0,$ the maximal speed is $c^*_0;$
- for other ranges of these parameters and some integral exchange $\up,$ a perturbation as above enhances the spreading for $\e$ small enough.
Such a difference between self-similar approximations and general approximations of a Dirac measure may be surprising, but a phenomenon of the same kind has already been observed by L. Glangetas in [@Glangetas_92] in a totally different context. We can also notice that these results underline how different are the influences of the two exchange functions.
#### Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi -Reaction-Diffusion Equations, Propagation and Modelling. I am grateful to Henri Berestycki and Jean-Michel Roquejoffre for suggesting me the model and many fruitful conversations. Part of the work was initiated by a relevant question pointed out by Grégoire Nadin, whom I thank for it. I also would like to thank the anonymous referees for their helpful comments.
Stationary solutions and long time behaviour
============================================
In this section, we are concerned with the well-posedness of the system (\[RPeq\]) combined with the initial condition $$\label{initial}
\begin{cases}
u|_{t=0}=u_0 & \in \R \\
v|_{t=0}=v_0 & \in \R^2.
\end{cases}$$
Existence, uniqueness and comparison principle
----------------------------------------------
The system (\[RPeq\]) is standard, in the sense that the coupling does not appear in the diffusion nor the reaction term. Anyway, well-posedness still has to be mentioned.
\[Cauchy\] Under the above assumptions on $f$, $\mu$ and $\nu$, the Cauchy problem (\[RPeq\])-(\[initial\]) admits a unique nonnegative bounded solution.
Using the formalism of [@henry], it is easy to show that the linear part on (\[RPeq\]) defines a sectorial operator, and that the non-linear is globally Lipschitz on $X:=\Cu(\R)\times\Cu(\R^2)$, which gives the existence and uniqueness of the solution of (\[RPeq\]). We can also derive the uniqueness of the solution of (\[RPeq\]) by showing that comparison between subsolutions and supersolutions is preserved during the evolution. Moreover, the following property will also be the key point in our later study of the spreading. Throughout this article, we will call a subsolution (resp. a supersolution) a couple satisfying the system (in the classical sense) with the equal signs replaced by $\leq$ (resp. $\geq$) signs, which is also continuous up to time 0.
\[comparaison\] Let $(\su,\sv)$ and $(\us,\vs)$ be respectively a subsolution bounded from above and a supersolution bounded from below of (\[RPeq\]) satisfying $\su\leq\us$ and $\sv\leq\vs$ at $t=0$. Then, either $\su<\us$ and $\sv<\vs$ for all $t>0$, or there exists $T>0$ such that $(\su,\sv)=(\us,\vs),\ \forall t\leq T.$
Once again, the proof is quite classical and omitted here. This comparison principle extend immediately to generalised sub and supersolutions given by the supremum of subsolutions and the infimum of supersolutions. For our spreading result, we will need a more general class of subsolutions, already used for several results in this context. See for instance Proposition 3.3 in [@BRR1].
Long time behaviour and stationary solutions
--------------------------------------------
The main purpose of this section is to prove that any (nonnegative) solution of (\[RPeq\]) converges locally uniformly to a unique stationary solution $(U_s,V_s)$, which is bounded, positive, $x$-independent, and solution of the stationary system of equations (\[stateq\]): $$\label{stateq}
\begin{cases}
-DU''(x) & = -\mub U(x)+\int \nu(y)V(x,y)dy \\
-d\Delta V(x,y)& = f(V)+\mu(y)U(x)-\nu(y)V(x,y).
\end{cases}$$ In the same way as above, we call a subsolution (resp. a supersolution) of (\[stateq\]) a couple satisfying the system (in the classical sense) with the equal signs replaced by $\leq$ (resp. $\geq$).
\[stationary1\] Let $(u,v)$ be the solution of (\[RPeq\]) starting from $(u_0,v_0)\not\equiv(0,0)$. then there exist two positive, bounded, x-independent, stationary solutions $(U_1,V_1)$ and $(U_2,V_2)$ such that $$U_1\leq \underset{t\to +\infty}{\liminf}\ u(x,t) \leq \underset{t\to +\infty}{\limsup}\ u(x,t)\leq U_2,$$ $$V_1(y)\leq \underset{t\to +\infty}{\liminf}\ v(x,y,t) \leq \underset{t\to +\infty}{\limsup}\ v(x,y,t)\leq V_2(y),$$ locally uniformly in $(x,y)\in \R^2$.
The proof is adapted from [@BRR2]. We first need a $L^\infty$ a priori estimate.
#### A priori estimate
Considering the hypothesis on the reaction term $f$, there exists $K>0$ such that $$\forall s\geq K,\ f(s)\leq s(\frac{\nub}{\mub}\mu(y)-\nu(y)),\ \forall y\in \R.$$ Thus, for all constant $V\geq K$, $V(\frac{\nub}{\mub},1)$ is a supersolution of (\[RPeq\]).
#### Construction of $(U_1,V_1)$
Let $R>0$ large enough in such a way that the first eigenvalue of the Laplace operator with Dirichlet boundary condition in $B_R\subset \R^2$ is less than $\frac{f'(0)}{3d}$, $\phi_R$ the associated eigenfunction. We extend $\phi_R$ to 0 outside $B_R$. $\phi_R$ is continuous, bounded, and satisfies $$-d\Delta\phi_R\leq\frac{1}{3}f'(0)\phi_R \ \textrm{in}\ \R^2.$$ Let us choose $\e>0$ such that if $0<x\leq \e,\ f(x)>\frac{2}{3}f'(0)x$. Then define $M>R$ such that $\forall y \ /\ |y|>M-R,\ \nu(y)\leq\frac{1}{3}f'(0)$. Since $(u_0,v_0)\not\equiv(0,0)$ and $(0,0)$ is a solution, the comparison principle implies that $u,v>0,\ \forall t>0.$ Now, let us define $\eta$ such that $\eta\phi_R(x,|y|-M)<v(x,y,1)$ and $\eta \|\phi_R\|_{\infty}\leq\e.$ Define $\underline{V}(x,y):=\eta\phi_R(x,|y|-M)$, and, up to a smaller $\eta,$ $(0,\underline{V})$ is a subsolution of (\[RPeq\]) which is strictly below $(u,v)$ at $t=1$. Let $(u_1,v_1)$ be the solution of (\[RPeq\]) starting from $(0,\underline{V})$ at $t=1$; $(u_1,v_1)$ is strictly increasing in time, bounded by $K(\frac{\nub}{\mub},1)$, and converges to a positive stationary solution $(U_1,V_1)$, satisfying $$U_1\leq \underset{t\to +\infty}{\liminf}\ u \qquad V_1\leq \underset{t\to +\infty}{\liminf}\ v$$ locally uniformly in $(x,y)\in\R^2$.
It remains to show that $(U_1,V_1)$ is invariant in $x$. For $h \in \R$, let us denote $\tau_h$ the translation by $h$ in the x-direction: $\tau_h w(x,y)=w(x+h,y)$. Since $\underline{V}$ is compactly supported, there exists $\e>0$ such that $$\forall h\in (-\e,\e),\ \tau_h\underline{V}<V_1 \textrm{ and } \tau_h\underline{V}<v \textrm{ at } t=1.$$ Thus, because of the $x$-invariance of the system (\[RPeq\]), the solution $(\ut_1,\vt_1)$ of (\[RPeq\]) starting from $(0,\tau_h\underline{V})$ at $t=1$ is equal to the translated $(\tau_h u_1, \tau_h v_1)$. So, $(\ut_1,\vt_1)$ converges to $(\tau_h U_1,\tau_h V_1)$. But, since $(\ut_1,\vt_1)$ is below $(U_1,V_1)$ at $t=1$ and $(U_1,V_1)$ is a (stationary) solution, from the comparison principle given by Proposition \[comparaison\] we deduce $(\ut_1,\vt_1)<(U_1,V_1),\ \forall t >1$, and then $$(\tau_h U_1,\tau_h V_1)\leq (U_1,V_1),\ \forall h\in (-\e,\e).$$ Namely, $(U_1,V_1)$ does not depend on $x$.
#### Construction of $(U_2,V_2)$
Let $\overline{V}=\max (\|v_0\|_\infty,K)$ and $\overline{U}=\max (\|u_0\|_\infty,\overline{V}\frac{\nub}{\mub})$. Let $(u_2,v_2)$ be the solution of (\[RPeq\]) with initial datum $(\overline{U},\overline{V})$. From the comparison principle (\[comparaison\]), $(u,v)$ is strictly below $(u_2,v_2)$, for all $t>0,\ (x,y)\in\R^2$. Moreover, since $(\overline{U},\overline{V})$ is a supersolution of (\[RPeq\]) it is clear that $\partial_t u_2,\ \partial_t v_2 \leq 0$ at $t=0$. Still using Proposition \[comparaison\], it is true for all $t\geq 0$, and $u_2$ and $v_2$ are nonincreasing in $t$, bounded from below by $(U_1,V_1)$. Thus, $(u_2,v_2)$ converges as $t\to \infty$ to a stationary solution $(U_2,V_2)$ of (\[RPeq\]) satisfying $$\underset{t\to +\infty}{\limsup}\ u(t,x)\leq U_2 \qquad \underset{t\to +\infty}{\limsup}\ v(t,x,y)\leq V_2(y),$$ locally uniformly in $(x,y)\in\R^2$. From the construction of $(U_2,V_2)$, which is totally independent of the $x$-variable, it is easy to see that $(U_2,V_2)$ does not depend in $x$.
#### Uniqueness of the stationary solution
The previous proposition provides a theoretical proof of the existence of stationary solutions. It also means that a solution is either converging to a stationary solution, or will remain between two stationary solutions. In order to obtain a more precise description of the long time behaviour, we need the following uniqueness result.
\[stationary2\] There is a unique positive, bounded, stationary solution of (\[RPeq\]), denoted $(U_s,V_s)$.
To prove the uniqueness, we first need the following intermediate lemma which is the key to all uniqueness properties in this kind of problem. The idea that a bound from below implies uniqueness appeared for the first time in [@BHR].
\[stationary3\] Let $(U,V)$ be a positive, bounded stationary solution of (\[RPeq\]). Then there exists $m>0$ such that $$\forall (x,y)\in \R^2,\ U(x)\geq m,\ V(x,y)\geq m.$$
Let $(U,V)$ be such a stationary solution.
*First step*: there exists $M>0$ such that $$m_1:=\inf \{V(x,y),\ |y|>M\}>0.$$ We will state the proof for positie $y.$ Let $R>0$ large enough in such a way that the first eigenvalue of the Laplace operator with Dirichlet boundary condition in $B_R\subset \R^2$ is less than $\frac{f'(0)}{3d}$, $\phi_R$ the associated eigenfunction. We extend $\phi_R$ to 0 outside $B_R$. $\phi_R$ is continuous, bounded in $\R^2$, positive in $B_R$. For $M>0,$ we define $\tau_M\phi_R(x,y)=\phi_R(x,y-M).$ As above, let us define $M_0>R$ such that $\forall y \ /\ |y|>M_0-R,\ \nu(y)\leq\frac{1}{3}f'(0).$ Then, there exists $\e>0$ such that $\forall M>M_0,$ $\lp 0,\e\tau_M\phi_R\rp$ is a subsolution of (\[stateq\]). As $V$ is positive, up to smaller $\e,$ we can suppose that $\e\tau_{M_0}\phi_R<V.$ Now, we claim that $$\forall y>M_0,\ V(0,y)>\e\phi_R(0,0).$$ Indeed, let us define $$M_1:=\sup \{M\geq M_0,\ \forall K\in [M_0,M],\ \e\tau_K\phi_R<V\}.$$ Since $V$ and $\phi_R$ are continuous, $M_1>M_0.$ Suppose that $M_1<+\infty.$ Then $\lp U,V\rp\geq \lp0,\e\tau_{M_1}\phi_R\rp$ and there exists $(x_0,y_0),$ $V(x_0,y_0)=\e\tau_{M_1}\phi_R(x_0,y_0).$ Considering that the dynamical system starting from $(0,\e\tau_{M_1}\phi_R),$ which is a subsolution, we get a contradiction from Proposition \[comparaison\]. Hence $M_1=+\infty$ and our claim is proved. Using the same argument in the $x$-direction, we get that $m_1\geq \e\phi_R(0,0).$
*Second step*: $$m_2:=\inf \{V(x,y),\ (x,y)\in\R^2\}>0.$$ If $m_2=m_1$, the assumption is proved. It is obvious that $m_2\geq0.$ Let us assume by way of contradiction that $m_2=0.$ We consider $(x_n,y_n)$ such that $V(x_n,y_n)\to 0$ with $n\to \infty$. Now, we set $$U_n:=U(.+x_n),\ V_n:=V(.+x_n,.+y_n),\ \mu_n:=\mu(.+y_n),\ \nu_n:=\nu(.+y_n).$$ Using the fact that $U$ and $V$ are smooth and bounded, by standard elliptic estimates (see [@GT] for example), there exists $\vp:\N\to\N$ strictly increasing such that $(U_{\vp(n)})_n,\ (V_{\vp(n)})_n$ converge locally uniformly to some functions $\tilde{U},\ \tilde{V}$ satisfying $$\begin{cases}
-D\tilde{U}''(x) & = -\mub \tilde{U}(x)+\int \tilde{\nu}(y)\tilde{V}(x,y)dy \\
-d\Delta \tilde{V}(x,y) & = f(\tilde{V})+\tilde{\mu}(y)\tilde{U}(x)-\tilde{\nu}(y)\tilde{V}(x,y)
\end{cases}$$ where $\tilde{\mu},\ \tilde{\nu}$ are some translated of $\mu,\ \nu$. Furthermore, $\tilde{V}\geq 0$ and $\tilde{V}(0,0)=0$. Thus in a neighbourhood of $(0,0)$ we have $$-d\Delta \tilde{V}(x,y)+\tilde{\nu}(y)\tilde{V}(x,y)\geq 0,\ \min(\tilde{V})=0.$$ From the strong elliptic maximum principle, we deduce $\tilde{V}\equiv 0.$ But by step 1 $\tilde{V}(.,2M)\geq m_1>0$, and we get a contradiction. Hence the result stated above, $m_2:=\inf(V)>0.$
Third step: $U$ is also bounded from below by a positive constant. Indeed, if we set $\phi(x)=\frac{1}{D}\int \nu(y)V(x,y)dy$, $U$ is solution of $$\label{eqU}
-U''+\frac{\mub}{D}U=\phi,$$ with $\phi$ continuous and $\phi\geq m_2\|\nu\|_{L^1}$. Using $\Phi(x)=\frac{D}{2\mub}\exp(-\sqrt{\frac{\mub}{D}}|x|)$ which is the fundamental solution of (\[eqU\]) we get $$U(x)=\phi*\Phi(x)\geq \|\Phi\|_{L^1}.m_2.\|\nu\|_{L^1}:=m_3>0.$$ Now, set $m=\inf(m_1,m_2,m_3)$ and the proof is concluded.
#### Proof of proposition \[stationary2\]
It remains now to prove the uniqueness of the stationary solution of (\[RPeq\]). The difficulties come from the fact that it is a coupled system in an unbounded domain: for bounded domains, uniqueness was proved in [@Berestycki1]. Let $(U_1,V_1)$, $(U_2,V_2)$ be two bounded, positive solutions of (\[stateq\]), and let us show that $(U_1,V_1)=(U_2,V_2).$ From Lemma \[stationary3\], there exists $m>0$ such that $(U_i,V_i)\geq m,\ i=1..2.$ Hence, for $T$ large enough, $T(U_1,V_1)>(U_2,V_2).$ Let $$T_1=\inf\{T,\ \forall T'>T,\ T'(U_1,V_1)>(U_2,V_2)\}>0,$$ and $$(\d U,\d V)=T_1(U_1,V_1)-(U_2,V_2).$$ Up to take $T_1(U_2,V_2)-(U_1,V_1)$ if needed, we can suppose $T_1\geq1.$ The couple $(\d U,\d V)$ satisfies the following system: $$\begin{cases}
-D\d U''(x) & = -\mub \d U(x)+\int \nu(y)\d V(x,y)dy \\
-d\Delta \d V(x,y)& = T_1 f(V_1)-f(V_2)+\mu(y)\d U(x)-\nu(y)\d V(x,y)
\end{cases}$$ and $\inf(\d U)=0\ \underline{\textrm{or}} \ \inf(\d V)=0.$ In order to show that $(\d U,\d V)\equiv 0$ we have to distinguish five cases.
Case 1: there exists $(x_0,y_0)\in\R^2,\ \d V(x_0,y_0)=0.$ Then, using the fact that $f(0)=0$ and that $f$ is strictly concave, we can easily check that $T_1f(V_1)-f(V_2)\geq 0$ in a neighbourhood of $(x_0,y_0).$ Thus, because $\d U\geq 0$, $\d V$ is solution of the inequality system $$\begin{cases}
-d\Delta \d V +\nu \d V & \geq 0 \\
\d V \geq 0, & \d V(x_0,y_0)=0.
\end{cases}$$ From the elliptic maximum principle, we infer $\d V\equiv 0$. Because $\mu\not\equiv 0,$ we immediately get $\d U\equiv 0.$ So $(U_2,V_2)=T_1(U_1,V_1)$ ; subtracting the two systems (\[stateq\]) in $(U_1,V_1)$ and $T_1(U_1,V_1)$ yields $T_1 f(V_1)=f(V_1)$ and $V_1>0$. So $T_1=1$, and $(U_2,V_2)=(U_1,V_1)$.
Case 2: there exists $x_0$ such that $\d U(x_0)=0.$ In the same way we infer $\d U\equiv 0$. Then, $\forall x\in\R,\ \int\nu\d V=0.$ In particular, there exists $y_0$ such that $\d V(x_0,y_0)=0$, and the problem is reduced to the (solved) first case: $T_1=1$, and $(U_2,V_2)=(U_1,V_1)$.
Case 3: there is a contact point for $U$ at infinite distance. Formally, there exists $(x_n)_n,\ |x_n|\to\infty$ such that $\d U(x_n)\to 0$ with $n\to\infty.$ We set $$U_i^n:=U_i(.+x_n),\ V_i^n:=V_i(.+x_n,.),\ i=1,2.$$ In the same way as above, there exist $\tilde{U}_i,\ \tilde{V}_i$ such that, up to a subsequence, $(U_i^n,V_i^n)$ converges locally uniformly to $(\tilde{U}_i,\ \tilde{V}_i)$, and the couples $(\tilde{U}_1,\tilde{V}_1)$ and $(\tilde{U}_2,\tilde{V}_2)$ both satisfy (\[stateq\]) and $$\begin{cases}
T_1=\inf\{T,\ \forall T'>T,\ T'(\tilde{U}_1,\tilde{V}_1)>(\tilde{U}_2,\tilde{V}_2)\}, \\
(T_1 \tilde{U}_1-\tilde{U}_2)(0)=0.
\end{cases}$$ The problem is once again reduced to the first case, and $T_1=1.$
Case 4: there is a contact point for $V$ at infinite distance in $x$, finite distance in $y$, say $y_0$. We use the same trick as above, the limit problem is this time reduced to the second case, and we still get $T_1=1.$
Case 5: there is a contact point for $V$ at infinite distance in $y$. That is to say there exist $(x_n)_n,\ (y_n)_n$, with $|y_n|\to \infty$ such that $\d V (x_n,y_n)\underset{n\to\infty}{\longrightarrow}0.$ Once again, we set $$V_i^n:=V_i(.+x_n,.+y_n),\ i=1,2.$$ Now, considering that $U_1,\ U_2$ are bounded and that $\mu,\nu\underset{|y|\to\infty}{\longrightarrow}0$, $(V_1^n)_n,\ (V_2^n)_n$ converge locally uniformly to some functions $\tilde{V}_1,\ \tilde{V}_2$ which satisfy $$\begin{cases}
-d\Delta \tilde{V}_i & = f(\tilde{V}_i) \\
(T_1 \tilde{V}_1-\tilde{V}_2)(0,0) & = 0
\end{cases}$$ and $(T_1 \tilde{V}_1-\tilde{V}_2) \geq 0$ in a neighbourhood of $(0,0)$. Thus, using the concavity of $f$ as in the first case, we get $T_1=1.$
From the five cases considered above, whatever may happen, $T_1 = 1,$ and the proof is complete.
The proof of Proposition \[liouville\] is now a consequence of Propositions \[stationary1\] and \[stationary2\].
Exponential solutions of the linearised system
==============================================
Looking for supersolution of the system (\[RPeq\]) lead us to search positive solutions of the linearised system (\[RPli\]), hence we are looking for solutions of the form: $$\label{solexp}
\begin{pmatrix}
u(x,t) \\
v(x,y,t)
\end{pmatrix}
= e^{-\la(x-ct)} \begin{pmatrix}
1 \\ \phi(y)
\end{pmatrix},$$ where $\la, c$ are positive constants, and $\phi$ is a nonnegative function in $H^1(\R)$. The system on $(\la, \phi)$ reads: $$\label{eq1}
\begin{cases}
-D\la^2+\la c+\mub = \int \nu(y)\phi(y)dy \\
-d\phi''(y)+(\la c-d\la^2-f'(0)+\nu(y))\phi(y) = \mu(y).
\end{cases}$$
The first equation of (\[eq1\]) gives the graph of a function $\la\mapsto\Psi_1(\la,c):= -D\la^2+\la c+\mub$, which, if (\[solexp\]) is a solution of (\[RPli\]), is equal to $\int \nu(y)\phi(y)dy$.\
The second equation of (\[eq1\]) gives, under some assumptions on $\la$, a unique solution $\phi=\phi(y;\la,c)$ in $H^1(\R)$. To this unique solution we associate the function $\Psi_2(\la,c):=\int \nu(y)\phi(y)dy$. Let us denote $\Gamma_1$ the graph of $\Psi_1$ in the $(\la, \Psi_1(\la))$ plane, and $\Gamma_2$ the graph of $\Psi_2$. So, (\[eq1\]) amounts to the investigation of $\la,\ c>0$ such that $\Gamma_1$ and $\Gamma_2$ intersect.
The graph of $\la \mapsto \Psi_1(\la)$ is a parabola. As we are looking for a nonnegative function $\phi$, we are interested in the positive part of the graph. The function $\la\mapsto\Psi_1(\la)$ is nonnegative for $\la \in [\la_1^-(c) , \la_1^+(c)]$, with $\la_1^\mp(c) = \frac{c\mp\sqrt{c^2+4D\mub}}{2D}$.\
It reaches its maximum value in $\la=\frac{c}{2D}$, with $\Psi_1(\frac{c}{2D})=\mub+\frac{c^2}{4D} > \mub$.\
We also have $$\Psi_1(0)=\Psi_1(\frac{C}{D})=\mub,$$ which will be quite important later.\
We may observe that: with $D$ fixed, $(\la_1^-(c) , \la_1^+(c)) \underset{c\to +\infty}{\longrightarrow} (0^-, +\infty)$; $\la \mapsto \Psi_1(\la)$ is strictly concave; $\displaystyle\frac{d\Psi_1}{d\la}_{|\la=c/D}=-c$.We can summarize it in fig. (\[parabole\]).
Study of $\Psi_2$
-----------------
The study of $\Psi_2$ relies on the investigation of the solution $\phi=\phi(\la;c)$ of $$\label{eqgeneralesurphi}
\begin{cases}
-d\phi''(y)+(\la c-d\la^2-f'(0)+\nu(y))\phi(y) = \mu(y) \\
\phi \in H^1(\R) \ \phi\geq0.
\end{cases}$$ Since $\mu$ is continuous and decays no slower than an exponential, $\mu$ belongs to $L^2(\R)$. Since $\nu$ is nonnegative and bounded, the Lax-Milgram theorem assures us that (\[eqgeneralesurphi\]) admits a unique solution if $\la c-d\la^2-f'(0) >0$, that is to say if $\la$ belongs to $]\la_2^-(c),\la_2^+(c)[$, where $$\la_2^\mp(c) = \frac{c\mp\sqrt{c^2-c_{KPP}^2}}{2d},$$ with $$c_{KPP}=2\sqrt{df'(0)}.$$ As in [@BRR1], the KPP-asymptotic spreading speed will have a certain importance in the study of the spreading in our model. Moreover, since $\nu, \mu$ tend to $0$ with $|y|\to\infty$, an easy computation will show that, for $\la<\la_2^-$ or $\la>\la_2^+$, equation (\[eqgeneralesurphi\]) cannot have a constant sign solution. Morever, we look for $H^1$ solutions. We will see in Lemma \[lemhomo\] that it prevents the existence of a solution for $c=c_{KPP}.$ Thus, $$\label{existencegamma2}
\Gamma_2 \textrm{ exists if and only if } c>c_{KPP}.$$
The main properties of $\Psi_2$ are the following:
\[sourire\] If $c>c_{KPP}$, then:
1. $\la \mapsto \Psi_2(\la)$ defined on $]\la_2^-,\la_2^+[$ is positive, smooth, strictly convex and symmetric with respect to the line $\{\la=\frac{c}{2d}\}$. With $\la$ fixed we also have $\frac{d}{dc}\Psi_2(\la;c)<0$.
2. $\Psi_2(\la) \underset{\la\to \la_2^\mp}{\longrightarrow} \mub$.
3. $\frac{d\Psi_2}{d\la}(\la)\underset{\underset{\la>\la_2^-}{\la\to \la_2^-}}{\longrightarrow} -\infty$.
The graph $\Gamma_2$ looks like fig. (\[Sourire\]).
[**Proof of the first part of proposition (\[sourire\])**]{}
#### Positivity, smoothness
For all $\la$ in $]\la_2^-,\la_2^+[$, $$P(\la):=\la c-d\la^2-f'(0) >0.$$ Consequently, $\forall \la \in ]\la_2^-,\la_2^+[, \ \forall y\in \R,\ P(\la)+\nu(y)>0$. From the elliptic maximum principle, as $\mu$ is nonnegative, we deduce that $\phi(y)>0,\ \forall y \in \R$. Hence, since $\nu$ is nonnegative, we have $\Psi_2(\la)= \int \phi(y;\la)\nu(y)dy >0$, and $\Psi_2$ is positive.\
Considering that $\la\mapsto P(\la)$ is polynomial, with the analytic implicit function theorem, we see immediately that $\la\mapsto \phi(y;\la)$ is analtytic (see [@Cartan], Theorem 3.7.1). Since $\nu$ is integrable, $\la\mapsto\Psi_2(\la)$ is also analytic.\
From the symmetry of $\la\mapsto P(\la)$ and the uniqueness of the solution, we deduce the symmetry of $\Gamma_2$ with respect to the line $\{\la=\frac{c}{2d}\}$.
#### Monotonicity, convexity
Denote by $\pl$ the derivative of $\phi$ with respect to $\la$. Then, if we differentiate (\[eqgeneralesurphi\]) with respect to $\la$, we can see that $\pl$ satisfies: $$\label{eqpl}
-d\pl''(y)+(P(\la)+\nu(y))\pl(y) = (2d\la-c)\phi(y).$$ In the same way as equation (\[eqgeneralesurphi\]), (\[eqpl\]) has a unique solution in $H^1(\R)$ for all $\la \in ]\la_2^-,\la_2^+[$. Since $\phi$ is positive, $\pl$ is of constant sign, with the sign of $(2d\la-c)$. Hence we have that $\Psi_2$ is decreasing on $]\la_2^-,\frac{c}{2d}[$ and increasing on $]\frac{c}{2d},\la_2^+[$.\
Differentiating once again (\[eqpl\]) with respect to $\la$, the second derivative of $\phi$ with respect to $\la$ satisfies: $$\label{eqpll}
-d\pll''(y)+(P(\la)+\nu(y))\pll(y) = 2d\phi(y)+2(2d\la-c)\pl(y).$$ In the same way, $\phi$ is positive for all $\la\in ]\la_2^-,\la_2^+[$, and $\pl(\la)$ has the positivity of $(2d\la-c)$. Hence the left term of equation (\[eqpll\]) is positive, for all $\la\in ]\la_2^-,\la_2^+[$, and $\Psi_2$ is strictly convex on $]\la_2^-,\la_2^+[$.\
With the same arguments we see that $\phi_c$, the derivative of $\phi$ with respect to $c$, satisfies $$-d\phi_c''+(P(\la)+\nu)\phi_c = -\la\phi <0,$$ and then we get $\int_{\R}\phi_c(y) \nu(y)dy = \frac{d}{dc}\Psi_2(\la;c)<0.$
In order to end the proof of the proposition (\[sourire\]), we need to study behaviour of $\Psi_2$ near $\la_2^-.$ Setting $\e=P(\la),$ it is sufficient to study the behaviour of the solution $\phi=\phi(y;\e)$ of $$\label{epseq1}
\begin{cases}
-\phi''(y)+(\e+\nu(y))\phi(y)=\mu(y) \\
\phi\in H^1(\R),\ \e>0,\ \e\to 0.
\end{cases}$$ The main lemma here is the following, which will evidently conclude Proposition \[sourire\]:
\[lemeps\]
1. If $\phi$ is solution of (\[epseq1\]) then $\int_{\R}\phi(y)\nu(y)dy
\underset{\underset{\e>0}{\e\to 0}}{\longrightarrow} \mub$ holds true. Moreover, $\|\phi\|_{L^{\infty}}$ is uniformly bounded on $\e$.
2. The derivative of $\phi$ with respect to $\e$, denoted $\phi_{\e}$, satisfies $\int_{\R}\phi_{\e}(y)\nu(y)dy
\underset{\underset{\e>0}{\e\to 0}}{\longrightarrow} -\infty. $
#### Proof of the first part of the Lemma \[lemeps\]
An explcit computation is needed. We use a boxcar function for this. Under the assumptions on $\nu$ and $\mu$, there exist $\a,\ M,\ m_1>0$ such that:
- $\nu(y) \geq \a \mathbf{1}_{[-m_1,m_1]},\ \forall y\in \R$ (because $\nu(0)>0$, and $\nu$ is continuous);
- $\mu(y)\leq M e^{-a|y|},\ \forall y\in \R$ (from the exponential decay of $\mu$).
Denoting $\psi=\psi(y;\e)$ the solution of $$\label{eqpsi1}
-\psi''+(\e+\a \mathbf{1}_{[-m_1,m_1]})\psi=M e^{-a|y|},$$ $\psi$ is a supersolution for (\[epseq1\]) and $$\label{majphipsi1}
\forall \e >0,\ \forall y\in \R,\ 0<\phi(y;\e)\leq\psi(y;\e).$$ We have already seen that $\forall \e>0,\ \int_{\R}\phi''(y;\e)dy=0$. Consequently, the assumption $\int_{\R}\phi(y)\nu(y)dy\underset{\e\to 0}{\longrightarrow} \mub$ is equivalent to $\e\int_{\R}\phi(y;\e)dy\underset{\e\to 0}{\longrightarrow} 0$. To conclude, it remains to compute the solution $\psi$ and to show that $\e\int_{\R}\psi(y;\e)dy\underset{\e\to 0}{\longrightarrow} 0$. But the solution of (\[eqpsi1\]) can be explicitly computed, which gives that $\|\phi(\e)\|_{L^{\infty}(\R)}$ is uniformly bounded on $\e$ and that ther exists $C>0$ such that for $\e>0$ small and $y>m_1$, $$\psi(y;\e)< C e^{-\sqrt{\e}y},$$ so $$\int_\R \psi(y;\e)dy = O(\frac{1}{\sqrt{\e}}) \textrm{ as } \e\to 0$$ and $$\e\int_\R \psi(y;\e)dy \underset{\e\to 0}{\longrightarrow}0,$$ which concludes the proof of the first statement in Lemma \[lemeps\]. Notice that we also get that there exist two constant $C_1,C_2$ not depending on $\e$ such that for all $y$ in $\R$, $\psi(y;\e)\leq C_1 e^{-\sqrt{\e}|y|}+C_2 e^{-a|y|},$ that will be useful later.
Let us prove the second part of Lemma (\[lemeps\]). In order to prove it, we will first deal with the study of the homogeneous limit differential equation.
\[lemhomo\] Let us consider the scalar homogeneous equation (\[homogeneous\]): $$\label{homogeneous}
-\psi''+\nu.\psi=0.$$ Under the assumptions on $\nu$, there exist $\phi_1$, $\phi_2$ satisfying
- $\phi_1(x)\underset{x\to+\infty}{\longrightarrow}0$, and, for x large enough, $\phi_1(x)\geq0$ ;
- $\exists C_1,C_2>0$ such that $C_1x \leq \phi_2(x)\leq C_2x$ when $x$ goes to $+\infty$ (notation: $\phi_2(x)=\varTheta(x)$ ) ;
such that $$\begin{cases}
\psi_1:=1+\phi_1 \\
\psi_2:=\phi_2(1+\phi_1)
\end{cases}$$ is a fundamental system of solutions of (\[homogeneous\]).
[**Construction of $\phi_1$**]{}: let $\psi:=1+\phi_1$ be a solution of (\[homogeneous\]). Thus, $\phi_1$ must satisfy $$\label{homogeneous2}
-\phi_1''+\nu+\nu.\phi_1 = 0.$$ Let us show that there exists a solution of (\[homogeneous2\]) which is nonnegative for $x$ large enough and tends to $0$ as $x$ goes to $+\infty$. Let $M\geq0$ such that $\int_M^{\infty}\int_x^{\infty}\nu(y)dydx<1$ which is possible thanks to the assumption (\[nucond\]) on $\nu$. Now, define $$\mathcal{E}:=\{\phi\in C([M,+\infty[)/ \forall x\geq M,\phi(x)\geq0 \textrm{ and }\phi(x)\underset{x\to\infty}{\longrightarrow}0 \}$$ and $$F\: \ \begin{cases}
\mathcal{E} \to & \mathcal{E} \\
\phi \mapsto & F\phi: x\mapsto \int_x^{\infty}\int_y^{\infty}(1+\phi(z))\nu(z)dzdy.
\end{cases}$$ From the hypothesis on $\mathcal{E}$ and $\nu,$ $F$ is well defined. $\mathcal{E}$ is a closed subset of the Banach space $C_0([M,\infty[)$. The choice of $M$ implies that $F$ is a contraction. From a classical Banach fixed point argument, there exists a unique positive solution $\phi_1$ in $C([M,+\infty[)$ of $\ref{homogeneous2}$ satisfying $\phi(x)\underset{x\to+\infty}{\longrightarrow}0.$
Moreover, without loss of generality, we can only consider the case $M=0$.
[**Construction of $\phi_2$**]{}: we are looking for a second solution of (\[homogeneous\]) in the form $\psi_2=\phi_2.\psi_1$. Integrating the equation we get for $x\geq 0$: $$\phi_2(x)=\int_0^x \frac{dy}{(1+\phi_1(y))^2},$$ and $\psi_2:=\phi_2(1+\phi_1)$ is a second solution of the homogeneous equation (\[homogeneous\]). Finally, considering that $\phi_1(x)\to 0$ with $x\to +\infty$, we get the desired estimate for $\phi_2$.
Of course, we have a similar result for $x\to-\infty.$ This lemma first allows us to give a useful lower bound of $\phi(y;\e)$ at the limit $\e=0.$
\[minoration\] Let $\phi=\phi(y;\e)$ be the solution of (\[epseq1\]). There exists $k>0$ such that, $\forall y\in\R$, $\exists \e_y,\ \forall\e<\e_y$, $\phi(y;\e)\geq k,$ and this uniformly on every compact set in $y$.
Since $\mu\not\equiv 0$ there exists a nonnegative compactly supported function $\mu_c\not\equiv 0$ such that $0\leq\mu_c \leq \mu.$ Let us now consider the (unique) solution $\phis=\phis(y;\e)$ of $$\label{epseq2}
\begin{cases}
-\phis''(y)+(\e+\nu(y))\phis(y)=\mu_c(y) \\
\phis\in H^1(\R),\ \e>0.
\end{cases}$$ From the first part of Lemma \[lemeps\], we know that $\exists K>0,\forall y\in\R,\forall \e> 0$, $0<\phis(y;\e)\leq \phi(y;\e)<K.$ Let us recall that for fixed $y\in\R,\ \phis(y;\e)$ is increasing with $\e\to 0$ and bounded by $K$. Hence there exists a positive function $\phis_0$ such that $\phis(y;\e)\underset{\e\to 0}{\longrightarrow}\phis_0(y)$. Moreover, from the uniform boundedness of $\phis(\e)$ and Ascoli’s theorem, the convergence is uniform for $\phis$ and $\phis'$ in every compact set. Thus, $\phis_0$ satisfies in the classical sense $$\begin{cases}
-\phis_0''(y)+\nu(y)\phis_0(y)=\mu_c(y) \\
0<\phis_0\leq K.
\end{cases}$$ As $\mu_c$ is compactly supported, for $|y|$ large enough, let us say greater than $A>0$, $\phis_0$ is a solution of (\[homogeneous\]), that is to say, in the positive semi-axis $$\begin{cases}
-\phis_0''(y)+\nu(y)\phis_0(y)=0, & y>A \\
0<\phis_0(y)\leq K<+\infty & y>A.
\end{cases}$$ Thus, there exist $\a^+,\b^+$ such that $$\forall y>A,\ \phis_0(y)=\a^+(1+\phi_1(y))+\b^+\phi_2(y)(1+\phi_1(y)),$$ where $\phi_1$ and $\phi_2$ are defined in Lemma \[lemhomo\]. Now considering that $\phi_1(y)=o(1)$ and $\phi_2(y)=\varTheta(y)$ in $y\to+\infty$, as $\phis_0$ is bounded, $\b^+=0.$ Then, as $\phis_0>0,$ $\a^+>0$. We have a similar result for $y<-A,$ with $\b^-=0$ and $\a^->0$. Finally, define $$k=\frac{1}{2}\min(\a^-,\a^+,\min\{\phis_0(y),y\in[-A,A]\})>0$$ and the proof is concluded.
#### Proof of the second part of Lemma \[lemeps\]
Differentiating equation (\[epseq1\]) with respect to $\e$, we get for the derivative $\phi_{\e}$ $$\label{eqpes2}
-\phi_{\e}''(y;\e)+(\e+\nu(y))\phi_{\e}(y;\e)=-\phi(y;\e).$$ Since $\phi$ is positive, we get that $\phi_{\e}$ is negative. Let us denote $$\vp(y)=\vp(y;\e):=-\phi_\e (y;\e)>0.$$ We have previously seen (in the proof of the first part of Proposition \[sourire\]) that $\forall y\in\R,$ $\frac{d}{d\e}\vp(y;\e)<0$, *i.e.* $\vp$ is increasing with $\e\to 0,\e>0.$ Our purpose is to show that in a neighbourhood of $0,$ $\inf(\vp(\e))\underset{\e\to0}{\longrightarrow}+\infty.$ For all $\e>0,$ define the function $\vps=\vps(y;\e)$ as the unique solution of $$\label{soussolderivee}
\begin{cases}
-\vps''(y;\e)+(\e+\nu(y))\vps(y;\e)=\min(k,\phi(y;\e)) \\
\vps\in H^1(\R).
\end{cases}$$ The function $\vps$ is obviously well-defined. By its definition, the elliptic maximum principle ensures us that $0<\vps\leq \vp,\ \forall y\in\R,\e>0.$ We have also to notice that uniformly on every compact set in $y,$ $\min(k,\phi(y;\e))=k$ for $\e$ small enough (consequence of corollary \[minoration\]). Assume by way of contradiction that $$\label{hypothesecontr}
\left(\underset{y\in[-1,1]}{\min}(\vps(y;\e))\right)_\e \textrm{ is bounded.}$$ Let us show that it is inconsistent with the fact that $\vps>0,\forall \e>0.$ As $\min(k,\phi(y;\e))$ is uniformly bounded, from Harnack inequalities (see [@GT], Theorem 8.17 and 8.18) we know that for all $R>0,$ there exist $C_1=C_1(R),C_2=C_2(R),$ independent of $\e$, such that for all $\e>0,$ $$\underset{[-R,R]}{\sup}\vps \leq C_1\underset{[-R,R]}{\inf}(\vps+C_2).$$ Combining this and hypothesis (\[hypothesecontr\]), we get that $\left(\vps(y;\e) \right)_{\e>0}$ is increasing with $\e\to0$ and uniformly in every compact set in $y$. Using the same argument as in the proof of Corollary \[minoration\], $(\vps(\e))_\e$ converges locally uniformly to some function $\vps_0$ which satisfies in the classical sense $$\label{eqlimite2}
\begin{cases}
-\vps_0''(y)+ \nu(y)\vps_0(y) = k \\
\vps_0(y)\geq 0,\ \forall y\in\R.
\end{cases}$$ So there exist $\a,\b\in\R$ such that $\vps_0=\a(1+\phi_1)+\b\phi_2(1+\phi_1)+\phi_s,$ where $\phi_1,\phi_2$ are defined in Lemma \[lemhomo\] and $\phi_s$ is a particular solution of (\[eqlimite2\]). Thus, for $x\geq0,$ $$\phi_s(y) = -k\left(1+\phi_1(y)\right)\left(1+\phi_1(0)\right)\left( \int_0^y\int_z^y \frac{1+\phi_1(z)}{(1+\phi_1(t))^2}dtdz \right).$$ Now, recall that $\phi_1>0,\phi_1(y)=o(y)$ as $y$ goes to $+\infty$. So there exists $\gamma>0,$ $\phi_s(y)\underset{y\to\infty}{\sim}-\gamma.y^2.$ As a result, for $y\to \infty$, $$\begin{cases}
\vps_0(y) = -\gamma.y^2+o(y^2)\underset{y\to+\infty}{\longrightarrow}-\infty \\
\vps_0\geq0,\ \forall y\in\R,
\end{cases}$$ which is obviously a contradiction. So the first hypothesis (\[hypothesecontr\]) is false, which gives, combined with the monotonicity in $\e,$ $$\underset{y\in[-1,1]}{\min}(\vps(y;\e))\underset{\e\to0}{\longrightarrow}+\infty,$$ and then, as $\nu$ is continuous and $\nu(0)>0,$ $$\int_\R \phi_\e(y;\e)\nu(y)dy \underset{\e\to0}{\longrightarrow} -\infty,$$ and the proof of the main Lemma \[lemeps\] is complete.
Intersection of $\Gamma_1$ and $\Gamma_2$, supersolution
--------------------------------------------------------
#### First case: $D>2d$.
If $D>2d$, we have of course $\frac{c}{D}<\frac{c}{2d},\ \forall c\geq c_{KPP}.$ Thus, for $c$ close enough to $c_{KPP}$, $\Gamma_2$ does not intersect the closed convex hull of $\Gamma_1$. But since $$\frac{c}{D} \underset{c\to +\infty}{\longrightarrow} +\infty\textrm{ and }\la_2^-(c)\underset{c\to +\infty}{\longrightarrow} 0^+,$$ there exists $$c_*=c_*(D)>c_{KPP}$$ such that $\forall c>c_*,\ \Gamma_1$ and $\Gamma_2$ intersect, and $\forall c<c_*$, $\Gamma_2$ does not intersect the closed convex hull of $\Gamma_1$. Moreover, the strict concavity of $\Gamma_1$ and the strict convexity of $\Gamma_2$ allow us to assert that for $c=c_*,\ \Gamma_1 \textrm{ and }\Gamma_2$ are tangent on $\la=\la(c_*)$ and for $c>c_*,\ c$ close to $c_*$, $\Gamma_1$ and $\Gamma_2$ intersect twice, at $\la(c)^+$ and $\la(c)^-$. The different situations are illustrated in fig. (\[Dsup2d\]).\
When $c$ is such that $\la_2^-\leq \frac{c}{D}$, i.e.$\displaystyle c\geq \frac{D}{2\sqrt{dD-d^2}}c_{KPP}$ there is only one solution for $\la=\la(c).$
#### Second case: $D=2d$.
If $D=2d$, then the point $(\frac{c}{2d},\mub)$ belongs to $\Gamma_1$. Therefore, for all $c>c_{KPP}$, $\Gamma_1$ and $\Gamma_2$ intersect once at $\la=\la(c)$. We set: $$c_*(2d):=c_{KPP}.$$
#### Third case: $D<2d$.
If $D<2d$, we have $\frac{c}{D}>\frac{c}{2d}$. Then, $\forall c>c_{KPP},\ \la_2^-(c)<\frac{c}{D}$, the left part of $\Gamma_2$ is strictly below $\Gamma_1$, and every $c>c_{KPP}$ gives a super-solution. We set again: $$c_*(D):=c_{KPP}.$$ All of this concludes the proof of Proposition \[spreading\_speed\]. Moreover, we can assert from geometrical considerations that $$\label{inegaliteD}
\frac{c_*}{D}\leq \frac{c_*-\sqrt{c_*^2-c_{KPP}^2}}{2d}\leq \frac{c_*+\sqrt{c_*^2+4D\mub}}{2D}.$$ It was proved in [@BRR1] that (\[inegaliteD\]) implies that $$\sqrt{4\mub^2+f'(0)^2}-2\mub\leq \underset{D\to+\infty}{\lim\inf}\frac{c_*^2}{D}\leq
\underset{D\to+\infty}{\lim\sup}\frac{c_*^2}{D}\leq f'(0).$$
[Explicit computation of $\Psi_2:=\Psi_2^0$ in the reference case (\[BRReq2\])]{} In the limit case, (\[eqgeneralesurphi\]) can be written as follows, setting $P(\la)=-d\la^2+c\la-\fp$: $$\label{eqsurphiBRR}
-d\phi''(y)+\lp P(\la)+\nub\d_0\rp\phi(y)=\mub\d_0.$$ Thus, an explicit computation (see [@BRR1] or [@Pauthier2]) gives $$\label{solPsi2BRR}
\Psi_2^0(\la) := \nub\phi(0) = \frac{\nub\mub}{\nub+2\sqrt{dP(\la)}}.$$ Notice that this function satisfies all properties given by Proposition \[sourire\].
Spreading
=========
In order to prove that solutions spread at least at speed $c_*$, we are looking for compactly supported general stationary subsolution in the moving framework at velocity $c<c_*$, arbitrarily close to $c_*.$ We consider the linearised system penalised by $\d>0$ in the moving framework :
$$\label{pendelta}
\begin{cases}
\partial_t u-D \partial_{xx} u+c\partial_x u = -\mub u+\int \nu(y)v(t,x,y)dy & x \in \R,\ t>0 \\
\partial_t v-d\Delta v+c\partial_x v = (f'(0)-\d)v +\mu(y)u(t,x)-\nu(y)v(t,x,y) & (x,y)\in \R^2,\ t>0.
\end{cases}$$
The main result is here the following:
\[soussol\] Let $c_*=c_*(D)$ be as in the previous section. Then, for $c<c_*$ close enough to $c_*$, there exists $\d>0$ such that (\[pendelta\]) admits a nonnegative, compactly supported, generalised stationary subsolution $(\underline{u},\underline{v})\not\equiv (0,0)$.
As in the previous section, we will study separately the case $D>2d$, which is the most interesting, and the case $D\leq 2d$.
Construction of subsolutions: $D>2d$
------------------------------------
In order to keep the notation as light as possible, we will use the notation $\ft:=f'(0)-\d$ and $\Pt(\la):=-d\la^2+c\la-\ft$, because all the results will perturb for small $\d>0$. We just have to keep in mind that $\ft<f'(0)$ and $\d\ll1$, hence $\Pt(\la)>P(\la)$ and $\Pt(\la)-P(\la)\ll 1$.\
Our method is to devise a stationary solution of (\[pendelta\]) not in $\R^2$ anymore, but in the horizontal strip $\Omega^L = \R \times (-L,L)$, with $L>0$ large enough. Thus, we are solving $$\label{stationnaire}
\begin{cases}
-DU''+cU'=-\mub U+\int_{(-L,L)}\nu(y)V(x,y)dy & x\in \R \\
-d\Delta V+c\partial_x V=\ft V+\mu(y)U(x)-\nu(y)V(x,y) & (x,y)\in \Omega^L \\
V(x,L)=V(x,-L) = 0 & x\in \R.
\end{cases}$$ In a similar fashion as in the previous section, we are looking for solutions of the form $$\label{soussolexp}
\begin{pmatrix}
U(x) \\
V(x,y)
\end{pmatrix}
= e^{\la x} \begin{pmatrix}
1 \\ \vp(y)
\end{pmatrix},$$ where $\vp$ belongs to $H_0^1(-L,L)$. The system on $(\la,\vp)$ reads: $$\label{eqsoussol}
\begin{cases}
-D\la^2+\la c+\mub = \int_{(-L,L)} \nu(y)\phi(y)dy \\
-d\vp''(y)+(\Pt(\la)+\nu(y))\vp(y) = \mu(y) & \vp(-L) = \vp(L)=0.
\end{cases}$$ The first equation of (\[eqsoussol\]) gives a function $\la \mapsto \Psi_1(\la;c)=-D\la^2+\la c+\mub$. The second equation of (\[eqsoussol\]) gives a unique solution $\vp=\vp(y;\la,c;L)\in H^1_0(-L,L)$. We associate this unique solution with the function $\Psi_2^L(\la;c)=\int_{(-L,L)}\nu(y)\vp(y)dy$. A solution of the form (\[soussolexp\]) exists if and only if $\Psi_1(\la;c)=\Psi_2^L(\la;c)$ for some $\la,c$, that is to say if and only if $\Gamma_1$ and $\Gamma_2^L$ intersect (with straightforward notations). In this section, the game is to make them intersect not with real but with complex $\la$.
#### Study of $\Gamma_1$
The function $\la\mapsto \Psi_1$ is exactly the same as in the search for supersolutions. In particular, it does not depend in $L$. Thus, the curve $\Gamma_1$ is the same as in the previous section: it is a parabola, symmetric with respect to the line $\{\la=\frac{c}{2D}\}$. Notice that being a parabola, its curvature is positive at any point ; it will be important later.
#### Study of $\Gamma_2^L$
The study of $\Gamma_2^L$ is quite similar to that of $\Gamma_2$. It amounts to studying the solutions of $$\label{eqsoussol2}
\begin{cases}
-d\vp''(y)+(\Pt(\la)+\nu(y))\vp(y) = \mu(y) & y\in (-L,L)\\
\vp \in H^1_0(-L,L).
\end{cases}$$ For real $\la$, (\[eqsoussol2\]) admits solution for $\la \in [\la_{2,\d}^-,\la_{2,\d}^+]$, with $\la_{2,\d}^\pm = \frac{c\pm\sqrt{c^2-4d\ft}}{2d}$. We may notice that $\la_{2,\d}^-<\la_2^-,\ \la_{2,\d}^+>\la_2^+$, and of course $\la_{2,\d}^\pm \longrightarrow \la_2^\pm$ as $\d\to 0$. With a simple study of (\[eqsoussol2\]) and using what we proved in proposition (\[sourire\]), we can assert : $$\label{convergencegamma}
\underset{L\to\infty}{\lim} \underset{\d\to 0}{\lim} \Psi_2^L(\la;c) = \underset{\d\to 0}{\lim} \underset{L\to\infty}{\lim} \Psi_2^L(\la;c)
=\Psi(\la;c),$$ and this uniformly on every compact set in $]\la^-_2,\la^+_2[\times ]2\sqrt{df'(0)},+\infty[.$
As a consequence, the picture is analogous to the case described in fig. (\[Dsup2d\]): there exists a unique $c_*^L$ (which depends on $\d$) such that $\Gamma_2^L$ intersects $\Gamma_1$ twice if $c>c_*^L,$ close to $c_*^L$, once if $c=c_*^L$, and never if $c<c_*^L$ (for real $\la$).\
Moreover, since $\Gamma_2^L$ is below $\Gamma_2$, we have $c_{KPP}<c_*^L<c_*$. We also have $c_*^L\longrightarrow c_*$ as $L\to\infty,\ \d\to 0$.
#### Complex solutions
We use the same argument as in [@BRR1]. Let us call $\b$ the ordinate of the plane $(\la,\Psi_{1,2}^L)$. For $c=c_*^L$, call $(\la^L_*,\b^L_*)$ the tangent point between $\Gamma_1$ et $\Gamma_2^L$. The functions $\Psi_1$ and $\Psi_2^L$ are both analytical in $\la$ at this point, and $\frac{d}{d\la}\Psi_1(\la),\quad\frac{d}{d\la}\Psi_2^L(\la)\neq0,$ for $(c,\la)$ in a neighbourhood of $(c_*^L,\la^L_*)$. Due to the implicit function theorem, there exist $\la_1(c,\b),\ \la_2^L(c,\b)$ defined in a neighbourhood $V_1$ of $(c_*^L,\b^L_*)$, analytical in $\b$, such that $$\label{implicite}
\begin{cases}
\Psi_1(\la_1(c,\b);c)=\b & \forall (c,\b)\in V_1 \\
\Psi_2^L(\la_2^L(c,\b);c)=\b & \forall (c,\b)\in V_1.
\end{cases}$$
Then, set $$h^L(c,\b)=\la_2^L(c,\b)-\la_1(c,\b),\qquad \textrm{for} \ (c,\b)\in V_1,$$ and we get: $$\label{derivees}
\begin{cases}
\partial_\b h^L(c_*^L,\b_*^L)=0. \\
\partial_{\b\b} h^L(c_*^L,\b_*^L):=2a>0. \\
\partial_c h^L(c_*^L,\b_*^L):=-e<0.
\end{cases}$$ The first point is obvious. The second comes from the fact that $\Gamma_2^L$ is concave and $\Gamma_1$ has a positive curvature at any point. The third is obvious given the first equation of (\[eqsoussol\]).
Now, because we are working in a vicinity of $(c_*^L,\b_*^L)$, set : $$\xi:=c_*^L-c,\qquad \tau=\b-\b_*^L.$$ Call $b:=\partial_{c\b}h^L(c_*^L,\b_*^L)$. From (\[implicite\]) and (\[derivees\]), we can assert that there exists a neighbourhood $V_2\subseteq V_1$ of $(c_*^L,\b_*^L)$, there exists $\eta=\eta(\tau,\xi)$ analytical in $\tau$ in $V_2-(c_*^L,\b_*^L)$, vanishing at $(0,0)$ like $|\tau|^3+\xi^2$, such that $$\label{analytical}
(h^L(c,\b)=0,\ (c,\b)\in V_2)\Leftrightarrow (a\tau^2+b\xi\tau+e\xi=\eta(\tau,\xi)).$$ Recall that $a$ and $e$ are positive, so the discriminant $\Delta=(b\xi)^2-4ae\xi$ is negative for $\xi>0$ small enough. The trinomial $a\tau^2+b\xi\tau+e\xi$ has two roots $\tau_\pm=\frac{-b\xi\pm i\sqrt{4ea\xi-(b\xi)^2}}{2a}$. Then, from an adaptation of Rouché’s theorem (see [@BRR1]), the right handside of (\[analytical\]) has two roots, still called $\tau_\pm$, satisfying $\tau_\pm=\pm i\sqrt{(e/a)\xi}+O(\xi)$. Reverting to the full notation, we can see that for $c$ strictly less than and close enough to $c_*^L$, there exist $\b,\la \in \mathbb{C},\ \vp \in H^1_0((-L,L),\mathbb{C})$ satisfying (\[eqsoussol\]). Since $\b=\Psi_1(\la)=-D\la^2+c\la+\mub$ and $\b$ has nonzero imaginary part, $\la$ has also nonzero imaginary part. We can therefore write $(\la,\b)=(\la_1+i\la_2,\b_1+i\b_2)$ and: $$\begin{pmatrix}
U \\ V
\end{pmatrix}
=e^{(\la_1+i\la_2)x}
\begin{pmatrix}
1 \\ \vp_1(y)+i\vp_2(y)
\end{pmatrix}$$ with $$\begin{cases}
\la_2,\b_2 \neq 0 \\
\int\nu(y)\vp_1(y)dy=\b_1=\b_*^L+O(c_*^L-c) \\
\int\nu(y)\vp_2(y)dy=\b_2=O(\sqrt{c_*^L-c}).
\end{cases}$$ Thus :
- ${{\cal R}e \left ( U \right )} >0$ on $(-\frac{\pi}{2\la_2},\frac{\pi}{2\la_2})$ and vanishes at the ends ;
- ${{\cal R}e \left ( V \right )}>0 \Leftrightarrow \vp_1\cos(\la_2x)>\vp_2\sin(\la_2x)$.
The set where ${{\cal R}e \left ( V \right )} >0$ is periodic of period $\frac{2\pi}{\la_2}$ in the direction of the road. Its connected components intersecting the strip $\R\times (-L,L)$ are bounded. The function $\vp_2$ is continuous in $c$, hence the functions $y\mapsto\vp(y;c)$ are uniformly equicontinuous for $c$ near $c_*^L$. Since $\nu(0)>0$ and $\int \nu\vp_2=O(\sqrt{c_*^L-c})$, we have $\vp_2(0)=O(\sqrt{c_*^L-c})$, and we can make one of the connected components of $\{{{\cal R}e \left ( V \right )} >0\}$, denoted by $F$, satisfy the property that $\{(x,0)\in \overline{F}\}$ is arbitrary close to $[-\frac{\pi}{2\la_2},\frac{\pi}{2\la_2}]$. We can now define the following functions: $$\begin{array}{lcl}
\underline{u}(x) & := & \begin{cases}
\max ({{\cal R}e \left ( U(x) \right )},0) & \textrm{ if } |x|\leq \frac{\pi}{2\la_2} \\
0 & \textrm{ otherwise }
\end{cases}
\\
\underline{v}(x,y) & := & \begin{cases}
\max ({{\cal R}e \left ( V(x,y) \right )},0) & \textrm{ if } (x,y)\in \overline{F} \\
0 & \textrm{ otherwise }.
\end{cases}
\end{array}$$ The choice of $F$ implies that $(\underline{u},\underline{v})$ is a subsolution of (\[pendelta\]).
Subsolution: case $D\leq 2d$
----------------------------
Now assume that $0\leq D \leq 2d$. In the previous section, we define $c_*(D)=c_{KPP}=2\sqrt{df'(0)}$. Let $c\leq c_{KPP}$. Thus, $4df'(0)-c^2>0$. Let $\d$ be such that $0<2\d<\frac{4df'(0)-c^2}{4d}=f'(0)-\frac{c^2}{4d}$. With $\omega=\frac{\sqrt{4d(f'(0)-2\d)-c^2}}{2d}$, we define $$\phi(x)=e^{\frac{c}{2d}x}cos(\omega x)\mathbf{1}_{(-\frac{\pi}{2\omega},\frac{\pi}{2\omega})}.$$ The function $\phi$ is continuous and satisfies $$-d\phi''+c\phi=(f'(0)-2\d)\phi \qquad \textrm{on} \ (-\frac{\pi}{2\omega},\frac{\pi}{2\omega}).$$ Then, let us choose $R>0$ such that the first eigenvalue of $-\partial_{yy}$ in $(-R,R)$ is equal to $\frac{\d}{d}-\a$, and $\psi_R$ an associated nonnegative eigenfunction in $H_0^1(-R,R)$, where $0<\a<\d$. The function $\psi_R$ satisfies $$-d\psi_R''=(\d-\a)\psi_R \ \textrm{in}\ (-R,R),\ \psi_R(y)>0,\ \forall |y|<R,\ \psi_R(R)=\psi_R(-R)=0.$$ We extend $\psi_R$ by 0 outside $(-R,R)$. Let $M>0$ such that $\forall |y|>M-R,\ \nu(y)\leq \a,$ which is possible since $\nu(y)\to0$ with $y\to\pm\infty.$ The function $$\underline{V}(x,y):=\phi(x)\psi_R(|y|-M)$$ is a solution of $$\begin{cases}
-d\Delta V+c\partial_x V=(f'(0)-\d)V-\a V \\
x\in (-\frac{\pi}{2\omega},\frac{\pi}{2\omega}), \ |y|\in(M-R,M+R),
\end{cases}$$ vanishing on the boundary. Hence, from the choice of $M$ and $\a,$ $(0,\underline{V})$ is a nonnegative compactly supported subsolution of (\[pendelta\]), non identically equal to $(0,0)$ ; which concludes the proof of Proposition \[soussol\]. The proof of the main Theorem \[spreadingthm\] follows as in [@BRR1].
The intermediate model (\[RPSL\])
=================================
#### Formal derivation of the semi-limit model
Starting from the full model (\[RPeq\]), we consider normal (i.e. integral) exchange from the field to the road but localised exchange from the road to the field. Formally, we define $\mu_\e = \frac{1}{\e}\mu(\frac{y}{\e})$ and take the limit with $\e\to 0$ of the system (\[RPepsSL\]) : $$\label{RPepsSL}
\begin{cases}
\partial_t u-D \partial_{xx} u = -\mub u+\int \nu(y)v(t,x,y)dy & x \in \R,\ t>0 \\
\partial_t v-d\Delta v = f(v) +\mu_\e(y)u(t,x)-\nu(y)v(t,x,y) & (x,y)\in \R^2,\ t>0.
\end{cases}$$ There is no influence in the first equation (the dynamic on the road), which is the same in the limit system. Though the second equation in (\[RPepsSL\]) tends to $$\partial_t v-d\Delta v = f(v)-\nu(y)v(t,x,y),\qquad (x,y)\in \R\times\R\diagdown\{0\},\ t>0.$$ It remains to determine the limit condition between at the road. We may assume that for $\e=0$ $v$ is still continuous at $y=0$. Now set $\xi=y/\e$ and $\vt(t,x,\xi):=v(t,x,y)$. The second equation in (\[RPepsSL\]) becomes in the $(t,x,\xi)$-variables $$\e^2\left(\partial_t \vt-d\partial_{xx}\vt-f(\vt)+\nu(\xi)\vt(t,x,\xi)\right)-d\partial_{\xi\xi}\vt = \e\mu(\xi)u(t,x).$$ Passing to the limit, it yields, in the $y$-variable: $$-d\left(\partial_y v(t,x,0^+)-\partial_y v(t,x,0^-) \right) = \mub u(t,x).$$ Consequently, the formal limit system of (\[RPepsSL\]) should be (\[RPSL\]) presented in the Introduction, which is the system we will study from now. Our assumptions on $\nu$ and $f$ are the same as above. The investigation is similar to the one done for the model (\[RPeq\]), and we will only develop the parts which differ.
#### Comparison principle
Throughout this section, we will call a supersolution of (\[RPSL\]) a couple $(\us,\vs)$ satisfying, in the classical sense, the following system: $$\label{RPSLsuper}
\begin{cases}
\partial_t \us-D \partial_{xx} \us \geq v(x,0,t)-\mub u +\nu(y)v(t,x,y) & x\in\R,\ t>0\\
\partial_t v-d\Delta v\geq f(v) -\nu(y)v(t,x,y) & (x,y)\in\R\times\R^*,\ t>0\\
v(t,x,0^+)=v(t,x,0^-), & x\in\R,\ t>0 \\
-d\left\{ \partial_y v(t,x,0^+)-\partial_y v(t,x,0^-) \right\}\geq \mub u(t,x) & x\in\R,\ t>0,
\end{cases}$$ which is also continuous up to time $0.$ Similarly, we will call a subsolution of (\[RPSL\]) a couple $(\su,\sv)$ satisfying (\[RPSLsuper\]) with the inverse inequalities (*i.e.* the $\geq$ signs replaced by $\leq$. We now need a comparison principle in order to get monotonicity for solutions :
\[comparaisonSL\] Let $(\su,\sv)$ and $(\us,\vs)$ be respectively a subsolution bounded from above and a supersolution bounded from below of (\[RPSL\]) satisfying $\su\leq\us$ and $\sv\leq\vs$ at $t=0$. Then, either $\su<\us$ and $\sv<\vs$ for all $t>0$, or there exists $T>0$ such that $(\su,\sv)=(\us,\vs),\ \forall t\leq T.$
We omit the proof.
#### Long time behaviour and stationary solutions
We want to show that any (nonnegative) solution of (\[RPSL\]) converges locally uniformly to a unique stationary solution $(U_s,V_s)$, which is bounded, positive, $x$-independent, and of course is solution of the stationary system of equations (\[stateqSL\]): $$\label{stateqSL}
\begin{cases}
-D U''(x) = -\mub U +\int\nu(y)V(x,y) \\
d\Delta V(x,y) = f(V) -\nu(y)V(x,y) \\
V(x,0^+) = V(x,0^-) \\
-d\left\{ \partial_y V(x,0^+)-\partial_y V(x,0^-) \right\}=\mub U(x).
\end{cases}$$ Proofs of Propositions \[stationary1\] and \[stationary2\] can be easily adapted to this new system. The only nontrivial point lies in the existence of an $L^\infty$ a priori estimate. Set $\la=\frac{\nub}{d}$. From conditions on the reaction term, there exists $M_1$ such that $\forall s>M_1,\ f(s)<-\frac{\nub^2}{d}s.$ Now, set $$M=\max (M_1,\frac{\nub}{\mub}\|u_0\|_\infty,\|v_0\|_\infty)$$ and the couple $(\overline{U},\overline{V})$ given by $$\overline{V}(y)=M(1+e^{-\la|y|}),\ \overline{U}=\frac{1}{\mub}\int_\R \nu(y)\overline{V}(y)dy$$ is a supersolution of (\[stateqSL\]) which is above $(u_0,v_0)$.
The proof of the corresponding Proposition \[liouville\] follows easily.
#### Exponential solutions, spreading
We are looking for solutions of the linearised system: $$\label{RPSLli}
\begin{cases}
\partial_t u-D \partial_{xx} u= v(x,0,t)-\mub u +\nu(y)v(t,x,y) & x\in\R,\ t>0\\
\partial_t v-d\Delta v=f'(0)v -\nu(y)v(t,x,y) & (x,y)\in\R\times\R^*,\ t>0\\
v(t,x,0^+)=v(t,x,0^-), & x\in\R,\ t>0 \\
-d\left\{ \partial_y v(t,x,0^+)-\partial_y v(t,x,0^-) \right\}=\mub u(t,x) & x\in\R,\ t>0,
\end{cases}$$
and these solutions will be looked for under the form $$\label{solexpSL}
\begin{pmatrix}
u(t,x) \\
v_1(t,x,y) \\
v_2(t,x,y)
\end{pmatrix}
= e^{-\la(x-ct)} \begin{pmatrix}
1 \\ \phi_1(y) \\ \phi_2(y)
\end{pmatrix}$$ where $\la,c$ are positive constants and $\phi$ is a nonnegative function in $H^1(\R)$, with $v=v_1,\ \phi=\phi_1$ for $y\geq 0$ and $v=v_2, \phi=\phi_2$ for $y\leq0$. The system in $(\la,\phi)$ reads $$\label{laphi}
\begin{cases}
-D\la^2+\la c+\mub = \int \nu(y)\phi(y)dy \\
-d\phi_1''(y)+(\la c-d\la^2-f'(0)+\nu(y))\phi_1(y) = 0 & y\geq 0. \\
-d\phi_2''(y)+(\la c-d\la^2-f'(0)+\nu(y))\phi_2(y) = 0 & y\leq 0. \\
\phi_1(0)=\phi_2(0) & i.e. \ \phi \textrm{ is continuous.} \\
-\phi'_1(0)+\phi'_2(0) = \frac{\mub}{d}.
\end{cases}$$ The study is exactly the same as in the third section. The only point which deserves some explanation is the well-posedness of (\[laphi\]). For $M>0$ let us consider $\vp_M$ the unique solution of $$\label{eqM}
\begin{cases}
-d\vp_M''(y)+(P(\la)+\nu(y))\vp_M(y) = 0 & y\in]0,+\infty[ \\
\vp_M(0)=M & \vp_M\in H^1(\R^+).
\end{cases}$$ Let us show the following lemma, which will prove the well-posedness of (\[laphi\]):
\[wellposed\]
1. $M\mapsto \vp_M'(0)$ is decreasing ;
2. $ \vp_M'(0) \underset{M\to 0}{\longrightarrow}0$ ;
3. $ \vp_M'(0) \underset{M\to +\infty}{\longrightarrow}-\infty$.
Let us consider $M_1,M_2$ with $0<M_1<M_2$, $\vp_{M_1},\vp_{M_2}$ the associated solutions of (\[eqM\]). The elliptic maximum principle yields $0<\vp_{M_1}(y)<\vp_{M_2}(y),\ \forall y\geq0$ and Hopf’s lemma gives $0>\vp_{M_1}'(0)>\vp_{M_2}'(0),$ which proves the first point.
Then, if we integrate (\[eqM\]) we get $$\vp_M'(0)=-\frac{1}{d}\int_0^\infty (P(\la)+\nu(y))\vp_M(y)dy.$$ Let us now consider $\vps_M$ the (unique) solution of $$\begin{cases}
-d\vps_M''(y)+P(\la)\vps_M(y) = 0 & y\in]0,+\infty[ \\
\vps_M(0)=M & \vps_M\in H^1(\R^+).
\end{cases}$$ $\vps_M$ is a supersolution of (\[eqM\]). Thus, $\vps_M(y)\geq\vp_M(y),\ \forall y\geq0.$ Moreover we have an explicit expression for $\vps_M$: $\vps_M(y)=M\exp(-\sqrt{\frac{P(\la)}{d}}y)$. Hence, $$0\leq-\vp_M'(0)\leq\frac{M}{d}\int_0^\infty(P(\la)+\nu(y))e^{-\sqrt{\frac{P(\la)}{d}}y}dy$$ and $$-\vp_M'(0) \underset{M\to0}{\longrightarrow}0$$ uniformly in $\la$, which proves the second point.
In the same way, the unique solution $\svp$ of $$\begin{cases}
-d\svp_M''(y)+(P(\la)+\|\nu\|_\infty)\svp_M(y) = 0 & y\in]0,+\infty[ \\
\svp_M(0)=M & \svp_M\in H^1(\R^+).
\end{cases}$$ is a subsolution of (\[eqM\]), and $\svp_M(y)\leq\vp_M(y),\ \forall y\geq0.$ Hence,
$$\begin{array}{lcl}
-\vp_M'(0) & \geq & \frac{1}{d}\int_0^\infty (P(\la)+\nu(y))\svp_M(y)dy \\
& \geq & \frac{M}{d}\int_0^\infty (P(\la)+\nu(y))e^{-\sqrt{\frac{P(\la)+\|\nu\|_\infty}{d}}y}du \\
-\vp_M'(0) & \to & +\infty \textrm{ as }M\to +\infty,
\end{array}$$
which concludes the proof of Lemma \[wellposed\].
The corresponding Proposition \[spreading\_speed\] and Theorem \[spreadingthm\] follows as in the previous part.
The large diffusion limit $D\to+\infty$
=======================================
The behaviour of the spreading speed $c^*$ as $D$ goes to $+\infty$ has already been investigated in [@BRR1] for the initial model (\[BRReq2\]). It has been shown that there exists $c_\infty>0$ such that $$\frac{c^*(D)}{\sqrt{D}}\underset{D\to+\infty}{\longrightarrow}c_\infty.$$ In the following Proposition, we show the robustness of this result and extend it to the general cases (\[BRReq2\])-(\[RPSL2\]). We also give an asymptotic behaviour as $\fp$ tends to $+\infty.$
\[asymptoticDfp\] Let us consider any of the systems (\[BRReq2\])-(\[RPSL2\]) with fixed parameters $d,\nub,\mub.$ Let $c^*(D,\fp)$ be the associated spreading speed given by Theorem \[spreadingthm\].
1. There exists $c_\infty,$ $\displaystyle \frac{c^*(D,\fp)}{\sqrt{D}}\underset{D\to+\infty}{\longrightarrow}c_\infty.$
2. $c_\infty$ satisfies $\displaystyle c_\infty\underset{\fp\to+\infty}{\sim}\sqrt{\fp}.$
That is, with $D\to+\infty$ and $\fp\to+\infty,$ we have $c_0^*\sim\sqrt{\fp D},$ *i. e.* half of the KPP spreading speed for a reaction-diffusion on the road.
#### Proof of Proposition \[asymptoticDfp\]
We prove the result for the nonlocal system (\[RPeq\]), the other cases being similar. We set $$\ut(t,x)=u(t,\sqrt{D}x),\quad \vt(t,x,y) = v(t,\sqrt{D}x,y).$$ The system in the rescaled variables becomes $$\label{systemDinfty}
\begin{cases}
\partial_t\ut-\partial_{xx}\ut = -\mub\ut+\int\nu(y)\vt(t,x,y)dy \\
\partial_t\vt-d\lp\partial_{yy}\vt+\frac{1}{D}\partial_{xx}\vt\rp = f(\vt)+\mu(y)\ut(t,x)-\nu(y)\vt(t,x,y).
\end{cases}$$ The $(c,\la,\phi)-$system associated to (\[systemDinfty\]) is then $$\begin{cases}
\la c-\la^2+\mub = \int \nu\phi \\
-d\phi''(y)+\lp\la c-\frac{d}{D}\la^2-\fp+\nu(y)\rp\phi(y) = \mu(y).
\end{cases}$$ Hence we get that $c^* = \sqrt{D}\ct$ where $\ct$ is the first $c$ such that the graphs of $\Psit_1$ and $\Psit_2$ intersect, where $\Psit_1$ and $\Psit_2$ are defined as follows: $$\Psit_1 : \la \longmapsto \la c -\la^2+\mub$$ and $$\Psit_2 : \begin{cases}
]\tilde{\la}^-,\tilde{\la}^+[ & \longrightarrow \R \\
\la & \longmapsto \int\nu\phi
\end{cases}$$ where $\phi$ is the unique $H^1$ solution of $$\label{eqphirescalee}
-d\phi''(y)+\lp\la c-\frac{d}{D}\la^2-\fp+\nu(y)\rp\phi(y) = \mu(y)$$ and $\displaystyle\tilde{\la}^\pm = \frac{D}{2d}\lp c\pm\sqrt{c^2-4\frac{d\fp}{D}}\rp.$ We can see that as $D$ tends to $+\infty,$ $\displaystyle\tilde{\la}^-=\frac{\fp}{c}+o(1)$ and $\displaystyle\tilde{\la}^+\to+\infty.$ Behaviours of $\Psit_{1,2}$ have already been studied above. $\Psit_1$ is a concave parabola, $\Psit_2$ is strictly convex, symmetric with respect to $\{\la=\frac{cD}{2d}\}.$ Moreover, it has been showed that the solution $\phi$ of (\[eqphirescalee\]) is bounded in $L^\infty,$ uniformly in $\la,c,D.$ It is also pointwise strictly decreasing in $\lp\la c-\frac{d}{D}\la^2-\fp\rp.$ Now, let $\vp$ be the $H^1$ solution of the limit system defined for $\la>\frac{\fp}{c}$ $$\label{eqphiDinfty}
-d\vp''(y)+\lp\la c-\fp+\nu(y)\rp\vp(y) = \mu(y).$$ From the maximum principle and the monotonicity of $\phi$ with respect to the nonlinear eigenvalue, we can easily see that $\displaystyle \lV\vp-\phi\rV_{L^\infty}\to 0$ as $D\to\infty,$ locally uniformly in $\la,c.$ Hence, $\Psit_2$ tends to $\Psit_{2,\infty}$ defined by $$\Psit_2 : \begin{cases}
]\frac{\fp}{c},+\infty[ & \longrightarrow \R \\
\la & \longmapsto \int\nu\vp
\end{cases}$$ where $\vp$ is the unique solution of (\[eqphiDinfty\]), and $\ct$ tends to $c_\infty,$ where $c_\infty$ is the first $c$ such that the graphs of $\Psit_1$ and $\Psit_{2,\infty}$ intersect. This concludes the proof of the first part of Propostion \[asymptoticDfp\].
(-0.1,0) – (2.5,0); (0,-0.1) – (0,1.4); plot (,[(0.8\*-\^2+1)]{}); plot(,[1/(1+2\*sqrt(0.8\*-1))]{}); (0.5,1.3) node [$\Psit_1$]{}; (2,0.2) node [$\Psit_{2,\infty}$]{}; (1.48,0) node\[below right\] [$\frac{c+\sqrt{c^2+4\mub}}{2}$]{}; (0.8,0) – (0.8,1); (0.8,0) node\[below\] [$c$]{}; (0,1) – (1.25,1); (1.25,1) – (1.25,0); (1.25,0) node\[below\] [$\frac{\fp}{c}$]{}; (0,1) node\[left\] [$\mub$]{};
For the second part of Proposition \[asymptoticDfp\], we can see from geometric considerations (see figure \[FigDinfini\]) that $c_\infty$ must satisfy $$\label{ineqcinfty}
c_\infty \leq \frac{\fp}{c_\infty} \leq \frac{c_\infty+\sqrt{c_\infty^2+4\mub}}{2}.$$ Passing to the limit $\fp\to+\infty$ in (\[ineqcinfty\]) yields the expected result.
Enhancement of the spreading speed in the semi-limit case (\[RPSL2\])
=====================================================================
This section is devoted to the semi-limit model (\[RPSL2\]) and the proof of Proposition \[vitessemaxRPSL2\]. For $\mub>0,$ let $$\Lambda_{\mub} = \{\mu\in C_0(\R),\mu\geq 0, \int\mu=\mub,\mu \textrm{ is even} \}.$$ Now, for fixed constants $d,D,\nub,\fp,$ for any function $\mu\in\Lambda_{\mub},$ let $c^*(\mu)$ be the spreading speed associated to the semi-limit system (\[RPSL2\]) with exchange function from the road to the field $\mu.$ Let $c^*_0$ the spreading speed associated with the limit system (\[BRReq2\]) with the same parameters and exchange rate from the road to the field $\mub.$
##### Proof of Proposition \[vitessemaxRPSL2\]
If $D\leq 2d,$ then for all systems, $c^*=2\sqrt{d\fp}=c_K$ and the result is obvious. We consider only the case $D>2d.$ Let $c>2\sqrt{d\fp},$ $\displaystyle\la_2^\pm=\frac{c\pm\sqrt{c^2-c_K^2}}{2d}.$ Then, for all $\la\in]\la_2^-,\la_2^+[,$ the $(c,\la,\phi)-$equation (\[eqgeneralesurphi\]) associated to the semi-limit system (\[RPSL2\]) can be written as follows: $$\label{eqphiRPSL2}
\begin{cases}
-d\phi''(y)+\lp\la c-d\la^2+\fp\rp\phi(y) = \mu(y) \qquad y>0\\
-2d\phi'(0)=-\nub\phi(0), \qquad \phi\in H^1(\R^+).
\end{cases}$$ We keep in mind that we are interested in the behaviour of $$\Psi_2(\la;\mu):=\nub\phi(0)$$ where $\phi$ is the unique solution of (\[eqphiRPSL2\]). For the sake of simplicity, we set $$P(\la)=\la c-d\la^2-\fp,\qquad \a^2=\frac{P(\la)}{d}.$$ From the variation of constants method and the boundary conditions in 0 and $+\infty$ we have $$\phi(y) = e^{\a y}\lp K_1-\frac{1}{2\a}\int_0^y e^{-\a z}\frac{\mu(z)}{d}dz\rp +
e^{-\a y}\lp K_2+\frac{1}{2\a}\int_0^y e^{\a z}\frac{\mu(z)}{d}dz\rp.$$ where $$K_1=\frac{1}{2\a}\int_0^\infty e^{-\a z}\frac{\mu(z)}{d}dz,\ K_2 = \frac{2\a d-\nub}{2\a(2\a d+\nub)}\int_0^\infty e^{-\a z}\frac{\mu(z)}{d}dz.$$ We finally get, returning in the $(\la,c)-$variables, $$\label{formulePsi2RPSL2}
\Psi_2(\la;\mu):=\nub\phi(0)=\frac{2\nub}{\nub+2\sqrt{dP(\la)}}\int_0^\infty e^{-\sqrt{\frac{P(\la)}{d}}z}\mu(z)dz.$$ Now, since $e^{-\sqrt{\frac{P(\la)}{d}}z}\leq1$ for all $z\geq0$ and $\mu$ being nonnegative and even, it is easily seen that $$\label{ineqRPSL2}
\Psi_2(\la;\mu)\leq \Psi_2^0(\la;\mub)$$ where $\Psi_2^0$ is given by the limit model (\[BRReq2\]) associated to the same constants and exchange term $\mub.$ Hence, the above inequality (\[ineqRPSL2\]) allows us to assert that $$\forall \mu\in\Lambda_{\mub},\ c^*(\mu)\leq c^*_0.$$ Then, stating $c=c^*_0$, let us consider any approximation to the identity sequence in (\[formulePsi2RPSL2\]). For any $\mu\in\Lambda_{\mub},\e>0,$ set $\mu_\e(y)=\frac{1}{\e}\mu\lp\frac{y}{\e}\rp$. Then we get that $\Psi_2(\la;\mu_\e)$ converges to $\Psi_2^0(\la;\mub)$ as $\e$ goes to 0, uniformly in any compact set in $]\la_2^-,\la_2^+[$ in $\la.$ Hence, $$c^*(\mu_\e)\underset{\e\to 0}{\longrightarrow}c_0^*$$ and the proof of Proposition \[vitessemaxRPSL2\] is concluded.
Self-similar exchanges for the semi-limit case (\[RPSL\])
=========================================================
Considering the above result, it may seem natural that in the opposite case (\[RPSL\]), that is when exchanges from the road to the field are localised on the road, the spreading speed would also be maximum for localised exchange from the field to the road. In order to compare the spreading speed associated to the initial model (\[BRReq2\]) and the one given by an integral model (\[RPSL\]), it is first natural to look for the behaviour of the spreading speed when replacing the exchange function $\nu$ by a self-similar approximation of a Dirac mass $\frac{1}{\e}\nu(\frac{y}{\e}).$ Hence, for a fixed constant rate $\nub,$ we will consider an exchange function of the form $$\nu\in\Lambda_{\nub}:=\{\nu\in C_0(\R),\nu\geq 0, \int\nu=\nub,\nu \textrm{ is even} \}.$$ For fixed constant $\fp,d,D,\mub,$ and $\nu\in\Lambda_{\nub}$ let $c^*_0$ be the spreading speed associated to the limit system (\[BRReq2\]), and $c^*(\e)$ the spreading speed associated to the semilimit model (\[RPSL\]) with exchange term $$\nu_\e : y\longmapsto \frac{1}{\e}\nu\lp\frac{y}{\e}\rp.$$ The $(c,\la,\phi)-$equation (\[eqgeneralesurphi\]) associated is $$\label{eqsurphiRPSLlim}
\begin{cases}
-d\phi''(y)+(P(\la)+\nu_\e(y))\phi(y) = \mub\d_0 \\
\phi \in H^1(\R),\ \phi \textrm{ is continuous}.
\end{cases}$$ The $\Psi_2$ function is given by $$\Psi_2(\la,c;\e)=\int_\R \nu_\e(y)\phi(y)dy$$ where $\phi$ is the unique solution of (\[eqsurphiRPSLlim\]) and $P(\la)=\la c-d\la^2-\fp.$ An integration of (\[eqsurphiRPSLlim\]) yields the following expression for $\Psi_2$ $$\label{eqPsi2}
\Psi_2(\la,c;\e)=\mub-P(\la)\int_\R \phi(y;\la,c,\e)dy$$ from which we get the next proposition.
\[deriveePsi2positive\] The function $\Psi_2,$ defined by (\[eqPsi2\]) and (\[eqsurphiRPSLlim\]), is continuously differentiable in all variables $\la,c,\e$ up to $\e=0$ and satisfies for all $\la,c$ $$\frac{d}{d\e}{\Psi_2}_{|\e=0}>0.$$
Considering the monotonicity of $\Psi_2$ with respect to $c,$ this provides the corollary
\[corollaireVitesse\] Let us consider $c^*$ as a function of the $\e$ variable. Then there exists $\e_0,$ $$\forall \e<\e_0,\ c^*(\e)>c^*_0$$ In other words, the Dirac mass is a local minimizer for the spreading speed when considering approximation of Dirac functions.
#### Proof of Proposition \[deriveePsi2positive\] {#proof-of-proposition-deriveepsi2positive .unnumbered}
Throughout the proof, the function $\phi,$ depending on $\e,$ will be the solution of (\[eqsurphiRPSLlim\]), and we will denote $$\vp:=\frac{d}{d\e}\phi$$ its derivative with respect to $\e.$ Moreover, once again, we set $d=1$ for the sake of simplicity, and consider an exchange function $\nu$ with support in $[-1,1].$ Differentiating (\[eqsurphiRPSLlim\]) we obtaint that $\vp$ is the unique solution of $$\label{deriveephiSL}
\begin{cases}
-\vp''(y)+\lp P(\la+\frac{1}{\e}\nu\lp\frac{y}{\e}\rp)\rp\vp(y) = \frac{1}{\e^2}g\lp\frac{y}{\e}\rp\phi(y) \\
\vp\in H^1(\R)
\end{cases}$$ where the function $g,$ with compact support in $[-1,1],$ is defined by $$\label{defg}
g:z\mapsto \frac{d}{dz}[z.\nu(z)].$$
Thanks to (\[eqPsi2\]), it is enough to prove that $\vp$ tends to a negative function as $\e$ goes to 0, uniformly $L^1$ near $\e=0.$ The proof is divided in four steps. We first recall the convergence of $\phi$ as $\e$ goes to 0. Then, the most important step is the convergence of the righthandside of (\[deriveephiSL\]) to a Dirac measure of negative mass. The third step is to find some uniform boundedness for the sequence $\lp\vp\rp_\e,$ in order to finally pass to the limit and conclude the proof.
##### Convergence of $\phi$ {#convergence-of-phi .unnumbered}
It has been proved in [@Pauthier2] that $\phi$ converges in the $C^1$ norm to $$\label{phi0}
\phi_0:y\mapsto \frac{\mub}{\nub+2\sqrt{P(\la)}}e^{-\sqrt{P(\la)}|y|}$$ as $\e$ goes to 0, and this convergence is locally uniform in $\la,c.$ Actually, the monotonicity of $\phi$ with respect to $y$ makes the proof easier.
##### Convergence of the righthandside of (\[deriveephiSL\]) to a Dirac mass {#convergence-of-the-righthandside-of-deriveephisl-to-a-dirac-mass .unnumbered}
As its support shrinks to 0, and thanks to the reguarity of $g$ and $\phi$ uniformly in $\e,$ it is enough to prove the convergence of the mass to get the convergence in the sense of distribution. Let us consider the integral $$I(\e):=\int_\R \frac{1}{\e^2}g\lp\frac{y}{\e}\rp\phi(y;\e)dy.$$ Evenness of $g$ and $\phi,$ compact support of $g,$ and a Taylor formula yield $$\label{int1}
\frac{1}{2}I(\e)=\frac{1}{\e}\int_0^1 g(z)\lp\phi(0)+\e z\phi'(0)+\int_0^{\e z}(\e z-t)\phi''(t)dt\rp dz.$$ Recall that $g$ is defined by (\[defg\]), so $\displaystyle \int_0^1 g(z)dz=0.$ An integration by parts gives $$\int_0^1 z.g(z)dz=-\int_0^1z.\nu(z)dz>0.$$ It remains to determine the last term in (\[int1\]) given by $$I_2=\frac{1}{\e}\int_0^1 g(z)\int_0^{\e z}(\e z-t)\phi''(t)dtdz.$$ Recall that $\phi$ is a solution of (\[eqsurphiRPSLlim\]) and we get locally in $\la,c$ $$\begin{aligned}
I_2 & = \e\int_0^1 g(z)\int_0^z \lp z-u\rp\lp P(\la)+\frac{1}{\e}\nu(u)\rp\phi(\e u)dudz \nonumber \\
& = \int_0^1 g(z)\int_0^z \lp z-u\rp\nu(u)\phi(\e u)dudz +O(\e). \label{intC}\end{aligned}$$ With the uniform boundedness of $\lV\phi'\rV_{L^\infty}$ in $\e,$ (\[intC\]) becomes $$\begin{aligned}
I_2 & = \phi(0)\lp \int_0^1 z.g(z)\int_0^z \nu(u)dudz - \int_0^1 g(z)\int_0^z u.\nu(u)dudz \rp + O(\e)\nonumber \\
& = -\phi(0)\int_0^1 z\nu(z)\int_0^z \nu(u)dudz+ O(\e). \label{intC12}\end{aligned}$$ Inserting (\[intC12\]) in (\[int1\]), and with the convergence together to its derivative of $\phi$ to $\phi_0$ defined by (\[phi0\]), we get, as $\e$ tends to 0, $$\label{integralelimite}
I \underset{\e\to 0}{\longrightarrow}\mub\int_0^1\lp \frac{1}{\nub+2\sqrt{P(\la)}}\int_{-z}^z\nu(u)du - 1 \rp z.\nu(z)dz:=I_0.$$ We notice that $I_0<0.$
##### Uniform boundedness for $\lV\vp\rV_{L^\infty}$ {#uniform-boundedness-for-lvvprv_linfty .unnumbered}
Once again, let us set $\a^2:=P(\la).$ As $g$ is compactly supported and even, for all $\e>0,$ there exists $K(\e)$ such that $$\label{yplusgdqueeps}
\forall |y|>\e,\qquad \vp(y)=K(\e) e^{-\a|y|}.$$ We do the change of variable $\xi=\frac{y}{\e}$ and use the same notation $\vp(\xi)=\vp(y)$ for the sake of simplicity. Equation (\[eqsurphiRPSLlim\]) becomes in the $\xi-$variable $$\label{eqenxi}
\begin{cases}
-\vp''(\xi)+\lp \e^2\a^2+\e\nu(\xi) \rp\vp(\xi)=g(\xi)\phi(\e\xi) \\
\vp(\pm1)=K(\e)e^{-\a\e}.
\end{cases}$$ As for Theorem \[thmdenonacceleration\], let us set $$\vp(\xi)=\vp_0(\xi)+\e\vp_1(\xi)$$ where $\vp_0$ is the unique solution of $$\begin{cases}
-\vp_0''(\xi)=g(\xi)\phi(\e\xi) \\
\vp_0(\pm1)=K(\e)e^{-\a\e}.
\end{cases}$$ This yields the following explicit formula for $\vp_0$ $$\label{varpi0}
\vp_0(\xi)=K(\e)e^{-\a\e}-\int_{-1}^\xi\int_0^z g(u)\phi(\e u)dudz.$$ Now we introduce the operator $$\ML : \begin{cases}
X & \longrightarrow X \\
\psi & \longmapsto \left\{ \xi\mapsto \int_{-1}^\xi\int_0^z\lp\e\a^2+\nu(u) \rp\psi(u)dudz \right\}
\end{cases}$$ where $X=\{\psi\in C^1(-1,1),\psi \textrm{ is even}\}$ endowed with the $C^1$ norm. $\ML$ is obviously a bounded operator and $\vp_1$ satisfies $$\lp I-\e\ML\rp\vp_1=\ML\vp_0.$$ Hence there exists a constant $C,$ for $\e$ small enough, $\displaystyle \lV\vp_1\rV_{C^1(-1,1)}\leq C\lV\vp_0\rV_{C^1(-1,1)}.$ We also have the integral equation for $\vp_1$ $$\vp_1(\xi)=\int_{-1}^\xi\int_0^z\lp\e\a^2+\nu(u) \rp\lp \vp_0+\e\vp_1\rp(u) dudz.$$ The continuity of the derivative in 1 gives $$\label{continuitederivee}
\vp_0'(1)+\e\vp_1'(1)=-\e\a K(\e)e^{-\a\e}.$$ The computation done in the previous paragraph yields: $$\begin{aligned}
\vp_0'(1) & = -\int_0^1 g(u)\phi(\e u)du \nonumber \\
\vp_0'(1) & = -\e\frac{1}{2}I_0 + o(\e) \label{phi0prime1}\end{aligned}$$ where $I_0$ is defined by (\[integralelimite\]). Using the integral equation for $\vp_1,$ the previous domination, (\[varpi0\]) and at last the convergence of $\phi$ as $\e$ goes to 0, there exists a constant $M$ such that $$\begin{aligned}
\vp_0'(1) & = \int_0^1\lp\e\a^2+\nu(u) \rp\lp \vp_0+\e\vp_1\rp(u) du \nonumber \\
& = \frac{\nub}{2}K(\e)-\int_0^1\nu(\xi)\int_{-1}^\xi\int_0^z g(u)\phi(\e u)dudzd\xi+O\lp\e.K(\e)\rp \nonumber \\
\vp_0'(1) & = \frac{\nub}{2}K(\e)-M+O\lp\e\lp1+K(\e)\rp\rp. \label{phi1prime1}\end{aligned}$$ Insert (\[phi0prime1\]) and (\[phi1prime1\]) in (\[continuitederivee\]) and we get $$\lim_{\e\to0}\sup \ |K(\e)|<+\infty,$$ which provides with (\[varpi0\]) and (\[yplusgdqueeps\]) the boundedness of $\lV\vp\rV_{L^\infty(\R)}$ as $\e$ goes to 0. Moreover, the bound is locally uniform on $\la,c.$
##### Convergence of $\vp,$ conclusion of the proof {#convergence-of-vp-conclusion-of-the-proof .unnumbered}
We return to the initial variable. Let $K_0$ be any limit point of $\lp K(\e)\rp_\e.$ Then a subsequence of $\lp \vp\rp_\e$ converges in the sense of distributions to $$\vp_l(y)=K_0 e^{-\sqrt{P(\la)}|y|}$$ and $\vp_l$ satisfies in the sense of distributions $$-\vp_l''(y)+\lp P(\la)+\nub\d_0\rp\vp_l(y)=I_0\d_0$$ whose unique solution is $$\label{limitevp}
\vp_l:y\longmapsto \frac{I_0}{\nub+2\sqrt{P(\la)}}e^{-\sqrt{P(\la)}|y|}.$$ Being the only possible limit point, (\[limitevp\]) is the limit of $(\vp)$ as $\e$ goes to 0. $I_0$ is negative, and so is $\vp_l.$ The uniform boundedness allows the derivation in (\[eqPsi2\]), and the proof is concluded.
Proof of Corollary \[corollaireVitesse\]
----------------------------------------
Let $c^*_0$ the spreading speed associated to the limit model (\[BRReq2\]), $(c^*_0,\la^*_0,\phi_0(c^*_0,\la^*_0))$ the corresponding linear travelling wave. We consider $\Psi_2$ as a function of $(\la,c;\e),$ $\Psi_1$ as a function of $(\la,c).$ We have $\Psi_1(\la^*_0,c^*_0)=\Psi_2(\la^*_0,c^*_0;0)$ and $\Psi_1(\la^-_2,c^*_0)<\Psi_2(\la^-_2,c^*_0;0).$ Hence ,there exists $\las,$ $\la_2^-<\las<\la_0^*,$ $\Psi_2(\las,c^*_0;0)=\Psi_1(\la_2^-,c^*_0).$ Let $V$ be any open set in $]\la_2^-,\la_2^+[$ containing $\las$ and $\la_0^*.$ From Proposition \[deriveePsi2positive\], there exists $\e_0$ such that $\forall \e<\e_0,$ $\forall \la\in V,$ $\Psi_2(\la,c^*_0;\e)>\Psi_2(\la,c^*_0;0).$ From the definition of $\las,$ it yields $\Psi_2(\la,c^*_0;\e)>\Psi_1(\la,c^*_0),$ $\forall \la.$ The monotonicity of $\Psi_1$ and $\Psi_2$ with respect to $c$ concludes the proof.
The semi-limit case (\[RPSL\]): non optimality of concentrated exchanges
========================================================================
Considering the above result, it may seem natural that in the case (\[RPSL\]), that is when exchanges from the road to the field are localised on the road, the spreading speed would be minimal for localised exchange from the field to the road. The purpose of this section is the proof of Theorem \[thmdenonacceleration\] in which we show that any behavour may happen in the neighbourhood of a Dirac measure. For the sake of convenience, throughout this section we set $$d=\nub=1.$$ Let us recall that we consider exchange terms $\nu$ of the form $$ \nu(y)=(1-\e)\d_0+\e\up(y)$$ where $$\up\in\Lambda_1:=\{\up\in C_0(\R),\up\geq 0, \int\up=1,\up \textrm{ is even} \}.$$
Let $c^*_0$ the spreading speed associated to the limit model (\[BRReq2\]), $(c^*_0,\la^*_0,\phi_0(c^*_0,\la^*_0))$ the corresponding linear travelling waves. The $(c,\la,\phi)-$equation associated to the system (\[RPSL\]) with exchange term of the form (\[nuperturbe\]) is as follows, completed by evenness: $$\label{eqphinupert}
\begin{cases}
-\phi''(y)+\lp-\fp+\la c-\la^2+\up(y)\rp\phi(y) = 0 \qquad y>0\\
\phi'(0)=\frac{1}{2}\lp(1-\e)\phi(0)-\mub\rp, \qquad \phi\in H^1(\R^+).
\end{cases}$$ The associated function $\Psi_2$ is given by $$\label{Psi2pert}
\Psi_2(\la,c)=(1-\e)\phi(0)+\e\int_\R \up\phi$$ where $\phi$ is the unique solution of (\[eqphinupert\]). What we have to show is that, in a neighbourhood of $(\la^*_0,c^*_0),$ the difference $\lp\Psi_2^0(\la,c)-\Psi_2(\la,c)\rp$ is of constant sign for $\e$ small enough, and that this sign can be different depending on the parameters $D,\mub,\fp.$ Once again, for the sake of simplicity and as long as there is no possible confusion, we set $$P(\la)=-\fp+\la c-\la^2, \qquad \a=\sqrt{P(\la)}.$$
Of course, we are looking for function $\phi$ of the form $$\label{ansatz}
\phi = \phi_0+\e\phi_1$$ where $\phi_0$ is solution of (\[eqsurphiBRR\]). Hence, $\phi_1$ satisfies $$\label{eqphi1}
-\phi_1''+(\d_0+\a^2)\phi_1 = \lp\d_0-\up\rp\lp\phi_0+\e\phi_1\rp.$$
\[bornephi1\] Let $\a_0>0.$ There exist $\e_0>0,$ $K>0,$ depending only on $\a_0,$ such that $\forall \e<\e_0,$ $$\lV\phi_1\rV_{L^\infty} \leq K \lV\phi_0\rV_{L^\infty}$$ where $\phi_1$ is the solution of (\[eqphi1\]). We may also keep in mind that $\lV\phi_0\rV_{L^\infty}=\phi_0(0).$ We can see in (\[solPsi2BRR\]) that it is uniformly bounded in $\a,D,\fp.$
We introduce the operator $$\ML : \begin{cases}
X & \longrightarrow X \\
\psi & \longmapsto \vp
\end{cases}$$ where $X=\{\psi\in BUC(\R),\psi \textrm{ is even}\}$ and $\vp$ is the only bounded solution of $$-\vp''+(\a^2+\d_0)\vp = \lp\d_0-\up\rp\psi.$$ From (\[eqphi1\]), it is easy to see that $\phi_1$ satisfies $
\phi_1=\ML\phi_0+\e\ML\phi_1.
$ As $\up$ and $\phi_0$ are even, we focus on $\ML$ defined for bounded, uniformly continuous even functions. Let $\psi\in BUC(\R)$ be any even function, and $\vp:=\ML\psi.$ That is, $\vp$ satisfies $$\begin{cases}
-\vp''+\a^2\vp = -\up\psi \qquad y>0 \\
\vp'(0)=\frac{1}{2}\lp\vp(0)-\psi(0)\rp.
\end{cases}$$ As in the previous section, a simple computation gives $$\begin{aligned}
\vp(y) = & -\frac{e^{\a y}}{2\a}\int_y^\infty e^{-\a z}(\up\psi)(z)dz \label{calculL} \\
& + e^{-\a y}\lp \frac{\psi(0)}{1+2\a}+\frac{1-2\a}{2\a(1+2\a)}\int_0^\infty e^{-\a z}(\up\psi)(z)dz -\frac{1}{2\a}\int_0^y e^{\a z}(\up\psi)(z)dz \rp. \nonumber\end{aligned}$$ Recall that $\up$ is nonnegative and of weight 1, and $\a=\sqrt{P(\la)}>0$. A rough majoration in (\[calculL\]) yields $$\label{normeoperator}
\lV\vp\rV_{L^\infty} \leq \lV\psi\rV_{L^\infty}\lp\frac{1}{1+2\a}+\frac{\lb 1-2\a\rb}{4\a(1+2\a)}+\frac{1}{4\a}\rp.$$ Hence $\ML$ is a bounded linear operator, with norm $\lV\ML\rV$ depending on $\a,$ and uniformly bounded on $\a>\a_0>0.$ For $\e$ small enough, $(I-\e\ML)$ is invertible with bounded inverse and $$\label{operator}
\phi_1 = \lp I-\e\ML\rp^{-1}\ML \phi_0.$$ Moreover, $\phi_1$ satisfies the integral equation $\phi_1=\ML(\phi_0+\e\phi_1)$ given by (\[calculL\]). Combining (\[operator\]) with (\[normeoperator\]) concludes the proof of Lemma \[bornephi1\].
##### The difference $\Psi_2^0-\Psi_2$
The function $\Psi_2$ is given by (\[Psi2pert\]) with $\phi$ of the form (\[ansatz\]). Then, using Lemma \[bornephi1\], for all $\a>\a_0,$ $$\begin{aligned}
\Psi_2^0-\Psi_2 & = \phi_0(0)-(1-\e)\lp\phi_0(0)+\e\phi_1(0)\rp-\e\int_\R\lp\phi_0+\e\phi_1\rp\up \nonumber \\
& = \e\lp\phi_0(0)-\phi_1(0)-\int_\R\up\phi_0\rp+o(\e). \label{difference1}\end{aligned}$$ It appears necessary to compute $\phi_1(0).$ Equation (\[calculL\]) gives $$\begin{aligned}
\phi_1(0) = & \lp\frac{1-2\a}{2\a(1+2\a)}-\frac{1}{2\a}\rp\int_\R^\infty e^{-\a y}\up(y)\lp\phi_0(y)+\e\phi_1(y)\rp+
\frac{1}{1+2\a}\lp\phi_0(0)+\e\phi_1(0)\rp \nonumber \\
= & \frac{\phi_0(0)}{1+2\a}-\frac{2}{1+2\a}\int_0^\infty e^{-\a y}\up(y)\phi_0(y)dy + O(\e) \nonumber \\
= & \frac{\phi_0(0)}{1+2\a}\lp 1- \int_\R e^{-2\a|y|}\up(y)dy\rp +O(\e). \label{phi1zero}\end{aligned}$$ Now recall that $\up$ is of mass 1 and, using (\[phi1zero\]) in (\[difference1\]), $$\begin{aligned}
\Psi_2^0-\Psi_2 = & \e\phi_0(0)\int_\R\up(y)\lp 1-e^{-\a|y|}-\frac{1}{1+2\a}\lp 1-e^{-2\a|y|}\rp\rp dy+o(\e) \nonumber \\
= & \e\phi_0(0)\int_\R\up(y)g(\a,y) dy+o(\e). \label{difference2}\end{aligned}$$ The function $g$ is obviously even in $y$, and smooth on ${\R_*^+}^2.$ We can easily see that:
- if $\a\geq\frac{1}{2},$ then $\forall y>0,\ g(\a,y)>0.$
- If $\a<\frac{1}{2},$ then there exists $y(\a)$ such that, in a neighbourhood of $\a,$ $\forall|y|<y(\a),$ $g(.,y)<0.$
We are interested in the local behaviour near $(\la^*,c_0^*).$ Hence, $g(\a,y)$ has to be considered near $\a^*:=\sqrt{-\fp+c^*_0\la^*-{\la^*}^2}=P(\la^*).$
##### Perturbation enhancing the velocity: $\a^*<1/2$
$P$ achieves its maximum at $\la=\frac{c}{2}$ and $c\mapsto P(\frac{c}{2})$ is nondecreasing. From [@BRR1] we know that $c^*_0$ satisfies $$\frac{c^*_0}{D}\leq \frac{c^*_0-\sqrt{{c^*_0}^2-c_K^2}}{2}$$ where $c_K=2\sqrt{\fp}$ is the classical spreading speed for KPP-type reaction-diffusion. It follows easily that $c^*_0\leq \frac{D\sqrt{\fp}}{\sqrt{D-1}}$ which, combined with the two upper remarks, yields the following sufficient condition for $\a^*=\sqrt{P(\la^*)}$ to be less than $1/2$: $$\label{conditionm1}
D < 2+\frac{1}{2\fp}+\frac{1}{2}\sqrt{12+(\frac{1}{\fp})^2+\frac{7}{\fp}}:=m_1.$$ Hence, provided the condition (\[conditionm1\]) holds, $\a^*<1/2$ and there exists $y(\a^*)$ and a neighbourhood $\MV$ of $\a^*$ such that $g(\a,y)<0$ for $|y|<y(\a^*)$ and $\a\in\MV.$ Take $\up$ such that $\textrm{supp}(\up)\subset]-y(\a^*),y(\a^*)[$ and, for all $\a\in\MV,$ that is for all $\la$ in a neighbourhood of $\la^*,$ for $\e$ small enough, $$\lp\Psi_2^0-\Psi_2\rp(\la,c^*_0)<0.$$ The result follows from the monotonicity of $\Psi_2$ with respect to $c.$
##### Locally maximal velocity for for $\nu=\d_0:$ $\a^*>1/2$: proof of Theorem \[thmdenonacceleration\]
It remains to show that $\a^*$ can be greater than $\frac{1}{2}.$ We will need the second part of Proposition \[asymptoticDfp\]. From now, we fix an exchange rate $\mub>4.$ We will use the fact that, at $(c^*_0,\la^*),$ $$\label{egalitederivees}
\frac{d}{d\la}\lp\Psi_1-\Psi_2^0\rp(\la)=0.$$ Explicit computation gives $$\label{deriveesPsi}
\begin{cases}
\frac{d}{d\la}\Psi_1(\la) & = -2D\la+c \\
\frac{d}{d\la}\Psi_2^0(\la) & = -\frac{\mub(c-2\la)}{\sqrt{P(\la)}\lp 1+2\sqrt{P(\la)}\rp^2}.
\end{cases}$$ Recall that $\la^*$ has to satisfy $$\frac{c^*_0}{D}\leq \la^*\leq \la_1^+:=\frac{c^*_0+\sqrt{{c^*_0}^2+4D\mub}}{2D}.$$ Now applying Lemma \[asymptoticDfp\], for all $\d>0$ there exists $M>0,$ $\fp,D>M$ entails $\lb\Psi_1'(\la^*)-c^*_0\rb<\d$ and $\lb \la_2^-\rb<\d$ (recall that $\la_2^-=\frac{c-\sqrt{c^2-c_K^2}}{2}$). To prove that $\a^*=\sqrt{P(\la^*)}>1/2,$ we distinguish two cases.
First case: $\displaystyle\la^*>\frac{1}{2}\lp \la_2^-+\frac{c^*_0}{2}\rp.$ Thus $\displaystyle\la^*>\frac{c^*_0}{4}-\d$ which yields with Lemma \[asymptoticDfp\] $$P(\la^*)=D\fp\frac{3}{16}-\fp+O(\d D\fp)>\frac{1}{4}.$$
Second case: $\displaystyle\la_2^-<\la^*<\frac{1}{2}\lp \la_2^-+\frac{c^*_0}{2}\rp.$ Thus, from (\[egalitederivees\]), (\[deriveesPsi\]) and the above inequalities given by Lemma \[asymptoticDfp\], $$c^*_0+\d>\lb{\Psi_2^0}'(\la^*)\rb=\lb\frac{\mub(c^*_0-2\la^*)}{\a^*(1+2\a^*)^2}\rb
>\frac{2(c^*_0-2\d)}{\a^*(1+2\a^*)}$$ which implies $\a*>1/2$ for $\d$ small enough. This concludes the proof of Theorem \[thmdenonacceleration\].
[^1]: e-mail: `antoine.pauthier@math.univ-toulouse.fr`
|
---
abstract: 'Enhancing motivation and learning attitudes in an introductory physics course is an important but difficult task that can be achieved through class blogging. We incorporated into an introductory course a blog operated by upper-level physics students. Using the Colorado Learning Attitudes about Science Survey (CLASS), periodic in-class surveys, analysis of student blog comments, and post-instructional interviews, we evaluate how the blog combined with class instruction provided the students with a better sense of relevance and confidence and outline recommendations for future use of this strategy.'
author:
- 'Ashley A. August'
- 'Kenneth C. Bretey'
- 'Bryant T. Cory'
- 'Elliott R. Finkley III'
- 'Robbie D. Jones'
- 'Dennis W. Marshall'
- 'Phillip C. Rowley'
- 'W. Brian Lane'
title: 'Enhancing Introductory Student Motivation with a Major-Managed Course Blog: A Pilot Study'
---
INTRODUCTION
============
The Importance of Motivation in Physics Education
-------------------------------------------------
Physics education research has identified a variety of student learning attitudes that shape and are shaped by the learning experience,[@Hammer:00] many of which are not explicitly addressed in typical introductory courses.[@Redish]^,^[@Tarshis] Similarly, learning theory describes the importance of considering student motivation in curriculum design.[@Ames]^,^[@Keller2] In particular, it is important for science instructors to consider the motivational factors of relevance and confidence,[@Hammer:94]^,^[@Redish] which have a significant impact on student performance in both class and lab[@Lynch] since they determine the amount and quality of effort that the students put forth.[@Burke]
Instruments that measure student learning attitudes and motivational factors reveal that introductory physics courses typically result in notable declines.[@CLASS]^,^[@Redish] The challenge for instructors, therefore, is to design instructional strategies that can encourage, develop, and reward these learning attitudes and motivational factors.
Course Blogs: A Possible Remedy
-------------------------------
Weblogs can be used in an introductory physics course to improve attitudes and expectations by providing an opportunity for the students to relate to the course material and to develop their confidence. One study[@Duda_a]^,^[@Duda_b] that used an extra-credit course blog written by the instructors to supplement class discussion found that students that actively participated in the blog maintained more positive attitudes towards physics (particularly a stronger sense of relevance of physics concepts) throughout the course than those who did not actively participate. A second study[@Harrison] that required the students to compose articles on a course blog found that students engaged at a deeper level on the blog than they did in class or traditional homework because of the freedom to pursue their interests and because of their sense of involvement in an online community.
Course blogs offer a number of useful benefits to instruction because of their ability to$\ldots$
- Encourage a combination “of solitary thought and social interaction to engage students and reinforce learning,”[@Harrison] which can result in improved confidence.
- Create a community of learning and collaboration that benefits students of all personality types and learning styles.[@Harrison]
- Help students develop critical thinking and reflection skills.[@Harrison]
- Assess and capitalize on students’ pre-instructional interests,[@Harrison] thereby improving their sense of relevance.
- Incorporate into the learning community outsiders at different stages along the novice-to-expert journey.[@Sprague]^,^[@Daley]^,^[@Dreyfus]
- Complement and enhance other instructional strategies.[@Higdon]
- Provide students with real-world experience working with Web media.[@Harrison]
Course blogs can also present certain challenges, such as students’ participating under the pressures of the last minute or distractions,[@Harrison] which can have a detrimental impact on the students’ learning experience and motivation. When using a course blog, the instructor may need to make explicit connections in class between the course material and blog content.[@Harrison] Doing so can encourage students to participate and successfully develop their sense of relevance.
In Section II, we describe the implementation and mixed-methods assessment strategy of an introductory course blog managed by upper-level physics majors. In Section III, we present the results of the instructional assessment and in Section IV we present a set of recommendations based on that assessment. After wrapping up with a few conclusions in Section V, we present the assessment materials in Appendix A.
THE PRESENT STRATEGY: A MAJOR-MANAGED INTRODUCTORY COURSE BLOG
==============================================================
We conducted a semester-long implementation of a blog in a spring 2011 introductory physics course for Aviation majors operated by upper-level physics majors in order to improve the learning attitudes and motivational factors of the introductory physics students. The upper-level physics majors were selected as blog authors to function as “mediators” of the course material between the instructor and the introductory students, since the upper-level physics majors are presumably found between the extremes of novice and expert.[@Dreyfus]^,^[@Daley]
The Introductory Students of Interest
-------------------------------------
These Aviation majors train to become commercial or military pilots while pursuing an undergraduate business degree. Typically, Aviation Physics is the only lab science course these students take. The course is designed to provide the students with some understanding of how aircraft and aircraft systems physically operate, producing well-rounded pilots who are more aware and more adaptive. As was revealed by student blog comments and post-instructional interviews, many of these students have not had much, if any, exposure to physics in a formal learning environment.
Insight into these Aviation majors’ motivation, learning attitudes toward physics, and experience in the Aviation Physics course can be gained by examining the pre- and post-instructional results of the Colorado Learning Attitudes about Science Survey (CLASS), which assesses student epistemological beliefs about physics and learning physics along various categories by reporting a percent favorable score that indicates the students’ agreement with expert-like beliefs.[@CLASS] In this study, we focus on the percent favorable score of the overall survey and of the categories of Personal Interest, Real World Connection, Applied Conceptual Understanding, and Problem Solving Confidence. The first three of these categories are related to the motivational factor of relevance; the fourth category is related the motivational factor of confidence.[@CLASS]
Figure 1 depicts the CLASS results for the fall 2010 semester (during which the present instructional strategy was not implemented), indicating that the Aviation majors enter the course with novice-like approaches to physics, and that these beliefs do not make strong shifts toward expert-like approaches. In particular, the Real World Connection percent favorable score showed a statistically significant decrease from pre- to post-instruction of 10%, illustrating that the course did not positively impact the students’ senses of relevance. Such decline is a common occurrence in introductory physics courses.[@Redish]^,^[@CLASS] The only percent favorable score to show a statistically significant increase was that for Problem Solving Confidence, likely due to the fact that problem solving practice is heavily emphasized in the Aviation Physics course.
Blog Design
-----------
Because of its ability to link to external sources and provide multimedia-based presentation of concepts, the major-managed course blog was conceived as an opportunity to develop the spring 2011 students’ sense of relevance. In the first week of running the course blog, the introductory students were asked what benefits they hoped to gain from the course blog; the majority of the students indicated that they would like the blog to serve as a venue for seeing demonstrations of and practicing the types of problems they would be expected to solve in the course, indicating a felt need for a better sense of confidence.
The blog structure was therefore designed by the authors[@PEER] to focus on developing the students’ motivational factors of relevance and confidence.[@Keller1]^,^[@Scherr] The blog articles were categorized into six topic sequences, each of which was authored by one or more upper-level physics majors or the instructor:
1. *Physics in the News*. This topic sequence showed the introductory students how physics relates to the real world, and how physics is an ever-evolving field of study.
2. *Physics in Aviation*. This topic sequence directly related physics to the Aviation majors’ primary field of study.
3. *Physics in Entertainment*. This sequence provided a discussion of physics principles (both accurate and inaccurate) in movies and television shows, which has been demonstrated to be an effective instructional supplement.[@Hadz]
4. *Physics Humor*. This topic sequence asked the students to apply the concepts of the course to understand physics-related humor. Such humor-related content has been demonstrated to be an effective instructional supplement.[@Tatalovic]
5. *Sample Problems*. Usually, students only encounter sample problems in the textbook (which is archived but not interactive) and in the classroom (which is interactive, but not always well archived); these articles provided an interactive and automatically archived learning experience.
6. *Graphing Project Head Start*. The introductory students were given a weekly assignment that asked them to investigate a theoretical model of interest, involving the creation and discussion of a graph depicting the results of their model. This topic sequence helped them begin these assignments by including tips, reminders of previously learned techniques, and further explanations of the model.
In the established motivational framework,[@Keller1]^,^[@Hammer:94]^,^[@Redish] topic sequences 1 through 4 were designed to enhance the introductory students’ sense of relevance, while topic sequences 5 and 6 were designed to enhance the introductory students’ sense of confidence. Topic sequence 1 was written by the instructor. Topic sequences 2 through 4 were written by one upper-level major each. Topic sequences 5 and 6 were written by two upper-level majors each. Each upper-level major was required to post an article to the blog once a week, following a set of posting guidelines designed by the upper-level majors to ensure professionalism, consistency, effective writing and multimedia use, and engaging the introductory students in the on-line conversation.[@PEER] The upper-level physics majors participated as blog authors in partial fulfillment of a senior-level physics seminar course to develop their skills in communication, collaboration, research, and technology. In the five-stage novice-to-expert transition scheme of Dreyfus & Dreyfus,[@Dreyfus]^,^[@Daley] these upper-level physics majors exhibit characteristics of the third stage called “competent” (able to plan and adapt) and are advancing to the fourth stage of “proficient” (able to see the big picture). Thus, they served as helpful mediators between the instructor (at the fifth stage of “expert”) and the introductory students (at the first stage of “novice”).
Means of Assessment
-------------------
The impact of the major-managed course blog on introductory student motivation was assessed by the following mixed-methods approach:
1. The pre-to-post-instruction shifts in the introductory students’ responses to the CLASS[@CLASS] assessed the general impact of the course.
2. Short, periodic in-class surveys were administered throughout the semester to monitor the introductory students’ senses of relevance and confidence.[@Keller1] These surveys (found in Appendix A) were based on established design principles for course-related self-confidence surveys and interest checklists.[@CAT]
3. The frequency and content of the introductory students’ comments on the blog were analyzed to further elaborate on the survey data and to explore to what degree they engaged in each of the course blog’s topic sequences. Duda & Garrett[@Duda_b] identify five dimensions for evaluating student comments on a course blog: (1) student interactivity, (2) students’ introduction of new knowledge, (3) students’ relating the post to “the course material, real-life, or other disciplines,” (4) “self-disclosure of prior knowledge or admission of learning,” and (5) expression of fascination or interest. Similarly, Harrison[@Harrison] describes quality blog posts as those that have substantial content and that relate to the principles of the course.
4. End-of-semester interviews with the introductory students were conducted to further investigate their responses to the in-class surveys and the CLASS, to explore how those responses were impacted by the course blog, and to receive feedback from the students about the blog.
ASSESSMENT RESULTS
==================
CLASS Shifts
------------
Figure 2 shows the results of the pre- and post-instructional CLASS, indicating that, on the whole, these students did not exhibit the decline in percent favorable scores that are typical in the literature,[@CLASS] a similar feature to those students who participated in the course blog in Duda & Garrett’s study.[@Duda_a] In fact, each category (including those not reported here) saw an increase in the average percent favorable score, though not all increases were statistically significant.
The one category score to display a statistically significant shift was Personal Interest percent favorable. The Applied Conceptual Understanding percent favorable score showed an increase of comparable size (though not statistically significant). The Real-World Connection percent favorable score showed only a slight increase, but when compared with the shifts from the previous semester (which showed a statistically significant decline of 10%), it would seem that the spring 2011 course was successful in at least maintaining the students’ strong pre-instructional sense of Real-World Connection. These results all indicate that, on the whole, the students’ motivational factor of relevance seems to have been positively impacted by the course.
Examining the Problem Solving Confidence category, we see that the students’ average percent favorable score increased somewhat. These results are not as strong as in the previous semester, but still represent a positive impact on the students’ sense of confidence in applying the course material.
The comparison of the pre- and post-instructional CLASS results indicates that the course made a positive impact on the students’ senses of relevance and confidence in the course material. Such impact is notable when compared with results typical in the literature[@CLASS] which exhibit a notable decline across CLASS categories. The task that remains is to identify if and how the course blog contributed to these changes; to do so, we turn to the in-class surveys, the blog comments, and the post-instructional interviews.
In-Class Surveys
----------------
Short in-class surveys (found in Appendix A) were given every two weeks to determine shifts in the students’ motivational factors of relevance and confidence and what they perceived to cause those shifts. In general, students reported improvement in their self-confidence in each of the identified skills. When asked which learning activities helped to improve their confidence, students cited the laboratory exercises and in-class activities (particularly the group problem-solving sessions) as being particularly helpful, with a few of the students citing the course blog as being helpful.
When asked about their change in interest on the various topics of the course, student responses were mixed. Students who did report an increase in topics interest primarily identified the lab and class discussions as leading to their increased interest. When describing their experience in the lab, students indicated that the hands-on nature of lab work was very important to their improved interest. Two of the students credited the blog as improving their interest, though one of those students never commented on the blog, which we will discuss further below).
Blog Comment Analysis
---------------------
The frequency of student comments on the blog was somewhat minimal. Three of the seven introductory students posted a total of 15 comments. However, one student who cited the blog as contributing to his level of interest on the in-class surveys never commented on the blog. This piece of feedback indicates that the students may have read the blog more frequently than they commented on it. We will see this conclusion confirmed in the student interviews. The introductory students’ comments did progress from surface-level to substantive over the course of the semester, eventually demonstrating fluency with terminology, precision, and a willingness to apply the concepts of the course to new situations.
Examining the patterns of the introductory students’ blog comments yields a number of interesting observations. First, we find that students commented more frequently on articles written by authors with whom they had established a face-to-face relationship. For example, one of the blog authors also worked as a physics tutor on campus, and spent a few tutoring sessions with one of the introductory students; after this tutoring relationship began, the introductory student began commenting on the tutor’s articles. Another blog author saw an increase in comments on her articles after she visited one of the introductory students’ class sessions. Finally, we note that the instructor’s blog articles received the greatest number of comments. From these observations, we tentatively conclude that a face-to-face relationship between the blog participants and authors can be important in encouraging responses. We will see this conclusion confirmed in the student interviews.
Second, the students seemed to primarily respond to articles that were most directly related to what they were studying in class at the time.
Finally, the topic sequences that received the most number of comments were the Graphing Project Head Start, Physics in the News, and Physics Humor. We will further examine the popularity of these topic sequences in the student interviews.
Student Interviews
------------------
The post-instruction student interviews largely confirmed many of the findings discussed above.
When asked about how their self-confidence changed over the semester, students identified that their confidence level increased as they applied themselves to the course assignments. “After a while,” one student said, “I got used to$\ldots$ how \[the instructor\] wanted the assignments turned in, so the assignments became easier.” “At the time,” said another, “I thought$\ldots$ ‘This is a lot of work!’ but looking back in retrospect, \[the assignments\] really increased my confidence level.” “My confidence level has increased$\ldots$. As we progressed, I have more of a base, so I’m able to figure out things on my own.”
When asked about how their interest level had changed, most students replied that their interest increased most significantly when they had a sense of the applicability of the material, especially when that applicability extended to aviation. One student reported, “Physics is pretty interesting. You can apply everything we learn in class to something we do in life.” Another student said that his interest improved “when we started working with$\ldots$ all that stuff that had to deal with aviation.” One student also reported that he was excited to see the principles of mathematics applied to real-world situations, which indicates an important lesson about student motivation: They may not see the relevance of what they learn in a course until they are immersed in a subsequent course. Another student related her sense of interest to a growing sense of intrigue with the course material. One student also indicated that his interest was lowest in topics whose problems required many laborious mechanical steps, of which he gave the unit on vectors as an example.
The student interviews revealed that all of the students read the blog, and that they all read more often than they posted comments. A couple students visited just a few times during the semester, while one student reported visiting the blog daily (even though he did not comment on any of the articles). When asked what prevented them from commenting, many students cited busyness and time constraints. “I did not give as much time as I wished that I could have to the blog,” one such student regretted. Another student said, “I think I probably went to the course blog like three times out of the whole semester. I think I was just caught up in all the other stuff, like the assignments.” One student even stated that when he was able to peruse the blog, he was usually “burned out on physics.” Such comments indicate that forgoing blog participation (and the accompanying extra credit) was a necessary sacrifice as the students sought to manage their time and their cognitive load,[@de; @Jong] similar to behavior observed by Harrison.[@Harrison]
Some students indicated that they did prefer to interact on the blog with authors with whom they previously had a face-to-face relationship, even if that relationship consisted of only one encounter. The student who participated in out-of-class tutoring with one of the blog authors confirmed that he visited and commented on the blog thanks to this relationship and that, in fact, the author had encouraged him during tutoring to visit the blog. Another student indicated that she felt more encouraged to comment on articles written by one of the authors who had visited the introductory class, even though that one class visit was their only face-to-face contact. These observations confirm the above hypothesis that the students preferred to interact with blog authors whom they had met in person.
Overall, the introductory students reported a positive experience on the blog when they visited it. One student indicated that he was challenged to think critically, while another indicated that it was nice to discuss and work problems with others, reflecting many of Harrison’s observations about the benefits of course blogs.[@Harrison] Another student indicated that interacting with “empathetic” upper-level students helped to change his perceptions of physicists, an excellent benefit derived from the learning community created by the blog. One student who commented frequently confirmed that he felt that his comments became more sophisticated as the semester progressed, thanks to an increase in comfort and knowledge level. “As the class went on,” he said, “I learned more about physics and got more in depth$\ldots$. \[My responses\] definitely changed to a more analytical way of thinking.”
It seems the variety of topic sequences was valuable to the students. When asked which topic sequences were most helpful, students identified the Physics Humor, Physics in Entertainment, Graphing Project Head Start, and Sample Problems topic sequences. One student also indicated that he would like to see the Physics in Aviation sequence expanded, particularly to include engineering and design topics.
RECOMMENDATIONS
===============
Overall, it seems that the students’ experience of the blog was positive, even when they did not participate in the online discussion. Many students reported that they read the blog regularly, and that the blog did help their learning motivation, if not as significantly as the class activities, lab, and assignments. They particularly reported many of the benefits described in the literature, including a development of critical thinking, engagement, and reinforced learning,[@Harrison] and found the blog to complement other learning activities.[@Higdon] The interaction with physics majors also seems to have caused a change in at least one student’s perception of physicists. The introductory students seem to have appreciated and benefited from nearly all of the topic sequences.
We particularly noted a correlation between increased student participation and established face-to-face relationships, even if that relationship was a one-time classroom visit. We therefore recommend developing face-to-face relationships between blog authors and participant students near the beginning of the blog implementation. For example, the authors may be able to visit the introductory test students’ class meeting to deliver a presentation (perhaps in partial fulfillment of a scientific communications course). An instructor could also encourage blog authors and participants to attend a social event outside of class. The blog authors could also conduct demonstrations in or help proctor the introductory course’s lab, help the instructor conduct office hours, or serve as private tutors for the introductory course.
As with any instructional strategy, we found it vital to assess the introductory students’ needs to determine the focus and direction of the course blog. The pre-instructional CLASS, the periodic in-class surveys, and the initial query of student wishes for the blog were very helpful in fulfilling this task.
Even though its impact on motivation was not formally assessed, the introductory students did report that the hands-on nature of the lab activities significantly helped their motivation. Similarly, when describing which in-class activities helped their motivation, students cited the hands-on nature of in-class problem sessions as important to them. This common characteristic would suggest that the blog should, likewise, include hands-on features. Such features could include computer simulations or instructions for at-home demos.
As described above, this instructional strategy also offers learning benefits to the upper-level majors, including development of their skills in communication, collaboration, technology, and research. These benefits are briefly explored in the reflective section of the students’ account of the implementation[@PEER] and warrant formal assessment in future implementations of this strategy.
CONCLUSION
==========
We implemented a course blog managed by upper-level physics majors in an introductory physics course. We evaluated the impact of the course blog using the CLASS, periodic in-class surveys, the student blog comments, and post-instruction student interviews. The blog articles focused on fostering the motivational factors of relevance and confidence in the introductory students. The pre- to post-instructional CLASS shifts show a positive impact on the students’ learning attitudes and motivation, especially in the area of Personal Interest; these results were confirmed by the periodic in-class surveys and the interviews. The blog comments revealed an increase in the depth of student engagement on the blog, even though commenting was infrequent. The interviews revealed the students’ positive experience with the blog, and have resulted in a number of recommendations for future use.
In-Class Surveys
================
The table below contains a list of skills that you are expected to develop to succeed at this course. For each skill, please circle your current level of self-confidence in that skill.
---------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
1\. Read and understand a physics problem. [& Very & Somewhat & Not very & Not at all\
]{} 2. Select a physics equation to apply to a problem. [& Very & Somewhat & Not very & Not at all\
]{} 3. Use algebra to solve a physics equation for a desired unknown quantity. [& Very & Somewhat & Not very & Not at all\
]{} 4. Use multiple physics equations in combination to solve a single problem. [& Very & Somewhat & Not very & Not at all\
]{} 5. Apply the concepts of trigonometry to a right triangle. [& Very & Somewhat & Not very & Not at all\
]{} 6. Apply the concepts of trigonometry to a vector. [& Very & Somewhat & Not very & Not at all\
]{} 7. Work with a vector equation. [& Very & Somewhat & Not very & Not at all\
]{} 8. Create a graph to depict the relationship between two physical quantities based on predetermined data. [& Very & Somewhat & Not very & Not at all\
]{} 9. Create an electronic graph to depict the relationship between two physical quantities based on an equation. [& Very & Somewhat & Not very & Not at all\
]{} 10. Examine and draw conclusions from a graph depicting the relationship between two physical quantities. [& Very & Somewhat & Not very & Not at all\
]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
The table below contains a list of topics that we will cover in this course. Please circle the number after each topic below that best represents your level of interest in that topic. The numbers stand for the following responses:\
0 = No interest at all.\
1 = Interested in an overview of this topic.\
2 = Interested in reading about and discussing this topic.\
3 = Interested in applying ideas about this topic in problems or experiments.
-------------------------------------------------------- -- -- -- --
1\. Forces and motion [& 0 & 1 & 2 & 3\
]{} 2. Energy [& 0 & 1 & 2 & 3\
]{} 3. The forces of flight [& 0 & 1 & 2 & 3\
]{} 4. Cruising flight [& 0 & 1 & 2 & 3\
]{} 5. Constant-velocity flight [& 0 & 1 & 2 & 3\
]{} 6. Constant-speed turning flight [& 0 & 1 & 2 & 3\
]{} 7. Constant-velocity gliding [& 0 & 1 & 2 & 3\
]{} 8. Electric and magnetic fields [& 0 & 1 & 2 & 3\
]{} 9. DC circuits [& 0 & 1 & 2 & 3\
]{} 10. Resistors [& 0 & 1 & 2 & 3\
]{} 11. Capacitors [& 0 & 1 & 2 & 3\
]{} 12. Inductors [& 0 & 1 & 2 & 3\
]{} 13. AC circuits [& 0 & 1 & 2 & 3\
]{} 14. Thermodynamics [& 0 & 1 & 2 & 3\
]{}
-------------------------------------------------------- -- -- -- --
The table below contains a list of skills that you are expected to develop to succeed at this course. For each skill, please circle how your self-confidence in that skill has changed in the last two weeks of the course.
-------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
1\. Read and understand a physics problem. [& Improved & Stayed the same & Declined\
]{} 2. Select a physics equation to apply to a problem. [& Improved & Stayed the same & Declined\
]{} 3. Use algebra to solve a physics equation for a desired unknown quantity. [& Improved & Stayed the same & Declined\
]{} 4. Use multiple physics equations in combination to solve a single problem. [& Improved & Stayed the same & Declined\
]{} 5. Apply the concepts of trigonometry to a right triangle. [& Improved & Stayed the same & Declined\
]{} 6. Apply the concepts of trigonometry to a vector. [& Improved & Stayed the same & Declined\
]{} 7. Work with a vector equation. [& Improved & Stayed the same & Declined\
]{} 8. Create a graph to depict the relationship between two physical quantities based on predetermined data. [& Improved & Stayed the same & Declined\
]{} 9. Create an electronic graph to depict the relationship between two physical quantities based on an equation. [& Improved & Stayed the same & Declined\
]{} 10. Examine and draw conclusions from a graph depicting the relationship between two physical quantities. [& Improved & Stayed the same & Declined\
]{}
-------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- --
This week, we discussed the topic of \_\_\_\_. How do you feel your interest in this topic has changed (increased, decreased, or remained the same) since the beginning of the week? Please explain.
Please describe what activities led to your change (or lack of change) in interest in this topic. In particular, please describe the role (if any) that class discussions and the course blog played to change your interest in this topic. (If you feel that they did not contribute to your change in interest, please indicate.)
We thank the Aviation Physic students for their participation in this pilot study. We also thank the 2010-2011 Scholarship of Teaching & Learning Faculty Learning Community (FLC): Heather Downs, Michelle Edmonds, Thomas Harrison (who graciously recorded the student interviews), Andre Megerdichian, Rob Tudor, Colleen Wilson, and Kathy Ingram (who organized the FLC). This work was supported by the 2010-2011 CTL SoTL Fellowship.
[5]{}
D. Hammer, “Student resources for learning introductory physics,” Am. J. Phys. **68** S52-S59 (2000). E. F. Redish, J. M. Saul, J. M., & R. N. Steinberg, “Student expectations in introductory physics,” Am. J. Phys. **66** (3), 212-224 (1998). D. Tarshis, “Measuring What’s Hidden: How College Physics Courses Implicitly Influence Student Beliefs,” Physics Honors Thesis, University of Colorado at Boulder (2008). C. Ames, “Motivation: What teachers need to know,” Teach. Coll. Rec. **91**, 409-472 (1990). J. M. Keller, “Strategies for stimulating the motivation to learn,” Performance and Instruction, **26** (8), 1-7 (1987). D. Hammer, “Epistemological Beliefs in Introductory Physics,” Cognition and Instruction **12** (2), 151-183 (1994). D. Lynch, “Motivational beliefs and learning strategies as predictors of academic performance in college physics,” College Student Journal **44** (4), 920-927 (2010). L. Burke & J. Moore, “A Perennial Dilemma in OB Education: Engaging the Traditional Student,” *Academy of Management Learning and Education* **2** (1), 37-52 (2003). W. K. Adams, K. K. Perkins, N. S. Podolefsky, M. Dubson, N. D. Finkelstein, & C. E. Wieman, “New instrument for measuring student beliefs about physics and learning physics,” Phys. Rev. S.T. - P.E.R. **2**, 010101 (2006). G. Duda & K. Garrett, “Blogging in the physics classroom: A research-based approach to shaping students’ attitudes toward physics,” Am. J. Phys. **76** (11), 1054-1065 (2008). G. Duda & K. Garrett, “Probing Student Online Discussion Behavior with a Course Blog in Introductory Physics,” (C. Henderson, M. Sabella,& L. Su, Eds.), AIP Conference Proceedings, **1064**, 111-114 (2008). D. Harrison, “Can Blogging Make a Difference?” Retrieved March 2, 2011, from Campus Technology: http://campustechnology.com/Articles/2011/01/12/Can-Blogging-Make-a-\
Difference.aspx?Page=1 (2011, January 12). J. Sprague, & D. Stuart, *The speaker’s handbook*, (Harcourt College Publishers, Fort Worth, 2000). B. Daley “Novice to expert: an exploration of how professionals learn,” *Adult Education Quarterly* **49** (4), 133-147 (1999). S. Dreyfus, S. & H. Dreyfus, “A five-stage model of the mental activities involved in directed skill acquisition,” unpublished report supported by the Air Force Office of Scientific Research No. Contract F49620-79-C-0063, University of California at Berkeley (1980). J. Higdon & C. Topaz, “Blogs and Wikis as Instructional Tools,” College Teaching **57** (2), 105-109 (2009). A. A. August, K. C. Bretey, B. T. Cory, E. R. Finkley III, R. D. Jones, D. W. Marshall, et al, “Enhancing introductory student motivation and learning attitudes with a major-managed course blog,” *Journal of Research Across the Disciplines*, under review (2011). J. M. Keller, “Development and use of the ARCS model of instructional design,” J. Inst. Dev. **10** (3), 2-10 (1987). R. E. Scherr, & D. Hammer, “Student Behavior and Epistemological Framing: Examples from Collaborative Active-Learning Activities in Physics,” Cognition and Instruction **27** (2), 147-168 (2007). Y. Hadzigeorgiou, T. Kodakos, & V. Garganourakis, “Using a feature film to promote scientific inquiry.” *Phys. Ed.* **45** (1), 32-36 (2010). M. Tatalovic, “Science comics as tools for science education and communication: a brief, exploratory study,” J. Sci. Comm. **08** (04), A02 (2009). T. Angleo, & K. Cross, *Classroom Assessment Techniques*, (Jossey-Bass, San Francisco, CA, 1993). T. de Jong, “Cognitive load theory, educational research, and instructional design: some food for thought,” *Inst. Sci.* **38**, 105-134 (2010).
Figures {#figures .unnumbered}
=======
![\[fig:fall10\]Fall 2010 CLASS results (N = 12 students), exhibiting typical declines in many categories, though a statistically significant increase in Problem Solving Confidence.](Fall_10_CLASS.png){width="\textwidth"}
![\[fig:spring11\]Spring 2011 CLASS results (N = 7 students), without the typical declines.](Spring_11_CLASS.png){width="\textwidth"}
|
---
abstract: 'We present the results from the Tevatron on the direct searches for Standard Model Higgs boson produced in $p\bar{p}$ collisions at a center of mass energy of 1.96 TeV, using the data corresponding to the integrated luminosity of 10fb$^{-1}$. The searches are performed in the Higgs boson mass range from 100 to 200 GeV/c$^{2}$. The dominant production channels, $H\rightarrow b\bar{b}$ and $H \rightarrow WW$, are combined with all the secondary channels and significant analysis improvements have been implemented to maximize the search sensitivity. We observe a significant excess of data events compared to background predictions with the local significance of 3.0 standard deviations. The global significance for such an excess anywhere in the full mass range investigated is approximately 2.5 standard deviations.'
author:
- Yuri Oksuzian On behalf of the CDF and D0 collaborations
title: Searches for the Higgs boson at the Tevatron
---
[ address=[University of Virginia, Charlottesville, Virginia 22904, USA]{} ]{}
Introduction
============
The Higgs boson is a crucial element of the standard model (SM) of elementary particles and interactions. Within the SM, vector boson masses arise from the spontaneous breaking of electroweak symmetry due to the existence of the Higgs particle. The winter results from the LHC and the Tevatron experiments have excluded wide regions of the possible Higgs mass ranges. The most interesting region to search for the Higgs is the mass range between 115 and 127 GeV/$c^2$ where the both the ATLAS and the CMS experiments have found some excesses [@atlas; @cms]. The Tevatron experiments can contribute to the understanding of this region by analyzing the data collected through the years of 2001-2011.
Higgs Search Channels at the Tevatron
=====================================
Low Mass Channels
-----------------
The SM Higgs boson $H$ is predicted to be produced in association with a $W$ or $Z$ boson at the Fermilab Tevatron $p{\bar{p}}$ collider and its dominant decay mode is predicted to be into a bottom-antibottom quark pair ($b{\bar{b}}$), if its mass $m_H$ is less than 135 GeV/$c^2$ (low Higgs mass region). The searches use the complete Tevatron data sample of $p{\bar{p}}$ collisions at a center of mass energy of 1.96 TeV collected by the CDF and D0 detectors at the Fermilab Tevatron, with an integrated luminosity of 9.45 fb$^{-1}$ – 9.7 fb$^{-1}$. The CDF and D0 detectors are multipurpose solenoidal spectrometers surrounded by hermetic calorimeters and muon detectors and are designed to study the products of 1.96 TeV proton-antiproton collisions [@cdfdetector; @d0detector].
The online event selections (triggers) rely on fast reconstruction of combinations of high-$p_T$ lepton candidates, jets, and $\mbox{$\not\!\!E_T$}$. Event selections are similar in the CDF and D0 analyses, consisting typically of a preselection based on event topology and kinematics, and a subsequent selection using $b$-tagging. Each channel is divided into exclusive sub-channels according to various lepton, jet multiplicity, and $b$-tagging characterization criteria aimed at grouping events with similar signal-to-background ratio and so optimize the overall sensitivity.
Due to the importance of $b$-tagging, both collaborations have developed multivariate approaches to maximize the performance of the $b$-tagging algorithms. A boosted decision tree algorithm is used in the D0 analysis, which builds and improves upon the previous neural network $b$-tagger [@Abazov:2010ab], giving an identification efficiency of $\approx 80\%$ for $b$ jets with a mis-identification rate of $\approx 10\%$. The CDF $b$-tagging algorithm has been recently augmented with an MVA [@tagging], providing a $b$-tagging efficiency of $\approx 70$% and a mis-identification rate of $\approx 5$%.
In $H\rightarrow b\bar{b}$ final states, the single most sensitive observable to distinguish between the Higgs signal and various types of background is the invariant mass of dijet system, $m_{jj}$, which approximately accounts for 75% of analysis sensitivity. In all low mass Higgs searches at Tevatron, we include additional variables through the multivariate analysis techniques. Dedicated studies have been performed to improve the search sensitivity through the improvements in dijet mass resolution, lepton identification algorithm, $b$-tagging, multijet background suppression and modeling, final discriminant optimization. The detailed information on low mass Higgs channels is present in Ref. [@cdfwh2012; @cdfzh2012; @cdfzhll2012; @dzwh2012; @dzzh2012; @dzzhll2012]
To validate our background modeling and search methods, we perform a search for SM diboson production in the same final states used for the SM $H\rightarrow b\bar{b}$ searches. The data sample, reconstruction, process modeling, uncertainties, and sub-channel divisions are identical to those of the SM Higgs boson search. The measured cross section for $WZ$ and $ZZ$ production is [@mor12tevdibosons]. This is consistent with SM prediction of $\sigma(WW+WZ)=\vznlo\pm\vznloe$ pb [@dibo] and corresponds to a significance of standard deviations above the background-only hypothesis.
Other Complimentary Channels
----------------------------
Even though $H\rightarrow b\bar{b}$ final states are the most sensitive channels at the Tevatron below 135 GeV/c$^2$, in the final combination we consider all the complimentary channels to improve the Higgs search sensitivity. The complete list of channels that goes into the Higgs Tevatron combination is given in Ref. [@higgscombo]. One of the channels that needs to be mentioned is $H \rightarrow WW$. Being the most sensitive channel for the high mass Higgs region, it has significant contribution to the low mass region as well. For the $H \rightarrow WW$ analyses, signal events are characterized by large $\mbox{$\not\!\!E_T$}$ and two opposite-signed, isolated leptons. The presence of neutrinos in the final state prevents the accurate reconstruction of the candidate Higgs boson mass. The most sensitive variable for Higgs signal is the opening angle, $\Delta R$, between the outgoing leptons. Both CDF and D0 include additional event properties and their correlations through multi variate algorithms. CDF uses neural network outputs, including likelihoods constructed from calculated matrix element probabilities and D0 uses boosted decision trees outputs.
Tevatron Combination
====================
In the Tevatron combination, we combine all the major low mass Higgs channels with the complimentary final state searches from CDF and D0. In this section we report the results presented at CIPANP conference, which are based on analyses presented for the Winter conferences and described in more detailes in Ref. [@higgscombo].
To determine the estimates of the interest like the upper limits on SM Higgs production at 95% C.L. and to gain confidence that the final result does not depend on the details of the statistical formulation, we perform two types of combinations: Bayesian approach where the nuisance parameters are integrated out to determine posterior probabilities; and Modified Frequentist approach where the minimum of the likelihood is used to determine the nuisance parameters. Both approaches yield limits on the Higgs boson production rate that agree within 10% at each value of $m_H$, and within 1% on average. Systematic uncertainties enter on the predicted number of signal and background events as well as on the distribution of the discriminants in each analysis (“shape uncertainties”).
The 95% C.L. limits on Higgs production are shown in Fig. \[fig:TevAll\], along with the significance of the excess in the data over the background prediction, assuming a signal is truly absent. The regions of Higgs boson masses excluded at the 95% C.L. are $100<m_H<106$ GeV/$c^{2}$ and $147<m_{H}<179$ GeV/$c^{2}$. The expected exclusion regions are $100<m_H<119$ GeV/$c^{2}$ and $141<m_{H}<184$ GeV/$c^{2}$. There is an excess of data events with respect to the background estimation in the mass range 115<$m_H$<135 GeV/$c^2$. The observed $p$-value as a function of $m_H$ exhibits a broad minimum, and the maximum local significance corresponds to 2.7 standard deviations at $m_H=120$ GeV/$c^2$. Correcting for the Look-Elsewhere Effect (LEE), which accounts for the possibility of a background fluctuation affecting the local $p$-value anywhere in the search region, yields a global significance of 2.2 standard deviations.
![ \[fig:TevAll\] Final Tevatron combination for the winter conferences: (Left) The observed 95% credibility level upper limits on SM Higgs boson production as a function of Higgs boson mass. The dashed line indicates the median expected value in the absence of a signals. (Right) The $p$-value as a function of $m_H$ under the background-only hypothesis. The associated dark and light-shaded bands indicate the 1 s.d. and 2 s.d. fluctuations of possible experimental outcomes.](442_Oksuzian-f1 "fig:"){height=".3\textheight"} ![ \[fig:TevAll\] Final Tevatron combination for the winter conferences: (Left) The observed 95% credibility level upper limits on SM Higgs boson production as a function of Higgs boson mass. The dashed line indicates the median expected value in the absence of a signals. (Right) The $p$-value as a function of $m_H$ under the background-only hypothesis. The associated dark and light-shaded bands indicate the 1 s.d. and 2 s.d. fluctuations of possible experimental outcomes.](442_Oksuzian-f2 "fig:"){height=".277\textheight"}
Updated results
===============
We have recently updated and combined the results in $H\rightarrow b\bar{b}$ final states at CDF and D0 [@TevBBprl]. An observation of this process would support the SM prediction that the mechanism for electroweak symmetry breaking, which gives mass to the weak vector bosons, is also the source of fermionic mass in the quark sector.
The broad observed excess in the low mass range, shown on Fig. \[fig:TevBB\], results in a minimum $p$-value of 3.3 standard deviations away from the background-only hypothesis at a Higgs mass of $m_H$ = 135 GeV/c$^2$. The global $p$-value is 3.1 standard deviation. We interpret this result as evidence for the presence of a particle that is produced in association with a $W$ or $Z$ boson and decays to a bottom-antibottom quark pair. The excess seen in the data is most significant in the mass range between 120 and 135 GeV/$c^2$, and is consistent with production of the SM Higgs boson.
The updated Tevatron combination [@higgscomboNew] across all channels on CDF and D0 yields the local(global) significance for such an excess of 2.5(3.0) standard deviations.
![ \[fig:TevBB\] Updated Tevatron combination for $H\rightarrow b\bar{b}$ channels: (Left) The observed 95% credibility level upper limits on SM Higgs boson production as a function of Higgs boson mass. The dashed line indicates the median expected value in the absence of a signals. (Right) The $p$-value as a function of $m_H$ under the background-only hypothesis. The associated dark and light-shaded bands indicate the 1 s.d. and 2 s.d. fluctuations of possible experimental outcomes.](442_Oksuzian-f3 "fig:"){height=".3\textheight"} ![ \[fig:TevBB\] Updated Tevatron combination for $H\rightarrow b\bar{b}$ channels: (Left) The observed 95% credibility level upper limits on SM Higgs boson production as a function of Higgs boson mass. The dashed line indicates the median expected value in the absence of a signals. (Right) The $p$-value as a function of $m_H$ under the background-only hypothesis. The associated dark and light-shaded bands indicate the 1 s.d. and 2 s.d. fluctuations of possible experimental outcomes.](442_Oksuzian-f4 "fig:"){height=".277\textheight"}
Conclusions
===========
We combine all available CDF and D0 results on SM Higgs boson searches. A broad excess is observed in data with respect to the background estimation, corresponding to a 2.5 standard deviations. Considering only the $H\rightarrow b\bar{b}$ final state searches yields an excess, corresponding to a 3.1 standard deviations. The excess is observed to be consistent with SM Higgs boson production.
The author thanks the CIPANP conference organizers, conveners, colleagues from CDF and D0 experiments, and acknowledges the support from DOE and visiting scholar award from URA.
[9]{}
ATLAS Collaboration, “Combined search for the Standard Model Higgs Boson using up to 4.9 fb$^{-1}$ of pp collision data at $\sqrt{s}$ = 7 TeV with the ATLAS detector at the LHC”, Phys. Lett. B [**710**]{}, 49 (2012)
CMS Collaboration, “Combined results of searches for the standard model Higgs boson in pp collisions at $\sqrt{s}$ = 7 TeV”, Phys. Lett. B [**710**]{}, 26 (2012).
The CDF and D0 Collaborations and the TEVNPH Working Group, “Combined CDF and D0 Measurement of $WZ$ and $ZZ$ Production in $b$-tagged Channels with up to 9.5 fb-1 of Data” arXiv:1203.3782v1 (2012).
J. M. Campbell and R. K. Ellis, Phys. Rev. D [**60**]{}, 113006 (1999). We used [MCFM]{} v6.0. Cross sections are computed using a choice of scale $\mu_0^2=M_V^2+p_T^2(V)$, where $V$ is the vector boson, and the MSTW2008 PDF set.
The CDF and D0 Collaborations and the TEVNPH Working Group, “Combined CDF and Upper Limits on Standard Model Higgs Boson Production with up to 10.0 fb$^{-1}$ of Data,”, FERMILAB-CONF-12-065-E, CDF Note 10806, D0 Note 6303, arXiv:1203.3774v1 (2012);
The CDF and D0 Collaborations and the TEVNPH Working Group, “Updated Combination of CDF and Searches for Standard Model Higgs Boson Production with up to 10.0 fb$^{-1}$ of Data”, FERMILAB-CONF-12-318-E; CDF Note 10884; D0 Note 6348 arXiv:1207.0449v2 (2012);
|
---
abstract: 'In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is an axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $\frac{u^{r}}{r}$ and $\frac{\omega_\theta}{r}$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of entails the desired regularity.'
author:
- |
Changxing Miao$^1$ and Xiaoxin Zheng$^2$\
\
\
\
\
\
[P.O. Box 2101, Beijing 100088, P.R. China.]{}\
title: 'On the global well-posedness for the Boussinesq system with horizontal dissipation'
---
[**Mathematics Subject Classification (2000):**]{}76D03, 76D05, 35B33, 35Q35\
[**Keywords:**]{}Boussinesq system, losing estimate, horizontal dissipation, anisotropic inequality, global well-posedness.
Introduction
============
The Boussinesq system describes the influence of the convection phenomenon in the dynamics of the ocean or atmosphere. In fact, it is used as a toy model for geophysical flows whenever rotation and stratification play an important role (see [@J-P]). This system is described by the following equations: $$\label{full}
\begin{cases}
(\partial_{t}+u\cdot\nabla)u-\kappa\Delta u+\nabla p=\rho e_{n},\quad(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{n},\quad n=2,3,\\
(\partial_{t}+u\cdot\nabla)\rho-\nu\Delta\rho=0,\\
\text{div}u=0,\\
(u,\rho)|_{t=0}=(u_{0},\rho_{0}),
\end{cases}$$ where, the velocity $u=(u^1,\cdots,u^n)$ is a vector field with zero divergence and $\rho$ is a scalar quantity such as the concentration of a chemical substance or the temperature variation in a gravity fields, in which case $\rho e_n$ represents the buoyancy force. The nonnegative parameters $\kappa$ and $\nu$ denote the viscosity and the molecular diffusion respectively. In addition, the pressure $p$ is a scalar quantity which can be expressed by the unknowns $u$ and $\rho$.
In the case where $\nu$ and $\kappa$ are nonnegative constants, the local well-posedness of can be easily established by using the energy method. When variables $\kappa$ and $\nu$ are both positive, the classical methods allow to establish the global existence of regular solutions in dimension two and for three dimension with small initial data. Unfortunately, for the inviscid Boussinesq , whether or not smooth solution for some nonconstant $\rho_0$ blows up in finite time is still an open problem. The intermediate situation has been attracted considerable attentions in the past years and important progress has been made. When $\nu$ is a positive constant and $\kappa=0$; or $\nu=0$ and $\kappa$ is a positive constant, D. Chae [@ha], and T.Y. Hou and C. Li [@hou-li] proved the global well-posedness independently for the two-dimensional Boussinesq system. It is also shown the global well-posedness in the critical spaces, see [@ah]. In addition, C. Miao and L. Xue [@CMX] proved the global well-posedness of the two-dimensional Boussinesq equations with fractional viscosity and thermal diffusion when the fractional powers obey mild condition. Other interesting results on the two-dimensional Boussinesq equations can be found in [@acw; @acw1; @HKR1; @HKR2].
Recently, there are many works devoted to the study of the tridimensional axisymmetric Boussinesq system without swirl for different viscosities. In [@A-H-K0], a global result was established but under some restrictive conditions on the initial density, namely it does not intersect the axis $r=0$. Subsequently, T. Hmidi and F. Rousset [@hrou1] removed the assumption on the support of the density and proved the global well-posedness for the Navier-Stokes-Boussinesq system by virtue of the structure of the coupling between two equations of with $\nu=0$. In [@hrou], they also proved the global well-posedness for the tridimensional Euler-Boussinesq system with axisymmetric initial data without swirl.
In the present paper, we consider the case that the diffusion and the viscosity only occur in the horizontal direction. More precisely, $$\label{eq1.1}
\begin{cases}
(\partial_{t}+u\cdot\nabla)u-\Delta_{h}u+\nabla p=\rho e_{z},\quad(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{3},\\
(\partial_{t}+u\cdot\nabla)\rho-\Delta_{h}\rho=0,\\
\text{div}u=0,\\
(u,\rho)|_{t=0}=(u_{0},\rho_{0}),
\end{cases}$$ Here $\Delta_{h}=\partial^{2}_{1}+\partial_{2}^{2}$. Let us point out that the anisotropic dissipation assumption is natural in the studying of geophysical fluids. It turns out that, in certain regimes and after suitable rescaling, the vertical dissipation (or the horizontal dissipation) is negligible as compared to the horizontal dissipation (or the vertical dissipation). In fact, there are several works devoted to study of the two-dimensional Boussinesq system with anisotropic dissipation. In [@dp2], R. Danchin and M. Paicu proved the global existence for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. They mainly exhibited a polynomial control of ${\left\|\nabla u\right\|}_{\sqrt{L}}$, where the space $
\sqrt{L}$ stands for the space of functions $f$ in $\cap_{2\leq p<\infty}L^p$ such that $$\|f\|_{\sqrt{L}}:=\sup_{2\leq p<\infty}{p}^{-\frac{1}{2}}\|f\|_{L^{p}}\leq \infty.$$ Combining this with the following estimate $${\left\|\nabla u\right\|}_\infty\leq C\big(1+{\left\|\nabla u\right\|}_{\sqrt{L}}\log(e+{\left\|u\right\|}_{H^{s}})\big),\quad s>2$$ yields the global well-posedness of smooth solutions. Next, they observed the fact ${\left\|\nabla u\right\|}_{\sqrt{L}}$ implies that $u\in L^{2}_{\rm loc}({\mathbb{R}}^+,{\rm LogLip^{\frac12}})$, where ${\rm LogLip^{\frac12}}$ stands for the set of bounded functions $f$ such that $$\sup_{x\neq y;|x-y|\leq\frac12}\frac{|f(y)-f(x)|}{|x-y|\log^{\frac12}(|x-y|)^{-1}}\leq +\infty.$$ And then they established the global existence with uniqueness for rough data with the help of a losing estimate. Recently, A. Adhikari, C. Cao and J. Wu also established some global results for different model under various assumption on dissipation in a series of recent papers, see in particular [@acw; @acw1; @cwMhd; @cw]. In [@cw], proved the global well-posedness for the two-dimensional Boussinesq system with vertical viscosity and vertical diffusion in terms of a Log-type inequality. In their proof, they first find that ${L^{p}}$-norm on vertical component of velocity with $2\leq p<\infty$ at any time does not grow faster than $\sqrt{p~ {\rm\log}p}$ as $p$ increase by means of the low-high decomposition techniques.
To better understand the axisymmetric fields, let us recall some algebraic and geometric properties of the axisymmetric vector fields and discuss the special structure of the vorticity of system , . First, we give some general statement in cylindrical coordinates: we say that a vector field $u$ is axisymmetric if it satisfies $$\label{axisymmetric-def}
\mathcal{R}_{-\alpha}\{u(\mathcal{R}_{\alpha}x)\}=u(x),\quad\forall\alpha \in[0,2\pi],\quad \forall x\in \mathbb{R}^{3},$$ where $\mathcal{R}_{\alpha}$ denotes the rotation of axis $(Oz)$ and with angle $\alpha$. Moreover, an axisymmetric vector field $u$ is called without swirl if it has the form: $$u(t,x)=u^{r}(r,z)e_{r}+u^{z}(r,z)e_{z},\quad x=(x_1,x_2,x_3),\quad r=\sqrt{x^{2}_{1}+x^{2}_{2}}\quad\text{and }z=x_{3},$$ where $(e_{r},e_{\theta},e_{z})$ is the cylindrical basis of $\mathbb{R}^{3}$. Similarly, a scalar function $f:{\mathbb{R}}^3\to{\mathbb{R}}$ is called axisymmetric if the vector field $x\mapsto f(x)e_z$ is axisymmetric, which means that $$\label{scalar}
f(\mathcal{R}_\alpha x)=f(x),\quad\,\forall\, x\in{\mathbb{R}}^3, \quad\forall\,\alpha\in[0,2\pi].$$ This is equivalent to say that $f$ depends only on $r$ and $z$. Direct computations show us that the vorticity $\omega:=\text{curl} u$ of the vector field $u$ takes the form $$\omega=(\partial_{z}u^{r}-\partial_{r}u^{z})e_\theta:=\omega_\theta e_\theta.$$ On the other hand, we know that $$\label{axisy-1}
u\cdot\nabla=u^{r}\partial_{r}+u^{z}\partial_{z},\quad \text{div}u=\partial_{r}u^{r}+\frac{u^{r}}{r}+\partial_{z}u^{z}\quad \text{and}\quad\omega\cdot \nabla u=\frac{u^{r}}{r}\omega$$ in the cylindrical coordinates. Therefore, the vorticity $\omega$ satisfies $$\label{tourbillon-0}
\partial_t \omega +u\cdot\nabla\omega-\Delta_{h}\omega
=-\partial_{r}\rho e_{\theta}+\frac{u^r}{r}\omega.$$ Since the horizontal Laplacian operator has the form $\Delta_{h}=\partial_{rr}+\frac{1}{r}\partial_{r}$ in the cylindrical coordinates then the $\omega_
\theta$ satisfies $$\label{tourbillon}
\partial_t \omega_\theta +u\cdot\nabla\omega_\theta-\Delta_{h}\omega_\theta
+\frac{\omega_\theta}{r^2} =-\partial_{r}\rho+\frac{u^r}{r}\omega_\theta.$$ In this paper, we are going to establish the global well-posedness for the system corresponding to large axisymmetric data without swirl. Since the dissipation only occurs in the horizontal direction, it seems not obvious to get the global regularity of solutions following from [@A-H-K0] directly. Indeed, their proof relies on the smoothing effect on vertical direction. Also, we do not expect to obtain the growth estimate of $L^{p}$-norm about vertical component of velocity as in [@cw] for the tridimensional axisymmetric Boussinesq equations. Besides, as the space $H^1({\mathbb{R}}^3)$ fails to be embedded in $\sqrt{L}({\mathbb{R}}^3)$, it is impossible to obtain the bound of ${\left\|\nabla u\right\|}_{\sqrt{L}}$ in terms of ${\left\|\omega\right\|}_{\sqrt{L}}$ just as in [@dp2]. This requires us to further study the structure of axisymmetric flows and establish priori estimate to control the vorticity in $L^{1}_{\text{loc}}({\mathbb{R}}^{+},L^{\infty})$. Now, let us briefly to sketch the proof of results. According to and the properties of axisymmetric flows, we find that the quantity $\frac{\omega_\theta}{r}$ satisfies $$\label{ww}
\big(\partial_t+u\cdot\nabla\big)\frac{\omega_\theta}{r}-\big(\Delta_{h}+{{2 \over r}}\partial_r\big) \frac{\omega_\theta}{r} =-\frac{\partial_r\rho}{r}.$$ We observe that the main difficulty is the lack of information about the influence of the term in the right side of and how to use some priori estimates on $\rho$ to control it. Therefore we need to study the properties of the operator $\frac{\partial_r}{r}$ so as to analyze the influence of the forcing term $\frac{\partial_r\rho}{r}$ on the motion of the fluid. Indeed, the behavior of $\Delta_h+\frac{\partial_r}{r}$ is like that of $\Delta_h$, which be derived from the fact that $\frac{\partial_r}{r}$ is a part of the operator $\Delta_h=\partial^2_r+\frac{\partial_r}{r}$. This induces us to consider the structure of the coupling between two equation of . From this observation, we introduce a new quantity $\Gamma:= \frac{\omega_\theta}{r} -\frac{1}{2}\rho$ and then $\Gamma$ solves the following transport equation $$(\partial_{t}+u\cdot\nabla)\Gamma-(\Delta_{h}+\frac{2}{r}\partial_{r})\Gamma=0.$$ It follows that $${\left\|\Gamma(t)\right\|}_{L^p}\leq{\left\|\Gamma_0\right\|}_{L^p},\quad\forall p\in[1,\infty].$$ This together with the $L^{p}$-estimate of $\rho$ gives that $$\Big\|\frac{\omega_\theta}{r}(t)\Big\|_{L^p}\leq\Big\|\frac{\omega_\theta}{r}(0)\Big\|_{L^p},\quad\forall p\in[1,\infty].$$ This estimate enables us to establish a global $H^1$-bound of the velocity. Now, by taking the $L^{2}$-inner product of with $\omega_\theta$ and using the anisotropic inequality which will be described in Appendix \[appendix\], we obtain $$\label{H1}
\begin{split}
&\frac12\frac{\rm d}{{\rm d}t}\|\omega_\theta(t)\|_{L^2}^2+\|\nabla_{h}\omega_\theta(t)\|_{L^2}^2+\Big\|\frac{\omega_\theta}{r}(t)\Big\|_{L^2}^2\\
\leq&\Big\|\frac{u^{r}}{r}\Big\|^{\frac{3}{4}}_{L^{6}}\Big\|\partial_{z}\Big(\frac{u^{r}}{r}\Big)\Big\|^{\frac{1}{4}}_{L^{2}}
{\left\|\omega_\theta\right\|}^{\frac{1}{2}}_{2}{\left\|\nabla_{h}\omega_\theta\right\|}^{\frac{1}{2}}_{2}{\left\|\omega_\theta\right\|}_{2}+{\left\|\rho\right\|}^{2}_{L^{2}}
+\frac14\|\nabla_{h}\omega_\theta(t)\|_{L^2}^2+\frac14\left\|\frac{\omega_\theta}{r}(t)\right\|_{L^2}^2.
\end{split}$$ As a consequence, it is impossible to use the information of $\frac{\omega_\theta}{r}$ to control the quantity ${\left\|\partial_z(u^r/r)\right\|}_{L^{2}}$ via the following pointwise estimate established by T. Shirota and T. Yanagisawa (abbr. S-Y) $$\Big|\frac{u^{r}}{r}\Big|\leq C\frac{1}{|x|^{2}}\ast\Big|\frac{\omega_\theta}{r}\Big|.$$ This forces us to establish the new relationship between $\frac{u^{r}}{r}$ and $\frac{\omega_\theta}{r}$ instead of the S-Y estimate. To fulfill the goal, we find the following algebraic identity deduced from the geometric structure of axisymmetric flows and the Biot-Savart law: $$\frac{u^{r}}{r}=\partial_{z}\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)-2\frac{\partial_r}{r}\Delta^{-1}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)\cdot$$ This identity allows us to conclude the S-Y estimate, one can see Propositions \[prop-identity\] and \[prop-sy\] for more details.
Before stating our results, let us introduce the space $L$ of those functions $f$ which belong to every space $L^p$ with $2\leq p<\infty$ and satisfy $$\|f\|_{L}:=\sup_{2\leq p<\infty}{p}^{-1}\|f\|_{L^{p}}\leq \infty.$$Our results are stated as follows.
\[thm1\] Let $u_0\in H^1$ be an axisymmetric divergence free vector field without swirl such that $\frac{{\omega_0}}{r}\in L^2$ and $\partial_{z}\omega_{0}\in L^{2} $. Let $\rho_0\in H^{0,1}$ be an axisymmetric function. Then there is a unique global solution $(u,\rho)$ of the system such that $$u\in\mathcal{C}(\mathbb{R}_+;H^1)\cap L^2_{\textnormal{loc}}(\mathbb{R}_+;H^{1,2}\cap H^{2,1}),\quad \partial_z\omega\in \mathcal{C}({\mathbb{R}}_+;L^{2})\cap L^2_{\textnormal{loc}}(\mathbb{R}_+;H^{1,0}),$$ $$\frac{\omega}{r}\in L^\infty_{\textnormal{loc}}(\mathbb{R}_+;L^2)\cap L^2_{\textnormal{loc}}(\mathbb{R}_+;H^{1,0}),\quad
\rho\in \mathcal{C}(\mathbb{R}_+; H^{0,1})\cap L^2_{\textnormal{loc}}(\mathbb{R}_+;H^{1,1}).$$ Here and in what follows, we can refer to Section \[space\] for the definition of spaces such as $H^{1}$, $H^{0,1}$, etc.
The main difficulty is how to establish the $H^1$-estimates of velocity due to the lack of dissipation in the vertical direction. To overcome this difficulty, we explore an algebraic identity between $\frac{u^{r}}{r}$ and $\frac{\omega_\theta}{r}$, which strongly rely on the geometric structure of axisymmetric flows, and control the stretching term in vorticity equation $$\partial_t \omega +u\cdot\nabla\omega-\Delta_{h}\omega
=-\partial_{r}\rho e_{\theta}+\frac{u^r}{r}\omega.$$ We observe the diffusion in a direction perpendicular to the buoyancy force, and this helps us to control the source term $\partial_{r}\rho e_{\theta}$ by virtue of the horizontal smoothing effect.
\[lose-global\] Let $u_0\in H^{1}$ be an axisymmetric divergence free vector field without swirl such that $\frac{{\omega_0}}{r}\in L^2$ and $\omega_{0}\in L^\infty$. Let $\rho_{0}\in H^{0,1}$ be an axisymmetric function. Then the system admits a unique global solution $(\rho,u)$ such that $$u\in\mathcal{C}_{w}(\mathbb{R}_+;H^1)\cap L^2_{\textnormal{loc}}(\mathbb{R}_+;H^{1,2}\cap H^{2,1}),\quad \nabla u\in L_{\rm loc}^{\infty}({\mathbb{R}}_{+};{L}),$$ $$\frac{\omega}{r}\in L^\infty_{\textnormal{loc}}(\mathbb{R}_+;L^2)\cap L^2_{\textnormal{loc}}(\mathbb{R}_+;H^{1,0}),\quad\rho\in\mathcal{C}_{w}(\mathbb{R}_{+};H^{0,1})\\
\cap
L_{\textnormal{loc}}^{2}({\mathbb{R}}_{+};H^{1,1})\cap\mathcal{C}_{b}(\mathbb{R}_{+};L^2).$$
Compared with Theorem \[thm1\], the condition $\partial_z\omega_{0}\in L^2$ has been replaced by $\omega_{0}\in L^{\infty}$ in Theorem \[lose-global\]. It enables us to extend the global well-posedness theory to vector-field lying in space $L$ (which ensures the vector-field belongs to LogLip space) instead of being Lipschitz. Our choice is motivated by the well-known result that the velocity in the LogLip space $LL$ (see in Appendix \[appendix\]) seems to be the minimal requirement for uniqueness to the incompressible Euler equations. Indeed, the vorticity equation can provides us the $L^{p}$-norms of vorticity with $2\leq p<\infty$. This allows us to get that $\nabla
u\in L^{\infty}_{\rm loc}({\mathbb{R}}_+;L)$ by means of the relation ${\left\|\nabla u\right\|}_{L^p}\leq Cp{\left\|\omega\right\|}_{L^p} $. Furthermore, we can obtain the global well-posedness by exploring losing estimates.
The paper is organized as follows. In Section \[section-pre\] we shall give the definitions of the functional spaces that we shall use and state some useful propositions and algebraic identity. Next, we shall obtain a priori estimate for sufficiently smooth solutions of in Section \[section-priori\]. The last two sections will be devoted to proving Theorems \[thm1\] and \[lose-global\]. In Appendix, we shall give a few technical lemmas used throughout the paper. We shall also prove an existence result and a losing estimate for the anisotropic equations with a convection term, which are the key ingredients in the proof of the results.
**Notations:** Throughout the paper, we write $\mathbb{R}^{3}=\mathbb{R}^{2}_{h}\times\mathbb{R}_{v}$ The tridimensional vector field $u$ is denoted by $(u^h,u^z)$, and we agree that $\nabla_h=(\partial_1,\partial_2)$. Finally, the $X_{h}$ (resp.,$X_v$) stands for that $X_{h}$ is a function space over $\mathbb{R}_{h}^2$ (resp.,${\mathbb{R}}_v$).
Preliminaries {#section-pre}
==============
Littlewood-Paley Theory and Besov spaces {#space}
----------------------------------------
In this subsection, we provide the definition of some function spaces based on the so-called Littlewood-Paley decomposition.
Let $(\chi,\varphi)$ be a couple of smooth functions with values in $[0,1]$ such that $\chi$ is supported in the ball $\big\{\xi\in\mathbb{R}^{n}\big||\xi|\leq\frac{4}{3}\big\}$, $\varphi$ is supported in the shell $\big\{\xi\in\mathbb{R}^{n}\big|\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\big\}$ and $$\begin{aligned}
\label{one}
\chi(\xi)+\sum_{j\in \mathbb{N}}\varphi(2^{-j}\xi)=1\quad {\rm for \ each\ }\xi\in \mathbb{R}^{n}.\end{aligned}$$ For every $u\in \mathcal{S}'(\mathbb{R}^{n})$, we define the dyadic blocks as $$\Delta_{-1}u=\chi(D)u\quad\text{and}\quad {\Delta}_{j}u:=\varphi(2^{-j}D)u\quad {\rm for\ each\ }j\in\mathbb{N}.$$ We shall also use the following low-frequency cut-off: $${S}_{j}u:=\chi(2^{-j}D)u.$$ One can easily show that the formal equality $$\label{eq2.1}
u=\sum_{j\geq-1}{\Delta}_{j}u$$ holds in $\mathcal{S}'(\mathbb{R}^{n})$, and this is called the *inhomogeneous Littlewood-Paley decomposition*. It has nice properties of quasi-orthogonality: $$\label{eq2.2}
{\Delta}_{j}{\Delta}_{j'}u\equiv 0\quad \text{if}\quad |j-j'|\geq 2.$$ $$\label{eq2.22}
{\Delta}_{j}({S}_{j'-1}u{\Delta}_{j'}v)\equiv0\quad \text{if}\quad |j-j'|\geq5.$$ Next, we first introduce the Bernstein lemma which will be useful throughout this paper.
\[bernstein\] There exists a constant $C$ such that for $q,k\in{\mathbb{N}},$ $1\leq a\leq b$ and for $f\in L^a({\mathbb{R}}^n)$, $$\begin{aligned}
\sup_{|\alpha|=k}\|\partial ^{\alpha}S_{q}f\|_{L^b}&\leq& C^k\,2^{q(k+n(\frac{1}{a}-\frac{1}{b}))}\|S_{q}f\|_{L^a},\\
\ C^{-k}2^
{qk}\|{\Delta}_{q}f\|_{L^a}&\leq&\sup_{|\alpha|=k}\|\partial ^{\alpha}{\Delta}_{q}f\|_{L^a}\leq C^k2^{qk}\|{\Delta}_{q}f\|_{L^a}.\end{aligned}$$
\[def2.2\] For $s\in \mathbb{R}$, $(p,q)\in [1,+\infty]^{2}$ and $u\in \mathcal{S}'(\mathbb{R}^{n})$, we set $${\left\|u\right\|}_{{B}^{s}_{p,q}(\mathbb{R}^{n})}:=
\Big(\sum_{j\geq-1}2^{jsq}{\left\|{\Delta}_{j}u\right\|}_{L^{p}(\mathbb{R}^{n})}^{q}\Big)^{\frac{1}{q}}
\quad\text{if}\quad r<+\infty$$ and $${\left\|u\right\|}_{{B}^{s}_{p,\infty}(\mathbb{R}^{n})}:=\sup_{j\geq-1}2^{js}{\left\|{\Delta}_{j}u\right\|}_{L^{p}(\mathbb{R}^{n})}.$$ Then we define the *inhomogeneous Besov spaces* as $${B}^{s}_{p,q}(\mathbb{R}^{n}):=\big\{u\in\mathcal{S}'(\mathbb{R}^{n})\big|{\left\|u\right\|}_{{B}^{s}_{p,q}(\mathbb{R}^{n})}<+\infty\big\}.$$ We also denote ${B}^{s}_{2,2}$ by $H^{s}$.
Since the dissipation only occurs in the horizontal direction, it is natural to introduce the following definition.
\[def-anisotropic\] For $s,t\in \mathbb{R}$, $(p,q)\in [1,+\infty]^{2}$ and $u\in \mathcal{S}'(\mathbb{R}^{3})$, we set $${\left\|u\right\|}_{{B}^{s,t}_{p,q}(\mathbb{R}^{3})}:=
\Big(\sum_{j,k\geq-1}2^{jsq}2^{ktq}{\left\|{\Delta}^{h}_{j}{\Delta}^{v}_{k}u\right\|}_{L^{p}(\mathbb{R}^{3})}^{q}\Big)^{\frac{1}{q}}
\quad\text{if}\quad r<+\infty$$ and $${\left\|u\right\|}_{{B}^{s,t}_{p,\infty}(\mathbb{R}^{3})}:=\sup_{j,k\geq-1}2^{js}2^{kt}{\left\|{\Delta}^{h}_{j}{\Delta}^{v}_{k}u\right\|}_{L^{p}(\mathbb{R}^{3})}.$$ Then we define the *anisotropic Besov spaces* as $${B}^{s,t}_{p,q}(\mathbb{R}^{3}):=\big\{u\in\mathcal{S}'(\mathbb{R}^{3})\big|{\left\|u\right\|}_{{B}^{s,t}_{p,q}(\mathbb{R}^{3})}<+\infty\big\}.$$ We also denote ${B}^{s,t}_{2,2}$ by $H^{s,t}$.
Let us now state some basic properties for $H^{s,t}$ spaces which will be useful later.
\[properties\] The following properties of anisotropic Besov spaces hold:
1. Inclusion realtion: ${\left\|u\right\|}_{H^{s_{2},t_{2}}(\mathbb{R}^{3})}\subseteq{\left\|u\right\|}_{H^{s_{1},t_{1}}(\mathbb{R}^{3})}$ if $s_{2}\geq s_{1}$ and $t_{2}\geq t_{1}.$
2. Interpolation: for $s_{1},s_{2},t_{1},t_{2}\in \mathbb{R}$ and $\theta\in[0,1]$, we have $${\left\|u\right\|}_{H^{\theta s_{1}+(1-\theta)s_{2},\theta t_{1}+(1-\theta)t_{2}}(\mathbb{R}^{3})}\leq
{\left\|u\right\|}^{\theta}_{H^{s_{2},t_{2}}(\mathbb{R}^{3})}{\left\|u\right\|}^{1-\theta}_{H^{s_{1},t_{1}}(\mathbb{R}^{3})}.$$
3. For $s,t\geq 0$, ${\left\|u\right\|}_{H^{s,t}(\mathbb{R}^{3})}$ is equivalent to $${\left\|u\right\|}_{L^{2}(\mathbb{R}^{3})}+{\left\|\Lambda^{s}_{h}u\right\|}_{L^{2}(\mathbb{R}^{3})}
+{\left\|\Lambda^{t}_{v}u\right\|}_{L^{2}(\mathbb{R}^{3})}+{\left\|\Lambda^{t}_{v}\Lambda^{s}_{h}u\right\|}_{L^{2}(\mathbb{R}^{3})}.$$
4. ${\left\|u\right\|}_{H^{s,t}(\mathbb{R}^{3})}\simeq\big\|\|u\|_{{H}^{s}(\mathbb{R}_{h}^{2})}\big\|_{{H}^{t}(\mathbb{R}_{v})}\simeq
\big\|\|u\|_{{H}^{t}(\mathbb{R}_{v})}\big\|_{{H}^{s}(\mathbb{R}_{h}^{2})}.$
5. Algebraic properties: for $s>1$ and $t>\frac 12$, ${\left\|u\right\|}_{H^{s,t}(\mathbb{R}^{3})}$ is an algebra.
We first point out that $(i)$ and $(ii)$ are obviously true. We only need to prove $(iii)$, $(iv)$ and $(v)$. From the definition of *anisotropic Besov Spaces* ${\left\|u\right\|}_{H^{s,t}(\mathbb{R}^{3})}$ and the Plancherel theorem, we can conclude that $$\begin{split}
{\left\|u\right\|}^{2}_{H^{s,t}(\mathbb{R}^{3})}=&
\sum_{j,k\geq-1}2^{2js}2^{2kt}{\left\|{\Delta}^{h}_{j}{\Delta}^{v}_{k}u\right\|}_{L^{2}(\mathbb{R}^{3})}^{2}
= \sum_{j,k\geq-1}2^{2js}2^{2kt}{\left\|{\varphi}^{2}_{hj}{\varphi}^{2}_{vk}\hat{u}\right\|}_{L^{2}(\mathbb{R}^{3})}^{2}\\
=& \sum_{j,k\geq-1}2^{2js}\sum_{j,k\geq-1}2^{2kt}\int_{\mathbb{R}^{3}}{\varphi}^{2}_{hj}{\varphi}^{2}_{vk}|\hat{u}|^{2}(\xi)\mathrm{d}\xi\\
=& \int_{\mathbb{R}_{h}^{2}}\sum_{j\geq-1}2^{2js}{\varphi}^{2}_{hj}\int_{\mathbb{R}_{v}}\sum_{k\geq-1}2^{2kt}
{\varphi}^{2}_{vk}|\hat{u}|^{2}(\xi_{h},\xi_3)\mathrm{d}\xi_3\mathrm{d}\xi_{h}.
\end{split}$$ This together with Equality yields that $$\label{eqmi}
\begin{split}
{\left\|u\right\|}^{2}_{H^{s,t}(\mathbb{R}^{3})} \simeq&\int_{\mathbb{R}^{3}}(1+\xi^2_h)^{s}(1+\xi^2_v)^{t}|\hat{u}|^{2}(\xi)\mathrm{d}\xi\\
\simeq&\int_{\mathbb{R}^{3}}|\hat{u}|^{2}(\xi)\mathrm{d}\xi+\int_{\mathbb{R}^{3}}\xi^{2s}_h|\hat{u}|^{2}(\xi)\mathrm{d}\xi
+\int_{\mathbb{R}^{3}}\xi_v^{2t}|\hat{u}|^{2}(\xi)\mathrm{d}\xi+\int_{\mathbb{R}^{3}}\xi_h^{2s}\xi_v^{2t}|\hat{u}|^{2}(\xi)\mathrm{d}\xi\\
=&{\left\|u\right\|}^{2}_{L^2(\mathbb{R}^{3})}+ {\left\|\Lambda^{s}_h u\right\|}^{2}_{L^{2}(\mathbb{R}^{3})} +{\left\|\Lambda^{t}_v u\right\|}^{2}_{L^{2}(\mathbb{R}^{3})} +{\left\|\Lambda^{s}_{h}\Lambda^{t}_v u\right\|}^{2}_{L^{2}(\mathbb{R}^{3})}.
\end{split}$$ This implies the desired result $(iii)$.
From , it is clear that $$\begin{split}
{\left\|u\right\|}^{2}_{H^{s,t}(\mathbb{R}^{3})}\simeq&\int_{\mathbb{R}^{3}}(1+\xi^2_h)^{s}(1+\xi^2_v)^{t}|\hat{u}|^{2}(\xi)\mathrm{d}\xi
\simeq\int_{\mathbb{R}^{3}}|(1+\Lambda^{s}_h)(1+\Lambda^{t}_v)u|^{2}(x)\mathrm{d}x\\
\simeq&{\left\|(1+\Lambda^{2}_v)^\frac{t}{2}\|(1+\Lambda^{2}_h)^\frac{s}{2}u\|_{L_{h}^{2}}\right\|}^{2}_{L_{v}^{2}}
\simeq\big\|\|u\|_{H^{s}(\mathbb{R}_{h}^{2})}\big\|^{2}_{H^{t}(\mathbb{R}_{v})}.
\end{split}$$ Similarly, we can show that ${\left\|u\right\|}_{H^{s,t}(\mathbb{R}^{3})}\simeq\big\|\|u\|_{{H}^{t}(\mathbb{R}_{v})}\big\|_{{H}^{s}(\mathbb{R}_{h}^{2})}.$
Finally, according to the fact that $H^{s}(\mathbb{R}^{n})(s>\frac n2)$ is an algebra and $(iv)$. Thus, for any $u,v\in H^{s,t}(s>1,t>\frac12)$, $$\begin{aligned}
{\left\|uv\right\|}_{H^{s,t}}\leq& C\big\|\|uv\|_{{H}^{s}(\mathbb{R}_{h}^{2})}\big\|_{{H}^{t}(\mathbb{R}_{v})}\leq C\big\|\|u\|_{{H}^{s}(\mathbb{R}_{h}^{2})}\|v\|_{{H}^{s}(\mathbb{R}_{h}^{2})}\big\|_{{H}^{t}(\mathbb{R}_{v})}\\
\leq&C{\left\|u\right\|}_{H^{s,t}}{\left\|v\right\|}_{H^{s,t}}.\end{aligned}$$ This is exactly the last result.
Heat kernel and Algebraic Identity
----------------------------------
In this subsection, we first review the properties of the heat equation. Next, we give two useful algebraic identities and its properties.
[@che1; @Lemar] \[heat\] There exist $c$ and $C>0$ such that for every $u$ solution of $$\begin{cases}
\partial_t u-\Delta u=0,\quad x\in \mathbb{R}^{n},\\
u|_{t=0}=u_{0},
\end{cases}$$ the following estimates hold true
1. $
\|u (t) \|_{L^{p}(\mathbb{R}^{n})}=\|e^{t\Delta }u_{0}\|_{L^{p}(\mathbb{R}^{n})}\leq Ct^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\|u_{0}\|_{L^{q}(\mathbb{R}^{n})}
$, for $1\leq q\leq p\leq\infty$.
2. $
\|\Delta_{j}u (t) \|_{L^{p}(\mathbb{R}^{n})}=\|e^{t\Delta }\Delta_{j}u_{0}\|_{L^{p}(\mathbb{R}^{n})}\leq Ce^{-ct2^{2j}}\|u_{0}\|_{L^{p}(\mathbb{R}^{n})},
$ for $j\geq 0$.
Next, we intend to recall the behavior of the operator $\frac{\partial_r}{r}\Delta^{-1}$ over axisymmetric functions.
\[prop1\][@hrou] If $u$ is an axisymmetric smooth scalar function, then we have $$\label{prop1-12}
\Big(\frac{\partial_r}{r}\Big)\Delta^{-1}u(x)=\frac{x_2^2}{r^2}\mathcal{R}_{11}u(x)+\frac{x_1^2}{r^2}\mathcal{R}_{22}u(x)-2\frac{x_1x_2}{r^2}\mathcal{R}_{12}u(x),$$ with $\mathcal{R}_{ij}=\partial_{ij}\Delta^{-1}$. Moreover, for $p\in]1,\infty[$ there exists $C>0$ such that $$\label{prop1-2}
\|({\partial_r}/{r})\Delta^{-1}u\|_{L^{p}}\le C\| u\|_{L^{p}}.$$
\[prop-identity\] Let $u$ be a free divergence axisymmetric vector-field without swirl and $\omega=\text{\rm curl}u$. Then $$\label{identity}
\frac{u^{r}}{r}=\partial_{z}\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)-2\frac{\partial_r}{r}\Delta^{-1}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right).$$ Besides, there hold that $$\label{ansitro}
\Big\|\partial_{z}\Big(\frac{u^{r}}{r}\Big)\Big\|_{L^{p}}\leq C{\left\|\frac{\omega_{\theta}}{r}\right\|}_{L^{p}},\quad 1<p<\infty$$ and $$\label{qianru}
\Big\|\frac{u^{r}}{r}\Big\|_{L^{\frac{3q}{3-q}}}\leq C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^{q}},\quad 1<q<3.$$
Using Biot-Savart law and the fact $\omega=\omega_\theta e_\theta$, we have $$\label{eq-2.5-1}
u^{1}=\Delta^{-1}\left((\partial_z \omega_\theta)\cos\theta\right)=\partial_z\Delta^{-1}\left(x_{1}\frac{\omega_{\theta}}{r}\right)$$ and $$\label{eq-2.5-2}
u^{2}=\Delta^{-1}\left((\partial_z \omega_\theta)\sin\theta\right)=\partial_z\Delta^{-1}\left(x_{2}\frac{\omega_{\theta}}{r}\right).$$ On the other hand, we observe that $$\label{eq2.14}
\Delta^{-1}\left(x_{i}\frac{\omega_{\theta}}{r}\right)=x_{i}\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)-\big[x_{i},\Delta^{-1}\big]
\left(\frac{\omega_{\theta}}{r}\right),\quad \text{for }i=1,2.$$ Applying the Laplace operator to the commutator $\big[x_{i},\Delta^{-1}\big]
\left(\frac{\omega_{\theta}}{r}\right)$, we get $$\begin{split}
\Delta\big[x_{i},\Delta^{-1}\big]\left(\frac{\omega_{\theta}}{r}\right)=&-x_i\frac{\omega_\theta}{r}+\Delta\left(x_{i}\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)\right)\\
=&-x_i\frac{\omega_\theta}{r}+x_i\frac{\omega_\theta}{r}+2\partial_i x_i\partial_i\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)\\
=&2\partial_i\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right).
\end{split}$$ This means that $$\big[x_{i},\Delta^{-1}\big]\left(\frac{\omega_{\theta}}{r}\right)=2\partial_i\Delta^{-2}\left(\frac{\omega_{\theta}}{r}\right)=2x_{i}\frac{\partial_r}{r}\Delta^{-2}\left(\frac{\omega_{\theta}}{r}\right).$$ Inserting this estimate in gives $$\label{eq-2.5-3}
\Delta^{-1}\left(x_{i}\frac{\omega_{\theta}}{r}\right)=x_{i}\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)-2x_{i}\frac{\partial_r}{r}\Delta^{-2}\left(\frac{\omega_{\theta}}{r}\right).$$ Plugging in and , respectively, we get $$\begin{split}
\frac{u^{r}}{r}=\frac{x_{1}u^{1}+x_{2}u^{2}}{r^{2}}=&\frac{x^{2}_{1}+x^{2}_{2}}{r^{2}}\partial_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)
-2\frac{x^{2}_{1}+x^{2}_{2}}{r^{2}}\frac{\partial_r}{r}\Delta^{-1}\partial_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)\\
=&\partial_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)
-2\frac{\partial_r}{r}\Delta^{-1}\partial_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right).
\end{split}$$ Furthermore, $$\partial_z\Big(\frac{u^{r}}{r}\Big)
=\partial^{2}_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)
-2\frac{\partial_r}{r}\Delta^{-1}\partial^{2}_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right).$$ Combining this with Proposition \[prop1\] ensures us to get the estimate .
Finally, by using Proposition \[prop1\], $L^p$-boundedness of Riesz operator and the Sobolev embedding theorem, $$\begin{aligned}
\Big\|\frac{u^{r}}{r}\Big\|_{L^{\frac{3q}{3-q}}}\leq C{\left\|\partial_z\Delta^{-1}\left(\frac{\omega_{\theta}}{r}\right)\right\|}_{L^{\frac{3q}{3-q}}}
\leq C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^{q}}.\end{aligned}$$ This completes the proof.
Next, we will give a precise expression about $u^r\over r$ and $\omega_\theta\over r$ by virtue of the algebraic identity and the harmonic analysis tools. More precisely:
\[prop-sy\] Let $u$ be a free divergence axisymmetric vector-field without swirl and $\omega=\text{\rm curl}u$. Then $$\label{eq.2.14}
\frac{u^{r}}{r}=(c_{1}-4\gamma_{1}i)\frac{x_{3}}{|x|^{3}}\ast\frac{\omega_\theta}{r}+6\gamma_{1}i\frac{x^{2}_{2}
}{r^{2}}\frac{x^{2}_{1}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r}+6\gamma_{1}i\frac{x^{2}_{1}
}{r^{2}}\frac{x^{2}_{2}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r}-12\gamma_{1}i\frac{x_{1}
x_{2}}{r^{2}}\frac{x_{1}x_{2}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r},$$ where $c_1=2\pi^{\frac{3}{2}}\frac{\Gamma(\frac 12)}{\Gamma(1)}$ and $\gamma_1=i\pi^{\frac 32}\frac{\Gamma(\frac{1}{2})}{\Gamma(2)}$.
From the algebraic identities and , we obtain $$\label{pidenity}
\frac{u^{r}}{r}=\partial_{z}\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)-2\frac{x^{2}_{2}
}{r^{2}}\mathcal{R}_{11}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)-2\frac{x^{2}_{1}
}{r^{2}}\mathcal{R}_{22}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)+4\frac{x_{1}
x_{2}}{r^{2}}\mathcal{R}_{12}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right).$$ On the one hand, $$\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)=-\mathcal{F}^{-1}\Big(\frac{1}{|\xi|^{2}}
\Big(\frac{\omega_\theta}{r}\widehat{\Big)}(\xi)\Big)=-c_{1}\frac{1}{|x|}\ast\frac{\omega_\theta}{r}.$$ Thus, $$\partial_{z}\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)=c_{1}\frac{x_{3}}{|x|^{3}}\ast\frac{\omega_\theta}{r}.$$ On the other hand, $$\label{eq.2.15}
\mathcal{R}_{kj}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)=-i\partial_z\partial_{j}\mathcal{F}^{-1}\Big(\frac{\xi_{k}}{|\xi|^{4}}{
\Big(\frac{\omega_\theta}{r}}\widehat{\Big)}(\xi)\Big).$$ Since $\xi_{k}$ is a harmonic polynomial of order one, then we can get by using Theorem 5 of [Chap-4]{} in [@S] that $$\mathcal{F}^{-1}\Big(\frac{\xi_{k}}{|\xi|^{4}}\Big)=\gamma_{1}\frac{x_{k}}{|x|}.$$ Inserting this equality to , we have $$\begin{split}
\mathcal{R}_{kj}\partial_z\Delta^{-1}\left(\frac{\omega_\theta}{r}\right)=&-i\gamma_{1}\partial_z\partial_{j}
\Big(\frac{x_{k}}{|x|}\ast\frac{\omega_\theta}{r}\Big)(x)\\
=&i\gamma_{1}\delta_{kj}\frac{x_{3}}{|x|^{3}}\ast\frac{\omega_\theta}{r}(x)-3i\gamma_{1}\frac{x_{k}x_{j}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r}(x).
\end{split}$$ Plugging these equalities in gives the desired result.
Let us point out that the equality implies the S-Y estimate of [@Taira]. More precisely, by virtue of and the triangle inequality, we can conclude that $$\begin{split}
\Big|\frac{u^{r}}{r}\Big|=&\Big|(c_{1}-4\gamma_{1}i)\frac{x_{3}}{|x|^{3}}\ast\frac{\omega_\theta}{r}+6\gamma_{1}i\frac{x^{2}_{2}
}{r^{2}}\frac{x^{2}_{1}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r}+6\gamma_{1}i\frac{x^{2}_{1}
}{r^{2}}\frac{x^{2}_{2}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r}-12\gamma_{1}i\frac{x_{1}
x_{2}}{r^{2}}\frac{x_{1}x_{2}x_{3}}{|x|^{5}}\ast\frac{\omega_\theta}{r}\Big|\\
\leq&C\frac{1}{|x|^{2}}\ast\Big|\frac{\omega_\theta}{r}\Big|.
\end{split}$$
\[partial\] Let $u$ be a smooth axisymmetric vector field with zero divergence and we denote $\omega=\omega_\theta e_\theta$. Then $$\Big\|\frac{u^r}{ r}\Big\|_{L^\infty}\le C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^{2}}^{\frac12}\Big\|\nabla_h\Big(\frac{\omega_\theta}{r}\Big)\Big\|_{L^{2}}^{\frac12}.$$
From , it is clear that $$\label{234}
\Big\|\frac{u^{r}}{r}\Big\|_{L^\infty(\mathbb{R}^{3})}\leq C\Big\|\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|_{L^\infty(\mathbb{R}^{3})}
+C\sum^{2}_{k,j}\Big\|\mathcal{R}_{kj}\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|_{L^\infty(\mathbb{R}^{3})}.$$ For the first term in the right side of , by using Lemma \[sharp\], we obtain $$\begin{split}
\Big\|\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|_{L^\infty(\mathbb{R}^{3})}\leq & \Big\|\nabla\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|^{\frac12}_{L^2(\mathbb{R}^{3})}
\Big\|\nabla_h\nabla\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|^{\frac12}_{L^2(\mathbb{R}^{3})}\\
\leq &C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^{2}(\mathbb{R}^{3})}^{\frac12}\Big\|\nabla_h\Big(\frac{\omega_\theta}{r}\Big)\Big\|_{L^{2}(\mathbb{R}^{3})}^{\frac12}.
\end{split}$$ Using Lemma \[sharp\] again and applying the $L^p$-boundedness of Riesz operator, the second term can be bounded by $$\Big\|\nabla\mathcal{R}_{kj}\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|^{\frac12}_{L^2(\mathbb{R}^{3})}
\Big\|\nabla_h\nabla\mathcal{R}_{kj}\partial_z\Delta^{-1}\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|^{\frac12}_{L^2(\mathbb{R}^{3})}\leq C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^{2}(\mathbb{R}^{3})}^{\frac12}\Big\|\nabla_h\Big(\frac{\omega_\theta}{r}\Big)\Big\|_{L^{2}(\mathbb{R}^{3})}^{\frac12}.$$ This ends the proof.
The priori estimate {#section-priori}
====================
In this section, we will give some useful priori estimates.
Energy estimate and higher-order estimate
-----------------------------------------
We start with $L^{2}$ energy estimates and the maximum principle.
\[Prop-Energy\] Let $(u,\rho)$ be a solution of , then $$\label{energy}
{\left\|u(t)\right\|}^{2}_{L^2}+{\left\|\nabla_{h}u\right\|}_{L_{t}^2L^{2}}^{2}\leq ({\left\|u_{0}\right\|}_{L^2}+t{\left\|\rho_{0}\right\|}_{L^2})^{2}$$ and $$\label{eq3.2}
{\left\|\rho(t)\right\|}^{2}_{L^2}+{\left\|\nabla_{h}\rho\right\|}_{L_{t}^2L^{2}}^{2}\leq {\left\|\rho_{0}\right\|}^{2}_{L^2}.$$ Besides, for $2\leq p\leq\infty$, $$\label{eq3.3}
{\left\|\rho(t)\right\|}_{L^p}\leq{\left\|\rho_{0}\right\|}_{L^p}.$$
The proof is standard, we also give the proof for reader convenience. We first prove the estimate . Multiplying the second equation of by $|\rho|^{p-2}\rho$ and integrating by parts yields that $$\frac{1}{p}\frac{\rm d}{{\rm d}t}{\left\|\rho(t)\right\|}^{p}_{L^{p}}+(p-1)\int |\rho|^{p-2}|\nabla_{h}\rho|^{2}{\rm d}x=0.$$ Thus we obtain $$\frac{\rm d}{{\rm d}t}{\left\|\rho(t)\right\|}^{p}_{L^{p}}\leq 0,$$ which implies immediately $${\left\|\rho(t)\right\|}_{L^{p}}\leq{\left\|\rho_{0}\right\|}_{L^{p}}.$$ For $p=\infty$, it is just the maximum principle.
For the first one we take the $L^2$-inner product of the velocity equation with $u$. From integration by parts and the fact that $u$ is divergence free, we obtain $$\label{eqs1}
\frac12\frac{\rm d}{{\rm d}t}\|u(t)\|_{L^2}^2+\|\nabla_{h} u(t)\|_{L^2}^2\le\|u(t)\|_{L^2}\|\rho(t)\|_{L^2}.$$ Furthermore, we conclude that $$\frac{\rm d}{{\rm d}t}\|u(t)\|_{L^2}\le\|\rho(t)\|_{L^2}.$$ By integration in time, we get that $$\|u(t)\|_{L^2}\le\|u_0\|_{L^2}+\int_0^t\|\rho(\tau)\|_{L^2}{ \rm d}\tau\le\|u_0\|_{L^2}+t\|\rho_{0}\|_{L^2},$$ where we used the fact ${\left\|\rho(t)\right\|}_{L^2}\leq{\left\|\rho_{0}\right\|}_{L^2}$. Plugging this estimate into yields $$\frac12\|u(t)\|_{L^2}^2+\int_0^t\|\nabla_{h} u(\tau)\|_{L^2}^2{\rm d}\tau
\le
\frac12\|u_0\|_{L^2}^2+\big(\|u_0\|_{L^2}+t\|\rho_0\|_{L^2}\big)\|\rho_0\|_{L^2} t.$$ This implies the first result.
Finally, by the same argument as in proof of , we obtain the estimate .
Subsequently, we will establish the estimate of the quantities $\frac{\omega_\theta}{r}$ and $\omega$ which enable us to get the global existence of axisymmetric system .
\[Strong\] Assume that $u_0\in H^1,$ with $\frac{\omega_0}{r}\in L^2$ and $\rho_0\in L^{2}$. Let $(u,\rho)$ be a smooth axisymmetric solution $(u,\rho)$ of without swirl, then we have
$$\Big\| {\omega \over r} (t)\Big\|^{2}_{L^2}+\int_{0}^{t}\Big\| \nabla_{h}\left({\omega \over r}\right) (\tau)\Big\|^{2}_{L^2}{\rm d}\tau\le 2 \left({\left\|\frac{\omega_{0}}{r}\right\|}_{L^2}+{\left\|\rho_{0}\right\|}_{2}\right)^{2},$$ and $$\|u(t)\|_{H^1}^2+\int_0^t\|\nabla_{h}u(\tau)\|_{H^1}^2{\rm d}\tau\le C_{0} e^{C_0 t},$$ where $C_0$ depends only on the norm of the initial data.
According to the equation , it is clear that $\frac{\omega_{\theta}}{r}$ satisfies the following equation $$\label{equation_i}
\big(\partial_t+u\cdot\nabla\big)\frac{\omega_\theta}{r}-\big(\Delta_{h}+{{2 \over r}}\partial_r\big) \frac{\omega_\theta}{r} =-\frac{\partial_r\rho}{r}\cdot$$ Now we recall that $(\partial_{t}+u\cdot\nabla)\rho-\Delta_{h}\rho=0$, which can be rewritten as $$\label{eq-rtem}
(\partial_{t}+u\cdot\nabla)\rho-(\Delta_{h}+\frac{2}{r}\partial_{r})\rho=-\frac{2}{r}\partial_{r}\rho.$$ In view of and , we can set $\Gamma:= \frac{\omega_\theta}{r}-\frac{1}{2}\rho$ and then $\Gamma$ solves the equation $$(\partial_{t}+u\cdot\nabla)\Gamma-(\Delta_{h}+\frac{2}{r}\partial_{r})\Gamma=0.$$ Taking the $L^2$-inner product with $\Gamma$ and integrating by parts, we have $$\label{idI}
\frac12\frac{\rm d}{{\rm d}t}\|\Gamma(t)\|^{2}_{L^2}+\|\nabla_{h}\Gamma(t)\|_{L^2}^2\leq0,$$ where we used the facts that $u$ is divergence free and $-\int\frac{\partial_{r}\Gamma}{r}\Gamma {\mathrm d}x\geq 0$. By integration in time, we obtain that $$\|\Gamma(t)\|^{2}_{L^2}+\int_{0}^{t}\|\nabla_{h}\Gamma(\tau)\|_{L^2}^2\mathrm{d}\tau\leq\|\Gamma_{0}\|^{2}_{L^2}.$$ This together with the estimate yield that $$\begin{split}
&{\left\|\frac{\omega_\theta}{r}(t)\right\|}^{2}_{L^{2}}+\int_{0}^{t}\Big\|\nabla_{h}\Big(\frac{\omega_\theta}{r}\Big)(\tau)\Big\|_{L^2}^2\mathrm{d}\tau\\
\leq&\big({\left\|\Gamma(t)\right\|}_{L^{2}}+{\left\|\rho(t)\right\|}_{L^{2}}\big)^{2}+\big(\|\nabla_{h}\Gamma(t)\|_{L^{2}_{t}L^2}
+\|\nabla_{h}\rho(t)\|_{L^{2}_{t}L^2}\big)^2\\
\leq&2\big({\left\|\Gamma_{0}\right\|}_{L^{2}}+{\left\|\rho_{0}\right\|}_{L^{2}}\big)^{2}.
\end{split}$$ This gives the first claimed estimate.
To prove the second estimate. By taking the $L^2$-inner product of with $\omega_\theta$ we get $$\frac12\frac{\rm d}{{\rm d }t}\|\omega_\theta(t)\|_{L^2}^2+\|\nabla_{h}\omega_\theta(t)\|_{L^2}^2+\left\|\frac{\omega_\theta}{r}(t)\right\|_{L^2}^2
=\int_{{\mathbb{R}}^3}\frac{u^r}{r}\omega_\theta \omega_\theta \mathrm{d}x-\int_{{\mathbb{R}}^3}\partial_r\rho\omega_\theta \mathrm{d}x.$$ Integrating by parts, $$\begin{split}
\int_{{\mathbb{R}}^3}\partial_r\rho\omega_\theta \mathrm{d}x&=2\pi\int\partial_r\rho\omega_\theta r \mathrm{d}r\mathrm{d}z=2\pi\int\rho\partial_r\omega_\theta r \mathrm{d}r\mathrm{d}z+2\pi\int\rho\omega_\theta \mathrm{d}r\mathrm{d}z\\
&=\int_{\mathbb{R}^{3}}\rho\partial_r\omega_\theta \mathrm{d}x+\int_{\mathbb{R}^{3}}\rho\frac{\omega_\theta}{r} \mathrm{d}x.
\end{split}$$ Thus, by the Hölder inequality, we have that $$\begin{split}
\Big|\int_{{\mathbb{R}}^3}\partial_r\rho\omega_\theta \mathrm{d}x \Big| \leq& \|\rho \|_{L^2}\, \big( \|\nabla_{h} \omega_{\theta} \|_{L^2}+
\| \omega_{\theta}/r \|_{L^2}\big)\\
\leq & 2{\left\|\rho_0\right\|}^{2}_{L^{2}}+\frac{1}{4}\big( \|\nabla_{h} \omega_{\theta} \|^{2}_{L^2}+
\| \omega_{\theta}/r \|^{2}_{L^2}\big).
\end{split}$$ Next, by virtue of the equality , and the Young inequality, we obtain that $$\begin{aligned}
\left|\int_{{\mathbb{R}}^3}\frac{u^{r}}{r}\omega_\theta\omega_\theta \mathrm{d}x\right|\leq & \Big\|\frac{u^{r}}{r}\Big\|^{\frac{3}{4}}_{L^{6}}\Big\|\partial_{z}\Big(\frac{u^{r}}{r}\Big)\Big\|^{\frac{1}{4}}_{L^{2}}
{\left\|\omega_\theta\right\|}^{\frac{1}{2}}_{2}{\left\|\nabla_{h}\omega_\theta\right\|}^{\frac{1}{2}}_{2}{\left\|\omega_\theta\right\|}_{2}\\
\leq &C\Big\|\frac{\omega_{\theta}}{r}\Big\|_{L^{2}}{\left\|\omega_\theta\right\|}^{\frac{3}{2}}_{2}{\left\|\nabla_{h}\omega_\theta\right\|}^{\frac{1}{2}}_{2}\\
\leq &C\Big\|\frac{\omega_{\theta}}{r}\Big\|^{\frac{4}{3}}_{L^{2}}{\left\|\omega_\theta\right\|}^{2}_{2}+\frac{1}{4}{\left\|\nabla_{h}\omega_\theta\right\|}^{2}_{2}.\end{aligned}$$ Collecting these estimates with yield $$\label{fin1}
\begin{split}
\frac{\rm d}{{\rm d}t}\|\omega_\theta(t)\|_{L^2}^2+ \|\nabla_{h}\omega_\theta\|_{L^2}^2+\left\|\frac{\omega_\theta}{r}\right\|_{L^2}^2 \lesssim{\left\|\rho_{0}\right\|}^{2}_{2}+\Big\|\frac{\omega_{\theta}}{r}\Big\|^{\frac{4}{3}}_{L^{2}}{\left\|\omega_\theta\right\|}^{2}_{2}.
\end{split}$$ Therefore we get by the Gronwall inequality that $$\begin{aligned}
&\|\omega_\theta(t)\|_{L^2}^2+\int_0^t\Big(\|\nabla_{h}\omega_\theta(\tau)\|_{L^2}^2+\left\|\frac{\omega_\theta}{r}(\tau)\right\|_{L^2}^2\Big)\mathrm{d}\tau \\
\leq& Ce^{\int^{t}_{0}\|\frac{\omega_\theta}{r}(\tau)\|^{\frac{4}{3}}_{L^{2}}{\rm d}\tau}\Big(\Big\|\frac{\omega_\theta}{r}(0)\Big\|_{L^{2}}+{\left\|\rho_0\right\|}_{L^{2}}t\Big).\end{aligned}$$ Since $\| \omega \|_{L^2}= \| \omega_{\theta} \|_{L^2}$ and $\|\nabla_{h}\omega\|_{L^2}^2=\|\nabla_{h}\omega_\theta\|_{L^2}^2+\left\|\frac{\omega_\theta}{r}\right\|_{L^2}^2.
$ So, we finally obtain that $$\|\omega(t)\|_{L^2}^2+\int_0^t\|\nabla_{h}\omega(\tau)\|_{L^2}^2\mathrm{d}\tau \le C_0e^{C_0 t}.$$ This together with the energy estimates yields the second desired estimate. This ends the proof.
One derivative estimate on vertical variable
--------------------------------------------
In the absence of dissipation on vertical variable, we need to establish the following estimate on vertical variable in order to compensate this deficiency.
\[vertical\] Assume that $\partial_{z}\rho_0\in L^{2}$ and $\partial_{z}\omega_0\in L^{2}$, then we have $$\label{eqv1}
{\left\|\partial_{z}\rho(t)\right\|}^{2}_{L^{2}}+\int_{0}^{t}{\left\|\nabla_{h}\partial_{z}\rho(\tau)\right\|}_{L^{2}}^{2}\mathrm{d}\tau\leq C_{1}e^{
\exp{C_{1}t}} ,$$ and $$\label{eqv2}
{\left\|\partial_{z}\omega(t)\right\|}^{2}_{L^{2}}+\int_{0}^{t}{\left\|\nabla_{h}\partial_{z}\omega(\tau)\right\|}_{L^{2}}^{2}\mathrm{d}\tau\leq C_{2}e^{\exp{C_{2}t}},$$ where $C_1$ and $C_2$ depend only on the norm of the initial data $\rho_0$ and $\omega_0$.
Applying the operator $\partial_{z}$ to the second equation of , we obtain $$\label{eq-vertical}
(\partial_{t}+u\cdot\nabla)\partial_{z}\rho-\Delta_{h}\partial_{z}\rho=-\partial_{z}u^{r}\partial_{r}\rho-\partial_{z}u^{z}\partial_{z}\rho.$$ Taking the $L^{2}$-inner product of the equation with $\partial_{z}\rho$ and integrating by parts, we get $$\begin{split}
&\frac{1}{2}\frac{\rm d}{{\rm d}t}{\left\|\partial_z\rho(t)\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h}\partial_z\rho(t)\right\|}^{2}_{L^{2}}=-\int\partial_{z}u^{r}\partial_{r}\rho\partial_z\rho \mathrm{d}x
-\int\partial_{z}u^{z}\partial_{z}\rho\partial_z\rho \mathrm{d}x\\
=&-\int\partial_{z}u^{r}\partial_{r}\rho\partial_z\rho \mathrm{d}x+\int\frac{u^r}{r}\partial_z \rho\partial_z \rho\mathrm{d}x+\int\partial_{r}u^r\partial_z \rho\partial_z \rho\mathrm{d}x\\
:=& I+II+III,
\end{split}$$ where we used the fact $\text{div}u=\partial_ru^{r}+\frac{u^r}{r}+\partial_zu^{z}=0.$
For the first term $I$, by using , the Hölder and the Young inequalities, we have $$\begin{aligned}
I\leq&{\left\|\partial_zu^r\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\partial_zu^r\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_r\rho\right\|}_{L^{2}}^{\frac{1}{2}}
{\left\|\partial_z\partial_r\rho\right\|}_{L^{2}}^{\frac{1}{2}}{\left\|\partial_z\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_h\partial_z\rho\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq&2{\left\|\partial_zu^r\right\|}_{L^{2}}{\left\|\nabla_{h}\partial_zu^r\right\|}_{L^{2}}{\left\|\partial_r\rho\right\|}_{L^{2}}
{\left\|\partial_z\rho\right\|}_{L^{2}}+\frac{1}{4}{\left\|\nabla_h\partial_z\rho\right\|}^{2}_{L^{2}}\\
\leq&{\left\|\nabla_{h}\partial_zu^r\right\|}^{2}_{L^{2}}+{\left\|\omega\right\|}^{2}_{L^{2}}{\left\|\partial_r\rho\right\|}^{2}_{L^{2}}
{\left\|\partial_z\rho\right\|}^{2}_{L^{2}}+\frac{1}{4}{\left\|\nabla_h\partial_z\rho\right\|}^{2}_{L^{2}}.\end{aligned}$$ We now turn to bound the term $II$, by using , Proposition \[prop-identity\] and the Young inequality, we have $$\begin{split}
II\leq & {\left\|u^r/r\right\|}^{\frac{3}{4}}_{L^{6}}{\left\|\partial_z(u^r/r)\right\|}^{\frac{1}{4}}_{L^{2}}
{\left\|\partial_z\rho\right\|}^{\frac12}_{L^{2}}{\left\|\nabla_h\partial_z\rho\right\|}^{\frac12}_{L^{2}}{\left\|\partial_z\rho\right\|}_{L^{2}}\\
\leq &C{\left\|\omega_\theta/r\right\|}_{L^{2}}{\left\|\partial_z\rho\right\|}^{\frac32}_{L^{2}}{\left\|\nabla_h\partial_z\rho\right\|}^{\frac12}_{L^{2}}
\leq C{\left\|\omega_\theta/r\right\|}^{\frac{4}{3}}_{L^{2}}{\left\|\partial_z\rho\right\|}^{2}_{L^{2}}+\frac{1}{4}{\left\|\nabla_h\partial_z\rho\right\|}^{2}_{L^{2}}.
\end{split}$$ Similarly, the term $III$ can be bounded by $$\begin{split}
& {\left\|\partial_r u^{r}\right\|}^{\frac{3}{4}}_{L^{6}}{\left\|\partial_z\partial_r u^{r}\right\|}^{\frac{1}{4}}_{L^{2}}{\left\|\partial_z\rho\right\|}^{\frac32}_{L^{2}}
{\left\|\nabla_h\partial_z\rho\right\|}^{\frac12}_{L^{2}}\\
\leq &C{\left\|\partial_r\nabla u\right\|}_{L^{2}}{\left\|\partial_z\rho\right\|}^{\frac32}_{L^{2}}{\left\|\nabla_h\partial_z\rho\right\|}^{\frac12}_{L^{2}}
\leq C{\left\|\nabla_h\omega\right\|}^{\frac{4}{3}}_{L^{2}}{\left\|\partial_z\rho\right\|}^{2}_{L^{2}}+\frac{1}{4}{\left\|\nabla_h\partial_z\rho\right\|}^{2}_{L^{2}}.
\end{split}$$ Combining these estimates, we have $$\begin{split}
&\frac{\rm d}{{\rm d}t}{\left\|\partial_z\rho(t)\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h}\partial_z\rho(t)\right\|}^{2}_{L^{2}}\\ \lesssim&{\left\|\nabla_{h}\partial_zu^r\right\|}^{2}_{L^{2}}+{\left\|\omega\right\|}^{2}_{L^{2}}{\left\|\partial_r\rho\right\|}^{2}_{L^{2}}
{\left\|\partial_z\rho\right\|}^{2}_{L^{2}}+{\left\|\nabla_h\omega\right\|}^{\frac{4}{3}}_{L^{2}}{\left\|\partial_z\rho\right\|}^{2}_{L^{2}}+\Big\|\frac{\omega_{\theta}}{r}\Big\|^{\frac{4}{3}}_{L^{2}}{\left\|\partial_z\rho\right\|}^{2}_{L^2}.
\end{split}$$ Since ${\left\|\nabla_h\nabla u\right\|}_{L^{2}}\simeq{\left\|\nabla_h\omega\right\|}_{L^{2}}$. By the Gronwall inequality and Proposition \[Strong\], we obtain the first desired result .
Applying $\partial_{z}$ to the equation , we get $$\label{tourbillon-vertical}
\partial_t \partial_{z}\omega +u\cdot\nabla\partial_{z}\omega-\Delta_{h}\partial_{z}\omega
=-\partial^{2}_{zr}\rho e_{\theta}+\frac{\partial_{z}u^r}{r}\omega +\frac{u^r}{r}\partial_{z}\omega-\partial_{z}u^{r}\partial_{r}\omega-\partial_{z}u^{z}\partial_{z}\omega.$$ Taking the $L^{2}$-inner product to the above equation with $\partial_{z}\omega$ and integrating by parts, we obtain $$\begin{aligned}
&\frac{1}{2}\frac{\rm d}{{\rm d}t}{\left\|\partial_z\omega(t)\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h}\partial_z\omega(t)\right\|}^{2}_{L^{2}}\\
=&-\int\partial^{2}_{zr}\rho e_{\theta}\partial_{z}\omega \mathrm{d}x+\int\frac{\partial_{z}u^r}{r}\omega \partial_{z}\omega \mathrm{d}x+\int \frac{u^r}{r}\partial_{z}\omega\partial_z\omega \mathrm{d}x\\
&-\int\partial_{z}u^{r}\partial_{r}\omega\partial_z\omega \mathrm{d}x
-\int\partial_{z}u^{z}\partial_{z}\omega\partial_z\omega \mathrm{d}x\\
=&-\int\partial^{2}_{zr}\rho e_{\theta}\partial_{z}\omega \mathrm{d}x+\int\partial_{z}\left(\frac{u^{r}}{r}\right)\omega \partial_{z}\omega \mathrm{d}x+2\int \frac{u^r}{r}\partial_{z}\omega\partial_z\omega \mathrm{d}x
\\&
-\int\partial_{z}u^{r}\partial_{r}\omega\partial_z\omega \mathrm{d}x
+\int\partial_{r}u^{r}\partial_{z}\omega\partial_z\omega \mathrm{d}x\\
:=&\sum^{5}_{i=1}J_{i}.\end{aligned}$$ Here we used the fact ${\rm div}u=\partial_ru^r+\frac{u^r}{r}+\partial_zu^z=0$.
By the Hölder inequality and the Cauchy-Schwarz inequality, we know $$J_1 \leq{\left\|\partial^{2}_{zr}\rho\right\|}_{L^{2}}{\left\|\partial_z \omega\right\|}_{L^{2}}\leq {\left\|\partial_z \omega\right\|}^{2}_{L^{2}}+{\left\|\partial^{2}_{zr}\rho\right\|}^{2}_{L^{2}}.$$ By the inequality and Proposition \[prop-identity\], $$\begin{aligned}
J_{2}\leq&\Big\|\partial_z\Big(\frac{u^{r}}{r}\Big)\Big\|_{L^2}{\left\|\omega\right\|}^{\frac{3}{4}}_{L^{6}}{\left\|\partial_z\omega\right\|}^{\frac{1}{4}}_{L^{2}}
{\left\|\partial_z\omega\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_h\partial_z\omega\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq&C\Big\|\partial_z\Big(\frac{u^{r}}{r}\Big)\Big\|_{L^2}{\left\|\nabla\omega\right\|}^{\frac{3}{4}}_{L^{2}}{\left\|\partial_z\omega\right\|}^{\frac{3}{4}}_{L^{2}}
{\left\|\nabla_h\partial_z\omega\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq&C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^2}{\left\|\nabla_{h}\omega\right\|}^{\frac{3}{4}}_{L^{2}}{\left\|\partial_z\omega\right\|}^{\frac{3}{4}}_{L^{2}}
{\left\|\nabla_h\partial_z\omega\right\|}^{\frac{1}{2}}_{L^{2}}+C\Big\|\frac{\omega_\theta}{r}\Big\|_{L^2}{\left\|\partial_z\omega\right\|}^{\frac{3}{2}}_{L^{2}}
{\left\|\nabla_h\partial_z\omega\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq &C\Big\|\frac{\omega_\theta}{r}\Big\|^{\frac{4}{3}}_{L^2}{\left\|\nabla_{h}\omega\right\|}^{2}_{L^{2}}+
C\Big\|\frac{\omega_\theta}{r}\Big\|^{\frac{4}{3}}_{L^2}{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}+\frac{1}{8}{\left\|\nabla_h\partial_z\omega\right\|}^{2}_{L^{2}}.\end{aligned}$$ For the third term $J_3$, we get by virtue of the inequality and Proposition \[prop-identity\] that $$\begin{aligned}
J_3 \leq&2\Big\|\frac{u^{r}}{r}\Big\|^{\frac{3}{4}}_{L^{6}}\Big\|\partial_{z}\Big(\frac{u^{r}}{r}\Big)\Big\|^{\frac{1}{4}}_{L^{2}}
{\left\|\partial_{z}\omega\right\|}^{\frac{3}{2}}_{L^{2}}
{\left\|\nabla_{h}\partial_{z}\omega\right\|}_{L^{2}}\\
\leq&C{\left\|\frac{\omega}{r}\right\|}^{\frac{4}{3}}_{L^{2}}
{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}
+\frac{1}{8}{\left\|\nabla_{h}\partial_{z}\omega\right\|}^{2}_{L^{2}}.\end{aligned}$$ Arguing as for proving $J_2$, the term $J_4$ can be bounded as follows. $$\begin{aligned}
J_{4}\leq&{\left\|\partial_r\omega\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}\partial_r\omega\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}u^{r}\right\|}^{\frac{1}{2}}_{L^{2}}
{\left\|\nabla_{h} \partial_{z}u^{r}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}\omega\right\|}^{\frac{1}{2}}_{L^{2}}
{\left\|\nabla_{h}\partial_{z}\omega\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq&{\left\|\omega\right\|}^{\frac{1}{2}}_{L^2}{\left\|\partial_r\omega\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h} \partial_{z}u^{r}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}\omega\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\partial_{z}\omega\right\|}_{L^{2}}\\
\leq&C{\left\|\omega\right\|}_{L^2}{\left\|\partial_r\omega\right\|}_{L^{2}}{\left\|\nabla_{h} \partial_{z}u^{r}\right\|}_{L^{2}}{\left\|\partial_{z}\omega\right\|}_{L^{2}}+\frac{1}{8}{\left\|\nabla_{h}\partial_{z}\omega\right\|}^{2}_{L^{2}}\\
\leq&C{\left\|\omega\right\|}^{2}_{L^2}{\left\|\nabla_{h} \partial_{z}u^{r}\right\|}^{2}_{L^{2}}{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}+C{\left\|\partial_r\omega\right\|}^{2}_{L^{2}}+\frac{1}{8}{\left\|\nabla_{h}\partial_{z}\omega\right\|}^{2}_{L^{2}}.\end{aligned}$$ We turn to bound the term $J_5$, by Lemma \[lema.1\] and the Young inequality, we obtain $$\begin{split}
J_{5}\leq&{\left\|\partial_1 u^{1}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}\partial_1 u^{1}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}\omega\right\|}_{L^{2}}
{\left\|\nabla_{h}\partial_{z}\omega\right\|}_{L^{2}}\\
\leq&{\left\|\omega\right\|}_{L^{2}}{\left\|\partial_{z}\partial_1 u^{1}\right\|}_{L^{2}}{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}
+\frac{1}{8}{\left\|\nabla_{h}\partial_{z}\omega\right\|}_{L^{2}}.
\end{split}$$ Putting this all together and using the fact that ${\left\|\nabla_h\nabla u\right\|}_{L^{2}}\simeq{\left\|\nabla_h\omega\right\|}_{L^{2}}$, we get $$\begin{aligned}
&\frac{\rm d}{{\rm d}t}{\left\|\partial_z\omega(t)\right\|}^{2}_{L^{2}}+\frac{3}{4}{\left\|\nabla_{h}\partial_z\omega(t)\right\|}^{2}_{L^{2}}\\
\leq&{\left\|\partial_z \omega\right\|}^{2}_{L^{2}}+{\left\|\partial^{2}_{zr}\rho\right\|}^{2}_{L^{2}}+C\Big\|\frac{\omega_\theta}{r}\Big\|^{\frac{4}{3}}_{L^2}{\left\|\nabla_{h}\omega\right\|}^{2}_{L^{2}}+
C\Big\|\frac{\omega_\theta}{r}\Big\|^{2}_{L^2}{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}\\
&+C{\left\|\omega\right\|}^{2}_{L^2}{\left\|\nabla_{h} \partial_{z}u^{r}\right\|}^{2}_{L^{2}}{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}+C{\left\|\partial_r\omega\right\|}^{2}_{L^{2}}+{\left\|\omega\right\|}_{L^{2}}{\left\|\nabla_{h}\nabla u\right\|}_{L^{2}}{\left\|\partial_{z}\omega\right\|}^{2}_{L^{2}}\\
\lesssim &\Big(1+\Big\|\frac{\omega_\theta}{r}\Big\|^{\frac{4}{3}}_{L^2}+{\left\|\omega\right\|}^{2}_{L^2}{\left\|\nabla_{h} \omega\right\|}^{2}_{L^{2}}\Big){\left\|\partial_z \omega\right\|}^{2}_{L^{2}}+{\left\|\partial^{2}_{zr}\rho\right\|}^{2}_{L^{2}}+\Big\|\frac{\omega_\theta}{r}\Big\|^{\frac{4}{3}}_{L^2}{\left\|\nabla_{h}\omega\right\|}^{2}_{L^{2}}
+{\left\|\partial_r\omega\right\|}^{2}_{L^{2}}.\end{aligned}$$ This together with Proposition \[Strong\] and the Gronwall inequality yields the desired the result.
Strong a priori estimate
------------------------
In the following, our target is to establish the global estimate about Lipschitz norm of the velocity which ensures the global existence of solution.
Let us first give a useful lemma which provides the maximal smooth effect of the velocity in horizontal direction.
\[smoothing\] Let $s_{1},s_{2}\in {\mathbb{R}}$ and $p\in[2,\infty[$. Assume that $(u, \rho)$ be a smooth solution of the system , then there holds that $${\left\|u\right\|}_{L^{1}_{t}B^{s_{1}+2,s_{2}}_{p,1}}
\lesssim {\left\|u_{0}\right\|}_{B_{p,1}^{s_{1},s_{2}}}+{\left\|u\right\|}_{L^{1}_{t}B_{p,1}^{s_{1},s_{2}}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}B_{p,1}^{s_{1}+1,s_{2}}\cap L^{1}_{t}B_{p,1}^{s_{1},s_{2}+1}}+{\left\|\rho\right\|}_{L^{1}_{t}B_{p,1}^{s_{1},s_{2}}}.$$
Applying the operator $\Delta^{h}_{q}\Delta^{v}_{k}$ to and using Duhamel formula we get $$\begin{split}
u_{q,k}(t)=&e^{t\Delta_{h}}u_{q,k}(0)-\int_0^te^{(t-\tau)\Delta_{h}}\Delta^{h}_q\Delta^{v}_k\mathcal{P}(u\cdot\nabla u)(\tau,x)\mathrm{d}\tau-\int_0^te^{(t-\tau)\Delta_{h}}\Delta^{h}_q\Delta^{v}_k\mathcal{P}\rho(\tau,x)e_z\mathrm{d}\tau,
\end{split}$$ where $u_{q,k}=\Delta^{h}_q\Delta^{v}_k u$ and $\mathcal{P}$ is the Leray projection on divergence free vector fields.
According to Proposition \[heat\], we have the following estimate for $q\geq0$ $$\|e^{t\Delta_{h}}\Delta^{h}_q \Delta^{v}_k f\|_{L^p({\mathbb{R}}^3)}\leq{\left\|\|e^{t\Delta_{h}}\Delta^{h}_q \Delta^{v}_k f\|_{L^p({\mathbb{R}}_{h}^2)}\right\|}_{L^{p}({\mathbb{R}}_{v})}\le Ce^{-ct2^{2q}}\|\Delta^{h}_q \Delta^{v}_k f\|_{L^p({\mathbb{R}}^3)}.$$ Therefore, for $q\geq 0$, we have $$\|u^{r}_{q,k}\|_{L^1_tL^p}\lesssim 2^{-2q}\|u_{q,k}(0)\|_{L^p}+2^{-2q}\int_0^t\|\Delta^{h}_q \Delta^{v}_k(u\cdot\nabla u)(\tau)\|_{L^p}\mathrm{d}\tau+2^{-2q}\|\rho_{q,k}\|_{L^1_t L^p}$$ Multiplying $2^{q(s_1+2)}2^{ks_2}$ and summing over $q,k$, we obtain $$\begin{aligned}
&\sum^{+\infty}_{q=0,k=-1}2^{q(s_{1}+2)}2^{ks_{2}}\|u_{q,k}\|_{L^1_tL^p}\\
\lesssim & \sum^{+\infty}_{q=0,k=-1}2^{qs_{1}}2^{ks_{2}}\|u_{q,k}(0)\|_{L^p}+\int_0^t\sum^{+\infty}_{q=0,k=-1}2^{q(s_{1}+1)}2^{ks_{2}}\|\Delta^{h}_q \Delta^{v}_k(u\otimes u)(\tau)\|_{L^p}\mathrm{d}\tau\\&+\int_0^t\sum^{+\infty}_{q=0,k=-1}2^{qs_{1}}2^{k(s_{2}+1)}\|\Delta^{h}_q \Delta^{v}_k(u\otimes u)(\tau)\|_{L^p}\mathrm{d}\tau+\int_{0}^{t}\sum^{+\infty}_{q=0,k=-1}2^{qs_{1}}2^{ks_{2}}\|\rho_{q,k}(\tau)\|_{ L^p}\mathrm{d}\tau\end{aligned}$$ It follows that, $$\begin{split}
{\left\|u\right\|}_{L^{1}_{t}B^{s_{1}+2,s_{2}}_{p,1}}\lesssim & {\left\|u\right\|}_{L^{1}_{t}B_{p,1}^{s_{1},s_{2}}}+\sum^{+\infty}_{q=0,k=-1}2^{q(s_{1}+2)}2^{ks_{2}}\|u_{q,k}\|_{L^1_tL^p}\\
\lesssim & {\left\|u\right\|}_{L^{1}_{t}B_{p,1}^{s_{1},s_{2}}}+{\left\|u_{0}\right\|}_{B_{p,1}^{s_{1},s_{2}}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}B_{p,1}^{s_{1}+1,s_{2}}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}B_{p,1}^{s_{1},s_{2}+1}}+{\left\|\rho\right\|}_{L^{1}_{t}B_{p,1}^{s_{1},s_{2}}}.
\end{split}$$ This ends the proof.
\[Lipschitz\] Let $u_0\in H^1$ be a divergence free axisymmetric without swirl vector field such that $\frac{\omega_0}{r}\in L^2,~\partial_z\omega_{0}\in L^{2}$ and $\rho_0\in H^{0,1}$ an axisymmetric function. Then any smooth solution $(u, \rho)$ of the system satisfies $$\|\nabla u\|_{L^1_tL^\infty}\le C_0e^{\exp{C_0 t}}.$$ Here, the constant $C_{0}$ depends on the initial data.
According to the structure of axisymmetric flows and the incompressible property of velocity, we know that ${\rm div }u=\partial_ru^r+\frac{u^r}{r}+\partial_zu^z=0$ and $\omega_\theta=\partial_zu^r-\partial_ru^z$. Therefore $$\begin{split}
\|\nabla u\|_{L^1_tL^\infty}\leq& \|\partial_r u^{r}\|_{L^1_tL^\infty}+\|\partial_z u^{r}\|_{L^1_tL^\infty}+\|\partial_r u^{z}\|_{L^1_tL^\infty}+\|\partial_z u^{z}\|_{L^1_tL^\infty}\\
\lesssim & \Big\|\frac{u^{r}}{r}\Big\|_{L^1_tL^\infty}+\|\partial_r u^{r}\|_{L^1_tL^\infty}+\|\partial_z u^{r}\|_{L^1_tL^\infty}+\|\partial_r u^{z}\|_{L^1_tL^\infty}.
\end{split}$$ For the quantity $\big\|\frac{u^{r}}{r}\big\|_{L^1_tL^\infty}$. By virtue of and , we get that $$\Big\|\frac{u^{r}}{r}\Big\|_{L^1_tL^\infty}\leq C\Big\|\frac{\omega_{\theta}}{r}\Big\|^{\frac12}_{L^\infty_tL^2}\Big\|\nabla_h\Big(\frac{\omega_{\theta}}{r}\Big)\Big\|^{\frac12}_{L^1_tL^2}
\leq Ce^{Ct}.$$ Next, we turn to bound the quantity $\|\partial_z u^{r}\|_{L^1_tL^\infty}$, by using and the Bernstein inequality, we have $$\|\partial_z u^{r}\|_{L^1_tL^\infty}\leq C\int_{0}^{t}\|\partial_z \nabla u^{r}\|^{\frac12}_{L^2}\|\nabla_h\partial_z \nabla u^{r}\|^{\frac12}_{L^2}{\mathrm d}\tau
\leq C\|\partial_z \omega\|^{\frac12}_{L^\infty_tL^2}\|\nabla_h\partial_z \omega\|^{\frac12}_{L^1_tL^2}.$$ For the quantity $\|\partial_r u^{r}\|_{L^1_tL^\infty}$ and $\|\partial_r u^{z}\|_{L^1_tL^\infty},$ by taking advantage of Lemma \[bernstein\], we know $$\|\partial_r u^{r}\|_{L^1_tL^\infty}+\|\partial_r u^{z}\|_{L^1_tL^\infty}\leq C{\left\|u\right\|}_{L^1_tB_{2,1}^{2,\frac12}}.$$ Furthermore, by virtue of Lemma \[smoothing\], we get $$\begin{aligned}
{\left\|u\right\|}_{L^1_tB_{2,1}^{2,\frac12}}\lesssim&{\left\|u_{0}\right\|}_{B^{0,\frac12}_{2,1}}+{\left\|u\right\|}_{L^{1}_{t}B_{2,1}^{0,\frac12}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}B_{2,1}^{1,\frac12}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}B_{2,1}^{0,\frac32}}+{\left\|\rho\right\|}_{L^{1}_{t}B_{2,1}^{0,\frac12}}\\
\lesssim&{\left\|u_{0}\right\|}_{H^{1,1}}+{\left\|u\right\|}_{L^{1}_{t}H^{1,1}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}H^{2,1}}+{\left\|u\otimes u\right\|}_{L^{1}_{t}{H}^{\frac54,\frac74}}+{\left\|\rho\right\|}_{L^{1}_{t}H^{1,1}}\\
\lesssim&{\left\|u_{0}\right\|}_{H^{1,1}}+{\left\|u\right\|}_{L^{1}_{t}H^{1,1}}+{\left\|u\right\|}^{2}_{L^{2}_{t}H^{2,1}}+
{\left\|u\right\|}^{2}_{L^{2}_{t}H^{\frac54,\frac74}}+{\left\|\rho\right\|}_{L^{1}_{t}H^{1,1}}.\end{aligned}$$ On the other hand, from the definition of space, we have $${\left\|u\right\|}_{L^2_tH^{2,1}}\lesssim{\left\|u\right\|}_{L^2_tL^{2}}+{\left\|\partial_zu\right\|}_{L^2_tL^{2}}+{\left\|\nabla^{2}_hu\right\|}_{L^2_tL^{2}}+{\left\|\nabla^{2}_h\partial_zu\right\|}_{L^2_tL^{2}}$$ and $$\begin{split}
{\left\|u\right\|}_{L^2_tH^{\frac54,\frac74}}\lesssim&{\left\|u\right\|}_{L^2_tL^{2}}+\big\|\Lambda^{\frac54}_{h}u\big\|_{L^2_tL^{2}}+\big\|\Lambda^{\frac74}_{v}u\big\|_{L^2_tL^{2}}
+\big\|\Lambda^{\frac54}_{h}\Lambda^{\frac74}_{v}u\big\|_{L^2_tL^{2}}\\
\lesssim&{\left\|u\right\|}_{L^2_tL^{2}}+\big\|\nabla_{h}u\big\|_{L^2_tL^{2}}+\big\|\nabla^{2}_{h}u\big\|_{L^2_tL^{2}}+\big\|\partial_{z}u\big\|_{L^2_tL^{2}}+\big\|\partial^{2}_{z}u\big\|_{L^2_tL^{2}}
+\big\|\nabla_{h}\partial_{z}\omega\big\|_{L^2_tL^{2}}.
\end{split}$$ It remains to bound the norm of $\rho$. By the first estimate of and , we have $${\left\|\rho\right\|}_{L^1_tH^{1,1}}\lesssim \big\|\rho\big\|_{L^1_tL^{2}}+
\big\|\partial_z\rho\big\|_{L^1_tL^{2}}+\big\|\nabla_h\rho\big\|_{L^1_tL^{2}}+\big\|\nabla_h\partial_z\rho\big\|_{L^1_tL^{2}}\leq C e^{\exp{Ct}}.$$ Collecting these estimates with , and yields that $$\label{line}
\|\nabla u\|_{L^1_tL^\infty}\le C_0 e^{\exp{C_0 t}}.$$ This ends the proof.
Proof of Theorem \[thm1\] {#sectionproof}
=========================
Here we use the Friedrichs method (see [@dp2] for more details): For $n\geq 1$, let $J_n$ be the spectral cut-off defined by $$\widehat{J_{n}f}(\xi)=1_{[0,n]}(|\xi|)\widehat{f}(\xi), \quad \xi \in{\mathbb{R}}^3.$$ We consider the following system in the spaces $L^{2}_{n}:=\{f\in L^2({\mathbb{R}}^{3})|\text{ supp} f\subset B(0,n)\}$: $$\begin{cases}
\partial_tu+\mathcal{P}J_{n}\text{div}(\mathcal{P}J_nu\otimes \mathcal{P}J_nu)-\Delta_{h}\mathcal{P}J_nu=\mathcal{P}J_n(\rho e_3),\\
\partial_t\rho +J_n\text{div}(J_nu J_n\rho)-\Delta_{h}J_n\rho=0,\\
(\rho,u)|_{t=0}=J_{n}(\rho_0,u_0).
\end{cases}$$ The Cauchy-Lipschitz theorem entails that this system exists a unique maximal solution $(\rho_n,u_n)$ in $\mathcal{C}^{1}([0,T^{*}_{n}[;L^{2}_{n})$. On the other hand, we observe that $J^{2}_n=J_n, \mathcal{P}^{2}=\mathcal{P}$ and $J_n\mathcal{P}=\mathcal{P}J_n$. It follows that $(\rho_n,\mathcal{P}u_n)$ and $(J_n\rho_n,J_n\mathcal{P}u_n)$ are also solutions. The uniqueness gives that $\mathcal{P}u_n=u_n, J_nu_n=u_n$ and $J_n\rho_n=\rho_n$. Therefore $$\label{approx}
\begin{cases}
\partial_tu_{n}+\mathcal{P}J_{n}\text{div}(u_{n}\otimes u_{n})-\Delta_{h}u_{n}=\mathcal{P}J_n(\rho_{n} e_3),\\
\partial_t\rho_{n} +J_n\text{div}(u_{n} \rho_{n})-\Delta_h\rho_n=0,\\
\text{div}u_{n}=0,\\
(\rho_{n},u_{n})|_{t=0}=J_{n}(\rho_0,u_0).
\end{cases}$$ As the operators $J_n$ and $\mathcal{P}J_n$ are the orthogonal projectors for the $L^2$-inner product, the above formal calculations remain unchanged. We will start with the following stability results.
\[lem5-1\] Let $u_0$ be a free divergence axisymmetric vector-field without swirl and $\rho_0$ an axisymmetric scalar function. Then
1. for every $n\in{\mathbb{N}}$, $u_{0,n}$ and $\rho_{0,n}$ are axisymmetric and $\textnormal{div}u_{0,n}=0.$
2. If $u_0\in H^1$ is such that $({\textnormal{curl }u_0})/{r}\in L^2$ and $\rho_{0}\in H^{0,1}$. Then there exists a constant $C$ independent of $n$ such that $$\|u_{0,n}\|_{H^1}\le \|u_0\|_{H^1},\quad
\big\| (\textnormal{curl }u_{0,n})/r \big\|_{L^2}\le C\big\| {(\textnormal{curl }u_0)}/{r} \big\|_{L^2},$$ $$\|\rho_{0, n}\|_{L^2} \leq \|\rho_{0}\|_{L^2}, \quad
\|\rho_{0, n} \|_{H^{0,1}} \leq C \|\rho_{0} \|_{H^{0,1}}.$$
The proof of $\big\| (\textnormal{curl }u_{0,n})/r \big\|_{L^2}\le C\big\| {(\textnormal{curl }u_0)}/{r} \big\|_{L^2}$ is subtle, one can see [@rd] for more details. Other estimates can be proved by the standard methods.
Now, we come back to the proof of the existence parts of Theorem \[thm1\]. From Lemma \[lem5-1\], we observe that the initial structure of axisymmetry is preserved for every $n$ and the involved norms are uniformly controlled with respect to this parameter $n$. This ensures us to construct locally in time a unique solution $(u_n,\rho_n)$ to the approximate system . On the other hand, we have seen in that the Lipschitz norm of the velocity keeps bounded in finite time. Therefore, this solution is globally defined. By standard compactness arguments and Lions-Aubin Lemma we can show that this family $(u_n,\rho_n)_{n\in{\mathbb{N}}}$ converges to $(u,\rho)$ which satisfies in turn our initial problem. And the Fatou Lemma ensures $(u,\rho)\in\mathcal{X}$, where $$\begin{aligned}
\mathcal{X}:=&\big(L^\infty_{\rm loc}({\mathbb{R}}_{+};H^1)\cap L^2_{\rm loc}({\mathbb{R}}_{+};H^{2,1})\cap L^\infty_{\rm loc}({\mathbb{R}}_+;H^{1,1}\cap H^{0,2})\cap L^2_{\rm loc}({\mathbb{R}}_+;H^{2,1}\cap H^{1,2})\\&\cap L^1_{\rm loc}({\mathbb{R}}_{+};{\rm Lip})\big)
\times \big(L^\infty_{\rm loc}({\mathbb{R}}_{+}; H^{0,1})\cap L^2_{\rm loc}({\mathbb{R}}_{+}; H^{1,1})\big).\end{aligned}$$
It remains to prove the time continuity of the solution $(u,\rho)$. We only show that $u$ belongs to $ \mathcal{C}({\mathbb{R}}_{+};H^{1})$, the other terms can be treated the same way. First we show the continuity of $u$ in $H^{1}$. Indeed, we just need to show that $\omega\in \mathcal{C}({\mathbb{R}}_{+};L^{2})$. Let us recall the vorticity equation $$\partial_t \omega +u\cdot\nabla\omega-\Delta_{h}\omega
=-\partial_{r}\rho e_{\theta}+\frac{u^r}{r}\omega.$$ It is easy to check that the source terms belong to $L^{2}_{\rm loc}({\mathbb{R}}_+;L^2)$. Using the fact $\nabla u\in L^{1}_{\rm loc}({\mathbb{R}}_+;{\rm Lip})$ and applying Proposition \[prop-con\], we get the desired result $\omega\in \mathcal{C}({\mathbb{R}}_{+};L^{2})$.
Next, let us turn to prove the uniqueness. We assume that $(u_{i},\rho_{i})\in \mathcal{X}, 1\leq i\leq 2$ be two solutions of the system with the same initial . Then the difference $(\delta\rho,\delta u,\delta p)$ between two solutions $(\rho_1,u_1,p_1)$ and $(\rho_2,u_2,p_2)$ satisfies $$\label{diff}
\begin{cases}
\partial_t\delta u+\text{div}({u_{2}\otimes\delta u})-\Delta_{h}\delta u+\nabla\delta p=-\delta{u}\cdot\nabla u_{1}+\delta\rho e_{z},\\
\partial_t\delta\rho+\text{div}(u_{2}\delta\rho)-\Delta_h\delta\rho=-\delta u\cdot\nabla\rho_1.
\end{cases}$$ Taking the $L^{2}$-inner product to the first equation of with $ u$, we obtain $$\label{diff-velo}
\begin{split}
\frac{1}{2}\frac{\rm d}{{\rm d} t}{\left\| \delta u(t)\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h} \delta u\right\|}^{2}_{L^{2}}=&-\int\delta u\nabla u_{1}\delta u {\rm d}x+\int\delta\rho e_{z}\delta u {\rm d}x\\
\leq&{\left\|\nabla u_{1}\right\|}_{L^{\infty}}{\left\|\delta u\right\|}^{2}_{L^{2}}+{\left\|\delta \rho\right\|}_{L^2}{\left\|\delta u\right\|}_{L^2}.
\end{split}$$ On the other hand, by the same computation, we get $$\begin{split}
\frac{1}{2}\frac{\rm d}{{\rm d} t}{\left\| \delta \rho(t)\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h} \delta \rho\right\|}^{2}_{L^{2}}=-\int\delta u\nabla \rho_1\delta \rho {\rm d}x
=-\int(\delta u)^{r}\partial_{r} \rho_1\delta \rho {\rm d}x-\int(\delta u)^{z}\partial_{z} \rho_1\delta \rho {\rm d}x.
\end{split}$$ By Lemma \[lema.2\] and the Young inequality, $$\begin{split}
\int(\delta u)^{r}\partial_{r} \rho_1\delta \rho {\rm d}x\leq& {\left\|(\delta u)^{r}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}(\delta u)^{r}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{r} \rho_1\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}\partial_{r} \rho_1\right\|}^{\frac{1}{2}}_{L^{2}}
{\left\|\delta\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq&C{\left\|\partial_{r} \rho_{1}\right\|}_{L^{2}}{\left\|\partial_{z}\partial_{r} \rho_{1}\right\|}_{L^{2}}
{\left\|\delta u\right\|}_{L^{2}}{\left\|\delta\rho\right\|}_{L^{2}}+\frac12{\left\|\nabla_{h}\delta u\right\|}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}_{L^{2}}.
\end{split}$$ Using Lemma \[lema.2\] and $\text{div}\delta u=0$, we have $$\begin{aligned}
\int(\delta u)^{z}\partial_{z} \rho_1\delta \rho {\rm d}x\leq& {\left\|(\delta u)^{z}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z}(\delta u)^{z}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z} \rho_1\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\partial_{z} \rho_1\right\|}^{\frac{1}{2}}_{L^{2}}
{\left\|\delta\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq& {\left\|(\delta u)^{z}\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}(\delta u)\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{z} \rho_1\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\partial_{z} \rho_1\right\|}^{\frac{1}{2}}_{L^{2}}
{\left\|\delta\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}^{\frac{1}{2}}_{L^{2}}\\
\leq&C{\left\|\partial_{z} \rho_{1}\right\|}_{L^{2}}{\left\|\nabla_{h}\partial_{z} \rho_{1}\right\|}_{L^{2}}
{\left\|\delta u\right\|}_{L^{2}}{\left\|\delta\rho\right\|}_{L^{2}}+\frac12{\left\|\nabla_{h}\delta u\right\|}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}_{L^{2}}.\end{aligned}$$ The combination of these estimates yield $$\begin{split}
&\frac{1}{2}\frac{\rm d}{{\rm d} t}{\left\| \delta \rho(t)\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h} \delta \rho\right\|}^{2}_{L^{2}}\\
\leq&C({\left\|\nabla_{h} \rho_{1}\right\|}_{L^{2}}+{\left\|\partial_{z} \rho_{1}\right\|}_{L^{2}}){\left\|\nabla_{h}\partial_{z} \rho_{1}\right\|}_{L^{2}}
{\left\|\delta u\right\|}_{L^{2}}{\left\|\delta\rho\right\|}_{L^{2}}+{\left\|\nabla_{h}\delta u\right\|}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}_{L^{2}}.
\end{split}$$ This together with yields that $$\begin{split}
&\frac{1}{2}\frac{\rm d}{{\rm d} t}\left({\left\| \delta \rho(t)\right\|}^{2}_{L^{2}}+{\left\| \delta u(t)\right\|}^{2}_{L^{2}}\right)+{\left\|\nabla_{h} \delta \rho\right\|}^{2}_{L^{2}}+{\left\|\nabla_{h} \delta u\right\|}^{2}_{L^{2}}\\
\leq&C\big({\left\|\nabla_{h} \rho_{1}\right\|}_{L^{2}}+{\left\|\partial_{z} \rho_{1}\right\|}_{L^{2}}\big){\left\|\nabla_{h}\partial_{z} \rho_{1}\right\|}_{L^{2}}
{\left\|\delta u\right\|}_{L^{2}}{\left\|\delta\rho\right\|}_{L^{2}}+{\left\|\nabla_{h}\delta u\right\|}_{L^{2}}{\left\|\nabla_{h}\delta\rho\right\|}_{L^{2}}\\
&+{\left\|\nabla u_1\right\|}_{L^{\infty}}{\left\|\delta u\right\|}^{2}_{L^{2}}+{\left\|\delta \rho\right\|}_{L^2}{\left\|\delta u\right\|}_{L^2}.
\end{split}$$ Consequently, $$\frac{\rm d}{{\rm d} t}\left({\left\| \delta \rho(t)\right\|}^{2}_{L^{2}}+{\left\| \delta u(t)\right\|}^{2}_{L^{2}}\right)\le C F(t) \left({\left\| \delta \rho(t)\right\|}^{2}_{L^{2}}+{\left\| \delta u(t)\right\|}^{2}_{L^{2}}\right),$$ where $$F(t)=({\left\|\nabla_{h} \rho_{1}\right\|}_{L^{2}}+{\left\|\partial_{z} \rho_{1}\right\|}_{L^{2}}){\left\|\nabla_{h}\partial_{z} \rho_{1}\right\|}_{L^{2}}+{\left\|\nabla u_1\right\|}_{L^{\infty}}+1.$$ By Proposition \[Prop-Energy\] and Proposition \[vertical\], we know that $F(t)$ is integrable. Therefore, we obtain the uniqueness by using the Gronwall inequality.
Proof of Theorem \[lose-global\] {#sectionproof-2}
================================
In this section, we intend to prove the global existence and the uniqueness of Theorem \[lose-global\] for another class of initial data.
\[log\] Assume that $u_0\in H^{1},$ with $\frac{\omega_0}{r}\in L^2$ and $\omega_{0}\in L^{\infty}$. Let $\rho_0\in H^{0,1}$. Then any smooth axisymmetric solution $(u,\rho)$ of without swirl satisfies $${\left\|\nabla u(t)\right\|}_{L}\leq Ce^{\exp{Ct}}\big({\left\|\omega_0\right\|}_{L^2\cap L^\infty}+1\big).$$ Here constant $C$ depends on the initial data.
Multiplying the vorticity equation with $|\omega_\theta|^{p-2}\omega_\theta$ and performing integration in space, we get $$\begin{split}
&\frac1p\frac{\rm d}{{\rm d}t}\int |\omega_\theta|^p{\rm d}x+(p-1)\int|\nabla_h\omega_\theta|^{2}|\omega_\theta|^{p-2}{\rm d}x+\int|\omega_\theta|^{p-2}\frac{\omega^{2}_\theta}{r^{2}}{\rm d}x\\
=&\int\frac{u^r}{r} |\omega_\theta|^p{\rm d}x-\int\partial_{r}\rho|\omega_\theta|^{p-2}\omega_\theta {\rm d}x.
\end{split}$$ We consider the case $p\geq 4$. For the first term in the last line, we deuce by the Hölder inequality that $$\int\frac{u^r}{r} |\omega_\theta|^p{\rm d}x\leq \Big\|\frac{u^{r}}{r}\Big\|_{L^{\infty}}{\left\|\omega_\theta\right\|}^{p}_{L^p}.$$ Since ${\left\|u\right\|}_{L^{p-2}}\leq{\left\|u\right\|}^{\alpha}_{L^{2}}{\left\|u\right\|}^{1-\alpha}_{L^{p}}
$ with $\alpha=\frac{4}{(p-2)^2}$ and $$\int|\omega_\theta|^{p-4}|\nabla_{h}\omega_\theta|^{2}{\rm d}x=\int|\omega_\theta |^{p-4}|\nabla_{h}\omega_\theta|^{\frac{2(p-4)}{p-2}}|\nabla_{h}\omega_\theta|^{\frac{4}{p-2}}{\rm d}x
\leq{\left\|\nabla_{h}\omega_\theta\right\|}^{\frac{4}{p-2}}_{L^{2}}\big\||\omega_\theta|^{\frac{p-2}{2}}\nabla_h\omega_\theta\big\|^{\frac{2(p-4)}{p-2}}_{L^{2}}.$$ By using Lemma \[lema.2\] and some based inequalities, the second term can be bounded as follows $$\begin{aligned}
&\int\partial_{r}\rho|\omega_\theta|^{p-2}\omega_\theta {\rm d}x\\
\leq&{\left\|\partial_r\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial^2_{rz}\rho\right\|}^{\frac{1}{2}}_{L^2}{\left\|\omega^{\frac{p-2}{2}}_\theta\right\|}^{\frac{1}{2}}_{L^2}
{\left\|\nabla_h\big(\omega^{\frac{p-2}{2}}_\theta\big)\right\|}^{\frac12}_{L^2}
{\left\|\omega^{\frac{p}{2}}_\theta\right\|}^{\frac{1}{2}}_{L^2}{\left\|\nabla_h\big(\omega^{\frac{p}{2}}_\theta\big)\right\|}^{\frac{1}{2}}_{L^2}\\
\leq&C\sqrt{p}{\left\|\partial_r\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial^2_{rz}\rho\right\|}^{\frac{1}{2}}_{L^2}{\left\|\omega_\theta\right\|}^{\frac{p-2}{4}}_{L^{p-2}}
{\left\|\omega^{\frac{p}{2}-2}_\theta\nabla_h\omega_\theta\right\|}^{\frac{1}{2}}_{L^2}
{\left\|\omega_\theta\right\|}^{\frac{p}{4}}_{L^p}{\left\|\nabla_h\big(\omega^{\frac{p}{2}}_\theta\big)\right\|}^{\frac{1}{2}}_{L^2}\\
\leq&Cp^{\frac{1}{p-2}}{\left\|\partial_r\rho\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial^2_{rz}\rho\right\|}^{\frac{1}{2}}_{L^2}{\left\|\omega_\theta\right\|}^{\frac{1}{p-2}}_{L^2}
{\left\|\omega_\theta\right\|}^{\frac{p(p-4)}{4(p-2)}}_{L^{p}}
{\left\|\omega_\theta\right\|}^{\frac{p}{4}}_{L^p}{\left\|\nabla_h\omega_\theta\right\|}^{\frac{1}{p-2}}_{L^{2}}
{\left\|\nabla_h\big(\omega^{\frac{p}{2}}_\theta\big)\right\|}^{\frac12+\frac{(p-4)}{2(p-2)}}_{L^2}\\
\leq&Cp^{\frac{2}{p-1}}{\left\|\partial_r\rho\right\|}^{\frac{p-2}{p-1}}_{L^{2}}{\left\|\partial^2_{rz}\rho\right\|}^{\frac{p-2}{p-1}}_{L^2}
{\left\|\nabla_h\omega_\theta\right\|}^{\frac2{p-1}}_{L^2}
{\left\|\omega_\theta\right\|}^{\frac{2}{p-1}}_{L^2}
{\left\|\omega_\theta\right\|}^{p-2-\frac{2}{p-1}}_{L^p}+\frac14{\left\|\nabla_h\big(\omega^{\frac{p}{2}}_\theta\big)\right\|}^{2}_{L^2}\end{aligned}$$ Without loss of generality, we assume that ${\left\|\omega_\theta\right\|}_{L^{p}}\geq 1$, thus $$\frac{\rm d}{{\rm d}t}{\left\|\omega_\theta\right\|}^{2}_{L^{p}}\leq C \Big\|\frac{u^{r}}{r}\Big\|_{L^{\infty}}{\left\|\omega_\theta\right\|}^{2}_{L^p}+Cp^{\frac{2}{p-1}}F(t)\leq C \Big\|\frac{u^{r}}{r}\Big\|_{L^{\infty}}{\left\|\omega_\theta\right\|}^{2}_{L^p}+4CF(t),$$ where $F(t):={\left\|\partial_r\rho\right\|}^{\frac{p-2}{p-1}}_{L^{2}}{\left\|\partial^2_{rz}\rho\right\|}^{\frac{p-2}{p-1}}_{L^2}
{\left\|\nabla_h\omega_\theta\right\|}^{\frac2{p-1}}_{L^2}
{\left\|\omega_\theta\right\|}^{\frac{2}{p-1}}_{L^2}$. According to Proposition \[Strong\] and Proposition \[vertical\], we know that $F(t)$ is integrable. Therefore, by the Gronwall inequality and the relation ${\left\|\omega\right\|}_{L^{p}}={\left\|\omega_\theta\right\|}_{L^{p}}$, we obtain that $${\left\|\omega\right\|}^{2}_{L^{p}}\leq Ce^{\exp{Ct}}\Big({\left\|\omega_0\right\|}^{2}_{L^{p}}+\int^{t}_{0}F(\tau){\rm d}\tau\Big).$$ This together with the second estimate of Proposition \[Strong\] yields $${\left\|\omega\right\|}^{2}_{L^{p}}\leq Ce^{\exp{Ct}}\quad {\rm for}\quad 2\leq p<\infty.$$ Since ${\left\|\nabla u\right\|}_{L^{p}}\leq C\frac{p^{2}}{p-1}{\left\|\omega\right\|}_{L^{p}}$ . So, we finally obtain that $
{\left\|\nabla u\right\|}_{L}\leq Ce^{\exp{Ct}}.$ This ends the proof.
Let us first focus on the existence part of Theorem \[lose-global\]. Let $$\begin{aligned}
\mathcal{Y}:=&\big(L^\infty_{\rm loc}({\mathbb{R}}_+;H^1)\cap L^2_{\rm loc}({\mathbb{R}}_+;H^{1,1})\cap L^\infty_{\rm loc}({\mathbb{R}}_+;H^{1,1}\cap H^{0,2})\cap L^2_{\rm loc}({\mathbb{R}}_+;H^{2,1}\cap H^{1,2})\\&\cap L^1_{\rm loc}({\mathbb{R}}_+;{\rm L})\big)
\times\big( L^\infty_{\rm loc} ({\mathbb{R}}_+;H^{0,1})\cap L^2_{\rm loc}({\mathbb{R}}_+;H^{1,1})\big).\end{aligned}$$To prove existence, we smooth out the initial data $(u_{0},\rho_{0})$ so as to obtain a sequence $(u_{0,n},\rho_{0,n})_{n\in{\mathbb{N}}}$ of smooth functions which converges to $(u_0,\rho_0)$. From Lemma \[lem5-1\], it is clear that the initial structure of axisymmetry is preserved for every $n$ . By preceding argument, it is easy to check that $(u_n,\rho_n)\in\mathcal{Y}$. Combining this with the equations , one may conclude that $\partial_{t}\rho_n\in L_{\text{loc}}^{2}({\mathbb{R}}_+;H^{-1})$ and $\partial_{t}u_n\in L_{\text{loc}}^{2}({\mathbb{R}}_+;L^2)$. On the other hand, we know that $L^{2}\hookrightarrow H^{-1}$ and $H^{1}\hookrightarrow L^{2}$ are locally compact. Therefore, by the classical Aubin-Lions argument and Cantor’s diagonal process, we can deduce that, up to extraction, family $(u_n,\rho_n)_{n\in{\mathbb{N}}}$ has a limit $(u,\rho)$ satisfying the equations and that $(u,\rho)\in\mathcal{Y}$. The same arguments as used in allows us to show that the time continuity of $(u,\rho)$ in low norms and the weak time continuity. In addition, in a similar way as used in , we can conclude that $\rho\in \mathcal{C}_{b}({\mathbb{R}}_+;L^2)$.
Now let us turn to the prove the uniqueness. We assume that $(u_{i},\rho_{i})\in \mathcal{Y}, 1\leq i\leq 2$ be two solutions of the system with the same initial . One can write $$\begin{cases}
\partial_t\delta u+\text{div}({u_{2}\otimes\delta u})+\text{div}({\delta u\otimes u_{1}})-\Delta_{h}\delta u+\nabla\delta p=\delta\rho e_{z},\\
\partial_t\delta\rho+\text{div}(u_{2}\delta\rho)-\Delta_h\delta\rho=-\text{div}(\delta u\rho_1).
\end{cases}$$ For the sake of convenience, let $(\alpha,\beta,\gamma)$ such that $\frac12<\alpha<\beta<\gamma\leq 1.$ Note that $$\frac{{\left\|S_{q}u\right\|}_{L^{\infty}}}{q}\leq 2^{\frac{3}{q}}\frac{{\left\|S_{q}u\right\|}_{L^{q}}}{q}\leq 2^{\frac{3}{2}}{\left\|u\right\|}_{L}.$$ By the same argument as in Proposition \[lostest\] with the vector-field $u_2$, then there exists $T_{1}$ such that $${\left\|\delta\rho\right\|}_{H^{\beta-1}}\leq C\int_{0}^{t}{\left\|\text{div}(\delta u\rho_1)\right\|}_{H^{\gamma-1}}{\rm d}\tau\quad\text{for all } t\in [0,T_{1}].$$ The term on the right side can be bounded as follows. By virtue of the Bony decomposition: $$\label{para}
\text{div}(\delta u\rho_1)=\text{div}\big(T_{\delta u}\rho_1 +R(\delta u,\rho_1)\big)+\sum^{2}_{i=1}T_{\partial_i\rho_1}\delta u^i,$$ where we have used the condition $\text{div}\delta u=0$.
From standard continuity results for operators $T$ and $R$ , we have $${\left\|T_{\delta u}\rho_1+R(\delta u,\rho_1)\right\|}_{H^{\gamma}}\leq C{\left\|\delta u\right\|}_{L^\infty}{\left\|\rho_1\right\|}_{H^{\gamma}}.$$ As for the last term, since $\gamma-1<0$, we infer that $${\left\|T_{\partial_i\rho_1}\delta u^i\right\|}_{H^{\gamma-1}}\leq C{\left\|\nabla\rho_1\right\|}_{H^{\gamma-1}}{\left\|\delta u\right\|}_{L^{\infty}}.$$ We eventually get $$\label{de-p}
{\left\|\delta\rho\right\|}_{L_{t}^{\infty}(H^{\beta-1})}\leq C{\left\|\rho_1\right\|}_{L_{t}^{2}(H^{\gamma})}{\left\|\delta u\right\|}_{L_{t}^{2}L^{\infty}}.$$ Now we turn to bound the term $\delta u$. By using Proposition \[lostest\], there exists $T_{2}$ such that for all $t\in [0,T_{2}],$ $${\left\|\delta u\right\|}_{L^{\infty}_{t}(H^\alpha)}+{\left\|\nabla_h\delta u\right\|}_{L^{2}_{t}(H^\alpha)}\leq C\big({\left\|\delta\rho\right\|}_{L^{2}_{t}(H^\beta)}+{\left\|\delta u\cdot\nabla u_1\right\|}_{L^{2}_{t}(H^\beta)}\big)$$ for some constant $C$ depending only on $\alpha,\beta$ and $u_2.$ Using again the Bony decomposition and arguing exactly as for proving , we get $${\left\|\delta u\cdot \nabla u_1\right\|}_{H^{\beta-1}}\leq C{\left\|\delta u\right\|}_{L^{\infty}}{\left\|u_1\right\|}_{H^{\beta}}.$$ Therefore, given that $u_1\in L_{{\rm loc}}^{\infty}({\mathbb{R}};H^{\beta})$, $${\left\|\delta u\right\|}_{L^{\infty}_{t}(H^{\alpha})}+{\left\|\nabla _h \delta u\right\|}_{L^{2}_{t}(H^{\alpha})}\leq C({\left\|\delta\rho\right\|}_{L^{2}_{t}(H^{\beta-1})}+
{\left\|\delta u\right\|}_{L^{2}_{t}(L^{\infty})}).$$ Next, our task is to show that ${\left\|\delta u\right\|}_{L_{t}^{2}L^{\infty}}$ may be bounded in terms of ${\left\|\delta u\right\|}_{L^{\infty}_{t}H^{\alpha}}$ and of ${\left\|\nabla_h\delta u\right\|}_{L^{2}_{t}H^{\alpha}}$.
According to the assumption $\alpha\in ]\frac12,1]$, we have (see the proof in $$\label{appen-l}{\left\|\delta u\right\|}_{L^{\infty}({\mathbb{R}}^{3})}\leq C{\left\|\delta u\right\|}^{\alpha-\frac12}_{H^\alpha({\mathbb{R}}^{3})}{\left\|\nabla_h\delta u\right\|}^{\frac32-\alpha}_{H^\alpha({\mathbb{R}}^{3})}.$$ Combining these estimates, we can deduce that for some constant $C$ depending only on $T=\min\{T_1,T_2\}$ and on the norms of $(\rho_1,u_1)$ and $(\rho_2,u_2)$, we have $${\left\|\delta \rho\right\|}_{L^{\infty}_{t}H^{\beta-1}}\leq Ct^{\frac\alpha2-\frac14}\delta U(t),
\quad \delta U(t)\leq C\big(t^{\frac12}{\left\|\delta\rho\right\|}_{L^{\infty}_{t}H^{\beta-1}}+t^{\frac\alpha2-\frac14}\delta U(t)\big)$$ with $$\delta U(t):={\left\|\delta u\right\|}_{L^{\infty}_{t}H^{\alpha}}+{\left\|\nabla_h\delta u\right\|}_{L^{2}_{t}H^{\alpha}}.$$ It follows that $\delta u\equiv 0$ (and thus $\delta\rho\equiv 0$) on a suitably small time interval. Finally, let us notice that our assumptions on the solutions ensure that $\delta\rho\in \mathcal{C}([0,T];H^{\beta-1})$ and $\delta u\in \mathcal{C}([0,T];H^{\alpha})$. Using a classical connectivity argument, it is now easy to get the uniqueness on the whole interval $[0,\infty[$.
Appendix
========
In this section, we first give some useful inequalities which have been used throughout the paper.
\[sharp\] There exists a constants $C$ such that $${\left\|u\right\|}_{L^\infty({\mathbb{R}}^{3})}\leq C{\left\|\nabla u\right\|}^{\frac12}_{L^{2}({\mathbb{R}}^{3})}{\left\|\nabla_{h}\nabla u\right\|}^{\frac12}_{L^{2}({\mathbb{R}}^{3})}.$$
By using the interpolation theorem, we get $${\left\|u(x_h,\cdot)\right\|}_{L^{\infty}({\mathbb{R}}_{v})}\leq C\|u(x_h,\cdot)\|^{\frac12}_{L^{6}({\mathbb{R}}_{v})}\|\Lambda^{\frac23}_{v}u(x_h,\cdot)\|^{\frac12}_{L^{2}({\mathbb{R}}_{v})}.$$ This together with the Minkowski inequality and the embedding theorem gives $$\label{sharp-1}
\begin{split}
{\left\|u\right\|}_{L^\infty({\mathbb{R}}^{3})}\leq& {\left\|\|u\|_{L^{\infty}({\mathbb{R}}_{v})}\right\|}_{L^{\infty}(\mathbb{R}_{h}^{2})}\\\leq & C\big\|\|u\|^{\frac12}_{L^{6}({\mathbb{R}}_{v})}\|\Lambda^{\frac23}_{v}u\|^{\frac12}_{L^{2}({\mathbb{R}}_{v})}\big\|_{L^{\infty}(\mathbb{R}_{h}^{2})}\\
\leq &\big\|\|\Lambda^{\frac13}_{v}u\|_{L^{\infty}({\mathbb{R}}_{h}^{2})}\big\|^{\frac12}_{L^{2}({\mathbb{R}}_{v})}
\big\|\|\Lambda_{v}^{\frac23}u\|_{L^{\infty}({\mathbb{R}}_{h}^{2})}\big\|^{\frac12}_{L^{2}({\mathbb{R}}_{v})}.
\end{split}$$ On the other hand, using again the interpolation theorem and the embedding theorem, we have $$\label{sharp-2}
\begin{split}
\|\Lambda^{\frac13}_{v}u(\cdot,z)\|_{L^{\infty}({\mathbb{R}}_{h}^{2})}\leq & C\|\Lambda^{\frac13}_{v}u(\cdot,z)\|^{\frac23}_{L^{6}({\mathbb{R}}_{h}^{2})}\|\Lambda^{\frac53}_{h}\Lambda^{\frac13}_{v}u(\cdot,z)\|^{\frac13}_{L^{2}({\mathbb{R}}_{h}^{2})}\\
\leq& C\|\Lambda^{\frac23}_{h}\Lambda^{\frac13}_{v}u(\cdot,z)\|^{\frac23}_{L^{2}({\mathbb{R}}_{h}^{2})}\|\Lambda^{\frac53}_{h}
\Lambda^{\frac13}_{v}u(\cdot,z)\|^{\frac13}_{L^{2}({\mathbb{R}}_{h}^{2})}
\end{split}$$ and $$\label{sharp-3}
\begin{split}
\|\Lambda^{\frac23}_{v}u(\cdot,z)\|_{L^{\infty}({\mathbb{R}}_{h}^{2})}\leq & C\|\Lambda^{\frac23}_{v}u(\cdot,z)\|^{\frac13}_{L^{3}({\mathbb{R}}_{h}^{2})}\|\Lambda^{\frac43}_{h}\Lambda^{\frac23}_{v}u(\cdot,z)\|^{\frac23}_{L^{2}({\mathbb{R}}_{h}^{2})}\\
\leq& C\|\Lambda^{\frac13}_{h}\Lambda^{\frac23}_{v}u(\cdot,z)\|^{\frac13}_{L^{2}({\mathbb{R}}_{h}^{2})}\|\Lambda^{\frac43}_{h}
\Lambda^{\frac23}_{v}u(\cdot,z)\|^{\frac23}_{L^{2}({\mathbb{R}}_{h}^{2})}.
\end{split}$$ Inserting and into , and using the Hölder inequality, we get $$\begin{aligned}
{\left\|u\right\|}_{L^\infty({\mathbb{R}}^{3})}\leq&\big\|\|\Lambda^{\frac23}_{h}\Lambda^{\frac13}_{v}u\|^{\frac23}_{L^{2}({\mathbb{R}}_{h}^{2})}\|\Lambda^{\frac53}_{h}
\Lambda^{\frac13}_{v}u\|^{\frac13}_{L^{2}({\mathbb{R}}_{h}^{2})}\big\|^{\frac12}_{L^{2}({\mathbb{R}}_{v})}
\big\|\|\Lambda^{\frac13}_{h}\Lambda^{\frac23}_{v}u\|^{\frac13}_{L^{2}({\mathbb{R}}_{h}^{2})}\|\Lambda^{\frac43}_{h}
\Lambda^{\frac23}_{v}u\|^{\frac23}_{L^{2}({\mathbb{R}}_{h}^{2})}\big\|^{\frac12}_{L^{2}({\mathbb{R}}_{v})}\\
\leq&\|\Lambda^{\frac23}_{h}\Lambda^{\frac13}_{v}u\|^{\frac13}_{L^{2}({\mathbb{R}}^{3})}\|\Lambda^{\frac53}_{h}
\Lambda^{\frac13}_{v}u\|^{\frac16}_{L^{2}({\mathbb{R}}^{3})}
\|\Lambda^{\frac13}_{h}\Lambda^{\frac23}_{v}u\|^{\frac16}_{L^{2}({\mathbb{R}}^{3})}\|\Lambda^{\frac43}_{h}
\Lambda^{\frac23}_{v}u\|^{\frac13}_{L^{2}({\mathbb{R}}^{3})}\\
\leq &C{\left\|\nabla u\right\|}^{\frac12}_{L^{2}({\mathbb{R}}^{3})}{\left\|\nabla_{h}\nabla u\right\|}^{\frac12}_{L^{2}({\mathbb{R}}^{3})}.\end{aligned}$$ This completes the proof.
\[lema.1\] Let $q\in]2,\infty[$, there holds that $$\label{a.1}
\begin{split}
\int_{\mathbb{R}^{3}}fgh{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}
\leq C{\left\|f\right\|}^{\frac{q-1}{q}}_{L^{2(q-1)}}{\left\|\partial_{x_{1}}f\right\|}^{\frac{1}{q}}_{L^{2}}{\left\|g\right\|}^{\frac{q-2}{q}}_{L^{2}}{\left\|\partial_{x_2}g\right\|}^{\frac{1}{q}}_{L^{2}}
{\left\|\partial_{x_3}g\right\|}^{\frac{1}{q}}_{L^{2}}{\left\|h\right\|}_{L^{2}}.
\end{split}$$ In particular, if we take $q=4$ in , we have $$\label{a.11}
\begin{split}
\int_{\mathbb{R}^{3}}fgh{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}
\leq C{\left\|f\right\|}^{\frac{3}{4}}_{L^{6}}{\left\|\partial_{x_{3}}f\right\|}^{\frac{1}{4}}_{L^{2}}{\left\|g\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}g\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|h\right\|}_{L^{2}}.
\end{split}$$
We only just to show the inequality for functions $f,g,h\in C^{\infty}_{0}(\mathbb{R}^{3})$ and then pass to the limit by virtue of the density argument.
Using some basic inequalities, we have $$\begin{aligned}
&\int_{\mathbb{R}^{3}}fgh{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}\\
\leq&C\int_{\mathbb{R}^{2}}\left[\max_{x_{1}}|f|\left(\int_{\mathbb{R}}g^{2}{\rm d}x_{1}\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}}h^{2}{\rm d}x_{1}\right)^{\frac{1}{2}}\right]{\rm d}x_{2}{\rm d}x_{3}\\
\leq&C\left[\int_{\mathbb{R}^{2}}\max_{x_{1}}|f|^{q}{\rm d}x_{2}{\rm d}x_{3}\right]^{\frac{1}{q}}\left[\int_{\mathbb{R}^{2}}\left(\int_{\mathbb{R}}g^{2}{\rm d}x_{1}\right)^{\frac{q}{q-2}}
{\rm d}x_{2}{\rm d}x_{3}\right]^{\frac{q-2}{2q}}\left(\int_{\mathbb{R}^{3}}h^{2}{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}\right)^{\frac{1}{2}}\\
\leq&C\left[\int_{\mathbb{R}^{2}}\int_{\mathbb{R}}|f|^{q-1}|\partial_{x_{1}f}|{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}\right]^{\frac{1}{q}}\left[\int_{\mathbb{R}^{2}}
\left(\int_{\mathbb{R}}g^{2}{\rm d}x_{1}\right)^{\frac{q}{q-2}}
{\rm d}x_{2}{\rm d}x_{3}\right]^{\frac{q-2}{2q}}{\left\|h\right\|}_{L^{2}}\\
\leq& C{\left\|f\right\|}^{\frac{q-1}{q}}_{L^{2(q-1)}}{\left\|\partial_{x_{1}}f\right\|}^{\frac{1}{q}}_{L^{2}}{\left\|g\right\|}^{\frac{q-2}{q}}_{L^{2}}{\left\|\partial_{x_2}g\right\|}^{\frac{1}{q}}_{L^{2}}
{\left\|\partial_{x_3}g\right\|}^{\frac{1}{q}}_{L^{2}}{\left\|h\right\|}_{L^{2}}.\end{aligned}$$ Indeed, by imbedding theorem, H$\rm\ddot{o}$lder’s inequality and Plancherel theorem, we obtain that $$\begin{aligned}
\big\|{\left\|g\right\|}_{L_{2,3}^{\frac{2q}{q-2}}(\mathbb{R}^{2})}\big\|_{L_{1}^{2}(\mathbb{R})}\leq &C\big\|\big\|\|\Lambda^{\frac{1}{q}}_{2}g\|_{L_{2}^{2}(\mathbb{R}^{1})}\big\|_{L_{3}^{\frac{2q}{q-2}}(\mathbb{R}^{1})}\big\|_{L_{1}^{2}(\mathbb{R})}
\leq C\big\|\big\|\|\Lambda^{\frac{1}{q}}_{2}g\|_{L_{3}^{\frac{2q}{q-2}}(\mathbb{R}^{1})}\big\|_{L_{2}^{2}(\mathbb{R}^{1})}\big\|_{L_{1}^{2}(\mathbb{R})}\\
\leq &C\big\|\big\|\|\Lambda^{\frac{1}{q}}_{3}\Lambda^{\frac{1}{q}}_{2}g\|_{L_{3}^{2}(\mathbb{R}^{1})}\big\|_{L_{2}^{2}(\mathbb{R}^{1})}\big\|_{L_{1}^{2}(\mathbb{R})}=
C\big\|\Lambda^{\frac{1}{q}}_{3}\Lambda^{\frac{1}{q}}_{2}g\big\|_{L^{2}}\\
=&C\Big(\int_{\mathbb{R}^{3}}|\xi_{2}|^{\frac{2}{q}}|\xi_{3}|^{\frac{2}{q}}\hat{g}^{2}(\xi){\rm d}\xi_{1} {\rm d}\xi_{2} {\rm d}\xi_{3}\Big)^{\frac{1}{2}}\leq {\left\|\hat{g}\right\|}^{\frac{q-2}{q}}_{L^{2}}{\left\|\xi_{2}\hat{g}\right\|}^{\frac{1}{q}}_{L^{2}}{\left\|\xi_{3}\hat{g}\right\|}^{\frac{1}{q}}_{L^{2}}\\
\leq& C{\left\|g\right\|}^{\frac{q-2}{q}}_{L^{2}}{\left\|\partial_{x_2}g\right\|}^{\frac{1}{q}}_{L^{2}}
{\left\|\partial_{x_3}g\right\|}^{\frac{1}{q}}_{L^{2}}.\end{aligned}$$ This completes the proof.
\[lema.2\][@Cp] A constant $C$ exists such that $$\label{a.5}
\begin{split}
\int_{\mathbb{R}^{3}}fgh{\rm d}x_{1}{\rm d}x_{2}{\rm d}x_{3}
\leq C{\left\|f\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\partial_{x_{3}}f\right\|}^{\frac{1}{2}}_{L^{2}}
{\left\|g\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}g\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|h\right\|}^{\frac{1}{2}}_{L^{2}}{\left\|\nabla_{h}h\right\|}^{\frac{1}{2}}_{L^{2}}.
\end{split}$$
As for $\alpha\in]\frac12,\frac32[$, by the interpolation theorem and the embedding theorem, we get that for any $z\in{\mathbb{R}}$ $$\begin{aligned}
\label{ad-1}
{\left\|u(\cdot,z)\right\|}_{L_{h}^{\infty}({\mathbb{R}}^{2})}\leq& C{\left\|u(\cdot,z)\right\|}^{\alpha-\frac{1}{2}}_{L_{h}^{\frac{4}{3-2\alpha}}({\mathbb{R}}^{2})}{\left\|\Lambda^{\alpha+\frac12}_{h}u(\cdot,z)\right\|}^{\alpha+\frac{1}{2}}_{L_{h}^{2}({\mathbb{R}}^{2})}\nonumber\\
\leq &C{\left\|\Lambda^{\alpha-\frac12}_{h}u(\cdot,z)\right\|}^{\alpha-\frac{1}{2}}_{L_{h}^{2}({\mathbb{R}}^{2})}{\left\|\Lambda^{\alpha+\frac12}_{h}u(\cdot,z)\right\|}^{\frac{3}{2}-\alpha}_{L_{h}^{2}({\mathbb{R}}^{2})}\\
\leq &C{\left\|u(\cdot,z)\right\|}^{\alpha-\frac{1}{2}}_{H^{\alpha-\frac12}({\mathbb{R}}^{2})}{\left\|\nabla_h u(\cdot,z)\right\|}^{\frac{3}{2}-\alpha}_{H^{\alpha-\frac12}({\mathbb{R}}^{2})}.\nonumber\end{aligned}$$ On the other hand, by the trace theorem that $H^{\alpha}({\mathbb{R}}^{3})\hookrightarrow L^{\infty}({\mathbb{R}}_{v};H^{\alpha-\frac12}({\mathbb{R}}_{h}^{2}))$, $${\left\|u\right\|}_{L_{v}^{\infty}(H^{\alpha-\frac12})}\leq C{\left\|u\right\|}_{(H^{\alpha})}\quad{\rm and }\quad{\left\|\nabla_hu\right\|}_{L_{v}^{\infty}(H^{\alpha-\frac12})}\leq C{\left\|\nabla_hu\right\|}_{(H^{\alpha})}.$$ Inserting these inequalities in , we get the desired result .
For the sake of completeness, we give an existence result for the anisotropic equations with a convection term, which is similar to the case of the transport equation in [@BCD11; @CM-1].
\[prop-con\] Let $s\geq-1$, $1\leq p< \infty$ and $1\leq r\leq \infty$. Assume that $f_0\in B^{s}_{p,r},g\in L^{1}([0,T];B^{s}_{p,r})$, and that $u$ be a divergence free vector-field satisfying $u\in L^{\sigma}([0,T];B^{-m}_{\infty,\infty})$ for some $\sigma>1$ and $m>0$, and $\nabla u\in L^{1}([0,T];L^\infty)$. Then the following equations $$\label{continuce}
\begin{cases}
\partial_tf+u\cdot\nabla f-\Delta_{h} f=g,\\
f|_{t=0}=f_{0}
\end{cases}$$admits a unique solution $f$ in
- the space $\mathcal{C}([0,T];B^{s}_{p,r})$, if $r<\infty$,
- the space $\big(\cap_{s'<s}\mathcal{C}([0,T];B^{s'}_{p,\infty})\big)\cap\mathcal{C}_{w}([0,T];B^{s}_{p,\infty})$, if $r=\infty$.
Moreover, we have $$\label{guji}
{\left\|f(t)\right\|}_{B^{s}_{p,r}}\leq e^{C\int_{0}^{t}V(\tau){\rm d}\tau}\Big({\left\|f_0\right\|}_{B^{s}_{p,r}}+\int^{t}_{0}e^{-C\int_{0}^{\tau}V(s){\rm d}s}{\left\|g(\tau)\right\|}_{B^{s}_{p,r}}{\rm d}\tau\Big).$$ Here $V(t):=\|\nabla u(t)\|_{ L^{\infty}}$.
Without loss of generality, we assume that $u$ and $g$ are defined on ${\mathbb{R}}\times{\mathbb{R}}^{n}$. We first construct the approximate solutions $f_n$ of as follows. $$\label{continuce-approx}
\begin{cases}
\partial_tf_{n}+u_{n}\cdot\nabla f_{n}-\Delta_{h} f_{n}=g_{n},\\
u_{n}:=\varphi_{n}\ast_{t}S_{n}u,\quad g_{n}:=\varphi_{n}\ast_{t}S_{n}g,\\
f_{n}|_{t=0}=f_{0,n}:=S_{n}f_{0},
\end{cases}$$ where, $\varphi_{n}$ denotes a family of mollifiers with respect to $t$. Thanks to the properties of mollifier and the operator $S_j$, it is clear that $f_{0,n}\in B^{\infty}_{p,r}$ and $u_{n},g_{n}\in \mathcal{C}([0,T];B^{\infty}_{p,r})$ with $B^{\infty}_{p,r}:=\cap_{s\in{\mathbb{R}}}B^{s}_{p,r}$. Moreover, $(f_{0,n})_{n\in{\mathbb{N}}}$ is bounded in $B_{p,r}^{s}$, $(g_{n})_{n\in{\mathbb{N}}}$ is bounded in $L^{1}([0,T];B_{p,r}^{s})$, $(u_{n})_{n\in{\mathbb{N}}}$ is bounded in $L^{\sigma}([0,T];B_{\infty,\infty}^{-m})$, and $(\nabla u_{n})_{n\in{\mathbb{N}}}$ is bounded in $L^{1}([0,T]; L^{\infty})$.
Applying $\Delta_{k}$ to , we have $$\begin{cases}
\partial_t\Delta_{k}f_{n}+u_{n}\cdot\nabla \Delta_{k}f_{n}-\Delta_{h} \Delta_{k}f_{n}=\Delta_{k}g_{n}+R^{k}_{n},\\
\Delta_{k}f_{n}|_{t=0}=\Delta_{k}f_{0,n},
\end{cases}$$ where $R^{k}_{n}:=u_n\cdot\nabla\Delta_{k}f_n-\Delta_{k}(u\cdot\nabla f_{n})$. Note that $$-\int \Delta_{h}(\Delta_{k}f_n)|\Delta_{k}f_n|^{p-2}\Delta_{k}f_n{\rm d}x\geq 0,$$ it is easy to conclude that $$\label{frequency}{\left\|\Delta_{k}f_n(t)\right\|}_{L^{p}}\leq{\left\|\Delta_{k}f_0\right\|}_{L^{p}}+\int_{0}^{t}{\left\|\Delta_{k}g_n(\tau)\right\|}_{L^{p}}{\rm d}\tau+\int_{0}^{t}{\left\|R^{k}_{n}(\tau)\right\|}_{L^{p}}{\rm d}\tau.$$ This together with the commutator estimate (see e.g. [@CM-1], Chap-2 ) $${\left\|R^{k}_{n}(t)\right\|}_{L^{p}}\leq Cc_k(t)2^{-ks}V_{n}(t){\left\|f_{n}(t)\right\|}_{B_{p,r}^{s}}\quad{\rm with }\quad {\left\|c_k(t)\right\|}_{l^{r}}=1$$ leads to $${\left\|f_n(t)\right\|}_{B^{s}_{p,r}}\leq{\left\|f_0\right\|}_{B^{s}_{p,r}}+\int_{0}^{t}\Big({\left\|g_n(\tau)\right\|}_{B^{s}_{p,r}}+CV_{n}(\tau){\left\|f(\tau)\right\|}_{B_{p,r}^{s}}\Big){\rm d}\tau,$$ where $V_{n}(t):=\|\nabla u_{n}(t)\|_{ L^{\infty}}$. Applying the Gronwall inequality, we obtain $${\left\|f_n(t)\right\|}_{B^{s}_{p,r}}\leq e^{C\int_{0}^{t}V_{n}(\tau){\rm d}\tau}\Big({\left\|f_0\right\|}_{B^{s}_{p,r}}+\int^{t}_{0}e^{-C\int_{0}^{\tau}V_{n}(s){\rm d}s}{\left\|g_n(\tau)\right\|}_{B^{s}_{p,r}}{\rm d}\tau\Big).$$ In the following, we shall show that, up to an subsequence, the sequence $(f_n)_{n\in{\mathbb{N}}}$ converges in $\mathcal{D}'({\mathbb{R}}_+\times{\mathbb{R}}^{n})$ to a solution $f$ of which has the desired regularity properties. First, one may write $$\partial_tf_n-g_n=-u_n\cdot\nabla f_n+\Delta_h f_n.$$ Hence $u_n\in L^{\sigma}([0,T];B_{\infty,\infty}^{-m})$ enables us to conclude that $\partial_tf_n-g_n$ is bounded in $L^{\sigma}([0,T];B_{p,\infty}^{-M})$ for some sufficiently large $M>0$. For the sake of convenience, let $$\bar f_n(t):=f_n(t)-\int_0^tg_n(\tau){\rm d}\tau.$$ Thanks to the imbedding theorem, one may deduce that $(\bar f_n)_{n}$ belongs to $ \mathcal{C}^{\beta}([0,T];B_{p,\infty}^{-M})$ with $\beta>0$ and hence uniformly equicontinuous with value in $B_{p,\infty}^{-M}$. Now, let $(\chi_{l})_{l\in{\mathbb{N}}}$ be a sequence of $C^{\infty}_{0}({\mathbb{R}}^{n})$ cut-off functions supported in the ball $B(0,l+1)$ of ${\mathbb{R}}^{n}$ and equal to 1 in a neighborhood of $B(0,l)$. On the other hand, by Theorem 2.94 of [@BCD11], we know that the map $u\mapsto \chi_{l}u$ is compact from $B_{p,r}^{s}$ to $B_{p,\infty}^{-M}$. By using Ascoli’s theorem and the Cantor diagonal process, there exists a subsequence which we still denote by $(\bar f_{n})_{n\in{\mathbb{N}}}$ such that, for all $l\in{\mathbb{N}}$, $$\chi_{l}\bar f_{n}\rightarrow_{n\rightarrow\infty}\chi_l \bar f\quad {\rm in }\quad \mathcal{C}([0,T];B_{p,\infty}^{-M}).$$ It follows that the sequence $(\bar f_{n})_{n\in{\mathbb{N}}}$ converges to some distribution $\bar f$ in $\mathcal{D}'({\mathbb{R}}_+\times{\mathbb{R}}^{n})$.
The only problem is to pass to the limit in $\mathcal{D}'({\mathbb{R}}_+\times{\mathbb{R}}^{n})$ for the convection term. Let $\psi\in C^{\infty}_{0}({\mathbb{R}}_+\times{\mathbb{R}}^{n})$ and $l\in{\mathbb{N}}$ be such that ${\rm supp}\psi\subset[0,T]\times B(0,l)$, we have the decomposition $$\label{decomp}\psi u_{n}\cdot\nabla f_{n}-\psi u\cdot\nabla f=\psi u_{n}\cdot\big(\nabla( \chi_{l}f_{n}-\chi_{l}f)\big)-\psi \chi_{l}(u_n-u)\cdot\nabla f$$Coming back to the uniform estimates of $f_n\in L^{\infty}([0,T];B_{p,r}^{s})$, the Fatou properties of Besov space ensures $\bar f$ belong to $L^{\infty}([0,T];B_{p,r}^{s})$. By preceding argument, we find that $\chi_{l}\bar f_{n}$ tends to $\chi_l \bar f$ in $\mathcal{C}([0,T];B^{s-\varepsilon}_{p,\infty})$ for all $\varepsilon>0$ and $l\in{\mathbb{N}}$. Therefore, both two terms in the right of tend to zero in $L^{\infty}([0,T];B^{s-1-\varepsilon}_{p,\infty})$. On the other hand, the sequences $(f_{0,n})_{n\in{\mathbb{N}}},(g_{n})_{n\in{\mathbb{N}}}$ and $(u_{n})_{n\in{\mathbb{N}}}$ converges to $f_0, g$ and $u$, respectively. So, we finally conclude that $f:=\bar f+\int_0^tg(\tau){\rm d}\tau$ is a solution of .
It remains to prove that $f\in \mathcal{C}([0,T];B^{s}_{p,r})$, when $r<\infty$. Making use of uniform estimates of $\bar f_{n}$, one can deduce that $\partial_tf$ belongs to $L^{1}([0,T];B_{p,\infty}^{-M})$. Obviously, for fixed $k$, $\partial_t\Delta_kf$ belongs to $L^{1}([0,T];L^{p})$ so that each $\Delta_kf$ is continuous in time with value in $L^p$. This implies $S_{k}f\in\mathcal{C}([0,T];B^{s}_{p,r})$ for all $k\in{\mathbb{N}}$. Since $$\Delta_{k'}(f-S_{k}f)=\sum_{|k''-k'|\leq1,k''\geq k}\Delta_{k'}(\Delta_{k''}f),$$ then we have $${\left\|f-S_{k}f\right\|}_{B_{p,r}^{s}}\leq C\Big(\sum_{k'\geq k-1}2^{k'sr}{\left\|\Delta_{k'}f\right\|}_{L^{p}}\Big)^{\frac{1}{r}}.$$ By the same argument as in proof of , one may conclude that $${\left\|\Delta_{k}f(t)\right\|}_{L^{p}}\leq{\left\|\Delta_{k}f_0\right\|}_{L^{p}}+\int_{0}^{t}{\left\|\Delta_{k}g(\tau)\right\|}_{L^{p}}{\rm d}\tau+C\int_{0}^{t}c_k(t)2^{-ks}V(t){\left\|f(\tau)\right\|}_{B_{p,r}^{s}}{\rm d}\tau.$$ It follows that $$\begin{aligned}
{\left\|f-S_kf\right\|}_{L^{\infty}_{T}(B_{p,r}^{s})}\leq &C\Big(\sum_{k'\geq k-1}\big(2^{k's}{\left\|\Delta_{k'}f_{0}\right\|}_{L^{p}}\big)^{r}\Big)^{\frac{1}{r}}\\
&+C\int_{0}^{T}\Big(\sum_{k'\geq k-1}\big(2^{k's}{\left\|\Delta_{k'}g(\tau)\right\|}_{L^{p}}\big)^{r}\Big)^{\frac{1}{r}}{\rm d}\tau\\
&+C{\left\|f\right\|}_{L^{\infty}_{T}(B_{p,r}^{s})}\int_{0}^{T}\Big(\sum_{k'\geq k-1}c_{k'}^{r}(\tau)\Big)^{\frac{1}{r}}V(\tau){\rm d}\tau\end{aligned}$$ The fact $f_{0}\in B_{p,r}^{s}$ ensures that the first term tends to zero as $k$ goes to infinity. Since $g,V\in L^{1}_{T}(B_{p,r}^{s})$, one conclude that the terms in the integrals also tends to zero for almost every $t$. This together with the Lebesgue dominated convergence theorem entails ${\left\|f-S_kf\right\|}_{L^{\infty}_{T}(B_{p,r}^{s})}$ tends to zero as $k$ goes to infinity. Thus, we can conclude that $f$ belongs to $\mathcal{C}([0,T];B^{s}_{p,r})$.
For the case $r=\infty$, by using the interpolation theorem, we deduce that for any $t_0\in[0,T]$ and $s'\in]-M,s[$, there exists a constant $\theta\in]0,1[$ depending on $s'$ such that $$\begin{aligned}
{\left\|u(t)-u(t_0)\right\|}_{B_{p,\infty}^{s'}}\leq&{\left\|u(t)-u(t_0)\right\|}^{\theta}_{B_{p,\infty}^{-M}}{\left\|u(t)-u(t_0)\right\|}^{1-\theta}_{B_{p,\infty}^{s}}\\
\leq& 2{\left\|u(t)-u(t_0)\right\|}^{\theta}_{B_{p,\infty}^{-M}}{\left\|u\right\|}^{1-\theta}_{L^{\infty}_{T}(B_{p,\infty}^{s})}.\end{aligned}$$ This together with the fact $f\in \mathcal{C}([0,T];B^{-M}_{p,\infty})$ yields $f\in \mathcal{C}([0,T];B^{s'}_{p,\infty})$ for all $s'<s$. Now, we only need to prove that $f\in \mathcal{C}_{w}([0,T];B^{s}_{p,\infty})$. Indeed, for fixed $\phi\in\mathcal{S}({\mathbb{R}}^n)$, the low-high decomposition technique leads to $$\begin{aligned}
\langle f(t),\phi\rangle=\langle S_{k}f(t),\phi\rangle+\langle({\rm Id}- S_{k})f(t),\phi\rangle=\langle S_{k}f(t),\phi\rangle+\langle f(t),({\rm Id}- S_{k})\phi\rangle.\end{aligned}$$ Combining this with $f\in \mathcal{C}([0,T];B^{s'}_{p,\infty})$ gives that the function $t\mapsto\langle S_{k}f(t),\phi\rangle$ is continuous. As for the second term, we have $$|\langle f(t),({\rm Id}- S_{k})\phi\rangle|\leq{\left\|f\right\|}_{B_{p,\infty}^{s}}{\left\|\phi-S_k\phi\right\|}_{B_{p',1}^{-s}}.$$It follows that $\langle f(t),({\rm Id}- S_{j})\phi\rangle$ tends to zero uniformly on $[0,T]$ as $k$ goes to infinity. This means that $f(t)\in \mathcal{C}_{w}([0,T];B^{s}_{p,\infty})$.
Now, we focus on the proof of the uniqueness. Let $f_{1}$ and $f_2$ solve with the same initial datum. If we define $\delta f=f_{1}-f_{2}$, then $\delta f$ solves $$\partial_t\delta f+u\cdot\nabla\delta f-\Delta_h\delta f=0.$$ This together with the estimate ensures the uniqueness of solution of .
The last part of the appendix is devoted to the proof of losing a priori estimate for with $\nabla u\in L^1([0,T];LL)$, where the LogLip space $LL$ is the set of those functions $f$ which belong to $\mathcal{S}'~$¡¡and satisfy $$\label{LL}
{\left\|f\right\|}_{LL}:=\sup_{2\leq q<\infty} \frac{{\left\|\nabla S_{q}f\right\|}_{L^{\infty}}}{q+1}<\infty.¡¡$$ This estimate is the cornerstone to the proof of uniqueness in Theorem \[lose-global\]. In the sprite of [@BCD11; @dp2], we prove linear losing a priori estimates for the general anisotropic system with convection. More precisely, we have:
\[lostest\] Let $s_{1}\in[-\frac12,1[$ and assume that $s\in]s_{1},1[$. Let $v$ satisfies the following system $$\label{lost}
\begin{cases}
\partial_tv+u\cdot\nabla v-\Delta_hv+\nabla p=f+ge_{3},\\
\text{\rm div}v={\rm div}u=0
\end{cases}$$ with initial data $v_0\in H^{s}$ and source terms $f\in L^{1}([0,T];H^{s})$, $g\in L^{2}([0,T];H^{s-1})$. Assume in addition that, for some $h(t)\in L^{1}[0,T]$ satisfying $$\label{trans}
{\left\| u\right\|}_{LL}\leq h(t).$$ Then there exists a constant $C$ such that for any $\lambda>C,T>0$ and $$s_{t}:=s-\lambda\int_{0}^{t}h(\tau){\rm d}\tau,$$ the following estimate holds $$\begin{aligned}
{\left\|v(t)\right\|}_{H^{s_{t}}}+{\left\|\nabla_hv\right\|}_{L^{2}_{t}H^{s_{t}}}
\leq C(1+\sqrt{t})\exp\Big({\frac{C}{\lambda}\int_0^t h(\tau){\rm d}\tau}\Big)\big({\left\|\rho_0\right\|}_{H^{s}}+{\left\|f\right\|}_{L^1_{t}H^{s}}
+{\left\|g\right\|}_{L^2_{t}H^{s-1}}\big).\end{aligned}$$
Applying the operator $\Delta_{q}$ to the system , we find that for all $q\geq-1$, the $f_{q}$ solves the following equations $$\partial_tv_q+ S_{q-1}u\cdot v_q-\Delta_hv_q+\nabla p_q=f_q+g_qe_3+F_q(u,v)$$ with $F_q(u,v)= S_{q-1}u\cdot\nabla v_q-\Delta_q(u\cdot\nabla v).$
Taking the $L^2$-inner product to the above equation with $v_q$ and using $\text{\rm div}u=0$, we see that $$\label{lost-1}
\frac12\frac{\rm d}{{\rm d}t}{\left\|v_q\right\|}^{2}_{L^{2}}+{\left\|\nabla_hv_q\right\|}^{2}_{L^{2}}=\int f_qv_q {\rm d}x+\int g_qv^{3}_q {\rm d}x+\int F_q(u,v)v_q {\rm d}x.$$ Assume that $q\geq 0$. Applying the Bernstein and the Young inequalities, we can deduce that $$\begin{aligned}
\int g_qv^{3}_q {\rm d}x\leq &C2^{-q}{\left\|g_{q}\right\|}_{L^{2}}{\left\|\nabla v^{3}_{q}\right\|}_{L^{2}}\leq\frac14{\left\|\nabla v^{3}_{q}\right\|}^{2}_{L^{2}}+C2^{-2q}{\left\|g_{q}\right\|}^{2}_{L^{2}}\\
\leq& \frac14{\left\|\nabla_{h} v^{3}_{q}\right\|}^{2}_{L^{2}}+\frac14{\left\|\partial_{3} v^{3}_{q}\right\|}^{2}_{L^{2}}+C2^{-2q}{\left\|g_{q}\right\|}^{2}_{L^{2}}\\
\leq&\frac12{\left\|\nabla_{h} v_{q}\right\|}^{2}_{L^{2}}+C2^{-2q}{\left\|g_{q}\right\|}^{2}_{L^{2}},\end{aligned}$$ in the last line we have used the fact $\text{div} v=0.$
Integrating the both sides of with respect to $t$, we get for all $q\geq 0$, $${\left\|v_{q}\right\|}^{2}_{L^{\infty}_{t}L^{2}}+{\left\|\nabla_hv_{q}\right\|}^{2}_{L^{2}_{t}L^{2}}
\leq{\left\|v_{q}(0)\right\|}^{2}_{L^{2}}+2{\left\|f_{q}\right\|}^{2}_{L^{1}_{t}L^{2}}+C2^{-2q}{\left\|g_{q}\right\|}^{2}_{L^{2}_{t}L^{2}}
+2{\left\|F_{q}(u,v)\right\|}^{2}_{L^{1}_{t}L^{2}}.$$ For $q=-1$, we merely have $${\left\|v_{-1}(t)\right\|}_{L^{2}}\leq {\left\|v_{-1}(0)\right\|}_{L^{2}}+\int_{0}^{t}\big({\left\|f_{-1}(\tau)\right\|}_{L^{2}}+{\left\|g_{-1}(\tau)\right\|}_{L^{2}}+{\left\|F_{-1}(u,v)(\tau)\right\|}_{L^{2}}\big){\rm d}\tau.$$ On the other hand, by the Bernstein inequality, we know that $${\left\|\nabla_hv_{-1}\right\|}_{L^{2}_{t}L^{2}}\leq Ct^{\frac{1}{2}}{\left\|v_{-1}\right\|}_{L^{\infty}_{t}L^{2}}.$$ Therefore, for all $q\geq-1$, we have $$\label{equ-lose}
\begin{split}
&{\left\|v_q\right\|}_{L^{\infty}_{t}L^{2}}+{\left\|\nabla_hv_q\right\|}_{L^{2}_{t}L^{2}}\\
\leq&2(1+\sqrt{t})\Big({\left\|v_q(0)\right\|}_{L^{2}}+{\left\|f_q\right\|}_{L^{1}_{t}L^{2}}+C2^{-q}{\left\|g_q\right\|}_{L^{2}_{t}L^{2}}+{\left\|F_q(u,v)\right\|}_{L^{1}_{t}L^{2}}\Big).
\end{split}$$ From a standard commutator estimate , we know that for all $\varepsilon\in]0,\frac{s+1}{2}[,q\geq -1$ and $t\in [0,T]$ $$\label{tr-2}
2^{q(s-\varepsilon)}{\left\|F_q(u,v)(t)\right\|}_{L^2}\leq Cc_{q}{(2+q)}h(t){\left\|v(t)\right\|}_{H^{s-\varepsilon}}\quad \text{with } c_{q}\in l^{2}
\mathcal{}$$ for some constant $C$ depending only on $s.$
Set $s_{t}:=s-\lambda\int_{0}^{t}h(\tau){\rm d}\tau$ for $t\in[0,T]$. Collecting and yields that $$\label{tr-3}
\begin{split}
2^{(2+q)s_{t}}{\left\|v_{q}\right\|}_{L^{2}}\leq 2(1+\sqrt{t})&\Big( 2^{(2+q)s}{\left\|v_q(0)\right\|}_{L^{2}}2^{-\eta(2+q)\int_{0}^{t}h(\tau){\rm d}\tau}\\&+\int_{0}^{t}2^{(2+q)s_{\tau}}{\left\|f_q(\tau)\right\|}_{L^{2}}2^{-\eta(2+q)\int_{\tau}^{t}h(\tau'){\rm d}\tau'}{\rm d}\tau\\
&+\Big(\int_{0}^{t}2^{2(2+q)s_{\tau}}2^{-2q}{\left\|g_q(\tau)\right\|}^{2}_{L^{2}}2^{-2\lambda(2+q)\int_{\tau}^{t}h(\tau'){\rm d}\tau'}{\rm d}\tau\Big)^{\frac{1}{2}}\\
&+Cc_{q}(2+q)\int_{0}^{t}h(\tau)2^{-\lambda(2+q)\int_{\tau}^{t}h(\tau'){\rm d}\tau'}{\left\|f(\tau)\right\|}_{H^{s_{\tau}}}{\rm d}\tau\Big).
\end{split}$$ For the last term of , we observe that $$\begin{aligned}
&Cc_{q}(2+q)\int_{0}^{t}h(\tau)2^{-\lambda(2+q)\int_{\tau}^{t}h(\tau'){\rm d}\tau'}{\left\|f(\tau)\right\|}_{H^{s_{\tau}}}{\rm d}\tau\\
\leq&Cc_{q}\frac{1}{\lambda\log2}\int_{0}^{t}{\rm d}2^{-\lambda(2+q)\int_{\tau}^{t}h(\tau'){\rm d}\tau'}\sup_{\tau\in[0,t]}{\left\|f(\tau)\right\|}_{H^{s_{\tau}}}\\
=&c_{q}\frac{C}{\lambda\log2}\big(1-2^{-\lambda(2+q)\int_{0}^{t}h(\tau){\rm d}\tau}\big)\sup_{\tau\in[0,t]}{\left\|f(\tau)\right\|}_{H^{s_{\tau}}}.\end{aligned}$$ Thus, multiplying $2^{qs_{\tau}}$ and taking the $l^{2}$-norm of both sides of over $q\geq-1$, we get $$\begin{aligned}
\sup_{\tau\in[0,t]}{\left\|f(\tau)\right\|}_{H^{s_{\tau}}}\leq 2(1+\sqrt{t})&\Big({\left\|f_{0}\right\|}_{H^{s}}+{\left\|f(\tau)\right\|}_{L^{1}_{t}H^{s_{\tau}}}+{\left\|g(\tau)\right\|}_{L^{2}_{t}H^{s_{\tau}-1}}\\
&+\frac{C}{\lambda\log2}\sup_{\tau\in[0,t]}{\left\|f(\tau)\right\|}_{H^{s_{\tau}}}\Big).\end{aligned}$$ Choosing $\lambda_0$ such that $\frac{2C(1+\sqrt{t})}{\lambda_0\log2}=\frac12$, we get by the Gronwall inequality that for any $\lambda>\lambda_0$, $$\sup_{t\in[0,t]}{\left\|f(t)\right\|}_{H^{s_{\tau}}}\leq 2(1+\sqrt{t})e^{\frac{C}{\lambda}\int_0^t h(\tau){\rm d}\tau}\Big({\left\|f_{0}\right\|}_{H^{s}}+{\left\|f(\tau)\right\|}_{L^{1}_{t}H^{s_{\tau}}}+{\left\|g(\tau)\right\|}_{L^{2}_{t}H^{s_{\tau}-1}}\Big).$$ This implies the desired result.
Acknowledgements. {#acknowledgements. .unnumbered}
-----------------
The authors are partly supported by the NSF of China No. 11171033. The authors would like to thank Dr.L. Xue for his valuable discussion.
[9999]{}
H. <span style="font-variant:small-caps;">Abidi</span> and T. <span style="font-variant:small-caps;">Hmidi</span>, *On the global well-posedness for Boussinesq System,* J. Diff. Equa., **233, 1** (2007), 199-220.
H. <span style="font-variant:small-caps;">Abidi</span> and T. <span style="font-variant:small-caps;">Hmidi</span>, K. <span style="font-variant:small-caps;">Sahbi</span>, *On the global regularity of axisymmetric Navier-Stokes-Boussinesq system,* Discrete Contin. Dyn. Syst., [**29**]{} (2011), 737-756.
H. <span style="font-variant:small-caps;">Abidi</span>, T. <span style="font-variant:small-caps;">Hmidi</span> and K. <span style="font-variant:small-caps;">Sahbi</span>, *On the global well-posedness for the axisymmetric Euler equations,* Math. Ann., [**347**]{} (1) (2010), 15-41.
A. <span style="font-variant:small-caps;">Adhikari</span>, C. <span style="font-variant:small-caps;">Cao</span> and J. <span style="font-variant:small-caps;">Wu</span>, *The 2-D Boussinesq equations with vertiacal viscosity and vertical diffusivity,* J. Differential Equations, [**249**]{} (2010). 1078-1088.
A. <span style="font-variant:small-caps;">Adhikari</span>, C. <span style="font-variant:small-caps;">Cao</span> and J. <span style="font-variant:small-caps;">Wu</span>, *Global regularity results for the 2-D Boussinesq equations with vertical disspation,* J. Differential Equations, [**251**]{} (2011). 1637-1655.
J. <span style="font-variant:small-caps;">Ben</span> <span style="font-variant:small-caps;">Ameur</span> and R. <span style="font-variant:small-caps;">Danchin,</span> *Limite non visqueuse pour les fluids incompressibles axisymétriques.* Nonlinear partial differential equations and their apllations. Collége de France seminar (Pairs, France, 1997-1998), vol. XIV, Stud. Math. Appl., vol.31, North-Holland, Amsterdam (2002), 29-55.
H. <span style="font-variant:small-caps;">Bahouri</span>, J.-Y. <span style="font-variant:small-caps;">Chemin</span> and R. <span style="font-variant:small-caps;">Danchin</span>, *Fourier Analysis and Nonlinear Partial Differential Equations*, Grundlehren der mathematischen Wissenschaften **343**, Springer-Verlag, (2011).
M. <span style="font-variant:small-caps;">Cannone</span>, *Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations*, 161–244, in: Handbook of Mathematical Fluid Dynamics, vol. III, S. J. Friedlander, D. Serre, eds., Elsevier, (2004).
C. <span style="font-variant:small-caps;">Cao</span> and E. S. <span style="font-variant:small-caps;">Titi</span>, *Global regularity criterion for the 3-D Navier-Stokes equations involving one entry of the velocity gradient tensor*, to appear Arch. Ration. Mech. Anal.
C. <span style="font-variant:small-caps;">Cao</span> and J. <span style="font-variant:small-caps;">Wu</span>, *Global regularity for the 2-D MHD equations with mixed partial disspation and magnetic diffusion*, Advances in Math., [**226**]{} (2011), 1803-1822.
C. <span style="font-variant:small-caps;">Cao</span> and J. <span style="font-variant:small-caps;">Wu</span>, *Global regularity results for the 2-D anisotropic Boussinesq equations with vertical disspation,* arXiv:1108.2678v1
D. <span style="font-variant:small-caps;">Chae</span>, *Global regularity for the $2$-D Boussinesq equations with partial viscous terms,* Advances in Math., [**203**]{}, 2 (2006), 497-513.
J.-Y. <span style="font-variant:small-caps;">Chemin</span>, *Perfect Incompressible Fluids,* Oxford University Press, (1998).
J.-Y. <span style="font-variant:small-caps;">Chemin</span>, B. <span style="font-variant:small-caps;">Desjardins,</span> I. <span style="font-variant:small-caps;">gallagher</span> and E. <span style="font-variant:small-caps;">Grenier,</span> *Fluids with anistrophic viscosity*, Modélisation Mathématique et Analyse Numérique, **34** (2000), 315-335.
J.-Y. <span style="font-variant:small-caps;">Chemin</span>, B. <span style="font-variant:small-caps;">Desjardins,</span> I. <span style="font-variant:small-caps;">gallagher</span> and E. <span style="font-variant:small-caps;">Grenier,</span> *Mathematical Geophysics: An Introduction to Rotating Fluids and to the Navier-Stokes Equations*, Oxford Unversity Press, (2006).
R. <span style="font-variant:small-caps;">Danchin</span>, *Axisymmetric incompressible flows with bounded vorticity,* Russian Math., Surveys [**62**]{} (2007), no 3, 73-94.
R. <span style="font-variant:small-caps;">Danchin</span> and M. <span style="font-variant:small-caps;">Paicu</span>, *Global existence results for the anisotropic Boussinesq system in dimension two.* Mathematical Models and Methods in Applied Sciences, **21** (2011), 421-457.
R. J. <span style="font-variant:small-caps;">Diperna</span> and P. L. <span style="font-variant:small-caps;">Lions</span>, *Ordinary differnential equations, transport theory and Sobolev spaces* Inventiones mathematicae, **98** (1989), 511-547.
T. <span style="font-variant:small-caps;">Hmidi</span>, *Low Mach number limit for the isentropic Euler system with axisymmetric initial data*, arXiv:1109.5339v1
T. <span style="font-variant:small-caps;">Hmidi</span> and F. <span style="font-variant:small-caps;">Rousset</span>, *Global well-posedness for the Euler-Boussinesq system with axisymmetric data.* J. Functional Analysis, **260** (2011), 745-796.
T. <span style="font-variant:small-caps;">Hmidi</span> and F. <span style="font-variant:small-caps;">Rousset</span>, *Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data.* Ann. I. H. Poincaŕe-AN., **27** (2010), 1227-1246.
T. <span style="font-variant:small-caps;">Hmidi</span>, S. <span style="font-variant:small-caps;">Keraani</span> and F. <span style="font-variant:small-caps;">Rousset</span>, *Global well-posedness for Euler-Boussinesq system with critical disspation.* Comm, Partical Differential Equations, [**36**]{} (2011), 420-445.
T. <span style="font-variant:small-caps;">Hmidi</span>, S. <span style="font-variant:small-caps;">Keraani</span> and F. <span style="font-variant:small-caps;">Rousset</span>, *Global well-posedness for Boussinesq-Navier-Stokes system with critical disspation.* J. Differential Equations, [**249**]{} (2010), 2147-2174.
T. <span style="font-variant:small-caps;">Hou</span> and C. <span style="font-variant:small-caps;">Li</span>, *Global well-posedness of the viscous Boussinesq equations,* Discrete and Cont. Dyn. Syst., [**12**]{} (2005), 1-12.
P. G. <span style="font-variant:small-caps;">Lemarié</span>, *Recent Developments in the Navier-Stokes Problem,* CRC Press, (2002).
C. <span style="font-variant:small-caps;">Miao</span>, J. <span style="font-variant:small-caps;">Wu</span> and Z. <span style="font-variant:small-caps;">Zhang</span>, *Littlewood-Paley Theory and Applications to Fluid Dynamics Equations*, Monographs on Modern pure mathematics, No.142. Science Press, Beijing, (2012).
C. <span style="font-variant:small-caps;">Miao</span> and L. <span style="font-variant:small-caps;">Xue</span>, *On the golbal well-posedness of a class of Boussinesq-Navier-Stokes systems,* Nonlinear Differ. Equ. Appl., **18** (2011), 707-735.
J. Pedlosky, *Geophysical Fluid Dynsmics*, Springer Verlag, New-York, (1987).
T. <span style="font-variant:small-caps;">Shirota</span> and T. <span style="font-variant:small-caps;">Yanagisawa</span>, *Note on global existence for axially symmetric solutions of the Euler system.* Proc. Japan Acad. Ser. A Math. Sci., [**70**]{} (1994), no. 10, 299-304.
E. M. <span style="font-variant:small-caps;">Stein</span>, *Singular Integrals and Differentiability Properties of Functions*, Princeton University Press, Princeton, New Jersey, (1970).
M. R. <span style="font-variant:small-caps;">Ukhovskii</span> and V. I. <span style="font-variant:small-caps;">Yudovich</span>, *Axially symmetric flows of ideal and viscous fluids filling the whole space,* Prikl. Mat. Meh., [**32**]{} (1968), no. 1, 59-69.
|
Inelastic x-ray scattering (IXS) has shown promise as a practical probe of electronic excitations in condensed matter because of its broad kinematic range and direct coupling to the electron charge. However, since x-rays are strongly absorbed in high density materials, successful applications of the technique have been limited to low-Z systems[@krisch; @johnhillli; @schulke1; @benlarson; @v2o3].
Several recent studies[@eric; @chichang; @johnhill; @butorin], have shown that by that by tuning the incident energy near an x-ray absorption edge a Raman effect could be measured, despite the high absorption, because of the resonant enhancement. These studies have emphasized the role of coulomb interactions in the scattering process. Since it involves coupling between highly excited virtual states and strongly correlated valence states, it is important to characterize the resonance process well for the technique to be useful.
With emphasis on systematics, we have measured resonant inelastic x-ray scattering at moderate resolution ($\Delta E$=0.9 eV) near the CuK absorption edge in the high-T$_c$ parent insulator La$_2$CuO$_4$ (LCO) as a function of incident photon energy. Based on the changes of inelastic intensity and peak position with incident energy we show that the scattering is well described in a “shakeup" picture in 3rd order perturbation threoy[@phil]. We also present higher resolution measurements ($\Delta E$=0.45 eV) on another insulator, Sr$_2$CuO$_2$Cl$_2$ (SCOC), as a function of momentum transfer, [**q**]{}, which show some new features, such as the 2 eV optical charge transfer gap.
Experiments were carried out at the X21 wiggler line at the National Synchrotron Light Source and the 3ID (SRI-CAT) undulator line at the Advanced Photon Source. At X21 the energy resolution was 0.45 eV and typical count rates were 0.4 Hz. At 3ID with 0.9 eV resolution 9 Hz was typical. The scattered light was collected with a spherical, diced, Ge(733) analyzer and imaged onto a detector. Energy analysis was done by rotating the analyzer and translating the detector in coincidence. The momentum transfer was varied by rotating the entire apparatus around the scattering center (exact experimental geometries are indicated in the figure captions).
The LCO and SCOC crystals were grown by techniques described previously [@lance; @cheong]. They were characterized with a spectroscopic ellipsometer to assure surface quality.
In the upper panel of Fig. 1 we show inelastic x-ray spectra from La$_2$CuO$_4$ at fixed [**q**]{} for 16 different incident x-ray energies across the Cu K absorption edge. Since the overall absorption change across the edge is only 14% (x-ray absorption being dominated by the La atoms) corrections for sample self-absorption did not significantly alter the line shapes. So shown here are the raw spectra. The strong peak at zero energy loss is elastic scattering. The energy gain side (negative energy loss) shows a background of 12 counts per minute. The principal feature of each curve is a single peak, seen previously in Nd$_2$CuO$_4$[@johnhill], whose position, intensity, and line shape vary greatly as a function of incident energy.
We summarize the resonant behavior in the lower frame of Fig. 1 where the inelastic peak height (open circles) and position (filled circles) are plotted against [*incident*]{} energy. The thick line is the CuK fluorescence yield which shows the location of the edge (a localized Cu$1s\rightarrow 4p$ transition). The peak height shows two maxima, the stronger of which correlates with the peak of the edge and the weaker with the pre-edge shoulder. In both cases the maximum is offset from the absorption peak by about 2.5 eV.
The peak [*position*]{} shifts nonlinearly with incident energy; below and above a resonance it is roughly linear while near a resonance it plateaus. This behavior differs fundamentally from a classic Raman effect, in which one expects a linear peak shift with unit slope with a gradual rise in intensity below a resonance, and saturation above [@eisenberger].
Independent of any model, this type of low energy loss scattering leads to excited states of the valence electron system in the absence of core excitations. However, because one is near the $1s$ absorption threshold of Cu the scattering proceeds through a set of almost real, highly excited (9 keV) intermediate states, which have a $1s$ core hole and an extra electron excited in a localized $4p$ state ($\bar{1s}4p$). When the highly excited intermediate state disappears it leaves behind low-lying valence excited states - in principle conserving energy and crystal momentum.
One must describe these many-body intermediate states, i.e. their matrix elements as well as their off-shell weight, in order to characterize the coherent, second order process. Different groups have resorted to different approximation schemes. Starting with N valence electrons in a small cluster, Tanaka and Kotani [@kotaniCeO] describe the intermediate states as a set of N+1 interacting electrons in the presence of a rigid impurity - the Anderson Impurity Model. This treatment assumes that the core state is suddenly created, and that it can be treated as a fixed, external potential. It emphasizes the multiplet coupling among the N+1 electrons, and is done numerically in exact diagonalization with a large number of basis functions.
Taking a more analytic approach, Platzman and Isaacs[@phil] treat the many-body problem by describing Coulomb interactions among electrons in a series of perturbation diagrams[@gelmukhanov]. This approach assumes that interactions can be taken to be weak for a suitable choice of basis functions [*or*]{} that one can sum enough terms in perturbation theory to include the important physics. They argue that near a sharp, dipole-allowed transition the dominant term occurs in 3rd order. Writing it out explicitly one arrives at
$$S_{f\leftarrow i}=\sum_{\bar{1s},4p} \frac{M_{em} \; M_{coul} \; M_{abs}}
{(\omega_s - E_{\bar{1s},4p} + i\gamma_{K})(\omega_i-E_{\bar{1s},4p}+i\gamma_K)}$$
In this expression the sum is on all possible states of the $1s$ hole and $4p$ electron, $E_{\bar{1s}4p}$ is their energy, and $\gamma_K$ is their inverse lifetime. The numerator contains matrix elements for absorption, M$_{abs}=(e/mc)\langle\bar{1s}\,4p|{\bf p}\cdot{\bf \hat{A}}|{\bf k}_i\rangle$, emission, M$_{em}=(e/mc)\langle {\bf k}_s|{\bf p}\cdot{\bf \hat{A}}
|\bar{1s}' \, 4p'\rangle$, and coulomb interaction between the core and valence states, M$_{coul}=\int{d{\bf x}\,d{\bf x'} \, \langle\bar{1s}' \, 4p';f|\hat{\rho}({\bf x})
\hat{\rho}({\bf x}')/|{\bf x}-{\bf x'}||\bar{1s} \, 4p;i\rangle}$. [**k**]{}$_i$ and [**k**]{}$_s$ are the incident and scattered photon momenta, (i.e. [**q**]{}=[**k**]{}$_i$-[**k**]{}$_s$) and $\omega=\omega_i-\omega_s$ is the energy loss.
Physically this expression represents the following. The incident photon, with energy tuned to the CuK absorption edge, creates a virtual $\bar{1s}4p$ pair on a Cu site. This pair is bound as an exciton by the coulomb interaction (not included in Eq. (1)) and so is non-dispersive. It takes up the momentum of the incident photon, 4.55 $\AA^{-1}$, and scatters off the valence electron system. When the exciton recombines, the emitted photon reflects the energy and momentum imparted to the system. This is commonly called a “shakeup" process, which to first order in the coulomb interaction is given by Eq. (1).
To get a transition rate we square the quantity (1) and perform an incoherent sum on final states. We postulate that the intermediate states are approximately degenerate with energy $E_{\bar{1s}4p}=E_K$ (since they are spacially localized), which allows factoring of the energy denominators from the sum. We arrive at
$$w=\frac{S_K({\bf q},\omega;\hat{\epsilon}_i,\hat{\epsilon}_s)}
{\left [ (\omega_i-E_K)^2+{\gamma_K}^2 \right ] \,
\left [ (\omega_s-E_K)^2+{\gamma_K}^2 \right]}$$
where
$$S_K=\frac{2\pi}{\hbar}\sum_{f}
\left |\sum_{\bar{1s},4p}\,M_{em}\,M_{abs}\,M_{coul}\right |^2
\delta(\omega-E_f+E_i).$$
and $\hat{\bf \epsilon}_i$ and $\hat{\bf \epsilon}_s$ are the polarizations of the incident and scattered photons. The two lorentzians in (2) are incoming and outgoing resonances in the photon frequency, so we see that this treatment is completely analogous to third-order optical Raman scattering from phonons in semiconductors, in which the scattering is described by a single operation of the [*electron-phonon*]{} interaction on a virtual [*valence*]{} electron-hole pair[@cardona]. The two resonances formally come about the same way.
The central result of this paper is Eq. (2). It says that, within our approximation, all the very different spectra in Figure 1 derive from the same fundamental quantity, $S_K$, which depends on the [*difference*]{} $\omega=\omega_i-\omega_s$ rather than on $\omega_i$ and $\omega_s$ independently. A way to test this result would be to take the curves from Fig. 1, divide each by its respective denominator from Eq. (2), and see if they all collapse to the same function.
In Eq. (2) we assumed that we are near a single resonance, so we take the nine highest curves from Fig. 1 (around the second peak in the resonance profile) and subtract their background and elastic scattering. We use $E_K$ and $\gamma_K$ as flexible parameters and divide each spectrum by its respective denominator. Using a nonlinear fitting algorithm, we adjust the values of $E_K$ and $\gamma_K$ to minimize the total variation (the $\chi^2$ summed over all points and all spectra) among the curves, irrespective of the resulting shape. For the values $E_K=(8995.14\pm 1.61)$ eV and $\gamma_K=(2.38\pm 0.542)$ eV we find that the spectra collapse onto a single curve, shown in Figure 2.
The result for $S_K({\bf q},\omega;\hat{\epsilon}_i,\hat{\epsilon}_s)$ is a single peak at 6.1 eV energy loss and width of 3.9 eV. Referring to the cluster calculations of Símon [@simon] we suggest that this feature is a transition from the $b_{1g}$ ground state to an $a_{1g}$ excitonic excited state composed of symmetric combinations of a central Cu3$d_{x^2-y^2}$ orbital and the surrounding O2$p_{\sigma}$ orbitals.
To illustrate what [*qualitative*]{} aspects of the data are captured by our fit, i.e. by the resonant denominators in (2), we take a single function for $S_K$, i.e. a fit to the collapsed data in Fig. 2 (thick line), combine it with the denominators in Eq. (2), and produce the model resonance profile shown in Fig. 3 (identical formatting to Fig. 1). The salient features are reproduced, including the peak shift with incident energy and the 2.5 eV offset. All this behavior comes from the energy denominators in (2) and is independent of the nature of the core state or the particular valence excitation.
In this simple shakeup description $S_K({\bf q},\omega;\hat{\epsilon}_i,\hat{\epsilon}_s)$ has an explicit relationship with $S({\bf q},\omega)$, the dynamical structure factor measured in nonresonant inelastic x-ray scattering[@schulke2]. This can be seen by writing out the matrix element $M_{coul}$ in momentum space, which (neglecting exchange between core and valence states) has the form
$$M_{coul}=\sum_{\bf G} \frac{4\pi e^2}{|{\bf q}+{\bf G}|^2}
F_{\bar{1s}4p}({\bf q}+{\bf G};\hat{\epsilon_i})
\langle f|\hat{\rho}_{v,{\bf q}+{\bf G}}|i\rangle.$$
Here $F_{\bar{1s}4p}(k)$ is the static x-ray form factor of the $\bar{1s}4p$ exciton. It is dependent implicitly on the incident polarization $\hat{\epsilon_i}$ since in the dipole approximation $M_{abs}$ determines the spacial orientation of the $4p$. $\hat{\rho}_{v,{\bf q}}$ is the valence part of the many body density operator $\hat{\rho}_{\bf q}$ and the sum in (4) is on all reciprocal lattice vectors, [**G**]{}.
The quantity $\langle f|\hat{\rho}_{v,{\bf q}}|i\rangle$, when squared, multiplied by $\delta(\omega-E_f+E_i)$, and summed on final states, is identically the valence part of the dynamic structure factor $S({\bf q},\omega)$. Doing the sum on [**G**]{} before squaring we find that $S_K({\bf q},\omega;\hat{\epsilon}_i,\hat{\epsilon}_s)$ is a superposition of many $S({\bf q}+{\bf G},\omega)$ functions, weighted by the form factor of the core states. Therefore, $S_K$ can be thought of as a response function similar to $S$ “projected" onto the form factor of the core state.
Finally we present some higher-resolution ($\Delta$E=0.45 eV), [**q**]{}-dependent spectra from the insulator Sr$_2$CuO$_2$Cl$_2$ (Fig. 4). The salient spectral features are three peaks: the same feature we saw in LCO (appearing here at about 5 eV), a feature at 2 eV which appears to shift and lose definition with [**q**]{}, and a peak at 10.5 eV which is absent at low-[**q**]{} but which gains intensity as [**q**]{} is increased. The 2 eV feature strongly resembles dipole-active charge transfer gap seen with optical reflectivity[@zibold].
To summarize, we measured RIXS spectra at the CuK edge in LCO and SCOC. From the resonance profile we deduce that the scattering can be described as a shakeup process in 3rd order which, analogous to optical Raman scattering from phonons, exhibits incoming and outgoing resonances. The scattered intensity has the form of a doubly resonant form factor multiplied by a response function, $S_K({\bf q},\omega;\hat{\bf \epsilon}_i,\hat{\bf \epsilon}_s)$, which can be thought of as the dynamical structure factor projected onto the form factor of the intermediate core state.
The advantages of this description are (i) that it makes no assumption that the core state is rigid and so momentum conservation enters naturally, and (ii) that it allows one, given certain knowledge of the intermediate state (i.e. $F_{\bar{1s}4p}({\bf k})$), to relate the scattering to a response function [*in terms of the valence electrons only*]{}. This description is useful when the core resonance is sharp and well isolated, and when one is mostly interested in the valence electron spectrum and is willing to sacrifice a detailed multiplet description of the core states.
We gratefully acknowledge E. E. Alp, Z. Hasan, C.-C. Kao, V. I. Kushnir, P. L. Lee, H. L. Liu, A. T. Macrander, G. A. Sawatzky, M. Schwoerer-Böhning, M. E. Símon, S. K. Sinha, J. P. Sutter, T. Toellner, and C. Varma. This work was supported by the NSF under grant DMR-9705131 and by the U.S. Department of Energy, Basic Energy Sciences, Office of Energy Research, under contract no. W-31-109-ENG-38.
M. H. Krisch, F. Sette, C. Masciovecchio, R. Verbeni, Phys. Rev. Lett., [**78**]{}, 2843 (1997)
J. P. Hill, C.-C. Kao, W. A. C. Caliebe, D. Gibbs, J. B. Hastings, Phys. Rev. Lett., [**77**]{}, 3665 (1996)
H. Nagasawa, S. Mourikis, W. Schülke, J. Phys. Soc. Jpn., [**66**]{}, 3139 (1997)
B. C. Larson, J. Z. Tischler, E. D. Isaacs, P. Zschack, A. Fleszar, and E. Eguiluz, Phys. Rev. Lett. [**77**]{}, 1346, (1996)
E. D. Isaacs, P. M. Platzman, P. Metcalf, and J. M. Honig, Phys. Rev. Lett., [**76**]{}, 4211 (1996)
E. D. Isaacs, Ferroelectrics, [**176**]{}, 249 (1996)
C. C. Kao, W. A. L. Caliebe, J. B. Hastings, and J.-M. Gillet, Phys. Rev. B, [**54**]{}, 16361 (1996)
J. P. Hill, C.-C. Kao, W. A. L. Caliebe, M. Matsubara, A. Kotani, J. L. Peng, and R. L. Greene, Phys. Rev. Lett., [**80**]{}, 4967 (1998)
S. M. Butorin, [*et. al*]{}, Phys. Rev. Lett., [**77**]{}, 547 (1996)
P. M. Platzman and E. D. Isaacs, Phys Rev. B [**57**]{}, 11107 (1998)
L. L. Miller, X. L. Wang, C. Stassis, D. C. Johnston, J. Faber, Jr., and C.-K. Loong, Phys. Rev. B [**41**]{}, 1921 (1990)
S.-W. Cheong, [*et. al.*]{}, Solid State Comm. [**65**]{}, 111 (1988)
P. Eisenberger, P. M. Platzman, H. Winick, Phys. Rev. B [**13**]{}, 2377 (1976)
S. Tanaka, Y. Kayanuma, and A. Kotani, J. Phys. Soc. Japan [**59**]{}, 1488 (1990)
For yet another treatment see Faris Gel’mukhanov and Hans Ågren, Phys. Rev. B [**57**]{}, 2780 (1998), Section IV-B
P. Y. Yu and M. Cardona, [*Fundamentals of Semiconductors*]{} (Springer-Verlag, Berlin, Heidelberg, New York, 1996), Section 7.2.8
M. E. Símon, A. A. Aligia, C. D. Batista, E. R. Gagliano, and F. Lema, Phys Rev. B [**54**]{}, R3780 (1996)
W. Schülke, [*Inelastic Scattering by Electronic Excitations*]{}, in [*Handbook on Synch. Rad.*]{}, V. 3, ed. G. Brown and D. E. Moncton, (Elsevier, 1991)
A. Zibold, H.L. Liu, S. W. Moore, J. M. Graybeal, and D. B. Tanner, Phys. Rev. B [**53**]{}, 11734 (1996)
|
---
abstract: 'We study and characterize the breather-induced quantized superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations. Such vortex filaments, emerging from an otherwise perturbed helical vortex, exhibit intriguing loop structures corresponding to the large amplitude of breathers due to the dual action of bending and twisting of the vortex. The loop induced by Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by super-regular breather—*the loop pair*—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of super-regular breather. In particular, we identify explicitly the generation conditions of these loop excitations by introducing a physical quantity—the integral of the relative quadratic curvature—which corresponds to the effective energy of breathers. Although the nature of nonlinearity, it is demonstrated that this physical quantity shows a *linear* correlation with the loop size. These results will deepen our understanding of breather-induced vortex filaments and be helpful for controllable ring-like excitations on the vortices.'
address:
- '$^1$School of Physics, Northwest University, Xi’an 710069, China'
- '$^2$Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China'
- '$^3$Institute of Modern Physics, Northwest University, Xi’an 710069, China'
- '$^4$Institute of Photonics and Photon-technology, Northwest University, Xi’an 710069, China'
author:
- 'Hao Li$^{1,2}$'
- 'Chong Liu$^{1,2}$'
- 'Wei Zhao$^{1,4}$'
- 'Zhan-Ying Yang$^{1,2}$'
- 'Wen-Li Yang$^{1,2,3}$'
title: Breather induced quantized superfluid vortex filaments and their characterization
---
INTRODUCTION
============
Quantum fluid [@superfluid1; @quantumfluid] has recently been the subject of extensive investigations that contains vortex generation, interaction, and reconnection of vortex lines influenced by vortices [@vortex; @reconnection; @vortices1]. The motion of quantum fluid is most succinctly illustrated by vortex filament due that it consists of vorticity of infinite strength concentrated along the filament and gives an intuitive geometric interpretation of the evolution of the vorticity field. In the case of ideal inviscid fluid, the motion of the fluid elements is constrained by Biot-Savart law which provides valuable information of vortex tangles [@BS; @vortex; @tangles1]. In particular, a variety of excitations are generated and evolve along the vortex filament due to the self-induced velocity [@LIA2; @AB; @multibreather; @excitations1; @excitations2]. Such excitations are physically important since they turn out to be the main degrees of freedom remaining in a superfluid in the ultra low temperature regime. Therefore, investigation of vortex structures of different fundamental excitations is both relevant and necessary.
The first prototype of such excitations is the so-called ‘Kelvin wave’ [@KW1]. The latter, which is originated from the small deformations of vortex lines, plays an important role in the decay of turbulence energy [@KW2]. However, one should note that Kelvin waves are low amplitude linear excitations of a straight vortex. In contrast, there are also some larger amplitude excitations propagating along the filament. Such large-amplitude excitations are induced by nonlinearity, i.e., ‘nonlinear excitations’.
There exists an interesting link between nonlinear excitations and vortex dynamics that has attracted considerable attention recently. It is presently known as ‘Hasimoto transformation’ [@LIA2] that allows us map the motion of vortex filament onto a scalar cubic nonlinear Schrödinger equation (NLSE) of the self-focusing type and the resulting loop structure of bright soliton on a vortex filament is demonstrated [@LIA2]. In addition to the classical solitons, the NLSE possesses rich ‘breathing’ excitations on a plane wave background which are known as ‘breathers’ [@Book97; @O1a; @SR1]. Such breathers are strongly associated with the modulation instability (MI) [@Book97; @O1a; @SR1; @OE; @MI1], where its nonlinear stage has been regarded as the prototype of rogue wave events [@Extreme09]. Surprisingly, although breather has been one of the center subjects in nonlinear physics and its observation has been realized widely in many nonlinear systems [@O1a; @W1b; @O1b; @Onorato; @Wabnitz], the link between one special type of breather—Akhmediev breather [@ABbreather] (as well as its multiple counterpart) and vortex filaments in a quantized superfluid has been revealed only recently [@AB; @multibreather]. In fact, the resulting new loop structure of Akhmediev breather, which differs from that of bright solitons [@LIA2], provides significant contributions to our understanding of quantum fluid and superfluid turbulence. This is therefore an interdisciplinary research—in the case of quantized superfluid vortex filaments, MI, and breathers—that needs more explorations.
However, the Akhmediev breather is merely the exact description for the MI emerging from a special purely periodic perturbations [@ABbreather]. There is another type of breathers describing the MI developing from localised perturbations that has not been studied in a quantized superfluid. This includes the Kuznetsov-Ma breather [@KMB] admitting localised single-peak perturbation and super-regular breather [@SR1] supporting localised multi-peak perturbation. Indeed, it is recently demonstrated that Kuznetsov-Ma breather describes not only the MI in the small amplitude regime but also the interference between bright soliton and plane wave in the large amplitude regime [@Mechanism; @of; @KMB]; while the super-regular breather admits the MI growth rate that coincides with the absolute difference of group velocities of the breather [@SR3; @SR4]. Given that these breathers are qualitatively different, two questions of fundamental importance now arise: How about the loop excitations triggered by these breathers? Is there a physical quantity to identify explicitly all these breather-induced loop excitations?
In this paper, we study the quantized superfluid vortex filaments induced by Kuznetsov-Ma breather and super-regular breather admitting localised perturbations. Such vortex filaments exhibit striking loop structures due to the dual action of bending and twisting of the vortex. Remarkably, an intriguing loop structure triggered by super-regular breather—the loop pair—exhibits spontaneous symmetry breaking, due to the broken reflection symmetry of the group velocities of super-regular breather. In particular, we identify explicitly these loop excitations by introducing the integral of the relative quadratic curvature, which corresponds to the effective energy of breathers. Although the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size.
Hasimoto transformation and inverse map
=======================================
For the incompressible and inviscid fluid, the Biot-Savart equation is reduced to a simpler local induction approximation (LIA) equation [@LIA1; @liq3; @liq4] by taking leading order $$\begin{aligned}
\label{equ:LIA}
\mathbf{v}=\left(\Gamma/4\pi\right)\ln\left(R/a_0\right)\kappa \mathbf{t}\times\mathbf{n}=\beta \kappa \mathbf{t}\times\mathbf{n}.\end{aligned}$$ Here $\Gamma$ is a circulation, $R$ is local radius of curvature and $a_0$ is the effective vortex core radius. $\mathbf{v}=\frac{d\mathbf{r}}{dt}$ is the flow velocity vector of the vortex filament, $\mathbf{t}$ and $\mathbf{n}$ are unit vectors corresponding to the tangent and principal normal directions, respectively. $\kappa$, as a real function of arc length variable $s$ and time $t$, represents the curvature distribution of the vortex filament. This equation makes us obtain more properties of the states related to the motion of vortex especially in the case of Hasimoto transformation. Assuming that $\beta$ is constant and making use of the Seret-Frenet equations given in [@SFe], $$\begin{aligned}
\label{equ:2}
\mathbf{r}'=\mathbf{t},~~\mathbf{t}'=\kappa\mathbf{n},~~\mathbf{n}'=\tau\mathbf{b}-\kappa\mathbf{t},~~\mathbf{b}'=-\tau\mathbf{n},\end{aligned}$$ where prime denotes a differential of arc length, $\mathbf{b}$ is binormal vector and $\tau$ is the torsion of the vortex filament, Eq. (\[equ:LIA\]) can be transformed into a 1D scalar cubic NLSE of self-focusing type [@LIA2] $$\begin{aligned}
\label{equ:NLSE}
\beta^{-1}\left(i\psi_t\right)=-\psi_{ss}-\frac{1}{2}|\psi|^2\psi.\end{aligned}$$ $\psi\left(s,t\right)$ is a complex function related with the local instantaneous geometric parameters curvature $\kappa\left(s,t\right)$ and torsion $\tau\left(s,t\right)$ in the context of vortices by the transformation $$\begin{aligned}
\psi\left(s,t\right)=\kappa\left(\sigma,t\right)e^{i\int_0^s \tau\left(\sigma,t\right)ds}.\end{aligned}$$
The NLSE (\[equ:NLSE\]) possesses rich ‘breathing’ excitations [@Book97], which provides a path for studying exactly breather-induced quantized superfluid vortex filaments. One can obtain the explicit configuration of these excitations by inverse map (see Appendix \[Appendix A\]).
Kuznetsov-Ma breather induced vortex filaments and exact characterization {#Sec3}
=========================================================================
We first consider the Kuznetsov-Ma breather that exhibits periodic pulsating dynamics along $t$. Its explicit expression for Eq. (\[equ:NLSE\]) is given by $$\begin{aligned}
\label{equ:KMB} \psi(s,t)=\left[1-2\frac{\chi^2\cos\left(\eta\beta t\right)+i\eta \sin\left(\eta\beta t\right)}{\kappa_0b~\cosh\left(\chi\xi\right)-\kappa_0^2~\cos\left(\eta\beta t\right)}\right]\psi_0,\end{aligned}$$ where $\chi=\sqrt{b^2-\kappa_0^2}$ with $b$ being a real constant ($b>\kappa_0$), $\eta=b\chi$, and $\xi=s-2\tau_0\beta t$. Physically, $b$ describes the oscillation period and amplitude of the Kuznetsov-Ma breather. $\kappa_0$ and $\tau_0$ are real constants which denote the amplitude and wave vector of the plane wave background $\psi_0$ respectively. The latter has the from $$\begin{aligned}
\label{equ:PWB} &&\psi_0=\kappa_0\exp{i(\tau_0s+\omega t)},~\omega=\beta\,\kappa_0^2/2-\beta\,\tau_0^2.\end{aligned}$$ This plane wave corresponds to a trivial uniform helical vortex without physical interest ($\kappa_0$ and $\tau_0$ describe the curvature and torsion of a uniform helical vortex, respectively). In contrast, the Kuznetsov-Ma breather describes nontrivial structure of vortex filament that has not been studied fully. From Eq. (\[equ:KMB\]), one can readily calculate the explicit expressions of the curvature and torsion of vortex filament induced by Kuznetsov-Ma breather, which are given respectively by $$\begin{aligned}
\label{equ:11}
\kappa=\left[\left(\kappa_0+\frac{2\chi^2\cos(\eta\beta t)}{n_1}\right)^2+\frac{4\eta^2\sin^2(\eta\beta t)}{\left(n_1\right)^2}\right]^{1/2},\end{aligned}$$ and $$\begin{aligned}
\label{equ:12}
\tau=\tau_0 \left[1+\frac{4\kappa_0\eta^2\sin\left(\eta\beta t\right)\sinh\left(\chi\xi\right)}{m_1+m_2+m_3+m_4}\right],\end{aligned}$$ with $$\begin{aligned}
\nonumber
&&n_1=\kappa_0\cos\left(\eta\beta t\right)-b\cosh\left(\chi\xi\right),\\\nonumber
&&m_1=\kappa_0^4-7\kappa0^2b^2+8b^4,~m_2=\kappa_0^4\cos\left(2\eta\beta t\right),\\\nonumber
&&m_3=4\kappa_0 b\left(\kappa_0^2-2b^2\right)\cos\left(\eta\beta t\right)\cosh\left(\chi\xi\right),\\\nonumber
&&m_4=\kappa_0^2b^2\cosh\left(2\chi\xi\right).\end{aligned}$$
![Temporal evolutions of curvature $\kappa\left(\xi,t\right)$ (a) and torsion $\tau\left(\xi,t\right)$ (c) corresponding to a Kuznetsov-Ma breather, see Eqs. (\[equ:11\]) and (\[equ:12\]). (b) and (d) are variations of $\kappa\left(\xi,t\right)$ and $\tau\left(\xi,t\right)$ at different times. The parameters are: $\kappa_0=1$, $\tau_0=0.05$, $b=1.2$, and $\beta=4\pi$.[]{data-label="fig1"}](fig1.jpg "fig:"){width="82mm"}\
The variations of curvature and torsion of Kuznetsov-Ma breather on the ($\xi$, $t$) plane (note here that $\xi=s-2\tau_0\beta t$ denotes the moving frame on the group velocity), with initial conditions $b=1.2$, $\kappa_0=1$, $\tau_0=0.05$, are shown in Fig. \[fig1\](a) and (c). As expected, the curvature of Kuznetsov-Ma breather, starting from a localised non-periodic (single-peak) perturbation, evolves gradually into its maximum at $t=0$ \[see the profiles in Fig. \[fig1\](b)\]. The curvature then exhibits periodic oscillation with the period $2\pi/(\eta\beta)$ as $t$ increases \[see Fig. \[fig1\](a)\].
A notable feature is that the torsion, as a function of arc length $s$ and time $t$, exhibits singular behavior as $t\rightarrow 2\pi/(\eta\beta)$. Figure \[fig1\](d) clearly indicates that the phase becomes ill defined at the point $\kappa=0$ near $t=0$, which leads to the severe twisting of the vortex filament. This is not surprising since the Kuznetsov-Ma breather admits a $\pi$ phase shift at the valleys. This phase shift results in the singular behavior of the torsion. Note that the singular does not make the Hasimoto transformation ill-defined due to the $\pi$ phase shift of nonlinear waves.
Figure \[fig2\] shows the corresponding vortex configuration of Kuznetsov-Ma breather within one growth-decay cycle. One can see clearly from the figure that the vortex filament, emerging from an otherwise perturbed helical vortex at $t=-0.314$ \[see Fig. \[fig2\] (a)\], exhibits a striking loop structure at $t=0$ due to the dual action of bending and twisting of the vortex. This loop structure disappears gradually as $t$ increases. At $t=0.314$, the vortex filament recovers the initial state. This process will emerge periodically as $t$ increases due to the feature of the Kuznetsov-Ma breather. One should note that when $\kappa_0\rightarrow0$, the loop of Kuznetsov-Ma breather reduces to the classical loop structure of bright solitons [@LIA2]; the periodic recurrence of the loop is gone.
{width="150mm"}\
The Kuznetsov-Ma breather can transform into the Peregrine rogue wave [@PRW] with double localization in the limit of $b\rightarrow \kappa_0$. The latter is also the limiting case of the Akhmediev breathers. All these breathers can induce loop-structure excitations on vortex filaments, as shown above and in Ref. [@AB; @multibreather].
One then wonder how to identify explicitly these vortex filaments induced by breathers, since each kind of vortex filament has a similar loop structure corresponding to the maximum curvature. This is the question of fundamental importance that has not been answered before. To do this, we introduce the following physical quantity—*the integral of the relative quadratic curvature* of the form $$\begin{aligned}
\Delta K=\int_{-\infty}^\infty \left[\kappa^2\left(s,t\right)-\kappa_0^2\left(s,t\right)\right]ds.\label{eqqc}\end{aligned}$$ Eq. (\[eqqc\]) corresponds to the *effective energy* of breathers in optics [@NA1; @NA2]. Namely, it coincides with the energy of breathers against plane wave, i.e., $\int_{-\infty}^\infty \left(\psi^2-\psi_0^2\right)ds$. For a quantum condensate fluid, Eq. (\[eqqc\]) stands for the effective atom numbers [@quantumfluid]. Generally, this is a quantity of physical importance which can be monitored effectively for localised nonlinear waves in experiments [@PN; @quantumfluid]. Here we highlight that Eq. (\[eqqc\]) can be used for characterising the breather induced vortex filaments in quantized superfluid.
It is interesting to note that for the Kuznetsov-Ma breather (\[equ:KMB\]), one obtains exactly $\Delta K=8\sqrt {b^2-\kappa_0^2}$, which indicates $\Delta K>0$; while for the Peregrine rogue wave and the Akhmediev breather, we find that $\Delta K=0$ (see Appendix \[Appendix C\]). This is the immanent reason why the loop structure induced by Kuznetsov-Ma breather exhibits periodic oscillation as $t$ increases, while the loop structure triggered by the Peregrine rogue wave and the Akhmediev breather appears only once during the time evolution. On the other hand, the condition $\Delta K=0$ indicates that the resulting vortex filament starts from a uniform helical vortex structure. This corresponds to the case of the Peregrine rogue wave and the Akhmediev breather. However, the uniform helical vortex structure will never appear for the vortex filament induced by the Kuznetsov-Ma breather.
![Relations between $\Delta K(\kappa_0, t)$ and $r_k(\kappa_0, t)$ on logarithmic coordinates ($\ln{\Delta K}$, $\ln{r_k}$) (a) as $\kappa_0$ varies with fixed $t=-0.01$; (b) as $t$ varies with fixed $\kappa_0=1$. The solid lines are precise description of relation between $\ln{\Delta K}$ and $\ln{r_k}$ as $b\rightarrow \infty$. The values of the parameter $b$ are random numbers in the region $b\in[2\kappa_0, 30\kappa_0]$. []{data-label="fig3"}](fig3.jpg "fig:"){width="82mm"}\
![Profile of $\Delta K\cdot r_k$, Eq. (\[eqlr2\]) as $b$ increases. Other parameters are $\kappa_0=1$, $t=2n\pi/(\eta\beta)$ ($n$ is an integer) and $\beta=4\pi$.[]{data-label="fig4"}](fig4.jpg "fig:"){width="80mm"}\
Let us take a closer look at Eq. (\[eqqc\]) by considering the relation between $\Delta K$ and the Kuznetsov-Ma breather-induced loop structure. To this end, we define the characteristic size of the loop structure, $r_k$, which describes the minimum radius of the structure.
Figure \[fig3\] shows the relation between $\Delta K(\kappa_0, t)$ and $r(\kappa_0, t)$ on logarithmic coordinates with *random values* of $b$ in the region $b\in[2\kappa_0,\infty]$. This parameter condition allows us to study the qualitative link between $\Delta K(\kappa_0, t)$ and $r_k(\kappa_0, t)$ from the vortex filaments induced by random Kuznetsov-Ma breathers (i.e., a series of Kuznetsov-Ma breathers with random period and amplitude).
For the fixed $t$ ($t=0$), we show the characteristics of $\ln(\Delta K)$ and $\ln(r_k)$ with increasing $\kappa_0$ in Fig. \[fig3\](a). It is interesting that, despite the random values of $b$, $\ln(\Delta K)$ decreases *linearly* as $\ln(r_k)$ increases and the corresponding rates $\alpha$ are exactly consistent at a fixed time ($\alpha=-1$). The similar linear relation also holds for the case with fixed $\kappa_0$ and $\tau_0$ and variational $t$, as shown in Fig. \[fig3\](b).
We then explain the linear relation above exactly. We note that for the Kuznetsov-Ma-breather-induced loop structure, the minimum loop radius $r_k$ is inversely proportional to the maximum curvature $\kappa_{m}$, i.e., $$r_k=\frac{1}{\kappa_{m}}.$$ It is given explicitly by Eq. (\[equ:11\]) at $\xi=0$: $$\begin{aligned}
\label{equ:13}
r_k=\left[\left(\kappa_0+\frac{2\chi^2\cos(\eta\beta t)}{n_2}\right)^2+\frac{4\eta^2\sin^2(\eta\beta t)}{\left(n_2\right)^2}\right]^{-1/2}\end{aligned}$$ with $n_2=\kappa_0\cos\left(\eta\beta t\right)-b$. Here $r_k$ is the function of $b$, $\kappa_0$ and $t$. Thus the accurate description of $\Delta K \cdot r_k$ reads $$\label{eqlr2}
\Delta K\cdot r_k=\frac{8\sqrt {b^2-\kappa_0^2}}{\left[\left(\kappa_0+\frac{2\chi^2\cos(\eta\beta t)}{n_2}\right)^2+\frac{4\eta^2\sin^2(\eta\beta t)}{\left(n_2\right)^2}\right]^{1/2}}.$$ Clearly, for the case of Kuznetsov-Ma breather, $\Delta K\cdot r_k\neq0$; while for case of the Peregrine rogue wave and Akhmediev breather, $\Delta K\cdot r_k=0$, since $\Delta K=0$.
We show the profile of Eq. (\[eqlr2\]) as $b$ increases in Fig. \[fig4\]. One can see that $\Delta K\cdot r_k$ increases monotonously with increasing $b$. Remarkably, as $b\rightarrow \infty$, we find $\Delta K\cdot r_k\rightarrow 4$. Indeed, one can readily obtain a simple relation from Eq. (\[eqlr2\]) as $b\rightarrow \infty$ $$\label{eqlre}
\Delta K\cdot r_k=4,$$ Namely, $$\ln\Delta K=-\ln r_k+\ln4,$$ on logarithmic coordinates.
We show the linear relation by the solid lines in Fig. \[fig3\]. Observably, the numerical results are in good agreement with the analytical relation (\[eqlre\]) (solid line). Physically, the Kuznetsov-Ma breather in the region $b\in[2\kappa_0,\infty]$ can be approximatively described by the *linear interference* between a bright soliton and a plane wave [@Mechanism; @of; @KMB]. As $b\rightarrow \infty$ ($b\gg\kappa_0$), i.e., the amplitude of the bright soliton is much bigger than that of the plane wave, the plane wave can be neglected. As a result, the effective energy $\Delta K$ is quadruple of the amplitude of the remaining bright soliton, which directly leads to Eq. (\[eqlre\]).
{height="95mm" width="115mm"}\
super-regular breather induced loop pair and symmetry breaking
==============================================================
Let us then consider the vortex filament induced by the super-regular breather. The latter, which recently serves as the exact MI scenario excited from localised multi-peak perturbations, is formed by the nonlinear superposition of two quasi-Akhmediev breathers [@SR1; @SR3; @SR4; @SR5; @SR6]. The exact solution of super-regular breather with $\tau_0=0$ is first provided in Ref. [@SR1]. However, the general solution with $\tau_0\neq0$ in the infinite NLSE is presented recently in Ref. [@SR3]. By using the transformation above and the super-regular solution in Ref. [@SR3], the corresponding properties of vortex filaments can be achieved effectively. Here we omit the tedious explicit expression but show the important and compact results. At the first step, the integral of the relative quadratic curvature of super-regular breather can be obtained explicitly from the exact solution in Appendix \[Appendix B\]. It reads, $$\label{eqqcsr}
\Delta K=16\kappa_0\left[\varepsilon~\cos\phi+\pi~\sin\phi~{\rm csch}\left(\frac{\pi~\sin\phi}{\varepsilon~\cos\phi}\right)\right],$$ where $\varepsilon=R-1$ and $\varepsilon\ll1$. $R(>1)$ and $\phi[\in(-\pi/2,\pi/2)]$ are two real parameters that denote respectively the radius and angle in polar coordinates (see Appendix \[Appendix B\]). Physically, $R$ (or $\varepsilon$) and $\phi$ are two important parameters that describe directly the amplitude and period of the super-regular breathers. It is therefore crucial to study the property of vortex filaments induced by super-regular breathers by the choice of parameters $R$ and $\phi$.
Just as the case of Kuznetsov-Ma breather, Eq. (\[eqqcsr\]) is also greater than zero, i.e., $\Delta K>0$. This indicates that the super-regular breather also admits long-time dynamics which is different from the Peregrine rogue wave and Akhmediev breather. Unlike the case of Kuznetsov-Ma breather, the evolutions of curvature and torsion of super-regular breather exhibit remarkably different characteristics. This stems from that the super-regular breather possesses a localised multi-peak perturbation rather than a localised single-peak perturbation.
Figure \[fig5\] shows the variation of curvature and torsion induced by super-regular breather with the initial parameters $\kappa_0=1$, $R=1.1$, and $\phi=\pi/8$. As can be seen from Fig. \[fig5\](a) that the curvature of a super-regular breather triggered from a localised multi-peak perturbation at $t=0$ \[see Fig. \[fig5\](b)\] increases gradually due to the exponential amplification of the MI at the linear stage. It reaches its maximum at $t=0.43$ and then splits into two quasi-Akhmediev breathers propagating along different directions during the nonlinear stage of MI. The corresponding torsion also suffers singular behavior starting from the maximum curvature point $t=0.43$ \[see Fig. \[fig5\](c)\]. Interestingly, the nonlinear propagation stage always holds the singular torsion at the maximum curvature point as $t>0.43$ \[see Fig. \[fig5\](d)\].
![Configuration of vortex filaments of super-regular breathers at different time (a) $t=0$, (b) $t=0.43$ and (c) $t=1.2$. Other parameters are the same as in Fig. \[fig4\].[]{data-label="fig6"}](fig6.jpg "fig:"){width="85mm"}\
![Top view of configuration of vortex filaments of super-regular breathers at a fixed time $t=1.2$ as $\tau_0$ increases. (a) $\tau_0=0.01$, (b) $\tau_0=0.17$ and (c) $\tau_0=0.24$. One can see clearly the reflection symmetry breaking of the loop pair with non-zero $\tau_0$. Other parameters are $\kappa_0=1$, $R=1.1$ and $\phi=\pi/8$.[]{data-label="fig7"}](fig7.jpg "fig:"){width="85mm"}\
Figure \[fig6\] displays the corresponding vortex structure at $t=0$, $t=0.43$, and $t=1.2$, respectively. One can see clearly that the vortex filament emerges from a perturbed helical vortex at $t=0$ and then exhibits a remarkable loop structure at $t=0.43$. This is the linear MI stage that corresponds to one loop excitation. Interestingly, once the vortex filament evolves into the nonlinear stage, the loop structure splits into a *loop pair* which corresponds to the two quasi-Akhmediev breathers propagation with different group velocities.
In particular, we find that the loop pair induced by super-regular breather at the nonlinear stage shows an interesting *reflection symmetry breaking*, as shown in Fig. \[fig7\]. We find that this remarkable feature comes from the asymmetry of the group velocities of the two quasi-Akhmediev breathers. Indeed, the group velocities of the super-regular breather are given by \[see Eq. (\[eqsr1\]) in Appendix \[Appendix B\]\] $$V_{g1}=2\beta\tau_0+d,~~V_{g2}=2\beta\tau_0-d,$$ where $d=\beta\kappa_0\frac{\left(R^4+1\right)}{R^3-R}\sin{\phi}$. Clearly, due to $\tau_0\neq0$, the absolute values of this two group velocities are always unequal. Once $\kappa_0$, $\varepsilon$, and $\phi$ are fixed, the degree of the asymmetry is proportional to the value of $|\tau_0|$.
Figure \[fig7\] shows the corresponding vortex structures induced by super-regular breather as $\tau_0$ increases. We see that as $\tau_0\rightarrow0$ the resulting loop pair exhibits quasi-reflection symmetry \[Fig. \[fig7\](a)\], while the reflection symmetry of the loop pair breaks greatly with increasing $\tau_0$ \[Figs. \[fig7\](b) and \[fig7\](c)\].
It is very interesting to note that, despite the broken reflection symmetry as $\tau_0\neq0$, the growth rate of modulation instability driven by the super-regular breather does not depend on $\tau_0$. Namely, this growth rate is only associated with the absolute difference of the group velocities, $G=\eta_r|V_{g1}-V_{g2}|$ with $\eta_r=\frac{a}{2}\left(R-1/R\right)\cos{\phi}$, as shown in Ref. [@SR3]. This result is physically important because that although the super-regular breather induced vortex structures can exhibit different loop pairs with symmetry breaking, the inherent MI property can remain invariable.
Finally, we consider the relation between $\Delta K$ and characteristic size $r_s$ of the super-regular breather induced vortex structures. Similar to the case of Kuznetsov-Ma breather, we define characteristic size $r_s$ as the minimum radius of the super-regular-breather induced vortex structure throughout the whole evolution. Thus, the characteristic size $r_s$, which is also inversely proportional to the maximum curvature $\kappa_{ms}$ (i.e., $r_s=1/\kappa_{ms}$), is given by $$\label{rs}
r_s=\left[\kappa_0+\kappa_0\left(1+\varepsilon+\frac{1}{1+\varepsilon}\right)\cos\phi\right]^{-1},$$ where $\varepsilon=R-1$ is a small value ($\varepsilon\ll1$) defined above.
Collecting Eq. (\[eqqcsr\]) and Eq. (\[rs\]), we obtain the explicit expression of $\Delta K\cdot r_s$ by omitting the high-order term $O(\varepsilon^2)$. It reads $$\label{Krs2}
\Delta K\cdot r_s=\alpha_s\varepsilon,$$ where $\alpha_s=16\cos\phi/\left(1+2\cos\phi\right)$.
In contrast to the case of Kuznetsov-Ma breather, Eq. (\[eqlr2\]), where only one parameter $b$ can be modulated when the plane wave parameters ($\kappa_0$, $\tau_0$) and the structural parameter $\beta$ are fixed, Eq. (\[Krs2\]) has two free physical parameters ($\varepsilon$ and $\phi$). But even so, we highlight that linear relations can also hold for the case of super-regular breather induced vortex structures.
![Relations between $\Delta K\cdot r_s$ and $\varepsilon$ of the vortex filament induced by super-regular breather as $\phi$ varies. Other parameters are $\kappa_0=1$. The discrete points are obtained with the high-order term $O(\varepsilon^2)$ considered, while the colored lines retain the first-order term only. []{data-label="fig8"}](fig8.jpg "fig:"){width="82mm"}\
![Relations between $\Delta K\cdot r_s$ and $\phi$ of the vortex filament induced by super-regular breather as $\varepsilon$ varies. Other parameters are $\kappa_0=1$.[]{data-label="fig9"}](fig9.jpg "fig:"){width="82mm"}\
Figure \[fig8\] shows the characteristics of $\Delta K\cdot r_s$ as $\varepsilon$ increases with different values of $\phi$. In particular, we compare the results obtained from the approximate expression Eq. (\[Krs2\]) (the solid lines) and the exact expression (the dotted lines), respectively. One can see that for each fixed $\phi$, $\Delta K\cdot r_s$ shows a linear relation with $\varepsilon$. The corresponding rate $\alpha_s$ decreases in the range of $[48,0]$ as $\phi$ increases from $0$ to $\pi/2$.
Figure \[fig9\] shows the characteristics of $\Delta K\cdot r_s$ as $\phi$ decreases with different values of $\varepsilon$. For each case with fixed $\varepsilon$, $\Delta K\cdot r_k$ increases monotonously with decreasing $\phi$. As $\phi\rightarrow0$, one obtains that $\Delta K\cdot r_k\rightarrow16\varepsilon/3$. This is similar with the case of Kuznetsov-Ma breather induced vortex structures shown in Fig. \[fig4\]. Physically, as $\phi\rightarrow0$, the super-regular breather transforms itself into two colliding Kuznetsov-Ma breathers, so that the similar linear relation can be maintained.
CONCLUSION
==========
In summary, we have investigated the superfluid vortex filaments induced by Kuznetsov-Ma breather and super-regular breather, which admit localised perturbations. We have shown that the loop structure induced by Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by super-regular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of super-regular breather. In particular, we have characterized and identified explicitly these loop excitations by introducing the integral of the relative quadratic curvature, which corresponds to the effective energy of breathers. Although the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
This work has been supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 11705145, 11875220, 11434013, and 11425522), Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JQ1003), and the Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2017KCT-12, 2017ZDJC-32).
THE POSITION VECTOR OF THE VORTEX FILAMENT {#Appendix A}
==========================================
We give the explicit expression of the position vector of the vortex filament by integrating the Frenet-Serret equations given in [@SFe] and the exact expression is formulated explicitly in Ref. [@model2], which should be represented as $$\begin{aligned}
\label{eqqa1}
\nonumber
\mathbf{r}\left(s,t\right)&&=
\begin{bmatrix}
x\left(s,t\right)\\
y\left(s,t\right)\\
z\left(s,t\right)
\end{bmatrix}\\\nonumber
&&=\begin{bmatrix}
x_0\left(t\right)+\displaystyle\sum_{k=1}^3c_{k1}\left(t\right)\int_0^sM_{k}\left(\sigma,t\right)\,d\sigma\\
y_0\left(t\right)+\displaystyle\sum_{k=1}^3c_{k2}\left(t\right)\int_0^sM_{k}\left(\sigma,t\right)\,d\sigma\\
z_0\left(t\right)+\displaystyle\sum_{k=1}^3c_{k3}\left(t\right)\int_0^sM_{k}\left(\sigma,t\right)\,d\sigma
\end{bmatrix}.\end{aligned}$$ Here, $x_0\left(t\right)$, $y_0\left(t\right)$, $z_0\left(t\right)$ are constants with respect to the initial position of vortex structures. $M_{k}\left(k=1,2,3\right)$ are $$\begin{aligned}
M_{1}=\frac{\gamma^2+\alpha^2\cos{\lambda}}{\lambda^2},M_{2}=\frac{\alpha\sin{\lambda}}{\lambda},
M_{3}=\frac{\alpha\gamma\left(1-\cos{\lambda}\right)}{\lambda^2}\nonumber,\end{aligned}$$ where $\alpha=\int_0^s\kappa\left(\sigma,t\right)\,d\sigma$, $\gamma=\int_0^s\tau\left(\sigma,t\right)\,d\sigma$, and\
$\lambda=\sqrt{\left(\int_0^s\kappa\left(\sigma,t\right)\,d\sigma\right)^2+\left(\int_0^s\tau\left(\sigma,t\right)\,d\sigma\right)^2}$.
EXPLICIT EXPRESSIONs OF $\Delta K$ OF AKHMEDIEV BREATHER AND PEREGRINE ROGUE WAVE {#Appendix C}
=================================================================================
$\Delta K$, *the integral of the relative quadratic curvature*, is expressed explicitly in the form $$\label{eqqc1}
\Delta K=\int_{-\infty}^\infty \left[\kappa^2\left(s,t\right)-\kappa_0^2\left(s,t\right)\right]ds.$$ Here, $\kappa\left(s,t\right)$ and $\kappa_0\left(s,t\right)$ represent the curvature distribution of the vortex filament and the curvature of the background uniform helical vortex corresponding to plane wave $\psi_0$ (\[equ:PWB\]) respectively.
As mentioned in Sec \[Sec3\], in addition to the Kuznetsov-Ma breather, plane wave (\[equ:PWB\]) also admits other breathing waves, including the Akhmediev breather and the Peregrine rogue wave. As comparison, we show here $\Delta K$ for the vortex filaments induced by Akhmediev breather and Peregrine rogue wave.
We first consider the Akhmediev breather that exhibits the explicit description for the MI emerging from periodic perturbations. Its exact expressions is given by $$\label{eqqc2} \psi_A(s,t)=\left[1-2\frac{\chi_1^2\cosh\left(\eta_1\beta t\right)+i\eta_1 \sinh\left(\eta_1\beta t\right)}{\kappa_0^2~\cos\left(\eta_1\beta t\right)-\kappa_0b~\cosh\left(\chi_1\xi\right)}\right]\psi_0,$$ where $\chi_1=\sqrt{\kappa_0^2-b^2}$ with $b<\kappa_0$, $\eta_1=b\chi_1$, and $\xi=s-2\tau_0\beta t$. The corresponding exact expression of the curvature is given by $$\label{eqqc3}
\kappa_A=\left[\left(\kappa_0-\frac{2\chi_1^2\cosh(\eta_1\beta t)}{A}\right)^2+\frac{4\eta_1^2\sinh^2(\eta_1\beta t)}{A^2}\right]^{1/2}$$ with $A=\kappa_0\cosh\left(\eta_1\beta t\right)-b\cos\left(\chi_1\xi\right)$. A substitution of Eq. (\[eqqc3\]) into Eq. (\[eqqc1\]) yields $\Delta K_A=0$.
We then consider the Peregrine rogue wave with double localization. The latter corresponds to the limiting case of Eq. (\[eqqc2\]) as $b\rightarrow \kappa_0$. Its exact expression is given by $$\psi_P\left(s,t\right)=\left[1-\frac{4i\kappa_0^2\beta t+4}{1+\kappa_0^4\beta^2 t^2+\kappa_0^2\left(s-2\beta\tau_0 t\right)^2}\right]\psi_0,$$ whose curvature is in the form of $$\label{eqqc4}
\kappa_P=\kappa_0\sqrt{\frac{16\kappa_0^4\beta^2 t^2}{a^2}+\left(1-\frac{4}{a}\right)^2}$$ with $a=1+\kappa_0^4\beta^2t^2+\kappa_0^2\left(s-2\beta\tau_0 t\right)^2$. By calculating Eq. (\[eqqc1\]), we demonstrate also that $\Delta K_P=0$.
As a result, both the Akhmediev breather and the Peregrine rogue wave share the vanishing $\Delta K$, which indicates that the corresponding vortex filaments start from a uniform helical vortex structure.
EXPLICIT EXPRESSION OF SUPER-REGULAR BREATHER {#Appendix B}
=============================================
The explicit expression of super-regular breather for Eq. (\[equ:NLSE\]) is given by the Darboux transformation [@SR3], where the spectral parameter $\lambda$ is parameterized by the Jukowsky transform [@SR1] as follows: $$\begin{aligned}
\label{eqlm}
\lambda=i\frac{\kappa_0}{2}\left(\Delta+\frac{1}{\Delta}\right)-\frac{\tau_0}{2},~\Delta=Re^{i\phi}.\end{aligned}$$ Here, $R$ and $\phi$ define the location of the spectral parameter $\lambda$ in the polar coordinates. They represent radius and angle respectively in the region $R>1$ and $\phi\in\left(-\pi/2,\pi/2\right)$. For $\tau_0=0$, Eq (\[eqlm\]) reduces to the spectral parameter used in Ref. [@SR1]. With different values of $R$ and $\phi$, the resulting exact solution can describe different breather dynamics [@SR1]. A more general phase diagram of breathers has been obtained recently in Ref. [@CLIP]. Here we consider the super-regular breather formed by two quasi-Akhmediev breathers with $R_1=R_2=R=1+\varepsilon$ ($\varepsilon\ll1$), $\phi_1=-\phi_2=\phi$. Its explicit expression of the solution is in the form: $$\begin{aligned}
\label{eqsr}
\psi\left(s,t\right)=\psi_0\left[1-4\rho\varrho\frac{\left(i\varrho-\rho\right)\Xi_1+\left(i\varrho+\rho\right)\Xi_2}{\kappa_0\left(\rho^2\Xi_3+\varrho^2\Xi_4\right)}\right].\end{aligned}$$ Here $$\begin{aligned}
&&\varrho=\frac{\kappa_0}{2}\left(R-\frac{1}{R}\right)\sin\phi, ~\rho=\frac{\kappa_0}{2}\left(R+\frac{1}{R}\right)\cos\phi\nonumber\\
&&\Xi_1=\varphi_{21}\phi_{11}+\varphi_{22}\phi_{21}, ~~~\Xi_2=\varphi_{11}\phi_{21}+\varphi_{21}\phi_{22},\nonumber\\
&&\Xi_3=\varphi_{11}\phi_{22}-\varphi_{21}\phi_{12}-\varphi_{12}\phi_{21}+\varphi_{22}\phi_{11},\nonumber\\
&&\Xi_4=\left(\varphi_{11}+\varphi_{22}\right)\left(\phi_{11}+\phi_{22}\right),\nonumber\end{aligned}$$ with $$\begin{aligned}
&&\phi_{jj}=\cosh\left(\Theta_2\mp i\psi\right)-\cos\left(\Phi_2\mp\phi\right),\nonumber\\
&&\varphi_{jj}=\cosh\left(\Theta_1\mp i\psi\right)-\cos\left(\Phi_1\mp\phi\right),\nonumber\\
&&\phi_{j3-j}=\pm i\cosh\left(\Theta_2\mp i\phi\right)-\cos\left(\Phi_2\mp\theta\right),\nonumber\\
&&\varphi_{j3-j}=\pm i\cosh\left(\Theta_1\mp i\phi\right)-\cos\left(\Phi_1\mp\theta\right),\nonumber\end{aligned}$$ where $\theta=\arctan\left[\left(1-iR^2\right)/\left(1+R^2\right)\right]$. $\Theta_j$ and $\phi_j$ are related with group and phase velocities respectively, which is in the form of $$\begin{aligned}
\Theta_j=2\eta_r\left(s-V_{gj}t\right),~\phi_j=2\eta_{ij}\left(s-V_{pj}t\right)\end{aligned}$$ where $$\begin{aligned}
&&\eta_{i1}=-\eta_{i1}=\frac{\kappa}{2}\left(R+\frac{1}{R}\right)\sin\phi,\nonumber\\ &&\eta_r=\frac{\kappa}{2}\left(R-\frac{1}{R}\right)\cos\phi,\nonumber\\ &&V_{p1}=2\beta\tau_0-d_1,~V_{p2}=2\beta\tau_0+d_2,\nonumber\\
&&V_{g1}=2\beta\tau_0+d,~V_{g2}=2\beta\tau_0-d.\label{eqsr1}\end{aligned}$$ with $d_1=\beta\kappa_0\left(R-\frac{1}{R}\right)\frac{\cos\left(2\phi\right)}{\sin\phi}$, $d_2=\beta\kappa_0\frac{\left(R-\frac{1}{R}\right)}{\sin\phi}$ and $d=\beta\kappa_0\frac{\left(R^4+1\right)}{R^3-R}\sin{\phi}$. The initial state of the super-regular breather can be extracted from the above solution at $t=0$. It reads, as $\varepsilon\rightarrow0$, $$\begin{aligned}
\label{eqsrin}
\psi\left(s,0\right)=\psi_0 \left(1-i\frac{4\varepsilon \cos\phi \cos\left(\kappa_0 s \sin\phi\right)}{\cosh\left(\kappa_0\varepsilon s \cos\phi\right)}\right).\end{aligned}$$ Note that for a given plane wave background (\[equ:PWB\]) (i.e., $\kappa_0$ and $\tau_0$ are fixed), $R$ and $\phi$ determine the amplitude and period of breathers. In particular, $R$ and $\phi$ effect the profile of the initial state (\[eqsrin\]) of super-regular breathers.
The integral of the relative quadratic curvature of super-regular breather $\Delta K$ \[see Eq. (\[eqqcsr\])\] is obtained explicitly from the initial state (\[eqsrin\]), since the NLSE shares the same $\Delta K$ at different times.
C. F. Barenghi and N. G. Parker, *A primer on quantum fluids* (Springer, 2016). I. Carusotto and C. Ciuti, [Rev. Mod. Phys. **85**, 299 (2013)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.85.299). S. Zuccher, M. Caliari, A. Baggaley, and C. Barenghi, [Phys. Fluids **24**, 125108 (2012)](https://aip.scitation.org/doi/abs/10.1063/1.4772198). S. K. Nemirovskii, [Phys. Rep. **524**, 85 (2013)](https://www.sciencedirect.com/science/article/pii/S037015731200350X). P. G. Saffman, *Vortex Dynamics* (Cambridge University Press, Cambridge, 1992). W. Gilpin, V. N. Prakash, M. Prakash, [Nat. Phys. **13**, 380 (2017)](https://www.nature.com/articles/nphys3981). H. Hasimoto, [J. Fluid Mech **51**, 477 (1972)](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/soliton-on-a-vortex-filament/1A2FF58DD1B8DE20509866A4713531F3). H. Salman, [Phys. Rev. Lett **111**, 165301 (2013)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.165301). H. Salman, [J. Phys.: Conf. Ser. **544**, 012005 (2014)](http://iopscience.iop.org/article/10.1088/1742-6596/544/1/012005/meta). B. K. Shivamoggi, [Phys. Rev. B **84**, 012506 (2011)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.84.012506). R. A. Van Gorder, [Phys. Rev. E **93**, 052208 (2016)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.052208); [Phys. Rev. E 91, 053201 (2015)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.053201); R. Shah, R. A. Van Gorder, [Phys. Rev. E 93, 032218 (2016)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.93.032218). W.T. Kelvin, Vibrations of a columnar vortex. Phil. Mag. Ser. 5, 155-168 (1880). E. Kozik and B. Svistunov, [Phys. Rev. Lett. 92, 035301 (2004)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.035301). N. Akhmediev and A. Ankiewicz, *Solitons: Nolinear Pulses and Beams* (Chapman and Hall, London, 1997). J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, [Nat. Photonics **8**, 755 (2014)](https://www.nature.com/articles/nphoton.2014.220). V. E. Zakharov and A. A. Gelash, [Phys. Rev. Lett. **111**, 054101 (2013)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.054101); A. A. Gelash and V. E. Zakharov, [Nonlinearity **27**, R1 (2014)](http://iopscience.iop.org/article/10.1088/0951-7715/27/4/R1/meta); B. Kibler, A. Chabchoub, A. Gelash, N. Akhmediev, and V. E. Zakharov, [Phys. Rev. X **5**, 041026 (2015)](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.041026). J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, [Opt. Express **17**, 21497 (2009)](https://www.osapublishing.org/oe/abstract.cfm?uri=oe-17-24-21497). L. C. Zhao and L. M. Ling, [J. Opt. Soc. **33**, 850 (2016)](https://www.osapublishing.org/josab/abstract.cfm?uri=josab-33-5-850). N. Akhmediev, J. Soto-Crespo, and A. Ankiewicz, [Phys. Lett. A **373**, 2137 (2009)](https://www.sciencedirect.com/science/article/pii/S0375960109004939); N. Akhmediev, J. Soto-Crespo, and A. Ankiewicz, [Phys. Rev. A **80**, 043818 (2009)](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.043818); N. Akhmediev, A. Ankiewicz, and M. Taki, [Phys. Lett. A **373**, 675 (2009)](https://www.sciencedirect.com/science/article/pii/S0375960108017945).
M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, [Phys. Rep. **528**, 47 (2013)](https://www.sciencedirect.com/science/article/pii/S0370157313000963). N. Akhmediev, [*et al*]{}, [J. Optics **18**, 063001 (2016)](http://iopscience.iop.org/article/10.1088/2040-8978/18/6/063001/meta). M. Onorato, S. Residori, and F. Baronio, *Rogue and Shock Waves in Nonlinear Dispersive Media* (Springer, 2016). S. Wabnitz, *Nonlinear Guided Wave Optics: a testbed for extreme waves* (IOP Publ., Bristol, UK, 2017). N. Akhmediev and V. I. Korneev, Theor. Math. Phys. **69**, 1089 (1986); N. Akhmediev, [Nature **413**, 267 (2001)](https://www.nature.com/articles/35095154). E. Kuznetsov, Sov. Phys. Dokl. **22**, 507 (1977); Y. C. Ma, Stud. Appl. Math. **60**, 43 (1979). L. C. Zhao, L. M. Ling, and Z. Y. Yang, [Phys. Rev. E **97**, 022218 (2018)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.97.022218). C. Liu, Z. Y. Yang, and W. L. Yang, [Chaos, **28**, 083110 (2018)](https://aip.scitation.org/doi/10.1063/1.5025632). Y. Ren, C. Liu, Z. Y. Yang, and W. L. Yang, [Phys. Rev. E **98**, 062223 (2018)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.062223). R. J. Arms, and F. R. Hama, [The Physics of fluids **8**, 553 (1965)](https://aip.scitation.org/doi/abs/10.1063/1.1761268). L. S. Da Rios, Rend. Circ. Mat. Palermo **22**, 117 (1906). R. Betchov, [J. Fluid Mech **22**, 471 (1965)](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/on-the-curvature-and-torsion-of-an-isolated-vortex-filament/28809E8B5A4890DEE004F4F161ECD893). L. M. Pismen, *Vortices in Nonlinear Fields* (Clarendon, Oxford, 1999). R. Shah, *Rogue waves on a vortex filament* (Oxford, 2015). D. H. Peregrine, [J. Aust. Math. Soc. Ser. B, Appl. Math. **25**, 16 (1983)](https://www.cambridge.org/core/journals/anziam-journal/article/water-waves-nonlinear-schrodinger-equations-and-their-solutions/D87F5416C657F3B5C35AE96DF9F73DD0). N. Akhmediev and A. Ankiewicz, (Eds.), *Dissipative solitons*, Lect. Notes Phys. 661 (Springer, Berlin Heidelberg 2005). N. Akhmediev and A. Ankiewicz, (Eds.), *Dissipative Solitons: From Optics to Biology and Medicine*, Lect. Notes Phys. 751 (Springer, Berlin Heidelberg 2008). P. Grelu and N. Akhmediev, Nat. Photonics 6, 84 (2012). C. Liu, Y. Ren, Z. Y. Yang, and W. L. Yang, [Chaos **27**, 083120 (2017)](https://aip.scitation.org/doi/abs/10.1063/1.4999916). Y. Ren, X. Wang, C. Liu, Z. Y. Yang, and W. L. Yang, [Commun. Nonlinear. Sci. Numer. Simulat. **63**, 161 (2018)](https://www.sciencedirect.com/science/article/pii/S1007570418300947). C. Liu, Z. Y. Yang, W. L. Yang, and N. Akhmediev, [J. Opt. Soc. Am. B 36, 1294 (2019)](https://www.osapublishing.org/josab/abstract.cfm?uri=josab-36-5-1294).
|
---
abstract: 'Processing sequential data of variable length is a major challenge in a wide range of applications, such as speech recognition, language modeling, generative image modeling and machine translation. Here, we address this challenge by proposing a novel recurrent neural network (RNN) architecture, the Fast-Slow RNN (FS-RNN). The FS-RNN incorporates the strengths of both multiscale RNNs and deep transition RNNs as it processes sequential data on different timescales and learns complex transition functions from one time step to the next. We evaluate the FS-RNN on two character level language modeling data sets, Penn Treebank and Hutter Prize Wikipedia, where we improve state of the art results to $1.19$ and $1.25$ bits-per-character (BPC), respectively. In addition, an ensemble of two FS-RNNs achieves $1.20$ BPC on Hutter Prize Wikipedia outperforming the best known compression algorithm with respect to the BPC measure. We also present an empirical investigation of the learning and network dynamics of the FS-RNN, which explains the improved performance compared to other RNN architectures. Our approach is general as any kind of RNN cell is a possible building block for the FS-RNN architecture, and thus can be flexibly applied to different tasks.'
author:
- |
Asier Mujika\
Department of Computer Science\
ETH Zürich, Switzerland\
`asierm@ethz.ch`\
Florian Meier\
Department of Computer Science\
ETH Zürich, Switzerland\
`meierflo@inf.ethz.ch`\
Angelika Steger\
Department of Computer Science\
ETH Zürich, Switzerland\
`steger@inf.ethz.ch`\
bibliography:
- 'main.bib'
title: 'Fast-Slow Recurrent Neural Networks'
---
Introduction {#sec:Introduction}
============
Related work {#sec:RelatedWork}
============
Fast-Slow RNN {#sec:Model}
=============
Experiments {#sec:Experiments}
===========
Conclusion {#sec:Conclusions}
==========
### Acknowledgments {#acknowledgments .unnumbered}
We thank Julian Zilly for many helpful discussions.
|
---
abstract: 'Asset monitoring in construction sites is an intricate, manually intensive task, that can highly benefit from automated solutions engineered using deep neural networks. We use Single-Shot Multibox Detector — SSD, for its fine balance between speed and accuracy, to leverage ubiquitously available images and videos from the surveillance cameras on the construction sites and automate the monitoring tasks, hence enabling project managers to better track the performance and optimize the utilization of each resource. We propose to improve the performance of SSD by clustering the predicted boxes instead of a greedy approach like non-maximum suppression. We do so using Affinity Propagation Clustering — APC to cluster the predicted boxes based on the similarity index computed using the spatial features as well as location of predicted boxes. In our attempts, we have been able to improve the mean average precision of SSD by 3.77% on custom dataset consist of images from construction sites and by 1.67% on PASCAL VOC Challenge.'
author:
- '\'
- '\'
- '\'
bibliography:
- 'IEEEabrv.bib'
- 'bibliography.bib'
title: 'Efficient Single-Shot Multibox Detector for Construction Site Monitoring'
---
Asset Monitoring, Automation in Construction, Single-shot Multibox Detector, Non-maximum Suppression, Clustering
Introduction
============
Construction sites forms an important parts of cities: they are intricate environments with a broad range of activities like clearing, dredging, excavating, and building [@batty2012smart]. These activities require large numbers of expensive equipments, and it is crucial to monitor their proper utilization. This monitoring is time-consuming, labor intensive, and prone to human errors by project managers. Smart Cities incorporate a large number of Internet-connected sensors and actuators. This paper considers the use of IP cameras such as sensors to automate monitoring on construction sites.
We propose an automated system to detect, localize and classify equipment from videos to generate real-time reports that facilitate decision-making. We do so using recently introduced computer vision algorithms [@lecun2015deep; @voulodimos2018deep] trained on surveillance videos from construction sites that are available but underutilized in computer vision research. We use Single-Shot MultiBox Detector — SSD with Non-Maximum Suppression — NMS [@DBLP:journals/corr/LiuAESR15] as base model, which reaches record performance for object detection, scoring over 74% mAP (mean Average Precision) at a real-time rate of 59 frames per second on the Pascal VOC Challenge [@everingham2010pascal].
SSD uses greedy NMS, where out of all the detected bounding boxes, the boxes with higher confidences are selected and the other boxes overlapping the selected boxes are suppressed subjected to an intersection over union ($iou$) threshold. NMS uses a static threshold, usually $0.5$, to winnow away candidate bounding boxes. But this very technique of NMS causes the detector to fail while looking for objects which appear smaller or have low resolution because of far away camera placement on construction sites. Fig.\[fig:ssdwonms\] and Fig.\[fig:ssdnms\] show this particular problem where SSD is not able to detect a small equipment.
We propose to replace NMS with Affinity Propagation Clustering [@frey2007clustering] to solve the drawback of greedy NMS. We choose APC over other clustering techniques, because it does not require the number of clusters to be determined a priori. Rather, it is an exemplar based algorithm that employs a simple message-passing technique to cluster based on similarity between bounding boxes generated by SSD. The algorithm progressively engenders communication between each bounding box and its exemplar to produce a high-quality set of clusters with the exemplars as the cluster centers.
In our technique, we cluster the bounding boxes on the basis of the location as well as the appearance-based spatial features of the pixels enclosed by those bounding boxes, hence distinguishing the objects based on both their location as well as appearance. This allows the selection of final detection to be adaptive in contrast to the hand-designed threshold values of NMS, and since there is no minimum value for the detection overlap, small objects are also detected successfully. Fig.\[fig:comparison\] depicts the performance comparison of SSD with NMS and with APC.
The paper is organized as follows. Section 2 describes the SSD and APC to provide intuition of the problem and capability of clustering to solve it. It also introduces the notation used in the paper. Section 3 describes the proposed algorithm. Section 4 covers the dataset used, experimental setup and result analysis under different situations.
Preliminaries
=============
Single Shot Multibox Detector
-----------------------------
[0.30]{} {width="\textwidth"}
[0.30]{} {width="\textwidth"}
[0.30]{} {width="\textwidth"}
The key feature of Single Shot Multibox Detector — SSD [@DBLP:journals/corr/LiuAESR15] is a feed-forward convolutional network, that can, in a single pass perform both the object classification, by predicting the class score and object localization, by performing bounding box regression followed by non-maximum suppression — NMS. Let $\chi$ be the domain set representing all the images from which objects are to be detected and $M = \{1, 2, ... l\}$ be the label set representing the labels for all the object-classes in those images. Let $D = \{d_1, d_2, ...d_n\}$ be the set of default boxes created using Algorithm 1 over different aspect ratios and regularly spaced scales. Each default box $d_i$ is a vector representing four values associated with the box $[c_x\ c_y\ w\ h]$, where $(c_x, c_y)$ are the coordinates of the centroid and $w$ and $h$ are respectively the width and height of the bounding box. SSD is a function $\varphi(x) = \hat{Y}$ that takes an arbitrary image $x \in \chi$ as an input and produces a matrix $\hat{Y} \in \mathbb{R}^{n \times (l + 4)}$ as the output. Each row of $\hat{Y}$ represents a $(l + 4)$ dimensional positive real valued vector, which contains $l$ per-class classification probabilities or confidences and four offsets in the default box dimensions. For simplicity consider $\hat{Y} = [\hat{Z}\ \hat{B}]$ where matrix $\hat{Z} \in \mathbb{R}^{n \times l}$ represents classification task and $\hat{B} \in \mathbb{R}^{n \times 4}$ represents localization task.
The training data available for object detection consists of images and the boxes circumscribing the objects (called ground-truth boxes) along with their class labels. Hence, for any arbitrary image $x \in \chi$, $G = \{g_1, g_2, ... g_h\}$ represents a set of ground-truth boxes, where $g_i \in \mathbb{R}^{(l+4)}$. The training objective of SSD is to learn the prediction rules to predict the object class present in each default box and the amount of offset required in the shape of the default boxes with respect to the ground-truth box. This correlation, associated with each default box is computed using Algorithm 2 and stored in a ground-truth matrix $Y \in \mathbb{R}^{n \times (l + 4)}$.
\
$p$ - Number of feature maps\
$f \in \mathbb{R}^p$ where $\forall k \in [1,p] : f[k]$ - Dimension of $k^{th}$ square feature map\
$s_{min}$ - Minimum Scale Value - Default 0.2\
$s_{max}$ - Maximum Scale Value - Default 0.9\
$D = \{\}$ - Set of Default Boxes\
$$\ \forall i,j \in [0, f[k]): C_k = \left\{\left(\frac{i + 0.5}{f[k]}, \frac{j + 0.5}{f[k]}\right)\right\}$$\
$$s_k = s_{min} + \frac{s_{max} - s_{min}}{p - 1}(k - 1),\ s'_k = \sqrt[]{s_k \cdot s_{k+1}}$$
\
$D$ - Set of Default Boxes
\
$\tau$ - Overlap threshold - Default 0.5\
$D = \{d_1, ..., d_n\}$ - Set of Default Boxes\
$G = \{g_1, ..., g_h\}$ - Set of Ground-truth Boxes\
$Y \in \mathbb{R}^{n \times (l + 4)}\ such\ that\ Y = [Z\ B] = [0]$\
$pos, neg = \{\}$ - Sets to store indexes of positively and negatively matched default boxes\
$Y = [Z\ B]$
SSD uses weighted sum of classification and localization loss as an overall loss function, minimize it using Adam Optimizer [@kingma2014adam] during the training. For classification, SSD calculates the multi-class softmax loss. If $N$ is the total number of positively matched default boxes in Algorithm 2 and $\sigma \in M$ is some label representing the background, the classification loss is: $$L_{clss}(\hat{Z}, Z) = - \sum_{c=1}^{l} \sum_{i \in pos} Z_{i,c} \cdot \log(\hat{Z}_{i,c}) - \sum_{j \in neg} \log(\hat{Z}_{j,\sigma})$$ Let $$smooth\_[L\_1]{}(x) = {
-------------------------------
$0.5 \times x^2$ if $|x| < 1$
$|x| - 0.5$ otherwise
-------------------------------
$$ denote the smooth L1 function [@DBLP:journals/corr/Girshick15]. The localization loss is: $$L_{loc}(\hat{B}, B) = \sum_{i \in pos} smooth_{L_1}(\hat{B}_{i,:} - B_{i,:})$$ Hence, the overall loss is: $$L(\hat{Y}, Y) = \frac{1}{N} \Big(L_{clss}(\hat{Z}, Z) + \alpha L_{loc}(\hat{B}, B)\Big)$$ where $\alpha$ is the weight value that controls the balance between the two losses.
The localization and classification tasks are followed by two post-processing steps. First, SSD creates a matrix $\Omega \in \mathbb{R}^{n \times (l+4)}$ representing an ordered set of predicted boxes given as: $$\Omega = [\hat{Z}\ \hat{B}+D]$$ Each row of matrix $\Omega$ represents the $l$ class probabilities with the exact bounding box circumscribing the object. Secondly, SSD performs per class non-maximum suppression to produce the final detection. For that, it removes all the predicted boxes which belong to the background class; and then, it iteratively performs non-maximum suppression for the other classes by selecting the most confident predicted box, and removing all other overlapping boxes with $iou > 0.5$, till there is no box overlapping the selected one.
Affinity Propagation Clustering
-------------------------------
Affinity Propagation Clustering clusters the data by exchanging certain real-valued similarity messages between the pairs of data points until convergence, producing a refined set of clusters and corresponding exemplars.
APC takes as inputs a set of data points $T = \{t_1, t_2, ... t_q\}$ and a similarity matrix $S \in \mathbb{R}^{q \times q}$, whose element $S_{i, j}$ is a measure of the similarity between data points $t_i$ and $t_j $, computed as per function $s(t_i, t_j)$. Additionally, it takes a preference vector $\rho = \{\rho_1, \rho_2, ... \rho_q\}$ , where $\rho_i$ is associated with each data point $t_i$ such that $t_i$ with larger $\rho_i$ are more likely to be the exemplars. The output of APC is largely influenced by the choice of $\rho$, as in, choosing a shared value (e.g. median) can result in moderate number of clusters, whereas choosing a minimum value can result in a small number of clusters.
Two types of messages are iteratively exchanged between the node pairs, that can be combined at any iteration to give the clusters and their respectively chosen exemplars. A *responsibility* message $r(i, j) \in \mathbb{R}$ from the data point $t_i$ to a potential exemplar $t_j$, reflecting the suitability of $t_j$ to be an exemplar for $t_i$, given the other potential exemplars; and an *availability* matrix $a(i, j) \in \mathbb{R}$ from a candidate exemplar $t_j$ to the data point $t_i$ reflecting how appropriate it would be for $t_j$ to serve as an exemplar of $t_i$, given the support from the other data points for $t_j$ to be an exemplar. In each iteration, these messages are updated as per Algorithm 3, until either the changes in the messages fall below some threshold, or there is no update in the computed clusters and corresponding exemplars over some iterations.
\
$T = \{t_1, t_2, \ldots, t_q\}$ - Set of data points\
$\rho \in \mathbb{R}^q$ - Preference vector\
$S$ - Set of pairwise similarities $$\forall (i, j) \in \{1, \ldots, q\}^2 : S_{i,j} = \begin{cases}s(t_i, t_j),\ i \neq j\\\\
\rho_j,\ i = j
\end{cases}$$\
$\forall (i, j) \in \{1, 2, ... q\}^2 : a(i, j) = 0$ $$\begin{aligned}
\forall(i, j) &\in \{1, 2, ... q\}^2 :\\
r(i, j) &= S_{i,j} - \max_{k : k \neq j}[S_{i,k} + a(i, k)]\\
a(i, j) &= \begin{cases}
\sum_{k : k \neq i} \max[0, r(k, j)],\quad \mbox{if }j = i\\\\
\min[0, r(j, j) + \sum_{k : k \notin \{i, j\}} \max[0, r(k, j)]], \mathrm{ow}
\end{cases}
\end{aligned}$$
\
Assignment vector $\hat{c} \in \mathbb{R}^q$ such that\
$$\forall i \in [1,q], \forall j \in [1,q] : \hat{c}_i = \operatorname*{argmax}_j [a(i, j) + r(i, j)]$$\
Set of exemplar data points\
$$\forall i \in [1, q] : E = {T[\hat{c}_i]}$$
Proposed Architecture
=====================
The proposed object detection pipeline for SSD with APC is shown in figure. To produce final predictions, instead of applying non-maximum suppression to the boxes predicted by SSD, we propose to cluster them based on their similarity using APC. We compute the preference vector $\rho$ of APC from the predicted matrix $\Omega \in \mathbb{R}^{n \times (l+4)}$ of SSD as $\forall i \in [1,n] : \rho_i = \max[\Omega_{i,1:l}]$. This allows APC to select boxes with high predicted confidences as exemplars. The similarity between two predicted boxes is calculated as a weighted sum of the location of the default box and its computed visual based features. The location based similarity is computed as $iou$ between two predicted boxes. The visual appearance based similarity is computed as euclidean distance between histogram of gradients [@dalal2005histograms] computed for those two predicted segments of the image.
Consider the label set $M = \{1, 2, ... l\}$, where 1 indicates the background and $\{2, 3 ... l\}$ indicates the labels for different object classes. For an arbitrary image $x$ with ground-truth boxes $G = \{g_1, g_2, ... g_h\}$, the predicted output of SSD $\Omega \in \mathbb{R}^{n \times (l + 4)}$ represents the $n$ segments of input image. Let $\Delta \in \mathbb{R}^{n \times 4}$ where $\forall i \in [1, n] : \Delta_{i,:} = \Omega_{i, l+1:l+4}$ represents the predicted bounding boxes. The location-based similarity can be calculated as: $$\forall (i, j) \in \{1, 2, ... n\}^2, i \neq j : \alpha_{i,j} = 1 - \frac{|\Delta_{i,:} \cap \Delta_{j,:}|}{|\Delta_{i,:} \cup \Delta_{j,:}}.$$ Let $\forall i \in [1,n] : \eta_i$ represents the histogram of gradient feature vector calculated using the method proposed in [@dalal2005histograms]. The visual appearance based similarity can be calculated as following: $$\forall (i, j) \in \{1, 2, ... n\}^2, i \neq j : \beta_{i, j} = - ||\eta_i - \eta_j||^2.$$
For real-valued weight factor $\lambda \in [0,1]$, the elements of the similarity matrix $S$ can be computed as: $$\forall (i, j) \in \{1, 2, ... q\}^2 : S_{i,j} = \begin{cases}\frac{\alpha_{i,j} + \lambda \beta_{i,j}}{2},\ i \neq j\\
\rho_i,\ i = j
\end{cases}.$$ Using similarity matrix from equation 7 with Algorithm 3 gives $E$, the set of exemplars of the predicted boxes representing the final detections of the objects. Ideally this $E$ should be close to the ground-truth $G$.
![Aspect Ratio Distribution of Train and Test Data[]{data-label="fig:ardistr"}](images/Training_Data_Distr.jpg "fig:"){width="\linewidth" height="4cm"} ![Aspect Ratio Distribution of Train and Test Data[]{data-label="fig:ardistr"}](images/Testing_Data_Distr.jpg "fig:"){width="\linewidth" height="4cm"}
----------------------- ---- ------- ------- ------- ------- ------- ------- ------- ------- --
**SSD-MobileNet** 32 59.12 70.45 44.62 69.41 15.50 53.14 48.56 51.24
**SSD-MobileNet-APC** 32 63.40 72.89 47.32 72.40 18.25 58.14 54.13 55.21
**SSD-Inception** 45 75.56 80.74 57.45 79.19 24.58 70.09 58.39 63.76
**SSD-Inception-APC** 46 79.12 83.80 60.01 82.11 29.34 74.27 64.23 67.55
----------------------- ---- ------- ------- ------- ------- ------- ------- ------- ------- --
------------------- ------ -------
Aeroplane 75.5 76.1
Bicycle 80.2 82.3
Bird 72.3 73.5
Boat 66.3 68.2
Bottle 46.6 48.7
Bus 83.0 85.12
Car 84.2 84.3
Cat 86.1 88.3
Chair 54.7 56.6
Cow 78.3 79.3
Dinning Table 73.9 76.2
Dog 84.5 85.2
Horse 85.3 85.5
Motor bike 82.6 83.3
Person 76.2 79.0
PottedPlant 48.6 49.6
Sheep 73.9 76.5
Sofa 76.0 77.2
Train 83.4 85.9
TV 74.0 76.2
mAP (%) 74.3 75.9
Time (ms) 38 38
% **Improvement**
------------------- ------ -------
: Average Precision per Class and Mean Average Precision Comparison for PASCAL VOC[]{data-label="pascaldata"}
Experimental Results and Comparison
===================================
As the goal of this research is to provide a better object detection algorithm to develop an automated asset monitoring and management system for the construction sites, we have created the dataset from images and videos, captured at different construction sites, having a wide range of equipments. The dataset has seven labels : Equipment-1 to Equipment-7. The images are taken from surveillance IP cameras placed at various construction sites with different angles and heights. This setup allows us to generate a dataset with objects ranging in different scales as well as aspect ratios. Figure \[fig:traindatastats\] and \[fig:testdatastats\] shows the details about training and testing dataset. Figure \[fig:ardistr\] shows the distribution of aspect ratios of objects present in the dataset. We also test the proposed approach on PASCAL VOC dataset [@everingham2010pascal] for fair comparison with SSD architecture proposed in [@DBLP:journals/corr/LiuAESR15]. The training and testing distribution of PASCAL VOC dataset is also the same as SSD-300 architecture in [@DBLP:journals/corr/LiuAESR15]. The performance evaluation of both the datasets is done and analyzed separately.
To understand the performance, we consider three variants of SSD based on different feature extraction network. i) SSD with Inception [@DBLP:journals/corr/SzegedyVISW15] ii) SSD with Mobilenet [@DBLP:journals/corr/HowardZCKWWAA17] and iii) SSD with VGG-16 [@DBLP:journals/corr/LiuAESR15; @simonyan2014very]. These different feature extraction networks cover almost all the state-of-the-art variants of SSD available and allow us to verify effectiveness of the proposed algorithm on different architectures. We evaluate the PASCAL VOC evaluation metrics and COCO evaluation metrics to compare performance of proposed algorithm.
### Evaluation on Custom Dataset
At first, we train SSD-Inception and SSD-Mobilenet with the custom dataset from construction sites. We have two versions of each architecture, one with NMS (SSD-Mobilenet and SSD-Inception) and another created using proposed algorithm (SSD-Mobilenet-APC and SSD-Inception-APC). We evaluate all the models for per-class average precision as well as mean average precision as suggested in PASCAL VOC evaluation metrics [@everingham2010pascal]. The evaluation of these models provide us an insight about the performance of all the variants of SSD on different custom object classes. The results in Table 1 indicate the per class average precision comparison of conventional SSD with NMS against proposed SSD with APC for each variant.
We observed that conventional SSD is fairly able to detect large objects, as in objects covering larger area with respect to total area of the image but, struggles against smaller objects. On the other hand, performance of SSD with APC is far better for smaller objects. It achieves this by detecting objects, which are rejected by conventional SSD during the NMS process. To verify the authenticity of proposed improvement and provide fair evaluation, we compared the conventional SSD model provided in [@DBLP:journals/corr/LiuAESR15] with the proposed algorithm on PASCAL VOC dataset. For this evaluation we used SSD with VGG-16 as base network. We kept all the default values as well as hyper parameters of the network same as the values mentioned in [@DBLP:journals/corr/LiuAESR15]. Table 2 provides the comparison for per class average precision and mean average precision. The real-time detection is the main reason of selecting SSD as base algorithm and replacing NMS with APC doesn’t affect the detection speed. Table 1 includes the time take to perform detection on single image.
Conclusion
==========
Our evaluation and analysis gives the significant drawbacks of using greedy non-maximum suppression approach and demonstrates how it restricts the performance of conventional SSD. We also provide the effect of object size on performance of conventional SSD. This paper highlights the use of affinity propagation clustering in SSD to overcome the drawbacks of NMS. Our evaluation and analysis strongly suggest the effectiveness of proposed approach and shows its potential to provide better object detection. We also cover the application of this improvement to automate the asset-monitoring in construction sites.
Acknowledgment {#acknowledgment .unnumbered}
==============
We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), \[funding reference number 396151363\].
|
---
abstract: 'We report Ferromagnetic Resonance Force Microscopy (FMRFM) experiments on a justaposed continuous films of permalloy and cobalt. Our studies demonstrate the capability of FMRFM to perform local spectroscopy of different ferromagnetic materials. Theoretical analysis of the uniform resonance mode near the edge of the film agrees quantitatively with experimental data. Our experiments demonstrate the micron scale lateral resolution in determining local magnetic properties in continuous ferromagnetic samples.'
author:
- 'E. Nazaretski'
- 'Yu. Obukhov'
- 'I. Martin'
- 'D. V. Pelekhov'
- 'K. C. Cha'
- 'E. A. Akhadov'
- 'P. C. Hammel'
- 'R. Movshovich'
title: 'Localized ferromagnetic resonance force microscopy in permalloy-cobalt films'
---
Magnetic resonance force microscopy (MRFM) is attracting increasing attention as a result of its high spin sensitivity and excellent spatial resolution in paramagnetic and nuclear spin systems.[@Rugar; @1992; @Rugar; @1994; @Zhang; @1996a; @Rugar; @2004; @Degen; @2005; @Rugar; @2009] MRFM studies on microfabricated and continuous ferromagnetic samples have been also performed. [@Zhang; @1996; @Loubens; @2007; @Nazaretski; @2006a; @LocalFMR:obukhov:prl:2008; @Nazaretski; @2009] Here we report FMRFM experiments performed on a non-overlapping permalloy (Py) and cobalt (Co) continuous films and demonstrate the capability of FMRFM to spectroscopically identify the distinct magnetic properties of two adjacent ferromagnetic films. We quantitatively model the resulting force signal strength and compare it with the experimental data.
The permalloy-cobalt sample is schematically shown in Fig. \[sample\]. A 20 nm thick Ti film was uniformly applied onto the surface of a 100 $\mu$m thick Si (100) wafer. 20 nm of Co was deposited into a rectangular area (2.5 $\times$ 5 mm) defined in photoresist followed by the lift-off. A complimentary rectangular area of 20 nm thick Py was similarly defined and deposited. The entire structure was then coated with a 20 nm thick layer of Ti. The interface between the Co and Py regions was examined in a scanning electron microscope (SEM) and revealed a gap whose width varies between 3 and 6 $\mu$m along the entire length of the sample (see SEM image in Fig. \[sample\]). An approximately 1.7 $\times$ 1.7 mm$^2$ piece was cut and glued to the stripline resonator of the FMRFM apparatus and the film plane was oriented perpendicular to the direction of the external magnetic field $H_{\rm ext}$. For FMRFM studies we used the cantilever with the spherical magnetic tip (see SEM image in Fig. \[sample\]) and its spatial field profile has been carefully characterized [@Nazaretski; @2008]. More details on the experimental apparatus can be found in Ref. [@Nazaretski; @2006]\
In Fig. \[Figure 1\] we show the evolution of the FMRFM signal as a function of the lateral position and applied magnetic field. The cantilever was scanned across the interface between Co and Py, in the region indicated by arrows in Fig. \[sample\]. The FMRFM signal was recorded in two different regions of $H_{ext}$ which correspond to Py and Co resonance fields for the microwave frequency of $f_{RF}$=9.35 GHz. Insets in Fig. \[Figure 1\] show the evolution of the FMRFM spectra as a function of lateral position. The signal, reminiscent of those reported earlier in [@Nazaretski; @2007], is comprised of two distinctive contributions. The first, a negative signal which occurs at lower values of $H_{ext}$ is a localized resonance originating from the region of the sample right under the cantilever tip where the probe field is strong and positive. The second contribution is positive and is observed at higher values of $H_{ext}$. This signal arises from a larger region of sample remote from the tip which, therefore, experience a weak negative tip field; we will label this the “uniform resonance”. As seen in Fig. \[Figure 1\], at the beginning of the lateral scan the negative (lower field) resonance structure is present only in the Co spectrum (see inset a)). Near a lateral position of 9 $\mu$m we see no localized signals (with negatively shifted $H_{ext}$) for either the Py or the Co signals. However upon scanning further over the Py film, the Py resonance begins to show a localized signal, while the Co signal continues to show only a uniform (positively shifted $H_{ext}$) signal (inset b)).
We analyze the uniform contribution to the FMRFM signals considering the case when the entire dynamic magnetization $m$ is constant and the resonance field is only weakly affected by the probe. This approximation is valid for the large probe-sample distances (insets b) and c) in Fig. \[Figure 1\]). The frequency of the uniform resonance in a thin film can be written as $\omega_{RF}/\gamma=H_{ext}-4\pi
M_s$, where $4\pi M_s$ is the saturation magnetization and $\gamma$ is the gyromagnetic ratio. FMRFM spectra shown in b) and c) insets in Fig. \[Figure 1\] yield the values of $4\pi M_s$ = 8052 G for Py and $4\pi M_s$ = 15013 G for Co respectively.
For quantitative analysis of the FMRFM data it is important to have an accurate estimate of the probe-sample separation. Magnetic Force Microscopy (MFM) measurements were used to calibrate the probe-sample separation. The cantilever was scanned across the Py - Co interface and changes in its resonance frequency were recorded. The gradient of the MFM force for a semi-infinite film can be written as follows: $$\frac{\partial F}{\partial
z}=4m_pM_sL\frac{x(x^2-3z^2)}{(x^2+z^2)^3},
\label{Eq:MFM_force_gardient}$$ where $m_p$ = 7$\times$10$^{-9}$ emu is the probe magnetic moment [@Nazaretski; @2008] and $L$ is the film thickness. $z$ is the probe-film distance and $x$ is the lateral position with respect to the film edge ($x\geq 0$). MFM data were acquired at H$_{ext}$ = 18255 G, thus, both films were saturated. The MFM data and the fit to Eq. \[Eq:MFM\_force\_gardient\] are shown in Fig. \[Figure 2\]a, yielding the tip-sample separation $z$ $\approx$ 4.4 $\mu$m and the films boundaries ($x$ $\leq$ 8 $\mu$m for Co and $x$ $\geq$ 11 $\mu$m for Py).
The tip field suppresses the uniform FMR mode in the region under the tip, and according to Obukhov [*et al.*]{} [@Obukhov:film; @2008] the magnitude of the suppression depends on the tip-sample separation. It is described as partial suppression at distances $z\gg\sqrt{\frac{2m_p}{\pi M_sL\alpha_0}}$ ($\alpha_0$ is the first zero of the Bessel function $J_0(\alpha_0)=0$) and full suppression at $z\ll\sqrt{\frac{2m_p}{\pi M_sL\alpha_0}}$. The region of suppressed magnetization is confined to a region of radius $r=\sqrt{2}z$. FMRFM data discussed here were taken at the boundary of these two regions, thus we consider the regime of full suppression, however we introduce the magnitude of the suppression as a fit parameter. Ferromagnetic resonance excitation generates a precessing transverse magnetization $m$, thus reducing $M_z$; the change of $M_z$ is $\delta M_z=\sqrt{M_s^2-m^2}-M_s \approx
-m^2/2M_s$. Here we modulate the amplitude of $m$ with a 100$\%$ modulation depth at the cantilever resonance frequency. The FMRFM force exerted on a cantilever is $F=-\int Lm^2/2M_s\cdot\partial
H_p/\partial z dr'$, where integration is performed over the entire film area. The total FMRFM force close the edge of the film is well approximated by $$F=-\frac{m^2}{2M_s}L\left(\frac{4xzm_p}{(x^2+z^2)^2}-\beta\int_S\theta(x')\frac{\partial H_p}{\partial z}
(x-x')dr'\right),
\label{Eq:FMRFM uniform}$$ where the first term describes the force between the probe and the semi-infinite film and the second term represents the force between the probe and the area of radius $r=\sqrt{2}z$ under the tip. The Heaviside function $\theta(x')$ represents the fact that the film is positioned at $x'$ $\geq$ 0 and the dimensionless parameter $\beta$ quantifies the degree of suppression of the uniform FMR mode. In Fig. \[Figure 2\]b we plot the experimental data extracted from Fig. \[Figure 1\] and corresponding fits using Eq. \[Eq:FMRFM uniform\]. Fig. \[Figure 2\]b demonstrates good qualitative and quantitative agreement between theory and experiment and demonstrates the validity of the model. It is important to mention that in our model we assume the dynamic magnetization $m$ to be constant throughout the film. However, $m$ may vary due to the change of the demagnetizing field e.g. $-4\pi M_s$ far from the film boundary and $-2\pi M_s$ at the film boundary. Our estimates show that $m$ changes from a constant value in the film down to zero at the film edge. The length scale of this change is $\pi M_sL/\Delta
H$ $\approx$ 1 $\mu$m ($\Delta H$ is the linewidth of the uniform resonance), small compared to the probe-sample distance thus only weakly affecting the fits shown in Fig. \[Figure 2\]b.
The spatial resolution of the uniform FMR mode shown in Fig. \[Figure 2\]b is comparable to the MFM lateral resolution depicted in Fig. \[Figure 2\]a and is determined by the probe-sample separation of z $\approx$ 4.4 $\mu$m. However, it can be further improved by tracking the intensity of the FMRFM signal at values of H$_{ext}$ lower than that of the uniform FMR mode (insets a) and d) in Fig. \[Figure 1\]). In Fig. \[Figure 2\]c we show the FMRFM force acquired at H$_{ext}$ = 17960 G for Co and H$_{ext}$ = 11150 G for Py respectively (values of H$_{ext}$ are schematically indicated with dotted lines in Fig. \[Figure 1\]). The contribution to the FMRFM signal at lower values of H$_{ext}$ originates from the localized region of the sample under the probe. As seen in Fig. \[Figure 2\]c the lateral resolution is on the order of 3 $\mu$m (10$\%$ - 90$\%$ change in localized signal intensity) and is determined by the FMR resonance linewidth and the spatial profile of the FMR mode under the tip. Further theoretical and numerical analysis is required to understand the evolution of the FMR modes excited under the probe in the presence of a strongly inhomogeneous tip field and boundaries of the sample.\
In conclusion, we have obtained local FMR spectra in justaposed ferromagnetic samples and our quantitative model for the spatial variation of the uniform mode agrees well with experimental data. We have demonstrated spectroscopic imaging of Py and Co semi-infinite films with the spatial resolution for the tip induced resonance of $\approx$ 3 $\mu$m.\
The work performed at Los Alamos was supported by the US Department of Energy, and Center for Integrated Nanotechnologies at Los Alamos and Sandia National Laboratories. The work at OSU was supported by the US Department of Energy through grant DE-FG02-03ER46054.
[99]{}
D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature [**360**]{}, 563 (1993)
D. Rugar and O. Züger and S. Hoen and C. S. Yannoni and H. M. Vieth and R. D. Kendrick, Science [**264**]{}, 1560 (1994)
Z. Zhang, M. L. Roukes, and P. C. Hammel, J. Appl. Phys. [**80**]{}, 6931 (1996)
D. Rugar, R. Budakian, H. J. Mamin, and W. Chui, Nature [**430**]{}, 329 (2004)
C. L. Degen, Q. Lin, A. Hunkeler, U. Meier, M. Tomaselli, and B. H. Meier, Phys. Rev. Lett. [**94**]{}, 207601 (2005)
C. L. Degen, M. Poggio, H. J. Mamin, C. T. Rettner, and D. Rugar, Proc. Natl. Acad. Sci. [**106**]{}, 1313 (2009)
Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys. Lett. [**68**]{}, 2005 (1996)
G. de Loubens, V.V. Naletov, O. Klein, J. Ben Youssef, F. Boust, and N. Vukadinovic, Phys. Rev. Lett. [**98**]{}, 127601 (2007)
Yu. Obukhov, D. V. Pelekhov, J. Kim, P. Banerjee, I. Martin, E. Nazaretski, R. Movshovich, S. An, T. J. Gramila, S. Batra, P. C. Hammel, Phys. Rev. Lett. [**100**]{}, 197601, (2008)
E. Nazaretski, D. V. Pelekhov, I. Martin, M. Zalalutdinov, D. Ponarin, A. Smirnov, P. C. Hammel, and R. Movshovich, Phys. Rev. B [**79**]{}, 132401 (2009)
E. Nazaretski, J. D. Thompson, M. Zalalutdinov, J. W. Baldwin, B. Houston, T. Mewes, D. V. Pelekhov, P. Wigen, P. C. Hammel, and R. Movshovich, J. Appl. Phys. [**101**]{}, 074905 (2007)
E. Nazaretski, E. A. Akhadov, I. Martin, D. V. Pelekhov, P. C. Hammel, and R. Movshovich, Appl. Phys. Lett. [**92**]{}, 214104 (2008).
E. Nazaretski, T. Mewes, D. Pelekhov, P. C. Hammel, and R. Movshovich, AIP Conf. Proc. [**850**]{}, 1641 (2006)
E. Nazaretski, D. V. Pelekhov, I. Martin, M. Zalalutdinov, J. W. Baldwin, T. Mewes, B. Houston, P. C. Hammel, and R. Movshovich, Appl. Phys. Lett. [**90**]{} 234105 (2007)
Yu. Obukhov, D. V. Pelekhov, E. Nazaretski, R. Movshovich, P. C. Hammel, Appl. Phys. Lett. [**94**]{}, 172508 (2008)
Figure Caption
Figure 1: Schematic of the Co - Py sample. The arrows mark the scan range for spectra shown in Fig. \[Figure 1\]. The SEM image on the right shows the gap between Py and Co. The SEM image on the left depicts the cantilever tip.
Figure 2: FMRFM force image as a function of $H_{ext}$ and the lateral position. We show the Co and Py forces in the upper and lower panels respectively. Insets a) - d) demonstrate the evolution of the FMRFM signal as a function of lateral position indicated on the left-hand side of each inset. Vertical dashed lines show the boundaries of the Co and Py films. The horizontal dashed-doted lines are drawn through the values of H$_{ext}$ = 18255 G for Co and H$_{ext}$ = 11296 G for Py respectively and correspond to the uniform resonance (see Fig. \[Figure 2\]b). The horizontal dotted lines at H$_{ext}$ = 17960 G and H$_{ext}$ = 11150 G for Co and Py respectively, mark the localized FMRFM signals. Experimental parameters: T = 11 K, $f_{RF}$=9.35 GHz, probe-sample distance $\approx$ 5.6 $\mu$m.
Figure 3: a) MFM data acquired at H$_{ext}$=18255 G, solid line is the fit to Eq. \[Eq:MFM\_force\_gardient\]. b) FMRFM force data for the uniform ferromagnetic resonance (FMR) modes. H$_{ext}$ = 18255 G for Co (squares) and H$_{ext}$ = 11296 G for Py (circles). Solid and dashed lines are fits of Eq. \[Eq:FMRFM uniform\] to the data. Fit parameters: $m/M_s$ = 0.0014, $\beta$ = 0.65 for Co and $m/M_s$ = 0.0028, $\beta$ = 0.5 for Py. c) FMRFM force for the localized (close to the probe) FMR mode acquired at H$_{ext}$ = 17960 G for Co (squares) and H$_{ext}$ = 11150 G for Py (circles). The lateral resolution is better than 3 $\mu$m.
![[]{data-label="sample"}](Figure1.eps){width="8.5cm"}
![[]{data-label="Figure 1"}](Figure2.eps){width="8.5cm"}
![[]{data-label="Figure 2"}](Figure3.eps){width="8.5cm"}
|
---
abstract: 'We report strongly non-reciprocal behaviour for quantum dot exciton spins coupled to nano-photonic waveguides under resonant laser excitation. A clear dependence of the transmission spectrum on the propagation direction is found for a chirally-coupled quantum dot, with spin up and spin down exciton spins coupling to the left and right propagation directions respectively. The reflection signal shows an opposite trend to the transmission, which a numerical model indicates is due to direction-selective saturation of the quantum dot. The chiral spin-photon interface we demonstrate breaks reciprocity of the system and opens the way to spin-based quantum optical components such as optical diodes and circulators in a chip-based solid-state environment.'
author:
- 'D.L. Hurst'
- 'D.M. Price'
- 'C. Bentham'
- 'M.N. Makhonin'
- 'B. Royall'
- 'E. Clarke'
- 'P. Kok'
- 'L.R. Wilson'
- 'M.S. Skolnick'
- 'A.M. Fox'
title: 'Non-Reciprocal Transmission and Reflection of a Chirally-Coupled Quantum Dot'
---
The deterministic coupling of a two-level system to a one-dimensional waveguide provides a near-ideal platform for demonstrating quantum-optical effects such as single-photon nonlinearities [@cite-key]. A key parameter for such “1-D atoms” is the $\beta$-factor, which quantifies the relative coupling to the waveguide compared to other optical modes. In the limit of $\beta\rightarrow1$ and with no decoherence, the scattering of a single photon results in its complete reflection, leading to a 100% dip in the transmission spectrum [@PhysRevA.82.063821]. Such effects have been observed in a variety of systems, notably semiconductor quantum dots (QDs) coupled to photonic crystal waveguides [@Javadi_2015; @Hallett:18] and SiV or GeV centres coupled to nanobeams [@Sipahigil847; @Bhaskar; @PhysRevApplied.8.024026], with transmission dips as large as 60% now reported [@Thyrrestrup_2017].
The recent discovery of non-reciprocal coupling between dipole emitters and nano-photonic structures [@Junge-2013; @Petersen; @Rodriguez; @Sollner; @Lodahl-review; @2040-8986-19-4-045001; @ncomms; @Kuipers] adds a new dimension to the system. These chiral effects arise from the spin-orbit interaction of light [@spin-orbit-light] and lead to directionality in the $\beta$-factor, with circular dipoles of opposite sense coupling to modes propagating in opposite directions. The result of a transmission-type experiment on a chirally-coupled emitter has to be different to the non-chiral case, as the emitter does not couple to the backward propagating mode and hence reflection is not possible. In the coherent, single-photon limit, with $\beta \rightarrow 1$, light is transmitted with 100% probability and so the transmission dip on resonance is now expected to be negligibly small.
The ideal behaviour is hard to observe in practice: the $\beta$-factor is never perfect and dephasing is always present to some extent. Moreover, the directional coupling efficiency is less than unity. In these non-ideal conditions, the behaviour is expected to lie somewhere between the limits of perfect reflection and perfect transmission for the non-chiral and chiral cases respectively. In this paper, we present experimental data on a single QD chirally coupled to a nanobeam waveguide and then use a theoretical model to describe the system. The key finding is the observation of a spin-dependent dip in the transmission, which depends strongly on the direction of propagation, thereby breaking reciprocity. We also present experimental data on directional spin-dependent reflectivity, where, unexpectedly, the more weakly coupled dipole gives the larger signal. The theoretical modelling shows that this counter-intuitive behaviour is caused by the increased saturation of the more strongly coupled QD spin at the power levels used in the experiment. The use of a semiconductor emitter fully integrated into a single-mode nanophotonic waveguide leads to a much larger overall $\beta$-factor than used in previous work on non-reciprocal transmission for cold atoms coupled to a nanofiber [@PhysRevX.5.041036], moving the system closer to the regime where the transmission dip on resonance is small.
\
\
The studies of non-reciprocal behaviour were carried out on a QD located at a chiral point (C-point) of a nanobeam waveguide, where opposite circular polarizations propagate in different directions [@ncomms]. The structure consisted of a single self-assembled InGaAs quantum dot embedded within a single-mode, suspended vacuum-clad GaAs waveguide with out-couplers at its ends for efficient photon extraction, as shown in Fig. \[fig:1a\]. (See *Methods* for further details of the sample.) The selection rules shown in Fig. \[fig:1b\] imply that opposite spin excitons couple to modes propagating in opposite directions. This applies both to emission, as shown schematically by the blue arrows in Fig. \[fig:1c\], and to resonant scattering of incoming photons, as represented by the red arrows.
QDs near C-points were identified by exciting from above the waveguide with a non-resonant laser at 808nm and collecting the photoluminescence (PL) from the left and right out-couplers [@ncomms; @Coles_2017]. The PL spectra, with a magnetic field of $B=1$ T applied out of the waveguide plane, for the QD employed in this work are shown in Fig. \[fig:1d\]. Clear evidence of directional emission is present with $\sigma^+$ light propagating predominantly to the left and $\sigma^-$ predominantly to the right, as in Fig. \[fig:1c\]. The large degree of directionality shows the strong chiral coupling for this particular QD. The unidirectional emission contrast was calculated as in [@ncomms; @Coles_2017] from the relative intensity of the Zeeman components $I^{\sigma +}$ and $I^{\sigma -}$ measured at a particular out-coupler: $$\begin{aligned}
C = \frac{I^{\sigma +} - I^{\sigma -}}{I^{\sigma +} + I^{\sigma -}} \label{eq:eq1} . \end{aligned}$$ Over 50 randomly positioned QDs were examined to find those with high spin-dependent directionality. For the QD employed for the data in Fig. \[fig:2\], the directional PL contrast ratios were $C_\text{L}=0.84$ and $C_\text{R}=-0.91$ for the left and right out-couplers, showing the strongly chiral coupling for this particular QD.
Having identified a chirally-coupled QD, the non-reciprocal behaviour in resonant transmission was probed. A tunable single-frequency laser was input to one of the out-couplers and the transmitted light detected from the opposite out-coupler. An 808 nm non-resonant repump laser was applied to stabilise the QD charge state [@Makhonin]; no resonant transmission dips were observed without the repump laser. The QD charge state was not known with certainty but this was not important as, under the applied magnetic field of $B=1$ T, both charged and neutral excitons emit circularly-polarised light that couples to chiral fields [@ncomms]. In fact, it is most likely that we observed a charged exciton, since a repump laser creates free electron-hole pairs. The use of the repump laser permits the measurement of differential transmission and reflectivity spectra, where the contribution of the resonant QD transition under study is clearly identified. (See *Methods*.)
Differential transmission spectra for L$\rightarrow$R propagation are shown in Fig. \[fig:2\]a and for the reverse case of R$\rightarrow$L propagation in Fig. \[fig:2\]b. Energies are measured as a function of detuning from the exciton transition energy at $B=0$ T. Clear transmission features from both the $\sigma^-$ and $\sigma^+$ exciton transitions are seen. However, on comparing Figs. \[fig:2\]a and \[fig:2\]b, it is apparent that the $\sigma^-$ transition is dominant for L$\rightarrow$R propagation, whereas $\sigma^+$ is dominant for R$\rightarrow$L propagation, providing clear evidence for non-reciprocal behaviour in resonant transmission. All spectra were collected with an incident laser power of 50 nW and weak saturation of the QD exciton transition is occurring at this power. We discuss the saturation in detail and its effect on the spectra when we go on to model the system but note now that the maximum transmission dip is reduced from more than 3% at lower powers to 2.5%.
The differential transmission spectra in Figs. \[fig:2\]a and \[fig:2\]b have dispersive Fano-like lineshapes which arise from the interaction of the QD with the weak Fabry-Pérot cavity formed by reflections from the out-couplers of the sample [@Javadi_2015]. The lineshape is determined by the phase difference between the QD optical response and continuum, and is highly sensitive to the wavelength of the transition and the precise position of the QD relative to the Fabry-Perot modes. This in turn depends on the propagation direction and the coupling of the incoming beam to the waveguide modes, a function of the measurement geometry. Positive signals occur when the QD resonance shifts the system to a point on the Fabry-Pérot mode with higher overall transmission, giving a larger increase in transmission than the drop caused by incoherent scattering. The contrast ratio was quantified by fitting the data to Fano line-shapes. (See *Methods*). The fits give directional contrast ratios, defined by Eq. \[eq:defC\], of $-0.86$ and $0.54$ for L$\rightarrow$R and R$\rightarrow$L propagation, respectively. On noting that L$\rightarrow$R propagation in transmission corresponds to R detection in PL, and vice versa for R$\rightarrow$L propagation, it is apparent that these contrasts correlate well with those obtained in PL, with the detailed differences likely originating from the different excitation regimes. The asymmetry in the directionality between the two propagation directions was observed previously in PL experiments [@ncomms] and is likely related to the intrinsic structural asymmetry of the QD.
[![Differential transmission and reflectivity spectra for the chirally-coupled QD at $B = 1$ T: (a) transmission change $\Delta T$, L$\rightarrow$R (left to right) propagation; (b) transmission change $\Delta T$, R$\rightarrow$L propagation; (c) reflectivity change $\Delta R$, input from left; (d) reflectivity change $\Delta R$, input from right. Differential spectra are used to isolate the resonant contribution of the QD transition. (See *Methods*.) The black solid lines show the results of Fano lineshape fits according to Eq. \[eq:defy\].[]{data-label="fig:2"}](fig2.pdf "fig:")]{}
Figures \[fig:2\]c and \[fig:2\]d present results obtained in the L$\rightarrow$L and R$\rightarrow$R reflection geometries respectively. As for the transmission, the normalized differential signal $\Delta R$ is plotted — see *Methods*, Eq. \[eq:Delta R\] — leading to the possibility of both positive and negative changes in the reflectivity. In Fig. \[fig:2\]c, the resonant laser is incident from the left grating-coupler and the signal is detected in back-scattering from the same grating. In marked contrast to the transmission experiment with the laser incident from the left (Fig. \[fig:2\]a), a stronger peak is seen in reflectivity for $\sigma^+$, with only a weak feature at $\sigma^-$. The opposite is observed when the laser is incident from the right coupler, as shown in Fig. \[fig:2\]d. The contrast ratios deduced from Fano fits to the differential reflectivity are $0.83$ and $-0.73$ respectively for L$\rightarrow$L and R$\rightarrow$R propagation. (See *Methods*.) The contrasts have opposite signs to those measured for the same direction of incidence in the transmission data.
As a control experiment, we repeated the measurements for a non-chiral system, where the QD is positioned close to the centre of the waveguide. We find that in both the transmission and reflection geometry, similar magnitude spectral features are observed for both spin states (see Supporting Information, Section S1). This provides strong evidence that the non-reciprocal effects we observe here are indeed due to chiral-coupling between the QD and waveguide.
The difference between the behaviour in transmission and reflection for the chirally-coupled QD, with opposite spins dominating in the two cases, is, at first, rather surprising; one might naively expect that the QD transition coupled most strongly to the mode would show the largest signals in both transmission and reflection. This would certainly be true for a non-chirally-coupled QD, but it is not the expected behaviour for a chirally-coupled QD, as we now discuss.
The complete system under consideration is shown schematically in Fig. \[fig:3a\]. A QD is coupled to the single optical mode of a nanobeam waveguide and driven by a resonant laser field. The laser scatters from the QD and is either transmitted through the waveguide, reflected back in the direction of the laser input or lost from the sample. The transmission of an ideal system with perfect directional coupling is 100% for both QD spin states, but the behaviour of a realistic system is more complicated, being highly sensitive to a number of key parameters that account for the effects of imperfect directional coupling, an emitter-waveguide coupling ($\beta$-factor) less than unity, dephasing, spectral wandering and blinking.
In the Supplemental Information we model a system such as that shown in Fig. \[fig:3a\] using the well-known Input-Output formalism [@PhysRevA.30.1386]. The magnitude of the transmission reduction and reflection due to the QD is then calculated given knowledge of the QD-waveguide coupling, spectral wandering, blinking and dephasing time of system. In practice, we do not have access to these values directly and, as many of them contribute to the spectrum of the QD in the same manner (spectral wandering and pure dephasing for example), they cannot be deduced from the data. Furthermore, Fig. \[fig:2\] shows highly Fano-type behaviour, which originates from reflections at the input and output couplers.
Owing to the number of free parameters, it is then not possible to perform a first-principles fitting of the theory to the experimental data. We can however use good estimates for these parameters, derived from both experimental data and the literature, to show that the observed behaviour of the system is both reasonable and expected. For instance the coupling between the QD and waveguide is deduced from simulations [@ncomms] and the QD lifetime is directly measurable. We define the quantity $\beta_{\text d}$ as the fraction of the QD emission directed into the waveguide which propagates in R$\rightarrow$L direction and use a value of $\beta_{\text d}=0.95$. This implies a PL contrast ratio of $[\beta_{\text d} - (1-\beta_{\text d})] = 0.90 $, in agreement with the results in Fig. \[fig:1d\]. A lower limit of the pure dephasing time $\tau_{\text d}>120$ps is set by the $8\,\mu$eV QD linewidth, but the actual value of $\tau_{\text d}$ is longer due to the inhomogeneous broadening caused by spectral wandering. In the model we use $\gamma_{\text d} = (\tau_{\text d})^{-1} = (800\,\text{ps})^{-1}$ as a reasonable semi-quantitative estimate for a quantum dot in a nano-photonic environment under resonant excitation [@Javadi_2015; @Sollner; @Hallett:18; @Makhonin]. The final parameters we require are estimates for the spectral wandering and blinking probability, $P_\text{dark}$. The spectral wandering is characterised by the parameter $\sigma$, the variance of the distribution, with $\sigma=4$ $\mu$eV giving a good fit to the measured 8 $\mu$eV QD linewidth. It is not possible to obtain a direct experimental estimate of $P_\text{dark}$ but previously reported values (e.g. in Refs. [@Hallett:18; @Javadi_2015]) fall within the range $0\leq P_\text{dark}\leq0.5$ and so we use $P_\text{dark}=0.25$ as a reasonable estimate. These parameters are summarised in Table \[parameters\]
Parameter Symbol Value Notes
----------------------------- ------------------- ----------- ---------------------------------------------------------------------
$ \beta$-factor $\beta$ 0.7 calculated in ref. [@ncomms]
Directionality $\beta_{\text d}$ 0.95 deduced from Fig. \[fig:1d\]
Radiative lifetime $\tau$ 1 ns 0.95 ns measured
Dephasing time $\tau_\text{d} $ 0.8 ns comparable to refs [@Javadi_2015; @Sollner; @Hallett:18; @Makhonin]
Spectral wandering variance $\sigma$ 4 $\mu$eV deduced from PL linewidth
Dark probability $P_\text{dark}$ 0.25 within range of refs [@Hallett:18; @Javadi_2015]
: Parameters used in the theoretical model.[]{data-label="parameters"}
The transmission spectra, calculated using the parameters of Table \[parameters\] are shown in Fig. \[fig:3b\]. The central energy of the QD is set at , and the splitting between the low and high energy Zeeman components is (as in the experimental data in Figs \[fig:1d\] and \[fig:2\]), with the higher frequency component having the stronger coupling. The transmission dips are asymmetric, with the dip being stronger for the component preferentially coupled to the QD, in agreement with the experimental data and our intuitive understanding. The depth of the dips are close to those observed in Fig. \[fig:2\], a maximum of 4% experimentally and 5% in the model, showing that the parameters used in the model are a reasonable approximation to the real system. We furthermore note that the size of the stronger dip is strongly dependent on the input power, which indicates that the system is saturated at powers of the order of $1$ nW impinging on the QD.
\
\
The qualitative behaviour in reflection is expected to be significantly different. Consider a R$\rightarrow$L input laser, coupling with relative efficiency of $\sim$95% to the $\sigma^+$ dipole, which is in turn coupled with $\sim$5% efficiency to the L$\rightarrow$R mode. By contrast, the $\sigma^-$ dipole couples with relative efficiency of 5% to the R$\rightarrow$L mode but 95% efficiency to the L$\rightarrow$R mode. As a first-order approximation and ignoring the interference effects that dominate in symmetrically-coupled systems, the fraction of the laser reflected into the L$\rightarrow$R mode is $\sim (95\% \times5\%)^2 \approx 0.2\%$ in *both* cases. (Note that the reflected and transmitted intensities are dependent on the *square* of the $\beta$-factor [@Thyrrestrup_2017]). This intuitive result with equal reflectivity peaks is reproduced by our numerical model provided that the power input to the system remains low, as shown in Fig. \[fig:3c\]. This low-power regime is characterised by the balancing of the stronger coupling to the laser with weaker back-scatter coupling, and vice versa. At higher powers asymmetry develops, as the more strongly coupled transition saturates first.
In order to obtain a more thorough comparison of experiment and theory, we need to relate the power levels used in the model to those for the measured spectra. The powers used numerically are those *within* the waveguide—after unknown coupling losses—and this makes direct comparison difficult. We can, however, calibrate the external power relative to that within the waveguide by analysing the predicted power dependence of the stronger transmission dip and comparing with experiment. In the main part of Fig. \[fig:4\], we plot the power saturation dependence predicted by the model and show as an inset the experimentally determined power dependence. The spectra used to determine the experimental power dependence can be found in the Supplemental Information, Fig. S2.
![Theoretical power dependence of the main transmission dip on resonance for the preferentially coupled component. Inset: the experimentally measured dependence. The power employed for the resonant transmission and reflection experiments is indicated by the grey star marker. The spectra used to determine the experimental power dependence can be found in the Supplemental material.[]{data-label="fig:4"}](fig4.pdf)
At low powers, below 10 pW, the main part of Fig. \[fig:4\] (the theory, the blue curve) confirms that the magnitude of the transmission dip is independent of incident laser power: fewer than one photon is interacting with the QD within its lifetime. As the power is increased up to 10 nW, the magnitude of the dip decreases as the QD can only interact with a certain fraction of the input light. At powers above 10 nW, the QD scatters an insignificant fraction of the incident photon flux and the fully saturated regime is entered. Experimentally we see very little reduction in transmission dip between 5 and 20nW and a marked reduction in transmission dip thereafter. By comparing points with the same 30% reduction in transmission dip and cross-correlating, we are able to deduce that the power of 50 nW incident on the sample corresponds to a power of 100 pW to 1 nW within the waveguide. Having semiquantitatively calibrated the power, and returning to the theory curves of Fig. \[fig:model\], we see that in this power range (represented by the green and yellow curves), the low frequency component still dominates in transmission, but the reflectivity has developed an asymmetry, with the higher frequency component being the stronger. The model thus qualitatively predicts the asymmetry in reflection observed in Figs \[fig:2\]c and \[fig:2\]d through the different saturation powers for the two transitions.
To take a specific example; in Fig. \[fig:2\]d we observe a contrast of -0.73 between the strongly and weakly coupled transitions. With the knowledge that the power incident on the QD lies in the range 100 pW to 1 nW, we now deduce this ratio from the reflectivity predictions of Fig. \[fig:3c\]. We see that for the green curve (100 pW), the contrast is -0.11 and -0.67 is predicted for the yellow curve (1 nW). The magnitude of experimental asymmetry in reflectivity is thus reproduced semi-quantitatively by the theory, providing good evidence for its origin in the direction-dependent saturation of the QD. Furthermore, we see that the green and yellow curves of Fig. \[fig:3b\] show that the transmission dips have contrasts of 0.5 and 0.33 for the more strongly and weakly coupled components respectively. This is in good agreement with the experimental data of Fig. \[fig:2\]b, which shows a ratio of 0.43.
Finally we note that a key parameter that can be calculated is the maximum phase shift, $ \Delta \phi$, that is imparted to a single photon as it is transmitted past the QD. The value of $\Delta \phi$ is $\pi$ for an ideal system with $ \beta \rightarrow 1$, $\beta_\text{d} \rightarrow 1$, and $(\tau_\text{d})^{-1} \rightarrow 0$. Since the transmission probability of the ideal system is 100%, a scalable quantum network can be implemented using this spin-dependent phase-shift[@Chiral-networks]. In our system, $ \Delta \phi$ is calculated to be of the order of 0.4 rads if we ignore spectral wandering, which occurs on time scales longer than the emitter lifetime. The actual ‘useful’ phase shift that could be extracted from an experimental sample would, of course, be lower, owing to spectral wandering and blinking. If we moved from a simple nanobeam to a photonic crystal platform, we could expect an increase in the $\beta$-factor from $\sim0.7$ to $\sim0.9$ and this would potentially boost $ \Delta \phi$ to $>0.6$ rads. The limiting factor at this point would then be the pure dephasing time, with a 3 ns time (as opposed to the 800 ps used for the modelling) giving $ \Delta \phi \sim 2$ rads. Since the dephasing time is an intrinsic property of the QD, a more realistic way to engineer this enhancement would instead be to reduce the radiative lifetime via a Purcell enhancement.
In conclusion, we have reported non-reciprocal transmission for a QD chirally coupled to the electromagnetic field supported by a nano-photonic waveguide. The key experimental result is the observation of a spin-dependent dip in the transmission spectrum, varying with the direction of propagation. The results observed in reflection geometry are initially counter-intuitive, with the more weakly-coupled transition giving a larger signal. We have shown that this is caused by partial saturation of the more-strongly coupled transition. We also show that the modelling of a realistic QD leads to a good understanding of the experimental data and what could be expected in non-ideal conditions. Further work with narrower-linewidth QDs in charge-stabilised structures [@Hallett:18; @Thyrrestrup_2017] is expected to lead to the observation of deeper transmission dips down to $\sim 30\%$ limited by the $\beta$-factor, so that the power dependence of the reflectivity could be explored in more detail. Alternatively, the use of dots with Purcell enhancement [@Mahmoodian_2017] and higher coherence could take us closer to the regime where a single photon can be deterministically imparted with a $\pi$-phase shift on transmission.
The proof-of-principle results demonstrated in the paper have the potential to pave the way towards a spin-photon interface that would have applications in communication and quantum information technologies. For example, the use of QDs with high directionality but low $\beta$-factors could open the way to the realisation of on-chip, compact optical diodes operating at the single-photon level [@PhysRevX.5.041036], or single-photon logic devices where the spin state is switched by external laser control [@Lodahl-review], while moving to higher $\beta$ could lead to spin-based quantum networks [@Chiral-networks], where quantum information is transmitted by emitted photons in a scalable, on-chip geometry.
Methods
=======
***Sample***. The experiments were carried out on single QDs embedded in vacuum-clad single-mode waveguides. The InGaAs quantum dots were grown by the Stranski-Krastanov technique and were embedded in 140 nm thick GaAs regions, grown on top of a 1 $\mu$m thick AlGaAs sacrificial layer. Single-mode nanobeam waveguides of thickness 280 nm and height 140 nm were produced by a combination of electron-beam-lithography and wet and dry etching. Second-order Bragg-grating in/out-couplers [@Faraon; @Luxmoore] were added on both ends of the waveguides for coupling to external laser fields. A scanning electron microscope image of a typical structure is shown in Fig. \[fig:1a\]. Further details of the sample structure and fabrication may be found in Ref. [@Makhonin].
***Experimental set-up***. The measurements were made at 4 K in a confocal system with separate control of the excitation and detection spots. The spatial resolution was 1–2 $\mu$m [@Luxmoore-APL] and a Faraday-geometry magnetic field $B=1$ T was applied to split the $\sigma^+$ and $\sigma^-$ Zeeman transitions, as shown in Fig. \[fig:1b\]. This provided a convenient method to observe the interactions of a resonant laser field with well-defined spin states of the QDs within the mode-hop-free scan range of the laser.
A weak non-resonant 808 nm repump laser with power 10 nW was used to stabilize the charge state of the dot [@Makhonin]. The repump laser beam was mechanically chopped at 500 Hz, and lock-in techniques were employed to maximise the signal to noise in the detection of the resonant laser transmitted to the out-coupler [@Nguyen_2012]. The normalized differential transmission spectrum $ \Delta T$ was obtained by finding the difference between the detected intensity with and without the repump laser: $$\Delta T=\frac{(I^T_{ON}-I^T_{OFF})}{I^T_{OFF}} \, ,$$ where $I^T_{ON}$ is the transmitted signal with the re-pump laser on and $I^T_{OFF}$ is the background signal with no re-pump laser. This differential signal gives the contribution of the quantum dot transition that is resonant with the laser. The differential reflectivity $ \Delta R$ was defined equivalently: $$\label{eq:Delta R}
\Delta R=\frac{(I^R_{ON}-I^R_{OFF})}{I^T_{OFF}} \, ,$$ where the superscript $R$ indicates that the reflected signal is measured.
***Fitting***. The fitting of the transmission and reflection data was performed using Fano lineshapes described by the following equation: $$y(\omega)=y_{0}+A\frac{(q\Gamma+\omega-\omega_{0})^2}{\Gamma^2+(\omega-\omega_{0})^2},
\label{eq:defy}$$ where $y_{0}$ is a background level, $A$ is the signal amplitude, $q$ is the Fano parameter, $\Gamma$ is the line broadening, and $\omega_{0}$ is the resonant frequency. The contrast ratio for the directional differential transmission and reflectivity were then calculated from the appropriate fitted amplitudes according to: $$\label{eq:defC}
C = \frac{A^{\sigma +} - A^{\sigma -}}{A^{\sigma +} + A^{\sigma -}} \, ,$$ where $A^{\sigma +}$ and $A^{\sigma -}$ are the Fano amplitudes for the $\sigma^+$ and $\sigma^-$ Zeeman components at the out-coupler under study.
Supporting Information
======================
The Supporting Information is available free of charge on the ACS Publications website at DOI:
The Supporting Information includes: PL, resonant transmission, and resonant reflectivity data for a symmetrically coupled QD; input-output formalism and discussion of the effects of spectral wandering and blinking.
Information
===========
The authors declare no competing financial interest. DLH and DMP contributed equally to the project and the manuscript was written with contributions from all authors. This work was funded by the EPSRC (UK) Programme Grants EP/J007544/1 and EP/N031776/1. DLH is supported by an EPSRC studentship.
@ifundefined
[28]{}
Chang, D. E.; Vuleti[ć]{}, V.; Lukin, M. D. *Nature Photonics* **2014**, *8*, 685 Fan, S.; Kocabas, S. E.; Shen, J. T. *Phys. Rev. A* **2010**, *82*, 063821 Javadi, A.; S[ö]{}llner, I.; Arcari, M.; Hansen, S. L.; Midolo, L.; Mahmoodian, S.; Kir[š]{}ansk[ė]{}, G.; Pregnolato, T.; Lee, E. H.; Song, J. D.; Stobbe, S.; Lodahl, P. *Nature Communications* **2015**, *6*, 8655 Hallett, D.; Foster, A. P.; Hurst, D. L.; Royall, B.; Kok, P.; Clarke, E.; Itskevich, I. E.; Fox, A. M.; Skolnick, M. S.; Wilson, L. R. *Optica* **2018**, *5*, 644–650 , *Science* **2016**, *354*, 847–850 Bhaskar, M. K.; Sukachev, D. D.; Sipahigil, A.; Evans, R. E.; Burek, M. J.; Nguyen, C. T.; Rogers, L. J.; Siyushev, P.; Metsch, M. H.; Park, H.; Jelezko, F.; Lon čar, M.; Lukin, M. D. *Phys. Rev. Lett.* **2017**, *118*, 223603 Burek, M. J.; Meuwly, C.; Evans, R. E.; Bhaskar, M. K.; Sipahigil, A.; Meesala, S.; Machielse, B.; Sukachev, D. D.; Nguyen, C. T.; Pacheco, J. L.; Bielejec, E.; Lukin, M. D.; Lon čar, M. *Phys. Rev. Applied* **2017**, *8*, 024026 Thyrrestrup, H.; Kirsanskė, G.; Le Jeannic, H.; Pregnolato, T.; Zhai, L.; Midolo, L.; Rotenberg, N.; Javadi, A.; Schott, R.; Wieck, A. D.; Ludwig, A.; Löbl, M. C.; Söllner, I.; Warburton, R. J.; Lodahl, P. *Nano Letters* **2018**, *18*, 1801–1806 Junge, C.; O’Shea, D.; Volz, J.; Rauschenbeutel, A. *Phys. Rev. Lett.* **2013**, *110*, 213604 Petersen, J.; Volz, J.; Rauschenbeutel, A. *Science* **2014**, *346*, 67–71 Rodríguez-Fortuño, F. J.; Barber-Sanz, I.; Puerto, D.; Griol, A.; Martínez, A. *ACS Photonics* **2014**, *1*, 762–767 Söllner, I.; Mahmoodian, S.; Hansen, S. L.; Midolo, L.; Javadi, A.; Kirsanskė, G.; Pregnolato, T.; El-Ella, H.; Lee, E. H.; Song, J. D.; Stobbe, S.; Lodahl, P. *Nature Nanotechnology* **2015**, *10*, 775 Lodahl, P.; Mahmoodian, S.; Stobbe, S.; Rauschenbeutel, A.; Schneeweiss, P.; Volz, J.; Pichler, H.; Zoller, P. *Nature* **2017**, *541*, 473 Lang, B.; Oulton, R.; Beggs, D. M. *Journal of Optics* **2017**, *19*, 045001 Coles, R. J.; Price, D. M.; Dixon, J. E.; Royall, B.; Clarke, E.; Kok, P.; Skolnick, M. S.; Fox, A. M.; Makhonin, M. N. *Nature Communications* **2016**, *7*, 11183 le Ferber, B.; Rotenberg, N.; Kuipers, L. *Nature Communications* **2015**, *6*, 6695 Bliokh, K. Y.; Rodríguez-Fortuño, F. J.; Nori, F.; Zayats, A. V. *Nature Photonics* **2015**, *9*, 796 Sayrin, C.; Junge, C.; Mitsch, R.; Albrecht, B.; O’Shea, D.; Schneeweiss, P.; Volz, J.; Rauschenbeutel, A. *Phys. Rev. X* **2015**, *5*, 041036 Coles, R. J.; Price, D. M.; Royall, B.; Clarke, E.; Skolnick, M. S.; Fox, A. M.; Makhonin, M. N. *Phys. Rev. B* **2017**, *95*, 121401 Makhonin, M. N.; Dixon, J. E.; Coles, R. J.; Royall, B.; Luxmoore, I. J.; Clarke, E.; Hugues, M.; Skolnick, M. S.; Fox, A. M. *Nano Letters* **2014**, *14*, 6997–7002 Collett, M. J.; Gardiner, C. W. *Phys. Rev. A* **1984**, *30*, 1386–1391 Mahmoodian, S.; Lodahl, P.; Sørensen, A. S. *Phys. Rev. Lett.* **2016**, *117*, 240501 Mahmoodian, S.; Prindal-Nielsen, K.; Söllner, I.; Stobbe, S.; Lodahl, P. *Opt. Mater. Express* **2017**, *7*, 43–51 Faraon, A.; Fushman, I.; Englund, D.; Stoltz, N.; Petroff, P.; Vučković, J. *Opt. Express* **2008**, *16*, 12154–12162 Luxmoore, I. J.; Wasley, N. A.; Ramsay, A. J.; Thijssen, A. C. T.; Oulton, R.; Hugues, M.; Kasture, S.; Achanta, V. G.; Fox, A. M.; Skolnick, M. S. *Phys. Rev. Lett.* **2013**, *110*, 037402 Luxmoore, I. J.; Wasley, N. A.; Ramsay, A. J.; Thijssen, A. C. T.; Oulton, R.; Hugues, M.; Fox, A. M.; Skolnick, M. S. *Applied Physics Letters* **2013**, *103*, 241102 Nguyen, H. S.; Sallen, G.; Voisin, C.; Roussignol, P.; Diederichs, C.; Cassabois, G. *Phys. Rev. Lett.* **2012**, *108*, 057401
|
---
abstract: |
We model a vortex system in a sample with bulk pinning and superficial pinning generated by a magnetic decoration. We perform a sequence of finite temperature numerical experiments in which external forces are applied to obtain a dynamically ordered vortex lattice. We analyze the final structures and the behavior of the total energy of the system.
PACS numbers: 74.60.Ge, 74.80.-g, 74.60.Jg
address: 'Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Río Negro, Argentina'
author:
- 'M. F. Laguna, P. S. Cornaglia, and C. A. Balseiro'
title: Simulations of Dynamical Ordering in Pinned Vortex Systems
---
The phase diagram of vortices in high temperature superconductors in the presence of pinning potentials has a rich variety of phases and transitions between them[@Nori; @Laguna]. Several techniques have been implemented to artificially create pinning centers[@tecnicas]. In particular, the structure of the vortex system in the presence of a superficial pinning potential has been studied by means of Bitter decorations[@Fasano]. Experiments show that there is no observable change in the critical current (the minimal current needed to depin the vortex lattice) after the decoration is performed. This indicates that the bulk pinning dominates the vortex dynamics. In a previous work we show that, for a set of parameters that are consistent with the experimentally observed structures, the depinning force is dominated by the bulk pinning[@Laguna2]. Here we present additional results in which a sequence of numerical experiments have been performed to show that: a) When the external force exceeds the depinning force, the vortex lattice becomes dynamically ordered, even with a quasi-periodical Bitter pinning. This order is similar to the one predicted for a vortex system displaced by high forces over a random potential[@Koshelev]. b) For a wide range of bulk pinning, polycrystalline structures are obtained after a gradual warming up from the dynamically ordered vortex lattice or by cooling down the system from high temperatures. c)These final structures are not reproducible due to the dense bulk pinning that randomly nucleates the solid phase. In spite of that, these low temperature structures have the same energy and melting temperature for the value of bulk pinning force used in this paper. Based on these results, we conclude that there is a numerically efficient way to treat the random bulk pinning present in the samples. This allows us to cover a wide range of magnetic fields and to study the interplay between the weak surface and bulk pinning potentials.
As in previous works, we model the vortex system as a set of point-like particles interacting with pinning potentials and an effective vortex-vortex interaction[@Laguna2]. We perform numerical simulations with two-dimensional Langevin dynamics. The dynamical equations are $\eta {\mathbf v}_{i} = {\mathbf f}_{i} = {\mathbf f}_{i}^{vv} + {\mathbf f}_{i}^{vp} + {\mathbf f}_{i}^{T}+ {\mathbf f}_{i}^{b}$, where $ {\mathbf v}_{i} = d {\mathbf r}_{i}/dt$ is the vortex velocity and $\eta$ is the Bardeen-Stephen friction. The logarithmic vortex-vortex interaction $ {\mathbf f}_{i}^{vv}$ has a cut off, the Bitter pinning ${\mathbf f}_{i}^{vp}$ is modeled as a set of attractive Gaussian wells and the effect of temperature ${\mathbf f}_{i}^{T}$ is added as a stochastic term[@Laguna]. To model the bulk pinning force ${\mathbf f}_{i}^{b}$ we compute the force acting on the [*i*]{}th vortex, which is allowed to move if the force is higher than a critical force $F_{c}$. We take $F_c$ time independent and uniform through all the sample. We simulate a magnetic decoration experiment by gradually cooling down a fixed number of vortices $N_{v}$. To simulate the double decoration experiment, we cool down the same number of vortices in the presence of a pinning potential located at the vortex positions of the first decoration. We take $N_{v}=1024$ in all the results of this paper. We simulate a dynamical ordering experiment adding to our system a homogeneous external force.
We perform a sequence of numerical experiments on a vortex system with a bulk pinning that leads to a polycrystalline low temperature structure. In our units, this is obtained with $F_{c} = 1$. In a first stage we cool down the vortex system on top of the first decoration, simulating a double decoration experiment. In a second stage, we simulate a dynamical ordering experiment at $T=0$. We calculate the average total energy $\left< E \right>$ of the vortex system in the stationary state. Note that the bulk pinning is so dense that its contribution is structure independent. In Fig. \[fig1\] (a) we plot $\left< E \right>$ as function of the external force $F_{ext}$. As a consequence of the lattice ordering, the elastic contribution and the total energy $\left< E \right>$ decrease. In the picture we do not observe an energy increase caused by de depinning of the vortices, because the Bitter pinning contribution to the energy is one order of magnitude lower than vortex-vortex interaction one. The decreasing of $\left< E \right>$ indicates a better ordered lattice. This coincides with the a reduction of the number of defects. The dynamically ordered lattice is shown in Fig. \[fig2\] (a). To study the topological order of the vortex solid, we make Voronoi cell constructions, where the gray (black) polygons corresponds to vortices with five (seven) nearest neighbors, and the white ones corresponds to vortices with coordination number equal to 6 (see Fig. \[fig2\] (b)). Before the application of the external force, the vortex system had several grains at random orientations, a defect fraction of $0.18$ and a high percentage of vortices at the Bitter pinning centers ($\sim 70 \%$). The vortex lattice after ordered has a very low defect fraction ($< 0.02$). The final state may depend on the direction of the external force due to finite size effects: The dynamically ordered lattice tend to be oriented along the force direction. However the sample is commesurated with a given orientation of the triangular lattice. When $F_{ext}$ is along the $y$ direction, the final state has typically a single grain closed to the optimum orientation. For $F_{ext}$ along the $x$ direction, the boundary condition leads to structures with either two grains (like in Fig \[fig2\]) or a single grain with crystalline axis forming a random angle with the direction of $F_{ext}$.
![Average energy of the vortex system. (a) An external force is applied at $T=0$ until the lattice becomes ordered. $\left< E \right>_0$ is the energy before the application of $F_{ext}$. (b) The temperature is slowly varied at $F_{ext}=0$, starting from the dynamically ordered structure. Arrows indicate the direction of the variation in temperature.[]{data-label="fig1"}](fig1.eps){width=".8\linewidth"}
Fig. \[fig1\] (b) shows the third stage of the experiment. The starting point is the vortex lattice dynamically ordered. We turn off the external force and increase the temperature from zero. When the temperature is low, the system remains frozen by the action of the bulk pinning. At a higher temperature, a few vortices start to move and get trapped at the Bitter pinning centers, improving the overlap with the first decoration. The deformation of the originally perfect elastic lattice gives rise to a increase in the energy $\left< E \right>$. When the temperature reaches a characteristic value $T \sim T(F_{c})$, the vortex system depins. At this temperature the energy $\left< E \right>$ coincides with $\left< E \right>_0$, the energy of the system at the initial time. If the system is cooled down from temperatures higher than $T(F_{c})$, the system energy converges to the original value $\left< E \right>_0$ (see Fig. \[fig1\] (b)). Conversely, if the system is cooled down from temperatures lower than $T(F_{c})$, the energy is keeping almost fixed, as can be observed in Fig. \[fig1\] (b) (open triangles).
In summary: [*i*]{}) When the system freezes, it has a characteristic energy $\left< E \right>_0$ corresponding to a wide set of metastable states. These states have a characteristic density of defects and overlap with the quasiperiodic Bitter pinning. [*ii*]{}) An external force can order the vortex structure decreasing its energy due to a reduction of the vortex lattice defects. When the first and second decorations are dynamically ordered, the overlap between them is not systematically improved for the range of parameters used. [*iii*]{}) The dynamically ordered lattices have a high orientational order which depends on the direction of $F_{ext}$.
We thank F. de la Cruz, Y. Fasano and M. Menghini for helpful discussions. We acknowledge financial support from CONICET, CNEA, Fundación Antorchas and ANPCyT Grant No. 99-6343.
[9]{}
C. Reichhardt [*et al.*]{}, Phys. Rev. Lett. [**78**]{}, 2648 (1997). M.F. Laguna [*et al.*]{}, Phys. Rev. B [**64**]{}, 104505 (2001). K. Harada [*et al.*]{}, Science [**274**]{}, 1167 (1996); A. Hoffmann [*et al.*]{}, Phys. Rev. B [**61**]{}, 6986 (2000); A. Soibel [*et al.*]{}, Nature [**406**]{}, 282 (2000). Y. Fasano [*et al.*]{}, Phys. Rev. B [**60**]{}, R15047 (1999); [**62**]{}, 15183 (2000). M.F. Laguna [*et al.*]{}, Phys. Rev. B [**66**]{}, 024522 (2002). A.E. Koshelev and V.M. Vinokur, Phys. Rev. Lett. [**73**]{}, 3580 (1994).
|
---
abstract: 'I explain the Sato-Lee (SL) model and its extension to the neutrino-induced pion production off the nucleon. Then I discuss applications of the SL model to incoherent and coherent pion productions in the neutrino-nucleus scattering. I mention a further extension of this approach with a dynamical coupled-channels model developed in Excited Baryon Analysis Center of JLab.'
author:
- 'Satoshi X. Nakamura'
bibliography:
- 'sample.bib'
title: 'Dynamical Model for Meson Production off Nucleon and Application to Neutrino-Nucleus Reactions'
---
[ address=[Excited Baryon Analysis Center (EBAC)\
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA]{} ]{}
Introduction
============
Neutrino oscillation experiments have been actively conducted in the last decade, and will be so in the forthcoming decade. Because those experiments detect the neutrino through the neutrino-nucleus ($\nu$-$A$) scattering, understanding of the $\nu$-$A$ scattering is a prerequisite for a successful interpretation of data. Some of the experiments measure the neutrino in the energy region of sub- and few-GeV where dominant processes are quasi-elastic nucleon knockout (QE) and single pion ($1\pi$) production via the $\Delta$-excitation. For QE, the elementary amplitude is reasonably well-known, and the challenge is to incorporate the nuclear correlation in the initial state and the final state interaction. Although there has been a reasonable success in describing QE in the electron-nucleus scattering, it was reported that the same framework does not work well for the $\nu$-$A$ scattering [@benhar1]. From here, I focus on the $1\pi$ productions in $\nu$-$A$ scattering in the $\Delta$ region, which constitute the dominant background in the neutrino oscillation experiments. In addition to the difficult problem of the nuclear effects, relevant elementary amplitudes, or dynamical models which generate them, also have to be carefully studied. Those dynamical models are developed through a careful analysis of data for electroweak $1\pi$ production off the nucleon. Actually, there have been active developments of such dynamical models, motivated by extensive experiments of photo- and electro meson-productions in the resonance region. These experiments aim to test resonance properties predicted by QCD-inspired models and Lattice QCD. A dynamical model for $1\pi$ production developed in this way provides a good starting point to study the neutrino-induced $1\pi$ production off the nucleon, because of the close relation between the weak and the electromagnetic currents. Furthermore, the dynamical model offers a good basis to study $1\pi$ production in the $\nu$-$A$ scattering.
Thus first, I give a brief description of a dynamical model, i.e., the Sato-Lee (SL) model [@sl], for the $1\pi$ photo-production off the nucleon. (For a fuller discussion, consult Ref. [@sl].) Then I discuss the extension of the SL model to the weak sector, done in Ref. [@SUL]. The elementary amplitudes generated by the SL model has been applied to pion productions in $\nu$-$A$ scattering. I describe the work done in Ref. [@sl-1pi] where the authors studied the quasi-free $\Delta$-excitation followed by the single pion production. I also discuss the coherent pion production off a nucleus studied with the SL model [@coh]. Finally, I discuss a possible future development.
Sato-Lee (SL) model
===================
In the SL model, one starts with a set of phenomenological Lagrangians, and derive an effective Hamiltonian using a unitary transformation. The effective Hamiltonian for pion photoproduction can be written as follows: $$\begin{aligned}
H_{eff} & = & H_0 + v_{\pi N} + v_{\gamma\pi}
+ \Gamma_{\pi N \leftrightarrow \Delta}
+ \Gamma_{\gamma N \leftrightarrow \Delta}, \label{hamile}\end{aligned}$$ where $H_0$ is the free Hamiltonian, $v_{\pi N}$ and $v_{\gamma\pi}$ are respectively non-resonant $\pi N\to \pi N$ and $\gamma N\to \pi N$ potentials, and are composed by the Born diagrams, $t$-channel $\rho$ and $\omega$ exchange terms, and the crossed $\Delta$ term. Bare vertices for $\pi N\leftrightarrow\Delta$ and $\gamma N\leftrightarrow\Delta$ transitions are respectively denoted by $\Gamma_{\pi N \leftrightarrow \Delta}$ and $\Gamma_{\gamma N \leftrightarrow \Delta}$. With the effective Hamiltonian, we can derive unitary pion photoproduction amplitude as $$\begin{aligned}
T_{\gamma\pi}(E) = t_{\gamma\pi}(E) +
\frac{
\bar{\Gamma}_{\Delta \rightarrow \pi N}(E)
\bar{\Gamma}_{\gamma N \rightarrow \Delta}(E)
}
{E - m_\Delta - \Sigma_\Delta(E)} , \label{tmatt}\end{aligned}$$ where the first (second) term is the nonresonant (resonant) amplitude, and $E$ is the total energy of the pion and nucleon; $m_\Delta$ is the $\Delta$ bare mass. The nonresonant amplitude is calculated by $$\begin{aligned}
t_{\gamma\pi}(E)= v_{\gamma\pi}
+ t_{\pi N}(E)G_{\pi N}(E)v_{\gamma\pi}, \label{tmatg}\end{aligned}$$ where $G_{\pi N}$ is the $\pi N$ free propagator, and $t_{\pi N}$ is obtained by solving Lippmann-Schwinger equation which includes $v_{\pi N}$. In Eq. (\[tmatt\]), the $\Delta$ vertices are dressed as $$\begin{aligned}
\bar{\Gamma}_{\gamma N \rightarrow \Delta}(E) &=&
\Gamma_{\gamma N \rightarrow \Delta} + v_{\gamma \pi} G_{\pi N}(E)
\bar{\Gamma}_{\pi N \rightarrow \Delta}(E) , \label{vertg} \\
\bar{\Gamma}_{\Delta \rightarrow \pi N}(E)
&=& [1+t_{\pi N}(E)G_{\pi N}(E)]\Gamma_{\Delta\rightarrow\pi N} \ , \label{vertp}\end{aligned}$$ so that dynamical pion cloud effect is taken into account as a consequence of the unitarity. The $\Delta$ self-energy in Eq. (\[tmatt\]) is given by $$\begin{aligned}
\Sigma_\Delta(E) =
\Gamma_{\pi N\rightarrow \Delta}
G_{\pi N}(E)\bar{\Gamma}_{\Delta \rightarrow \pi N}(E). \label{self} \end{aligned}$$ The $\pi N$ scattering amplitude is calculated similarly. Thus one first determine the strong interactions by analyzing $\pi N$ data in the $\Delta$-region. Then adjustable parameters relevant to electromagnetic interactions are fixed by analyzing $\gamma N\to \pi N$ data. With this approach, the SL model has been shown to give a reasonable description for the $\pi N$, $\gamma N\to \pi N$ [@sl] and $e N\to e'\pi N$ reactions [@sl2] in the $\Delta$-region.
1$\pi$ production in neutrino-nucleon scattering
================================================
With the SL model for pion photo-(and electro-) production discussed above, it is straightforward to extend it to the weak sector, as has been done in Ref. [@SUL]. One just need to replace the electromagnetic current with the with $V_\mu-A_\mu$ where $V_\mu$ ($A_\mu$) is the weak vector (axial-vector) current. The vector current conservation hypothesis tells us that the weak vector current is obtained from the isovector part of the electromagnetic current by the isospin rotation. Thus the remaining part is the axial-current. In Ref. [@SUL], the authors parametrized the least known axial-vector $N\Delta$ transition matrix element, more specifically form factors, as $$\begin{aligned}
g_{AN\Delta} (Q^2) =
g_{AN\Delta} (0) R(Q^2) G_A(Q^2) \ ,\end{aligned}$$ where $G_A(Q^2)=1/(1+Q^2/M_A^2)^2$ with $M_A=1.02$ GeV. The coupling $g_{AN\Delta} (0)$ is related to the nucleon axial coupling $g_A$ using the nonrelativistic constituent quark model. The remaining correction factor, $R(Q^2)=(1+aQ^2) e^{-bQ^2}$, is assumed to be the same as that used for the $\gamma N\to \Delta$ form factor, and thus is determined by analyzing the pion electroproduction data.
![\[fig:nu-tot\] Total cross sections for $\nu_\mu N \to \mu^- \pi N$. For an explanation of each curve, see the text. Data are from Ref. [@barish].](nu-tot1){width="70mm"}
![\[fig:nu-tot\] Total cross sections for $\nu_\mu N \to \mu^- \pi N$. For an explanation of each curve, see the text. Data are from Ref. [@barish].](nu-tot2){width="70mm"}
The total cross sections predicted by this model is compared with data in Fig. \[fig:nu-tot\]. It turns out that the full calculation (solid curves) shows a good consistency with the data. If we turn off the meson cloud effect, then we obtain the dotted curves, indicating the significant effect. We further turn off the contribution from the bare $N\Delta$ transition, then we obtain the dashed curve, showing the non-resonant contribution. Although the non-resonant contribution is smaller than the resonant one, it is still important to get a good agreement with data because it can interfere with the resonant amplitude.
1$\pi$ production in neutrino-nucleus scattering
================================================
The SL model discussed in the previous section can be applied to the neutrino-nucleus interaction in the $\Delta$-region, which has been conducted in Ref. [@sl-1pi] for the $^{12}$C target. A unique feature of this application is that the SL model treats resonant and non-resonant mechanism on the same footing so that the amplitude is unitary, while most previous works considered only resonant mechanisms. The challenge here is to incorporate elementary amplitudes generated by the SL model with various nuclear effects such as: the nuclear correlation effect in the initial state; the Pauli blocking on the final nucleons; the final state interactions (including pion absorption); the medium effect on the $\Delta$-propagation. The authors of Ref. [@sl-1pi] considered the initial nuclear correlation using the spectral function [@benhar], and the Pauli blocking using the Fermi gas model.
![\[fig:c12-1pi\] (Left) Nuclear effects on differential cross sections for $\nu_e + ^{12}{\rm C} \to e^- \pi X$ at $\theta=10^\circ$, normalized with the target mass number. (Right) Differential cross sections for $e^- + ^{12}{\rm C} \to e^- X$ at $E_e=1.1$ GeV with $\theta_e=37.5^\circ$. Data are from Ref. [@data]. For a description of each curve, see the text. ](c12-cc-10.eps){width="70mm"}
![\[fig:c12-1pi\] (Left) Nuclear effects on differential cross sections for $\nu_e + ^{12}{\rm C} \to e^- \pi X$ at $\theta=10^\circ$, normalized with the target mass number. (Right) Differential cross sections for $e^- + ^{12}{\rm C} \to e^- X$ at $E_e=1.1$ GeV with $\theta_e=37.5^\circ$. Data are from Ref. [@data]. For a description of each curve, see the text. ](c12em-1.1.eps){width="70mm"}
Left panel of Fig. \[fig:c12-1pi\] shows the nuclear effect on the differential cross sections for $\nu_e + ^{12}{\rm C} \to e^- \pi X$, normalized with the mass number, at the lepton scattering angle $\theta=10^\circ$ as a function of the final $\pi N$ invariant mass $W$. The dashed curve shows the differential cross sections for the neutrino-induced pion production off the free nucleon, averaged over the free proton and neutron. With the Fermi gas effect on the initial nucleon distribution, we obtain the dashed-double-dotted curve. We can see that the Fermi motion broaden the $\Delta$-peak. By considering the Pauli blocking in addition to the Fermi motion, we obtain the dash-dotted curve. The Pauli blocking reduces the forward cross sections by about 20%. Finally, the solid curve is obtained by replacing the Fermi gas model with the spectral function taken from Ref. [@benhar]. The spectral function further broadens the peak, and reduces the height of it by about 20%. For demonstrating the validity of the approach, the model prediction for $e^- + ^{12}{\rm C} \to e^- X$ is compared with data in Fig. \[fig:c12-1pi\] (right). The QE and $\Delta$ peaks reasonably reproduce data. However, the dip-region is rather underestimated, indicating the need of going beyond the impulse approximation, and/or more elaborate treatment of the nuclear correlation.
Coherent $\pi$ production
=========================
The SL model has been applied to the coherent pion production on $^{12}$C in Ref. [@coh]. The approach taken in Ref. [@coh] is to combine the elementary amplitudes from the SL model with the $\Delta$-hole model. This approach allows us not only to implement the nuclear effects such as modification of the $\Delta$-propagation and the pion absorption, but also to describe $\pi$-$A$ scattering, coherent $\pi$ photoproduction and coherent $\pi$ production in $\nu$-$A$ scattering in a unified manner. Thus, we can fix parameters relevant to the medium modification of the $\Delta$-propagation by analyzing $\pi$-$A$ (total and elastic) scattering, and then we can predict the coherent pion productions.
![\[fig:coh\] (Left) Differential cross sections for the coherent pion production off $^{12}$C. Data are from Ref. [@krusche]. (Right) The pion momentum distribution for the charged-current coherent pion production in $\nu-^{12}$C scattering. For a description of each curve, see the text. ](krusche-290){width="70mm"}
![\[fig:coh\] (Left) Differential cross sections for the coherent pion production off $^{12}$C. Data are from Ref. [@krusche]. (Right) The pion momentum distribution for the charged-current coherent pion production in $\nu-^{12}$C scattering. For a description of each curve, see the text. ](pmom.med.cc.1gev.eps){width="70mm"}
Figure \[fig:coh\] shows the nuclear effects and the predictive power of the model. The dashed curves in the left and right panels include neither the medium modification on the $\Delta$-propagation nor the final state interaction between the pion and nucleus. The shape of the curves are determined by the elementary amplitude, the nuclear form factor, and the phase-space factor. With the medium effect on the $\Delta$, we obtain the dotted curves. By further turning on the final state interaction, we obtain the solid curves. The nuclear effects are very significant, and bring the calculation into a good agreement with data for the photo-production. This is an important test of the model.
The dash-dotted curves are obtained by turning off the non-resonant amplitudes. By observing the difference between the solid and dash-dotted curves, we can see a significant contribution from the non-resonant amplitude, even in the $\Delta$-region. This is in contrast with the finding in Ref. [@amaro] that the non-resonant amplitude plays essentially no role. It is noted that Ref. [@amaro] used a tree-level elementary amplitude, while Ref. [@coh] used a unitary one from the SL model. The difference in the reaction mechanism may be responsible for differences observed in theoretical predictions of the pion momentum distributions and $E_\nu$-dependence of the total cross sections.
We can average the total cross sections using the neutrino flux from experiments. For the charged-current (CC) process, we use the flux from K2K, and obtain $6.3\times 10^{-40} {\rm cm}^2$ which is consistent with the report from K2K [@k2k], $< 7.7\times 10^{-40} {\rm cm}^2$. For the neutral-current (NC) process, we use the flux from MiniBooNE to obtain $2.8\times 10^{-40} {\rm cm}^2$ which is still consistent within the rather large error bar of the preliminary report [@raaf]: $ 7.7\pm 1.6\pm 3.6\times 10^{-40} {\rm cm}^2$. However, the CC/NC ratio is not in agreement with the recent report [@kurimoto], as no theoretical calculations are not.
Future development
==================
Having seen reasonable descriptions of pion productions in neutrino(photon, electron)-nucleus scattering with the SL model plus nuclear effects, it is highly hoped to extend this approach to higher mass resonance region. This is because a model that covers the region from the $\Delta$ to DIS is very useful for neutrino experiments. In this energy region, several hadronic channels couple, and $2\pi$ production reactions occupy quite a little portion of final states. Thus we need a dynamical model that takes care of channel-couplings, and treat the single and double meson productions on the same footing. In this context, continuous effort made at the Excited Baryon Analysis Center (EBAC) in JLab [@ebac] is quite encouraging. The EBAC has been analyzing world data of $\gamma N, \pi N\to \pi N, \pi\pi N, \eta N, KY$ reactions in the resonance region with a dynamical coupled-channels model (EBAC-DCC model), and aim to extract resonance information. The EBAC-DCC model is an extension of the SL model by extending the coupled-channels from $\pi N$ to $\pi N, \eta N, \pi\pi N(\pi\Delta, \sigma N, \rho N), K\Lambda, K\Sigma$, and also by including higher resonance states. It has been demonstrated that the EBAC-DCC model gives a reasonable description of pion- and photo-induced meson production reactions from the $\Delta$ to higher mass resonance region [@kamano]. An extension of the EBAC-DCC model to the weak sector and neutrino-nucleus reaction, as done with the SL model, seems a promising future direction.
The author would like to thank T. Sato, T.-S. H. Lee, B. Szczerbinska and K. Kubodera for their collaborations. This work is supported by the U.S. Department of Energy, Office of Nuclear Physics Division, under Contract No. DE-AC05-06OR23177 under which Jefferson Science Associates operates Jefferson Lab.
[9]{}
O. Benhar, P. Coletti and D. Meloni, *Phys. Rev. Lett.* [**105**]{}, 132301 (2010).
T. Sato and T.-S. H. Lee, *Phys. Rev. C* [**54**]{}, 2660 (1996).
T. Sato, D. Uno and T.-S. H. Lee, *Phys. Rev. C* [**67**]{}, 065201 (2003).
B. Szczerbinska, T. Sato, K. Kubodera and T.-S. H. Lee, *Phys. Lett.* [**B649**]{}, 132-138 (2007).
S. X. Nakamura, T. Sato, T.-S. H. Lee, B. Szczerbinska and K. Kubodera, *Phys. Rev. C* [**81**]{}, 035502 (2010).
T. Sato and T.-S. H. Lee, *Phys. Rev. C* [**63**]{}, 055201 (2001).
S. J. Barish et al., *Phys. Rev. D* [**19**]{}, 2521 (1979).
O. Benhar, A. Fabrocini, S. Fantoni and I. Sick, *Nucl. Phys.* [**A579**]{}, 493 (1994). R. M. Sealock, et al., *Phys. Rev. Lett.* [**62**]{}, 1350 (1989).
B. Krusche et al., *Phys. Lett.* [**B526**]{}, 287 (2002).
J.E. Amaro, E. Hernandez, J. Nieves and M. Valverde, *Phys. Rev. D* [**79**]{}, 013002 (2009).
M. Hasegawa et al. \[K2K Collaboration\], *Phys. Rev. Lett.* [**95**]{}, 252301 (2005).
J. L. Raaf, PhD thesis, University of Cincinnati, FERMILAB-THESIS-2007-20 (2005).
Y. Kurimoto et al., \[SciBooNE Collaboration\], *Phys. Rev. D* [**81**]{}, 111102(R) (2010).
http://ebac-theory.jlab.org/
H. Kamano, *Chin. Phys. C* [**33**]{}, 1077-1084 (2009).
|
---
abstract: |
In this paper we define the notion of semi-regular biorthogonal pairs what is a generalization of regular biorthogonal pairs in Ref. [@hiroshi1] and show that if $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. This result improves the results of Ref. [@h-t; @hiroshi1; @h-t2] in the regular case.\
---
[**Semi-regular biorthogonal pairs and generalized Riesz bases**]{}
\
Hiroshi Inoue\
\
Introduction
============
Let ${\cal H}$ be a Hilbert space with inner product $( \cdot | \cdot )$, $\bm{e}= \{ e_{n} \}$ an ONB in ${\cal H}$ and $\{ \phi_{n} \}$ a sequence in ${\cal H}$. In Ref. [@hiroshi1], the author has defined an operator $T_{\bm{e}}$ on $D_{\bm{e}} \equiv Span \{ e_{n} \}$ by $$\begin{aligned}
T_{\bm{e}} \left( \sum_{k=0}^{n} \alpha_{k}e_{k} \right)
= \sum_{k=0}^{n} \alpha_{k} \phi_{k} . \nonumber\end{aligned}$$ By using this operator $T_{\bm{e}}$, the author has investigated the relationship between a regular biorthogonal pair $( \{ \phi_{n} \} , \{ \psi_{n} \})$ and the notions of Riesz bases and semi-Riesz bases. Here, $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a pair of Riesz bases if there exists an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$ such that both $T_{\bm{e}}$ and $T_{\bm{e}}^{-1}$ are bounded, and $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a pair of semi-Riesz bases if there exists an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$ such that either $T_{\bm{e}}$ or $T_{\bm{e}}^{-1}$ are bounded. In this paper we consider the following operators in ${\cal H}$ defined by a sequence $\{ \phi_{n} \}$ in ${\cal H}$ and an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$: $$\begin{aligned}
T_{\phi, \bm{e}}
&\equiv& \sum_{k=0}^{\infty} \phi_{k} \otimes \bar{e}_{k} , \nonumber \\
T_{\bm{e},\phi}
&\equiv& \sum_{k=0}^{\infty} e_{k} \otimes \bar{\phi}_{k} , \nonumber\end{aligned}$$ where the tensor $x \otimes \bar{y}$ of elements $x, \; y$ of ${\cal H}$ is defined by $$\begin{aligned}
(x \otimes \bar{y} ) \xi = (\xi | y)x, \;\;\; \xi \in {\cal H}. \nonumber\end{aligned}$$ This is also denoted by the Dirac notation $|x >< y|$. Here we use the notation $x \otimes \bar{y}$.
In Section 2, we investigate the relationship between the operator $T_{\bm{e}}$ and the operators $T_{\phi, \bm{e}}$ and $T_{\bm{e},\phi}$. The operator $T_{\bm{e},\phi}$ is always closed, however $D(T_{\phi,\bm{e}}^{\ast})$ is not necessarily dense in ${\cal H}$, equivalently, $T_{\bm{e}}$ and $T_{\phi,\bm{e}}$ are not necessarily closable. Indeed, it is shown that the following statements are equivalent:
\(i) $T_{\bm{e}}$ is closable.
\(ii) $T_{\phi,\bm{e}}$ is closable.
\(iii) $D( T_{\bm{e},\phi} ) = D(\phi) \equiv \left\{ x \in {\cal H} ; \sum_{k=0}^{\infty} | (x | \phi_{k})|^{2} < \infty \right\} $ is dense in ${\cal H}$.\
If this holds, then $\bar{T}_{\bm{e}}=\bar{T}_{\phi,\bm{e}}=(T_{\bm{e},\phi})^{\ast}.$\
Furthermore we investigate the relationships between the notion of biorthogonal pairs and the operators $T_{\phi,\bm{e}}$, $T_{\bm{e},\phi}$. Indeed, if $D(\phi)$ is dense in ${\cal H}$, then $\bar{T}_{\phi,\bm{e}}$ has an inverse and $\bar{T}_{\phi,\bm{e}}^{-1} \subset T_{\bm{e},\psi}=(T_{\psi,\bm{e}})^{\ast}$. However, $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is not dense in ${\cal H}$ in general. And so we may give the conditions under what $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is dense in ${\cal H}$. In detail, the following statements are equivalent:
\(i) $D_{\phi} \equiv Span \{ \phi_{n} \}$ is dense in ${\cal H}$.
\(ii) $T_{\phi,\bm{e}}$ is closable and $\bar{T}_{\phi,\bm{e}}$ has a densely defined inverse.
\(iii) $T_{\phi,\bm{e}}^{\ast}(=T_{\bm{e},\phi})$ has a densely defined inverse.\
If this holds, then $T_{\bm{e},\phi}^{-1}= ( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast}$.\
In Section 3, we first investigate the relationship between semi-regular biorthogonal pairs and generalized Riesz bases. In Definition 2.1 in Ref [@h-t], the author has defined the notion of generalized Riesz bases under the assumption that $D_{\phi}$ and $D_{\psi}$ are dense in ${\cal H}$, and has shown that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. In this section, we redefine the notion of generalized Riesz bases, that is, $D_{\phi}$ and $D_{\psi}$ are not necessarily dense in ${\cal H}$ and show that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. This result improves the results of Ref. [@h-t; @hiroshi1; @h-t2]. Furthermore, we have the following results:
\(i) If $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair, then for any ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$. $\bar{T}_{\phi,\bm{e}}$ (resp. $\bar{T}_{\psi,\bm{e}}$) is the minimum among constructing operators of the generalized Riesz basis $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) and $T_{\bm{e},\psi}^{-1}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the maximum among constructing operator of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$). Furthermore, any cloesd operator $T$ (resp. $K$) satisfying $\bar{T}_{\phi,\bm{e}} \subset T \subset T_{\bm{e},\psi}^{-1}$ (resp. $\bar{T}_{\psi,\bm{e}} \subset K \subset T_{\bm{e},\phi}^{-1}$) is a constructing operator for $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).
\(ii) If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then $\bar{T}_{\phi,\bm{e}}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).
\(iii) If $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then $\bar{T}_{\psi,\bm{e}}$ (resp. $T_{\bm{e},\psi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$).\
We study the physical operators defined by the operators $T_{\phi, \bm{e}}$, $T_{\bm{e},\phi}$, $T_{\psi,\bm{e}}$ and $T_{\bm{e},\psi}$ and an ONB $\bm{e}= \{ e_{n} \}$. If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then lowering, raising and number operators $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$ and $N_{\phi,\bm{e}}$ for $\{ \phi_{n} \}$ are defined, respectively, and raising, lowering and number operators $A_{\bm{e},\phi}$, $B_{\bm{e},\phi}$ and $N_{\bm{e},\phi}$ for $\{ \psi_{n} \}$ are defined, respectively. Furthermore, if $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then lowering, raising and number operators $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$ and $N_{\psi,\bm{e}}$ for $\{ \psi_{n} \}$ are defined, respectively, and raising, lowering and number operators $A_{\bm{e},\psi}$, $B_{\bm{e},\psi}$ and $N_{\bm{e},\psi}$ for $\{ \phi_{n} \}$ are defined, respectively. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$, and its relatives. [@mostafazadeh; @bagarello11; @bagarello2013] Many researchers have investigated such operators mathematically. [@h-t; @h-t2; @hiroshi1; @b-i-t]
In Section 4, we shall show a method of constructing a semi-regular biorthogonal pair based on the following commutation rule under some assumptions. Here, the commutation rule is that a pair of operators $a$ and $b$ acting on a Hilbert space ${\cal H}$ satisfying $$\begin{aligned}
ab-ba=I. \nonumber\end{aligned}$$ The author has given assumptions to construct the regular biorthogonal pair in Ref. [@h-t2]. Indeed, the assumptions in Ref. [@h-t2] coincide with the definition of pseudo-bosons as originally given in Ref. [@bagarello10]. We shall give some assumptions to construct the semi-regular biorthogonal pair that connect with the definition of pseudo-bosons, and show that by using the results in Section 3 and Ref. [@h-t2], if $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then we may construct new pseudo-bosonic operators $\{ A_{\phi,\bm{e}}, B_{\phi, \bm{e}}, A_{\bm{e},\phi}, B_{\bm{e},\phi} \}$ and if $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then we may construct a new pseudo-bosonic operators $\{ A_{\psi,\bm{e}}, B_{\psi, \bm{e}}, A_{\bm{e},\psi}, B_{\bm{e},\psi} , \}$. Furthermore, we investigate the relationship between pseudo-bosonic operators $\{ a,b,a^{\dagger},b^{\dagger} \}$ satisfying some assumptions and the operators $\{ A_{\phi,\bm{e}}, B_{\phi, \bm{e}}, A_{\bm{e},\phi}, B_{\bm{e},\phi} \}$ and $\{ A_{\psi,\bm{e}}, B_{\psi, \bm{e}}, A_{\bm{e},\psi}, B_{\bm{e},\psi} , \}$.
This article is organized as follows. In Section 2, we define new operators $T_{\phi,\bm{e}}$ and $T_{\bm{e},\phi}$ and study the property of these operators. Furthermore, we study the relationship between the operator $T_{\bm{e}}$ and the operators $T_{\phi, \bm{e}}$ and $T_{\bm{e},\phi}$. In Section 3, we investigate the relationship between semi-regular biorthogonal pairs and generalized Riesz bases and give the physical operators defined by the operators $T_{\phi, \bm{e}}$, $T_{\bm{e},\phi}$, $T_{\psi,\bm{e}}$ and $T_{\bm{e},\psi}$ and an ONB $\bm{e}= \{ e_{n} \}$. In Section 4, we introduce a method of constructing a semi-regular biorthogonal pair based on the pseudo-bosonic operators $\{ a,b,a^{\dagger},b^{\dagger} \}$ under some assumptions and we investigate the relationship between pseudo-bosonic operators satisfying some assumptions and the physical operators $\{ A_{\phi,\bm{e}}, B_{\phi, \bm{e}}, A_{\bm{e},\phi}, B_{\bm{e},\phi} \}$ and $\{ A_{\psi,\bm{e}}, B_{\psi, \bm{e}}, A_{\bm{e},\psi}, B_{\bm{e},\psi} , \}$. In Section 5, we describe future issue with respect to biorthogonal pairs $( \{ \phi_{n} \} , \{ \psi_{n} \})$ and generalized Riesz bases.
Some operators defined by biorthogonal sequences and ONB
========================================================
Let ${\cal H}$ be a Hilbert space with inner product $( \cdot | \cdot )$. We consider the following operators in ${\cal H}$ defined by a sequence $\{ \phi_{n} \}$ in a Hilbert space ${\cal H}$ and an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$: $$\begin{aligned}
T_{\phi, \bm{e}}
&\equiv& \sum_{k=0}^{\infty} \phi_{k} \otimes \bar{e}_{k} , \nonumber \\
T_{\bm{e},\phi}
&\equiv& \sum_{k=0}^{\infty} e_{k} \otimes \bar{\phi}_{k} . \nonumber\end{aligned}$$ In Ref. [@hiroshi1], the author have defined an operator $T_{\bm{e}}$ on $D_{\bm{e}} \equiv Span \{ e_{n} \}$ by $$\begin{aligned}
T_{\bm{e}} \left( \sum_{k=0}^{n} \alpha_{k} e_{k} \right)
= \sum_{k=0}^{n} \alpha_{k} \phi_{k}. \nonumber\end{aligned}$$ For the operators $T_{\phi,\bm{e}}$, $T_{\bm{e},\phi}$ and $T_{\bm{e}}$ we have the following\
[*Lemma 2.1.*]{}
*The following statements hold.*
\(1) $T_{\phi, \bm{e}}$ is a densely defined linear operator in ${\cal H}$ such that $$\begin{aligned}
T_{\phi,\bm{e}} \supset T_{\bm{e}} \;\;\; {\rm and} \;\;\; T_{\phi, \bm{e}} e_{n} =\phi_{n}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$
\(2) $$\begin{aligned}
D( T_{\bm{e},\phi} ) = D(\phi) \equiv \left\{ x \in {\cal H} ; \sum_{k=0}^{\infty} | (x | \phi_{k})|^{2} < \infty \right\}
\;\;\; {\rm and} \;\;\; T_{\bm{e}}^{\ast}=T_{\phi, \bm{e}}^{\ast} =T_{\bm{e}, \phi}. \nonumber\end{aligned}$$
[*Proof.*]{} The statements (1) and (2) are easily proved by the definitions of $T_{\phi, \bm{e}}$, $T_{\bm{e}, \phi}$ and $T_{\bm{e}}$.\
\
By Lemma 2.1, (2), $T_{\bm{e},\phi}$ is closed. However $D(T_{\phi,\bm{e}}^{\ast})$ is not necessarily dense in ${\cal H}$, equivalently, $T_{\bm{e}}$ and $T_{\phi,\bm{e}}$ are not necessarily closable. Thus we investigate the conditions under what $T_{\phi,\bm{e}}$ is closable.\
[*Lemma 2.2.*]{}
*The following statements are equivalent:*
\(i) $T_{\bm{e}}$ is closable.
\(ii) $T_{\phi,\bm{e}}$ is closable.
\(iii) $D(\phi)$ is dense in ${\cal H}$.\
If this holds, then $$\begin{aligned}
\bar{T}_{\bm{e}}=\bar{T}_{\phi,\bm{e}}=(T_{\bm{e},\phi})^{\ast}. \nonumber\end{aligned}$$
[*Proof.*]{} This follows from Lemma 2.1, (2).\
\
Next we study the relationships between the notion of biorthogonal pairs and the operators $T_{\phi,\bm{e}}$, $T_{\bm{e},\phi}$. Then we have the following statements.\
[*Lemma 2.3.*]{} [*Suppose that $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ is a biorthogonal pair such that $D(\phi)$ is dense in ${\cal H}$, then $\bar{T}_{\phi,\bm{e}}$ has an inverse and $\bar{T}_{\phi,\bm{e}}^{-1} \subset T_{\bm{e},\psi}=(T_{\psi,\bm{e}})^{\ast}$.*]{}\
\
[*Proof.*]{} By the definitions of $T_{\phi,\bm{e}}$ and $T_{\bm{e},\psi}$, we have $$\begin{aligned}
T_{\bm{e},\psi}T_{\phi,\bm{e}} e_{n}= T_{\bm{e},\psi} \phi_{n}=e_{n}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$ Hence we have $$\begin{aligned}
T_{\bm{e}, \psi}T_{\phi,\bm{e}}=I \;\;\; {\rm on} \;\;\; D_{\bm{e}}. \nonumber\end{aligned}$$ Thus we have $$\begin{aligned}
T_{\bm{e},\psi} \bar{T}_{\phi, \bm{e}}= I. \nonumber\end{aligned}$$ This completes the proof.\
\
\
In general, $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is not necessarily dense in ${\cal H}$. We investigate the conditions under what $D(\bar{T}_{\phi,\bm{e}}^{-1})$ is dense in ${\cal H}$.\
[*Lemma 2.4.*]{}
*Suppose that $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ is a biorthogonal pair such that $D(\phi)$ is dense in ${\cal H}$. Then the following statements are equivalent:*
\(i) $D_{\phi} \equiv Span \{ \phi_{n} \}$ is dense in ${\cal H}$.
\(ii) $T_{\phi,\bm{e}}$ is closable and $\bar{T}_{\phi,\bm{e}}$ has a densely defined inverse.
\(iii) $T_{\phi,\bm{e}}^{\ast}(=T_{\bm{e},\phi})$ has a densely defined inverse.\
If this holds, then $T_{\bm{e},\phi}^{-1}= ( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast}$.\
\
[*Proof.*]{} (i)$\Rightarrow$(ii) Since $D(\phi)$ is dense in ${\cal H}$, by Lemma 2.2 and Lemma 2.3 we have $T_{\phi,\bm{e}}$ is closable and $\bar{T}_{\phi,\bm{e}}$ has an inverse. Furthermore, since $D(\bar{T}_{\phi,\bm{e}}^{-1})=\bar{T}_{\phi,\bm{e}}D(\bar{T}_{\phi,\bm{e}}) \supset D_{\phi}$ and $D_{\phi}$ is dense in ${\cal H}$, $\bar{T}_{\phi,\bm{e}}^{-1}$ is densely defined.\
(ii)$\Rightarrow$(iii) By Ref. [@hiroshi1], Lemma 2.2 and Lemma 2.3, we have $$\begin{aligned}
(\bar{T}_{\phi,\bm{e}}^{-1})^{\ast}
= (\bar{T}_{\phi,\bm{e}}^{\ast})^{-1}
=( T_{\bm{e},\phi})^{-1}. \nonumber\end{aligned}$$ Hence we have $$\begin{aligned}
D \left( ( \bar{T}_{\phi,\bm{e}}^{\ast})^{-1} \right)
= T_{\bm{e},\phi} D(T_{\bm{e},\phi}) \supset T_{\bm{e},\phi} D_{\psi} =D_{\bm{e}}. \nonumber\end{aligned}$$ Thus we have $( T_{\phi,\bm{e}}^{\ast} )^{-1}$ is densely defined.\
(iii)$\Rightarrow$(i) Take an arbitrary $x \in D_{\phi}^{\perp}$. Then, $$\begin{aligned}
0=(\phi_{n}|x) =( T_{\phi,\bm{e}} e_{n}|x)=(T_{\bm{e}} e_{n}|x), \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$ Hence, by Lemma 2.1, (2) we have $$\begin{aligned}
x \in D(T_{\bm{e}}^{\ast})=D\left( T_{\phi,\bm{e}}^{\ast} \right) \;\;\; {\rm and} \;\;\; T_{\bm{e}}^{\ast}x=T_{\phi,\bm{e}}^{\ast}x=0. \nonumber\end{aligned}$$ By (iii), it follows that $$\begin{aligned}
x= \left( T_{\phi,\bm{e}}^{\ast} \right)^{-1} T_{\phi,\bm{e}}^{\ast} x =0. \nonumber\end{aligned}$$ Thus, $D_{\phi}$ is dense in ${\cal H}$. This completes the proof.\
\
\
Similarly we have the following statements.\
[*Lemma 2.5.*]{}
*Suppose $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ is a biorthogonal pair such that $D(\psi)$ is dense in ${\cal H}$. Then the following statements are equivalent:*
\(i) $D_{\psi} \equiv Span \{ \psi_{n} \}$ is dense in ${\cal H}$.
\(ii) $T_{\psi,\bm{e}}$ is closable and $\bar{T}_{\psi,\bm{e}}$ has a densely defined inverse.
\(iii) $T_{\psi,\bm{e}}^{\ast}(=T_{\bm{e},\psi})$ has a densely defined inverse.\
If this holds, then $T_{\bm{e},\psi}^{-1}= ( \bar{T}_{\psi,\bm{e}}^{-1})^{\ast}$.\
\
[*Proof.*]{} This is shown similarly to Lemma 2.4.\
\
Semi-regular biorthogonal pairs and generalized Riesz bases
===========================================================
In Ref. [@h-t], the author has defined the notion of generalized Riesz bases. First we redefine the notion of generalized Riesz bases.\
[*Definition 3.1.*]{} [*If there exists a densely defined closed operator $T$ in ${\cal H}$ with a densely defined inverse and there exists an ONB $\bm{e}= \{ e_{n} \}$ in ${\cal H}$ such that $$\begin{aligned}
\{ e_{n} \} \subset D(T) \cap D \left( (T^{-1})^{\ast} \right) \;\;\; {\rm and} \;\;\; Te_{n} = \phi_{n}, \;\;\; n=0,1, \cdots , \nonumber\end{aligned}$$ then a sequence $\{ \phi_{n} \}$ in ${\cal H}$ is called a generalized Riesz basis with a constructing pair $( \bm{e} , T )$.*]{}\
\
Here, we delete the conditions of Definition 2.1, (ii) and (iii) in Ref. [@h-t], that is, $D_{\phi}$ and $D_{\psi}$ are not necessarily dense in ${\cal H}$. Then we have the following\
[*Lemma 3.2.*]{}
*Let $\{ \phi_{n} \}$ be a generalized Riesz basis. Then, we have the following statements.*
\(1) $T^{\ast}$ has a densely defined inverse and $(T^{\ast})^{-1}= (T^{-1})^{\ast}$.
\(2) $\psi_{n} \equiv (T^{-1})^{\ast} e_{n}$, $n=0,1, \cdots$. Then, $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are biorthogonal and $(T^{-1})^{\ast}$ is a densely defined closed operator in ${\cal H}$ with densely defined inverse $T^{\ast}$. Hence $\{ \psi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e} , (T^{-1})^{\ast} )$.
\(3) $D(\phi) \cap D(\psi)$ is dense in ${\cal H}$.\
[*Proof.*]{} (1) and (2) are easily shown.\
(3) We first show that $$D(T^{\ast}) \subset D(\phi) \;\;\; {\rm and} \;\;\; R(T)=D(T^{-1}) \subset D(\psi). \tag{2.1}$$ Indeed, this follows from $$\begin{aligned}
\sum_{k=0}^{\infty} |(x| \phi_{k})|^{2}
&=& \sum_{k=0}^{\infty} |(T^{\ast}x |e_{k})|^{2} \nonumber \\
&=& \| T^{\ast}x \|^{2}, \;\;\; x \in D(T^{\ast}) \nonumber\end{aligned}$$ and $$\begin{aligned}
\sum_{k=0}^{\infty} |( y| \psi_{k} )|^{2}
&=& \sum_{k=0}^{\infty} |(T^{-1}y |e_{k})|^{2} \nonumber \\
&=& \| T^{-1} y \|^{2}, \;\;\; y \in D(T^{-1}). \nonumber\end{aligned}$$ Since $D(T^{\ast})$ and $R(T)$ are dense in ${\cal H}$, it follows that $D(\phi)$ and $D(\psi)$ are dense in ${\cal H}$. Next we show that $D(\phi) \cap D(\psi)$ is dense in ${\cal H}$. Take an arbitrary $x \in D(T)$. Let $|T|= \int_{0}^{\infty} \lambda d E_{T}( \lambda)$ be the spectral resolution of the absolute $|T| \equiv (T^{\ast}T)^{\frac{1}{2}}$ of $T$. Then we have $TE_{T}(n) x \in D(T^{\ast}) \cap R(T)$, $n=0,1, \cdots$ and $\lim_{n \rightarrow \infty} TE_{T}(n) x= Tx$. Hence $D(T^{\ast}) \cap R(T)$ is dense in $R(T)$, and since $R(T)$ is dense in ${\cal H}$, it follows from (2.1) that $D(\phi) \cap D(\psi)$ is dense in ${\cal H}$. This completes the proof.\
\
In Ref. [@hiroshi1], we have shown that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. In order to generalize this result, we define the notion of semi-regular biorthogonal pair as follows:\
[*Definition 3.3.*]{} [*A pair $( \{ \phi_{n} \} , \{ \psi_{n} \})$ of biorthogonal sequences in ${\cal H}$ is said to be semi-regular if either $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$ or $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$.*]{}\
\
\
We give a concrete example[@bagarello13] of semi-regular and non regular biorthogonal bases. Let $\{ e_{n} \}$ be an ONB in ${\cal H}$ and put $\phi_{n}=e_{n}+e_{0}$ and $\psi_{n}=e_{n}$, $n=1,2, \cdots$. Then it is easily shown that $\{ \phi_{n} \}$ and $ \{ \psi_{n} \}$ are biorthogonal bases such that $D_{\phi}$ and $D(\phi)$ are dense in ${\cal H}$, but $D_{\psi}$ is not dense in ${\cal H}$. We show that if $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then both $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases. In detail, we have the following\
[*Theorem 3.4.*]{}
*Let $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ be biorthogonal sequences in ${\cal H}$, and let $\bm{e}= \{ e_{n} \}$ be an arbitrary ONB in ${\cal H}$. Then the following statements hold:*
\(1) Suppose that $( \{ \phi_{n} \} , \{ \psi_{n} \})$ is a regular biorthogonal pair. Then $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) is a generalized Riesz basis with constructing pairs $(\bm{e}, \bar{T}_{\phi,\bm{e}})$ and $(\bm{e}, T_{\bm{e},\psi}^{-1})$ (resp. $(\bm{e}, \bar{T}_{\psi,\bm{e}})$ and $(\bm{e}, T_{\bm{e},\phi}^{-1})$), and $\bar{T}_{\phi,\bm{e}}$ (resp. $\bar{T}_{\psi,\bm{e}}$) is the minimum among constructing operators of the generalized Riesz basis $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$), and $T_{\bm{e},\psi}^{-1}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the maximal among constructing operators of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$). Furthermore, any closed operator $T$ (resp. $K$) satisfying $\bar{T}_{\phi,\bm{e}} \subset T \subset T_{\bm{e},\psi}^{-1}$ (resp. $\bar{T}_{\psi,\bm{e}} \subset K \subset T_{\bm{e},\phi}^{-1}$) is a constructing operator for $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).
\(2) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then $ \{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) is a generalized Riesz basis with a constructing pair $(\bm{e}, \bar{T}_{\phi,\bm{e}})$ (resp. $( \bm{e} , T_{\bm{e}, \phi}^{-1} )$) and the constructing operator $\bar{T}_{\phi ,\bm{e}}$ (resp. $T_{\bm{e},\phi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$).
\(3) Suppose that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then $ \{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$) is a generalized Riesz basis with a constructing pair $(\bm{e}, \bar{T}_{\psi,\bm{e}})$ (resp. $( \bm{e} , T_{\bm{e}, \psi}^{-1} )$) and the constructing operator $\bar{T}_{\psi ,\bm{e}}$ (resp. $T_{\bm{e},\psi}^{-1}$) is the minimum (resp. the maximum) among constructing operators of $\{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$).\
\
[*Proof.*]{} Let $\bm{e}= \{ e_{n} \}$ be any ONB in ${\cal H}$.\
(1) Since $D(\phi)$ is dense in ${\cal H}$, it follows from Lemma 2.3 that $\bar{T}_{\phi, \bm{e}}$ has an inverse. Since $D_{\phi}$ is also dense in ${\cal H}$, it follows from Lemma 2.4 that the inverse $\bar{T}_{\phi,\bm{e}}^{-1}$ of $\bar{T}_{\phi,\bm{e}}$ is densely defined. Furthermore, since $\bar{T}_{\phi,\bm{e}}^{\ast} \psi_{n}=T_{\bm{e},\phi} \psi_{n}=e_{n}$, $n=0,1, \cdots $, we have $ \bm{e} \subset D \left( (\bar{T}_{\phi,\bm{e}}^{\ast})^{-1} \right) =D( T_{\bm{e}, \phi}^{-1})$. Thus $\{ \phi_{n} \}$ is a generalized Riesz basis with a constructing pair $( \bm{e}, \bar{T}_{\phi,\bm{e}} )$, and $\{ \psi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e} , T_{\bm{e}, \phi}^{-1})$. Similarly, $\{ \psi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e},\bar{T}_{\psi,\bm{e}})$, and $\{ \phi_{n} \}$ is a generalized Riesz basis with a constructing pair $(\bm{e},T_{\bm{e},\psi}^{-1})$. Hence $\{ \phi_{n} \}$ (resp. $\{ \psi_{n} \}$) is a generalized Riesz basis with constructing pairs $(\bm{e},\bar{T}_{\phi,\bm{e}})$ and $(\bm{e},T_{\bm{e},\psi}^{-1})$ (resp. $(\bm{e},\bar{T}_{\psi,\bm{e}})$ and $(\bm{e},T_{\bm{e},\phi}^{-1})$).
Take an arbitrary constructing operator $T$ of the generalized Riesz basis $\{ \phi_{n} \}$. Since $Te_{n}=\phi_{n}$ and $(T^{-1})^{\ast}e_{n}=\psi_{n}$, $n=0,1, \cdots $, we have $\bar{T}_{\phi,\bm{e}} \subset T$ and $\bar{T}_{\psi,\bm{e}} \subset (T^{-1})^{\ast}$, which implies that $T^{-1} \subset T_{\psi,\bm{e}}^{\ast}=T_{\bm{e},\psi}$. Hence, we have $T \subset T_{\bm{e},\psi}^{-1}$. Thus, $\bar{T}_{\phi,\bm{e}}$ and $T_{\bm{e},\psi}^{-1}$ are the minimum and the maximum among constructing operators of $\{ \phi_{n} \}$, respectively. Furthermore, suppose that $T$ is a closed operator in ${\cal H}$ such that $\bar{T}_{\phi,\bm{e}} \subset T \subset T_{\bm{e},\psi}^{-1}$. Then, since $D(T) \supset D_{\bm{e}}$, $TD(T) \supset T_{\phi,\bm{e}} D_{\bm{e}} = \{ \phi_{n} \}$ and $D((T^{\ast})^{-1}) \supset D(T_{\psi,\bm{e}}) \supset D_{\bm{e}}$, it follows that $T$ is a constructing operator for $\{ \phi_{n} \}$. Similar results for $\{ \psi_{n} \}$ are obtained.\
The statements (2) and (3) are shown similarly to (1). This completes the proof.\
[*Remark.*]{}
*Theorem 3.4 means the following:*
\(1) Suppose $D(\phi)$ and $D_{\phi}$ (resp. $D(\psi)$ and $D_{\psi}$) are dense in ${\cal H}$. Even if $D_{\psi}$ (resp. $D_{\phi}$) is not dense in ${\cal H}$, $\{ \psi_{n} \}$ (resp. $\{ \phi_{n} \}$) becomes a generalized Riesz basis.
\(2) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, but $D_{\psi}$ is not dense in ${\cal H}$. As shown in Theorem 3.4, $\bar{T}_{\phi,\bm{e}}$ is the minimum among constructing operators of $\{ \phi_{n} \}$, however the maximal constructing operator of $\{ \phi_{n} \}$ does not necessarily exist because $T_{\bm{e},\psi}^{-1}$ is not a constructing operator of $\{ \phi_{n} \}$ different to the case of regular biorthogonal pair. Furthermore, $T_{\bm{e},\phi}^{-1}$ is the maximum among constructing operators of $\{ \psi_{n} \}$, however the minimal constructing operator of $\{ \psi_{n} \}$ does not necessarily exist because $\bar{T}_{\psi,\bm{e}}$ is not a constructing operator of $\{ \psi_{n} \}$. Similar results for the case that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, but $D_{\phi}$ is not dense in ${\cal H}$ are obtained.
\
\
By Theorem 3.4, Ref. [@h-t] and [@h-t2], we can define the physical operators as follows:\
\(1) Suppose $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then, we put $$\begin{aligned}
A_{\phi,\bm{e}}
&=& \bar{T}_{\phi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) \bar{T}_{\phi,\bm{e}}^{-1}, \nonumber \\
B_{\phi,\bm{e}}
&=& \bar{T}_{\phi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) \bar{T}_{\phi,\bm{e}}^{-1}, \nonumber \\
N_{\phi,\bm{e}}
&=& \bar{T}_{\phi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) \bar{T}_{\phi,\bm{e}}^{-1}, \nonumber \end{aligned}$$ $$\begin{aligned}
A_{\bm{e},\phi}
&=& T_{\bm{e} , \phi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) T_{\bm{e}, \phi}, \nonumber \\
B_{\bm{e},\phi}
&=& T_{\bm{e} , \phi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \phi}, \nonumber \\
N_{\bm{e},\phi}
&=& T_{\bm{e} , \phi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \phi}. \nonumber \end{aligned}$$
\(2) Suppose $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then, we put $$\begin{aligned}
A_{\psi,\bm{e}}
&=& \bar{T}_{\psi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) \bar{T}_{\psi,\bm{e}}^{-1}, \nonumber \\
B_{\psi,\bm{e}}
&=& \bar{T}_{\psi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) \bar{T}_{\psi,\bm{e}}^{-1}, \nonumber \\
N_{\psi,\bm{e}}
&=& \bar{T}_{\psi,\bm{e}} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) \bar{T}_{\psi,\bm{e}}^{-1}, \nonumber \\
A_{\bm{e},\psi}
&=& T_{\bm{e} , \psi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right) T_{\bm{e}, \psi}, \nonumber \\
B_{\bm{e},\psi}
&=& T_{\bm{e} , \psi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \psi}, \nonumber \\
N_{\bm{e},\psi}
&=& T_{\bm{e} , \psi}^{-1} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right) T_{\bm{e}, \psi}. \nonumber \end{aligned}$$ Then we have the following\
[*Theorem 3.5.*]{}
*The following statements hold.*
\(1) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then we have $$\begin{aligned}
A_{\phi,\bm{e}} \phi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \phi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
B_{\phi,\bm{e}} \phi_{n} &=& \sqrt{n+1} \phi_{n+1} \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
N_{\phi,\bm{e}} \phi_{n}
&=& n \phi_{n} , \nonumber \\
\nonumber \\
A_{\bm{e},\phi} \psi_{n}
&=& \sqrt{n+1} \psi_{n+1} \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
B_{\bm{e},\phi} \psi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \psi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
N_{\bm{e},\phi} \psi_{n}
&=& n \psi_{n} . \nonumber\end{aligned}$$ Hence $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$ and $N_{\phi,\bm{e}}$ are lowering, raising and number operators for $\{ \phi_{n} \}$, respectively, and $A_{\bm{e},\phi}$, $B_{\bm{e},\phi}$ and $N_{\bm{e},\phi}$ are raising, lowering and number operators for $\{ \psi_{n} \}$, respectively.
\(2) Suppose that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then we have $$\begin{aligned}
A_{\psi,\bm{e}} \psi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \psi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
B_{\psi,\bm{e}} \psi_{n} &=& \sqrt{n+1} \psi_{n+1} \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
N_{\psi,\bm{e}} \psi_{n}
&=& n \psi_{n} , \nonumber \\
\nonumber \\
A_{\bm{e},\psi} \phi_{n}
&=& \sqrt{n+1} \phi_{n+1} \;\;\;\; ,n=0,1, \cdots , \nonumber \\
\nonumber \\
B_{\bm{e},\psi} \phi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \phi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
N_{\bm{e},\psi} \psi_{n}
&=& n \psi_{n} . \nonumber\end{aligned}$$ Hence $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$ and $N_{\psi,\bm{e}}$ are lowering, raising and number operators for $\{ \psi_{n} \}$, respectively, and $A_{\bm{e},\psi}$, $B_{\bm{e},\psi}$ and $N_{\bm{e},\psi}$ are raising, lowering and number operators for $\{ \phi_{n} \}$, respectively.\
[*Remark.*]{}
**
\(i) In case of (1), since $$\begin{aligned}
A_{\bm{e},\phi}
&=& \left( T_{\phi,\bm{e} }^{-1} \right)^{\ast} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k} \otimes \bar{e}_{k+1} \right)^{\ast} T_{\phi, \bm{e}}^{\ast}, \nonumber \\
B_{\bm{e},\phi}
&=& \left( T_{\phi,\bm{e} }^{-1} \right)^{\ast} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k} \right)^{\ast} T_{\phi, \bm{e}}^{\ast}, \nonumber \\
N_{\bm{e},\phi}
&=& \left( T_{\phi,\bm{e} }^{-1} \right)^{\ast} \left( \sum_{k=0}^{\infty} \sqrt{k+1} e_{k+1} \otimes \bar{e}_{k+1} \right)^{\ast} T_{\phi, \bm{e}}^{\ast}, \nonumber \end{aligned}$$ the author has denoted $A_{\bm{e}}^{\dagger}$, $B_{\bm{e}}^{\dagger}$ and $N_{\bm{e}}^{\dagger}$ in Ref. [@h-t2].
\(ii) Suppose that $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$. Then the number operators $N_{\phi,\bm{e}}$ and $N_{\bm{e},\phi}( \equiv N_{\phi,\bm{e}}^{\dagger})$ for $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$, respectively have the relation: $( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast} \bar{T}_{\phi,\bm{e}}^{-1} N_{\phi,\bm{e}}= N_{\phi,\bm{e}}^{\dagger} (\bar{T}_{\phi,\bm{e}}^{-1})^{\ast} \bar{T}_{\phi,\bm{e}}^{-1}$. This is called that $N_{\phi,\bm{e}}$ is a ${\it quasi}$-${\it Hermitian \; operator}$ [@s-k2012; @s-g-h1992; @j-d1961] and positive self-adjoint operator $( \bar{T}_{\phi,\bm{e}}^{-1})^{\ast} \bar{T}_{\phi,\bm{e}}^{-1}$ is often called a metric operator for the ${\it quasi}$-${\it Hermitian \; operator}$ $N_{\phi,\bm{e}}$. Suppose that $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$. Then the number operator $N_{\bm{e},\psi}$ is a ${\it quasi}$-${\it Hermitian \; operator}$ for the metric operator $( \bar{T}_{\psi,\bm{e}}^{-1})^{\ast} \bar{T}_{\psi,\bm{e}}^{-1}$. The results on generalized Riesz bases is related to the problem of finding metric operators for quasi-Hermitian operators.
Semi-regular biorthogonal pairs and Psuedo-bosonic operators
============================================================
In this section, we introduce a method of constructing a semi-regular biorthogonal pair based on the following commutation rule under some assumptions. Here, the commutation rule is that a pair of operators $a$ and $b$ acting on a Hilbert space ${\cal H}$ with inner product $( \cdot | \cdot )$ satisfies $$\begin{aligned}
ab-ba=I. \nonumber\end{aligned}$$ In particular, this collapses to the canonical commutation rule (CCR) if $b= a^{\dagger}$. In Ref. [@h-t2] the author has shown assumptions to construct the regular biorthogonal pair. Indeed, the assumptions in Ref. [@h-t2] coincide with the definition of pseudo-bosons as originally given in Ref. [@bagarello10], where in the recent literature many researchers have investigated. [@bagarello13; @bagarello10; @bagarello11; @mostafazadeh; @d-t]. In this section, we introduce that some assumptions to construct the semi-regular biorthogonal pair connect with the definition of pseudo-bosons. At first, we construct semi-regular biorthogonal pairs on the above commutation rule. We assume the following statements:\
[*Assumption 1.*]{}
*There exists a non-zero element $\phi_{0}$ of ${\cal H}$ such that*
\(i) $a \phi_{0}=0$,
\(ii) $\phi_{0} \in D^{\infty}(b) \equiv \cap_{k=0}^{\infty} D( b^{k})$,
\(iii) $b^{n} \phi_{0} \in D(a)$, $n=0,1, \cdots$.
\
\
Then, we may define a sequence $\{ \phi_{n} \}$ in ${\cal H}$ by $$\begin{aligned}
\phi_{n}
&\equiv& \frac{1}{\sqrt{n!}} \; b^{n} \phi_{0}, \;\;\; n =0,1, \cdots \nonumber \\
&=& \frac{1}{\sqrt{n}} \; b \phi_{n-1}, \;\;\; n =1,2, \cdots . \nonumber \end{aligned}$$ Furthermore, we have the following\
[*Proposition 4.1.*]{}
*The following statements hold.*
\(1) $b^{n} \phi_{0} \in D(a^{m})$ and $$\begin{aligned}
a^{m} b^{n} \phi_{0}
&=& \left\{
\begin{array}{cl}
_{n}P_{m} b^{n-m} \phi_{0} \;\;\;\;\;\;\;\;\;\; &,m\leq n, \\
\nonumber \\
0 \;\;\; &,m > n.
\end{array}
\right. \nonumber \end{aligned}$$
\(2) $\phi_{n} \in D(N^{m})$ and $N^{m} \phi_{n}=n^{m} \phi_{n}$, $n,m=0,1, \cdots$. In particular, $N\phi_{n}=n \phi_{n}$, $n=0,1, \cdots$.
\(3) $$\begin{aligned}
a \phi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \phi_{n-1}, \;\;\; &,n=1,2, \cdots,
\end{array}
\right. \nonumber \\
\nonumber \\
b \phi_{n} &=& \sqrt{n+1} \phi_{n+1} \;\;\;\; ,n=0,1, \cdots .\nonumber \end{aligned}$$
[*Proof.*]{} These proofs follow from Ref. [@h-t2].\
[*Assumption 2.*]{}
*There exists a non-zero element $\psi_{0}$ of ${\cal H}$ such that*
\(i) $b^{\dagger} \psi_{0}=0$,
\(ii) $\psi_{0} \in D^{\infty}(a^{\dagger}) \equiv \cap_{k=0}^{\infty} D( (a^{\dagger})^{k})$,
\(iii) $(a^{\dagger})^{n} \psi_{0} \in D(b^{\dagger})$, $n=0,1, \cdots$.
\
\
Then, we may define a sequence $\{ \psi_{n} \}$ in ${\cal H}$ by $$\begin{aligned}
\psi_{n}
&\equiv& \frac{1}{\sqrt{n!}} \; (a^{\dagger})^{n} \psi_{0}, \;\;\; n= 0,1, \cdots \nonumber \\
&=& \frac{1}{\sqrt{n}} \; a^{\dagger} \psi_{n-1}, \;\;\; n =1,2, \cdots . \nonumber \end{aligned}$$ And we put an operator $N^{\dagger} \equiv a^{\dagger} b^{\dagger}$. Furthermore we have the following\
[*Proposition 4.2.*]{}
*The following statements hold.*
\(1) $(a^{\dagger})^{n} \psi_{0} \in D((b^{\dagger})^{m})$ and $$\begin{aligned}
(b^{\dagger})^{m} (a^{\dagger})^{n} \psi_{0}
&=& \left\{
\begin{array}{cl}
_{n}P_{m} (a^{\dagger})^{n-m} \psi_{0} \;\;\;\;\;\;\;\;\;\; &,m\leq n, \\
\nonumber \\
0 \;\;\; &,m > n.
\end{array}
\right. \nonumber \end{aligned}$$
\(2) $\psi_{n} \in D((N^{\dagger})^{m})$ and $(N^{\dagger})^{m} \psi_{n}=n^{m} \psi_{n}$, $n,m=0,1, \cdots$. In particular, $N^{\dagger} \psi_{n}=n \psi_{n}$, $n=0,1, \cdots$.
\(3) $$\begin{aligned}
a^{\dagger} \psi_{n} &=& \sqrt{n+1} \psi_{n+1} \;\;\;\;\;\;\;\;\;\;\; ,n=0,1, \cdots ,\nonumber \\
\nonumber \\
b^{\dagger} \psi_{n}
&=& \left\{
\begin{array}{cl}
0 \;\;\;\;\;\;\;\;\;\; &,n=0, \\
\nonumber \\
\sqrt{n} \psi_{n-1}, \;\;\; &,n=1,2, \cdots .
\end{array}
\right. \nonumber \end{aligned}$$
\
[*Proof.*]{} These proofs follow from Ref. [@h-t2].\
\
The above Assumption 1 and Assumption 2 coincide with the assumptions of Ref. [@h-t2]. We weaken the assumption of Ref. [@h-t2] to the next assumption in order to construct semi-regular biorthogonal pairs.\
[*Assumption 3.*]{}
**
Either $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$ or $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$.
\
\
Then, if a pair of operators $a$ and $b$ acting on ${\cal H}$ satisfies Assumption 1-3, $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ becomes a semi-regular biorthogonal pair.
By Section 2, Section 3 and Ref. [@hiroshi1], in case of $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$ (resp. $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$), $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$ and $N_{\phi,\bm{e}}$ are lowering, raising and number operators for $\{ \phi_{n} \}$, respectively, and $A_{\bm{e},\phi}$, $B_{\bm{e},\phi}$ and $N_{\bm{e},\phi}$ are raising, lowering and number operators for $\{ \psi_{n} \}$, respectively. (resp. $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$ and $N_{\psi,\bm{e}}$ are lowering, raising and number operators for $\{ \psi_{n} \}$, respectively, and $A_{\bm{e},\psi}$, $B_{\bm{e},\psi}$ and $N_{\bm{e},\psi}$ are raising, lowering and number operators for $\{ \phi_{n} \}$, respectively.). And we have $$\begin{aligned}
A_{\phi,\bm{e}}B_{\phi,\bm{e}} - B_{\phi,\bm{e}}A_{\phi,\bm{e}} \subset I \;\;\; &{\rm and}& \;\;\;
B_{\bm{e},\phi} A_{\bm{e},\phi} -A_{\bm{e},\phi}B_{\bm{e},\phi}\subset I . \nonumber \\
({\rm resp.} \;\;\; A_{\psi,\bm{e}}B_{\psi,\bm{e}} - B_{\psi,\bm{e}}A_{\psi,\bm{e}} \subset I \;\;\; &{\rm and}& \;\;\;
B_{\bm{e},\psi} A_{\bm{e},\psi} -A_{\bm{e},\psi}B_{\bm{e},\psi}\subset I .) \nonumber \end{aligned}$$ Furthermore, we have the following statements with respect to the operators $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$, $A_{\bm{e},\phi}$ and $B_{\bm{e},\phi}$ (resp. $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$, $A_{\bm{e},\psi}$ and $B_{\bm{e},\psi}$). The proofs are easily shown.\
[*Proposition 4.3.*]{}
*If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then the following statements hold.*
\(1) $$\begin{aligned}
\phi_{n}&=& \frac{1}{\sqrt{n !}} B_{\phi, \bm{e}}^{n} \phi_{0}, \;\;\; n=0,1, \cdots , \nonumber \\
\psi_{n}&=& \frac{1}{\sqrt{n!}} A_{\bm{e}, \phi}^{n} \psi_{0}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$
\(2) $$\begin{aligned}
A_{\phi, \bm{e}} D_{\phi}
= D_{\phi}, && B_{\phi, \bm{e}}D_{\phi} =D_{\phi} , \nonumber \\
& {\rm and}& \nonumber \\
A_{\bm{e}, \phi} D_{\psi}
= D_{\psi}, &&
B_{\bm{e}, \phi} D_{\psi}
=D_{\psi} . \nonumber\end{aligned}$$
[*Proposition 4.4.*]{}
*If $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then the following statements hold.*
\(1) $$\begin{aligned}
\psi_{n}&=& \frac{1}{\sqrt{n !}} B_{\psi, \bm{e}}^{n} \psi_{0}, \;\;\; n=0,1, \cdots , \nonumber \\
\phi_{n}&=& \frac{1}{\sqrt{n!}} A_{\bm{e}, \psi}^{n} \phi_{0}, \;\;\; n=0,1, \cdots . \nonumber\end{aligned}$$
\(2) $$\begin{aligned}
A_{\psi, \bm{e}} D_{\psi}
= D_{\psi}, && B_{\psi, \bm{e}}D_{\psi} =D_{\psi} , \nonumber \\
& {\rm and}& \nonumber \\
A_{\bm{e}, \psi} D_{\phi}
= D_{\phi}, &&
B_{\bm{e}, \psi} D_{\phi}
=D_{\phi} . \nonumber\end{aligned}$$
Next we investigate the relationship between pseudo-bosonic operators $\{ a,b,a^{\dagger},b^{\dagger} \}$ satisfying Assumption 1-3 and the operators $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$, $A_{\bm{e},\phi}$ and $B_{\bm{e},\phi}$ ($A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$, $A_{\bm{e},\psi}$ and $B_{\bm{e},\psi}$).\
By Proposition 4.1, Proposition 4.2 and Theorem 3.5 we have the following\
[*Lemma 4.5.*]{}
*The following statements hold.*
\(1) If $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$, then $D(a) \cap D(b) \supset D_{\phi}$, $$\begin{aligned}
a \lceil_{D_{\phi}} \subset A_{\phi,\bm{e}} \;\;\; {\rm and} \;\;\; b\lceil_{D_{\phi}} \subset B_{\phi,\bm{e}}. \nonumber\end{aligned}$$
\(2) If $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$, then $D(a^{\dagger}) \cap D(b^{\dagger}) \supset D_{\psi}$, $$\begin{aligned}
a^{\dagger} \lceil_{D_{\psi}} \subset B_{\psi,\bm{e}} \;\;\; {\rm and} \;\;\; b^{\dagger} \lceil_{D_{\psi}} \subset A_{\psi,\bm{e}}. \nonumber\end{aligned}$$\
[*Proposition 4.6.*]{}
*The following statements hold.*
\(1) Suppose that $D(\phi)$ is dense in ${\cal H}$ and $D_{\phi}$ is a core for $\bar{a}$ and $\bar{b}$, then $\bar{a} \subset \bar{A}_{\phi, \bm{e}}$ and $\bar{b} \subset \bar{B}_{\phi,\bm{e}}$. In particular, if $\bar{T}_{\phi,\bm{e}}^{-1}$ is bounded, then $\bar{a} \subset A_{\phi,\bm{e}}=\bar{A}_{\phi, \bm{e}}$ and $\bar{b} \subset B_{\phi,\bm{e}}=\bar{B}_{\phi,\bm{e}}$, and if $\bar{T}_{\phi,\bm{e}}$ is bounded, then $\bar{a} = \bar{A}_{\phi,\bm{e}}$ and $\bar{b} = \bar{B}_{\phi,\bm{e}}$.
\(2) Suppose that $D(\psi)$ is dense in ${\cal H}$ and $D_{\psi}$ is a core for $\overline{a^{\dagger}}$ and $\overline{b^{\dagger}}$, then $\overline{a^{\dagger}} \subset \bar{B}_{\psi,\bm{e}}$ and $\overline{b^{\dagger}} \subset \bar{A}_{\psi,\bm{e}}$. In particular, if $\bar{T}_{\psi,\bm{e}}^{-1}$ is bounded, then $\overline{a^{\dagger}} \subset B_{\psi,\bm{e}} = \bar{B}_{\psi,\bm{e}}$ and $\overline{b^{\dagger}} \subset A_{\psi,\bm{e}}=\bar{A}_{\psi,\bm{e}}$, and if $\bar{T}_{\psi, \bm{e}}$ is bounded, then $\overline{a^{\dagger}} = \bar{B}_{\psi,\bm{e}}$ and $\overline{b^{\dagger}} = \bar{A}_{\psi,\bm{e}}$.
\
[*Proof.*]{} This is shown similarly to Proposition 2.5 in Ref. [@h-t2] by using Lemma 4.5.
Discussions
===========
As shown in Theorem 3.4, if $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is a semi-regular biorthogonal pair, then $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases, and so the physical operators (lowering, raising and number operators) are constructed. In case that $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is not a semi-regular biorthogonal pair, that is, both $D_{\phi}$ and $D_{\psi}$ are not dense in ${\cal H}$, it is meaningful to consider the following question:\
[*Question.*]{} [*Under what conditions is a biorthogonal pair $(\{ \phi_{n} \} , \{ \psi_{n} \})$ a generalized Riesz basis?* ]{}\
\
We have estimated that if a biorthogonal pair $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is a ${\cal D}$-quasi basis[@bagarello13; @bagarello2013], then $\{ \phi_{n} \}$ and $\{ \psi_{n} \}$ are generalized Riesz bases, where ${\cal D}$ is a dense subspace in ${\cal H}$ and $(\{ \phi_{n} \} , \{ \psi_{n} \})$ is a ${\cal D}$-quasi basis if $$\begin{aligned}
(f,g)
=\sum_{k=0}^{\infty} (f,\psi_{k})(\phi_{k},g)
= \sum_{k=0}^{\infty} (f,\phi_{k})(\psi_{k},g), \nonumber\end{aligned}$$ for all $f,g \in {\cal D}$
[99]{} H. Inoue and M. Takakura, Non-self-adjoint hamiltonians defined by generalized Riesz bases, e-print., arXiv:math-ph/1604.00161
H. Inoue, General theory of regular biorthogonal pairs and its physical applications, e-print., arXiv:math-ph/1604.01967
H. Inoue and M. Takakura, Regular biorthogonal pairs and Psuedo-bosonic operators, J. Math. Phys., [**57**]{}(2016), 083503
F. Bagarello, A. Inoue and C. Trapani, Non-self-adjoint hamiltonians defined by Riesz bases, J. Math. Phys., [**55**]{}(2014), 033501
F. Bagarello, From self to non self-adjoint harmonic oscillators: physical consequences and mathematical pitfalls, Phys. Rev. A., [**88**]{}(2013), 032120
F. Bagarello, More mathematics for pseudo-bosons, J. Math. Phys., [**54**]{}(2013), 063512
F. Bagarello, (Regular) pseudo-bosons versus bosons, J. Phys. A., [**44**]{}(2011), 015205
F. Bagarello, Pseudobosons, Riesz bases, and coherent states, J. Math. Phys., [**51**]{}(2010), 023531
J. Dieudonn[é]{}, Quasi-hermitian operators, Proc. Int. Symp. Lin. Spaces, (1961), 115
A. Mostafazadeh, Pseudo-Hermitian representation of Quantum Mechanics, Int. J. Geom. Methods Mod. Phys., [**7**]{}(2010), 1191-1306
P. Siegl and D. Krej[č]{}i[ř]{}[í]{}k, On the metric operator for the imaginary cubic oscillator, Phys. Rev. D, [**86**]{}(2012), 121702(R)
F.G. Scholtz, H.B. Geyer and F.J.W. Hahne, Quasi-[H]{}ermitian operators in quantum mechanics and the variational principle, Ann. Phys., [**213**]{}(2012), 74-101
D.A. Trifonov, Pseudo-boson coherent and Fock states, e-print., arXiv:quant-ph/0902.3744
\
Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan\
h-inoue@math.kyushu-u.ac.jp,\
|
---
abstract: 'In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov ([@SS]) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa ([@ENR]). In particular, we show that for any finite set $A\subset{\mathbb{R}}$ and any strictly convex or concave function $f$, $$|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}$$ and $$\max\{|A-A|,\ |f(A)+f(A)|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.$$ For the latter of these inequalities, we go on to consider the consequences for a sum-product type problem.'
address:
- 'Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK'
- 'Department of Mathematics, University Walk, Bristol, BS8 1TW, UK'
author:
- 'Liangpan Li and Oliver Roche-Newton'
bibliography:
- 'reviewbibliography.bib'
title: 'Convexity and a sum-product type estimate'
---
[Introduction]{} Given a finite set $A\subset{\mathbb{R}}$, the elements of $A$ can be labeled in ascending order, so that $$a_1<a_2<\cdots<a_n.$$ $A$ is said to be *convex*, if $$a_i-a_{i-1}<a_{i+1}-a_i,$$ for all $2\leq{i}\leq{n-1}$, and it was proved by Elekes, Nathanson and Ruzsa ([@ENR]) that $|A\pm{A}|\geq{|A|^{3/2}}$, an estimate which stood as the best known for a decade, under various guises. Schoen and Shkredov ([@SS]) recently made significant progress by proving that for any convex set $A$, $$|A-A|\gg{\frac{|A|^{8/5}}{(\log|A|)^{2/5}}},$$ and $$|A+A|\gg{\frac{|A|^{14/9}}{(\log|A|)^{2/3}}}.$$ See [@SS] and the references contained within for more details on this problem and its history.
In [@ENR], a number of other results were proved connecting convexity with large sumsets. In particular, it was shown that, for any convex or concave function $f$ and any finite set $A\subset{\mathbb{R}}$, $$\max\{|A+A|,|f(A)+f(A)|\}\gg{|A|^{5/4}}, \label{Aorf(A)}$$ and $$|A+f(A)|\gg{|A|^{5/4}}.
\label{A+f(A)}$$ By choosing particularly interesting convex or concave functions $f$, these results immediately yield interesting corollaries. For example, if we choose $f(x)=\log x$, then (\[Aorf(A)\]) immediately yields a sum-product estimate. Furthermore, if $f(x)=1/x$, then (\[A+f(A)\]) gives information about another problem posed by Erdős and Szemerédi ([@ES]) .
In this paper, the methods used by Schoen and Shkredov ([@SS]) are developed further in order to improve on some other results from [@ENR]. In particular, the bounds in (\[Aorf(A)\]) and (\[A+f(A)\]) are improved slightly, in the form of the following results.
\[theorem:main\] Let $f$ be any continuous, strictly convex or concave function on the reals, and $A,C\subset\mathbb{R}$ be any finite sets such that $|A|\approx|C|$. Then $$|f(A)+C|^6|A-A|^5\gg{\frac{|A|^{14}}{(\log|A|)^2}}.$$ In particular, choosing $C=f(A)$, this implies that $$\max\{|f(A)+f(A)|,|A-A|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.$$
\[theorem:main2\]Let $f$ be any continuous, strictly convex or concave function on the reals, and $A,C\subset\mathbb{R}$ be any finite sets such that $|A|\approx|C|$. Then $$|f(A)+C|^{10}|A+A|^9\gg{\frac{|A|^{24}}{(\log|A|)^2}}.$$ In particular, choosing $C=f(A)$, this implies that $$\max\{|f(A)+f(A)|,|A+A|\}\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}.$$
\[theorem:main3\]Let $f$ be any continuous, strictly convex or concave function on the reals, and $A\subset\mathbb{R}$ be any finite set. Then $$|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}.$$
Applications to sum-product estimates {#applications-to-sum-product-estimates .unnumbered}
-------------------------------------
By choosing $f(x)=\log x$ and applying Theorems \[theorem:main\] and \[theorem:main2\], some interesting sum-product type results can be specified, especially in the case when the productset is small. A sum-product estimate is a bound on $\max\{|A+A|,|A\cdot{A}|\}$, and it is conjectured that at least one of these sets should grow to a near maximal size. Solymosi ([@solymosi]) proved that $\max\{|A+A|,|A\cdot{A}|\}\gg\frac{|A|^{4/3}}{(\log|A|)^{1/3}}$, and this is the current best known bound. See [@solymosi] and the references contained therein for more details on this problem and its history.
In a similar spirit, one may conjecture that at least one of $|A-A|$ and $|A\cdot{A}|$ must be large, and indeed this is somewhat true. In an earlier paper of Solymosi ([@solymosi2]) on sum-product estimates, it was proved that $$\max\{|A+A|,|A\cdot{A}|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{3/11}}}.$$ It is easy to change the proof slightly in order to replace $|A+A|$ with $|A-A|$ in the above, however, in Solymosi’s subsequent paper on sum-product estimates, this was not the case. So, $\max\{|A-A|,|A\cdot{A}|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{3/11}}}$ represents the current best known bound of this type. Applying Theorem \[theorem:main\] with $f(x)=\log x$, and noting that $|f(A)+f(A)|=|A\cdot{A}|$, we get the following very marginal improvement.
$$|A\cdot{A}|^6|A-A|^5\gg{\frac{|A|^{14}}{(\log|A|)^2}}.
\label{differenceproduct}$$
In particular, this implies that $$\max\{|A\cdot{A}|, |A-A|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.$$
By applying Theorem \[theorem:main2\] in the same way, we establish that $$|A\cdot{A}|^{10}|A+A|^9\gg{\frac{|A|^{24}}{(\log|A|)^2}}.
\label{sumproduct}$$ In the case when the product set is small, then (\[differenceproduct\]) and (\[sumproduct\]) show that the sumset and difference set grow non-trivially. This was shown in [@Liconvex], and here we get a more explicit version of the same result.
[Notation and Preliminaries]{}
Throughout this paper, the symbols $\ll,\gg$ and $\approx$ are used to suppress constants. For example, $X\ll{Y}$ means that there exists some absolute constant $C$ such that $X<CY$. $X\approx{Y}$ means that $X\ll{Y}$ and $Y\ll{X}$. Also, all logarithms are to base 2.
For sets $A$ and $B$, let $E(A,B)$ be the additive energy of $A$ and $B$, defined in the usual way. So, defining $\delta_{A,B}(s)$ (and respectively $\sigma_{A,B}(s)$) to be the number of representations of an element $s$ of $A-B$ (respectively $A+B$), and $\delta_A(s)=\delta_{A,A}(s)$, we define $$E(A,B)=\sum_s\delta_A(s)\delta_B(s)=\sum_s\delta_{A,B}(s)^2=\sum_s\sigma_{A,B}(s)^2.$$ Given a set $A\subset{\mathbb{R}}$ and some $s\in{\mathbb{R}}$, define $A_s:=A\cap(A+s)$. A crucial observation to make is that $|A_s|=\delta_A(s)$. In this paper, following [@SS], the third moment energy $E_3(A)$ will also be studied, where $$E_3(A)=\sum_s\delta_{A}(s)^3.$$ In much the same way, we define $$E_{1.5}(A)=\sum_s\delta_{A}(s)^{1.5}.$$ Later on, we will need the following lemma, which was proven in [@Liconvex].
\[E1.5\] Let $A,B$ be any sets. Then $$E_{1.5}(A)^2\cdot|B|^2\leq E_3(A)^{2/3}\cdot E_3(B)^{1/3}\cdot E(A,A+B).$$
[Some consequences of the Szemerédi-Trotter theorem]{}
The main preliminary result is an upper bound on the number of high multiplicity elements of a sumset, a result which comes from an application of the Szemerédi-Trotter incidence theorem ([@ST]).
Let $\mathcal{P}$ be a set of points in the plane and $\mathcal{L}$ a set of curves such that any pair of curves intersect at most once. Then, $$|\{(p,l)\in{\mathcal{P}\times\mathcal{L}}:p\in{l}\}|\leq{4(|\mathcal{P}||\mathcal{L}|)^{2/3}+4|\mathcal{P}|+|\mathcal{L}|}.$$
**Remark**. While this paper was in the process of being drafted, a very similar result to the following lemma was included in a paper of Schoen and Shkredov ([@SSlong], Lemma 24) which was posted on the arXiv. See their paper for an alternative description of this result and proof. Note also that a weaker version of this result was also proven in [@Liconvex].
\[lemma:main\] Let $f$ be a continuous, strictly convex or concave function on the reals, and $A,B,C\subset\mathbb{R}$ be finite sets such that $|B||C|\gg{|A|^2}$. Then for all $\tau\geq{1}$, $$\label{ST1}
\big|\{x:\sigma_{f(A),C}(x)\geq\tau\}\big|\ll\frac{|A+B|^2|C|^2}{|B|\tau^3},$$ and $$\label{ST2}
\big|\{y:\sigma_{A,B}(y)\geq\tau\}\big|\ll\frac{|f(A)+C|^2|B|^2}{|C|\tau^3}.$$
Let $G(f)$ denote the graph of $f$ in the plane. For any $(\alpha,\beta)\in\mathbb{R}^2$, put $L_{\alpha,\beta}=G(f)+(\alpha,\beta)$. Define a set of points $\mathcal P=(A+B)\times(f(A)+C)$, and a set of curves $\mathcal
L=\{L_{b,c}: (b,c)\in B\times C\}$. By convexity or concavity, $|\mathcal L|=|B||C|$, and any pair of curves from $\mathcal L$ intersect at most once. Let ${\mathcal P}_{\tau}$ be the set of points of $\mathcal P$ belonging to at least $\tau$ curves from $\mathcal L$. Applying the aforementioned Szemerédi-Trotter theorem to ${\mathcal P}_{\tau}$ and $\mathcal L$, $$\tau|{\mathcal P}_{\tau}|\leq 4(|{\mathcal P}_{\tau}||B||C|)^{2/3}+4|{\mathcal P}_{\tau}|+|B||C|.$$ Now we claim for any $\tau>0$ one has $$\label{formula 41}|{\mathcal
P}_{\tau}|\ll\frac{|B|^2|C|^2}{\tau^3}.$$ The reason is as follows. Firstly, since there is no point of $\mathcal P$ belonging to at least $\min\{|B|+1,|C|+1\}$ curves from $\mathcal
L$, to prove (\[formula 41\]) we may assume that $\tau\leq\sqrt{|B||C|}$. Secondly, if $\tau< 8$, then (\[formula 41\]) holds true since $$|{\mathcal
P}_{\tau}|\leq|{\mathcal
P}|=|(A+B)\times(f(A)+C)|\leq|A|^2|B||C|\ll|B|^2|C|^2\leq64\frac{|B|^2|C|^2}{\tau^2}.$$ Finally, we may assume that $8\leq\tau\leq\sqrt{|B||C|}$. In this case we have $$\frac{\tau|{\mathcal P}_{\tau}|}{2}\leq4(|{\mathcal
P}_{\tau}||B||C|)^{2/3}+|B||C|.$$ Thus $$|{\mathcal
P}_{\tau}|\ll\max\{\frac{|B|^2|C|^2}{\tau^3},\frac{|B||C|}{\tau}\}=\frac{|B|^2|C|^2}{\tau^3}.$$ This proves the claim (\[formula 41\]).
Next, suppose $\sigma_{f(A),C}(x)\geq\tau$. There exist $\tau$ distinct elements $\{a_i\}_{i=1}^{\tau}$ from $A$, $\tau$ distinct elements $\{c_i\}_{i=1}^{\tau}$ from $C$, such that $x=f(a_i)+c_i\ (\forall i).$ Now we define $B_i\triangleq a_i+B \
(\forall i)$ and ${\mathcal
M}_x(s)\triangleq\sum_{i=1}^{\tau}\chi_{B_i}(s)$, where $\chi_{B_i}(\cdot)$ is the characteristic function of $B_i$. Since $$(a_i+b,x)=(a_i+b,f(a_i)+c_i)=\big(a_i,f(a_i)\big)+(b,c_i)\in L_{b,c_i}\ \ (\forall b, \forall i),$$ we have $(s,x)\in {\mathcal P}_{{\mathcal M}_x(s)}$. Note also $$\sum_{s\in A+B}{\mathcal M}_x(s)=\sum_{i=1}^{\tau}\sum_{s\in A+B}\chi_{B_i}(s)=\tau|B|.$$ Let $M\triangleq\frac{\tau|B|}{2|A+B|}$. Then $$\sum_{s\in A+B:\mathcal{M}_x(s)<M}\mathcal{M}_x(s)<|A+B|M=\frac{\tau|B|}{2},$$ and hence $$\sum_{s\in A+B: {\mathcal M}_x(s)\geq M }{\mathcal
M}_x(s)\geq\frac{\tau |B|}{2}.$$ Dyadically decompose this sum, so that $$\sum_{j}X_j(x)\gg{\tau|B|},
\label{dyadicdyadic}$$ where $$\begin{aligned}
X_j(x)&\triangleq\sum_{s:M2^j\leq {\mathcal
M}_x(s)<M2^{j+1}}{\mathcal M}_x(s),\\
Y_j(x)&\triangleq\big|\big\{s\in A+B:M2^j\leq {\mathcal
M}_x(s)<M2^{j+1}\big\}\big|.\end{aligned}$$ By (\[formula 41\]), $$\sum_{x:\sigma_{f(A), C}(x)\geq\tau}Y_j(x)\leq|{\mathcal P}_{M2^j}|\ll\frac{|B|^2|C|^2}{M^32^{3j}}.$$ Noting that $ X_j(x)\approx Y_j(x)M2^j$, thus $$\sum_{x:\sigma_{f(A), C}(x)\geq\tau}X_j(x)\ll\frac{|B|^2|C|^2}{M^22^{2j}},$$ which followed by first summing all $j$’s, then applying (\[dyadicdyadic\]), gives $$\tau|B|\cdot\big|\{x:\sigma_{f(A),C}(x)\geq\tau\}\big|\ll\frac{|B|^2|C|^2}{M^2}.$$ Equivalently, $$\big|\{x:\sigma_{f(A),C}(x)\geq\tau\}\big|\ll\frac{|A+B|^2|C|^2}{|B|\tau^3}.$$ This finishes the proof of (\[ST1\]).
In the same way one can prove (\[ST2\]). We only sketch the proof as follows and leave the details to the interested readers: Suppose $\sigma_{A,B}(y)\geq\tau$. There exist $\tau$ distinct elements $\{a_i\}_{i=1}^{\tau}$ from $A$, $\tau$ distinct elements $\{b_i\}_{i=1}^{\tau}$ from $B$, such that $y=a_i+b_i.$ Then we define $C_i\triangleq f(a_i)+C$ and ${\mathcal
M}_y(s)\triangleq\sum_{i=1}^{\tau}\chi_{C_i}(s)$, and as before, $(y,s)\in {\mathcal P}_{{\mathcal M}_y(s)}$. In precisely the same way as the proof of (\[ST1\]), one can prove that $$\begin{aligned}
\sum_{s\in f(A)+C: {\mathcal M}_y(s)\geq M }{\mathcal
M}_y(s)&\geq\frac{\tau |C|}{2},\\
\sum_{y:\sigma_{A,B}(y)\geq\tau}Y_j(y)\leq|{\mathcal
P}_{M2^j}|&\ll\frac{|B|^2|C|^2}{M^32^{3j}},\\
\sum_{y:\sigma_{A,B}(y)\geq\tau}X_j(y)&\ll\frac{|B|^2|C|^2}{M^22^{2j}},\\
\tau|C|\cdot\big|\{y:\sigma_{A,B}(y)\geq\tau\}\big|&\ll\frac{|B|^2|C|^2}{M^2},\\
\big|\{y:\sigma_{A,B}(y)\geq\tau\}\big|&\ll\frac{|f(A)+C|^2|B|^2}{|C|\tau^3},\end{aligned}$$ where $M\triangleq\frac{\tau|C|}{2|f(A)+C|}$, $X_j(y)\triangleq\sum_{s:M2^j\leq {\mathcal
M}_y(s)<M2^{j+1}}{\mathcal M}_y(s)$, $Y_j(y)\triangleq\big|\big\{s\in f(A)+C:M2^j\leq {\mathcal
M}_y(s)<M2^{j+1}\big\}\big|.$ This finishes the whole proof.
\[E3A\] Let $f$ be a continuous, strictly convex or concave function on the reals, and $A,C,F\subset\mathbb{R}$ be finite sets such that $|A|\approx|C|\ll|F|$. Then $$\begin{aligned}
E(A,A)&\ll E_{1.5}(A)^{2/3}|f(A)+C|^{2/3}|A|^{1/3},\label{99}\\
E(A,F)&\ll |f(A)+C||F|^{3/2}, \label{1010}\\
E_3(A)&\ll|f(A)+C|^2|A|\log|A|,\label{1111}\\
E(f(A),f(A))&\ll E_{1.5}(f(A))^{2/3}|A+C|^{2/3}|A|^{1/3},\label{1212}\\
E(f(A),F)&\ll
|A+C||F|^{3/2}, \label{1313}\\
E_3(f(A))&\ll|A+C|^2|A|\log|A|.\label{1414}\end{aligned}$$
Let $\triangle>0$ be an arbitrary real number. First decomposing $E(A)$, then applying Lemma \[lemma:main\] with $B=-A$, gives $$\begin{aligned}
E(A,A)&=\sum_{s:\delta_A(s)<{\triangle}}\delta_A(s)^2+\sum_{j=0}^{\lfloor\log|A|\rfloor}\sum_{s:2^j\triangle\leq\delta_A(s)<2^{j+1}\triangle}\delta_A(s)^2
\\&\ll{\sqrt{\triangle}\cdot E_{1.5}(A)+\sum_{j=0}^{\lfloor\log|A|\rfloor}\frac{|f(A)+C|^2|A|}{2^{3j}\triangle^{3j}}}\cdot2^{2j}\triangle^{2j}\\
&\ll \sqrt{\triangle}\cdot
E_{1.5}(A)+\frac{|f(A)+C|^2|A|}{\triangle}\ \ \ \
\big(\triangleq\Psi(\triangle)\big).\end{aligned}$$ Thus $E(A)\ll\min_{\triangle>0}\Psi(\triangle)\approx
E_{1.5}(A)^{2/3}|f(A)+C|^{2/3}|A|^{1/3}$, which proves (\[99\]). Similarly, applying Lemma \[lemma:main\] with $B=-F$, gives $$\begin{aligned}
E(A,F)&=\sum_{s:\delta_{A,F}(s)<{\triangle}}\delta_{A,F}(s)^2+\sum_{j=0}^{\lfloor\log|A|\rfloor}\sum_{s:2^j\triangle\leq\delta_{A,F}(s)<2^{j+1}\triangle}\delta_{A,F}(s)^2
\\&\ll\triangle\cdot E_{1}(A,F)+\sum_{j=0}^{\lfloor\log|A|\rfloor}\frac{|f(A)+C|^2|F|^2}{|C|2^{3j}\triangle^{3j}}\cdot2^{2j}\triangle^{2j}\\
&\ll \triangle|A||F|+\frac{|f(A)+C|^2|F|^2}{|C|\triangle}\ \ \ \
\big(\triangleq\Phi(\triangle)\big).\end{aligned}$$ Thus $E(A,F)\ll\min_{\triangle>0}\Phi(\triangle)\approx
|f(A)+C||F|^{1.5}$, which proves (\[1010\]). Once again applying Lemma \[lemma:main\] with $B=-A$, gives $$\begin{aligned}
E_3(A)&=\sum_{j=0}^{\lfloor\log|A|\rfloor}\sum_{s:2^j\leq\delta_A(s)<2^{j+1}}\delta_A(s)^3\\
&\ll\sum_{j=0}^{\lfloor\log|A|\rfloor}\frac{|f(A)+C|^2|A|}{2^{3j}\triangle^{3j}}\cdot2^{3j}\triangle^{3j}\approx|f(A)+C|^2|A|\log|A|,\end{aligned}$$ which proves (\[1111\]). (\[1212\])$\sim$(\[1414\]) can be established by the same way. This concludes the whole proof.
[Proofs of the main results]{}
[Proof of Theorem \[theorem:main\]]{} First, apply Hölder’s inequality as follows to bound $E_{1.5}(A)$ from below: $$|A|^6=\left(\sum_{s\in{A-A}}\delta_A(s)\right)^3\leq{\left(\sum_{s\in{A-A}}\delta_A(s)^{1.5}\right)^2|A-A|}=E_{1.5}(A)^2|A-A|.$$ Therefore, using the above bound and Lemma \[E1.5\] with $B=-A$ gives $$\frac{|A|^8}{|A-A|}\leq{E_{1.5}(A)^2|A|^2}\leq{E_3(A)E(A,A-A)}.$$ Finally, apply (\[1111\]), and (\[1010\]) with $F=A-A$, to conclude that $$\frac{|A|^8}{|A-A|}\ll{|f(A)+C|^3|A-A|^{3/2}|A|\log|A|},$$ and hence $$|f(A)+C|^6|A-A|^5\gg{\frac{|A|^{14}}{(\log|A|)^2}},$$ as required.
[Proof of Theorem \[theorem:main2\]]{} Using the standard Cauchy-Schwarz bound on the additive energy, and then (\[99\]), we see that $$\begin{aligned}
\frac{|A|^{12}}{|A+A|^3}&\leq{E(A,A)^3}
\\&\ll{E_{1.5}(A)^2|f(A)+C|^2|A|}
\\&=\left(\frac{|f(A)+C|^2}{|A|}\right)E_{1.5}(A)^2|A|^2.\end{aligned}$$ Next, apply Lemma \[E1.5\], with $B=A$, to get $$\frac{|A|^{12}}{|A+A|^3}\ll\left(\frac{|f(A)+C|^2}{|A|}\right)E_3(A)E(A,A+A),$$ and then apply (\[1111\]), and (\[1010\]) with $F=A+A$, to get $$\frac{|A|^{12}}{|A+A|^3}\ll{\frac{|f(A)+C|^2}{|A|}|f(A)+C|^3|A+A|^{3/2}|A|\log|A|},$$ which, after rearranging, gives $$|f(A)+C|^{10}|A+A|^9\gg{\frac{|A|^{24}}{(\log|A|)^2}}.$$
[Proof of Theorem \[theorem:main3\]]{} Observe that the Cauchy-Schwarz inequality applied twice tells us that $$\frac{|A|^{24}}{|A+f(A)|^6}\leq{E(A,f(A))^6}\leq{E(A,A)^3E(f(A),f(A))^3},$$ so that after applying (\[99\]) and (\[1212\]), with either $C=A$ or $C=f(A)$, $$\begin{aligned}
\frac{|A|^{26}}{|A+f(A)|^6}&\leq|A|^2\cdot{E_{1.5}(A)^2|A+f(A)|^2|A|\cdot
E_{1.5}(f(A))^2|A+f(A)|^2|A|}
\\&=(E_{1.5}(A)^2|f(A)|^2)\cdot(E_{1.5}(f(A))^2|A|^2)\cdot|A+f(A)|^4
\\&\leq{E_3(A)E_3(f(A))E(A,A+f(A))E(f(A),A+f(A))|A+f(A)|^4},\end{aligned}$$ where the the last inequality is a consequence of two applications of Lemma \[E1.5\]. Next apply (\[1111\]) and (\[1414\]), again with either $C=A$ or $C=f(A)$, to get $$\frac{|A|^{26}}{|A+f(A)|^6}\leq{|A+f(A)|^8|A|^2(\log|A|)^2E(A,A+f(A))E(f(A),A+f(A))}.$$ Finally, apply (\[1010\]) and (\[1313\]), still with either $C=A$ or $C=f(A)$, so that $$\frac{|A|^{26}}{|A+f(A)|^6}\leq{|A+f(A)|^{13}|A|^2(\log|A|)^2}.$$ Then, after rearranging, we get $$|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}.$$
**Acknowledgements.** This first listed author was supported by the NSF of China (11001174). The second listed author would like to thank Misha Rudnev for many helpful conversations.
|
---
abstract: 'We investigate the behaviour of exact closed bouncing Friedmann universes in theories with varying constants. We show that the simplest BSBM varying-alpha theory leads to a bouncing universe. The value of alpha increases monotonically, remaining approximately constant during most of each cycle, but increasing significantly around each bounce. When dissipation is introduced we show that in each new cycle the universe expands for longer and to a larger size. We find a similar effect for closed bouncing universes in Brans-Dicke theory, where $G$ also varies monotonically in time from cycle to cycle. Similar behaviour occurs also in varying speed of light theories.'
address: |
$^1$DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WA\
$^2$Theoretical Physics, The Blackett Laboratory, Imperial College, Prince Consort Road, London, SW7 2BZ, U.K.
author:
- 'John D. Barrow$^1$, Dagny Kimberly$^2$ and João Magueijo$^2$'
title: Bouncing Universes with Varying Constants
---
The “bouncing universe” is a modern cosmological reincarnation of an ancient fascination with the cyclic patterns of nature and the myth of the “eternal return” [@return; @BT; @tip]. Gravitation theories like general relativity allow us to make precise models of this popular conception of a phoenix cosmology, in which a universe periodically collapses to a Big Crunch, only to rebound into a new state of expansion, as if emerging from a unique Big Bang [@tolman]. Multiple bounces are possible but each cycle lasts longer and expands to a larger maximum size than the previous one, a consequence of a simple application of the second law of thermodynamics [@tolman; @tolman1; @tolman2; @zeld], unless there is a finite positive cosmological constant, in which case the oscillations must eventually cease [@bd] and are replaced by eternal de Sitter expansion. A sequence of many oscillations will drive the bouncing closed universe closer and closer to flatness.
Quantum gravity effects are invariably invoked to justify the bounce; possible detailed calculations, however, have only recently emerged. In loop quantum gravity, the semi-classical Friedmann equations receive corrections that produce a bounce [@singh; @lid]. The ekpyrotic model of the universe, inspired by string/M-theory, is another possible realization of phoenix cosmology [@ekpy]. It is also possible that ghost fields – fields endowed with negative energy – are capable of producing a *classical* bounce (this idea has been often rediscovered; see [@gary] for a good review). Classical bounces produced by conventional scalar fields with potentials which only violate the strong energy condition are difficult to produce in universes that grow large enough to be realistic: typically the probability of bounce is of order the ratio of the minimum to the maximum expansion size [@star; @BM].
It has been speculated that whatever causes a collapsing universe to bounce can reprocesses some aspects of physics, either randomly [@mtw], or systematically [@smol], by changing the particle spectrum or resetting the dimensionless constants of Nature. Both of these options are severely constrained by anthropic requirements but it is interesting to ask whether there are monotonic or asymptotic trends in the values of some quantities, as seems to be the case for the degree of flatness of the universe, over many bounces. This matter is clearly of great importance in the context of varying-constant theories [@uzan; @vslreview]. Here, quantities which are traditionally constants become space-time variables and if singularities are avoided in the bounce then their evolution from cycle to cycle is predictable by the field equations rather than the outcome of effectively random reprocessing. The values of any dimensionless ’constants’ of Nature could evolve towards asymptotic attractors if they are allowed to be variables in a self-consistent theory. Studies of these theories are also important in assessing the stability and level of fluctuations in a bouncing universe. It has been suggested that thermal fluctuations in bouncing models could be the origin of the cosmic structure [@pogo; @vslreview]; this would provide a distinct alternative to an origin from vacuum quantum fluctuations in a de Sitter phase of cosmological expansion.
Although these claims are intriguing, it is difficult to evaluate them in the absence of a concrete model for the bounce, which is usually viewed as a black box from which anything can emerge [@mtw]. In this paper we examine some exactly soluble examples. First, we consider a simple bouncing cosmology following from the simplest BSBM varying-constant theory we developed [@bsm] from Bekenstein’s varying-alpha model [@bek]. We derive the result that for suitable couplings (indeed those favoured by observations [@murphy; @webb]; see however [@vlt]) the theory leads to a bouncing universe. We derive similar behaviour in the Brans-Dicke theory of a varying gravitational constant $G$ [@bdicke], and in a class of varying speed of light theories [@covvsl; @moffat93; @am; @ba; @rev]. In each case we are able to find exact solutions in the absence of dissipation and compute the evolutionary trends in the fine structure constant and $G$ from cycle to cycle when dissipation occurs in accord with the Second Law. We find the interesting result that the varying constants in these theories change monotonically from cycle to cycle when the scale factor oscillates: the scalar fields determining the constants in each cycle do not oscillate.
Crucial to our models is the idea that cosmological fields may have a negative energy. Such fields are called ghosts [@gary], and are far from new, having found widespread application in the study of steady-state cosmology [@stst], phantom dark matter [@ph; @phbounce], and $\kappa $-essence [@kappa]. Bekenstein’s theory [@bek] is rooted in the use of a real scalar dielectric field $\psi $, representing the allowed variation of the electron charge according to $e=e_{0}e^{\psi }$, where $e_{0}$ is the present-day value of the electron charge, and so the fine structure ’constant’ evolves with respect to its present-day value, $\alpha
_{0},$ as $\alpha =\alpha _{0}e^{2\psi }$. By redefining the electromagnetic gauge field [@bsm], $A_{\mu }$, as $a_{\mu }=e^{\psi }A_{\mu }$, and the electromagnetic field tensor, $F_{\mu \nu }$, as $f_{\mu \nu }=e^{\psi
}F_{\mu \nu }$, the action may be written as $$S=\int d^{4}x\sqrt{-g}\left( \mathcal{L}_{g}+\mathcal{L}_{mat}+\mathcal{L}_{\psi }+\mathcal{L}_{em}e^{-2\psi }\right) , \label{Sbsm}$$where $\mathcal{L}_{\psi }=-{\frac{\omega }{2}}\partial _{\mu }\psi \partial
^{\mu }\psi $, $\mathcal{L}_{em}=-\frac{1}{4}f_{\mu \nu }f^{\mu \nu }$, and $\mathcal{L}_{mat}$ does not depend on $\psi $. The gravitational Lagrangian is the usual $\mathcal{L}_{g}=\frac{1}{16\pi G}R$, with $R$ being the curvature scalar. If the scalar coupling satisfies $\omega <0$, then $\psi $ has a negative kinetic energy term, and is thus a ghost field.
This theory can fit the observational constraints on varying alpha reported by [@murphy; @webb] as well as others [@bsm; @wep]. However for this to be possible in the ghost-free case with $\omega >0$ one has to choose a very special type of dark matter, in which the magnetostatic energy $B^{2}$ dominates over the electrostatic energy $E^{2}$. Even though a dark-matter candidate was found satisfying this condition (superconducting cosmic strings), most types of matter, including baryonic matter, are $E^{2}$ dominated, for which observational data implies $\omega <0$, so that $\psi $ is a ghost field.
Ghosts have been criticized on a variety of grounds. Classically they are a source of instabilities if coupled to other forms of matter, since they will try to off-load an infinite amount of positive energy into them. This is not necessarily cataclysmic if the rate of these processes is sufficiently slow. For instance, in steady-state cosmology to negative probabilities. At the quantum level, ghosts lead to negative norm states and so negative probabilities. The quantum instabilities are also much more severe and are present even without direct coupling to matter, for example in runaway particle production via the graviton vertex. Hence, at the quantum level ghosts *are* pathological.
However, we know that quantisation of the field $\psi $ is pathological even for $\omega >0$. For one thing, the theory is non-renormalisable. The attitude to $\psi $ should therefore be similar to that with regards to gravity: don’t quantise. General relativity is also, at face value, non-renormalisable: for instance, the quantum corrections to the relativistic precession of the perihelion of Mercury are infinite. This doesn’t stop the classical theory from being very successful. It may be that the quantisation of ghosts is simply more subtle; ghosts have been found as type $II^{\ast }$ string theories [@hull].
Non-relativistic matter and the cosmological constant may be neglected near a bounce, so let us consider a Friedmann-Robertson-Walker (FRW) universe filled with radiation and a dielectric field, $\psi $. The cosmological equations for BSBM varying-$\alpha $ theory (\[Sbsm\]) are $$\begin{aligned}
H^{2} &=&\frac{1}{3}\left( \rho _{r}e^{-2\psi }+\rho _{\psi }\right) -{\frac{K}{a^{2}}}, \label{fried1} \\
{\frac{\ddot{a}}{a}} &=&-{\frac{1}{6}}\left( 2\rho _{r}e^{-2\psi }+4\rho
_{\psi }\right) , \label{fried2}\end{aligned}$$where we have set $8\pi G=1$, $H\equiv \dot{a}/a$[ is the Hubble expansion rate, ]{}$K$ is the 3-curvature constant, and $\rho _{\psi }=\omega \dot{\psi}^{2}/2$. For the scalar field, in the absence of non-relativistic matter, we have $$\ddot{\psi}+3{H}\dot{\psi}=0. \label{fried3}$$so $\dot{\psi}\propto a^{-3}.$ We should not ignore the possibility that at high curvatures quantum processes may allow the conversion of $\psi $ energy into radiation. We take this into account by introducing variable ${\tilde{\rho}}_{r}=\rho _{r}e^{-2\psi }$ and rewriting the conservation equations as: $$\begin{aligned}
\dot{\rho}_{\psi }+6H\rho _{\psi } &=&-s({\tilde{\rho}}_{r},\dot{\psi},a),
\label{cons1} \\
\dot{\tilde{\rho}_{r}}+4H{\tilde{\rho}_{r}} &=&s({\tilde{\rho}}_{r},\dot{\psi},a). \label{cons2}\end{aligned}$$In this case, the equation of motion for $\psi $ will contain an additional $s$ term which models energy transfer between the $\psi $ field and the radiation sea in accord with the second law of thermodynamics. We shall consider the implications of such a process, in all its generality, below, but let us look first at the equations neglecting this coupling function.
Consider first a model $s=0$ bouncing universe which is exactly soluble. Taking $K=+1$ and $\omega <0,$ Eqn. (\[fried1\]) is $$\frac{\dot{a}^{2}}{a^{2}}=-\frac{S}{a^{6}}+\frac{\Gamma }{a^{4}}-\frac{1}{a^{2}}, \label{frw}$$where $S$ and $\Gamma $ are positive constants. In terms of conformal time $d\eta =a^{-1}dt$, this can be integrated to give $$a^{2}(\eta )=\frac{1}{2}\left[ \Gamma +\sqrt{\Gamma ^{2}-4S}\sin \{2(\eta
+\eta _{0})\}\right] \label{eta}$$when $\Gamma ^{2}>4S$. Identifying the expansion maximum and minimum, we see that $a(\eta )$ is given by $$a^{2}=\frac{1}{2}\left[ a_{\max }^{2}+a_{\min }^{2}+(a_{\max }^{2}-a_{\min
}^{2})\sin \{2(\eta +\eta _{0})\}\right] , \label{sol2}$$where $a_{\max }$ is global expansion maximum and $a_{\min }$ is the global minimum of $a(\eta )$, defined by
$$\begin{array}{c}
a_{\max }^{2} \\
a_{\min }^{2}\end{array}\equiv \frac{\Gamma \pm \sqrt{\Gamma ^{2}-4S}}{2}\ \label{min}$$
Since $\dot{\psi}=Ca^{-3}$ we have, for $\omega <0$, that $S=-\omega C^{2}/2$ and the scalar field driving time-variation of the fine structure constant is given by
$$\psi =\pm \frac{2}{\left\vert \omega \right\vert }\tan ^{-1}\left\{ \frac{\Gamma \tan (\eta +\eta _{0})+\sqrt{\Gamma ^{2}-4S}}{2\sqrt{S}}\right\} .$$In Fig. \[fig1\] we plot these solutions and show $a(t)$ and $\alpha (t)$ (recall that $\alpha \propto e^{2\psi }$), as functions of *proper* time, $t$. We note the steady increase of $\alpha $ with time despite the oscillatory behaviour of the expansion scale factor.
When $s=0$ we have a variety of oscillating solutions whose characteristics depend on the initial conditions. For bouncing solutions $\alpha $ remains nearly constant during each cycle but changes sharply, but still monotonically, at the bounce. There is no significant change of behaviour at the expansion maximum which also implies that there should be no gross difference in evolution inside and outside spherical overdensities far from the bounce. With $a_{max}\gg a_{min}$, and setting $\Gamma ={\tilde{\rho}_{r}}a^{4}/3$ and $S=-\rho _{\psi }a^{6}/3$, we have $a_{min}=\sqrt{S/\Gamma }$ and $a_{max}=\sqrt{\Gamma }$. We can then see that the bounce duration is $\Delta t\sim a_{min}^{2}/a_{max}$. Since $\dot{\psi}\sim \sqrt{6}\Gamma
^{3/2}/(S\left\vert \omega \right\vert ^{1/2})$ near the bounce, we find that $\Delta \psi \sim \sqrt{6/\left\vert \omega \right\vert }$, independently of initial conditions, during each bounce.
The extreme case is a stable static universe. Setting $\dot{a}=0$ and $\ddot{a}=0$, we can see that this case is realized when $\rho _{\psi }=-{\tilde{\rho}}_{r}/2$, giving $a=\sqrt{6/{\tilde{\rho}_{r}}}$. For such a universe $\psi $ evolves linearly in $t$, and since we have $\alpha \propto e^{2\psi }$ there is exponentially rapid increase [@bmota]. Even though such a universe is static, the rulers and clocks of observers change as alpha changes, so that they actually observe a Milne universe. We can see that the solution is stable because homogeneous and isotropic perturbations lead to a universe with regular sinusoidal oscillations as can be seen in Fig. [fig1]{}. Such solutions are described by (\[sol2\]) in the case where $a_{max}\approx a_{min}$. This situation differs from that found in general relativity in the absence of ghost fields [@btme].
If $s$ is a non-vanishing then, regardless of its exact functional form, there are two type of solutions. If $s\neq 0$ at all times, then sooner or later the universe enters a steady-state evolution with exponential expansion and constant overall energy density ensured by the appropriate transfer of energy between the $\psi $ field and radiation. However, we expect that these energy-transport processes will switch off at low curvatures, when the universe expands to a sufficiently large size ($a>>a_{\min }$) and transport processes become collisionless and far slower than the expansion rate. Then, the typical evolution is as plotted in Fig. \[fig2\]. Again, $\psi $ is approximately constant during each cycle and changes dramatically at the bounce. In addition, each cycle is now bigger than the previous one, because $\Gamma $ increases at each bounce. This is an interesting realisation of the standard Tolman scenario. Cycles get bigger (and entropy is generated near the bounce) specifically because radiation is produced from the scalar field close to each bounce. In producing Fig. \[fig2\] we have used $s\propto \rho _{\psi }$ ($\psi $ decays into radiation), but other functional forms may be used with similar effects.
We now examine similar solutions for the Brans-Dicke (BD) theory of varying $G$ [@bdicke]. In the Einstein frame, *if the matter content is pure radiation*, we recover the same equations. All we need to do, then, is convert the above results into the Jordan frame: the results found for $a$, $\rho $, and $\psi $ should then be translated into variables $a_{J}=a/\sqrt{\phi }$, $\rho _{J}={\tilde{\rho}}\phi ^{2}$, and $\phi =e^{\psi }$ (the latter corresponding roughly to $1/G$). Under this transformation we obtain $\omega =\omega _{BD}+{\frac{3}{2}}$, where $\omega _{BD}$ in the Brand-Dicke coupling parameter. Thus we need $\omega _{BD}<-3/2$ for the Brans-Dicke field to behave like a proper ghost in the Einstein frame. The resulting dynamics is plotted in Fig. \[fig3\].
These results may be understood analytically. The essential BD field equations in the Jordan frame are $$\begin{aligned}
\ddot{\phi}+3H\dot{\phi} &=&0 \\
H^{2} &=&\frac{8\pi \rho }{3\phi }-H\frac{\dot{\phi}}{\phi }+\frac{w_{BD}\dot{\phi}^{2}}{6\phi ^{2}}-\frac{K}{a_{J}^{2}}\end{aligned}$$where overdots now refer to derivatives with respect to $t_{J}$ and $H=\dot{a}_{J}/a_{J}$. Hence $$\begin{aligned}
\dot{\phi} &=&\frac{A}{a_{J}^{3}} \label{phi} \\
\frac{\dot{a}_{J}^{2}}{a_{J}^{2}} &=&\frac{\lambda }{a_{J}^{4}\phi }-\frac{\dot{a}_{J}}{a_{J}}\frac{\dot{\phi}}{\phi }+\frac{w_{BD}\dot{\phi}^{2}}{6\phi ^{2}}-\frac{K}{a_{J}^{2}} \nonumber\end{aligned}$$with $A$ and $\lambda >0$ constants. We are interested in negative $\omega $ solutions with $K=1$, which give oscillating closed universes (note that $\omega _{BD}<0$ is not sufficient for an expansion minimum, we need $\omega
_{BD}<-3/2$). Following the techniques of [@jbBD] we put $\frac{A^{2}}{3}(2\omega +3)\equiv -C,$ and in the bouncing case $\lambda ^{2}-C>0$, we have simple exact solution in terms of conformal Jordan time: $$\phi a_{J}^{2}=\frac{\lambda }{2}+\frac{1}{2}\sqrt{\lambda ^{2}-C\ }\sin
\{2(\eta +\eta _{0})\}. \label{y}$$We see the same behaviour as displayed by the radiation-scalar universe given above with $s=0$. The minimum value of $\phi a_{J}^{2}$ is $\lambda -\sqrt{\lambda ^{2}-C}$ and the maximum is $\lambda +\sqrt{\lambda ^{2}-C}.$ So in conformal time $\phi a_{J}^{2}\approx a_{J}^{2}/G$ undergoes oscillations of increasing amplitude as the entropy increases (ie if $\lambda $ increases in value to model increasing radiation density); that is, the horizon area (’entropy’) in Planck units increases from cycle to cycle. The full solution for $a(\eta )$ and $\phi (\eta )$ is then obtained using eqns. (\[phi\]) and (\[y\]): $$\phi =\phi _{1}\exp \left[ \frac{2A}{\sqrt{C}}\arctan \left[ \frac{\lambda
\tan \{\eta +\eta _{0}\}+\sqrt{\lambda ^{2}-C}}{\sqrt{C}}\right] \right] ,
\label{phisol}$$with $\phi _{1}$ constant and $$\begin{aligned}
a_{J}^{2} &=&\ \phi _{1}^{-1}\left[ \frac{\lambda }{2}+\frac{1}{2}\sqrt{\lambda ^{2}-C\ }\sin \{2(\eta +\eta _{0})\}\right] \times \nonumber \\
&&\exp \left[ -\frac{2A}{\sqrt{C}}\arctan \left[ \frac{\lambda \tan \{\eta
+\eta _{0}\}+\sqrt{\lambda ^{2}-C}}{\sqrt{C}}\right] \right] . \label{a}\end{aligned}$$The effect of increasing radiation entropy can be seen by increasing the constant $\lambda $ in these expressions. In this model the field $\phi $ and radiation are fully decoupled. Although the size of the universe increases in each successive cycle, its size with respect to Planck units remains the same, unless of course we consider a model in which the field $\phi $ and radiation may exchange energy.
We found similar solutions in the BSBM and BD theories because the dynamics in the Einstein frame is very similar in the absence of non-relativistic matter. Likewise, one can find identical solutions for the covariant varying speed of light (VSL) theories described in Refs. [@covvsl; @rev]. This does not imply that the VSL, Brans-Dicke and BSBM theories are equivalent; merely that one needs to add more general matter (even if only as test matter in a radiation-dominated universe) for their differences to become obvious. Specifically, the coupling to $\mathcal{L}_{em}$ in (\[Sbsm\]) is replaced in Brans-Dicke theory by non-minimally coupling the matter fields to the metric.
In summary, we have considered some simple exactly soluble models for closed bouncing universes in theories with varying $\alpha $ and varying $G$ and examined the effects of simple non-equilibrium behaviour. Even though the expansion scale factor of the universes undergoes periodic oscillations about a finite non-singular expansion minimum, we find steady monotonic change in the values of $\alpha $ and $G$ from cycle to cycle both in the presence and absence of non-equilibrium behaviour. In the non-equilibrium case the oscillations of the expansion scale factor grow monotonically in amplitude and period in accord with the Second Law, and the expansion dynamics approaches flatness.
*Acknowledgements* We would like to thank B. Carr and C. Hull for helpful comments.
References {#references .unnumbered}
==========
[99]{}
Eliade M 1954 *The Myth of the Eternal Return* (New York: Pantheon)
Barrow J D and Tipler F J 1986 *The Anthropic Cosmological Principle* (Oxford: Oxford UP)
Tipler F J 1979 *Nature* **280** 203
Tolman R C 1934 *Relativity, Thermodynamics and Cosmology* (Oxford: Clarendon Press)
Tolman R C 1931 *Phys. Rev.* **37** 1639
Tolman R C 1931 *Phys. Rev.* **38** 1758
Zeldovich Y B and Novikov I D 1983 *The Structure and Evolution of the Universe* (Chicago: University of Chicago Press)
Barrow J D and Dabrowskii M 1995 *Mon. Not. R. astron. Soc.* **275** 850
Singh P and Toporensky A gr-qc/0312110
Lidsey J E et al gr-qc/0406042
Khoury J et al 2001 *Phys. Rev*. D **64** 123522
Gibbons G hep-th/0302199
Starobinskii A A 1978 *Sov. Astron. Lett.* **4** 82
Barrow J D and Matzner R A 1980 *Phys. Rev.* D**21**, 336
Misner C, Thorne K and Wheeler J A 1973 *Gravitation* (New York: W. H. Freeman & Co.) p 1215
Smolin L 1992 *Class. Quantum Grav.* **9**, 173
Uzan J 2003 *Rev. Mod. Phys*. **75** 403
Magueijo J 2003 *Rep. Prog. Phys*. **66** 2025
Magueijo J and Pogosian L *Phys. Rev.* D **67** 043518, 2003
Sandvik H, Barrow J D and Magueijo J 2002 *Phys. Rev. Lett*. **88** 031302,
Bekenstein J D 1982 *Phys. Rev.* D **25** 1527
Murphy M T et al, 2001 *Mon Not. R. astron. Soc.* **327**, 1208
Webb J K, FlambaumV V, Churchill C W, Drinkwater M J and Barrow J D 1999 *Phys. Rev. Lett*. **82** 884
Srianand R et al 2004 *Phys. Rev. Lett.* **92** 121302
Brans C and Dicke R H, 1961 *Phys. Rev.* **124** 925
Magueijo J 2000 *Phys. Rev. D* **62** 103521
Moffat J W 1993 *Int. J. Mod. Phys. D* **2** 351
Albrecht A and Magueijo J 1999 *Phys. Rev. D* **59** 043516
Barrow J D 1999 *Phys. Rev. D* **59** 043515
Magueijo J 2003 *Reports of Progress in physics* **66** 2025
Hoyle F and Narlikar J V 1963 *Proc. Roy. Soc.* **273** 1
Caldwell R 2002* Phys. Lett.* B **545** 23
Brown M G and Freese K 2004 astro-ph/0405353
Armendariz-Picon C, Mukhanov V and Steinhardt P 2001 *Phys. Rev*. D** 63** 103510
Magueijo J, Barrow J D and Sandvik H 2002 *Phys. Lett*. B **549** 284
Hull C 1998 JHEP **9807** 021
Barrow J D and Mota D 2002 *Class. Quantum Gravity* **19** 6197
Barrow J D, Ellis G F R, Maartens R and Tsagas C 2003 *Class. Quantum Gravity* **20** L155
Barrow J D 1992 *Phys. Rev. D* **47** 5329
|
---
abstract: 'The exceptionally strong coupling realizable between superconducting qubits and photons stored in an on-chip microwave resonator allows for the detailed study of matter-light interactions in the realm of circuit quantum electrodynamics (QED). Here we investigate the resonant interaction between a single transmon-type multilevel artificial atom and weak thermal and coherent fields. We explore up to three photon dressed states of the coupled system in a linear response heterodyne transmission measurement. The results are in good quantitative agreement with a generalized Jaynes-Cummings model. Our data indicates that the role of thermal fields in resonant cavity QED can be studied in detail using superconducting circuits.'
address:
- '$^1$ Department of Physics, ETH Zurich, CH-8093, Zurich, Switzerland.'
- '$^2$ Département de Physique, Université de Sherbrooke, Québec, J1K 2R1 Canada.'
author:
- 'J. M. Fink$^1$, M. Baur$^1$, R. Bianchetti$^1$, S. Filipp$^1$, M. Göppl$^1$, P. J. Leek$^1$, L. Steffen$^1$, A. Blais$^2$ and A. Wallraff$^1$'
bibliography:
- 'QudevRefDB.bib'
title: |
Thermal Excitation of Multi-Photon Dressed States\
in Circuit Quantum Electrodynamics
---
Introduction
============
In cavity quantum electrodynamics [@Raimond2001; @Haroche2007; @Walther2006; @Ye2008] the fundamental interaction of matter and light is studied in a well controlled environment. On the level of individual quanta this interaction is governed by the Jaynes-Cummings Hamiltonian [@Jaynes1963]. In early experiments the quantization of the electromagnetic field was observed in cavity QED with Rydberg atoms by measurements of collapse and revival [@Rempe1987]. Similarly, the $\sqrt{n}$ scaling of the atom/photon coupling strength with the number of photons $n$ has been observed in the time domain by measuring $n$ photon Rabi oscillations using coherent states [@Brune1996] and Fock states [@Varcoe2000; @Bertet2002]. In the frequency domain the $\sqrt{n}$ scaling can be extracted from spectroscopic cavity transmission measurements. Initial attempts employing pump and probe spectroscopy with alkali atoms [@Thompson1998] were inconclusive. In a recent experiment however the two-photon vacuum Rabi resonance was resolved using high power nonlinear spectroscopy [@Schuster2008]. At the same time, we observed the quantum nonlinearity by measuring the spectrum of two photons and one artificial atom in circuit QED [@Fink2008]. In these measurements the originally proposed pump and probe spectroscopy scheme [@Thompson1998] was used. Similarly the $n$ photon Rabi mode splitting was studied using multi-photon transitions up to $n=5$ [@Bishop2009a]. In the dispersive regime the photon number splitting of spectroscopic lines [@Schuster2007a] provides similar evidence for the quantization of microwave radiation in circuit QED. Further experimental progress in the time domain has enabled the preparation and detection of coherent states [@Johansson2006], photon number states [@Hofheinz2008] and arbitrary superpositions of photon number states [@Hofheinz2009] in circuits. Using Rydberg atoms quantum jumps of light have been observed [@Gleyzes2007; @Guerlin2007] and Wigner functions of Fock states have been reconstructed [@Deleglise2008]. These experiments demonstrate the quantum nature of light by measuring the nonlinear $\sqrt{n}$ scaling of the dipole coupling strength with the discrete number of photons $n$ in the Jaynes-Cummings model [@Carmichael1996].
In this letter we extend our earlier spectroscopic study of the $\sqrt{n}$ scaling [@Fink2008] in the circuit QED architecture [@Blais2004; @Wallraff2004b]. We present a measurement of the photon/atom energy spectrum up to $n=3$ excitations using both thermal and coherent fields. A solid state qubit with a large effective dipole moment plays the role of the atom. It is coupled to weak microwave radiation fields in a coplanar waveguide resonator which provides a very large electric field strength per photon. The coupling strength of a few hundred MHz [@Schoelkopf2008], much larger than in implementations with natural atoms, can freely be chosen over a wide range of values and remains fixed in time. The qubit level spectrum is in situ tunable by applying flux to the circuit and the internal degrees of freedom can be prepared and manipulated with microwave fields.
Similar solid state circuit implementations of cavity QED have enabled a remarkable number of novel quantum optics [@Wallraff2004b; @Astafiev2007; @Houck2007; @Schuster2007a; @Fink2008; @Hofheinz2008; @Fragner2008; @Deppe2008; @Bishop2009a; @Hofheinz2009; @Baur2009; @Fink2009] and quantum computation [@Majer2007; @Sillanpaa2007; @DiCarlo2009] experiments.
Sample and experimental setup
=============================
For the experiments presented here we use a transmon type qubit [@Koch2007; @Schreier2008], which is a charge-insensitive superconducting qubit design derived from the Cooper pair box [@Bouchiat1998], as the artificial atom. Its transition frequency is given by $\omega_{g,e}/2\pi\simeq\sqrt{8 E_{C} E_{J}(\Phi)}-E_{C}$ with the single electron charging energy $E_{C}/h \approx0.232 \, ~\textrm{GHz}$, the flux controlled Josephson energy $E_{J}(\Phi)=E_{{J,max}}|\cos{(\pi \Phi/\Phi_{0})}|$ and $E_{{J,max}}/h \approx 35.1 \, \textrm{GHz}$, as determined by spectroscopic measurements. The two characteristic energies $E_{J}$ and $E_{C}$ define the full level spectrum of the qubit where the eigenenergy of level $l$ is approximately given as $E_l\simeq-E_J+\sqrt{8 E_C E_J} (l+1/2) - E_C/12 (6l^2+6l+3)$ [@Koch2007]. The cavity is realized as a coplanar resonator with bare resonance frequency $\nu_{r}\approx6.44 \, \textrm{GHz}$ and photon decay rate $\kappa/2\pi\approx1.6 \, \textrm{MHz}$. Details of the resonator design and fabrication can be found in Ref. [@Goppl2008]. Optical microscope images of the sample are shown in Fig. \[fig1\]a. A simplified electrical circuit diagram of the setup is shown in Fig. \[fig1\]b.
Generalized Jaynes-Cummings model
=================================
The transmon qubit is a superconducting circuit with a nonlinear energy level spectrum. In many experiments the nonlinearity, which can be adjusted by circuit design and fabrication, is sufficient to correctly model it as a two level system. For the experimental results presented in this work it is however essential to treat the qubit as a multilevel system taking into account the coupling of all relevant transmon levels to the cavity photons. The physics of this multilevel artificial atom strongly coupled to a single mode of the electromagnetic field is described by a generalized Jaynes-Cummings model with the Hamiltonian $$\label{jch}
\hat{\mathcal{H}}= \hbar\omega_{r}\, \hat{a}^\dagger \hat{a}+ \sum_{l=e,f,h,...}\left( \hbar\omega_{l}\,\hat{\sigma}_{l,l}\, +\hbar g_{l-1,l}\,(\hat{\sigma}^\dagger_{l-1,l}\,\hat{a} +\hat{a}^\dagger\hat{\sigma}_{l-1,l}\,)\right) \, .$$ Here, $\hbar\omega_{{l}}$ is the energy of the $l$’th excited state $|l\rangle$ of the multilevel artificial atom, $\omega_{r}$ is the frequency of the resonator field and $g_{l-1,l}$ is the coupling strength of the transition $l-1 \rightarrow l$ and one photon. $\hat{a}^{\dagger}$ and $\hat{a}$ are the raising and lowering operators acting on the field with photon number $n$ and $\hat{\sigma}_{i,j}=|i\rangle\langle j|$ are the corresponding operators acting on the qubit states. In Fig. \[fig2\], a sketch of the energy level diagram of the resonantly coupled qubit-resonator system ($\nu_{r}=\nu_{g,e}$) is shown for up to three photons $n=0,1,2,3$ and the first four transmon levels $l=g,e,f$ and $h$.
Considering the first two levels of the artificial atom at zero detuning ($\Delta\equiv\nu_{r}-\nu_{g,e}=0$) the eigenstates of the coupled system are the symmetric $(|g,n\rangle+|e,n-1\rangle)/\sqrt{2}\equiv|n+\rangle$ and antisymmetric $(|g,n\rangle-|e,n-1\rangle)/\sqrt{2}\equiv|n-\rangle$ qubit-photon superposition states, see Fig. \[fig2\]. For $n=1$, the coupled one photon one atom eigenstates are split due to the dipole interaction [@Agarwal1984; @Sanchez1983]. These states have been observed in hallmark experiments demonstrating the strong coupling of a single atom and a single photon both spectroscopically as a vacuum Rabi mode splitting with on average a single atom [@Thompson1992] and also without averaging using alkali atoms [@Boca2004], in superconducting circuits [@Wallraff2004b] and semiconducting quantum dot systems [@Reithmaier2004; @Yoshie2004a], or in time resolved measurements as vacuum Rabi oscillations at frequency $2g_{g,e}$ with Rydberg atoms [@Brune1996; @Varcoe2000] and superconducting circuits [@Johansson2006; @Hofheinz2008]. At least in principle however the observation of the first doublet $|1\pm\rangle$ alone, could also be explained as normal mode splitting in a classical linear dispersion theory [@Zhu1990]. The Jaynes-Cummings model however predicts a characteristic nonlinear scaling of this frequency as $\sqrt{n}\, 2 g_{g,e}$ with the number of excitations $n$ in the system. In the general multilevel case, the higher energy atomic levels renormalize the dipole coupled dressed state energies. This causes frequency shifts in the excitation spectrum as indicated in Fig. \[fig2\] and the simple $\sqrt{n}$ scaling is slightly modified.
Vacuum Rabi mode splitting in the presence of a thermal field
=============================================================
In this section we investigate vacuum Rabi resonances both in the presence of a weak thermal background field and also in the presence of externally applied quasi-thermal fields. To our knowledge this is the first experiment where thermally excited multi-photon dressed states are studied systematically in a cavity QED system.
Experimentally, the coupled circuit QED system is prepared in its ground state $|g,0\rangle$ by cooling it to temperatures below $<20\, \textrm{mK}$ in a dilution refrigerator. Measuring the cavity transmission spectrum $T$ in the anti-crossing region of the qubit transition frequency $\nu_{g,e}$ and the cavity frequency $\nu_r$ yields transmission maxima at frequencies corresponding to transitions from $|g,0\rangle$ to the first doublet $|1\pm\rangle$ of the Jaynes-Cummings ladder, see Fig. \[fig3\]a. On resonance, see Fig. \[fig3\]b, we extract the coupling strength of $g_{{g,e}}/2\pi = 133 \, \rm{MHz}$. This is a clear indication that the strong coupling limit $g_{g,e} \gg \kappa, \gamma$, with photon decay and qubit decoherence rates $\kappa/2\pi, \gamma/2\pi \sim 1\, \rm{MHz}$, is realized. Solid lines in Fig. \[fig3\]a (and Fig. \[fig5\]a) are numerically calculated dressed state frequencies, see Fig. \[fig3\]c, with the qubit and resonator parameters as stated above. For the calculation, the qubit Hamiltonian is diagonalized exactly in the charge basis. The qubit states $|g\rangle$ and $|e\rangle$ and the flux dependent coupling constant $g_{g,e}$ are then incorporated in the Jaynes-Cummings Hamiltonian Eq. (\[jch\]). Its numerical diagonalization yields the dressed states of the coupled system without any fit parameters.
In addition to the expected spectral lines corresponding to the transition from the ground state $|g,0\rangle$ to the first doublet states $|1\pm\rangle$, we observe three lines with very low intensities, see Figs. \[fig3\]a and b. These additional transitions are visible because the system is excited by a small thermal background field with a cavity photon number distribution given by the Bose-Einstein distribution. This thermal field is a consequence of incomplete thermalization of the room temperature black-body radiation at the input and output ports of the resonator. A quantitative analysis taking into account the two photon states $|2\pm\rangle$ and the presence of the higher energy qubit levels $f$ and $h$ in Eq. (\[jch\]) yields the transition frequencies indicated by yellow and red solid lines in Fig. \[fig3\]. For this analysis the coupling constants $g_{e,f}=184$ and $g_{f,h}=221$ of higher energy qubit levels to the cavity mode, obtained from exact diagonalization of the qubit Hamiltonian, have been included. We thus identify two of the additional spectral lines as transitions between the first $|1\pm\rangle$ and second $|2\pm\rangle$ doublet states of the resonant Jaynes Cummings ladder, see Fig. \[fig3\]c. The lowest frequency additional spectral line corresponds to a transition from the antisymmetric doublet state with one photon $|1-\rangle$ to the qubit $f$ level without a photon $|f,0\rangle$. The details of the thermally excited transmission spectrum can be used as a sensitive probe for the cavity field temperature. Analyzing the amplitudes of the Rabi splitting spectrum with a quantitative master equation model [@Bishop2009a], leads to an estimated photon temperature of $T_r\simeq 0.2$ which corresponds to a relatively high mean thermal occupation number of $\bar{n}_{th}\simeq 0.3$ photons for the data presented in Fig. 3a. Careful filtering and thermalization at the input and output ports results in a typical cavity photon temperature of $ < 90\, \textrm{mK}$ and $ < 54\, \textrm{mK}$ ($\bar{n}_{th} < 0.03$ and $\bar{n}_{th} < 0.003$) as reported in Refs. [@Fragner2008] and [@Bishop2009a].
In order to access the three photon doublet states $|3\pm\rangle$ of the coupled multi-photon multilevel-atom system we use externally applied broadband quasi-thermal fields. In this new approach the dressed eigenstates are populated according to a thermal distribution depending on the chosen thermal field temperature. This allows to investigate the flux dependence of all resolvable spectral lines for a given effective resonator mode temperature $T_r$ in a single experimental run. The spectrum of a one dimensional black body such as the considered cavity is given as $S_{1D}(\nu)=h\nu/[\exp{(h\nu/k_B T})-1]$. At a temperature of $T_r > h \nu_r/k_B\simeq300$ this energy spectrum is flat with a deviation of $<5\%$ within a 500 band centered at the cavity frequency $\nu_r\simeq6.44$ . It is therefore a very good approximation to make use of a white noise spectrum in the narrow frequency band of the experimentally investigated transition frequencies centered at $\nu_r$, in order to generate a quasi-thermal field of temperatures $T_r>300$ and populate the considered cavity mode with thermal photons.
In order to generate such a spectrum a carrier microwave tone at frequency $\nu_r$ is modulated with a low frequency large bandwidth quasi-random noise spectrum $S_\textrm{white}$ using a mixer, see Fig. \[fig1\]b. This approximately yields a microwave frequency white noise spectrum with a bandwidth of $500$ centered symmetrically around the cavity frequency $\nu_r$. Using tunable attenuators, we can adjust the noise power spectral density over a wide range of values. For this experiment we adjust it such that the thermal population of the cavity mode is on the order of $\bar{n}_{th} \sim 0.9$ corresponding to a temperature of $T_r \sim 0.4$ . This noise spectrum constitutes a sufficient approximation of a black body thermal noise source for the considered 1D cavity mode, temperature and frequency. At the same time, the chosen mean thermal population $\bar{n}_{th}\sim 0.9$ allows to observe all allowed transitions between the ground state $|g\rangle$ and the three photon doublet states $|3\pm\rangle$.
In the presence of the thermal field, we probe the cavity transmission spectrum as a function of flux in the anticrossing region, see Fig. \[fig5\]a, with a weak coherent probe tone $\nu_\textrm{rf}$ in the linear response limit. In this limit the weak probe tone is only a small perturbation to the field and no multi-photon transitions are induced. In this measurement we resolve all allowed transitions between the thermally occupied dipole coupled states in the generalized Jaynes-Cummings ladder. The solid lines are again the calculated dressed state transition energies which agree well with the observed spectral lines. In Fig. \[fig5\]b, a cavity transmission measurement at flux $\Phi/\Phi_0=0.25$ is shown. We identify 8 allowed transitions, compare with Fig. \[fig5\]c. It follows that the states $|1\pm\rangle$, $|2\pm\rangle$ and also $|f,0\rangle$ are thermally populated.
In the two-level-atom approximation, transitions between symmetric and antisymmetric doublet states are forbidden at degeneracy. In the generalized Jaynes-Cummings model the dressed state transition matrix elements are renormalized due to higher qubit levels. Numerical diagonalization however shows that the matrix elements squared, which are related to the amplitude of the expected spectral lines, are 140 (6) times smaller for the symmetry changing transitions $|1-\rangle\rightarrow|2+\rangle$ ($|1+\rangle\rightarrow|2-\rangle$) than for the observed symmetry preserving transitions $|1-\rangle\rightarrow|2-\rangle$ ($|1+\rangle\rightarrow|2+\rangle$) at degeneracy. Similarly, for the transitions $|2-\rangle\rightarrow|3+\rangle$ ($|2+\rangle\rightarrow|3-\rangle$) the matrix elements squared are 235 (16) times smaller than the measured transitions $|2-\rangle\rightarrow|3-\rangle$ ($|2+\rangle\rightarrow|3+\rangle$) at degeneracy. Therefore transitions between symmetric and antisymmetric doublet states are not resolved in our experiment. Symmetry changing transitions populating the antisymmetric states $|2-\rangle$ and $|3-\rangle$ have larger matrix elements than symmetry changing transitions populating the symmetric states $|2+\rangle$ and $|3+\rangle$ because the former are closer in frequency to the qubit levels $|f,0\rangle$ and $|h,0\rangle$. Similarly, the matrix element for the transition $|1-\rangle\rightarrow|f,0\rangle$ is 34 times larger than for the transition $|1+\rangle\rightarrow|f,0\rangle$ at degeneracy. The latter is therefore also not observed in the experimental data. In addition to the transition $|1-\rangle\rightarrow|f,0\rangle$, also seen in the data presented in Fig. \[fig3\], we observe a transmission line which corresponds to the transition $|f,0\rangle\rightarrow|f,1\rangle$, see Fig. \[fig5\]b. A numerical calculation shows that the matrix element is 5 times larger at degeneracy than $|2-\rangle\rightarrow|f,1\rangle$ and 7 times larger than $|f,0\rangle\rightarrow|h,0\rangle$ which in principle could also have been observed. All transitions observed in the experimental data are in qualitative agreement with the calculated matrix elements stated above.
In Fig. \[fig6\] the complete measured level spectrum of the bound photon/atom system up to the third excitation is shown. To calculate the absolute energies of the levels (blue dots) we extract the transition frequencies from data presented in Fig. \[fig5\] with Lorentzian line fits and add them accordingly. For the first doublet states $|1\pm\rangle$ we find excellent agreement with both a simple two-level atom Jaynes-Cummings model (dotted red lines) as well as the generalized multilevel Jaynes-Cummings model (solid red lines). In the case of the second $|2\pm\rangle$ and third doublet states $|3\pm\rangle$ we find considerable frequency shifts with regard to the two level model (compare dotted and solid red lines) but excellent agreement with the generalized model taking account the additional qubit levels. Furthermore it can be seen in Fig. \[fig6\] that the negative anharmonicity of the transmon qubit, together with the strong dipole coupling, causes large frequency shifts of the antisymmetric dressed levels $|2-\rangle$ and $|3-\rangle$ since they are closer in frequency to the qubit levels $|f,0\rangle$ and $|f,1\rangle$, $|h,0\rangle$. This leads to a small reduction of the $\sqrt{n}$ nonlinearity which is in agreement with the numerical results.
Vacuum Rabi mode splitting with two pump and one probe tone
===========================================================
In order to probe the excitation spectrum of the two and three photon doublet states we can also follow the pump and probe spectroscopy scheme similar to the one presented in Refs. [@Thompson1998; @Fink2008], where one of the first doublet states $|1\pm\rangle$ is coherently pumped and the transition to the states $|2\pm\rangle$ is probed. This technique avoids large intra-cavity photon numbers which are needed in high-drive and elevated temperature experiments. In analogy to the previous section we wait for the system to equilibrate with the cold environment to prepare the ground state $|g,0\rangle$. The qubit is then tuned close to degeneracy where $\Phi/\Phi_0\approx0.25$. We weakly probe the resonator transmission spectrum as shown in Fig. \[fig4\] (blue lines). In a second step we apply a pump tone at frequency $\nu_{g0,1-}$ ($\nu_{g0,1+}$) occupying the dressed state $|1-\rangle$ ($|1+\rangle$) and probe the system again, see Fig. \[fig4\]a (b) yellow line. Clearly the transitions $|1\pm\rangle\rightarrow|2\pm\rangle$ become visible at the calculated eigenenergy which is indicated with yellow vertical arrows in Figs. \[fig4\]a, b and c. We also note that the state $|f,0\rangle$ is populated by the probe tone via the transition $|1-\rangle\rightarrow|f,0\rangle$, see red arrows in Figs. \[fig4\]a and c. In a last step we apply two pump tones at frequencies $\nu_{g0,1-}$ and $\nu_{1-,2-}$, see Fig. \[fig4\] a (green line), or at frequencies $\nu_{g0,1+}$ and $\nu_{1+,2+}$, see Fig. \[fig4\] b (green line) respectively. The three-photon one-qubit dressed state transitions $|2\pm\rangle\rightarrow|3\pm\rangle$ become visible in the spectrum, see green vertical arrows. At the same time, transitions from the ground state are found to saturate considerably when the pump tones are turned on, compare the amplitudes of the spectral lines at the frequency indicated by the left blue arrow in Fig. \[fig4\] a, or similarly by the right blue arrow in figure Fig. \[fig4\] b. This is expected since the occupation probability of the ground state is reduced and the transition starts to become saturated when the pump tones are turned on.
Again, the observed transition frequencies are in good agreement with the calculated dressed state transition energies indicated by vertical arrows in Figs. \[fig4\]a and b. Similar to the experiment described in the last section, additional spectral lines with low intensity, see Fig. \[fig4\]a and b blue lines, occur because of a small probability of occupation of the first doublet due to the residual thermal field. In comparison to the data shown in Fig. \[fig5\] no transition $|f,0\rangle\rightarrow|f,1\rangle$ is observed because the level $|f,0\rangle$ is neither thermally, nor coherently populated here.
Summary
=======
We extended our previous work [@Fink2008] by introducing thermal fields to populate the dressed eigenstates in a resonant cavity QED system. In addition to the one and two photon/atom superposition states we report a measurement of the three photon doublet using both thermal and coherent fields. The results are in good agreement with a generalized multilevel-atom Jaynes-Cummings Hamiltonian without any fit parameters. Related results have been reported in an atomic system [@Schuster2008] and in circuit QED [@Bishop2009a]. Similar effects may be interesting to approach also in semiconducting cavity QED systems. It has been shown that cavity QED with superconducting circuits can be a sensitive probe for thermal fields. A more detailed quantitative analysis of the thermally excited vacuum Rabi spectra could be of interest in the context of environmentally induced dissipation and decoherence, thermal field sensing and the cross-over from the quantum to the classical regime of cavity QED.
Acknowledgements
================
This work was supported by EuroSQIP, SNF Grant No. 200021-111899 and ETHZ. A. B. was supported by NSERC, CIFAR and the Alfred P. Sloan Foundation.
References {#references .unnumbered}
==========
|
---
abstract: 'In the paper we consider the problem of estimating parameters entering the drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case, when the underlying stochastic integral is of Young type. We consider the sampling scheme that the process is observed continuously on $[0,T]$ and $T\to\infty$. For known Hurst parameter $H\in(0.5, 1)$, i.e. the long range dependent case, we construct a least-squares type estimator and establish strong consistency. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic function with a rate depending on $H$ and a non-central Cauchy limit result for the mean reverting parameter with exponential rate. For the special case that the periodicity parameter is the weight of a periodic function, which integrates to zero over the period, we can even improve the rate to $\sqrt{T}$.'
address: 'Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany'
author:
- 'Radomyra Shevchenko and Jeannette H.C. Woerner'
title: 'Inference for fractional Ornstein-Uhlenbeck type processes with periodic mean in the non-ergodic case'
---
Ł
Introduction
============
Parameter estimation in fractional diffusions has been actively studied in recent years, especially for equations of the Ornstein-Uhlenbeck type, i.e. of the form $$dX_t = \alpha X_t dt + dB^H_t,$$ where $B^H_t$ is a fractional Brownian motion (fBm for short), which is a centred Gaussian process with almost surely continuous paths defined via its covariance structure $$\mathbb E [B^H_t B^H_s]=\frac{1}{2} \left(t^{2H}+s^{2H}-|t-s|^{2H} \right)$$ for $H\in (0,\,1)$.
With no initial condition imposed, the equation has an ergodic solution for $\alpha <0$, which is why this case is often called ergodic, as opposed to the non-ergodic case $\alpha >0$.
Several approaches are known for the estimation of $\alpha$, among them the MLE approach in [@KLB] which uses the so called fundamental martingales related to the underlying fBm and a minimum $L_1$-norm estimation in [@Rao] based on the techniques from [@KPi].
Another possibility for the estimation is offered by the least squares approach, for which (following a heuristic notation) the term $\int_0^n (\dot{X}_t+ \alpha X_t)^2 dt$ is minimised, leading to the estimator $$\tilde{\alpha}_n:=\frac{\int_0^n X_t dX_t}{\int_0^n X_t^2 dt}.$$ For $H=\frac{1}{2}$ the process $B^H$ is the classical Brownian motion, which allows for Itō integration in this definition. However, for $H\neq \frac{1}{2}$ fBm is not a semimartingale, so in order to define such an estimator one has to find a different (and suitable) kind of a stochastic integral, possible choices including pathwise (or Young type) and Skorokhod (or divergence type) integrals. Form the practical point of view pathwise integrals are preferred, however, for $\alpha<0$ this choice does not yield a consistent estimator. This was shown in [@HNu] alongside with consistency and asymptotic normality for the estimators defined with Itō integrals for $H=\frac{1}{2}$ and for those defined with Skorokhod integrals for $H\in \Big(\frac{1}{2},\,\frac{3}{4}\Big)$. The non-ergodic case was treated in [@BESO] for $H\in \Big(\frac{1}{2},\,1\Big)$, where the same estimator defined with Young-type integrals is shown to be consistent and asymptotically Cauchy distributed. Note that for $H>1/2$ the processes possess long range dependence, which offers interesting possibilities for modelling.
In this paper we study Ornstein-Uhlenbeck type equations with an additional periodic mean term, i.e. equations of the form $$dX_t = (L(t)+\alpha X_t) dt + dB^H_t$$ with a periodic, parametric function $L$, which can be used for modelling seasonalities, and assume continuous observations. In this case a consistent and asymptotically normal least squares-type estimator (coinciding with the MLE) for the drift parameters (including $\alpha <0$) was constructed for $H=\frac{1}{2}$ in [@DFK], and it was later shown in [@DFW] (again, for $\alpha <0$) that a similarly defined estimator with divergence integrals has the properties of weak consistency and asymptotic normality for $H\in \Big(\frac{1}{2},\,\frac{3}{4}\Big)$. Strong consistency was then proved in [@BESV].
We will consider the same construction for the non-ergodic case and $H\in \Big(\frac{1}{2},\,1\Big)$ and investigate its asymptotic properties. We will prove strong consistency for our proposed estimator. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic functions and a non-central Cauchy limiting result for the mean reverting parameter. Both limits are uncorrelated. This means in particular, we will show that the asymptotics is partly inherited from the ergodic case treated in [@DFW] and partly follows the results for the non-ergodic case in [@BESO]. In addition we show that in the special case that the periodicity parameter is the weight of a periodic function, whose integral over the period is zero, we get a faster rate of convergence independent of $H$, namely $\sqrt{T}$. In this case the component is uncorrelated to all other components. Note, that this case was not yet treated both in the ergodic and non-ergodic case.
The structure of the paper is as follows. In Section \[sec:Setting\] the setting will be explained in more detail, such that we can proceed with the motivation and definition of the estimator in Section \[sec:Constr\]. Section \[sec:Aux\] contains auxiliary convergence statements, and finally in Section \[sec:Asympt\] the main results and their proofs are presented. The Appendix provides some further technical results.
Setting {#sec:Setting}
=======
Let $(B^H_t)_{t\in\mathbb R^+}$ be a fractional Brownian motion with known Hurst index $H\in \big(\frac{1}{2},\,1\big)$. Consider a stochastic differential equation ( SDE) of the following form:$$\label{eq:1}
\begin{split}
X_t & = X_0 + \int_0^t L(s)+\alpha X_s ds + \int_0^t \sigma dB^H_s,\\
X_0 &= x_0 \in \mathbb R .
\end{split}$$
$L$ is assumed to be a bounded $1$-periodic function which can be written as a linear combination of $p$ known bounded $1$-periodic $L^2([0,\,1])$-orthonormal functions with unknown real coefficients, $\mu=(\mu_1, \cdots, \mu_p)$ i.e. $$L(s)=\sum_{i=1}^{p} \mu_i \phi_i (s)\text{ for all }s\in[0,\,1].$$ We assume that the mean-reverting parameter $\alpha>0$ is also unknown. As argued in [@Rao], $\sigma$ can be estimated with probability one on any finite time interval, therefore it can be assumed to be known.
Moreover, it is important to define the stochastic integral appearing in the equation. In this paper we will consider it to be defined in Young’s sense (cf. [@You]). The integrals are well defined due to Hölder continuity of paths of the fractional Brownian motion of order $H$. Note that for deterministic integrands stochastic integrals in Young’s sense almost surely coincide with Skorokhod integrals.
The equation has a solution with almost surely continuous paths, which can be written as $$X_t = e^{\alpha t}x_0 +e^{\alpha t}\int_0^t e^{-\alpha s} L(s) ds + \sigma e^{\alpha t}\int_0^t e^{-\alpha s}dB^H_s$$ for $\alpha >0$. Let us fix the notation $\xi_t:= e^{\alpha t}\int_0^t e^{-\alpha s}dB^H_s$, $\tilde{\xi}_t:= e^{-\alpha t}X_t$ as well as $$\xi_{\infty}:= \int_0^\infty e^{-\alpha s}dB^H_s$$ and $$\tilde{\xi}_{\infty}:=x_0 + \int_0^\infty e^{-\alpha s} L(s) ds + \sigma\int_0^\infty e^{-\alpha s}dB^H_s.$$ We assume to observe $X$ continuously on \[0,T\] and derive limits for $T\to\infty$.
Construction of the estimator {#sec:Constr}
=============================
The estimator that we will consider has the same structure as the estimator defined in [@DFW] for the ergodic case. We will briefly outline the motivation as it was given there. The construction follows the least squares method applied to a discretised version of a more general equation $$dX_t = \langle\theta,\,f(t,\,X_t)\rangle dt + \sigma dB^H_t,$$ where $\theta:=(\theta_1,\dots ,\theta_{p+1})$ is a parameter vector and $f(t,\,x):=(f_1(t,\,x),\dots ,f_{p+1}(t,\,x) )$ is a collection of known real-valued functions. For a time interval $[0,\,T]$ and a uniform mesh size $\Delta t:=T\slash N$ the least squares approach for the equations $$X_{(i+1)\Delta t}-X_{i\Delta t}=\sum_{j=1}^{p+1}f_j (i\Delta t,\, X_{i\Delta t})\theta_j \Delta t + \sigma (B^H_{(i+1)\Delta t}-B^H_{i\Delta t}),\,i\in \{1,\dots ,N \},$$ yields the estimator $\tilde{\theta}_{T,\,\Delta t}=Q_{T,\,\Delta t}^{-1}P_{T,\,\Delta t}$ with $$Q_{T,\,\Delta t}=\left( \sum_{i=0}^N f_j(i\Delta t,\, X_{i\Delta t})f_k (i\Delta t,\,X_{i\Delta t})\Delta t\right)_{j,\,k\in \{1,\dots , p+1 \}}$$ and $$\begin{aligned}
P_{T,\,\Delta t}=\Big(\sum_{i=0}^N f_1(i\Delta t,\, X_{i\Delta t})&(X_{(i+1)\Delta t} - X_{i\Delta t}), \dots , \\
& \sum_{i=0}^N f_{p+1}(i\Delta t,\, X_{i\Delta t})(X_{(i+1)\Delta t} - X_{i\Delta t}) \Big)^T.\end{aligned}$$ Replacing the sums by their continuous counterparts and considering the special case $\theta = \vartheta := (\mu_1,\dots ,\mu_p,\,\alpha)$, $f(t,\,x)=(\phi_1 (t),\dots ,\phi_p(t),\, x)$ as well as putting $T=n$ we obtain the estimator $\hat{\vartheta}:=Q_n^{-1}P_n$ with $$P_n=\left(\int_0^{n} \phi_1 (t) dX_t,\dots ,\int_0^{n} \phi_p (t) dX_t,\, \int_0^{n} X_t dX_t\right)$$ and $$Q_n=\begin{pmatrix}
n E_p & a_n\\
a_n^T & b_n
\end{pmatrix},$$ where $$a_n^T=\left(\int_0^{n} \phi_1 (t) X_t dt\dots , \int_0^{n} \phi_p (t) X_t dt\right),$$ $$b_n=\int_0^{n} X_t^2 dt.$$ The two following results are an immediate analogy to the calculations in [@DFW].
We have $\hat{\vartheta}_n = \vartheta + \sigma Q_n^{-1}R_n$, where $$R_n = \left(\int_0^n \phi _1 (t) dB^H_t,\dots , \int_0^n \phi_p (t)dB^H_t,\, \int_0^n X_t dB^H_t\right)^T.$$
Since $$\int_0^n \phi_i(t)dX_t= \sum_{j=1}^p \mu_j \int_0^n \phi_i(t)\phi_j(t)dt+\alpha \int_0^n \phi_i(t)X_t dt + \sigma \int_0^n \phi_i (t)dB^H_t$$ for $i\in \{1,\dots p \}$ and $$\int_0^n X_t dX_t= \sum_{j=1}^p \mu_j \int_0^n X_t\phi_j(t)dt+\alpha \int_0^n X^2_t dt + \sigma \int_0^n X_t dB^H_t,$$ we have $P_n = Q_n\vartheta + \sigma R_n$, and the claim follows.
We have an explicit representation for $Q_n^{-1}$, namely $$Q_n^{-1}=\frac{1}{n}\begin{pmatrix}
E_p + \gamma_n \Lambda_n \Lambda_n^t & -\gamma_n\Lambda_n\\
-\gamma_n \Lambda_n^t & \gamma_n
\end{pmatrix}$$ with $$\Lambda_n = (\Lambda_{n,\,1},\dots , \Lambda_{n,\,p})^t =\left(\frac{1}{n}\int_0^n \phi_1 (t)X_t dt,\dots , \frac{1}{n}\int_0^n \phi_p (t)X_t dt\right)$$ and $\gamma_n = D_n^{-1} = \bigg(\frac{1}{n}\int_0^n X_t^2 dt-\sum_{i=1}^p \Lambda_{n,\,i}^2\bigg)^{-1}$.
This is a consequence of the fact that $$\begin{pmatrix}
n E_p & -a_n\\
-a_n^T & b_n
\end{pmatrix}^{-1}= \frac{1}{n}\begin{pmatrix}
E_p + \gamma_n \Lambda_n \Lambda_n^t & \gamma_n\Lambda_n\\
\gamma_n \Lambda_n^t & \gamma_n
\end{pmatrix},$$ which was proved in [@DFW].
Auxiliary convergence results {#sec:Aux}
=============================
In this section we provide some convergence results for the different components of the estimators defined in the previous section. They help to identify the dominating terms for the asymptotic behaviour of the estimator. The first lemma in this section as well as its proof are motivated by analogous results in [@BESO].
\[l:1\] With the above notation we have $e^{-\alpha t}X_t \to \tilde{\xi}_{\infty}$ as well as $e^{-2\alpha t}\int_0^t X_s^2 ds \to \frac{\tilde{\xi}_{\infty}^2}{2\alpha}$ almost surely.
The first statement follows directly from the fact that $\xi_t\to\xi_\infty$ a.s. (shown in Lemma 2, [@BESO]): $$e^{-\alpha t}X_t= x_0 +\int_0^t e^{-\alpha s} L(s) ds + \sigma \xi_t\to x_0 + \int_0^\infty e^{-\alpha s} L(s) ds + \sigma \xi_\infty \text{ a.s.}$$ For the second statement we start by noticing that $\tilde{\xi}_t$ is a process with a.s. continuous paths. We have for each $t\in\mathbb R^+$: $$\int_0^t X_s^2 ds \geq \int_{t\slash 2}^t e^{2\alpha s}\tilde{\xi}_s^2 ds \geq \frac{t}{2}e^{\alpha t}\inf_{\frac{t}{2}\leq s\leq t}\tilde{\xi}_s^2.$$ Since $\tilde{\xi}_t \to \tilde{\xi}_{\infty}$ a.s., it follows that $$\lim_{t\to\infty } \inf_{\frac{t}{2}\leq s\leq t}\tilde{\xi}_s^2 = \tilde{\xi}_{\infty}^2 \text{ a.s.}$$ From the fact that $\xi_\infty \sim N(0,\, \frac{H\Gamma (2H)}{\alpha^{2H}})$ (shown in [@BESO]) we can conclude that $\tilde{\xi}_{\infty}$ also follows a (non-degenerate) normal distribution, and hence, $\lim_{t\to\infty} \int_0^t X_s^2 ds = \infty$ a.s. Therefore, we get by l’Hôpital’s rule $$\lim_{t\to\infty} \frac{ \int_0^t e^{2\alpha s}\tilde{\xi}_s^2 ds}{e^{2\alpha t}}=\lim_{t\to\infty}\frac{\tilde{\xi}_t^2}{2\alpha}=\frac{\tilde{\xi}_{\infty}^2}{2\alpha}.$$
\[l:2\] For $i\in \{1,\dots , p \}$ the following statements hold almost surely:
1. $\frac{1}{n}\int_0^n \phi_i(t)dB^H_t\to 0$,
2. $e^{-\alpha n}\Lambda_{ni}\sqrt{n}\to 0$,
3. $nD_n e^{-2\alpha n}\to \frac{\tilde{\xi}_{\infty}^2}{2\alpha}$,
4. $e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n X_t dB^H_t\to 0$.
<!-- -->
1. This is an application of Lemma \[l:4\]: We have $$\begin{aligned}
\operatorname{{\mathbb E}}[(\frac{1}{n}\int_0^n \phi_i(t)dB^H_t)^2]=\frac{1}{n^2}\int_0^n \int_0^n \phi_i (u)\phi_i (v)|u-v|^{2H-2}du dv\lesssim n^{2H-2},\end{aligned}$$ and the result follows for $k=2$.
2. We write $\Lambda_{ni}$ as a sum of a deterministic and of a centred Gaussian part and show convergence separately: $$\begin{aligned}
e^{-\alpha n}\Lambda_{ni}=\frac{1}{n}e^{-\alpha n}\int_0^n \phi_i (t)&(e^{\alpha t}x_0 +e^{\alpha t}\int_0^t e^{-\alpha s}L(s)ds)dt\\
&+\frac{1}{n}e^{-\alpha n}\int_0^n \phi_i(t)\sigma e^{\alpha t}\xi_t dt=:A+B,\end{aligned}$$ where $\xi_t=\int_0^t e^{-\alpha r}dB^H_r$. For the deterministic part we write $$\begin{aligned}
\sqrt{n}A = \frac{1}{\sqrt{n}}e^{-\alpha n}\int_0^n \phi_i (t)e^{\alpha t}x_0 dt &+ \frac{1}{\sqrt{n}}e^{-\alpha n}\int_0^n e^{\alpha t}\int_0^t e^{-\alpha s}L(s)dsdt\\
&=:A_1+A_2,\end{aligned}$$ and we can bound the two summands as follows: $$\begin{aligned}
|A_1|\lesssim \frac{1}{\sqrt{n}}e^{-\alpha n}\int_0^n e^{\alpha t}dt=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n}}e^{-\alpha n}\to 0\end{aligned}$$ as well as $$\begin{aligned}
|A_2|\lesssim \frac{1}{\sqrt{n}}e^{-\alpha n}\int_0^n e^{\alpha t}\int_0^t e^{-\alpha s}ds dt = \frac{1}{\sqrt{n}}e^{-\alpha n}\int_0^n e^{\alpha t}dt - \frac{1}{\sqrt{n}}e^{-\alpha n}\to 0.\end{aligned}$$ We have shown convergence for the deterministic part and now we will calculate the second moment of the Gaussian part in order to apply Lemma \[l:4\]. $$\begin{aligned}
\operatorname{{\mathbb E}}[(\sqrt{n}B)^2]=&\operatorname{{\mathbb E}}[(\frac{1}{\sqrt{n}}e^{-\alpha n}\int_0^n \phi_i(t)\sigma e^{\alpha t}\xi_t dt)^2]\\
=&\frac{1}{n}e^{-2\alpha n}\int_0^n\int_0^n \phi_i(t)\phi_i (s)\sigma^2 e^{\alpha t}e^{\alpha s}\operatorname{{\mathbb E}}[\xi_t\xi_s]ds dt\end{aligned}$$ and we get by treating the stochastic integrals as Skorokhod integrals $$\begin{aligned}
\operatorname{{\mathbb E}}[\xi_t\xi_s] = \int_0^t \int_0^s e^{-\alpha r}e^{-\alpha v}|r-v|^{2H-2}dvdr.\end{aligned}$$ In total, we obtain $$\begin{aligned}
&\operatorname{{\mathbb E}}[(\sqrt{n}B)^2]\\
&=\frac{1}{n}e^{-2\alpha n}\sigma^2\int_0^n\int_0^n \phi_i(t)\phi_i (s) \int_0^t \int_0^s e^{\alpha s-\alpha r}e^{\alpha t-\alpha v}|r-v|^{2H-2}dvdrdsdt\\
&=\frac{1}{n}e^{-2\alpha n}\sigma^2\int_0^n\int_0^n |r-v|^{2H-2}\int_v^n \int_r^n \phi_i(t)\phi_i (s) e^{\alpha s-\alpha r}e^{\alpha t-\alpha v} ds dt dv dr\\
&\lesssim \frac{1}{\alpha^2}\frac{1}{n}e^{-2\alpha n}\sigma^2\int_0^n\int_0^n |r-v|^{2H-2} (e^{\alpha n-\alpha v}-1)(e^{\alpha n-\alpha r}-1)drdv\\
&\simeq \frac{1}{n}\int_0^n\int_0^n |r-v|^{2H-2} (e^{-\alpha v}-e^{-\alpha n})(e^{-\alpha r}-e^{-\alpha n})drdv\\
&\leq \frac{1}{n}\int_0^n\int_0^n |r-v|^{2H-2} e^{-\alpha v}e^{-\alpha r}drdv\lesssim \frac{1}{n},\end{aligned}$$ because the last integral is bounded (this was shown in [@HNu]). Lemma \[l:4\] yields almost sure convergence to zero and hence the desired result.
3. This follows from the previous result and Lemma \[l:1\]: $$\begin{aligned}
D_nne^{-2\alpha n}=e^{-2\alpha n}\int_0^n X_t^2 dt- \underbrace{\sum_{i=1}^p (\sqrt{n}\Lambda_{ni}e^{-\alpha n})^2}_{\to 0\text{ by }(2)}\to \frac{\tilde{\xi}_{\infty}^2}{2\alpha}.\end{aligned}$$
4. We plug in the expression $X_t$ and get $$\begin{aligned}
e^{-\alpha n}&\frac{1}{\sqrt{n}}\int_0^n X_t dB^H_t=e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n e^{\alpha t}x_0 dB^H_t\\
&+ e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n e^{\alpha t} \int_0^t e^{-\alpha s}L(s)ds dB^H_t\\
&+ e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n \sigma e^{\alpha t}\int_0^t e^{-\alpha s}dB^H_s dB^H_t=:A+B+C.\end{aligned}$$ The integral in $A$ can again be interpreted as a Skorokhod integral (yielding a centred Gaussian random variable) which allows us the computation of its $L^2$ norm: $$\begin{aligned}
\operatorname{{\mathbb E}}[A^2] &= x_0^2\frac{1}{n}e^{-2\alpha n}\int_0^n\int_0^n e^{\alpha u}e^{\alpha v}|u-v|^{2H-2}du dv\\
&=x_0^2\frac{1}{n}\underbrace{\int_0^n\int_0^n e^{-\alpha (n-u)}e^{-\alpha (n-v)}|u-v|^{2H-2}du dv}_{=:I_n}\lesssim \frac{1}{n},\end{aligned}$$ because $I_n$ is bounded as was shown in [@HNu]. Lemma \[l:4\] implies almost sure convergence. For $B$, which is also a centred Gaussian sequence, the calculation is similar: $$\begin{aligned}
&\operatorname{{\mathbb E}}[B^2]\\
&=\frac{1}{n}e^{-2\alpha n} \int_0^n\int_0^n e^{\alpha u} \int_0^u e^{-\alpha s}L(s)ds e^{\alpha v}\int_0^v e^{-\alpha r}L(r)dr|u-v|^{2H-2}du dv\\
&\lesssim \frac{1}{n}e^{-2\alpha n} \int_0^n\int_0^n e^{\alpha u} (1-e^{-\alpha u}) e^{\alpha v}(1-e^{-\alpha v})|u-v|^{2H-2}du dv\\
&=\frac{1}{n}e^{-2\alpha n} \int_0^n\int_0^n (e^{\alpha u}-1)(e^{\alpha v}-1)|u-v|^{2H-2}du dv\leq \frac{1}{n}I_n\lesssim \frac{1}{n},\end{aligned}$$ and the almost sure convergence follows. For $C$ we use Lemma 4 from [@BESO] to decompose the double integral: $$\begin{aligned}
C=e^{-\alpha n}&\frac{1}{\sqrt{n}}\sigma (\int_0^n e^{\alpha s}dB^H_s \int_0^t e^{-\alpha r}dB^H_r - \int_0^n e^{-\alpha s}\int_0^s e^{\alpha r}\delta B^H_r \delta B^H_s\\
& - H(2H-1)\int_0^n e^{-\alpha s}\int_0^s e^{\alpha r} |s-r|^{2H-2}dr ds)=:C_1-C_2-C_3,\end{aligned}$$ where $\delta$ stands for the Skorokhod integral. We show almost sure convergence for the three summands: $$\begin{aligned}
C_1=\sigma e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n e^{\alpha s}dB^H_s \xi_t,\end{aligned}$$ and hence we know from [@BESO] that $\xi_t\to\xi_{\infty}\sim N(0,\,\frac{H\Gamma (2H)}{\alpha^{2H}})$ a.s., it is enough to show that $e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n e^{\alpha s}dB^H_s \to 0$ almost surely. Since it is a centred Gaussian sequence, we again rely on Lemma \[l:4\] and compute the respective variances: $$\begin{aligned}
\operatorname{{\mathbb E}}[(e^{-\alpha n}\frac{1}{\sqrt{n}}&\int_0^n e^{\alpha s}dB^H_s)^2]\\
&\simeq e^{-2\alpha n}\frac{1}{n}\int_0^n\int_0^n e^{\alpha s}e^{\alpha r}|s-r|^{2H-2}dsdr = \frac{1}{n}I_n\lesssim \frac{1}{n}.\end{aligned}$$ In order to treat $C_2$ note that by Lemma 7 in [@BESO] $$Y_n:= e^{-\frac{\alpha n}{2}}\int_0^n e^{-\alpha s}\int_0^s e^{\alpha r}\delta B^H_r \delta B^H_s\stackrel{L^2}{\to} 0,$$ and consequently $\operatorname{{\mathbb E}}[Y_n^2]$ is bounded. Since, moreover, $Y_n$ is centred (as it is a Skorokhod integral), Markov inequality helps achieve the summability of tails: $$\begin{aligned}
\sum_{n=1}^{\infty} P(|C_2(n)|\geq \varepsilon)= \sum_{n=1}^{\infty} & P(|\frac{1}{\sqrt{n}}e^{-\frac{\alpha n}{2}} Y_n|\geq \varepsilon)\\
&\leq \sum_{n=1}^{\infty} \frac{\operatorname{{\mathbb E}}[Y_n^2]}{\varepsilon^2 n e^{\frac{\alpha n}{2}} }\lesssim \sum_{n=1}^{\infty} \frac{1}{n e^{\frac{\alpha n}{2}} } <\infty,\end{aligned}$$ and almost sure convergence to zero follows. Finally, Lemma 7 in [@BESO] ensures that $C_3 e^{\frac{\alpha n}{2}}\sqrt{n}$ converges to zero, which implies that also $C_3$ itself goes to zero as $n$ tends to infinity. This completes the proof of the initial claim.
\[cor:1\] For $\beta <\frac{1}{2}$ also $n^{\beta} e^{-\alpha n}\Lambda_{ni}\sqrt{n}\to 0 $ as well as $n^{\beta} e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n X_t dB^H_t\to 0$ almost surely.
The deterministic part of the sequence $n^{\beta} e^{-\alpha n}\Lambda_{ni}\sqrt{n}$ (i.e. $n^{\beta}\sqrt{n} A$, cf. the notation from the proof of (2) in \[l:2\]) is bounded up to a constant by $n^{\beta-0.5}$ and the variance of the random part by $n^{2\beta -1}$. This yields polynomial rates of convergence, thus Lemma \[l:4\] still can be applied and we obtain almost sure convergence. The same argument holds for the second convergence result. Lemma \[l:4\] can still be applied for $A$, $B$ and $C_1$ (from the proof of (4) in \[l:2\]), and for $C_2$ and $C_3$ the additional factor $n^\beta$ changes nothing in the structure of the arguments, so the proofs can be followed verbatim.
Asymptotic properties of $\hat{\vartheta}$ {#sec:Asympt}
==========================================
In this section we examine the asymptotic properties of our estimator. First we prove strong consistency, than we provide a second order limit result, which shows the substantially different behaviour of the parameters of the periodic function and the mean reverting parameter, namely we get both different limiting distributions and different rates. Finally, we will show that for basis functions $\phi$ with $\int_0^1 \phi(s) ds=0$ the rate in the central limit theorem improves to $\sqrt{n}$ independent of $H$. In this case we provide two representations of the asymptotic variance, one involving sums over the Riemann zeta function, the other an integral representation.
\[th:1\] $\hat{\vartheta}$ is strongly consistent, i.e.
1. for $i\in \{1,\dots , p\}$ $$\begin{aligned}
\hat{\mu_i}-\mu_i & = \sigma \frac{1}{n}(\int_0^n \phi_i (t)dB^H_t\\
&+ \frac{1}{D_n}\sum_{j=1}^p \Lambda_{ni}\Lambda_{nj}\int_0^n \phi_j (t)dB^H_t - \frac{1}{D_n}\Lambda_{ni}\int_0^n X_t dB^H_t)\to 0,\end{aligned}$$
2. $\hat{\alpha}-\alpha= -\sigma \frac{1}{nD_n}(\sum_{i=1}^p \Lambda_{ni}\int_0^n \phi_i (t)dB^H_t - \int_0^n X_t dB^H_t)\to 0$,
both almost surely.
We treat each summand separately and exploit Lemma \[l:2\].
1. Let us denote $M_1:= \frac{1}{n}\int_0^n \phi_i (t)dB^H_t$, $M_{2j}:= \frac{1}{n}\frac{1}{D_n}\Lambda_{ni}\Lambda_{nj}\int_0^n \phi_j (t)dB^H_t$, $M_3:= \frac{1}{n}\frac{1}{D_n}\Lambda_{ni}\int_0^n X_t dB^H_t $. In order to prove the claim we have to show that each of these summands converges to zero almost surely. For $M_1$ this was shown in Lemma \[l:2\] (1). To see this for $M_{2j}$ we rewrite it as follows: $$\begin{aligned}
M_{2j}&=\frac{1}{n}\frac{1}{D_n}\Lambda_{ni}\Lambda_{nj}\int_0^n \phi_j (t)dB^H_t\\
& = \underbrace{\frac{1}{n D_ne^{-2\alpha n}}}_{\to \frac{2\alpha}{\tilde{\xi}_{\infty}^2} \text{ by }\ref{l:2} (3)} \underbrace{(e^{-\alpha n}\Lambda_{ni}\sqrt{n})}_{\to 0 \text{ by }\ref{l:2} (2)}\underbrace{(e^{-\alpha n}\Lambda_{nj}\sqrt{n})}_{\to 0 \text{ by }\ref{l:2} (2)} \underbrace{\frac{1}{n}\int_0^n \phi_j (t)dB^H_t}_{\to 0 \text{ by }\ref{l:2} (1)},\end{aligned}$$ and since $\tilde{\xi}_{\infty}$ is almost surely nonzero, the whole expression converges a.s. to zero. $M_3$ can also be rewritten in a way that makes the convergence statement obvious: $$\begin{aligned}
M_3 &= \frac{1}{n}\frac{1}{D_n}\Lambda_{ni}\int_0^n X_t dB^H_t\\
& = \underbrace{\frac{1}{n D_ne^{-2\alpha n}}}_{\to \frac{2\alpha}{\tilde{\xi}_{\infty}^2} \text{ by }\ref{l:2} (3)} \underbrace{(e^{-\alpha n}\Lambda_{ni}\sqrt{n})}_{\to 0 \text{ by }\ref{l:2} (2)} \underbrace{(e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n X_t dB^H_t)}_{\to 0 \text{ by }\ref{l:2} (4)},\end{aligned}$$ the claim follows with the same argument as above and completes the proof of the theorem’s statement.
2. In this case we also start by introducing a notation for each type of summands. Let us denote $A_{1i}:=\frac{1}{nD_n} \Lambda_{ni}\int_0^n \phi_i (t)dB^H_t$ and $A_2:= \frac{1}{nD_n}\int_0^n X_t dB^H_t$. For the first type of summands we write $$\begin{aligned}
A_{1i}&= \frac{1}{nD_n} \Lambda_{ni}\int_0^n \phi_i (t)dB^H_t\\
&= \underbrace{\frac{1}{n D_ne^{-2\alpha n}}}_{\to \frac{2\alpha}{\tilde{\xi}_{\infty}^2} \text{ by }\ref{l:2} (3)} \underbrace{(e^{-\alpha n}\Lambda_{ni}\sqrt{n})}_{\to 0 \text{ by }\ref{l:2} (2)} \underbrace{\sqrt{n}e^{-\alpha n}}_{\to 0} \underbrace{\frac{1}{n}\int_0^n \phi_i (t)dB^H_t}_{\to 0 \text{ by }\ref{l:2} (1)}\end{aligned}$$ and for the second kind we obtain $$\begin{aligned}
A_2 &= \frac{1}{nD_n}\int_0^n X_t dB^H_t\\
&= \underbrace{\frac{1}{n D_ne^{-2\alpha n}}}_{\to \frac{2\alpha}{\tilde{\xi}_{\infty}^2} \text{ by }\ref{l:2} (3)} \underbrace{\sqrt{n}e^{-\alpha n}}_{\to 0} \underbrace{(e^{-\alpha n}\frac{1}{\sqrt{n}}\int_0^n X_t dB^H_t)}_{\to 0 \text{ by }\ref{l:2} (4)}.\end{aligned}$$ Both calculations yield almost sure convergence of the summands (again, using the argument given in (1)) and thus provide the proof for the initial claim.
The next lemma is an auxiliary result for the second order limit theorem that will be proved later.
\[l:3\] Let $F$ be any $\sigma (B^H)$-measurable random variable such that $P(F<\infty)=1$. Then, as $n\to\infty$, $$(n^{-H}\delta_n (\phi_1),\dots , n^{-H}\delta_n(\phi_p),\, F,\, e^{-\alpha n}\delta_n( e^{\alpha \cdotp}))\stackrel{d}{\to}(Z_1,\dots , Z_p,\, F,\,Z),$$ where $\delta_n$ is the integral over $[0,\,n]$ with respect to $B^H$, $Z_1,\dots , Z_p$ are centred and jointly normally distributed with the covariance matrix $(\int_0^1 \phi_i(x)dx\int_0^1 \phi_j(x)dx)_{i,\,j=1,\dots , p}$ and $((Z_1,\dots , Z_p),\, F,\, Z)$ are independent. Moreover, $\operatorname{Var}(Z)=\frac{H\Gamma (2H)}{\alpha^{2H}}$.
Due to an approximation argument rigorously explained in [@ESN] it is enough to show that for any $d\geq 1$, $s_1,\dots , s_d\in [0,\,\infty)$ $$\begin{aligned}
(n^{-H}\delta_n (\phi_1),\dots , n^{-H}\delta_n(\phi_p), &\, B^H_{s_1},\dots , B^H_{s_d} ,\, e^{-\alpha n}\delta_n( e^{\alpha \cdotp}))\\
& \stackrel{d}{\to}(Z_1,\dots , Z_p,\, B^H_{s_1},\dots , B^H_{s_d},\,Z)\end{aligned}$$ as $n\to\infty$. The left hand side is a Gaussian vector, and hence it suffices to determine the limits of the covariances. It was shown in [@BESO] that the limits of $\operatorname{Cov}(B^H_s,\, e^{-\alpha n}\delta_n( e^{\alpha \cdotp}))$ and $\operatorname{Var}( e^{-\alpha n}\delta_n( e^{\alpha \cdotp}))$ are as claimed. Moreover, in [@BESV] the joint limiting distribution of $(n^{-H}\delta_n (\phi_1),\dots , n^{-H}\delta_n(\phi_p))$ was established. Therefore, we only have to show that $\operatorname{Cov}( n^{-H}\delta_n (\phi_i), \, B^H_s )$ and $\operatorname{Cov}( n^{-H}\delta_n (\phi_i),\, e^{-\alpha n}\delta_n( e^{\alpha \cdotp}) )$ converge to zero. For the first statement recall that $B^H_s = \int_0^n 1_{[0,\, s]}dB^H_t$ for any $n\geq s$. Then we can write (for $n$ large enough) due to the isometry property of the integrals: $$\begin{aligned}
\operatorname{{\mathbb E}}&[ n^{-H}\delta_n (\phi_i) B^H_s] \lesssim n^{-H}\int_0^n \int_0^s |u-v|^{2H-2}du dv\\
& =n^{-H}\int_0^s \int_{-v}^{n-v}|z|^{2H-2}dz dv = n^{-H}\int_0^s \int_0^v z^{2H-2}dz + \int_0^{n-v}z^{2H-2} dz dv\\
& = \underbrace{n^{-H}\int_0^s v^{2H-1}dv}_{\to 0}+ n^{-H}\int_0^s (n-v)^{2H-1}dv\lesssim n^{-H}\int_{n-s}^n z^{2H-1}dz\\
& =n^{-H}(n^{2H}-(n-s)^{2H})\stackrel{\text{binom. series}}{=}n^{-H}O(n^{2H-1})=O(n^{H-1}),\end{aligned}$$ which tends to zero as $n$ tends to infinity.
For the second convergence refer to Proposition \[prop:2\] in the Appendix for the estimation $\int_0^t e^{\alpha u}u^{2H-2}du \lesssim t^{2H-2}e^{\alpha t}$. We use this for our calculation: $$\begin{aligned}
\operatorname{{\mathbb E}}&[ n^{-H}\delta_n (\phi_i) e^{-\alpha n}\delta_n( e^{\alpha \cdotp}) ] \lesssim n^{-H} e^{-\alpha n} \int_0^n \int_0^n e^{\alpha v} |u-v|^{2H-2}du dv\\
& = n^{-H} e^{-\alpha n} \int_0^n e^{\alpha u} \int_0^n e^{\alpha (v-u)}|v-u|^{2H-2}dv du\\
& =n^{-H}e^{-\alpha n} \int_0^n \left(\underbrace{\int_0^u e^{-\alpha z} z^{2H-2}dz}_{\text{bdd}}+ \int_0^{n-u} e^{\alpha z}z^{2H-2}dz\right) du\\
& \lesssim n^{-H}e^{-\alpha n} \int_0^n e^{\alpha u} e^{\alpha (n-u)}(n-u)^{2H-2}du = n^{-H}n^{2H-1}\to 0. \end{aligned}$$
Now we can proceed with the second order limit theorem for our estimator.
\[th:2\] $$(n^{1-H}(\hat{\mu}_1-\mu_1,\dots ,\hat{\mu}_p-\mu_p),\, e^{\alpha n}(\hat{\alpha}-\alpha))\stackrel{d}{\to}\sigma (Z_1,\dots ,\, Z_p,\, Z_{p+1})$$ with $Z_1,\dots ,Z_p$ as above and $Z_{p+1}= 2\alpha N\slash M$ with $N\sim \operatorname{N}(0,\,1)$ and $$M\sim \operatorname{N}\left(\frac{ \alpha^H}{\sqrt{H\Gamma(2H)}} \left(x_0+\int_0^{\infty}e^{-\alpha s}L(s)ds\right),\,1\right)$$ independent of $N$. Moreover, $(Z_1, \dots ,Z_p)$ and $Z_{p+1}$ also are independent.
This result reflects the structure of the estimator: In the first $p$ components the additive term $\sigma \frac{1}{n}\int_0^n \phi_i(t)dB^H_t$ is the slowest summand (note that it does not include the solution process $X$ and is, therefore, not influenced by its exponential growth), which yields the same rates of convergence as in the ergodic case. The estimator for $\alpha$, however, does not contain such a term; it converges with the same exponential rate as the estimator in [@BESO]. The limiting distribution is also structurally similar to the case $L\equiv 0$. As mentioned in [@Moe], if the estimator from [@BESO] is applied for an equation with a non zero starting value, the limiting distribution will also contain this value as an additional additive term in the denominator. Moreover, due to the possibility of considering Young integrals and exploiting different techniques in the proofs our results are valid for $H\in \Big(\frac{1}{2},\,1\Big)$ in contrast to only $H\in \Big(\frac{1}{2},\,\frac{3}{4}\Big)$ for the ergodic case in [@DFW].
First of all we divide the error into parts that contribute to the limit and the rest. We use the notation from the previous theorem and write: $n^{1-H}(\hat{\mu}_1-\mu_1)=\sigma (n^{1-H}M_1+n^{1-H}(\sum_{j=1}^p M_{2j}+M_3))$, $e^{\alpha n}(\hat{\alpha}-\alpha)=\sigma (-e^{\alpha n}\sum_{j=1}^p A_{1j}+ e^{\alpha n} A_2)$. Now we will identify the rest terms by showing: $n^{1-H}(\sum_{j=1}^p M_{2j}+M_3)$ and $e^{\alpha n}\sum_{j=1}^p A_{1j}$ converge to zero almost surely. For $M_{2j}$ and $M_3$ this follows from the fact that they contain the factor $(e^{-\alpha n}\Lambda_{nj}\sqrt{n})$ which would still converge to zero if multiplied by $n^{1-H}$, since $1-H<0.5$. $A_{1j}$ contains the factor $(e^{-\alpha n}\Lambda_{ni}\sqrt{n}) \sqrt{n}e^{-\alpha n} \frac{1}{n}\int_0^n \phi_j (t)dB^H_t$ converging to zero almost surely. The remainder $\frac{1}{n D_ne^{-2\alpha n}}$ tends almost surely to a random variable. We write $$\begin{aligned}
e^{\alpha n}&(e^{-\alpha n}\Lambda_{ni}\sqrt{n}) \sqrt{n}e^{-\alpha n} \frac{1}{n}\int_0^n \phi_j (t)dB^H_t = (e^{-\alpha n}\Lambda_{ni}\sqrt{n}) \sqrt{n}\frac{1}{n}\int_0^n \phi_j (t)dB^H_t\\
& = (e^{-\alpha n}\Lambda_{ni}\sqrt{n} n^{H-0.5})\left(n^{1-H} \frac{1}{n}\int_0^n \phi_j (t)dB^H_t\right).\end{aligned}$$ The factor $e^{-\alpha n}\Lambda_{ni}\sqrt{n} n^{H-0.5}$ converges to zero almost surely, because $H-0.5<0.5$ and the factor $n^{1-H} \frac{1}{n}\int_0^n \phi_j (t)dB^H_t$ converges in distribution to a normal random variable (this being a consequence of the previous lemma). In total we conclude that the above expression converges to zero in distribution and therefore in probability. Thus, also the whole term $e^{\alpha n}A_{1j}$ converges to zero in probability.
The next step is to consider and rewrite $A_2$. For this we apply the change of variables formula for Young integrals to the functions $e^{-\alpha n}X_n$ and $\int_0^n e^{\alpha t}dB^H_t$. We obtain the following formula: $$\begin{aligned}
\int_0^n X_s dB^H_s = \int_0^n & e^{\alpha t} dB^H_t \tilde{\xi}_n -\int_0^n e^{-\alpha t}L(t)\int_0^t e^{\alpha s}dB^H_s dt\\
& - \int_0^n \sigma e^{-\alpha t}\int_0^t e^{\alpha s}dB^H_s dB^H_t=:S_1+S_2+S_3,\end{aligned}$$ with which we can substitute the term $\int_0^n X_s dB^H_s$ in $A_2$. We will now show that only $S_1$ contributes to the convergence statement. Since $$e^{\alpha n}A_2 = \frac{1}{n D_ne^{-2\alpha n}} e^{-\alpha n} \int_0^n X_t dB^H_t$$ and the denominator converges almost surely, it is enough to show that $ e^{-\alpha n} (S_2+S_3)$ tend to zero in probability. For $S_3$ this has been shown in [@BESO], so we only show this for $S_2$. As a Lebesgue integral of a Gaussian process $e^{-\alpha n}S_2$ is again centred Gaussian, therefore showing convergence of the second moments will suffice: $$\begin{aligned}
\operatorname{{\mathbb E}}&\left[ \left( e^{-\alpha n} \int_0^n e^{-\alpha t}L(t)\int_0^t e^{\alpha s}dB^H_s dt \right)^2\right]\\
&\lesssim e^{-2 \alpha n}\int_0^n \int_0^n e^{-\alpha u} L(u)e^{-\alpha v} L(v)\int_0^u \int_0^v e^{\alpha s}e^{\alpha r}|s-r|^{2H-2}ds dr du dv\\
&\lesssim e^{-2 \alpha n}\int_0^n \int_0^n \int_0^u \int_0^v |s-r|^{2H-2}ds dr du dv \lesssim e^{-2 \alpha n} n^{2H+2}\to 0\end{aligned}$$ as $n$ tends to infinity.
For the last step of the proof we apply Lemma \[l:3\] to $F=\tilde{\xi}_{\infty}$ and obtain $$(n^{-H}\delta_n (\phi_1),\dots , n^{-H}\delta_n(\phi_p),\, \tilde{\xi}_{\infty},\, e^{-\alpha n}\delta_n( e^{\alpha \cdotp}))\stackrel{d}{\to}(Z_1,\dots , Z_p,\, \tilde{\xi}_{\infty},\,Z),$$ and consequently $$\left(n^{-H}\delta_n (\phi_1),\dots , n^{-H}\delta_n(\phi_p),\, \frac{e^{-\alpha n}\int_0^n e^{\alpha t}dB^H_t}{\tilde{\xi}_{\infty}} \right)\stackrel{d}{\to}\left(Z_1,\dots , Z_p,\, \frac{Z}{\tilde{\xi}_{\infty}}\right),$$ where $Z\sim \sqrt{\frac{H\Gamma (2H)}{\alpha^{2H}}}\operatorname{N}(0,\, 1)$ and $$\tilde{\xi}_{\infty}\sim \sqrt{\frac{H\Gamma (2H)}{\alpha^{2H}}} \operatorname{N}\left(\frac{ \alpha^H}{\sqrt{H\Gamma(2H)}} \left(x_0+\int_0^{\infty}e^{-\alpha s}L(s)ds\right),\,1\right) .$$ Now note additionally that $$(1,\dots, 1,\, \frac{\tilde{\xi}_n \tilde{\xi}_{\infty}}{n D_ne^{-2\alpha n}})\stackrel{\text{a.s.}}{\to} (1,\dots , 1,\, 2\alpha).$$ Multiplying both vectors elementwise using Slutsky’s lemma yields $$(n^{-H}\delta_n (\phi_1),\dots , n^{-H}\delta_n(\phi_p),\, \frac{e^{-\alpha n}S_1}{n D_ne^{-2\alpha n}} )\stackrel{d}{\to}(Z_1,\dots , Z_p,\, 2\alpha \frac{Z}{\tilde{\xi}_{\infty}}),$$ which is all that we needed to show, since all the other summands converge to zero in probability. Note that we inherit the independence statement directly from Lemma \[l:3\].
Consider the special case of a basis element $\phi_k$, $k\in \{1,\dots ,p\}$, which integrates to zero on $[0,\,1]$. The results of our theorems continue to hold, but the limiting vector $(Z_1,\dots ,Z_p)$ will have a zero entry at $Z_k$. This suggests that the convergence of the estimator’s $k$th component might be of a better order than $n^{H-1}$. Indeed, one obtains the following facts.
\[prop:1\] If $\phi_k$ for $k\in \{1,\dots ,p\}$ is such that $\int_0^1 \phi_k(t)dt=0$, then $$\sqrt{n}(\hat{\mu}_k-\mu_k)\stackrel{d}{\to}\sigma H(2H-1)\bar{Z}_k,$$ where $\bar{Z}_k$ is a zero mean Gaussian random variable with variance $$\begin{aligned}
\int_0^1 &\int_0^1 \phi_k (t)\phi_k (s)|t-s|^{2H-2}dt ds\\
&+ \sum_{l=1}^\infty 2 {{2H-2}\choose{2l}}\zeta (2l+2-2H) \int_0^1\int_0^1 \phi_k (t)\phi_k (s)(t-s)^{2l}dt ds,\end{aligned}$$ where $\zeta$ denotes the Riemann zeta function.
Recall that $$\sqrt{n}(\hat{\mu}_1-\mu_1)=\sigma \left(\sqrt{n} M_1+\sqrt{n}\left(\sum_{j=1}^p M_{2j}+M_3\right)\right)$$ with the notation from Theorem \[th:1\]. As in Theorem \[th:2\], Corollary \[cor:1\] ensures that $\sqrt{n}M_{2j}$ and $\sqrt{n}M_3$ converge to zero almost surely. Given that $$\sigma\sqrt{n}M_1=\sigma \frac{1}{\sqrt{n}}\int_0^n \phi_k(t)dB^H_t,$$ it is enough for our claim to investigate the term $\mathbb E \left[ \left(\frac{1}{\sqrt{n}}\int_0^n \phi_k(t)dB^H_t\right)^2\right]$.\
With $\alpha_H= H(2H-1)$ we have by isometry and periodicity: $$\begin{aligned}
\label{eq:2}\nonumber
\frac{1}{\alpha_H} & \mathbb E \left[ \left(\frac{1}{\sqrt{n}}\int_0^n \phi_k(t)dB^H_t\right)^2\right]\\ \nonumber
&=\frac{1}{n}\int_0^n \int_0^n \phi_k (t)\phi_k (s)|t-s|^{2H-2}dt ds\\ \nonumber
&=\frac{1}{n}\sum_{i,\,j =0}^{n-1}\int_0^1\int_0^1 \phi_k(t)\phi_k(s)|t+i-s-j|^{2H-2}dt ds\\ \nonumber
&= \frac{1}{n}\int_0^1 \int_0^1 \phi_k (t)\phi_k (s) n |t-s|^{2H-2}dtds\\ \nonumber
&\qquad +\frac{1}{n} \int_0^1\int_0^1 \phi_k(t)\phi_k(s)\sum_{i>j}|t-s+i-j|^{2H-2}dtds\\
&\qquad +\frac{1}{n} \int_0^1\int_0^1 \phi_k(t)\phi_k(s)\sum_{j>i}|s-t+j-i|^{2H-2}dtds.\end{aligned}$$ The first summand is constant with respect to $n$, hence, it remains to consider the second and the third one (which are equal for symmetry reasons). By rearranging the sum in the second summand, we obtain the following: $$\begin{aligned}
\frac{1}{n} & \int_0^1\int_0^1 \phi_k(t)\phi_k(s)\sum_{i>j}|t-s+i-j|^{2H-2}dtds\\
&=\frac{1}{n}\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}(n-m)|t-s+m|^{2H-2}dtds\\
&=\frac{1}{n}\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}(n-m)m^{2H-2}\left(\frac{t-s}{m}+1\right)^{2H-2}dtds\\
&=\frac{1}{n}\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}nm^{2H-2}\left(\frac{t-s}{m}+1\right)^{2H-2}dtds\\
&\qquad - \frac{1}{n}\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}m\cdotp m^{2H-2}\left(\frac{t-s}{m}+1\right)^{2H-2}dtds.\\\end{aligned}$$ Now we can use the binomial series expansion to get $$\left(\frac{t-s}{m}+1\right)^{2H-2} = \sum_{l=0}^\infty {2H-2 \choose l}(t-s)^lm^{-l}$$ and use the zero integral assumption in order to evaluate the above expression. We conclude: $$\begin{aligned}
\frac{1}{n}&\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}nm^{2H-2}\left(\frac{t-s}{m}+1\right)^{2H-2}dtds\\
&= \int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}m^{2H-2} \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^lm^{-l} dtds\\
&= \int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^l \sum_{m=1}^{n-1}m^{2H-2-l} dtds.\end{aligned}$$ By dominated convergence we now obtain $$\begin{aligned}
\lim_{n\to\infty} &\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^l \sum_{m=1}^{n-1}m^{2H-2-l} dtds\\
&= \int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^l \sum_{m=1}^{\infty}m^{2H-2-l} dtds\\
&=\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^l \zeta (l+2-2H) dtds,\end{aligned}$$ since the $m^{2H-2-l}$ are summable for $l\geq 1$.\
In a similar manner, we get $$\begin{aligned}
\frac{1}{n} &\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{m=1}^{n-1}m\cdotp m^{2H-2}\left(\frac{t-s}{m}+1\right)^{2H-2}dtds\\
& \int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^l \frac{1}{n}\sum_{m=1}^{n-1}m^{2H-1-l} dtds,\end{aligned}$$ which converges to zero, again, due to summability of $m^{2H-1-l}$.
In total, we conclude that the second summand in converges to $$\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(t-s)^l \zeta (l+2-2H) dtds,$$ and thus, with a symmetric calculation, the third summand tends to $$\int_0^1 \int_0^1 \phi_k(t)\phi_k(s) \sum_{l=2}^\infty {2H-2 \choose l}(s-t)^l \zeta (l+2-2H) dtds.$$ Adding up the two yields the desired result.
The next proposition provides some additional information about $\bar{Z}_k$ and provides a more concise form for its variance.
The variance of $\bar{Z}_k$ form the previous proposition can be simplified to $$\frac{1}{\Gamma (2-2H)}\int_0^1\int_0^1 \phi_k(t)\phi_k(s)\int_0^\infty \frac{u^{1-2H}}{e^u-1}(e^{u(1-|t-s|)}+e^{u|t-s|}-2)du dt ds.$$ This expression is positive for all bounded non zero $L^2$-functions $\phi_k$ with zero integrals.
Our goal is to show that $$\begin{aligned}
\int_0^1 &\int_0^1 \phi_k (t)\phi_k (s)|t-s|^{2H-2}dt ds\\
&+ \sum_{l=1}^\infty 2 {{2H-2}\choose{2l}}\zeta (2l+2-2H) \int_0^1\int_0^1 \phi_k (t)\phi_k (s)(t-s)^{2l}dt ds\end{aligned}$$ can be rewritten in the above integral form. For the first summand the definition of Gamma function provides the representation $$|t-s|^{2H-2}=\frac{1}{\Gamma (2-2H)} \int_0^\infty u^{1-2H}e^{-u|s-t|}du.$$ For the other summands we make use of the formula $\Gamma(z)\zeta (z)=\int_0^\infty \frac{u^{z-1}}{e^u-1}du$ for $z>1$ and rewrite them as follows: $$\begin{aligned}
2 & \sum_{l=1}^\infty {{2H-2}\choose{2l}}\zeta (2l+2-2H) \int_0^1\int_0^1 \phi_k (t)\phi_k (s)(t-s)^{2l}dt ds\\
&=2\int_0^1\int_0^1 \phi_k(t)\phi_k(s)\sum_{l=1}^\infty\frac{(2H-2)_{2l}}{(2l)!}\frac{1}{\Gamma (2l+2-2H)}\int_0^\infty \frac{u^{2l+1-2H}}{e^u-1}du (t-s)^{2l}ds dt\\
&=2\int_0^1\int_0^1 \phi_k(t)\phi_k(s)\int_0^\infty \sum_{l=1}^\infty \frac{(2H-2)_{2l}}{(2l)!\Gamma(2-2H)(2-2H)^{(2l)}}\frac{u^{2l+1-2H}(t-s)^{2l}}{e^u-1}dudsdt\\
&=\frac{1}{\Gamma (2-2H)}2 \int_0^1\int_0^1 \phi_k(t)\phi_k(s)\int_0^\infty \frac{u^{1-2H}}{e^u-1}\sum_{l=1}^\infty \frac{(u(t-s))^{2l}}{(2l)!}du ds dt\\
&=\frac{1}{\Gamma (2-2H)}2 \int_0^1\int_0^1 \phi_k(t)\phi_k(s)\int_0^\infty \frac{u^{1-2H}}{e^u-1}(\cosh (u(t-s))-1)du ds dt,\end{aligned}$$ where $(z)_{k}$ and $(z)^{(k)}$ denote the falling and rising factorials respectively. For even $k$ it follows from the definition that $(-z)_{k}=(z)^{(k)}$.
Recall that $$\cosh(u(t-s))-1=\frac{e^{u(t-s)}+e^{u(s-t)}-2}{2}=\frac{e^{u|t-s|}+e^{-u|t-s|}-2}{2}$$ for any $t,\,s$ and add up the summands of the variance expression in order to obtain $$\begin{aligned}
\int_0^1 &\int_0^1 \phi_k (t)\phi_k (s)\frac{1}{\Gamma (2-2H)}\int_0^\infty u^{1-2H} \left(e^{-u|s-t|}+ \frac{2}{e^u-1}(\cosh(u(t-s))-1)\right)dudsdt\\
&=\int_0^1 \int_0^1 \phi_k (t)\phi_k (s)\frac{1}{\Gamma (2-2H)}\int_0^\infty u^{1-2H} \left(e^{-u|s-t|}+ \frac{e^{u|t-s|}+e^{-u|t-s|}-2}{e^u-1}\right)dudsdt\\
&=\int_0^1 \int_0^1 \phi_k (t)\phi_k (s)\frac{1}{\Gamma (2-2H)}\int_0^\infty \frac{u^{1-2H}}{e^u-1}(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dudsdt,\end{aligned}$$ which was our claim.
Now let us prove that the obtained variance is indeed positive, thus confirming the rate of convergence suggested above. For elements of the real $L^2([0,\,1])$-Fourier basis this claim is shown (up to an application of Fubini’s Theorem) in Proposition \[prop:3\]. We also obtain from this proposition that in this particular case the variance simplifies to $$\frac{1}{\Gamma (2-2H)}\int_0^\infty \frac{u^{2-2H}}{(2\pi n)^2+u^2}du$$ for $\phi_k(x)=\sqrt{2}\sin(2\pi n)$ or $\phi_k(x)=\sqrt{2}\cos(2\pi n)$.
An arbitrary $L^2$-function $\phi_k$ with zero integral can be written as $\sum_{n\in\mathbb Z\backslash \{0\}}c_n f_n$, where $f_n$ are elements of the Fourier basis without the constant component and we have for such a decomposition: $$\begin{aligned}
\lefteqn{\int_0^1 \int_0^1 \phi_k (t)\phi_k (s)\frac{1}{\Gamma (2-2H)}\int_0^\infty \frac{u^{1-2H}}{e^u-1}(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dudsdt}&&\\
&=&\int_0^1 \int_0^1 \sum_{m,\,n\in\mathbb Z\backslash \{0\}}c_n f_n(t)c_m f_m(s)\frac{1}{\Gamma (2-2H)}\times\\
&&\int_0^\infty \frac{u^{1-2H}}{e^u-1}(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dudsdt\\
&=&\sum_{m,\,n\in\mathbb Z\backslash \{0\}}c_mc_n\frac{1}{\Gamma (2-2H)}\times\\
&&\int_0^\infty \frac{u^{1-2H}}{e^u-1} \int_0^1 \int_0^1 f_m (t)f_n (s)(e^{u(1-|t-s|)}+e^{u|t-s|}-2) dsdt du\\
&=&\sum_{n\in\mathbb Z\backslash \{0\}}c_n^2\frac{1}{\Gamma (2-2H)}\times\\
&&\int_0^\infty \frac{u^{1-2H}}{e^u-1} \int_0^1 \int_0^1 f_n (t)f_n (s)(e^{u(1-|t-s|)}+e^{u|t-s|}-2) dsdt du,\end{aligned}$$ since all the off-diagonal terms disappear, as was demonstrated in Proposition \[prop:3\]. We can now use the result for the Fourier basis and complete the calculations: $$\begin{aligned}
\int_0^1 & \int_0^1 \phi_k (t)\phi_k (s)\frac{1}{\Gamma (2-2H)}\int_0^\infty \frac{u^{1-2H}}{e^u-1}(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dudsdt\\
&=\frac{1}{\Gamma (2-2H)}\sum_{n\in\mathbb Z\backslash \{0\}}c_n^2 \int_0^\infty \frac{u^{2-2H}}{(2\pi n)^2+u^2}du,\end{aligned}$$ which is clearly positive if $\phi_k$ is nonzero.
In the context of a different scaling for some of the components there is an additional remark to be made.
Since for each $k\in \{1,\dots ,p\}$ the term $M_1$ in $\hat{\mu}_k-\mu_k$ is Gaussian, if different components of the vector $\mu$ are weighted differently (depending on whether the corresponding $\phi_k$ have zero integrals), the whole vector converges jointly to a multivariate Gaussian. With a calculation similar to those in \[prop:1\] one can show that the components with different weights are uncorrelated.
Appendix
========
In this chapter we have collected some technical results that are used in the proofs of this paper.
\[l:4\] For a centred normal sequence $(X_n)_{n\in\operatorname{{\mathbb N}}}$ of random variables we have: The squared $L^2$ norm of order at most $\frac{1}{n^\beta}$ for $\beta > 0$ implies almost sure convergence.
First note that the squared $L^2$ norm of a centred normal random variable is its variance. For $k\in\operatorname{{\mathbb N}}$ the $2k$-th moment is completely determined by it; we have $$\operatorname{{\mathbb E}}[X_n^{2k}]=C_k\operatorname{{\mathbb E}}[X_n^2]^k\lesssim \frac{1}{n^{\beta k}}$$ by assumption. If we now check the summability criterion, this consideration allows us to get the result by Markov’s inequality for $f(x)=x^{2k}$ and $k$ such that $\beta k>1$: $$\sum_{n=1}^{\infty}P(|X_n|>\varepsilon)\leq \sum_{n=1}^{\infty}\frac{\operatorname{{\mathbb E}}[X_n^{2k}]}{\varepsilon^{2k}}=\frac{1}{\varepsilon^{2k}}C_k\sum_{n=1}^{\infty}\operatorname{{\mathbb E}}[X_n^2]^k\lesssim \sum_{n=1}^{\infty}\frac{1}{n^{\beta k}}<\infty.$$
\[prop:2\] For $\alpha >0$ we have $\int_0^t e^{\alpha u} u^{2H-2}du\lesssim t^{2H-2}e^{\alpha t}$.
An analytic result from [@Abra] yields that the left-hand side is bounded by a constant times the right-hand side for large $t\in\mathbb R^+$. For smaller $t$, that is, for $t\leq t_0$ for some $t_0$, note that the left side is continuous while the right side has one discontinuity at $0$, where it tends to infinity. Therefore, it is also possible to find a constant for which the bound holds on the compact interval $[0,\,t_0]$. By taking the maximum of the two we obtain the result.
\[prop:3\] Let $(f_n)_{n\in\mathbb Z\backslash \{0\}}$ be the real $L^2([0,\,1])$-Fourier basis without the constant element, i.e. $f_n(x)=\sqrt{2}\sin (2\pi n x)$ and $f_{-n}(x)=\sqrt{2}\cos (2\pi n x)$ for $n\in\mathbb N$. Then for any $u>0$ the integral $$\int_0^1\int_0^1 f_n(t)f_m(s)(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dtds$$ is positive and equal to $\frac{2(e^u-1)u}{(2\pi n)^2+u^2}$ if $m=n$ and zero otherwise.
Let us wirte $z=e^u$ and calculate for $m,\,n\in\mathbb Z\backslash \{0\}$: $$\begin{aligned}
\int_0^1 &\int_0^1 f_n(t)f_m(s)(z^{(1-|t-s|)}+z^{|t-s|}-2)dtds\\
&=\int_0^1\int_{t-1}^t f_n(t)f_m(t-v)(z^{1-|v|}+z^{|v|})dvdt\\
&=\int_0^1 f_n(t) \int_{t-1}^t f_m(t-v)(z^{1-|v|}+z^{|v|})dvdt.\end{aligned}$$ By classical trigonometric identities we can decompose $f_m(t-v)$ as $$\sqrt{2}f_m(t-v) =f_m(t)f_{-m}(v)-f_{-m}(t)f_m(v)$$ if $m$ is positive and $$\sqrt{2}f_m(t-v)=f_m(t)f_m(v)+f_{-m}(t)f_{-m}(v)$$ if $m$ is negative. Thus, for the second part of the statement it suffices to show that the integral $$\int_{t-1}^t f_m(v)(z^{1-|v|}+z^{|v|})dv$$ is independent of $t$ for all $m\in \mathbb Z\backslash \{0\}$. This is indeed the case, because $$\int_{t-1}^0f_{m}(v)(z^{1+v}+z^{-v})dv=\int_t^1 f_m(v)(z^{1-v}+z^v),$$ and therefore, $$\int_{t-1}^t f_m(v)(z^{1-|v|}+z^{|v|})dv=\int_{0}^1 f_m(v)(z^{1-v}+z^{v})dv$$ is indeed independent of $t$. For symmetry reasons the integral vanishes for $m>0$.
If $n=m$, the same trigonometric identities can be used to show that $$\int_0^1\int_0^1 f_n(t)f_n(s)(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dtds=\frac{1}{\sqrt{2}}\int_0^1 f_{-n}(v)(z^{1-v}+z^v)dv$$ if $n$ is positive and $$\int_0^1\int_0^1 f_n(t)f_n(s)(e^{u(1-|t-s|)}+e^{u|t-s|}-2)dtds=\frac{1}{\sqrt{2}}\int_0^1 f_{n}(v)(z^{1-v}+z^v)dv$$ if $n$ is negative. Since $$\int_0^1 \cos(2\pi n v)(z^{1-v}+z^v)dv=\frac{2(z-1)\log(z)}{(2\pi n)^2+(\log(z))^2}=\frac{2(e^u-1)u}{(2\pi n)^2+u^2}$$ is positive for all $u>0$, the first part of the claim is proved.
[99999]{} M. Abramowitz and I. Stegun - Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington (1972). S. Bajja, K. Es-Sebaiy and L. Viitasaari - Least squares estimator of fractional Ornstein-Uhlenbeck processes with periodic mean\
*Journal of the Korean Statistical Society* (2017). R. Belfadli, K. Es-Sebaiy and Y. Ouknine - Parameter estimation for fractional Ornstein Uhlenbeck processes: non-ergodic case\
*Frontiers in Science and Engineering (An International Journal Edited by Hassan II Academy of Science and Technology)* (2011). H. Dehling, B. Franke and T. Kott - Drift estimation for a periodic mean reversion process\
*Statistical Inference for Stochastic Processes* (2010). H. Dehling, B. Franke and J.H.C. Woerner - Estimating drift parameters in a fractional Ornstein Uhlenbeck process with periodic mean\
*Statistical Inference for Stochastic Processes* (2016). K. Es-Sebaiy, I. Nourdin - Parameter estimation for $\alpha$-fractional bridges\
*Malliavin Calculus and Stochastic Analysis* (2011). Y. Hu and D. Nualart - Parameter estimation for fractional Ornstein Uhlenbeck processes\
*Statistics and Probability Letters Volume 80* (2010). M.L. Kleptsyna and A. Le Breton - Statistical analysis of the fractional Ornstein-Uhlenbeck type process\
*Statistical Inference for Stochastic Processes* (2002). Y. Kutoyants and P. Pilibossian - On minimum uniform metric estimate of parameters of diffusion-type processes\
*Stochastic Processes and their Applications* (1994). M. Moers - Hypothesis testing in a fractional Ornstein-Uhlenbeck model\
*International Journal of Stochastic Analysis* (2012). D. Nualart - The Malliavin Calculus and Related Topics. Springer, Heidelberg (2006). B. L. S. Prakasa Rao - Statistical Inference for Fractional Diffusion Processes. Wiley (2010). L. C. Young - An inequality of the Hölder type connected with Stieltjes integration\
*Acta Mathematica* (1936).
|
---
abstract: 'We calculate the effect of scattering on the static, exchange enhanced, spin susceptibility and show that in particular spin orbit scattering leads to a reduction of the giant moments and spin glass freezing temperature due to dilute magnetic impurities. The harmful spin fluctuation contribution to the intra-grain pairing interaction is strongly reduced opening the way for BCS superconductivity. We are thus able to explain the superconducting and magnetic properties recently observed in granular Pt as due to scattering effects in single small grains.'
address: 'I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstr. 9, 20355 Hamburg, Germany'
author:
- 'D. Fay and J. Appel'
title: 'Effect of spin orbit scattering on the magnetic and superconducting properties of nearly ferromagnetic metals: application to granular Pt'
---
The recent observation of superconductivity in Pt grains of $\approx 1 \mu$m size at $\approx 1 $mK [@HermPRL] motivated this theoretical study of the superconducting and magnetic properties of small grains taking account of the spin-orbit (s-o) scattering by external and internal surfaces[@Herm]. The importance of the s-o interaction at surfaces has been shown by Merservey and Tedrow[@Merservey] from a number of different measurements on superconductors. In Pb, for example,the probability that a conduction electron will change its spin direction in a surface collision is of the order 1. Taking into account s-o scattering, we show that the interplay between incipient magnetism and superconductivity in Pt is tilted towards BCS superconductivity because s-o scattering is inimical to magnetism and reduces the paramagon effects that inhibit singlet pairing in the late transition metals.
Why can s-o scattering generate superconductivity (sc) in Pt or other nearly ferromagnetic metals? In bulk Pt, no sc is observed despite a strong electron-phonon coupling; the BCS parameter is $\lambda^{Pt}_{Ph}\approx 0.4\,$ [@Pinski]. The reason for the absence of sc in bulk Pt is due to the strong exchange interactions between the itinerant 5d electrons. The resulting paramagnon effects suppress BCS s-wave sc. The Pt grains are small enough to have a large surface to volume ratio but are sufficiently large ($\ge 100\AA$) that the Bloch representation applies and we can ignore the Rashba effect [@GorRash]. Although the “lattice softening” near surfaces may enhance $\lambda^{Pt}_{Ph}$, more importantly, the s-o scattering at rough surfaces strongly reduces the harmful paramagnon effects. In the case of Pt grains, the extremely weak impurity magnetism observed at mK temperatures clearly points to an important role of the changed magnetic behavior for the occurence of sc [@HermPRL; @HermPR]. Independently of whether the sc extends throughout the grain or is restricted to a surface shell of thickness small compared with the grain size, s-o scattering at surfaces and defects will be important for the sc and magnetic properties. If shells exist in the compacted granules, they may (as in a thin film) consist of small crystallites large enough for bulk superconductivity but sufficiently small to limit the mean free path for s-o scattering. We find that, with reasonable values for the exchange and scattering parameters, sc in granular Pt is possible at the observed temperatures.
We first address the magnetic properties of small grains by calculating the static susceptibility $\chi(q)$ in the presence of ordinary and s-o scattering, taking the exchange enhancement effects into account as in Ref. . The Stoner factor $S$ accounts for the average exchange and the spin correlation range $\sigma$ measures the spatial range of the inter-atomic exchange. Scattering is included by considering the effect on $\chi_0\,$, the susceptibility without exchange enhancements. We find a significant effect of s-o scattering on $\chi_0$ that affects both $S$ and $\sigma$.
The susceptibility $\chi(r)$ is then calculated to determine how scattering as well as exchange enhancement affects the the RKKY (Ruderman-Kasuya-Kittel-Yoshida) oscillations occuring in $\chi_0(r)$. The short range and long range parts of $\chi(r)$ determine the two pertinent magnetic properties observed in dilute magnetic systems [@HermJLTP1; @HermJLTP2], namely, the magnitude of the giant moment $\mu_{gm}$ and the scale of the spin glass freezing temperature, $T_f\,/\,x$, in, e.g., $\mbox{PtFe}_x$. The exchange effects suppress the RKKY oscillations at small r yielding the ferromagnetic correlations responsible for $\mu_{gm}\,$; s-o scattering reduces $\mu_{gm}$. The spin glass transition observed in the bulk $\mbox{PtFe}_x$ systen is due to the long range oscillations of $\chi(r)$ relevant for the interaction between two impurity moments at a distance $r \gg a(=\mbox{lattice constant})$ in the absence of scattering effects. With scattering $\chi(r)$ at large $r$ is so strongly reduced that spin glass freezing would not be expected in the Pt grains where $x=4\,$ppm[@HermPRL].
We consider first the magnetic properties. The static q-dependent, exchange enhanced susceptibility $\chi(q)$ in the presence of ordinary and s-o scattering is appropriate to describe the giant moments and spin glass freezing. We assume a susceptibility of the RPA form: $$\chi(q)=\frac{2\mu^2_B \, \chi_0(q)}
{1- \chi_0(q) \frac{1}{3} \left[ U + 2J_H + 3J'(q) \right] } \, ,
\label{chi1}$$ where $\chi_0(q)=N(0)\,u(q)\,$, $N(0)$ is the density of states (DOS) per spin for all three 5d sub bands at the Fermi level, and $u(q)$ reduces to the Lindhard function for free electrons with no impurity scattering. $U$ is the intra-atomic self-exchange and $J_H$ is Hund’s rule exchange. Including up to second nearest neighbors we can define the inter-atomic exchange interaction $J'(q) \equiv J'(0)-(qa)^2J'_{eff}$ where $a$ is the lattice constant. We also define $\bar{U}=N(0)U/3\,$, $\bar{J}_H=N(0)J_H/3\,$, $\bar{J}'(q)=N(0)J'(q)/3\,$, $\bar{U}_{eff}=\bar{U} + 2\bar{J}_H + 3\bar{J}'(0)\,$, and $\bar{J}'_{eff}=N(0)J'_{eff}\,$ [@exchange]. Now $\chi(0)=2\mu^2_B \,S\,u_0$, where $S=1/(1-\bar{U}_{eff}\,u_0)\,$ and $u_0=u(0)\,$.
To define the spin correlation range, $\sigma$, we first expand Eq. (\[chi1\]) for small q . With $u(q)\approx u_0 + u_2 (qa)^2\,$, Eq. (\[chi1\]) becomes $$\chi(q)=\frac{2\mu^2_B S u_0}{1 + \sigma^2 q^2} \, ,
\label{chi2}$$ which yields a factor $\frac{1}{r}e^{-r/\sigma}$ where $\sigma^2/a^2=S\left[u_0\bar{J}'_{eff} - u_2 \bar{U}_{eff}
\right]-u_2/u_0\,$, in agreement with Clogston [@Clogston].
For arbitrary q we model the suceptibility with $$\chi(q)=\frac{2\mu^2_B \, \chi_0(q)}
{1-I(q)\, \chi_0(q)} \, ,
\label{chi3}$$ where $I(q)$ is a two parameter phenomenological interaction which is determined so that Eq.(\[chi1\]) reduces to Eq.(\[chi2\]) for small $q\,$. This yields $\bar{I}(q)=N(0)I(q)=\bar{U}_{eff}/\left[1+(qa)^2 (\bar{J}'_{eff}/\bar{U}_{eff})
\right] \,$. $\bar{U}_{eff}$ is determined directly by $S$. We take $S=3.8$ for Pt [@OKAnd] and find $\bar{U}_{eff}=0.737$. We fix $\bar{J}'_{eff}$ to provide a reasonable value for the spin fluctuation induced effective mass enhancement $\lambda_{SF}$ . As in the case of Pd, the problem here is to divide the effective mass enhancement $m^{\ast}/m=1+\lambda_{SF}+\lambda_{Ph}$ between the phonon and spin fluctuation contributions. We assume that $\lambda_{Ph}$ is about the same in Pt as in Pd and take the Pd value of $\lambda^{Pd}_{Ph}=0.41$ . Assuming $m^{\ast}/m-1=0.63$ for Pt [@OKAnd] we have $\lambda^{Pt}_{SF}=0.22$. Employing the standard calculation of $\lambda_{SF}$[@FayAppel], $$\lambda_{SF}=\frac{3}{2} \int_{0}^{2k_F}\,\frac{q\,dq}{2k_F^2}
\frac{[\bar{I}(q)]^2 u(q)}{1-\bar{I}(q)u(q)}\, ,
\label{lamsf}$$ we find $\bar{J}'_{eff}=0.163$ which yields $\sigma=3.21 \mbox{\AA}\,$. Physically, $\bar{J}'_{eff}$ is a measure of the range of $\bar{I}(r)$ in position space. Increasing $\bar{J}'_{eff}$ increases $\sigma$ and the range of $\bar{I}(r)$ but decreases the range of $\chi(q)$ in q-space yielding a smaller $\lambda_{SF}$ from Eq. (\[lamsf\]).
Assuming the RPA form of Eq. (\[chi1\]) is not changed by the presence of scattering centers, one need only consider the effect of scattering on $\chi_0\,$. This was first done by de Gennes [@deGennes] for ordinary scattering alone. He showed that $\chi_0(q)$ is not affected for $q=0\,$. Fulde and Luther (FL1) [@FL1] calculated $\chi(q,\omega)$ for small $q$ and Jullien [@Jullien] extended this work to arbitrary $q$ and $\omega$. Spin-orbit scattering was later added to the ordinary scattering in FL2 [@FL2]. We use the result of FL2 for the effect of s-o scattering on $\chi_0$. As FL2 were primarily interested in obtaining approximate analytic expressions for the $\omega$-dependent susceptibility we employ however the formalism of Julien [@Jullien] which is more suitable for computations. Equation (9) of Jullien for $\chi_0$ with ordinary scattering alone, can be generalized to include s-o scattering by comparison with Eq.(14) of FL2. The result is $$\chi_0({\bf q},\omega_0)=\frac{i}{2\,\pi} \int_{0}^{\infty}\,d\omega\,
\frac{Z(\omega) \left[ 1-\pi k_F \gamma_1 Z(\omega)/3\right] }
{1 - \pi k_F \gamma_0 Z(\omega)
\left[1-\pi k_F \gamma_1 Z(\omega)/3\right] \,} ,
\label{chi0}$$ where $Z(\omega)= \int \,\frac{d^3k}{(2\,\pi)^3}\,
G({\bf k},\omega)\,G({\bf k}+{\bf q},\omega + \omega_0 \,)$, $\gamma_0 = 1/k_F \ell_0\,$ and $\gamma_1 = 1/k_F \ell_1\,$ with $\ell_0$ and $\ell_1$ the mean free paths for ordinary and s-o scattering, respectively. The single particle propagator $G$ contains the scattering rates in the combination $\gamma = \gamma_0 + \gamma_1$. We set $\omega_0 = 0$ and consider from now on only $\chi(q)$. In their small-q approximation FL2 set $\chi_0(q)/N(0) = 1\,$. By doing this they neglected the $\gamma_1$ corrections to $\chi_0(0)$ that are crucial in the following considerations. The computation of $\chi$ proceeds as in Ref. leading to the results shown in Fig. (1) where we take for Pt, $k_F=0.642\,\mbox{ cm}^{-1}$ and $a=3.923\AA\,$. Fig. (1a) shows the large effect of s-o scattering on $\chi_0(0)$. In Fig. (1b) $S$, $u_0$, and $\sigma$ are shown vs $\gamma_1$. $S$ and $u_0$ do not depend on $\gamma_0$ and the dependence of $\sigma$ on $\gamma_0$ arises only through $u_2$ and is negligible.
We need the Fourier transform of $\chi(r)$ at small r to calculate the induced spin polarization around an impurity moment and at large r for the RKKY interaction responsible for spin glass transition. For $\chi(q) \rightarrow \chi_0(q)$ the Fourier transform can be done analytically yielding the usual RKKY oscillations. Integrating $\chi(r)$ over $d^3r$ leads directly to the sum rule $$\int\,d^3r\,\chi(r)=\chi(q=0)=2\mu^2_B N(0)Su_0\,.
\label{sumrule}$$ $\chi_0$ alone does not provide a reasonable induced moment since the RKKY oscillations do not yield the necessary short range ferromagnetic correlations. This problem does not occur in our two parameter model for $\chi$ as can be seen from Fig. (2a). Here we plot the dimensionless susceptibility $\bar{\chi}(r)$ which is defined by $\chi(r)=2\mu^2_B N(0) (\Omega/a^3)\, \bar{\chi}(r)\,$, where $\Omega$ is the atomic volume ($a^3/4$ for the fcc lattice). The first effect of $U_{eff}$ is to shift the curve to larger r increasing the spin correlation range $\sigma$, an effect discussed by Giovannini, Peter, and Schrieffer [@Peter]. Further increasing $U_{eff}$ pushes the curve above the axis for small r. Increasing $\bar{J}'_{eff}$ has a similar effect. The solid curve for the Pt parameters provides both the ferromagnetic short range correlations and the long range oscillations necessary for spin glass freezing. The effect of scattering at small r is shown in Fig. (2b). Ordinary scattering (dash curve) tends to smooth out the oscillations with little change in the area under the curve consistent with the sum rule, Eq. (\[sumrule\]). Spin orbit scattering (dot-dash curve) on the other hand reduces the magnitude of $\chi(r)$.
The giant moments observed in the bulk PtFe$_x$ are not seen in the Pt powders [@HermPRL] although, according to susceptibility measurements, the granules contain $x=(4\pm1)$ ppm of magnetic impurities. The giant moment consists of two parts, $\mu_{gm}=\mu(i) + \mu(h)\,$, where $\mu(i=\mbox{impurity})$ is the local moment of the 3d electrons of the Fe impurity atom and $\mu(h=\mbox{host})$ is the spin polarization of the 5d electrons of the Pt host matrix. We assume that $\mu(i)$ of Fe in Pt has approximately the same value as in Pd and take $\mu(i) \simeq 3\,\mu_B$. Using the experimental susceptibility value [@HermJLTP1], $\mu_{gm} \simeq 8\, \mu_B$, leads to $\mu(h) \simeq 5\,\mu_B$. We have $$\mu(h)=4\pi \int_{0}^{r_{gm}}r^2\,dr\,\sigma_s (r)\, ,
\label{muhost1}$$ where $r_{gm}$ is the giant moment radius and $\sigma_s (r)$ is the isotropic spin polarization induced by the Fe moment at $ {\bf{r}} =0$ due to the exchange interaction $V_{ex}$ between the 3d electrons of the impurity and the 5d electrons of the Pt host, $\sigma_s(r)=(V_{ex}/4)\, N(0)\, \mu_B\, \bar{\chi}(r)\,\,$[@Clogston]. N(0) is the DOS per spin and $\mbox{eV}\cdot \mbox{cm}^3\,$. In order to calculate $\mu(h)$ we need the parameters $r_{gm}$ and $V_{ex}$. $\mu(h)$ is not particularly sensitive to $r_{gm}$ and an upper limit can be obtained from the sum rule, Eq.(\[sumrule\]): $\mu(h)\mid_{r_{gm}\rightarrow\infty}\,=
V_{ex}\,N(0)\,a^3\,S\,u_0\,\mu_B\,/4\,$ . We take $r_{gm}\sim2.5\,a \sim 10 \mbox{\AA}$ as in Pd and then fix the value of the local exchange coupling $V_{ex}$ by requiring that Eq. (\[muhost1\]) yield $\mu(h)=5\,\mu_B$. We find $V_{ex}=2.504\,\mbox{eV}$ which is somewhat large but still seems reasonable. Here we have used $N(0)=0.386(m^{\ast}_{b}/m)\,$ states/eV/atom with band mass $m^{\ast}_{b} /m=3.36$[@OKAnd]. The effect of s-o scattering on $\mu(h)$ is shown in Fig. (3), where $\mu(h)$ from Eq. (\[muhost1\]) (solid curve) and for $r_{gm}\rightarrow\infty$ (dash curve) are shown versus the s-o scattering parameter $\gamma_1$. It turns out that $\mu(h)$ is practically independent of ordinary scattering, $\gamma_0$. This can be seen from the sum rule result, $r_{gm}\rightarrow\infty$, since $S$ and $u_0$ are only affected by s-o scattering. Due to the rapid decrease of $\chi(r)$ in the presence of scattering the sum rule is approximately exhausted for the experimental $r_{gm}$. The decrease of $\chi(r)$ at small r seen in Fig. (2b) leads to a reduction of $\mu(h)$ by a factor 2 for $\gamma_1 \simeq 0.2$ and can explain why giant moments are not observed in the Pt granules.
The spin glass freezing temperature $T_f$ in dilute impurity systems is determined by the long range spin polarization that provides the RKKY coupling between two magnetic impurities. At large $r$ and in the absence of scattering, $\chi(r)$ and $\chi_0(r)$ are nearly the same and proportional to $\cos(2\,k_F\,r)/r^3$. The scale of $T_f$ is set by the average RKKY coupling energy of a typical impurity atom pair. Although a correct calculation of $T_f$ requires evaluation of the second moment of the distribution of the couplings, an estimate can be obtained from the envelope of $\chi(r)$ determined by the peaks of the oscillations. Denoting this quantity by $<\bar{\chi}(r_{avg})>$ we take for Fe impurities in Pt $$k_B\,T_f \approx \mu^2_{\mbox{Fe}} \left( \frac{ V_{ex}}{2\mu_B} \right)^2\,2N(0)\,
\frac{\Omega}{a^3}<\bar{\chi}(r_{avg})>\,,
\label{Tf}$$ where $\mu_{\mbox{Fe}}$ is the bare Fe moment. Without scattering, $k_B\,T_f = \mu^2_{\mbox{Fe}}(V_{ex}/2\mu_B)^2\,2N(0)x/4\pi\,$, where $x=n_{\mbox{Fe}}\,/\,n_{\mbox{Pt}}$ with $n_{\mbox{Fe}}=1/r^3_{avg}$ and $n_{\mbox{Pt}}=4/a^3$. Using this equation with $x\approx5\,$ppm and $\mu_{\mbox{Fe}}=3\mu_B$, we obtain for bulk Pt a value for $T_f\,$ ($2.1\,\mbox{mK/ppm}$) that is almost an order of magnitude greater than the observed $0.26\,\mbox{mK/ppm}$. Our rather large value of $V_{ex}$ presumably contributes to this discrepency. Here however, we are concerned with the effect of scattering on $T_f\,$, Eq. (\[Tf\]). In the presence of either ordinary or s-o scattering, $\bar{\chi}(r)$ falls off rapidly at large r. A rough numerical fit gives $\bar{\chi}(r) \sim \exp(-5\gamma_i\,r/a)$ for $r/a\,>\,1/\gamma_i$ where $\gamma_i=\gamma_0\mbox{ or }\gamma_1$. Although a power law cannot be ruled out, the decrease is in any case much faster than $1/r^3$. We can thus conclude that the contributions to $\chi(r)$ we have calculated do not lead to a measurable $T_f$ in the presence of scattering in granular Pt. However, at large r, diffusion-type diagrams for $\chi$ may be dominant leading to a contribution proportional to $1/r^3$ and independent of ordinary scattering. In Ref. it was shown that these contributions are exponentially small in the presence of s-o scattering. Thus it seems quite reasonable that no spin glass transition was observed in the Pt granules.
The single grain superconducting transition temperature $T_c$ is affected by scattering only through the indirect effect on the spin fluctuation part pair interaction, $\lambda_{SF}$. There is no direct effect for ordinary scattering due to Anderson’s theorem which also holds for s-o scattering [@AppelOver], for other scattering processes that obey time-reversal symmetry, and in zero magnetic field. In order to estimate the indirect effect we calculate $T_c$ for s-wave pairing with the standard weak-coupling equation[@FayAppel; @Allen]: $$T_c = \Theta_D \, \exp\left[-\,\frac{1 + \lambda_{Ph} + \lambda_{SF}}
{\lambda_{Ph} - \lambda_{SF} - \mu^{\ast}}\right]\,.
\label{tc}$$ Here $\lambda_{SF}$ is given by Eq. (\[lamsf\]) but now with scattering included. We take $\Theta_D(Pt) = 234\, K$ and $\mu^{\ast}=0.1$ which is a standard estimate. We assume $\lambda_{Ph}\approx0.41$ is not affected by scattering and it turns out that $\lambda_{SF}$ is practically independent of $\gamma_0\,$, yielding a decrease of only a few percent in $T_c$. In Fig. (3) we plot $T_c$ versus $\gamma_1$ for $\gamma_0 = 0.01$. It is seen that the $T_c$’s observed in Pt powders [@HermPRL] are reached for a s-o scattering rate $\gamma_1$ less than 0.1 and that $T_c$ increases strongly with increasing $\gamma_1\,$.
In conclusion, we have shown that ordinary and s-o scattering affect the exchange enhanced magnetic properties of the itinerant d electrons so that $\mu_{gm}$ and $T_f$ of Pt$\mbox{Fe}_x\,$, e.g., are reduced. On the other hand, s-o scattering weakens the spin fluctuations to the extent that the phonons dominate and superconductivity with $T_c\approx 1$ mK can occur in single Pt granules. This is possible with moderate s-o scattering since the effective electron-electron interaction in bulk Pt is very close to zero [@Hauser]. Of the effects not considered here that could change $T_c\,$, phonon softening is probably the most important. To control surface phonon effects and to complement the studies of grains an experimental search for superconductivity in thin films of Pt is of interest. In films one must distinguish between thick ($\,>100\AA$) films where s-o scattering accounts for the surface effects on the 3D electron states and thin films with smooth surfaces where the Rashba s-o splitting occurs throughout the film thickness. How the spin fluctuations in very thin films of nearly ferromagnetic metals affect (spoil?) the Rashba effect is an open question. It would also be interesting to investigate thin films where the surface roughness suppresses the Rashba effect and s-o scattering reduces $\lambda_{SF}$. Finally, the films can become so thin, or the grains so small, that size quantization of electron states occurs that may lead to new effects in the interplay between magnetism and superconductivity.
We would like to thank P. Hertel for helpful discussions.\
\
R. König, A. Schindler, and T. Herrmannsdörfer, Phys. Rev. Lett. [**82**]{}, 4528 (1999). It is apparently not known whether the grains are single crystals or poly-crystallites. T. Herrmannsdörfer, private communication. R. Meservey and P. M. Tedrow, Phys. Rev. Lett. [**41**]{}, 805 (1978); ibid. [**43**]{}, 384 (1979). F. J. Pinski and W. H. Butler, Phys. Rev. B[**19**]{}, 6010 (1979). L. P. Gorkov and E. I. Rashba, Phys. Rev. Lett. [**87**]{}, 037004-1 (2001). A. Schindler, R. König, T. Herrmannsdörfer, and H. F. Braun Phys. Rev. B[**62**]{}, 14350 (2000). D. Fay and J. Appel, Phys. Rev. B[**16**]{}, 2325 (1977). T. Herrmannsdörfer, S. Rehmann, W. Wendler, and F. Pobell, J. Low Temp. Phys. [**104**]{}, 49 (1996). T. Herrmannsdörfer, S. Rehmann, and F. Pobell, J. Low Temp. Phys. [**104**]{}, 67 (1996). For the fcc lattice and up to 2nd n.n., we have $J'_{eff}=J'_1+3J'_2$, where $J'_1$ and $J'_2$ are the interatomic exchange integrals for 1st and 2nd n.n., respectively. A. M. Clogston, Phys. Rev. Lett. [**10**]{}, 583 (1967). O. K. Anderson, Phys. Rev. B[**2**]{}, 883 (1970). P. G. de Gennes, J. Phys. Radium [**23**]{}, 630 (1962). P. Fulde and A. Luther, Phys. Rev. [**170**]{}, 570 (1968). R. Jullien, J. Low Temp. Phys. [**42**]{}, 207 (1981). P. Fulde and A. Luther, Phys. Rev. [**175**]{}, 337 (1968). B. Giovannini, M. Peter, and J. R. Schrieffer, Phys. Rev. Lett. [**12**]{}, 736 (1964). A. Jagannathan, E. Abrahams, and M. J. Stephen, Phys. Rev. B[**37**]{}, 436 (1988) and references therein. J. Appel and A. W. Overhauser, Phys. Rev. B[**29**]{}, 99 (1984). P. B. Allen and B. Mitrovic, in [*Solid State Physics*]{} (F. Seitz, D. Turnbull, and H. Ehrenreich, eds.), Vol. 37, Academic Press, New York. J. J. Hauser, H. C. Theuerer, and N. R. Werthamer, Phys. Rev. [**136**]{}, A637 (1964).
-0.8cm
-1.5cm
-1cm
|
[**Adaptive Step Size for Hybrid Monte Carlo Algorithm**]{}
[*Swiss Center for Scientific Computing (SCSC)\
*ETH-Zürich, CH-8092 Zürich\
*Switzerland\
***]{}
Introduction
============
Simulations including dynamical fermions remain the most challenging ones for lattice QCD. The standard method to simulate dynamical fermions is, at the moment, the Hybrid Monte Carlo (HMC) algorithm[@HMC] although it still requires large amounts of computational time. An alternative method to simulate the dynamical fermions is a local multiboson algorithm based on a polynomial approximation of the fermion matrix, proposed by Lüscher[@Luscher]. Much interest has been recently devoted to this algorithm[@BOSON] to make it as efficient as the HMC algorithm.
The HMC simulation combines Molecular Dynamics (MD) evolution with a Metropolis test. In order to obtain the correct equilibrium, the integrator used in the MD evolution must satisfy two conditions; it must be:
- time-reversible
- area preserving
One such integrator satisfying these conditions is the leapfrog integrator, which is normally used in the HMC simulation. Errors of the leapfrog integrator start with $O(\Delta t^3)$, where $\Delta t$ is the step size of the integrator. These errors cause violation of the conservation of the total energy, which must be corrected by the Metropolis test at the end of the MD trajectory. Let $\Delta H$ be the energy violation: $$\Delta H = H_{end} - H_{begin}$$ where $H_{begin}(H_{end})$ is the total enery at the beginning(end) of the MD trajectory. The Metropolis test accepts a new configuration with a probability $P_{prob}$: $$P_{prob} \propto \min(1,\exp(-\Delta H)).$$ In order to maximize acceptance of the Metropolis test, it might be preferable to use a higher order integrator[@Creutz]. However higher order integrators do not appear practical for lattice QCD since they require more arithmetic operations (force evaluations coming from the fermionic action) than the simplest low-order integrator, and this overhead exceeds the gain in step-size.
So far conventional HMC simulations have been performed with a fixed step size $\Delta t$ during the MD simulation. The local integration error does not remain constant in this case. When the trajectory approaches an energy barrier ($S_{eff} = - Log det D$ large), it is repelled and bounces off. The curvature of the trajectory increases, and with it the integration error. This situation becomes more pronounced at small quark mass, since the height of the energy barriers diverges in the presence of zero modes. Therefore we expect a behavior of the MD trajectory similar to Fig.1. Varying the step size adaptively, keeping the local error constant, may be a good way to obtain a better integrated trajectory and may result in higher acceptance. Naively it would seem that this can be accomplished by calculating a local error at $(p, U)$, where $p=(p_1,p_2,...)$ and $U=(U_1,U_2,...)$ collectively represent momenta and link variables respectively, and then by keeping this local error constant. However this naive scheme is not applicable for the HMC simulation because it violates reversibility.
Recently an adaptive step size method with reversible structure was proposed by Stoffer[@Stoffer]. He constructed a symmetric error estimator which gives the same error value at a reflected point. The step size is then determined at every integration point by demanding that the symmetric error estimator remain constant. Stoffer implemented his method for the Kepler problem and obtained better results than the conventional ones. The possibility to apply this adaptive step size method to the HMC algorithm was stated in Ref[@FORC]. In this letter, we implement this method for the HMC simulation and examine its cost and its efficiency.
Construction of Adaptive Step Size
==================================
Here we construct an adaptive step size compatible with a time-reversible integrator. We follow the idea proposed by Stoffer[@Stoffer].
Let $H$ be the Hamiltonian of our system, $$H= \frac{1}{2}\sum p_i^2 + S(U).$$\[Hamilton\] where $p_i$ are momenta, $U$ are gauge links, and $S(U)$ consists of a gauge part $S_g(U)$ and a fermionic part $S_f(U)$, $$S(U)=S_g(U) + S_f(U),$$ $$S_g(U)= \beta \sum (1 - \frac{1}{N_c} ReTr U_{plaq}),$$ $$S_f(U)= \phi^\dagger (DD^\dagger)^{-1}\phi,$$ where $N_c$ is the number of colors, $\phi$ is a pseudo-fermion vector and $D = 1 + \kappa M$ is the Wilson fermion matrix, with $\kappa$ the hopping parameter.
Call $T(\Delta t)$ a one-step integrator. It maps momenta and link variables $(p,U)$ onto $(p^\prime,U^\prime)$, $$T(\Delta t):(p,U) \longrightarrow (p^\prime,U^\prime).$$ If this one-step integrator is reversible, then it satisfies $$T(-\Delta t):(p^\prime,U^\prime) \longrightarrow (p,U),$$ In this study, we use the leapfrog integrator as our one-step integrator. In terms of the time evolution operators[@Creutz; @SW], the one-step integrator $T(\Delta t)$ can be written as $$T(\Delta t) = exp( \frac{\Delta t}{2} L(\frac{1}{2}\sum p_i^2))
exp(\Delta t L(S(U))) exp( \frac{\Delta t}{2} L(\frac{1}{2}\sum p_i^2)),
\label{LEAP}$$ where $L(\bullet)$ is the linear operator which is given by the Poisson bracket[@SW]. The fermionic part of the force depends on the solution of a linear equation of the type $D x = \phi$, which is obtained iteratively at great expense of computer time. Thus force evaluations dominate the computation, and the cost of our algorithm can be measured in units of force evaluation.
Now we define a symmetric error estimator, $$E_S (p,U:\Delta t) = e(p,U:\Delta t) + e(p^\prime, U^\prime : -\Delta t),
\label{ES}$$ where $e(p,U:\Delta t)$ is a local error at $(p,U)$ when the system is integrated by some integrator with a step size $\Delta t$, and the integrator maps $(p,U)$ on $(p^\prime, U^\prime)$. The local error is assumed to increase monotonically with $\Delta t$. We will define the local error later. If the integrator is reversible, eq.(\[ES\]) is obviously symmetric under the exchange: $$(p,U,\Delta t) \longleftrightarrow (p^\prime, U^\prime,-\Delta t).$$ Namely, this means $$E_S(p,U:\Delta t) = E_S(p^\prime, U^\prime:-\Delta t).$$ The adaptive step size is then determined by solving a symmetric error equation, $$\label{tol}
E_S(p,U:\Delta t) = tolerance$$ The tolerance should be kept constant during the MD simulation. The adaptive step size determined by eq.(\[tol\]) takes the same value at the reflected point $(-p^\prime, U^\prime)$. Therefore we find that an integrator with the adaptive step size determined by eq.(\[tol\]) is reversible.
Any local error can be defined provided that it increases monotonically with $\Delta t$. Our local error is defined as follows.
First, we integrate $(p,U)$ by the two-step integrator $T^2(\Delta t)$ and the one-step integrator $T(2\Delta t)$. $$T^2(\Delta t):(p,U) \longrightarrow (p^\prime, U^\prime)
\label{def1}$$ $$T(2\Delta t):(p,U) \longrightarrow (\tilde{p^\prime}, \tilde{U^\prime})$$ If $\Delta t$ is not too large, $(p^\prime, U^\prime)$ and $(\tilde{p^\prime}, \tilde{U^\prime})$ should be close to each other. We define the local error at $(p,U)$ by $$e(p,U:\Delta t) = \sum_{\mu, x}^{4, V}(1-\frac{1}{N_c}Re Tr {U^\prime}^\dagger_\mu(x)
\tilde{U^\prime_\mu}(x))/(4 V)
\label{local}$$ where $V$ is the volume of the lattice. One could also use the momenta in the definition of the local error. Similarly, we integrate $(p^\prime, U^\prime)$ by $T^2(-\Delta t)$ and $T(-2\Delta t)$ in the inverse time direction, $$T^2(-\Delta t):(p^\prime, U^\prime) \longrightarrow (p, U)
\label{T2}$$ $$T(-2\Delta t):(p^\prime, U^\prime) \longrightarrow (\tilde{p}, \tilde{U}).$$ Since the integrator is reversible, the calculation of eq.(\[T2\]) is not needed. The local error at $(p^\prime, U^\prime)$ is also defined like eq.(\[local\]), $$e(p^\prime,U^\prime:-\Delta t) = \sum_{\mu, x}^{4, V}(1-\frac{1}{N_c}Re Tr {U}^\dagger_\mu(x)
\tilde{U_\mu}(x))/(4 V)
\label{def2}$$ In the case of the leapfrog integrator of eq.(\[LEAP\]), we need 4 force evaluations to construct the symmetric error estimator $E_S$, instead of just 2 for the evolution $T^2(\Delta t)$.
Eq.(\[tol\]) is a non-linear equation. In general, it should be solved numerically, eg. by iterative bisection. With our definition of the symmetric error estimator, however, we can anticipate the scaling behavior of eq.(\[tol\]) and use it to accelerate convergence. The vector potentials evolved by the leapfrog integrator have $O(\Delta t^3)$ errors, $$\tilde{A}_\alpha^\prime = A_\alpha^\prime + O(\Delta t^3).$$ Therefore, $$\begin{aligned}
{U^\prime}^\dagger \tilde{U^\prime} & = & exp( -i A^\prime_\alpha \lambda_\alpha)
exp( i \tilde{A^\prime_\alpha} \lambda_\alpha) \\
& \approx & 1 + i c_\alpha \lambda_\alpha O(\Delta t^3) + d_{\alpha\beta} \lambda_\alpha\lambda_\beta
O(\Delta t^6)
\label{RT}\end{aligned}$$ where $\lambda_\alpha$ are SU(3) generators, and $c_\alpha$ and $d_{\alpha\beta}$ are some real constants whose explicit values are not important here. Taking $Real$ and $Trace$ of eq.(\[RT\]) and substituting it to eq.(\[local\]) and (\[def2\]), we find that in the leading order the symmetric error estimator starts with $O(\Delta t^6)$. This behavior is verified numerically, as illustrated in Fig.2: if $\Delta t$ is not too large, the symmetric error estimator behaves like $E_S \propto \Delta t_A^6$. This property is used for solving eq.(\[tol\]). Choose some initial $\Delta {t_A}_1$ for the step-size and calculate ${E_S}_1=E_S(\Delta {t_A}_1)$. If $\Delta {t_A}_1$ does not satisfy the symmetric error equation eq.(\[tol\]) then input the second trial value $\Delta {t_A}_2$, which is the solution of $$log(\frac{tolerance}{{E_S}_1}) = 6 log(\frac{\Delta {t_A}_2}{\Delta {t_A}_1}).$$ If further trials are necessary, the following approximation can be used [@Stoffer], $$log(\frac{\Delta {t_A}_3}{\Delta {t_A}_2})=\frac{log(\Delta {t_A}_2/\Delta {t_A}_1)}{log({E_S}_2/{E_S}_1)}
log(\frac{tolerance}{{E_S}_2}).$$ This recurrence is continued until eq.(\[tol\]) is satisfied to sufficient accuracy, and then a new configuration $(p^\prime, U^\prime)$ is stored. Note that 2 strategies are available, just like for the stopping criterion of the linear solver in the force calculation: either the initial guess $\Delta {t_A}_1$ is invariant under time-reversal (eg. it is equal to the average step-size), and the accuracy to which eq.(\[tol\]) must be satisfied can be set arbitrarily low; or the initial guess makes use of past information (eg. it is equal to the previous step-size), and eq.(\[tol\]) must be satisfied exactly. We use the second method, and take the previous result as our initial guess. We then solve eq.(\[tol\]) to within 5%. Since we do not solve eq.(\[tol\]) exactly, we introduce a tiny, controllable source of irreversibility in the dynamics: the step size under time-reversal could be different by about $5/6 \sim 1 \%$. For this exploratory study, we have not considered this aspect further.
Efficiency
==========
listed in Table 1. We chose $\beta = 0$ in several instances, to eliminate the gauge part of the action and hopefully be more sensitive to the energy barriers coming from the fermionic part. The effect of changing the quark mass can be obtained by comparing cases A and B, that of changing the volume by comparing A and D. The adaptive step size is determined by the symmetric error equation eq.(\[tol\]), with the tolerance set as per Table 1. The adaptive step sizes are summed up from the beginning of the trajectory and when the total trajectory length becomes greater than the trajectory length of Table 1, a Metropolis test is performed.
Fig.3 shows histograms of the adaptive step size at $\beta=0.0$, $\kappa=0.215$ and 0.230 ( cases A and B in Table 1 ). The distribution remains strongly peaked. This is also true of cases C and D. The average step size $<\Delta t_A>$ and its relative variance are summarized in Table 2, where the relative variance $\sigma$ is defined by $$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (\frac{{\Delta t_A}_i}{<\Delta t_A>} - 1)^2.$$ As the quark mass $m_q$ decreases, the energy barriers in phase space become higher and sharper, so that one would expect the variation of the step size to increase. Indeed this is what happens, and the relative variance in cases A and B increases roughly like $1/m_q$, where $m_q \propto (\kappa^{-1} - \kappa_c^{-1})$ and $\kappa_c = 1/4$ at $\beta = 0$. On the other hand, as the volume increases, the relative variance seems to decrease sharply, like $1 / \sqrt V$ or faster (see cases A and D). Perhaps this can be explained by considering the relative fluctuations of the effective action $-Log~det~D$: as the volume increases, the relative fluctuations decrease, so that the system tends to stay at some average distance from the energy barriers, rather than bouncing off them.
From the schematic picture of Fig.1, it is expected that the approach of an energy barrier causes a reduction in the adaptive step size, and at the same time an increase in the number of iterations taken by the solver to converge. Fig.4 shows $\Delta t_A$ versus the number of iterations in the solver: the expected anti-correlation between them can be observed, and becomes more pronounced as the quark mass is reduced.
In order to compare the adaptive method with the conventional HMC algorithm, we define the efficiency of the adaptive method as follows.
First, find the fixed step-size $\Delta t_{HMC}$ of the HMC simulation which gives the same acceptance as the adaptive step-size method. The total trajectory length of the HMC simulation is set to the average total trajectory length $<$traj. length$>$ of the corresponding adaptive step size method’s case. We performed the HMC simulations with several step-sizes and determined the corresponding step-size of the HMC algorithm by interpolating those results. The results of the corresponding step-size are summarized in Table 2. For the acceptance of the adaptive step-size method, see Table 2.
Then, define the gain by $$g_A = <\Delta t_A> / \Delta t_{HMC}.$$ When $g_A > 1$, the adaptive step-size method really takes larger steps on average, without compromising the acceptance. However the real efficiency of the method can only be assessed by taking into account the overhead of determining the adaptive step-size, since additional force evaluations are necessary.
From the definition of our symmetric error estimator eq.(\[def1\])-(\[def2\]), we know that one construction of $E_S$ needs 4 force evaluations. Call $R_T$ the average number of trial steps needed to solve eq.(\[tol\]). After $4 R_T$ force evaluations, we have determined the step-size $\Delta t_A$ and use the integrator $T^2(\Delta t_A)$ (eq.\[def1\]) to advance the dynamics by 2 steps $\Delta t_A$. Therefore the cost of the method is $2 R_T$ force evaluations per step, compared with $1$ force evaluation per step for standard HMC.
Real efficiency will be achieved if the number of force evaluations per unit time decreases, ie. if $2 R_T < g_A$. Results for the gain $g_A$ and the cost $2 R_T$ are summarized in Table 3. For all cases we studied, real efficiency is not achieved.
It is disappointing to see how small the gains $g_A$ are. The reason for such small gains can be understood by considering the behavior of the Hamiltonian. The dependence of the energy violation at each step with the step-size is, in general, non-linear. Therefore it is not necessary that a small local error correspond to a proportionally small energy violation of the Hamiltonian. Fig.5 shows $|\Delta H|$, the absolute value of the energy violation after one integration step, versus the local error $E_S$. The 2 clusters of points correspond to fixed step sizes $\Delta t=0.04$ and $0.08$. No strong correlation between $E_S$ and $\Delta H$ can be observed. This is further evidenced by the dashed lines, which are the result of fitting to a scaling law $\sqrt{<\Delta H^2>} = E_S^b$: for the larger step-size, $b$ is almost zero. Therefore, it becomes clear that fixing $E_S$ and varying $\Delta t$ adaptively cannot have a strong effect on the acceptance, which solely depends on $\Delta H$.
Two approaches could be used to improve the efficiency of our scheme:\
i) decrease the overhead: instead of estimating the error by comparing $T^2(\Delta t)$ with $T(2 \Delta t)$, one could replace the latter by an Euler integrator, which requires no additional force evaluation. Note that the error eq.(\[ES\]) remains symmetric under the exchange $(p,U,\Delta t) \longleftrightarrow (p^\prime, U^\prime,-\Delta t)$ even though the Euler integrator is not time-reversible. The problem we found with that approach is that, for the large step-sizes used on our small lattices, the error eq.(\[ES\]) no longer obeyed a simple scaling law eq.(\[RT\]) as a function of the step-size. Then the number of iterations needed to solve eq.(\[tol\]) increased, defeating the expected reduction in overhead. On larger lattices with smaller step-sizes, this problem would be milder.\
ii) change the definition of the error eq.(\[def2\]), so that it is better correlated with $|\Delta H|$, the energy violation at each step. Note that $|\Delta H|$ itself cannot be chosen, because it does not increase monotonically with the step-size: in that case eq.(\[tol\]) admits multiple solutions; the overhead of converging to one of them, and the same one under time-reversal, increases considerably. With our definition eq.(\[def2\]), the error is only weakly correlated with $|\Delta H|$, but the situation again seems to improve with smaller step-sizes, on larger lattices (compare the 2 dashed lines in Fig.5). Nonetheless it would be desirable to control the step-size with a more relevant quantity than eq.(\[def2\]), since all that matters in the end is energy conservation.
Finally, instead of varying the step-size, one could vary adaptively the couplings of the Hamiltonian $H$, eq.(\[Hamilton\]), at each step, or even include some new operators in $H$, trying to tune them so as to best conserve energy. The general difficulty with that approach is to find an error eq.(\[def2\]) which varies monotonically with the couplings of $H$.
Conclusion
==========
We have implemented an adaptive step-size method for Hybrid Monte Carlo simulations, and tested it at several parameters $(\beta,\kappa, volume)$. The relative variance of the step-size increases for small quark masses and small volumes. The average step-size seems somewhat larger than the corresponding fixed step-size at same acceptance. But this gain is more than offset by the overhead of determining the adaptive step-size. It seems very difficult to achieve real gains in efficiency, because conservation of energy, which is necessary for high Metropolis acceptance in the HMC algorithm, is poorly correlated with the conventional error governing the adaptive step-size.
A plausible extrapolation from our results would indicate that the relative variance of the step-size scales like $m_q^{-1} V^{-1/2}$, ie. as $(m_{\pi} L)^{-2}$, where $m_{\pi}$ is the pion mass and $L$ the physical size of the lattice. This quantity normally remains constant as the continuum limit $a \rightarrow 0$ of the lattice theory is taken, so that the relative fluctuations in the adaptive step-size $\Delta t$ would tend to a constant. Even if this analysis is no more than plausible at this stage, it is clear that the two limits $m_q \rightarrow 0$, $V \rightarrow \infty$ have opposite effects on the fluctuations of $\Delta t$, making it unlikely that such fluctuations become very large on present lattice sizes. This observation is consistent with the limited fluctuations (a factor 2 or so) in the number of iterations needed by the solver to compute the force in the largest QCD simulations [@SESAM].
Thus it appears that QCD is much “easier” to simulate than the Kepler problem: in lattice QCD, the force on the gauge links varies little in magnitude, and the curvature of the Molecular Dynamics trajectory is rather small. One intuitive explanation is that the QCD force is dominated by short-range UV contributions, which drown the IR component sensitive to the energy barrier $det D \sim 0$.
We thank D. Stoffer for helpful discussions. T.T. is supported in part by the Japan Society for the Promotion of Science. Ph. de F. thanks Hiroshima Univ. and Tsukuba Univ., especially Profs. O. Miyamura and Y. Iwasaki, for hospitality during this project.
Figure 1: Schematic behavior of a Molecular Dynamics trajectory in configuration space. $detD$ is the determinant of the Dirac matrix.
Figure 2: Adaptive step size $\Delta t_A$ versus symmetric error $E_S$, for two configurations of size $4^4$ at $\kappa=0.230$. The straight line $E_S \propto \Delta t_A^6$ is shown for comparison.
Figure 3: Histograms of adaptive step size $\Delta t_A$, for $4^4$ lattices at $\kappa=0.215$ and 0.230. The logarithmic scale shows the increase with $\kappa$ of the relative variance.
Figure 4: Adaptive step size versus number of iterations in the solver (BiCG$\gamma_5$).
Figure 5: $|\Delta H|$ versus the local error $E_S$, on a $4^4$ lattice at $\kappa=0.215$. The step size is fixed at $\Delta t=0.04$ and $0.08$. $\Delta H$ is the change in the total energy after one integration step. The dotted lines result from fitting to the form $\sqrt{<\Delta H^2>} = E_S^b$, and show the correlation (or absence of) between the 2 quantities. $<\Delta H^2>$ is obtained by dividing the data in 10 bins and averaging the values $\Delta H^2$ in each bin.
case $\beta$ size $\kappa$ traj. length tolerance($\pm5\%$)
------ --------- ------- ---------- -------------- ---------------------
A 0.0 $4^4$ 0.215 0.8 $10^{-4}$
B 0.0 $4^4$ 0.230 0.4 $8\times 10^{-6}$
C 5.4 $4^4$ 0.162 1.0 $10^{-6}$
D 0.0 $8^4$ 0.215 0.3 $10^{-7}$
: Run parameters
case $<\Delta t_A>$ $\sigma(\%)$ $<$traj. length$>$ acceptance(%) $\Delta t_{HMC}$
------ ---------------- -------------- -------------------- --------------- ------------------
A 0.0911(3) 3.3 0.91 36(2) 0.0897(13)
B 0.0431(1) 5.6 0.44 57(2) 0.0419(09)
C 0.0673(1) 0.8 1.08 87(2) 0.0688(80)
D 0.03281(2) 0.6 0.33 63(2) 0.0328(10)
: Results of the adaptive step-size method and fixed step-size $\Delta t_{HMC}$ of the HMC algorithm. $<$traj. length$>$ stands for the average total trajectory length. The fixed step-size $\Delta t_{HMC}$ is determined so that it gives the same acceptance as the adaptive step size method.
case $g_A$ $R_T$ Cost per step $(=2R_T)$
------ ----------- ------- -------------------------
A 1.016(15) 2.25 4.50
B 1.029(22) 2.45 4.90
C 1.0(1) 1.13 2.26
D 1.00(3) 1.15 2.30
: Gain $g_A$, average number of trial steps $R_T$ and Cost per step
[99]{} S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. B195 (1987) 216 M. Lüscher, Nucl. Phys. B418 (1994) 637 B. Bunk, K. Jansen, B. Jegerlehner, M. Lüscher, H. Simma and R. Sommer, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 49; B. Jegerlehner, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 879; A. Boriçi and Ph. de Forcrand, Nucl. Phys. B 454 (1995) 645; C. Alexandrou, A. Borrelli, Ph. de Forcrand, A. Galli and F. Jegerlehner, Nucl. Phys. B 456 (1995) 296; B. Jegerlehner, hep-lat/9512001; K. Jansen, B. Jegerlehner and C. Liu, hep-lat/9512018; A. Borrelli, Ph. de Forcrand and A. Galli, hep-lat/9602016 M. Creutz and A. Gocksch, Phys. Rev. Lett. 63 (1989) 9 D. Stoffer, Computing 55 (1995) 1 Ph. de Forcrand, Nucl. Phys. B (Proc. Suppl.) 47 (1996) 228 J.C. Sexton and D.H. Weingarten, Nucl. Phys. B 380 (1992) 665 T. Lippert, SESAM collaboration, private communication
|
---
author:
- 'Mikio <span style="font-variant:small-caps;">Eto</span>[^1] and Tomohiro <span style="font-variant:small-caps;">Yokoyama</span>'
title: Quantum Dot Spin Filter in Resonant Tunneling and Kondo Regimes
---
The generation of spin current with no magnetic field or ferromagnets is an important issue for spin-based electronics, “spintronics.”[@Zutic] In this context, the spin-orbit (SO) interaction has attracted much interest. For conduction electrons in direct-gap semiconductors, an external potential $U({\bm r})$ results in the Rashba SO interaction[@Rashba; @Rashba2] $$H_\text{RSO} =
\frac{\lambda}{\hbar} {\bm \sigma} \cdot
\left[{\bm p} \times {\bm \nabla} U({\bm r}) \right],
\label{eq:SOorg}$$ where ${\bm p}$ is the momentum operator and ${\bm \sigma}$ is the Pauli matrices indicating the electron spin ${\bm s}={\bm \sigma}/2$. The coupling constant $\lambda$ is markedly enhanced by the band effect, particularly in narrow-gap semiconductors, such as InAs.[@Winkler; @Nitta] The spatial inversion symmetry is broken in compound semiconductors, which gives rise to another type of SO interaction, the Dresselhaus SO interaction.[@Dresselhaus] It is given by $$\begin{aligned}
H_\text{DSO}=\frac{\lambda'}{\hbar}
\bigl[ p_x(p_y^2-p_z^2)\sigma_x+
p_y(p_z^2-p_x^2)\sigma_y
\nonumber \\
+p_z(p_x^2-p_y^2)\sigma_z \bigr].
\label{eq:Dresselhaus}\end{aligned}$$
In the presence of SO interaction, the spin Hall effect (SHE) produces a spin current traverse to an electric field applied by the bias voltage. Two types of SHE have been intensively studied. One is an intrinsic SHE, which is induced by the drift motion of carriers in the SO-split band structures.[@Murakami; @Wunderlich; @Sinova] The other is an extrinsic SHE caused by the spin-dependent scattering of electrons by impurities.[@Dyakonov] Kato [*et al*]{}. observed the spin accumulation at sample edges traverse to the current,[@Kato] which is ascribable to the extrinsic SHE with $U({\bm r})$ being the screened Coulomb potential by charged impurities in eq. (\[eq:SOorg\]).[@Engel] In our previous studies,[@EY1; @YE1] we theoretically examined the extrinsic SHE in semiconductor heterostructures due to the scattering by an artificial potential created by antidots, STM tips, and others. The potential is electrically tunable. We showed that the SHE is significantly enhanced by the resonant scattering when the attractive potential is properly tuned. We proposed a three-terminal spin-filter including a single antidot.
In the present letter, we study the enhancement of the “extrinsic SHE” by resonant tunneling through a quantum dot (QD) with a strong SO interaction, e.g., InAs QD.[@Igarashi; @Fasth; @Pfund; @Takahashi] The QD is connected to $N$ external leads via tunnel barriers. In the QD, the number of electrons can be tuned, one by one, owing to the Coulomb blockade when the electrostatic potential is changed by the gate voltage $V_\text{g}$. The current through a QD shows a peak structure as a function of $V_\text{g}$ (Coulomb oscillation). We use the term SHE in the following meaning: For $N \ge 3$, when an unpolarized current is injected to the QD from a lead, polarized currents are ejected to the other leads. In other words, the QD works as a spin filter. First, we examine the SHE around the current peaks, where the resonant tunneling takes place. We show that the spin polarization is markedly enhanced when the energy-level spacing in the QD is smaller than the level broadening due to the tunnel coupling to external leads. Next, we examine the many-body resonance induced by the Kondo effect in the Coulomb blockade regime with spin 1/2 in the QD. We obtain a large spin current in the presence of the SU(4) Kondo effect when the level spacing is less than the Kondo temperature.
![Model for a quantum dot connected to $N$ external leads ($N \ge 2$). The quantum dot has two energy levels, $\varepsilon_j$ ($j=1,2$), which are coupled to lead $\alpha$ by $V_{\alpha,j}$ \[$\alpha=$S or D$n$, $n=1,\cdots, (N-1)$\]. An unpolarized current is injected from lead S and ejected to the other leads. The spin-orbit interaction is present in the quantum dot. ](Eto-Yokoyama-Fig1.eps){width="4.5cm"}
We assume that the SO interaction is present only in the QD and that the level spacing in the QD is comparable to the level broadening $\Gamma$ ($\sim 1$ $\mathrm{meV}$), in accordance with experimental situations.[@Igarashi; @Fasth; @Pfund; @Takahashi] The strength of SO interaction, $\Delta_\text{SO}$ in eq.(\[eq:Hdot\]), is approximately $0.2$ $\mathrm{meV}$.[@Fasth; @Pfund; @Takahashi] As a minimal model, we examine two levels in the QD. Note that previous theoretical papers[@Kiselev3; @Bardarson; @Krich1; @Krich2] concerned the spin-current generation in a mesoscopic region, or an open QD with no tunnel barriers, in which many energy levels in the QD participate in the transport.
We examine a two-level Anderson model with $N$ $(\ge 2)$ leads, shown in Fig. 1. The energy levels in the QD are denoted by $\varepsilon_1$ and $\varepsilon_2$ before the SO interaction is turned on. In the absence of magnetic field, wavefunctions of the states, i.e., $\langle {\bm r} |1 \rangle$ and $\langle {\bm r} |2 \rangle$, can be real. In the case of Rashba SO interaction, the orbital part in eq. (\[eq:SOorg\]) is a pure imaginary operator, and hence it has off-diagonal elements only; $\langle 2| H_\text{RSO} | 1 \rangle =
\text{i} {\bm h}_\text{SO} \cdot {\bm \sigma}/2$ with $\text{i} {\bm h}_\text{SO}/2= (\lambda/\hbar)
\langle 2 | ({\bm p} \times {\bm \nabla} U) | 1 \rangle$. If the quantization axis of spin is taken in the direction of ${\bm h}_\text{SO}$, the Hamiltonian in the QD reads $$\begin{aligned}
H_\text{dot}
&=& \sum_{\sigma=\pm 1}
(d_{1,\sigma}^{\dagger}, d_{2,\sigma}^{\dagger})
\left( \varepsilon_\text{d} - \frac{\Delta}{2}
\tau_z + \sigma \frac{\Delta_\text{SO}}{2} \tau_y
\right)
\left( \hspace*{-0.2cm}
\begin{array}{c}
d_{1,\sigma} \\ d_{2,\sigma}
\end{array}
\hspace*{-0.2cm} \right)
\nonumber \\
& & +H_\text{int},
\label{eq:Hdot}\end{aligned}$$ where $d_{j,\sigma}^{\dagger}$ and $d_{j,\sigma}$ are the creation and annihilation operators of an electron with orbital $j$ and spin $\sigma$, respectively. $\varepsilon_\text{d} =(\varepsilon_1 + \varepsilon_2 )/2$, $\Delta =\varepsilon_2 -\varepsilon_1$, and $\Delta_\text{SO}=|{\bm h}_\text{SO}|$. The Pauli matrices, $\tau_y$ and $\tau_z$, are introduced for the pseudo-spin representing level $1$ or $2$. $H_\text{int}$ describes the Coulomb interaction between electrons. The same form of the QD Hamiltonian is derived similarly in the case of Dresselhaus SO interaction in eq.(\[eq:Dresselhaus\]).[@com0] Note that the level spacing would be $\sqrt{\Delta^2+(\Delta_\text{SO})^2}$ in an isolated QD.
The state $|j \rangle$ in the QD is connected to lead $\alpha$ by tunnel coupling, $V_{\alpha, j}$ ($j=1,2$), which is real. The tunnel Hamiltonian is $$\begin{aligned}
H_\text{T} & = &
\sum_{j=1,2} \sum_{\alpha,k,\sigma}
(V_{\alpha, j} d_{j,\sigma}^{\dagger}c_{\alpha k,\sigma}+
\text{h.c}.)
\nonumber \\
& = &
\sum_{\alpha,k,\sigma}
V_{\alpha} [ (e_{\alpha,1} d_{1,\sigma}^{\dagger}+
e_{\alpha,2} d_{2,\sigma}^{\dagger}) c_{\alpha k,\sigma}+
\text{h.c}.],
\label{eq:Htunnel}\end{aligned}$$ where $c_{\alpha k,\sigma}$ annihilates an electron with state $k$ and spin $\sigma$ in lead $\alpha$. $V_{\alpha}=\sqrt{(V_{\alpha, 1})^2+(V_{\alpha, 2})^2}$ and $e_{\alpha,j}=V_{\alpha, j}/V_{\alpha}$. We introduce a unit vector, ${\bm e}_{\alpha}=(e_{\alpha,1},e_{\alpha,2})^T$. $V_{\alpha}$ is controllable by electrically tuning the tunnel barrier, whereas ${\bm e}_{\alpha}$ is determined by the wavefunctions $\langle {\bm r} | 1 \rangle$ and $\langle {\bm r} | 2 \rangle$ in the QD and hardly controllable for a given current peak. ($\{ {\bm e}_{\alpha} \}$ and $\Delta$ vary from peak to peak during the Coulomb oscillation. We can choose a peak with appropriate parameters for the SHE in experiments.)
We assume a single channel of conduction electrons in the leads. The total Hamiltonian is $$H=\sum_{\alpha}\sum_{k,\sigma} \varepsilon_k
c_{\alpha k,\sigma}^{\dagger} c_{\alpha k,\sigma}
+H_\text{dot}+H_\text{T}.
\label{eq:Hamiltonian}$$
The strength of the tunnel coupling is characterized by the level broadening, $\Gamma_\alpha = \pi \nu_\alpha
(V_{\alpha})^2$, where $\nu_\alpha$ is the density of states in lead $\alpha$. We also introduce a matrix of $\hat{\Gamma}=\sum_{\alpha} \hat{\Gamma}_{\alpha}$ with $$\begin{aligned}
\hat{\Gamma}_{\alpha}
=
\Gamma_\alpha
\left( \begin{array}{cc}
(e_{\alpha, 1})^2 & e_{\alpha, 1}e_{\alpha, 2}
\\
e_{\alpha, 1}e_{\alpha, 2} & (e_{\alpha, 2})^2
\end{array}
\right).
\label{eq:broadening-M}\end{aligned}$$
An unpolarized current is injected into the QD from a source lead ($\alpha=$S) and output to other leads \[D$n$; $n=1,\cdots,(N-1)$\]. The electrochemical potential for electrons in lead S is lower than that in the other leads by $-eV_\text{bias}$. The current with spin $\sigma=\pm$ from lead $\alpha$ to the QD is written as $$I_{\alpha,\sigma}=\frac{\text{i}e}{\pi \hbar} \int d\varepsilon
\text{Tr}\{
\hat{\Gamma}_{\alpha}
[f_{\alpha}(\varepsilon)
(\hat{G}^\text{r}_{\sigma}-\hat{G}^\text{a}_{\sigma})+
\hat{G}^{<}_{\sigma}]
\},
\label{eq:current}$$ where $\hat{G}^\text{r}_{\sigma}$, $\hat{G}^\text{a}_{\sigma}$, and $\hat{G}^{<}_{\sigma}$ are the retarded, advanced, and lesser Green functions in the QD, respectively, in $2\times 2$ matrix form in the pseudo-spin space.[@Meir] $f_{\alpha}(\varepsilon)$ is the Fermi distribution function in lead $\alpha$. In the absence of electron-electron interaction, $H_\text{int}$, the conductance into lead D$n$ with spin $\sigma$ is given by[@com1] $$G_{n,\sigma}=
\left.
-\frac{\text{d}I_{\text{D}n,\sigma}}{\text{d}V_\text{bias}}
\right|_{V_\text{bias} = 0}
=\frac{4e^2}{h} \text{Tr}
[\hat{G}^\text{a}_{\sigma}(\varepsilon_\text{F})
\hat{\Gamma}_{\text{D}n}
\hat{G}^\text{r}_{\sigma}(\varepsilon_\text{F})
\hat{\Gamma}_{\text{S}}],
\label{eq:conductance0}$$ where the QD Green function is $$\hat{G}^\text{r}_{\pm}(\varepsilon) =
\left[
\left( \begin{array}{cc}
\varepsilon-\varepsilon_\text{d} + \frac{\Delta}{2}
& \pm \text{i} \frac{\Delta_\text{SO}}{2}
\\
\mp \text{i} \frac{\Delta_\text{SO}}{2} &
\varepsilon-\varepsilon_\text{d} - \frac{\Delta}{2}
\end{array}
\right)
+\text{i} \hat{\Gamma}
\right]^{-1}
\label{eq:G0}$$ and $\varepsilon_\text{F}$ is the Fermi energy.
{width="7cm"}
Now, we discuss the SHE in the vicinity of the Coulomb peaks. The electron-electron interaction is neglected in this regime. From eqs. (\[eq:conductance0\]) and (\[eq:G0\]), we obtain $$\begin{aligned}
G_{n,\sigma} & = & \frac{e^2}{h}
\frac{4\Gamma_\text{S} \Gamma_{\text{D}n}}{|D|^2}
\left[ g_{n}^{(1)} + g_{n,\sigma}^{(2)} \right],
\label{eq:conductance}
\\
g_{n}^{(1)} & = &
\Biggl[
\left(
\varepsilon_\text{F}-\varepsilon_\text{d}-\frac{\Delta}{2}
\right)
e_{\text{D}n,1} e_{\text{S},1}
\nonumber \\
& &
+
\left(
\varepsilon_\text{F}-\varepsilon_\text{d}+\frac{\Delta}{2}
\right)
e_{\text{D}n,2} e_{\text{S},2}
\Biggr]^2,
\\
g_{n,\pm}^{(2)} & = &
\Biggl[
\pm \frac{\Delta_\text{SO}}{2}
({\bm e}_\text{S} \times {\bm e}_{\text{D}n})_z
\nonumber \\
& &
+
\sum_{\alpha} \Gamma_{\alpha}
({\bm e}_{\text{D}n} \times {\bm e}_{\alpha})_z
({\bm e}_\text{S} \times {\bm e}_{\alpha})_z
\Biggr]^2,\end{aligned}$$ where $D$ is the determinant of $[\hat{G}^\text{r}_{\sigma}(\varepsilon_\text{F})]^{-1}$ in eq. (\[eq:G0\]), which is independent of $\sigma$, and $({\bm a} \times {\bm b})_z = a_1 b_2 - a_2 b_1$. Let us consider two simple cases. (I) When $\Delta \gg \Gamma_{\alpha}$ and $\Delta_\text{SO}$, $G_{n,\sigma}$ consists of two Lorentzian peaks as a function of $\varepsilon_\text{d}$, reflecting the resonant tunneling through one of the energy levels, $\varepsilon_{1,2}=
\varepsilon_\text{d} \mp \Delta/2$: $$G_{n,\sigma} \approx \frac{4 e^2}{h}
\Gamma_\text{S} \Gamma_{\text{D}n}
\sum_{j=1,2}
\frac{(e_{\text{D}n,j} e_{\text{S},j})^2}
{(\varepsilon_j-\varepsilon_\text{F})^2+(\Gamma_{jj})^2},
\label{eq:G-largeD}$$ where $\Gamma_{jj}$ \[$jj$ component of matrix $\hat{\Gamma}$; $\sum_{\alpha} \pi \nu_{\alpha} (V_{\alpha,j})^2$\] is the broadening of level $j$. In this case, the spin current \[$\propto (G_{n,+}-G_{n,-})$\] is very small. $\Delta$ should be comparable to or smaller than the level broadening to observe a considerable spin current. (II) In a two-terminated QD ($N=2$), the second term in $g_{n,\pm}^{(2)}$ vanishes. Since $g_{n,+}^{(2)}=g_{n,-}^{(2)}$, no spin current is generated.[@Eto05b] Three or more leads are required to observe a spin current, as pointed out by other groups.[@Krich1; @Zhai; @Kiselev3]
We focus on $G_{1,\pm}$ in the three-terminal system ($N=3$). Then $g_{1,\pm}^{(2)} =
[\pm (\Delta_\text{SO}/2)
({\bm e}_\text{S} \times {\bm e}_{\text{D}1})_z +
\Gamma_{\text{D}2}
({\bm e}_{\text{D}1} \times {\bm e}_{\text{D}2})_z
({\bm e}_\text{S} \times {\bm e}_{\text{D}2})_z
]^2$. We exclude specific situations in which two out of ${\bm e}_\text{S}$, ${\bm e}_{\text{D}1}$, and ${\bm e}_{\text{D}2}$ are parallel to each other hereafter. The conditions for a large spin current are as follows: (i) $\Delta^{<}_{\sim}$ (level broadening), as mentioned above. Two levels in the QD should participate in the transport. (ii) The Fermi level in the leads is close to the energy levels in the QD, $\varepsilon_\text{F} \approx \varepsilon_\text{d}$ (resonant condition). (iii) The level broadening by the tunnel coupling to lead D$2$, $\Gamma_{\text{D}2}$, is comparable to the strength of SO interaction $\Delta_\text{SO}$.
Figures 2 and 3 show two typical results of the conductance $G_{1,\pm}$ as a function of $\varepsilon_\text{d}$ (Coulomb peak). In $g_{1}^{(1)}$, $e_{\text{D}1,1} e_{\text{S},1}$ and $e_{\text{D}1,2} e_{\text{S},2}$ have different (same) signs in Fig. 2 (Fig. 3): $g_{1}^{(1)}=0$ has no solution (a solution) in $-\Delta/2 <
\varepsilon_\text{d}-\varepsilon_\text{F}
< \Delta/2$.
In Fig. 2, the conductance shows a single peak. We set $\Gamma_\text{S}=\Gamma_{\text{D}1} \equiv \Gamma$. When $\Delta=0.2 \Gamma$ (main panels), we obtain a large spin current around the current peak, which clearly indicates an enhancement of the SHE by resonant tunneling \[conditions (i) and (ii)\]. With increasing $\Gamma_{\text{D}2}$ from (a) $0.2\Gamma$ to (d) $2\Gamma$, the spin current increases first, takes a maximum in panel (c), and then decreases \[condition (iii)\]. Therefore, the SHE is tunable by changing the tunnel coupling. When $\Delta=\Gamma$ (insets), the SHE is less effective, but we still observe a spin polarization of $P=(G_{1,+}-G_{1,-})/(G_{1,+}+G_{1,-}) \approx 0.25$ at the conductance peak in panel (c).
{width="5cm"}
In Fig. 3, the conductance $G_{1,\pm}$ shows a dip at $\varepsilon_\text{d} \approx \varepsilon_\text{F}$ for small $\Gamma_{\text{D}2}$.[@com3] Around the dip, the spin polarization is markedly enhanced: $|P|$ is close to unity in panel (a).
Next, we examine the Kondo effect in the Coulomb blockade regime with a single electron in the QD. The Kondo effect is not broken by the SO interaction since the time-reversal symmetry holds. For the electron-electron interaction in the QD, we assume that $H_\text{int}=U \sum_{j} n_{j,+} n_{j,-}
+U' \sum_{\sigma,\sigma'} n_{1,\sigma} n_{2,\sigma'}$, where $n_{j,\sigma}=d_{j,\sigma}^{\dagger}d_{j,\sigma}$, with infinitely large $U$ and $U'$. The Kondo effect creates the many-body resonant state at the Fermi level, and thus condition (ii) is always satisfied. The resonant width is given by the Kondo temperature $T_\text{K}$.[@Hewson] When $T_\text{K} < \Delta$, the upper level in the QD is irrelevant. The spin at the lower level is screened out by the conventional SU(2) Kondo effect. When $T_\text{K} > \Delta$, on the other hand, the pseudo-spin as well as the spin are screened by the SU(4) Kondo effect.[@com4] The latter situation is required for an enhanced SHE since two levels should be relevant to the resonance \[condition (i)\].
The crossover between the SU(2) and SU(4) Kondo effects can be semiquantitatively described by the slave-boson mean-field theory.[@Lim] The theory describes the Kondo resonant state on the assumption of its presence and Fermi liquid behavior and yields the conductance at temperature $T=0$. A boson operator $b$ is introduced to represent an empty state in the QD. $d_{j,\sigma}^{\dagger}=
f_{j,\sigma}^{\dagger}b$ and $d_{j,\sigma}=
b^{\dagger} f_{j,\sigma}$, with a fermion operator $f_{j,\sigma}$ representing the pseudo-spin $j$ and spin $\sigma$. $H_\text{int}$ is taken into account by the constraint of $Q \equiv \sum_{j,\sigma} f_{j,\sigma}^{\dagger}f_{j,\sigma}+
b^{\dagger}b-1=0$. $b$ is replaced with the mean field $\langle b \rangle$, which is determined by minimizing $\langle H+\lambda Q \rangle$ with the Lagrange multiplier $\lambda$.[@Hewson] The conductance is given by eq. (\[eq:conductance\]) if $\varepsilon_\text{d}$ and $\Gamma_\alpha$ are replaced with the renormalized ones, $\varepsilon_\text{d}+\lambda$ ($\sim \varepsilon_\text{F}$) and $\Gamma_\alpha \langle b \rangle^2$ ($\sim T_\text{K}$), respectively.
Figure 4 shows $G_{1,\pm}$ as a function of $\varepsilon_\text{d}$ in the three-terminal system. The parameters are the same as those in the main panels in Fig. 2. In the two-terminal situation (curve $a$; $G_{1,+}=G_{1,-}$), the conductance increases with decreasing $\varepsilon_\text{d}$ and saturates, indicating the charge fluctuation regime and Kondo regime, respectively. With three leads (curves $b$–$e$), we observe a spin current around the beginning of the Kondo regime. $P \approx 0.5$ in the case of curve $e$. As $\varepsilon_\text{d}$ is decreased further, $T_\text{K}$ decreases and becomes smaller than $\Delta$, which weakens the SHE. We obtain similar results using the parameters in Fig. 3.
{width="7cm"}
In summary, we have examined the SHE in a multiterminated QD with SO interaction. The spin polarization in the output currents is markedly enhanced by resonant tunneling if the level spacing in the QD is smaller than the level broadening. The spin current is also enlarged by the many-body resonance due to the SU(4) Kondo effect. The SHE is electrically tunable by changing the tunnel coupling to the leads.
Hamaya [*et al*]{}. fabricated InAs QDs connected to ferromagnets.[@Hamaya] If a ferromagnet is used as a source lead in our model, spin filtering is electrically detected through an “inverse SHE.” The current to lead D$1$ is proportional to $(1+p\cos\theta) G_{\text{D}1,+}+
(1-p\cos\theta) G_{\text{D}1,-}$, where $p$ is the polarization in the ferromagnet and $\theta$ is the angle between the magnetization and ${\bm h}_\text{SO}$.
The SHE in QDs is useful for the fundamental research as well as for the application to an efficient spin filter. The SHE enhanced by resonant scattering or Kondo resonance was examined for metallic systems with magnetic impurities.[@Fert; @Fert2; @Guo] In semiconductor QDs, we can observe the SHE due to the scattering by a single “impurity” with the tuning of various conditions.
The authors acknowledge fruitful discussion with G. Schön and Y. Utsumi. This work was partly supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science, and by the Global COE Program “High-Level Global Cooperation for Leading-Edge Platform on Access Space (C12).” T. Y. is a Research Fellow of the Japan Society for the Promotion of Science.
[99]{}
I. Žutić, J. Fabian, and S. Das Sarma: Rev. Mod. Phys. **76** (2004) 323. E. I. Rashba: Fiz. Tverd. Tela (Leningrad) **2** (1960) 1224 \[in Russian\]. Yu. A. Bychkov and E. I. Rashba: J. Phys. C **17** (1984) 6039. R. Winkler: [*Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems*]{} (Springer, Heidelberg, 2003). J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki: Phys. Rev. Lett. **78** (1997) 1335. G. Dresselhaus: Phys. Rev. **100** (1955) 580.
S. Murakami, N. Nagaosa, and S. C. Zhang: Science **301** (2003) 1348. J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth: Phys. Rev. Lett. **94** (2005) 47204. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald: Phys. Rev. Lett. **92** (2004) 126603. M. I. Dyakonov and V. I. Perel: Phys. Lett. **35A** (1971) 459. Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom: Science **306** (2004) 1910. H. Engel, B. I. Halperin, and E. I. Rashba: Phys. Rev. Lett.**95** (2005) 166605.
M. Eto and T. Yokoyama: J. Phys. Soc. Jpn. **78** (2009) 073710. T. Yokoyama and M. Eto: Phys. Rev. B **80** (2009) 125311.
Y. Igarashi, M. Jung, M. Yamamoto, A. Oiwa, T. Machida, K. Hirakawa, and S. Tarucha: Phys. Rev. B **76** (2007) 081303(R). C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and D. Loss: Phys. Rev. Lett. **98** (2007) 266801. A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq: Phys. Rev. B **76** (2007) 161308(R). S. Takahashi, R. S. Deacon, K. Yoshida, A. Oiwa, K. Shibata, K. Hirakawa, Y. Tokura, and S. Tarucha: Phys. Rev. Lett. **104** (2010) 246801.
A. A. Kiselev and K. W. Kim: Phys. Rev. B **71** (2005) 153315. J. H. Bardarson, I. Adagideli, and Ph. Jacquod: Phys. Rev. Lett. **98** (2007) 196601. J. J. Krich and B. I. Halperin: Phys. Rev. B **78** (2008) 035338. J. J. Krich: Phys. Rev. B **80** (2009) 245313.
For $H_\text{DSO}$, ${\bm h}_\text{SO}$ is replaced with ${\bm h}_\text{DSO}$, where $\text{i} h_{\text{DSO},x}/2=(\lambda'/\hbar)
\langle 2 | p_x (p_y^2 - p_z^2) | 1 \rangle$, etc. In InAs QDs, $H_\text{RSO}$ and $H_\text{DSO}$ may coexist. Then ${\bm h}_\text{SO}$ is changed to ${\bm h}_\text{SO}+{\bm h}_\text{DSO}$.
Y. Meir and N. S. Wingreen: Phys. Rev. Lett. **68** (1992) 2512.
For noninteracting electrons, $\hat{G}^\text{r}_{\sigma}-\hat{G}^\text{a}_{\sigma}
=-2\text{i} \hat{G}^\text{r}_{\sigma} \hat{\Gamma}
\hat{G}^\text{a}_{\sigma}$ and $\hat{G}^{<}_{\sigma}=2\text{i} \hat{G}^\text{r}_{\sigma}
(\sum_{\alpha} \hat{\Gamma}_{\alpha} f_{\alpha})
\hat{G}^\text{a}_{\sigma}$. The substitution of these relations into eq. (\[eq:current\]) yields $I_{\text{D}n,\sigma}=(4e/h) \int d\varepsilon
[f_\text{D}(\varepsilon)-f_\text{S}(\varepsilon)]
\text{Tr}(\hat{G}^\text{a}_{\sigma}
\hat{\Gamma}_{\text{D}n} \hat{G}^\text{r}_{\sigma}
\hat{\Gamma}_\text{S})$, where $f_{\text{D}n}(\varepsilon) \equiv f_\text{D}(\varepsilon)$.
In the presence of more than one channel in a lead, a spin current can be generated in two-terminal systems, e.g., see, M. Eto, T. Hayashi, and Y. Kurotani: J. Phys. Soc. Jpn. **74** (2005) 1934. F. Zhai and H. Q. Xu: Phys. Rev. Lett.**94** (2005) 246601.
The conductance dip is caused by the destructive interference between propagating waves through two orbitals in the QD. In the two-terminal system without SO interaction, the conductance $G_{1,\sigma} \propto g_{1}^{(1)}$ completely vanishes at the dip, where the “phase lapse” of the transmission phase takes place. See ref. 36 and related references cited therein.
A. C. Hewson: [*The Kondo Problem to Heavy Fermions*]{} (Cambridge University Press, Cambridge, 1993).
We have two sets of Kramers’ degenerate levels in the QD and $N$ channels in the leads. By the unitary transformation for the $N$ channels, we obtain two channels connected to the QD and $(N-2)$ channels disconnected from the QD. This is the situation of the full-screening Kondo effect.[@Nozieres] When $\Delta < T_\text{K}$, the SU(4) Kondo effect is realized even when the strengths of the tunnel coupling are not identical between the two levels in the QD. This is because the fixed point is marginal (and thus not unstable) in the Kondo scaling.[@Eto05]
J. S. Lim, M.-S. Choi, M. Y. Choi, R. López, and R. Aguado: Phys. Rev. B **74** (2006) 205119.
K. Hamaya, M. Kitabatake, K. Shibata, M. Jung, M. Kawamura, K. Hirakawa, T. Machida, T. Taniyama, S. Ishida, and Y. Arakawa: Appl. Phys. Lett. **91** (2007) 022107.
A. Fert and O. Jaoul: Phys. Rev. Lett. **28** (1972) 303. A. Fert, A. Friederich, and A. Hamzic: J. Magn. Magn. Mater. **24** (1981) 231. G. Y. Guo, S. Maekawa, and N. Nagaosa: Phys. Rev. Lett. **102** (2009) 036401.
C. Karrasch, T. Hecht, A. Weichselbaum, Y. Oreg, J. von Delft, and V. Meden: Phys. Rev. Lett. **98** (2007) 186802.
P. Nozières and A. Blandin: J. Phys. (Paris) **41** (1980) 193.
M. Eto: J. Phys. Soc. Jpn. **74** (2005) 95.
[^1]: E-mail address: eto@rk.phys.keio.ac.jp
|
---
author:
- 'David A. Green'
- Patrick Elwood
title: 'Arcminute MicroKelvin Imager observations at 15 GHz of the 2020 February outburst of Cygnus X-3'
---
The high mass X-ray binary shows occasional giant radio flares, with flux densities up to $\sim 10$ Jy. The first such burst was observed in 1972 [@1972Natur.239..440G], and subsequently there have been several similar flares, e.g. see @1986ApJ...309..707J [@1995AJ....110..290W; @1997MNRAS.288..849F; @2001ApJ...553..766M; @2012MNRAS.421.2947C; @2016MNRAS.456..775Z; @2017MNRAS.471.2703E] for some examples.
Here we report observations of a recent giant radio flare from Cygnus X-3 in 2020 February, made with the Arcminute MicroKelvin Imager (AMI, @2008MNRAS.391.1545Z [@2018MNRAS.475.5677H]). These observations were triggered by the detection of an increase in the emission from Cygnus X-3 seen at 37 GHz with the Metsähovi Radio Observatory (Karri Koljonen, private communication). The observations were made with the AMI ‘Small Array’ which is a radio interferometer consisting of ten 3.7-m diameter antennas. A single linear polarisation, Stokes parameter $I+Q$, was observed over a frequency range of 13.0 to 18.0 GHz. However, in practice the ends of this frequency range were not used (either due to poor sensitivity or, at the lower frequencies, satellite interference), and the observed band was 14.2 to 17.4 GHz.
{width="17.5cm"}
The observations consisted of 1-hour integrations on Cygnus X-3, interleaved with 400-s observations of a nearby, compact calibrator source J2052$+$3635. The first observation was made on 2020 Feb 7th, for $\approx 6.4$ hours. Long observations (up to $\approx 9.7$ hours, after flagging) were made on all but one of the following days up to and including 2020 Feb 15th; the exception was Feb 9th, when only a short observation ($\approx 1.5$ hours) was possible, due to extremely high winds, which meant the array had to be stowed. The number of antennas available during these observations was usually 8 (except on Feb 9th and 13th when the number was, respectively, 7 and 9).
The data were processed using standard procedures, with the flux density scale established from short observations of the standard calibrator source 3C286 which were made most days, together with the ‘rain gauge’ measurements made during the observations which were use to correct for varying atmospheric conditions (see @2008MNRAS.391.1545Z). The data were flagged: (i) automatically to eliminate bad data due to various technical problems, interference, and when some antennas were shadowed at the end of the observations (at low elevations); (ii) manually, to eliminate remaining interference, and some periods with very heavy rain. The interleaved observations of J2052$+$3635 provided the initial phase calibration of each antenna in the array throughout each observation. The amplitudes of the J2052$+$3635 observations were used to check the consistency of the flux density scale during the observations. It was found that the r.m.s. deviation of the J2052$+$3635 flux densities was $1.8$ per cent. Subsequently the observations were phase self-calibrated on a timescale of 10 min. Flux densities were derived for 10-min averages, for 5 broad frequency channels covering 14.2 to 17.4 GHz, and then a power law fit was made to obtain a flux density at 15 GHz.
Figure \[fig:ami-sa\] shows the 15-GHz light curve of Cygnus X-3 from these observations. This shows a rapid increase on Feb 7th, rising from $\approx 2$ Jy to a peak of $\approx 10$ Jy in just over 4 hours. During this period the radio spectrum across the AMI band changed from optically thick (flux density rising with increasing frequency) to optically thin (flux density falling with increasing frequency) synchrotron emission. In subsequent days Cygnus X-3 showed other peaks at $\approx 11$ Jy on Feb 8th, and $\approx 14$ Jy on Feb 11th, after which it faded, approximately exponentially (with a characteristic time of 1.7 days).
We thank the staff of the Mullard Radio Astronomy Observatory, University of Cambridge, for their support in the maintenance, and operation of AMI. We acknowledge support from the European Research Council under grant ERC-2012-StG-307215 LODESTONE.
Corbel, S., Dubus, G., Tomsick, J. A., et al. 2012, , 421, 2947
Egron, E., Pellizzoni, A., Giroletti, M., et al. 2017, , 471, 2703
Fender, R. P., Bell Burnell, S. J., Waltman, E. B., et al. 1997, , 288, 849
Gregory, P. C., & Kronberg, P. P. 1972, , 239, 440
Hickish, J., Razavi-Ghods, N., Perrott, Y. C., et al. 2018, , 475, 5677
Johnston, K. J., Spencer, J. H., Simon, R. S., et al. 1986, , 309, 707
Mioduszewski, A. J., Rupen, M. P., Hjellming, R. M., et al. 2001, , 553, 766
Waltman, E. B., Ghigo, F. D., Johnston, K. J., et al. 1995, , 110, 290
Zdziarski, A. A., Segreto, A., & Pooley, G. G. 2016, , 456, 775
Zwart, J. T. L., Barker, R. W., Biddulph, P., et al. 2008, , 391, 1545
|
---
abstract: 'The Hardy constant of a simply connected domain $\Omega\subset{{\mathbb{R}}}^2$ is the best constant for the inequality $$\int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \; , \;\;\quad u\in C^{\infty}_c(\Omega).$$ After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In this work we determine the Hardy constant of an arbitrary quadrilateral in the plane. In particular we show that the Hardy constant is the same as that of a certain infinite sectorial region which has been studied by E.B. Davies.'
author:
- 'G. Barbatis[^1]'
- 'A. Tertikas [^2]'
date:
-
-
title: |
On the Hardy constant of non-convex planar domains:\
the case of the quadrilateral
---
[**Keywords:**]{} Hardy inequality, Hardy constant, distance function.
[**2010 Mathematics Subject Classification:**]{} 35A23, 35J20, 35J75 (46E35, 26D10, 35P15)
Introduction
============
In the 1920’s Hardy established the following inequality [@h]: $$\int_0^{\infty} u'(t)^2dt \geq \frac{1}{4}\int_0^{\infty} \frac{u^2}{t^2}dt \; , \;\; \mbox{ for all }u\in C^{\infty}_c(0,\infty).
\label{or}$$ The constant $1/4$ is the best possible, and equality is not attained for any non-zero function in the appropriate Sobolev space.
Inequality (\[or\]) immediately implies the following inequality on ${{\mathbb{R}}}^N_+={{\mathbb{R}}}^{N-1}\times (0,+\infty)$: $$\int_{{{\mathbb{R}}}^N_+}|\nabla u|^2dx \geq \frac{1}{4}\int_{{{\mathbb{R}}}^N_+} \frac{u^2}{x_N^2}dx \; , \;\; \mbox{ for all }u\in C^{\infty}_c({{\mathbb{R}}}^N_+),
\label{or1}$$ where again the constant $1/4$ is the best possible. The analogue of (\[or1\]) for a domain $\Omega\subset{{\mathbb{R}}}^N$ is $$\int_{\Omega}|\nabla u|^2dx \geq \frac{1}{4}\int_{\Omega} \frac{u^2}{d^2}\, dx \; , \;\; \mbox{ for all }u\in C^{\infty}_c(\Omega),
\label{or2}$$ where $d=d(x)={{\rm dist}}(x,\partial\Omega)$. However, (\[or2\]) is not true without geometric assumptions on $\Omega$. The typical assumption made for the validity of (\[or2\]) is that $\Omega$ is convex [@da2]. A weaker geometric assumption introduced in [@bft] is that $\Omega$ is weakly mean convex, that is $$-\Delta d(x)\geq 0 \; , \quad \mbox{ in }\Omega,
\label{c}$$ where $\Delta d$ is to be understood in the distributional sense. Condition (\[c\]) is equivalent to convexity when $N=2$ but strictly weaker than convexity when $N\geq 3$ [@ak].
In the last years there has been a lot of activity on Hardy inequality and improvements of it under the convexity or weak mean convexity assumption on $\Omega$; see [@bm; @bft; @hhl; @fmt]. If no geometric assumptions are imposed on $\Omega$, then one can still obtain inequalities of similar type. If for example $\Omega$ is bounded with $C^2$ boundary then one can still have inequality (\[or2\]) for all $u\in C^{\infty}_c(\Omega_{\epsilon})$ where $\Omega_{\epsilon}=\{x\in\Omega : d(x)<\epsilon\}$, provided $\epsilon>0$ is small enough [@fmt]. In the same spirit, under the same assumptions on $\Omega$ it was proved in [@bm] that there exists $\lambda\in{{\mathbb{R}}}$ such that $$\int_{\Omega}|\nabla u|^2dx + \lambda \int_{\Omega}u^2 dx
\geq \frac{1}{4}\int_{\Omega} \frac{u^2}{d^2}\, dx \; , \;\; \mbox{ for all }u\in C^{\infty}_c(\Omega).
\label{or3}$$
More generally, it is well known that for any bounded Lipschitz domain $\Omega\subset{{\mathbb{R}}}^N$ there exists $c>0$ such that $$\label{hohi}
\int_{\Omega}|\nabla u|^2 dx\geq c\int_{\Omega}\frac{u^2}{d^2}dx \;\; , \qquad \mbox{ for all }u\in C^{\infty}_c(\Omega).$$ Following [@da1] we call the best constant $c$ of inequality (\[hohi\]) the Hardy constant of the domain $\Omega$.
In two space dimensions Ancona [@an] using Koebe’s 1/4 theorem discovered the following remarkable result: for any simply connected domain $\Omega\subset{{\mathbb{R}}}^2$ there holds $$\int_{\Omega}|\nabla u|^2dx
\geq \frac{1}{16}\int_{\Omega} \frac{u^2}{d^2}\, dx \; , \;\; \mbox{ for all }u\in C^{\infty}_c(\Omega).
\label{ancona}$$ This result is typical of two space dimensions: Davies [@da1] has proved that no universal Hardy constant exists in dimension $N\geq 3$.
From now on we concentrate on two space dimensions. Two questions arise naturally, and have already been posed in the literature [@la; @da1; @da2; @b; @laso]:
[(1)]{}
: Given a simply connected domain $\Omega\subset{{\mathbb{R}}}^2$ find (or obtain information about) the Hardy constant of $\Omega$.
[(2)]{}
: Find the best uniform Hardy constant valid for all simply connected domains $\Omega\subset{{\mathbb{R}}}^2$. Moreover, determine whether there are extremal domains, that is domains $\Omega$ whose Hardy constant coincides with the best uniform Hardy constant.
Laptev and Sobolev [@laso] established a more refined version of Koebe’s theorem and obtained a Hardy inequality which takes account of a quantitative measure of non-convexity. In particular they proved that if any $y\in\partial\Omega$ is the vertex of an infinite sector $\Lambda$ of angle $\theta\in [\pi,2\pi]$ independent of $y$ such that $\Omega\subset \Lambda$, then the constant $1/16$ of (\[ancona\]) can be replaced by $\pi^2/4\theta^2$. The convex case corresponds to $\theta=\pi$, in which case the theorem recovers the $1/4$ in the case of convexity. Analogous results were obtained recently in [@avla; @ab].
Davies [@da1] studied problem (1) in the case of an infinite sector of angle $\beta$. He used the symmetry of the domain to reduce the computation of the Hardy constant to the study of a certain ODE; see (\[ode\]) below. In particular he established the following two results, which are also valid for the circular sector of angle $\beta$:
\(a) The Hardy constant is $1/4$ for all angles $\beta\leq \beta_{cr}$, where $\beta_{cr} \cong 1.546\pi$.
\(b) For $\beta_{cr} \leq \beta\leq 2\pi$ the Hardy constant strictly decreases with $\beta$ and in the limiting case $\beta=2\pi$ the Hardy constant is $\cong 0.2054$.
Our aim in this work is to answer questions (1) and (2) in the particular case where $\Omega$ is a quadrilateral. Since the Hardy constant for any convex domain is $1/4$ we restrict our attention to non-convex quadrilaterals. In this case there is exactly one non-convex angle $\beta$, $\pi <\beta <2\pi$. As we will see, this angle plays an important role and determines the Hardy constant. Our result reads as follows:
Let $\Omega$ be a non-convex quadrilateral with non-convex angle $\pi<\beta <2\pi$. Then $$\int_{\Omega}|\nabla u|^2 dx \geq c_{\beta}\int_{\Omega} \frac{u^2}{d^2}dx \; , \quad u\in C^{\infty}_c(\Omega),
\label{mix}$$ where $c_{\beta}$ is the unique solution of the equation $$\label{tans1}
\sqrt{c_{\beta}}\tan\big( \sqrt{c_{\beta}}(\frac{\beta-\pi}{2})\big) = 2 \bigg(
\frac{\Gamma(\frac{3+\sqrt{1-4c_{\beta}}}{4})}
{\Gamma(\frac{1+\sqrt{1-4c_{\beta}}}{4})}
\bigg)^2,$$ when $\beta_{cr}\leq\beta < 2\pi$ and $c_{\beta}=1/4$ when $\pi <\beta\leq \beta_{cr}$. The constant $c_{\beta}$ is the best possible.
As we shall see, the constant $c_{\beta}$ is precisely the Hardy constant of the sector of angle $\beta$, so equation (\[tans1\]) provides an analytic description of the Hardy constant computed in [@da1] numerically. From (\[tans1\]) we also deduce that the critical angle $\beta_{cr}$ in (b) is the unique solution in $(\pi,2\pi)$ of the equation $$\label{tans2}
\tan\big( \frac{\beta_{cr}-\pi}{4}\big) = 4\bigg(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\bigg)^2.$$ Relation (\[tans2\]) was also obtained, amongst other interesting results, by Tidblom in [@ti]. We also note that the constant $c_{2\pi}$ is the uniform Hardy constant for the class of all quadrilaterals. The sharpness of the constant $c_{\beta}$ follows from the results of Davies [@da1].
An important ingredient in the proof of our theorem is the following elementary inequality valid on any domain $U$. Suppose $\partial U=\Gamma \cup\tilde\Gamma$. Then, under certain assumptions, for any function $\phi>0$ on $U\cup \Gamma$ we have $$\int_U |\nabla u|^2dx \geq -\int_U\frac{\Delta\phi}{\phi}u^2dx +\int_{\Gamma} u^2 \frac{\nabla\phi}{\phi} \cdot\vec{\nu} dS
\label{lib1}$$ for all smooth functions $u$ which vanish near $\tilde\Gamma$. Inequality (\[lib1\]) will be applied to suitable subdomains $U_i$ of $\Omega$ and for suitable choices of functions $\phi$. Roughly, each subdomain $U_i$ consists of points whose nearest boundary point belongs to a different part of $\partial\Omega$. The contribution along the boundary $\partial\Omega$ is zero because of the Dirichlet boundary conditions whereas there are non-zero interior boundary contributions that have to be taken into account.
The structure of the paper is simple: in Section \[sec:lemmas\] we establish a number of auxiliary results that are used in Section \[sec:quads\] where our theorem is proved.
Auxiliary estimates {#sec:lemmas}
===================
Let $\beta >\pi$ be fixed. We start by defining the potential $V(\theta)$, $\theta\in (0,\beta)$, $$V(\theta)=\tarr{\;\Frac{1}{\sin^2\theta} ,}{ 0< \theta <\frac{\pi}{2} ,}{1,}{ \frac{\pi}{2}<\theta <\beta-\frac{\pi}{2},}
{\Frac{1}{\sin^2(\beta-\theta)},}{\beta-\frac{\pi}{2} <\theta <\beta.}
\label{v}$$ For $c>0$ we consider the following boundary-value problem: $$\darr{ -\psi''(\theta) =c V(\theta)\psi(\theta),}{ 0\leq\theta\leq\beta ,}{ \psi(0)=\psi(\beta)=0\, }{}
\label{ode}$$ It was proved in [@da1] that the largest positive constant $c$ for which (\[ode\]) has a positive solution coincides with Hardy constant of the sector of angle $\beta$. Due to the symmetry of the potential $V(\theta)$ this also coincides with the largest constant $c$ for which the following boundary value problem has a solution: $$\darr{ -\psi''(\theta) =c V(\theta)\psi(\theta),}{ 0\leq\theta\leq\beta/2,}{ \psi(0)=\psi'(\beta/2)=0\, .}{}
\label{ode1}$$ Due to this symmetry, we shall identify the solutions of problems (\[ode\]) and (\[ode1\]).
The largest angle $\beta_{cr}$ for which the Hardy constant is $1/4$ for $\beta\in [\pi,\beta_{cr}]$ was computed numerically in [@da1] and analytically in [@ti] where (\[tans2\]) was established; the approximate value is $\beta_{cr} \cong 1.546\pi$.
We first study the following algebraic equation $$\sqrt{c}\tan\big( \sqrt{c}(\frac{\beta-\pi}{2})\big) = 2 \bigg(
\frac{\Gamma(\frac{3+\sqrt{1-4c}}{4})}
{\Gamma(\frac{1+\sqrt{1-4c}}{4})}
\bigg)^2.
\label{eq:alg}$$ We note that choosing in (\[eq:alg\]) $c=1/4$ we obtain $\beta_{cr}$ which is given by (\[tans2\]).
For any $\beta\geq \beta_{cr}$ there exists a unique $c=c_{\beta}$ satisfying (\[eq:alg\]). Moreover the function $\beta\mapsto c_{\beta}$ is smooth and strictly decreasing for $\beta\geq \beta_{cr}$. In particular we have $$c_{2\pi}<c_{\beta}<\frac{1}{4}\; \mbox{ for all }\; \beta_{cr} <\beta<2\pi.$$
[*Note.*]{} From (\[eq:alg\]) we obtain the numerical estimate $c_{2\pi}\cong 0.20536$ of [@da1].
[*Proof.*]{} Setting $x=\sqrt{1-4c}$ equation (\[eq:alg\]) takes the equivalent form´ $$G(x,\beta):= \frac{1}{2}(1-x^2)^{1/4} \tan^{1/2}\big( (1-x^2)^{1/2}\frac{\beta-\pi}{4}\big) -\frac{\Gamma(\frac{3+x}{4})}{\Gamma(\frac{1+x}{4})} =0,$$ where we are interested in the range $0\leq x <1$ and $\beta$ is such that $$(1-x^2)^{1/2}\frac{\beta-\pi}{4} <\frac{\pi}{2}.$$ For this range of $x$ and $\beta$ we can easily see that $G(x,\beta)$ is $C^{\infty}$. We will apply the Implicit Function Theorem. We first note that $G(0,\beta_{cr})=0$. Moreover a simple but tedious computation gives $$\begin{aligned}
\frac{\partial G}{\partial x}(x,\beta)&=& - \frac{x(\beta-\pi)}{16(1-x^2)^{1/4}}
\frac{1+ \tan^2\Big( (1-x^2)^{1/2}\frac{\beta-\pi}{4}\Big)}{\tan^{1/2}\Big( (1-x^2)^{1/2}\frac{\beta-\pi}{4}\Big)} \\
&& -\frac{x}{4(1-x^2)^{3/4}} \tan^{1/2}\Big( (1-x^2)^{1/2}\frac{\beta-\pi}{4}\Big)
-\frac{\Gamma( \frac{3+x}{4})}{4\Gamma( \frac{1+x}{4})} \bigg( \frac{\Gamma'(\frac{3+x}{4})}{\Gamma(\frac{3+x}{4})}
- \frac{\Gamma'(\frac{1+x}{4})}{\Gamma(\frac{1+x}{4})} \bigg).\end{aligned}$$ Since $$\frac{d}{dx}\big( \frac{\Gamma'(x)}{\Gamma(x)}\big) =\sum_{n=0}^{\infty}\frac{1}{(x+n)^2} >0,$$ we conclude that $\partial G/\partial x <0$ for all $(x,\beta)$ with $0\leq x <1$ and $$\beta_{cr} \leq \beta <\frac{2\pi}{\sqrt{1-x^2}} +\pi .$$ We also easily see that $\partial G/\partial \beta >0$ in the above range of $x$, $\beta$. This implies the existence and uniqueness locally near $\beta=\beta_{cr}$. A standard argument then gives the global existence of a smooth, strictly increasing function $x=x(\beta)$ for $\beta\geq \beta_{cr}$. The proof is concluding by substituting $c=\frac{1-x^2}{4}$. $\hfill\Box$
We next study the boundary value problem (\[ode1\]). The solution will be expressed using the hypergeometric function $$F(a,b,c ; z):=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty}\frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}\frac{z^n}{n!}.$$
Let $\beta>\beta_{cr}$. The boundary value problem (\[ode1\]) has a positive solution if and only if $c$ solves (\[eq:alg\]). In this case the solution is given by $$\psi(\theta)=\darr{ \Frac{\sqrt{2}\cos\big(\sqrt{c}(\beta-\pi)/2\big)\sin^{\alpha}(\theta/2)\cos^{1-\alpha}(\theta/2) }{F({\frac{1}{2}},{\frac{1}{2}}, \alpha +{\frac{1}{2}}; {\frac{1}{2}})}F({\frac{1}{2}},{\frac{1}{2}},\alpha +{\frac{1}{2}}; \sin^2(\frac{\theta}{2})),}
{0<\theta\leq \frac{\pi}{2},}{ \cos\big( \sqrt{c}(\frac{\beta}{2}-\theta) \big),}{ \frac{\pi}{2} <\theta \leq \frac{\beta}{2},}$$ where $\alpha$ is the largest solution of $\alpha(1-\alpha)=c$. Moreover $\psi\in H^1_0(0,\beta)$. \[lem:largeb\]
[*Proof.*]{} Clearly the function $$\psi(\theta)=\cos (\sqrt{c_{\beta}}(\frac{\beta}{2}-\theta))\, , \qquad \frac{\pi}{2}\leq \theta\leq \frac{\beta}{2} \, .$$ is a positive solution of the differential equation in $(\pi/2 ,\beta/2)$ and satisfies the boundary condition $\psi'(\beta/2)=0$. For $\theta\in (0,\pi/2)$ we set $\xi =\sin^2\theta/2$ and $y(\theta)=\sin^{\alpha}(\theta/2) \cos^{1-\alpha}(\theta/2) w(\xi)$ and we obtain after some computations that $w(\xi)$ solves the hypergeometric equation $$\xi(1-\xi)w_{\xi\xi} +(2\xi +\alpha -\frac{3}{2})w_{\xi} +\frac{1}{4}w=0 \; , \qquad 0<\xi<\frac{1}{2},$$ the general solution of which is described via hypergeometric functions $F(\alpha,\beta,\gamma,\xi)$ and is well-defined for $|\xi|<1$; see [@pz; @as] for details and various properties of the hypergeometric functions. We thus conclude that the general solution of the differential equation in (\[ode1\]) is $$\begin{aligned}
y(\theta) &=&c_1 \sin^{\alpha}(\frac{\theta}{2}) \cos^{1-\alpha}(\frac{\theta}{2})F(\frac{1}{2},\frac{1}{2},\alpha+\frac{1}{2};\sin^2(\frac{\theta}{2})) \\
&& +
c_2 \sin^{1-\alpha}(\frac{\theta}{2}) \cos^{1-\alpha}(\frac{\theta}{2})F(1-\alpha,1-\alpha,\frac{3}{2}-\alpha;\sin^2(\frac{\theta}{2})).\end{aligned}$$ In order to maximize $c$ we take $c_2=0$. The matching conditions at $\theta=\pi/2$ force $c$ to satisfy equation (\[eq:alg\]) and determine $c_1$. $\hfill\Box$
Let $\pi <\beta \leq \beta_{cr}$. The largest value of $c$ so that the boundary value problem (\[ode1\]) has a positive solution is $c=1/4$. For $\beta=\beta_{cr}$ the solution is $$\psi(\theta)=\darrsp{ \Frac{\cos\big(\frac{\beta_{cr}-\pi}{4}\big)\sin^{1/2}\theta }{F({\frac{1}{2}},{\frac{1}{2}}, 1 ; {\frac{1}{2}})}F({\frac{1}{2}},{\frac{1}{2}}, 1; \sin^2(\frac{\theta}{2})),}
{0<\theta\leq \frac{\pi}{2},}{ \cos\big( \frac{1}{2}(\frac{\beta_{cr}}{2}-\theta) \big),}{ \frac{\pi}{2} <\theta \leq \frac{\beta_{cr}}{2}.}$$ \[lem:smallb\]
[*Proof.*]{} Let $c=1/4$. Working as in the proof of Lemma \[lem:largeb\] we find that the general solution of the differential equation (\[ode1\]) in $(0,\pi/2)$ now is $$\begin{aligned}
y(\theta) &=&c_1 \sin^{1/2}(\frac{\theta}{2}) \cos^{1/2}(\frac{\theta}{2})F(\frac{1}{2},\frac{1}{2},1;\sin^2(\frac{\theta}{2})) \\
&& + c_2 \sin^{1/2}(\frac{\theta}{2}) \cos^{1/2}(\frac{\theta}{2})F(\frac{1}{2},\frac{1}{2},1;\sin^2(\frac{\theta}{2}))
\int_{\sin^2(\theta/2)}^{1/2}\frac{dt}{t(1-t)F^2(\frac{1}{2},\frac{1}{2},1;t)}.\end{aligned}$$ The matching conditions at $\theta=\pi/2$ determine $c_1$ and $c_2$. In order for $\psi$ to be positive it is necessary that $c_2\geq 0$. This turns out to be equivalent to $$4\frac{ \Gamma^2(\frac{3}{4})}{ \Gamma^2(\frac{1}{4})} \geq \tan(\frac{\beta -\pi}{4}).$$ This implies that $\beta\leq \beta_{cr}$ and in the case $\beta=\beta_{cr}$ we have $c_2=0$. $\hfill\Box$
For our purposes it is useful to write the solution of (\[ode1\]) in case $\beta\geq\beta_{cr}$ as a power series $$\label{ps}
\psi(\theta) =\theta^{\alpha}\sum_{n=0}^{\infty}a_n\theta^{n} \; ,$$ where $\alpha$ is the largest solution of the equation $\alpha(1-\alpha)=c$ in case $\beta>\beta_{cr}$ and $\alpha=1/2$ when $\beta=\beta_{cr}$. We normalize the power series setting $a_0=1$; simple computations then give $$a_1=0 \;\; , \qquad a_2 =-\frac{\alpha(1-\alpha)}{6(1+2\alpha)} .
\label{asymptotics}$$
For our analysis it will be important to study the following two auxiliary functions: $$f(\theta) =\frac{\psi'(\theta)}{\psi(\theta)} \; , \qquad \theta\in (0, \beta) \; ,
\label{f}$$ and $$g(\theta) =\frac{\psi'(\theta)}{\psi(\theta)}\sin\theta \; , \qquad \theta\in (0, \beta) \; ,
\label{g}$$ where $\psi$ is the normalized solution of (\[ode\]) described in Lemmas \[lem:largeb\] and \[lem:smallb\]. We note that these functions depend on $\beta$. Simple computations show that they respectively solve the differential equations $$f'(\theta) +f^2(\theta)+cV(\theta)=0 \; , \qquad 0<\theta <\beta
\label{def}$$ and $$\label{deg}
g'(\theta) =-\frac{1}{\sin\theta} \Big[ g(\theta)^2 -\cos\theta \, g(\theta) +c \Big] \; \; , \quad 0<\theta \leq\pi/2,$$ where $c=c_{\beta}$.
\[lem:g\] Let $\pi\leq\beta\leq 2\pi$. The function $g(\theta)$ is monotone decreasing on $(0,\pi/2]$.
[*Proof.*]{} In the case where $\pi\leq\beta\leq \beta_{cr}$ we have $c=1/4$ and therefore monotonicity follows at once from (\[deg\]). Suppose now that $\beta_{cr}\leq\beta\leq 2\pi$. Using the asymptotics (\[asymptotics\]) we obtain $$g(\theta)=\alpha + (2a_2 -\frac{\alpha}{6})\theta^2 +O(\theta^3) \;\; , \qquad \mbox{ as }\theta\to 0+.
\label{g:asymp}$$ Now, by (\[deg\]) $g(\theta)$ is monotone decreasing in $[\theta_0,\pi/2]$ where $\theta_0\in [0,\pi/2]$ is determined by $\cos^2\theta_0 =4c$. Let $\rho^{+}(\theta)$ denote the largest root of the equation $t^2 -\cos\theta \, t +c$, $0\leq\theta\leq\theta_0$. We note that $g(0)=\rho^+(0)$, $g'(0)=0$ and (by (\[g:asymp\])) $g''(0)<0$. Hence there exists an non-empty interval $(0,\theta^*)$ on which $g$ is strictly monotone decreasing and, therefore, $g(\theta)>\rho^+(\theta)$. To prove that $g$ is monotone decreasing on the whole $[0,\pi/2]$, let us assume that it is not. Then there exists a least positive $\theta_1$ such that $g'(\theta_1)=0$. We then have $g(\theta_1)=\rho^+(\theta_1)$. But $(\rho^+)'<0$, hence $g(\theta)<\rho^+(\theta)$ for $\theta<\theta_1$ close enough to $\theta_1$. This contradicts the definition of $\theta_1$. $\hfill\Box$
Let $\pi\leq\beta\leq 2\pi$. For $\pi/2\leq\gamma\leq\pi$ let $\theta_1$ be the angle in $[0,\pi/2]$ determined by the relation $$\label{thetasss}
\cot\theta_1 =\sin\gamma .$$ Then there holds $$\label{calll}
\frac{2+\cos\gamma}{1+\sin^2\gamma}f(\theta_1)\geq f(\frac{\pi}{2}) \; , \qquad \frac{\pi}{2}\leq\gamma\leq\pi.$$ \[lem:sss1\] \[lem:last\]
[*Proof.*]{} We define $$Q(\gamma)= \frac{2+\cos\gamma}{1+\sin^2\gamma}f(\theta_1).$$ We will establish that $Q$ is a decreasing function in $[\pi/2,\pi]$. An easy calculation gives $$Q'(\gamma) =\frac{\cos\gamma \; (2+\cos\gamma) }{ (1+\sin^2\gamma)^2}\bigg[ f(\theta_1)^2 -\frac{\sin\gamma (\cos^2\gamma +4\cos\gamma+2)}
{\cos\gamma (2+\cos\gamma)}f(\theta_1) +c(1+\sin^2\gamma) \bigg],$$ where $\theta_1=\theta_1(\gamma)$, $\pi/2\leq\gamma\leq\pi$.
We first consider the interval where $-2+\sqrt{2}\leq\cos\gamma\leq 0$. For such $\gamma$ we have $\cos^2\gamma +4\cos\gamma+2 \geq 0$ and the result follows at once.
We next consider the case where $-1\leq\cos\gamma \leq -2+\sqrt{2}$. The discriminant $\Delta$ of the quadratic polynomial above is $$\Delta =\frac{ \sin^2\gamma(\cos^2\gamma +4\cos\gamma+2)^2 -4c\cos^2\gamma(1+\sin^2\gamma)(2+\cos\gamma)^2}{\cos^2\gamma(2+\cos\gamma)^2}.$$ However, since $$\frac{d}{dt}( t^2-4t+2)^2 = 4( t^2-4t+2)(t-2)<0 \; , \qquad 2-\sqrt{2}\leq t\leq 1,$$ we conclude that $( t^2-4t+2)^2\leq 1$ for $2-\sqrt{2}\leq t\leq 1$ and therefore $$\Delta \leq \frac{ (1-\cos^2\gamma) -4c\cos^2\gamma (2-\cos^2\gamma)(2+\cos\gamma)^2}{\cos^2\gamma(2+\cos\gamma)^2} , \qquad \mbox{ for }
-1\leq\cos\gamma \leq -2+\sqrt{2}.$$ Next we shall prove that $(1-\cos^2\gamma) -4c\cos^2\gamma (2-\cos^2\gamma)(2+\cos\gamma)^2\leq 0$ for $-1\leq\cos\gamma \leq -2+\sqrt{2}$. For this we set $t=-\cos\gamma$ and we define $w(t)= 1-t^2 -4ct^2(2-t^2)(2-t)^2$, $t>0$. We have $$w'(t)=-2t\bigg( 1+4c [-3t^4+10t^3-4t^2-12t+8] \bigg).$$ Now, the function $p(t)=-3t^4+10t^3-4t^2-12t+8$ has derivative $$p'(t)=(t-1)(-12t^2 +18t +10) -2 \leq 0 \; , \qquad 0\leq t\leq 1.$$ Therefore $1+4cp(t) \geq 1+4cp(1) =1-4c\geq 0$ for $0\leq t\leq 1$. This in turn implies that $w(t)$ decreases in $[0,1]$. But $$w(2-\sqrt{2})=4\sqrt{2}-5 -64c(5\sqrt{2}-7) <0,$$ since $c> (4\sqrt{2}-5)/(64(5\sqrt{2} -7))\approx 0.1444$. We thus conclude that $w(t)\leq 0$ for $2-\sqrt{2}\leq t\leq 1$, which in turn implies that $\Delta\leq 0$ for $-1\leq\cos\gamma \leq -2+\sqrt{2}$. Therefore $Q(\gamma)$ is decreasing also in this this interval. Since $Q(\pi)=f(\pi/2)$, the proof is complete. $\hfill\Box$
Let $\pi\leq\beta\leq 2\pi$ and $\pi/2\leq\gamma\leq\pi$. For $\theta\in [\pi/2 , (3\pi/2)-\gamma]$ denote by $\theta_1=\theta_1(\theta)$ be the angle in $[0,\pi/2]$ uniquely determined by the relation $$\label{thetas}
\cot\theta_1 =-\cos(\theta+\gamma).$$ Then there holds $$\label{cal}
f(\theta_1) \geq f(\theta) \frac{1+\cos^2(\theta+\gamma)}{2+\sin(\theta+\gamma)} \, ,
\qquad \frac{\pi}{2}\leq \theta \leq\frac{3\pi}{2} -\gamma \, .$$ \[lem:sss\]
[*Proof.*]{} For $\theta =\pi/2$ the corresponding value $\theta_*=\theta_1(\pi/2)$ is the one given by (\[thetasss\]) hence the result is a consequence of Lemma \[lem:last\].
To prove (\[cal\]) we shall consider $\theta_1$ as the free variable so that $\theta=\theta(\theta_1)$ is given by (\[thetas\]). Since $f(\theta_1)$ satisfies $f'(\theta_1) +f^2(\theta_1) +c/\sin^2\theta_1=0$, it suffices to show that the function $$h(\theta_1) := f(\theta) \frac{1+\cos^2(\theta+\gamma)}{2+\sin(\theta+\gamma)} \;\qquad (\theta=\theta(\theta_1))$$ satisfies $$H(\theta_1):= h'(\theta_1) +h^2(\theta_1) +\frac{c}{\sin^2\theta_1} \leq 0\;\; , \qquad \theta_* \leq\theta_1\leq\frac{\pi}{2}\; ,
\label{nicesong}$$ where $\theta_*\in (0,\pi/2)$ is determined by $\cot\theta_* = \sin \gamma$.
We express $H(\theta_1)$ in terms of $f(\theta)$ and $f'(\theta)$; we also use the fact that, by (\[thetas\]), $$\frac{d\theta_1}{d\theta} =-\frac{ \sin(\theta+\gamma)}{1+\cos^2(\theta+\gamma)}.$$ Using (\[def\]) and setting $\omega=\theta+\gamma$ we obtain after some simple computations that $$\begin{aligned}
&& H(\theta_1)=\frac{1+\cos^2\omega}{\sin\omega (2+\sin\omega)^2} \bigg[ 2(1+\cos^2\omega)(1+\sin\omega)f^2(\theta) + \label{eis}\\
&&\qquad\qquad + \cos\omega(\sin^2\omega +4\sin\omega +2)f(\theta) +2c(1+\sin\omega)(2+\sin\omega)\bigg]. \nonumber\end{aligned}$$ In brackets we have a quadratic polynomial of $f(\theta)$ whose discriminant is itself a polynomial $P(t)$ of $t=-\sin\omega \in [-\cos \gamma,1]\subseteq [0,1]$, $$P(t)=(1-t) \Big[ t^5 +(16c-7)t^4 +12(1-4c)t^3 +4t^2 +12(8c-1)t +4(1-16c)\Big]=:(1-t)Q(t)\, .$$ We observe that $Q(0)<0$ and $Q(1)=2>0$; moreover $$Q'(t) =5t^4 +4(16c-7)t^3+36(1-4c)t^2 +8t +12(8c-1).
\label{q'}$$ Recall that $1/8 <c \leq 1/4$, hence all the summands in (\[q’\]) are non-negative in $[0,1]$ with the exception of $4(16c-7)$. Since $|4(16c-7)|=28-64c < 36(1-4c) + 8 + 12(8c-1)$, we conclude that $Q'>0$ in $[0,1]$.
The above considerations imply that there exists a unique $t_0\in (0,1)$ such that $P(t)<0$ in $(0,t_0)$ and $P(t)>0$ in $(t_0,1)$. This immediately implies that $H(\theta_1)\leq 0$ in the range $0<t<t_0$.
For $t_0<t<1$ the quadratic polynomial in (\[eis\]) has two roots of the same sign as the sign of $t^2-4t+2$. The equation $t^2-4t+2=0$ has solutions $2\pm\sqrt{2}$. It follows that the quadratic polynomial above has negative two roots when $\max\{ t_0,2-\sqrt{2}\}<t<1$. Since $f(\theta)>0$, $0<\theta<\beta/2$, we conclude once again that $H(\theta_1)\leq 0$ in this case as well. But we easily check that $Q(2-\sqrt{2})<0$, which implies that $\max\{ t_0,2-\sqrt{2}\}=t_0$. This completes the proof. $\hfill\Box$
Let $\pi\leq \beta\leq 2\pi$. The following inequalities hold: $$\begin{aligned}
{({\rm i})}&& \mbox{If $0\leq\omega\leq\pi/4$ then}\\
&& \hspace{1.5cm} f(\theta)\sin\theta\cos(\theta +\omega) + \alpha\cos\omega \geq 0 \; , \quad 0\leq\theta\leq\frac{\pi}{2}, \\[0.2cm]
{({\rm ii})}&& \mbox{If $3\pi/2 -\beta\leq \omega\leq 2\pi-\beta$ then}\\
&& \hspace{1.5cm} f(\theta)\cos(\theta+\omega)+\alpha[1+\sin(\theta+\omega)]\geq 0 \; ,\quad
\frac{\pi}{2}\leq\theta\leq\beta -\frac{\pi}{2} \, ,\\[0.2cm]
{({\rm iii})}&& \mbox{If $0\leq \omega\leq 2\pi-\beta$ then}\\
&& \hspace{1.5cm} -f(\theta)\cos(\theta+\omega) +\alpha[1-\sin(\theta+\omega)] \geq 0 \; ,
\quad\frac{\pi}{2}\leq\theta \leq\beta-\frac{\pi}{2}.\end{aligned}$$ \[lem:quad\]
[*Proof.*]{} (i) The inequality is trivially true for $0\leq\theta\leq\pi/2 -\omega$, so we restrict our attention to the interval $\pi/2 -\omega\leq\theta\leq\pi/2$. We must prove that $$f(\theta)\leq F(\theta) \; , \qquad \frac{\pi}{2} -\omega\leq\theta\leq\frac{\pi}{2},
\label{fg}$$ where $f$ is given by (\[f\]) and $$F(\theta)=-\alpha\frac{\cos\omega}{\sin\theta\cos(\theta+\omega)}.$$ Using the fact that $\sqrt{c}\leq\alpha$ we have $$\begin{aligned}
F(\frac{\pi}{2})-f(\frac{\pi}{2}) &=& \alpha \cot\omega-\sqrt{c}\tan[\sqrt{c}(\frac{\beta}{2}-\frac{\pi}{2})] \nonumber\\
&\geq& \alpha\Big\{ \cot\omega-\tan[\sqrt{c}(\frac{\beta}{2}-\frac{\pi}{2})]\Big\} \nonumber\\
&= & \frac{\alpha}{\sin\omega \cos[\sqrt{c}(\frac{\beta}{2}-\frac{\pi}{2})]}\cos\Big(\sqrt{c}(\frac{\beta}{2}-\frac{\pi}{2}) +\omega \Big) \nonumber\\
&\geq& 0,
\label{p2}\end{aligned}$$ since $0<\sqrt{c}(\frac{\beta}{2}-\frac{\pi}{2}) +\omega \leq \frac{\beta}{4}-\frac{\pi}{4}+\omega\leq \pi/2$.
We shall prove that $F'(\theta) +F(\theta)^2 +\frac{c}{\sin^2\theta} \leq 0$ in $[\pi/2-\omega,\pi/2]$. This, combined with (\[def\]) and (\[p2\]) will imply that $f(\theta)\leq F(\theta)$ in $[\pi/2-\omega,\pi/2]$.
Recalling that $c= \alpha(1-\alpha)$, we have for $\theta\in [\pi/2-\omega,\pi/2]$, $$\begin{aligned}
&&F'(\theta)+F^2(\theta) +\frac{c}{\sin^2\theta} \\
&=&\frac{\alpha\cos\omega\cos\theta}{\sin^2\theta\cos(\theta+\omega)} -
\frac{\alpha\cos\omega\sin(\theta+\omega)}{\sin\theta\cos^2(\theta+\omega)}
+\frac{\alpha^2\cos^2\omega}{\sin^2\theta\cos^2(\theta+\omega)} +\frac{c}{\sin^2\theta} \\
&=&\alpha\frac{\cos\omega\cos\theta\cos(\theta+\omega) -\cos\omega\sin(\theta+\omega)\sin\theta +\alpha\cos^2\omega +(1-\alpha)\cos^2(\theta+\omega)}{\sin^2\theta \cos^2(\theta+\omega)} \\
&=&\alpha\frac{ 2\cos\omega\cos\theta\cos(\theta+\omega) -(1-\alpha) [\cos^2\omega-\cos^2(\theta+\omega)]}{\sin^2\theta \cos^2(\theta+\omega)} \\
&=&\alpha\frac{ 2\cos\omega\cos\theta\cos(\theta+\omega) -(1-\alpha)\sin\theta \sin(\theta+2\omega)}{\sin^2\theta \cos^2(\theta+\omega)} \\
&\leq& 0,\end{aligned}$$ since the last term is the sum of two non-positive terms. Hence ${({\rm i})}$ has been proved.
\(ii) We first note that $$f(\theta)=\sqrt{c}\tan\Big( \sqrt{c}(\frac{\beta}{2}-\theta)\Big), \qquad\quad \frac{\pi}{2}\leq\theta\leq\beta -\frac{\pi}{2},$$ and $$-\frac{\pi}{4}\leq \sqrt{c}(\frac{\beta}{2}-\theta) \leq\frac{\pi}{4} \;, \qquad \quad \frac{\pi}{2}\leq\theta\leq\beta -\frac{\pi}{2}.$$ It follows that the required inequality is written equivalently, $$\alpha(1+\sin(\omega+\theta))\cos(\sqrt{c}( \frac{\beta}{2}-\theta)) +\sqrt{c}\sin(\sqrt{c}( \frac{\beta}{2}-\theta))\cos(\omega+\theta)\geq 0 \; ,
\;\; \frac{\pi}{2}\leq\theta\leq\beta -\frac{\pi}{2}.$$ Hence, since $\alpha\geq\sqrt{c}$, $$\begin{aligned}
&&\hspace{-2cm}\alpha(1+\sin(\theta+\omega))\cos(\sqrt{c}( \frac{\beta}{2}-\theta)) +\sqrt{c}\sin(\sqrt{c}( \frac{\beta}{2}-\theta))\cos(\theta+\omega) \nonumber\\
&\geq&\sqrt{c}\Big\{(1+\sin(\theta+\omega))\cos(\sqrt{c}( \frac{\beta}{2}-\theta)) +\sin(\sqrt{c}( \frac{\beta}{2}-\theta))\cos(\theta+\omega)\Big\} \nonumber\\
&=&\sqrt{c}\Big\{ \cos(\sqrt{c}( \frac{\beta}{2}-\theta)) +\sin[\sqrt{c}( \frac{\beta}{2}-\theta) +\theta+\omega]\Big\} \nonumber\\
&=&\sqrt{c}\Big\{\cos(\sqrt{c}( \frac{\beta}{2}-\theta)) -\cos[\frac{\pi}{2}+\sqrt{c}( \frac{\beta}{2}-\theta) +\theta+\omega]\Big\}.\nonumber\\
&=& 2\sqrt{c} \sin\Big[ \sqrt{c}(\frac{\beta}{2}-\theta)+\frac{\pi}{4}+\frac{\theta}{2}+\frac{\omega}{2}\Big] \sin(\frac{\pi}{4} +\frac{\theta}{2}+\frac{\omega}{2}).
\label{s1}\end{aligned}$$ But for the given range of $\omega$ and $\theta$ we have $$0\leq \frac{\pi}{4} +\frac{\theta}{2}+\frac{\omega}{2} \leq\pi \qquad
\mbox{ and }\qquad
0\leq \sqrt{c}(\frac{\beta}{2}-\theta)+\frac{\pi}{4}+\frac{\theta}{2}+\frac{\omega}{2} \leq \pi.$$ Hence the last quantity in (\[s1\]) is non-negative.
\(iii) We have $\cos(\theta+\omega)\leq 0$ for $\frac{\pi}{2}\leq\theta\leq\beta-\frac{\pi}{2}$, therefore the inequality is trivial for $\theta\in [\pi/2,\beta/2]$ (since $f \geq 0$ there). We now consider the complementary interval $\beta/2\leq\theta\leq\beta -\pi/2$. Arguing as in (\[s1\]) above we see that it suffices to prove that $$-\sin(\sqrt{c}( \frac{\beta}{2}-\theta))\cos(\theta+\omega) +[1-\sin(\theta+\omega) ]\cos(\sqrt{c}( \frac{\beta}{2}-\theta))\geq 0,$$ or equivalently, $$\cos\big(\sqrt{c}(\theta- \frac{\beta}{2})\big) \geq \sin \big(\sqrt{c}( \frac{\beta}{2}-\theta)+\theta+\omega\big) \; , \quad
\frac{\beta}{2}\leq\theta\leq\beta -\frac{\pi}{2}.
\label{ph}$$ We have $$\cos\big(\sqrt{c}(\theta- \frac{\beta}{2})\big) -\sin \big(\sqrt{c}( \frac{\beta}{2}-\theta)+\theta+\omega \big)=
-2\sin\big(\frac{\pi}{4} -\frac{\theta+\omega}{2} \big)\sin\big(\sqrt{c}(\frac{\beta}{2}-\theta) +\frac{\theta+\omega}{2} -\frac{\pi}{4}\big)$$ Since $\beta+\omega \leq 2\pi$, we have $$0\leq \frac{\theta}{2}+\frac{\omega}{2} -\frac{\pi}{4} \leq\frac{\pi}{2}$$ and $$0\leq \sqrt{c}(\frac{\beta}{2}-\theta) +\frac{\theta+\omega}{2} -\frac{\pi}{4}
\leq -\frac{\sqrt{c}(\beta-\pi)}{2} +\frac{\beta+\omega}{2} \leq\frac{\pi}{2},$$ hence (\[ph\]) is true. $\hfill\Box$
Proof of the Theorem {#sec:quads}
====================
In this section we will give the proof of our Theorem. We start with a lemma that plays fundamental role in our argument and will be used repeatedly. We do not try to obtain the most general statement and for simplicity we restrict ourselves to assumptions that are sufficient for our purposes.
Let $U$ be a domain and assume that $\partial U=\Gamma\cup\tilde\Gamma$ where $\Gamma$ is Lipschitz continuous. We denote by $\vec{\nu}$ the exterior unit normal on $\Gamma$.
Let $\phi\in H^1_{\rm loc}(U)$ be a positive function such that $\nabla\phi /\phi \in L^2(U)$ and $\nabla\phi /\phi$ has an $L^1$ trace on $\Gamma$ in the sense that $v \nabla\phi/\phi$ has an $L^1$ trace on $\partial U$ for every $v\in C^{\infty}(\overline{U})$ that vanishes near $\tilde\Gamma$. Then $$\int_U |\nabla u|^2dx\, dy \geq -\int_U\frac{\Delta\phi}{\phi}u^2dx\, dy +\int_{\Gamma} \frac{\nabla\phi}{\phi} \cdot\vec{\nu} u^2 dS
\label{lib}$$ for all smooth functions $u$ which vanish near $\tilde\Gamma$. Here $\Delta\phi$ is understood in the distributional sense. If in particular there exists $c\in{{\mathbb{R}}}$ such that $$-\Delta\phi\geq\frac{c}{d^2}\phi \; ,
\label{1}$$ in the weak sense in $U$, where $d={{\rm dist}}(x,\tilde\Gamma)$, then $$\int_U |\nabla u|^2dx\, dy \geq c\int_{U}\frac{u^2}{d^2}dx\, dy +\int_{\Gamma} u^2 \frac{\nabla\phi}{\phi} \cdot\vec{\nu} dS
\label{2}$$ for all functions $u\in C^{\infty}(\overline{U})$ that vanish near $\tilde\Gamma$. \[lem:1\]
[*Proof.*]{} Let $u$ be a function in $C^{\infty}(\overline{U})$ that vanishes near $\tilde\Gamma$. We denote $\vec{T}=-\nabla\phi/\phi$. Then $$\begin{aligned}
\int_U u^2 {{\rm div}}\vec{T}\, dx\, dy &=& -2\int_U u\nabla u\cdot \vec{T}\, dx\, dy +\int_{\Gamma}u^2 \vec{T}\cdot \vec{\nu} \, dS \\
&\leq &\int_U |\vec{T}|^2u^2 dx\, dy +\int_U |\nabla u|^2 dx\, dy + \int_{\Gamma}u^2 \vec{T}\cdot \vec{\nu} \, dS\, ,\end{aligned}$$ that is $$\int_U |\nabla u|^2 dx\, dy \geq \int_U ({{\rm div}}\vec{T} -|\vec{T}|^2) u^2 dx\, dy -\int_{\Gamma} \vec{T}\cdot \vec{\nu}u^2 dS\, .$$ Using assumption (\[1\]) we obtain (\[2\]). $\hfill\Box$
Let us now consider a non-convex quadrilateral $\Omega$, with vertices $O$, $A$, $B$ and $C$ (as in the diagrams) and corresponding angles $\beta$, $\gamma$, $\delta$ and $\zeta$. We assume that the non-convex vertex is $O$ and, is located at the origin, and that the side $OA$ lies along the positive $x$-axis and has length one.
Our argument depends fundamentally on two geometric features of the quadrilateral $\Omega$. While in all cases the methodology remains the same, the technical details are different. The first feature is whether or not one of the angles adjacent to the non-convex one is larger than $\pi/2$. The second one is related to the structure of the equidistance curve $$\Gamma=\{ P \in\Omega : {{\rm dist}}(P, OA \cup OC) ={{\rm dist}}(P, AB\cup BC)\}.$$ Clearly the curve $\Gamma$ consists of line and parabola segments. Taking also account of symmetries, each non-convex quadrilateral $\Omega$ fits within one of the following five types, each one of which will be dealt with separately:
[**Type A1.**]{} We have $\gamma\leq\pi/2$, $\zeta\leq\pi/2$ and the curve $\Gamma$ consists of two line and two parabola segments (Here we also include the special case where $\Gamma$ consists of two line segments and one parabola segment.)
[**Type A2.**]{} We have $\gamma\leq\pi/2$, $\zeta\leq\pi/2$ and the curve $\Gamma$ consists of three line segments and one parabola segment.
[**Type B1.**]{} $\gamma >\pi/2$ and the curve $\Gamma$ consists of two line segments and two parabola segments. (Here we also include the special case where $\Gamma$ consists of two line segments segments and one parabola segment.)
[**Type B2.**]{} $\gamma >\pi/2$ and the curve $\Gamma$ consists of three line and one parabola segment: starting from the point $A$ we first have two line segments, then a parabola segment and then a last line segment.
[**Type B3.**]{} $\gamma >\pi/2$ and the curve $\Gamma$ consists again of three line and one parabola segment: starting from the point $A$ we first have a line segment, then a parabola segment and then two more line segments.
In all cases the curve $\Gamma$ divides $\Omega$ into two parts $\Omega^-$ and $\Omega^+$ where points in $\Omega^-$ have nearest boundary point on $OA \cup OC$ and points on $\Omega^+$ have nearest boundary points on $AB\cup BC$. We denote by $\vec{\nu}$ the unit normal along $\Gamma$ which is outward with respect to $\Omega^-$. We also denote by $S$ the point where $\Gamma$ intersects the bisector at the vertex $B$.
We shall often make use of the following simple fact: let $P$ be the parabola determined by the origin and the line $x\sin\alpha +y\cos\alpha +l=0$, where $l>0$. The exterior (with respect to the convex component) unit normal along $\partial P$ is given in polar coordinates by $$\vec{\nu}=\frac{(\cos\theta -\sin\alpha, \sin\theta-\cos\alpha)}{\sqrt{2-2\sin(\theta+\alpha)}}.
\label{normal}$$
[***Proof of Theorem: type A1.***]{} We parametrize $\Gamma$ by the polar angle $\theta\in [0,\beta]$. For $\theta\in [0,\pi/2]$ $\Gamma$ is a straight line; the same is true for $\theta\in [\beta -\pi/2 ,\beta]$. Finally, for $\theta\in [\pi/2 ,\beta-\pi/2]$ $\Gamma$ consists of segments of two parabolas. These parabolas meet at the point $S$ which is equidistant from $AB$, $BC$ and the origin. Let $\theta_0$ be the polar angle of $S$. We assume without loss of generality that $\theta_0\leq\beta /2$. Hence $\Gamma$ consists of four segments which when parametrized by the polar angle $\theta$ are described as $$\Gamma_1=\{ 0\leq \theta\leq \pi/2\} , \;
\Gamma_2=\{ \pi/2 \leq \theta\leq \theta_0 \} , \;
\Gamma_3=\{ \theta_0\leq \theta\leq \beta-\frac{\pi}{2}\} , \;
\Gamma_4=\{\beta-\frac{\pi}{2}\leq\theta \leq\beta\}.$$ We shall apply Lemma \[lem:1\] with $U=\Omega_-$, $\tilde\Gamma=OA\cup OC$ and $\phi(x,y)=\psi(\theta)$, where $\psi(\theta)$ is the solution of (\[ode\]) described in Lemmas \[lem:largeb\] and \[lem:smallb\]. An easy computation shows that $$-\Delta \psi =\frac{c}{d^2}\psi \, .$$ We thus obtain that $$\int_{\Omega_-}|\nabla u|^2 dx\, dy \geq c\int_{\Omega_-}\frac{u^2}{d^2}dx\, dy +
\int_{\Gamma}\frac{\nabla\phi}{\phi}\cdot\vec{\nu} u^2 dS \, , \qquad u\in C^{\infty}_c(\Omega).
\label{pa}$$ We next apply Lemma \[lem:1\] for $\Omega_+$ and the function $\phi_1(x,y)=d(x,y)^{\alpha}$ (we recall that $\alpha$ is the largest solution of $\alpha(1-\alpha)=c$). We note that in $\Omega_+$ the function $d(x,y)$ coincides with the distance from $AB\cup BC$ and this implies that $$-\Delta d^{\alpha} \geq \alpha(1-\alpha)\frac{d^{\alpha}}{d^2}\; , \qquad \mbox{ on }\Omega_+\, .$$ (The difference of the two functions above is a positive mass concentrated on the bisector of the angle $B$). Applying Lemma \[lem:1\] we obtain that $$\begin{aligned}
\int_{\Omega_+}|\nabla u|^2 dx\, dy &\geq& c\int_{\Omega_+}\frac{u^2}{d^2}dx\, dy -\int_{\Gamma}\frac{\alpha\nabla d}{d}\cdot\vec{\nu}\, u^2 dS
\, , \qquad u\in C^{\infty}_c(\Omega).
\label{pa1}\end{aligned}$$ Adding (\[pa\]) and (\[pa1\]) we conclude that $$\int_{\Omega}|\nabla u|^2 dx\, dy \geq c\int_{\Omega}\frac{u^2}{d^2}dx\, dy +
\int_{\Gamma}\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu}\, u^2 dS \, , \qquad u\in C^{\infty}_c(\Omega).
\label{nd}$$ We emphasize that in the last integral the values of $\nabla\phi/\phi$ are obtained as limits from $\Omega_-$ while those of $\nabla d/d$ are obtained as limits from $\Omega_+$.
It remains to prove that the line integral in (\[nd\]) is non-negative. For this we shall consider the different segments of $\Gamma$.
\(i) The segment $\Gamma_1$ ($0\leq\theta\leq\pi/2$). Simple calculations give $$\frac{\nabla\phi}{\phi}=\frac{1}{r}\frac{\psi'(\theta)}{\psi(\theta)}(-\sin\theta , \cos\theta) \, , \quad \mbox{ in } \Omega_-\, .
\label{fl1}$$ The line $AB$ has equation $y+(x-1)\tan\gamma =0$, so $d(x,y)= (1-x)\sin\gamma -y\cos\gamma$ on $\{P\in\Omega : d(P)={{\rm dist}}(P,AB)\}$ and therefore $$\alpha\frac{\nabla d}{d}=-\alpha\frac{(\sin\gamma ,\cos\gamma) }{d} \; , \qquad \mbox{ on }\Gamma_1\cup\Gamma_2 .
\label{fl2}$$ Since $\vec{\nu}=(\sin(\gamma/2), \cos(\gamma/2))$ along $\Gamma_1$, (\[fl1\]) and (\[fl2\]) yield $$\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu} =\frac{1}{r}\frac{\psi'(\theta)}{\psi(\theta)}\cos(\theta +\frac{\gamma}{2}) +
\frac{\alpha \cos(\gamma/2)}{d} \; , \qquad \mbox{ on }\Gamma_1\, .$$ However $d(x,y)=y=r\sin\theta$ on $\Gamma_1$, so we conclude by (i) of Lemma \[lem:quad\] (with $\omega=\gamma/2$) that $$\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu} =\frac{1}{r\sin\theta}\Big(g(\theta)\cos(\theta +\frac{\gamma}{2}) +
\alpha \cos(\gamma/2)\Big)\geq 0 \; , \qquad \mbox{ on }\Gamma_1\, .
\label{fl4}$$
\(ii) The segment $\Gamma_2$ ($\pi/2\leq\theta\leq\theta_0$). This is (part of) the parabola determined by the origin and the side $AB$. Applying (\[normal\]) we obtain that the outward (with respect to $\Omega_-$) unit normal along $\Gamma_2$ is $$\vec{\nu} =\frac{(\cos\theta +\sin\gamma, \sin\theta +\cos\gamma)}{\sqrt{2+2\sin(\theta+\gamma)}}.
\label{fl5}$$ Combining (\[fl1\]), (\[fl2\]), (\[fl5\]) and (ii) of Lemma \[lem:quad\] (with $\omega=\gamma$) we obtain $$\Big(\frac{\nabla\phi}{\phi}-
\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu} =\frac{1}{r\sqrt{2+2\sin(\theta+\gamma)}}\Big(\frac{\psi'(\theta)}{\psi(\theta)}\cos(\theta+\gamma) +\alpha
[1+\sin(\theta +\gamma )]\Big)\geq 0\; , \quad \mbox{ on }\Gamma_2\, .
\label{fl6}$$ (iii) The segment $\Gamma_3$ ($\theta_0\leq\theta\leq\beta -\pi/2$). This is (part of) the parabola determined by the origin and the side $BC$. Now, the line $BC$ has equation $$(x+T) \sin(\gamma+\delta) +y\cos(\gamma+\delta) =0 \; ,$$ where $(-T,0)$ is the point where the side $BC$ intersects the $x$-axis. Applying (\[normal\]) we thus obtain that the outward unit normal is $$\vec{\nu}=\frac{(\cos\theta -\sin(\gamma+\delta),\sin\theta-\cos(\gamma+\delta))}{\sqrt{2-2\sin(\theta+\gamma+\delta)}}.$$ Hence, by (iii) of Lemma \[lem:quad\] (with $\omega=\gamma+\delta$), $$\Big(\frac{\nabla\phi}{\phi}-
\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu} =\frac{1}{r\sqrt{2-2\sin(\theta+\gamma)}}\Big(-\frac{\psi'(\theta)}{\psi(\theta)}\cos(\theta+\gamma+\delta) +\alpha [1-\sin(\theta +\gamma+\delta )]\Big)\geq 0\; , \;\; \mbox{ on }\Gamma_3\, .
\label{fl7}$$ (iv) The segment $\Gamma_4$ ($\beta-\pi/2\leq\theta\leq\beta$). Replacing $\theta$ by $\beta-\theta$, $\gamma$ by $2\pi -\beta-\gamma-\delta$ (the angle at $C$) and using the relation $\psi(\theta)=\psi(\beta-\theta)$, the computations become identical to those for the segment $\Gamma_1$; hence we obtain $$\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu}\geq 0 \; , \qquad \mbox{ on }\Gamma_4\, .
\label{fl8}$$ The proof of the theorem is completed by combining (\[nd\]), (\[fl4\]), (\[fl6\]), (\[fl7\]) and (\[fl8\]). $\hfill\Box$
In this case the curve $\Gamma$ consists of three line segments and one parabola segment. Without loss of generality we assume that starting from $\theta=0$ we first meet two line segments, then the parabola segment and then the last line segment. Then the first two line segments meet at the point $S$ with polar angle $\theta_0\leq\pi/2$ and the four components of $\Gamma$ are $$\Gamma_1=\{ 0\leq \theta\leq \theta_0\} , \;
\Gamma_2=\{ \theta_0\leq \theta\leq \frac{\pi}{2}\} , \;
\Gamma_3=\{ \frac{\pi}{2}\leq \theta\leq \beta-\frac{\pi}{2}\} , \;
\Gamma_4=\{\beta-\frac{\pi}{2}\leq\theta \leq\beta\}.$$ As in the case A1, we apply Lemma \[lem:1\] on $\Omega_-$ and $\Omega_+$ with the functions $\phi(x,y)=\psi(\theta)$ and $\phi_1(x,y)=d(x,y)^{\alpha}$ respectively. We arrive at an inequality similar to (\[nd\]) and we conclude that the result will follow once we prove that $$\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu}\geq 0 \; , \qquad \mbox{ on }\Gamma\, .
\label{fl88}$$
The computations along the segments $\Gamma_1$, $\Gamma_3$ and $\Gamma_4$ are identical to those for the type A1 considered above and are omitted.
For $\Gamma_2$ we consider the point $(-T,0)$, $T>0$, where the side $BC$ intersects the $x$-axis. The distance from the line $BC$ is $(x+T)\sin(\gamma+\delta)+y\cos(\gamma+\delta)$, therefore $\nabla d=(\sin(\gamma+\delta),\cos(\gamma+\delta))$ on $\Gamma_2$. Moreover along $\Gamma_2$ we have $\vec{\nu}=(-\cos((\gamma+\delta)/2),\sin((\gamma+\delta)/2))$. We also note on $\Gamma_2$ we have $d(x,y)=y=r\sin\theta$. Combining the above we obtain that $$\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu} =
\frac{1}{r\sin\theta}\Big[g(\theta)\sin\big(\theta +\frac{\gamma+\delta}{2}\big) +
\alpha\sin(\frac{\gamma+\delta}{2})\Big] \; , \qquad \mbox{ on }\Gamma_2,$$ which is non-negative for $\theta\in [0,\pi/2]$ since $\gamma+\delta\leq\pi$. $\hfill\Box$
We next consider the cases where one of the two angles that are adjacent to the non-convex angle exceeds $\pi/2$. Without loss of generality we assume that $\gamma\geq\pi/2$ (the angle at the vertex $A$). We note that since $\beta_{cr} > 3\pi/2$, in this case we have $\pi\leq \beta\leq \beta_{cr}$ hence the Hardy constant is $c=1/4$.
We now divide $\Omega_+$ in two parts, $\Omega_+^A$ and $\Omega_+^C$, the parts of $\Omega_+$ with nearest boundary points on $AB$ and $BC$ respectively. We denote by $\Gamma_*$ the common boundary of $\Omega_+^A$ and $\Omega_+^C$, that is the line segment $SB$. We also denote by $\vec{\nu}_*$ the normal unit vector along $\Gamma_*$ which is outward with respect to $\Omega_+^A$.
[***Proof of Theorem: type B1.***]{} As in the case A1, the curve $\Gamma$ is made up of four segments, $$\Gamma_1=\{ 0\leq \theta\leq \pi/2\} , \;
\Gamma_2=\{ \pi/2 \leq \theta\leq \theta_0 \} , \;
\Gamma_3=\{ \theta_0\leq \theta\leq \beta-\frac{\pi}{2}\} , \;
\Gamma_4=\{\beta-\frac{\pi}{2}\leq\theta \leq\beta\},$$ where $\theta_0$ is the polar angle of the point $S$. We use again Lemma \[lem:1\]. On $\Omega_-$ we use the function $\phi(x,y)=\psi(\theta)$, exactly as in types A1 and A2 and we obtain that $$\int_{\Omega_-}|\nabla u|^2 dx\, dy \geq \frac{1}{4}\int_{\Omega_-}\frac{u^2}{d^2}dx\, dy +
\int_{\Gamma}\frac{\nabla\phi}{\phi}\cdot\vec{\nu}u^2 dS \, , \qquad u\in C^{\infty}_c(\Omega).
\label{111}$$ On $\Omega_+^C$ again we work as in types A1 and A2: we use the function $\phi(x,y)=d(x,y)^{1/2}$ and we obtain $$\int_{\Omega_+^C}|\nabla u|^2 dx\, dy \geq \frac{1}{4}\int_{\Omega_+^R}\frac{u^2}{d^2}dx\, dy -\frac{1}{2}\int_{\Gamma_3\cup\Gamma_4}\frac{\nabla d}{d}\cdot\vec{\nu} u^2 dS - \frac{1}{2}\int_{\Gamma_*}\frac{\nabla d}{d}\cdot\vec{\nu}_* u^2 dS \, ,
\qquad u\in C^{\infty}_c(\Omega).
\label{112}$$ Concerning $\Omega_+^A$, we cannot use the test function $\phi=d^{1/2}$ since part (i) of Lemma \[lem:quad\] is not valid for the full range $\pi/4<\omega<\pi/2$. So we construct a different function $\phi$. To do this we consider a second orthonormal coordinate system with cartesian coordinates denoted by $(x_1,y_1)$ and polar coordinates denoted by $(r_1,\theta_1)$. The origin $O_1$ of this system is located on the extension of the side $AB$ from $A$ and at distance $-\cos\gamma$ from $A$, and the axes are chosen so that the point $A$ has cartesian coordinates $(-\cos\gamma, 0)$ with respect to the new system. We note that this choice is such that $$\label{113}
\mbox{the point on $\Gamma_1$ for which $\theta=\frac{\pi}{2}-\frac{\gamma}{2}$ satisfies also $\theta_1=\frac{\pi}{2}-\frac{\gamma}{2}$.}$$
We apply Lemma \[lem:1\] on $\Omega_+^A$ with the function $\phi_1(x,y)=\psi(\theta_1)$. This function clearly satisfies $-\Delta \phi_1 \geq \frac{1}{4}\, d^{-2}\phi_1$, hence we obtain $$\int_{\Omega_+^A}|\nabla u|^2 dx\, dy \geq \frac{1}{4} \int_{\Omega_+^A}\frac{u^2}{d^2}dx\, dy
-\int_{\Gamma_1\cup\Gamma_2}(\frac{\nabla\phi_1}{\phi_1}\cdot\vec{\nu})u^2 \,dS
+ \int_{\Gamma_*}(\frac{\nabla\phi_1}{\phi_1}\cdot\vec{\nu}_*)u^2 \,dS
\, \quad u\in C^{\infty}_c(\Omega).
\label{114}$$ Adding (\[111\]), (\[112\]) and (\[114\]) we conclude that $$\begin{aligned}
&&\int_{\Omega}|\nabla u|^2 dx\, dy \geq \frac{1}{4}\int_{\Omega}\frac{u^2}{d^2}dx\, dy + \int_{\Gamma_1\cup\Gamma_2}\Big(\frac{\nabla\phi}{\phi}-\frac{\nabla\phi_1}{\phi_1}\Big)\cdot\vec{\nu}\, u^2 dS \nonumber\\
&&\qquad + \int_{\Gamma_3\cup\Gamma_4}\Big(\frac{\nabla\phi}{\phi}-\frac{\nabla d}{2d}\Big)\cdot\vec{\nu}\, u^2 dS +
\int_{\Gamma_*}\Big(\frac{\nabla\phi_1}{\phi_1}-\frac{\nabla d}{2d}\Big)\cdot\vec{\nu}_*\, u^2 dS
\label{115}\end{aligned}$$ for any $u\in C^{\infty}_c(\Omega)$. So it remains to prove that the three line integrals in (\[115\]) are non-negative. For this we shall separately consider the different the segments $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ and $\Gamma_4$ and the segment $\Gamma_*$.
\(i) The segment $\Gamma_1$ ($0\leq\theta\leq\pi/2$). We have $$\frac{\nabla\phi}{\phi}\cdot\vec{\nu} = \frac{\psi'(\theta)}{r\psi(\theta)}\cos(\theta+\frac{\gamma}{2}) \; , \quad
\mbox{ on } \Gamma_1.$$ and similarly $$\frac{\nabla\phi_1}{\phi_1}\cdot\vec{\nu} = -\frac{\psi'(\theta_1)}{r_1 \psi(\theta_1)}\cos(\theta_1-\frac{\gamma}{2}) \; , \quad
\mbox{ on } \Gamma_1.$$ However we have $r_1\sin\theta_1 =r\sin\theta$ along $\Gamma_1$, so recalling definition (\[g\]) we see that it is enough to prove the inequality $$g(\theta)\cos(\theta+\frac{\gamma}{2}) +g(\theta_1)\cos(\theta_1-\frac{\gamma}{2})\geq 0 \; , \qquad \mbox{ on }\Gamma_1\, .
\label{222}$$ Recalling (\[113\]) and applying the sine law we obtain that along $\Gamma_1$ the polar angles $\theta$ and $\theta_1$ are related by $$\cot \theta_1 =-\cos\gamma \cot\theta +\sin\gamma \; .
\label{223}$$ [**Claim.**]{} There holds $$\label{claim1}
\theta_1\geq \theta +\gamma -\pi \; , \qquad \mbox{ on }\Gamma_1 \, .$$ [*Proof of Claim.*]{} We fix $\theta\in [0,\pi/2]$ and the corresponding $\theta_1=\theta_1(\theta)$. If $\theta+\gamma -\pi\leq 0$, then (\[claim1\]) is obviously true, so we assume that $\theta+\gamma -\pi\geq 0$. Since $0\leq\theta+\gamma -\pi\leq \pi/2$ and $0\leq\theta_1\leq\pi/2$, (\[claim1\]) is written equivalently $\cot\theta_1\leq\cot(\theta +\gamma -\pi)$; thus, recalling (\[223\]), we conclude that to prove the claim it is enough to show that $$-\cos\gamma\cot\theta +\sin\gamma\leq\cot(\theta+\gamma) \; , \quad
\pi-\gamma\leq\theta\leq\frac{\pi}{2},$$ or, equivalently (since $\pi\leq\theta+\gamma\leq 3\pi/2$), $$-\cos\gamma\cot^2\theta +(-\cos\gamma\cot\gamma -\cot\gamma+\sin\gamma)\cot\theta + 1+\cos\gamma \geq 0\; , \qquad
\pi-\gamma\leq\theta\leq\frac{\pi}{2}.
\label{eq:new}$$ The left-hand side of (\[eq:new\]) is an increasing function of $\cot\theta$ and therefore takes its least value at $\cot\theta =0$. Hence the claim is proved.
For $0\leq\theta\leq\pi/2-\gamma/2$ (\[222\]) is true since all terms in the left-hand side are non-negative. So let $\pi/2-\gamma/2\leq\theta\leq\pi/2$ and $\theta_1=\theta_1(\theta)$. From (\[223\]) we find that $$\begin{aligned}
\frac{d\theta_1}{d\theta}-1&=&-\frac{\cos\gamma (1+\cot^2\theta)+1+\cot^2\theta_1}{1+\cot^2\theta_1} \\
&=& -\frac{ 1+\sin^2\gamma +\cos\gamma -2\sin\gamma\cos\gamma\cot\theta +\cos\gamma(1+\cos\gamma)\cot^2\theta}{1+\cot^2\theta_1}.\end{aligned}$$ The function $$h(x):= 1+\sin^2\gamma +\cos\gamma -2\sin\gamma\cos\gamma x +\cos\gamma(1+\cos\gamma)x^2$$ is a concave function of $x$. We will establish the positivity of $h(\cot\theta)$ for $\pi/2-\gamma/2
\leq \theta\leq\pi/2$. For this it is enough to establish the positivity at the endpoints. At $\theta=\pi/2$ positivity is obvious, whereas $$h(\tan(\frac{\gamma}{2}))=1+\sin^2\gamma+\cos\gamma -2\cos\gamma\sin^2\frac{\gamma}{2}\geq 0.$$ From (\[113\]) we conclude that $\theta_1\leq\theta$ for $\pi/2-\gamma/2\leq\theta\leq\pi/2$.
We next apply Lemma \[lem:g\]. We obtain that for $\pi/2-\gamma/2\leq\theta\leq\pi/2$, $$\begin{aligned}
g(\theta)\cos(\theta+\frac{\gamma}{2}) +g(\theta_1)\cos(\theta_1-\frac{\gamma}{2})
&\geq& g(\theta) [\cos(\theta+\frac{\gamma}{2}) +\cos(\theta_1-\frac{\gamma}{2})]\\
&=&2g(\theta)\cos(\frac{\theta+\theta_1}{2}) \cos (\frac{\theta-\theta_1+\gamma}{2})\\
&\geq& 0,\end{aligned}$$ where for the last inequality we made use of the claim. Hence (\[222\]) has been proved.
\(ii) The segment $\Gamma_2$ ($\frac{\pi}{2}\leq\theta\leq\theta_0$). Computations similar to those that led to (\[fl6\]) together with the fact that $r=r_1\sin\theta_1$ on $\Gamma_2$ give that along $\Gamma_2$ we have $$\begin{aligned}
&&\Big(\frac{\nabla\phi}{\phi}-\frac{\nabla\phi_1}{\phi_1}\Big)\cdot\vec{\nu} \nonumber\\
&=&\frac{1}{\sqrt{2+2\sin(\theta+\gamma)}}\Big[ \frac{f(\theta)}{r} \cos(\theta+\gamma)
- \frac{f(\theta_1)}{r_1}[\sin(\theta_1-\theta -\gamma)-\cos\theta_1] \Big] \label{night}\\
&=&\frac{1}{r\sqrt{2+2\sin(\theta+\gamma)}}\Big[ f(\theta)\cos(\theta+\gamma)- f(\theta_1)\sin\theta_1[\sin(\theta_1-\theta -\gamma)-\cos\theta_1] \Big].\nonumber\end{aligned}$$ Now, simple geometry shows that along $\Gamma_2$ the angles $\theta$ and $\theta_1$ are related by $$\cot\theta_1 =-\cos(\theta+\gamma).
\label{224}$$ It follows that $$\sin\theta_1 [\sin(\theta_1-\theta-\gamma)-\cos\theta_1]= \frac{\cos(\theta+\gamma) [2+\sin(\theta+\gamma)]}{1+\cos^2(\theta+\gamma)},
\;\;\;\;\mbox{ along $\Gamma_2$}\, .$$ Since $\cos(\theta+\gamma)\leq 0$, (\[224\]) and Lemma \[lem:sss\] imply that $(\nabla\phi/\phi -\nabla\phi_1/\phi_1)\cdot\vec{\nu}\geq 0$ along $\Gamma_2$, as required.
\(iii) The segments $\Gamma_3$ and $\Gamma_4$ ($\theta_0\leq\theta \leq \beta$). Since $\zeta <\pi/2$, the change $\theta \leftrightarrow
\beta -\theta$ reduces this case to that of the segments $\Gamma_2$ and $\Gamma_1$ respectively for a quadrilateral of type A1, already considered above.
\(iv) The segment $\Gamma_*$. The contribution from $\Omega_+^A$ is $$\frac{\nabla\phi_1}{\phi_1}\cdot \vec{\nu}_* = \frac{f(\theta_1)}{r_1}\cos(\theta_1+\frac{\delta}{2})\geq 0\, , \qquad
\mbox{ on }\Gamma_*,$$ since $\theta_1\leq \gamma/2$, by construction of the new coordinate system and $\gamma+\delta<\pi$. Given that the contribution from $\Omega_+^C$ is positive, the proof is complete.
[***Proof of Theorem: type B2.***]{} As in the case of type A2, there exists an angle $\theta_0\leq\pi/2$ such that the four segments of $\Gamma$ are $$\Gamma_1=\{ 0\leq \theta\leq \theta_0\} , \;
\Gamma_2=\{ \theta_0\leq \theta\leq \frac{\pi}{2}\} , \;
\Gamma_3=\{ \frac{\pi}{2}\leq \theta\leq \beta-\frac{\pi}{2}\} , \;
\Gamma_4=\{\beta-\frac{\pi}{2}\leq\theta \leq\beta\}.$$ So $\Gamma_3$ is a parabola segment while $\Gamma_1$, $\Gamma_2$ and $\Gamma_4$ are line segments. We define the sets $\Omega_+^A$, $\Omega_+^C$ and the vector $\vec{\nu}_*$ as in the case of type B1 and apply Lemma \[lem:1\] with the same functions, that is $\psi(\theta)$ on $\Omega_-$, $d(x,y)^{1/2}$ on $\Omega_+^C$ and $\psi(\theta_1)$ on $\Omega_+^A$ (where we use exactly the some construction for the coordinate system $(x_1,y_1)$).
The computations along $\Gamma_1$, $\Gamma_3$ and $\Gamma_4$ are identical to those for the type B1 and are omitted. On $\Gamma_2$ we have, as in the case of subtype A2, $$\Big(\frac{\nabla\phi}{\phi}-\alpha\frac{\nabla d}{d}\Big)\cdot\vec{\nu} =
\frac{1}{r\sin\theta}\Big[g(\theta)\sin\big(\theta +\frac{\gamma+\delta}{2}\big) +{\frac{1}{2}}\sin(\frac{\gamma+\delta}{2})\Big] \geq 0\, ,$$ since $\gamma+\delta\leq\pi$. Finally, the computations along $\Gamma_*$ are identical to the corresponding computations for the case $B1$. This completes the proof.
[***Proof of Theorem: Type B3.***]{} In this case there exist angles $\theta_0,\theta_0'$ with $$\frac{\pi}{2}\leq \theta_0 <\theta_0' \leq \beta -\frac{\pi}{2}$$ such that the four segments of $\Gamma$ are $$\Gamma_1=\{ 0\leq \theta\leq \frac{\pi}{2}\} , \;
\Gamma_2=\{ \frac{\pi}{2}\leq \theta\leq \theta_0\} , \;
\Gamma_3=\{ \theta_0\leq \theta\leq \theta_0' \} , \;
\Gamma_4=\{\theta_0'\leq\theta \leq\beta\}.$$ So $\Gamma_2$ is a parabola segment while $\Gamma_1$, $\Gamma_3$ and $\Gamma_4$ are line segments. To proceed, we define the sets $\Omega_+^A$, $\Omega_+^C$ and the vector $\vec{\nu}_*$ as in the cases B1 and B2 and apply Lemma \[lem:1\] with the same functions, that is $\psi(\theta)$ on $\Omega_-$, $d(x,y)^{1/2}$ on $\Omega_+^C$ and $\psi(\theta_1)$ on $\Omega_+^A$, where again we use exactly the some construction for the coordinate system $(x_1,y_1)$.
The computations for the line segments $\Gamma_1$ and $\Gamma_4$ and for the parabola segment $\Gamma_2$ are identical to those for a quadrilateral of type B1 and are omitted. We next consider the line segment $\Gamma_3$ whose points are equidistant from the sides $AB$ and $OC$. Calculations similar to those above give that $$\Big(\frac{\nabla\phi}{\phi}-\frac{\nabla \phi_1}{\phi_1}\Big)\cdot\vec{\nu} =
\frac{1}{r\sin\theta}\Big[g(\theta)\sin\big(\frac{\beta-\gamma}{2}-\theta\big) +
g(\theta_1)\sin(\frac{\beta+\gamma}{2}-\theta_1)\Big] \; , \;\;\mbox{ on }\Gamma_3.$$ Now, it follows by construction that $$\theta \geq \frac{\pi}{2} \geq \frac{\beta+\gamma-\pi}{2} \geq\theta_1 \;\; , \qquad \mbox{ on }\Gamma_3.$$ Since $0 < (\beta+\gamma)/2 -\theta_1 <\pi$, by the monotonicity of $g$ we have $$\begin{aligned}
\Big(\frac{\nabla\phi}{\phi}-\frac{\nabla \phi_1}{\phi_1}\Big)\cdot\vec{\nu} &\geq& \frac{g(\theta)}{r\sin\theta}
\Big[\sin\big(\frac{\beta-\gamma}{2}-\theta\big) + \sin(\frac{\beta+\gamma}{2}-\theta_1)\Big] \\
&=& \frac{2g(\theta)}{r\sin\theta}\sin\big( \frac{\beta-\theta-\theta_1}{2}\big)\cos\big( \frac{\gamma+\theta -\theta_1}{2}\big).\end{aligned}$$ Since $0< \beta -\theta-\theta_1 <2\pi$, the last sine is positive. It is also clear that $\gamma +\theta-\theta_1 >0$. Hence the proof will be complete if we establish the following
[**Claim:**]{} There holds $$\theta_1 \geq \theta+\gamma-\pi \; \; , \qquad \mbox{ on }\Gamma_3.
\label{abcd}$$ [*Proof of Claim.*]{} Simple geometry shows that along $\Gamma_3$ the polar angles $\theta$ and $\theta_1$ are related by $$\cot\theta_1 =-\cos(\beta+\gamma) \cot(\beta-\theta) -\sin(\beta+\gamma) \; .$$ and $[\theta_0,\theta_0']\subset [\pi/2, \beta - \pi/2]\subset [\pi/2, (\beta -\gamma+ \pi)/2] $. We will actually establish (\[abcd\]) for the larger range $\pi/2\leq\theta \leq (\beta -\gamma+ \pi)/2$.
For this, we initially observe that for $\theta=(\beta-\gamma+\pi)/2$ inequality (\[abcd\]) holds as an equality. Therefore the claim will be proved if we establish that $$\frac{d\theta_1}{d\theta} -1 \leq 0 \; , \quad\quad \frac{\pi}{2}\leq \theta\leq \frac{\beta-\gamma+\pi}{2}.$$ However, we easily come up to $$\begin{aligned}
\frac{d\theta_1}{d\theta} -1 &=& -\frac{ \cos(\beta+\gamma)(\cos(\beta+\gamma) -1)\cot^2(\beta-\theta) +2\sin(\beta+\gamma)\cos(\beta+\gamma)\cot(\beta-\theta)}{1+\cot^2\theta_1} \\
&& -\frac{1 +\sin^2(\beta+\gamma)-\cos(\beta+\gamma)}{1+\cot^2\theta_1}.\end{aligned}$$ The function $$h(x):= \cos(\beta+\gamma)(\cos(\beta+\gamma) -1)x^2 +2\sin(\beta+\gamma)\cos(\beta+\gamma)x +1 +\sin^2(\beta+\gamma)-\cos(\beta+\gamma)$$ is a concave function of $x$. We will establish the positivity of $h(\cot(\beta-\theta))$, $\pi/2\leq \theta\leq (\beta-\gamma+\pi)/2$, and for this it is enough to establish positivity at the endpoints. A simple computation shows that $$h( \cot(\beta- \frac{\beta -\gamma+\pi}{2})) =2\tan^2(\frac{\beta+\gamma}{2}).$$ At the other endpoint we have $$\begin{aligned}
h(\cot(\beta-\frac{\pi}{2}))&=& \cos(\beta+\gamma)(\cos(\beta+\gamma) -1)\tan^2\beta -\\
&& -2\sin(\beta+\gamma)\cos(\beta+\gamma)\tan\beta+1+\sin^2(\beta+\gamma)-\cos(\beta+\gamma)\\
& =&\frac{2 \sin^2(\frac{\beta+\gamma}{2})}{\cos^2\beta}\big[ 1+ \cos(2\beta)\cos^2(\frac{\beta+\gamma}{2}) \big] \\
&& - \frac{\sin(\beta+\gamma)}{2\cos^2\beta}\big( \sin(\beta-\gamma) +\sin(2\beta) \cos(\beta+\gamma)\big)\\
&\geq& 0,\end{aligned}$$ since $3\pi/2 \leq \beta+\gamma\leq 2\pi$ and $0\leq \beta-\gamma\leq \pi$. Hence the claim is proved and therefore the total contribution along $\Gamma_3$ is non-negative.
It finally remains to establish that the total contribution along $\Gamma_*$ is non-negative. As in type B1 the contribution from $\Omega_+^A$ is $$\frac{\nabla\phi_1}{\phi_1}\cdot \vec{\nu}_* = \frac{f(\theta_1)}{r_1}\cos(\theta_1+\frac{\delta}{2}).$$ This is is non-negative since $\theta_1 <(\beta+\gamma-\pi)/2$ and $\beta+\gamma+\delta <2\pi$. This completes the proof. $\hfill\Box$
[RRR]{}
A. Ancona. On strong barriers and an inequality of Hardy for domains in ${{\mathbb{R}}}^n$. [*J. London Math. Soc.*]{} 34 (2) (1986), 274-290.
D. H. Armitage and U. Kuran. The convexity and the superharmonicity of the signed distance function. [*Proc. Amer. Math. Soc.*]{} 93 (4) (1985), 598-600.
F. Avkhadiev and A. Laptev. Hardy inequalities for nonconvex domains. In [*Around the research of Vladimir Maz’ya.*]{} I 1-12, Int. Math. Ser. (N.Y.), 11 [*Springer, New York,*]{} 2010.
R. Banũelos. Four unknown constants. Oberwolfach report no. 06, 2009.
G. Barbatis, S. Filippas and A. Tertikas. A unified approach to improved $L^p$ Hardy inequalities with best constants. Trans. Amer. Math. Soc. 356 (2004), 2169-2196.
H. Brezis and M. Marcus. Hardy’s inequalities revisited, Dedicated to Ennio De Giorgi. [*Ann. Scuola Norm. Sup. Pisa Cl. Sci.*]{} 25 (1997), 217-237.
E.B. Davies. The Hardy constant. [*Quart. J. Math. Oxford Ser.*]{} (2) 184 (1995) 417-431.
E.B. Davies. A review of Hardy inequalities. [*The Maz’ya anniversary collection, 55-67, Oper. Theory Adv. Appl.*]{}, 110, Birkhauser, Basel, 1999.
S. Filippas, V. Maz’ya and A. Tertikas. Critical Hardy-Sobolev inequalities. [*J. Math. Pures Appl.*]{} 87 (2007), 37-56.
G.H. Hardy. Note on a theorem of Hilbert. [*Math. Z.*]{}, 6 (1920), 314-317.
Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Laptev A. A geometrical version of Hardy’s inequality. [*J. Funct. Anal.*]{} 189 (2002), 539-548.
A. Laptev. Lecture Notes, Warwick, April 3-8, 2005 (unpublished) http://www2.imperial.ac.uk/ alaptev/Papers/ln.pdf
A. Laptev and A. Sobolev. Hardy inequalities for simply connected planar domains, [*Spectral theory of defferential operators*]{}, 133-140, Amer. Math. Soc. Transl. Ser. 2, 225, Amer. Math. Soc., Providence, RI, 2008.
A.D. Polyanin and V.F. Zaitsev. Handbook of exact solutions for ordinary differential equations, CRC Press, 1995.
J. Tidblom. Improved $L^p$ Hardy inequalities, PhD Thesis, Stockholm University, 2005.
[^1]: Department of Mathematics, University of Athens, 15784 Athens, Greece
[^2]: Department of Mathematics, University of Crete, 71409 Heraklion, Greece and Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece
|
---
abstract: 'A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g. in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behaviour of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, cross-over regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.'
author:
- 'A. Yu. Cherny$^{1,2}$ and J. Brand$^1$'
title: 'Self-consistent calculation of the coupling constant in the Gross-Pitaevskii equation'
---
\#1\#2 \#1 \#1[\_[\#1]{}]{} \#1[\_[\#1]{}\^]{} -2.5mm 7.5mm
Introduction {#sec:intro}
============
The Gross-Pitaevskii (GP) equation [@gp] is a powerful tool for describing most of the physical properties of Bose-Einstein condensates of trapped alkali atoms (see reviews [@walls; @dalfovo; @leggett; @pethick]). In the GP approach, the ground state energy of a trapped dilute Bose gas of atoms of mass $m$ is the functional $$E = \int \d{\bf r}\left[\frac{\hslash^2}{2m}\,|\nabla \phi|^2+
V_{\rm ext}({\bf r}) |\phi|^2 +\frac{g}{2} |\phi|^4 \right],
\label{gpfun}$$ of the order parameter $\phi=\phi({\bf r})=\langle \hat\Psi({\bf r})
\rangle$, where $\hat\Psi({\bf r})$ is the bosonic field operator and $V_{\rm ext}({\bf r})$ is an external trapping potential. The coupling constant $g$ in the GP functional (\[gpfun\]) is intimately related to the density expansion of the energy of the homogeneous Bose gas. Indeed, in the homogeneous case Eq. (\[gpfun\]) yields $E/N=gn/2$ where $n=N/V\simeq|\phi|^2$ is the 3D density. This expression should be equal to the known first term in the density expansion $E/N =2\pi\hslash^2a n/m$ in three dimensions [@bog47; @lee]. With $a$ being the 3D scattering length, the coupling constant $g =4\pi\hslash^2 a/m$ coincides with the zero momentum limit of the scattering amplitude, the two-body $T$-matrix, for two particles scattering in a vacuum. This standard approach based on the low-density expansion of the homogeneous gas neglects the influence of inhomogeneous trapping potentials which may require a renormalization of the coupling constant.
The situation is more complicated for two dimensional Bose gases, which can be regarded as the limiting case of a 3D gas with a highly inhomogeneous trapping potential. Kolomeisky [*et al.*]{} [@kolom] proposed that the form (\[gpfun\]) of the energy functional is still valid. However, in this case the coupling constant becomes dependent on the local density. Indeed, Schick’s result for the energy $E/N=2\pi\hslash^{2}n_{\rm 2D}/[-m\ln(n_{\rm
2D}a_{\rm 2D}^{2})]$ of a dilute 2D Bose gas [@schick] leads to the coupling constant $g=4\pi\hslash^{2}/(-m\ln |\phi|^{2}a_{\rm
2D}^{2})$ [@kolom] and further corrections were derived in Refs. [@popov; @kolom1; @ovchin; @cherny4]. Here, $n_{\rm 2D}$ and $a_{\rm 2D}$ denote the two dimensional density and scattering length, respectively. This generalization can be understood as a local density approximation, which yields consistent energy values in the homogeneous and inhomogeneous cases but does not reveal the nature of the additional non-linearity in the GP equation. Moreover, it is not clear what the nonlinearity should be in the crossover regimes from 2D to 3D and from 1D to 3D. A mathematically rigorous justification of the GP functional [@lieb] is of importance but hardly can help us in this situation.
The purpose of the present paper is to show how a density-dependent renormalized coupling constant emerges naturally in a simplified Hartree-Fock-Bogoliubov (HFB) approximation starting from the bare interaction potential $V(r)$. To this end, we derive generalized GP equations where the order parameter is coupled to the pair wave function of two bosons in the condensate. The latter has already been discussed in detail in Refs. [@cherny; @cherny2; @cherny4; @cherny1]. The generalized GP equations permit us to consider interaction potentials with a hard core directly without the $\delta$-function replacement by accurately accounting for the short-range spatial correlations of the particles. These correlations become essential in low dimensions since the Born approximation for two scattering particles fails at small momenta (see, e.g., Ref. [@landau], Sec. 45). We note that the correct treatment of the short-range correlations is also possible within the Jastrow[@cowell] and the Faddeev-Yakubovsky [@sorensen] approaches.
It is well known that the original HFB scheme leads to an artificial gap in the spectrum [@hohmartin; @Hugenholtz1959a]. Moreover, this scheme in conjunction with the $\delta$-function replacement has an ultraviolet divergence [@lee; @leggett1]. These problems then have to be cured by further approximations as classified by Griffin [@griffin]. Alternatively, complicated renormalization procedures [@Bijlsma1997a; @Hutchinson1998a] or pseudopotentials [@huang57; @olshanii:010402] have been suggested. In this paper we will discuss a novel approximation derived from the full HFB scheme where the use of the bare two-body potential provides an implicit renormalization, and the ultraviolet divergences are avoided. We will discuss the excitation spectrum and show that it is gapless in a reasonable approximation.
Low dimensional Bose systems are not only of general theoretical interest but also find the attention of current experimental exploration [@ketterle; @tolra; @paredes04; @moritz:250402]. In these experiments, the low dimensional condensate is realized by highly anisotropic 3D trapping potentials when the single-particle energy-level spacing in the tightly confined dimensions exceeds the interaction energy between atoms $\hslash\omega_{\rho,z} \gtrsim \mu$. Here the frequencies are associated with the axially symmetric harmonic potential, and $\mu$ is the part of the chemical potential coming from the particle interaction, which is of order of the mean interaction energy per particle. This criterion takes the form $l_{\rho,z}\lesssim \xi$ in terms of the coherence length $\xi=\hslash/\sqrt{\mu m}$ [@note1b] and the radial (axial) harmonic oscillator length $l_{\rho,z}=\sqrt{\hslash/(m\omega_{\rho,z})}$.
Theoretically, the 2D regime $l_z\ll\xi$, $l_\rho\gg\xi$ was investigated in detail by Petrov, Holzmann, and Shlyapnikov [@shlyap2D]. The coupling constant was assumed to be the $T$-matrix of two particles scattering in the harmonic trap with $l_\rho=\infty$ at the energy of relative motion $E=2\mu$. An additional nonlinearity is introduced, since the local value of $\mu$ depends on the density and the coupling constant in a self-consistent way. In this regime, the motion of particles is confined in $z$-direction to zero-point oscillation. This implies that the order parameter can be represented in the form $\phi(x,y,z)=\phi_0(z)\phi(x,y)$, where $\phi_0(z)$ is the ground state of the 1D harmonic oscillator and $\phi(x,y)$ is governed by the two dimensional GP equation resulting from the functional (\[gpfun\]) in two dimensions. So, in this regime, the behaviour of the condensate in $x$-$y$ plane is the same as in the “pure” 2D case with the 2D scattering length written in terms of the length $l_z$ of the tight confinement [@shlyap2D].
An improved many body $T$-matrix theory was developed by Stoof and coworkers [@stoof] in order to describe not only the homogeneous low-dimensional Bose gases but also the crossover from 3D to lower dimensions. The coupling constant in the inhomogeneous case is represented by the local value of the $T$-matrix, which depends on the local value of the chemical potential. The local $T$-matrix approximation was also used in Ref. [@burnett] and the microscopic approach of Ref. [@burnett1]. Thus, one can say, slightly simplifying the situation, that the common method of evaluating of the coupling constant in the above works is to determine first the $T$-matrix from the corresponding two-body Schrödinger equation supposing that the motion of the particles is infinite in some directions, and after that to replace the coupling constant by the local value of the $T$-matrix. In this paper we offer a method beyond the local $T$-matrix approximation. The coupling constant is determined self-consistently for a given 3D geometry from a unified variational scheme. As a result, we obtain a non-local term in the energy functional, which can be of practical importance if the external potential varies on the scale of the interaction potential. This may be realized, e.g., for condensates of loosely-bound molecules in tight or strongly oscillating potentials like optical lattices. We expect experiments to enter this regime in the near future as both atomic condensates in optical lattices [@tolra; @Greiner2002a] and molecular condensates [@jochim03; @zwierlein:250401; @greiner04] are currently under intense experimental investigation.
As a starting point of our approach we assume that we have a Bose-Einstein condensate or quasi-condensate with a well-defined order parameter. Long-range fluctuations of the phase, which become important for many physical properties in low-dimensional traps [@dettmer01], are beyond the scope of our scheme. They can be studied by means of the approaches of Refs. [@shlyap2D; @stoof; @shlyap1D; @ho]. Also the strongly-interacting fermionized regime of the Tonks-Girardeau gas [@paredes04], which was studied in Ref. [@brand04a], cannot be described with the methods of this paper. However, we note that our scheme, within its accuracy, is simple and physically transparent and able to reproduce not only the value of the coupling constant in 1D [@olshanii] and 2D [@shlyap2D] regimes but also to describe the 3D–2D and 3D–1D crossovers. Furthermore, it allows us to calculate directly the correct values of the kinetic and interaction energies of bosons in the trap, which are [*not*]{} given by the first and the third terms, respectively, in the GP functional (\[gpfun\]) [@cherny3].
The paper is organized as follows. In Sec. \[sec:gpgen\] we derive the generalized GP functional and corresponding equations from a simplified HFB approximation. In Sec. \[sec:cross\], a few specific cases are considered that admit analytical estimations, including the homogeneous and inhomogeneous Bose gases in low dimensions. In Sec. \[sec:kinpot\] we calculate the values of various contributions in the energy. In particular, the release energy of the low dimensional gases is estimated. In Sec. \[sec:vir\] we derive a useful virial theorem and a relation between the chemical potential and different parts of the energy functional. The eigenfunctions of the two-body density matrix and a relation between the normal and anomalous averages are obtained within the HFB approximation in Appendices \[sec:pwf\] and \[sec:hfrel\], respectively.
Generalized Gross-Pitaevskii equations {#sec:gpgen}
======================================
Failure of standard GP approach
-------------------------------
In the standard approach [@leggett; @dalfovo; @walls; @pethick], the equilibrium value of the order parameter $\phi$ is determined by minimization of the GP functional (\[gpfun\]) with the constraint of particle-number conservation $\delta(E-\mu' N) /\delta\phi^{*}({\bf r})=0$. Here $N\simeq N_0$ is the number of particles and the chemical potential $\mu'$ appears as a Lagrange multiplier. Introducing for later convenience $\mu=\mu'-E_0$ as the chemical potential due to interaction where $E_0$ is the ground state energy of a non-interacting particle, we arrive at $$(E_0+\mu)\phi=-\frac{\hslash^2}{2m}\nabla^{2}\phi+V_{\rm ext}({\bf r})\phi + g|\phi|^2\phi.
\label{gpeq}$$
The simplicity of this derivation is based on the simple form of the GP energy functional (\[gpfun\]), where the effects of the binary inter-particle interactions has been reduced to a single parameter given by the coupling constant. In order to determine the constant self-consistently, it should be examined carefully how the interaction term $(g/2)|\phi|^4$ appears in the GP functional (\[gpfun\]).
In a general many-body system with binary interactions, the expectation value of the interaction energy is a functional of the [*two-body*]{} density matrix $\langle{\hat\Psi}^{\dagger}(x_1){\hat\Psi}^{\dagger}(x_{2})
{\hat\Psi}(x'_{2}){\hat\Psi}(x'_1)\rangle$. For a pairwise interaction potential $V(x_1,x_2) =V({\bf r}_1- {\bf r}_2,
\sigma_1,\sigma_2)$ we thus obtain [@bogbog; @landau] $$\begin{aligned}
E_{\rm int}&=&\Bigl\langle\frac{1}{2}\sum_{i\not=j} V(x_i,x_j)\Bigr\rangle
=\frac{1}{2}\int\d x_1\d x_2\,V(x_1,x_2)\nonumber \\
&&\times\langle{\hat\Psi}^{\dagger}(x_1){\hat\Psi}^{\dagger}(x_{2})
{\hat\Psi}(x_{2}){\hat\Psi}(x_1)\rangle,
\label{eint} \end{aligned}$$ where $x=({\bf r},\sigma)$ stands for the coordinate and spin or sort indices of a particle, respectively and $\int\d x\cdots =
\sum_{\sigma}\int\d{\bf r}\cdots$. The kinetic energy and the energy of interaction with an external field are determined by the one-body matrix $\langle{\hat\Psi}^\dag(x){\hat\Psi}(x')\rangle$ $$\begin{aligned}
E_{\rm kin}\!&=&\!\Big\langle\sum_{i}\frac{p_{i}^{2}}{2m}\Big\rangle \nonumber\\
\!&=\!&-\frac{\hslash^2}{2m}\int\d x\,\left.\nabla^2_x\langle{\hat\Psi}^\dag(x'){\hat\Psi}(x)\rangle\right|_{x'=x},
\label{ekin}\\
E_{\rm ext}\!&=&\!\Big\langle\sum_{i} V_{\rm ext}(x_{i})\Big\rangle
\!=\!\int\! \d x\,V_{\rm ext}(x)\langle{\hat\Psi}^\dag(x){\hat\Psi}(x)\rangle.
\label{eext} \end{aligned}$$ The behaviour of the one- and two-body matrices is easy to understand in the dilute limit, when the condensate depletion $(N-N_0)/N$ is small. We note that the number of bosons in the condensate $N_0$ is defined as the macroscopic eigenvalue of the one-body density matrix $\langle \hat{\Psi}^{ \dag} (x) \hat{\Psi}(x')\rangle$, that is $\int
\d x'\,\langle \hat{\Psi}^{ \dag} (x') \hat{\Psi}(x)\rangle
\phi_{0}(x') = N_{0} \phi_{0}(x)$. The field operator can be expanded in the complete orthonormal set of eigenfunctions of the one-body matrix $\hat{\Psi}(x)=\hat{a}_0\phi_0(x)+\sumpr_\nu\hat{a}_\nu\phi_{\nu}(x)$, where the sum $\sumpr_\nu$ means $\sum_{\nu\not=0}$ and $\int \d
x\,|\phi_{\nu}(x)|^{2}=1$. Appearance of the Bose-Einstein condensate implies the macroscopic occupation of $N_0$, i.e. the ratio $N_0/N$ remains finite in the thermodynamic limit. Following Bogoliubov [@bog47; @bogquasi] we now replace the condensate operators by $c$-numbers $\hat{a}^\dag_{0}=\hat{a}_{0}\simeq \sqrt{N_0}$ and represent the Bose field operator in the form $\hat{\Psi}(x)=
\phi(x) +\hat{\vartheta}(x)$ [@castin98n]. Here $\phi(x)=
\sqrt{N_0}\phi_0(x)$ is the $c$-number part, and $\hat{\vartheta}(x)
=\sumpr_\nu\, \hat{a}_\nu \phi_\nu(x)$ the operator part, for which we have $\langle\hat{\Psi}(x)\rangle =\phi(x)$ and $\langle\hat{\vartheta}(x)\rangle=0$. Thus, the order parameter is nothing else but the non-normalized eigenfunction of the one-body density matrix.
In the original approach of Gross and Pitaevskii, the simplest mean-field approximation is used when the operator part is completely neglected: $\hat{\Psi}(x) \simeq\phi(x)$ and $\hat{\Psi}^\dag(x)
\simeq\phi^*(x)$. Assuming additionally that the order parameter does not change significantly at the distances of order of the radius $R_{\rm e}$ of the interaction potential, we obtain the GP energy functional (\[gpfun\]) for spinless bosons from Eqs. (\[eint\]–\[eext\]) with the coupling constant $$\label{eqn:couplingc}
g\simeq g_{\rm B}\equiv\int\d{\bf r}\,V(r).$$ This coupling constant can be identified with the two-body scattering amplitude at zero momentum in the Born approximation. The validity of the GP approach with the coupling constant $g_{\rm B}$ of Eq. (\[eqn:couplingc\]) is certainly linked to the validity condition of the Born approximation at zero momentum that the potential $V(r)$ be small and integrable.
Of the two assumptions mentioned above, namely the slow spatial variation of the order parameter and the validity of the Born approximation, the former is usually fulfilled as the healing length $\xi=\hslash/\sqrt{\mu m}$ as a lower bound of the length scale of the order parameter [@dalfovo; @leggett] is usually much larger than the effective range of the interaction. The latter assumption, however, is clearly not fulfilled for experiments with cold atomic gases as their interactions are of the hard-core type.
The validity of the GP approach can be extended to such systems by an argument attributed to Landau [@bog47; @cherny1]. He noted that at extremely low energies, as predominant in the dilute-gas BEC, the scattering properties are completely determined by only one single parameter, which is the 3D $s$-wave scattering length $a$. This allows us to replace the Born approximation for the scattering amplitude $g_{\rm
B}$ by its exact value $g=4\pi\hslash^2a/m$, which can be found from the two-body Schrödinger equation even for hard-core potentials. This indirect argument, however, cannot be used in one or two dimensions as there is no such simple relation between the integral in Eq. (\[eqn:couplingc\]) and the scattering amplitude as we have in three dimensions. Furthermore the Born series for the scattering amplitude diverges for small momenta in two dimensions and below (see, e.g., Ref. [@landau], Sec. 45).
Pair wavefunction in a medium
-----------------------------
The above described deficiencies of the naive GP approach may be remedied by accounting for the two-particle scattering processes explicitely. Within the HFB scheme this is possible through certain correlations introduced by the fluctuation operators $\hat{\vartheta}$. In order to see the relation between the two-particle scattering process and the correlation functions mentioned above it is useful to introduce the concept of a two-body or [*pair wave function in the medium*]{} of other particles [@bogquasi; @cherny]. The pair wave functions in the medium of the many-body system are defined as eigenfunctions of the two-body density matrix, as discussed in detail in Appendix \[sec:pwf\]. They should be distinguished from the two-body wave functions in the vacuum, which relate to a system of two particles. For the latter we will use the superscript $^{(0)}$.
Let us suppose that we know the exact eigenfunctions of the two-body density matrix. Then we can expect for the dilute gas, where low-momentum two-body processes dominate the behaviour of the system, that the pair wave functions in the medium should be very close [@cherny] to the two-body wave functions in the vacuum, which are the solutions of the two-body Schrödinger equation. This physical assumption was used to obtain the density expansions for the 3D [@cherny1; @cherny2] and 2D [@cherny4] homogeneous Bose gases in a very simple manner. However, various approximations in the many-body theory can break this relation.
A simplified HFB scheme
-----------------------
Within the HFB approximation for the homogeneous Bose gas, all eigenfunctions of the two-body density matrix except for one are plane waves and are thus treated in the Born approximation as shown in Appendix \[sec:pwf\]. This is an obvious drawback of the HFB scheme. It turns out that the two-body wave function that is not a plane wave is proportional to the anomalous average $$\varphi(x_1,x_2)\equiv\langle{\hat\Psi}(x_1) {\hat\Psi}(x_{2})\rangle$$ and corresponds to the macroscopic eigenvalue $N_0(N_0-1)$ in the limit of large $N$. It is the pair wave function that describes the two-particle scattering process in the medium of the Bose-Einstein condensate [@cherny; @note1]. Thus we can go beyond the Born approximation in the framework of the HFB method by keeping only the contribution of the anomalous average $\langle{\hat\Psi}(x_1)
{\hat\Psi}(x_{2})\rangle$ and neglecting the contribution of the other wave functions in the two-body density matrix. Due to small condensate depletion $(N-N_0)/N\ll 1$, one can expect that the contribution of only this eigenfunction will be sufficient for obtaining the coupling constant in the GP equation. In this simplified version of the HFB approximation we set $$\langle{\hat\Psi}^{\dagger}(x_1) {\hat\Psi}^{\dagger}(x_{2}) {\hat\Psi}(x'_{2}) {\hat\Psi}(x'_1)\rangle\simeq
\varphi^*(x_1,x_2)\varphi(x'_1,x'_2).
\label{2matr}$$ Extracting the $c$-number part of the field operator, the anomalous averages can be rewritten as $$\varphi(x_1,x_2)=
\phi(x_1)\phi(x_2)+\psi(x_1,x_2),
\label{phidef}$$ where we introduced the notation $\psi(x_1,x_2) \equiv
\langle\hat{\vartheta}(x_1) \hat{\vartheta}(x_2)\rangle$ for the anomalous two-boson correlation function associated with the scattering part of the two-body wave function. The functions $\varphi(x_1,x_2)$ and $\psi(x_1,x_2)$ are symmetric with respect to permutation of $x_1$ and $x_2$ due to the commutation relations for the Bose field operators.
For the one-body density matrix we find $\langle{\hat\Psi}^\dag(x){\hat\Psi}(x')\rangle= \phi^*(x)\phi(x')
+\langle\hat{\vartheta}^\dag(x) \hat{\vartheta}(x')\rangle$. We note that the normal $\langle\hat{\vartheta}^\dag(x)
\hat{\vartheta}(x')\rangle$ and anomalous $\langle\hat{\vartheta}(x)
\hat{\vartheta}(x')\rangle$ averages are not independent quantities as discussed in Appendix \[sec:hfrel\] and Refs. [@cherny1; @cherny2]. Within the Hartree-Fock-Bogoliubov scheme, the relations between them appear as a specific property of the HFB ground state (the quasiparticle vacuum) and do not contain parameters of the Hamiltonian in explicit form. We will use the approximate relation (\[thetaapp\]), which leads to $$\langle{\hat\Psi}^\dag(x){\hat\Psi}(x')\rangle= \phi^*(x)\phi(x')
+\int \d x_2\,\psi^*(x,x_2)\psi(x_2,x').
\label{onebody}$$ With the help of Eqs. (\[2matr\]), (\[phidef\]), and (\[onebody\]) we rewrite Eqs. (\[eint\]), (\[ekin\]), and (\[eext\]) in terms of the anomalous averages $$\begin{aligned}
E_{\rm int}&=&\frac{1}{2}\int \d x_1\d x_2\,V(x_1,x_2)|\varphi(x_1,x_2)|^2,
\label{eint1}\\
E_{\rm ext}&=&\frac{1}{2}\int \d x_1\d x_2\,[V_{\rm ext}(x_1)+V_{\rm ext}(x_2)]|\psi(x_1,x_2)|^2\nonumber\\
&&+\int \d x_1\,V_{\rm ext}(x_1)|\phi(x_1)|^2,
\label{eext1}\\
E_{\rm kin}&=&\frac{1}{2}\int \d x_1\d x_2\,\psi^*(x_1,x_2)(\hat{T}_1+\hat{T}_2)\psi(x_1,x_2)\nonumber\\
&&+\int \d x_1\,\phi^*(x_1)\hat{T}_1\phi(x_1),
\label{ekin1}\end{aligned}$$ where $\hat{T}_j=-\hslash^2\nabla^2_{j}/(2m)$ and $j=1,2$. The total number of particles is related directly to the one-body matrix: $N=\int\d x\,\langle{\hat\Psi}^\dag(x){\hat\Psi}(x)\rangle$. With Eq. (\[onebody\]) we find $$\label{N}
N=\int \d x_1\,|\phi(x_1)|^2 + \int \d x_1\d x_2\,|\psi(x_1,x_2)|^2.$$ The current approximations are useful for a variational scheme where the functions $\phi(x_1)$ and $\psi(x_1,x_2)$ are determined by minimization of the total energy with the constraint $N={\rm
const}$. Using the Lagrange method, we obtain the conditions $\delta
E/\delta\phi(x_1)= \delta E/\delta\phi^{*}(x_1) =\delta
E/\delta\psi(x_1,x_2) =\delta E /\delta\psi^*(x_1,x_2) =0$ for the energy functional $$E[\{\phi,\psi\},\mu']=E_{\rm kin}+E_{\rm ext}+E_{\rm int}-\mu'(N-{\cal N}),
\label{efunc}$$ given by Eqs. (\[eint1\])-(\[N\]). Here $\mu'=\mu +E_0$ is the chemical potential and ${\cal N}=\langle \hat{N}
\rangle$ is the average number of particles, i.e. the l.h.s. of Eq. (\[N\]) at the equilibrium values of the functions $\phi$ and $\psi$ corresponding to the minimum (ground state) of the functional (\[efunc\]). Note that the variation $\delta\psi(x_1,x_2)$ is symmetric under the permutation of $x_1$ and $x_2$, so, the equation $\int \d x_1\d
x_2\,g(x_1,x_2)\delta\psi(x_1,x_2)=0$ leads to $g(x_1,x_2)+g(x_2,x_1)=0$ for arbitrary functions $g(x_1,x_2)$.
This variational procedure yields the following system of equations for the one- and two-body functions $\phi(x_1)$ and $\varphi(x_1,x_2)$, respectively, $$\begin{aligned}
\mathcal{L}_1\phi(x_1)+ \int \d x_2\, \phi^{*}(x_2)V(x_1,x_2)\varphi(x_1,x_2)&=&0,
\label{phieq} \\
(\mathcal{L}_1+\mathcal{L}_2)\psi(x_1,x_2) + V(x_1,x_2)\varphi(x_1,x_2)&=&0,
\label{psieq} \end{aligned}$$ where the operators $\mathcal{L}_1$ and $\mathcal{L}_2$ stand for $$\mathcal{L}_j=-\hslash^2\nabla_{j}^2/(2m)-\mu -E_0 + V_{\rm ext}(x_j),\quad j=1,2,$$ and $\phi$, $\varphi$ and $\psi$ are simply related by Eq. (\[phidef\]). Due to this relation, Eq. (\[phieq\]) is nonlinear with respect to $\phi$ and can be associated with the GP equation. Equation (\[psieq\]) is the analogue of the two-particle Schrödinger equation and is linear with respect to $\varphi$, though not uniform. The obtained system of two equations allows us to determine the coupling constant self-consistently. A specific feature of Eq. (\[phieq\]) is the [*non-local*]{} nature of the last term, which can play a role when the radius of the interacting potential becomes of the order of the characteristic length of the anisotropic trapping potential in some direction, say, $R_{\rm e} \sim l_z\ll \xi$, or if the trapping potential has a strongly oscillating contribution with the scale of the order of $R_{\rm e}$. At the same time, Eqs. (\[phieq\]) and (\[psieq\]) indeed reduce to the GP equation with the 3D coupling constant $g =4\pi\hslash^2 a/m$ in the limit $R_{\rm e}\ll \xi \ll l_x,\ l_y,\ l_z$ as will be shown in Sec. \[sec:gpregime\].
When the external potential becomes independent of some coordinate, say $z$, particles can move freely in $z$-direction and we should impose the boundary conditions that follow from Bogoliubov’s principle of correlation weakening [@bogquasi]: $$\langle{\hat\Psi}(x){\hat\Psi}(x')\rangle \simeq\langle{\hat\Psi}(x)
\rangle\langle{\hat\Psi}(x')\rangle \quad\text{when}\quad
|z-z'|\to\infty .$$ Physically, this implies that the function $\psi(x_1,x_2) =\langle\hat{\vartheta}(x_1)
\hat{\vartheta}(x_2)\rangle$ vanishes at the spatial distances of order of the coherence (healing) length, $|{\bf r}_1-{\bf r}_2|\gtrsim
\xi$.
A time-dependent generalization of Eqs. (\[phieq\]) and (\[psieq\]) can in principle be derived from the equations of motion of the field operators. Here, however, we will not elaborate the full derivation but instead present a simple argument that leads to a useful time-dependent scheme. In the case of a time-independent Hamiltonian, it can be easily seen that the GP order parameter depends on time as $$\begin{aligned}
\phi(x,t)\!&=&\!\langle N-1|{\hat\Psi}(x,t)|N\rangle
\nonumber\\
\!&=&\!\phi(x)\exp[{-i(E_{N}-E_{N-1})t}/{\hslash}] \nonumber\\
\!&=&\!\phi(x)\exp[{-i\mu't}/{\hslash}] . \nonumber\end{aligned}$$ Here, $|N\rangle$ and $E_{N}$ are the ground state and energy of $N$ bosons, respectively. By analogy, we find $\psi(x_1,x_2,t)=\psi(x_1,x_2)\exp[{-i 2\mu' t/\hslash}]$ and $\varphi(x_1,x_2,t)=\varphi(x_1,x_2)\exp[{-i 2\mu't/\hslash}]$. We now argue that the chemical potential in Eqs. (\[phieq\]) and (\[psieq\]) arises due to time derivatives, which leads to the obvious generalization $$\begin{aligned}
i\hslash\frac{\partial}{\partial t}\phi(x_1,t)&=&\hat{H}_{1}\phi(x_1,t)+ E_{\rm nl}(x_1,t),
\label{phieqt} \\
i\hslash\frac{\partial}{\partial t}\varphi(x_1,x_2,t)&=&[\hat{H}_{1}+\hat{H}_{2}+V(x_1,x_2)]\varphi(x_1,x_2,t)
\nonumber\\&&
+ \phi(x_1,t)E_{\rm nl}(x_2,t)
\nonumber\\&&
+ \phi(x_2,t)E_{\rm nl}(x_1,t),
\label{psieqt} \end{aligned}$$ where we denote $$\begin{aligned}
\hat{H}_{j}&=&-\frac{\hslash^2\nabla_{j}^2}{2m} + V_{\rm ext}(x_j,t),\quad j=1,2,
\nonumber \\
E_{\rm nl}(x,t)&=&\int \d y\, \phi^{*}(y,t)V(x,y)\varphi(x,y,t).
\nonumber\end{aligned}$$ The functions $\phi=\langle{\hat\Psi}(x,t)\rangle$ and $\varphi=\langle{\hat\Psi}(x_{1},t){\hat\Psi}(x_{2},t)\rangle$ are normalized as $\int\d x\,|\phi(x,t)|^{2} =N_{0}$ and $\int\d
x_{1}\d x_{2}\, |\varphi(x_{1},x_{2},t)|^{2} =N_{0}(N_{0}-1)\simeq
N^{2}_{0}$, respectively. The time-dependent generalized GP equations (\[phieqt\]-\[psieqt\]) become the ordinary one- and two-body Schrödinger equations, respectively, in the limit $\xi \gg
l_{x},l_{y},l_{z}$ when we can neglect all the nonlinear terms, which are responsible for many-body effects. Therefore they are of slightly more general validity than the stationary equations (\[phieq\]-\[psieq\]), which imply a large particle number, since $E_{N}-E_{N-2}\simeq 2(E_{N}-E_{N-1})\simeq 2\mu'$ is valid only for $N\gg 1$. We notice that the time-dependent equations similar to that of (\[phieqt\]) and (\[psieqt\]) were derived in papwer [@kohler] by the method of noncommutative cumulants.
Properties and limits of validity
---------------------------------
The time-dependent Equations (\[phieqt\]-\[psieqt\]) give access to the elementary excitation spectrum. At the moment we cannot prove the gaplessness of the spectrum in the most general case, but we can solve for the excitation energies approximately. With the ansatz $\varphi({\rm r}_{1},{\rm r}_{2},t)=\phi({\rm
r}_{1},t)\phi({\rm r}_{2},t)[1+\psi(r)/n_0]$, we obtain the Bogoliubov excitation energy $\omega_{k}=\sqrt{T_k^2+2n_{0}U(k)T_k}$ with the $k$-dependent scattering amplitude $U(k)$ (for the notations see Sec. \[sec:hom\]). This form of the spectrum for a singular two-particle interaction was proposed without derivation by Bogoliubov in Ref. [@bog47]. For small $k$ we can replace $U(k=0) =
4\pi\hbar^2 a/m$ and obtain the usual (gapless) Bogoliubov dispersion. The additional features in the obtained spectrum at medium and high energies reflect the structure of the interaction potential neglected in the standard GP approach and present a clear advantage of our extended scheme. As a consequence, we can expect that Levinson’s theorem for quasi-particle scattering [@brand1] will be modified.
Let us discuss limits of validity of the generalized GP equations (\[phieq\]) and (\[psieq\]). First, we imply that the Bose-Einstein condensate (or quasi-condensate in low dimensions, see Sec. \[sec:nonhom\]) is developed strongly. This means that $r_0\ll\xi$, where $r_0$ is an average distance between bosons [@dalfovo; @leggett]. Second, the above derivation can be applied only to the short-range interaction potentials that decrease at least as fast as $V(r)\sim 1/r^{\varepsilon+D}$ for $r\to\infty$, where $D$ is dimension and $\varepsilon>0$ [@cherny1; @cherny2]. For a long-range interaction like Coulomb repulsion, the approximation (\[2matr\]) works badly. Third, the approximations (\[2matr\]) and (\[thetaapp\]) are insufficient to describe the long-range behaviour of the normal $\langle{\hat\vartheta}^{\dag}(x_1)
{\hat\vartheta}(x_{2})\rangle$ and anomalous $\langle{\hat\vartheta}(x_1){\hat\vartheta}(x_{2})\rangle$ correlation functions, which are governed by Bogoliubov’s “$1/q^{2}$” theorem [@bogquasi; @mullin; @fischer]. According to this theorem, the above correlation functions should decay as $1/|{\bf r}_1-{\bf
r}_2|^{2}$ when $|{\bf r}_1-{\bf r}_2|\gtrsim \xi$ at zero temperature if the Bose-Einstein condensate exists. Our scheme gives $1/|{\bf
r}_1-{\bf r}_2|$ decay, as we show in Sec. \[sec:hom\]. However, we stress that [*the long-range behaviour is not needed for obtaining the coupling constant*]{}, since the integral in Eq. (\[phieq\]) contains the anomalous correlation function multiplied by the short-range potential $V(x_1,x_2)$ with the characteristic radius $R_{\rm
e}\lesssim \xi$. Since the developed scheme describes well only the short-range behaviour of $\psi(x_1,x_2)$ for $|{\bf r}_1-{\bf
r}_2|\lesssim \xi$, the integration in the last term of Eq. (\[N\]) should be restricted to this region $$N=\int \d x_1\,|\phi(x_1)|^2 + \int_{|{\bf r}_1-{\bf
r}_2|\leqslant\xi} \d x_1\d x_2\,|\psi(x_1,x_2)|^2,
\label{13a}$$ otherwise we obtain formally divergent term. This modification of the original scheme, however, does not change the working equations (\[phieq\] - \[psieqt\]) in the region $|{\bf r}_1-{\bf r}_2|\leqslant \xi$, which is of sole interest for our purposes. We stress that Eq. (\[13a\]) is really needed only when minimizing the energy functional (\[efunc\]) directly. Furthermore, if we obtain the solutions of Eqs. (\[phieq\]) and (\[psieq\]) as functions of the chemical potential in the grand canonical ensemble, then the condition (\[N\]) or (\[13a\]) can be employed without the second term at all in order to rewrite the answer in terms of the total number of particles in the canonical ensemble, owing to small condensate depletion.
Note that the standard HFB approximation can be obtained by using the variational scheme if, first, one substitutes Eqs. (\[eint\])-(\[eext\]) into the energy functional (\[efunc\]), second, employ the restrictions Eqs. (\[fphicord1\]) and (\[fphicord\]), and third, retain all additional terms missing in Eq. (\[2matr\]), where the three- and four-boson averages of $\hat{\vartheta}$ and $\hat{\vartheta}^{\dag}$ ought to be evaluated by means of the Wick’s theorem and, consequently, the three-boson averages vanish.
Examples {#sec:cross}
========
In this section we restrict ourselves to spinless bosons with an isotropic short-range interaction $V=V(r)$, where $r=|{\bf r}_1 -{\bf
r}_2|$. Even after this simplification, the solution of the generalized GP equations (\[phieq\]) and (\[psieq\]) remains a rather complex problem. Nevertheless, in a number of specific limiting cases we are able to obtain analytic results.
The homogeneous case {#sec:hom}
--------------------
Let us investigate Eqs. (\[phieq\]) and (\[psieq\]) in three and two dimensions for the homogeneous Bose gas. In the homogeneous case $V_{\rm ext}=0$, hence we have $\psi=\psi(r)$, $E_0=0$, and Eq. (\[phieq\]) gives the trivial solution $\phi=\sqrt{n_0}={\rm const}$. In this subsection, we use the common notation $n_{0}$ for the condensate density in both 2D and 3D cases. Thus, Eqs. (\[phieq\]) and (\[psieq\]) read $$\begin{aligned}
\mu&=&\int\d{\bf r}\,V(r)[n_0+\psi(r)],
\nonumber\\
2\mu\psi(r)&=&-\frac{\hslash^2}{m}\nabla^2\psi(r) + V(r)[n_0+\psi(r)],
\nonumber\end{aligned}$$ and $\psi(r)\to0$ for $r\to\infty$ in accordance with Bogoliubov’s principle of correlation weakening. Taking the Fourier transformation of the last equation, we obtain $$\begin{aligned}
\mu&=&n_0U(0),
\label{muhom} \\
\frac{\psi(k)}{n_0}&=&-{\rm P.P.}\frac{U(k)}{2(T_k-\mu)},
\label{psik} \end{aligned}$$ where we denote $U(k)= \int\d{\bf r}\, V(r)e^{-i{\bf k} \cdot{\bf
r}}[1+\psi(r)/n_0]$, $T_k= \hslash^2k^2/(2m)$, and the symbol P.P. stands for the principle value of the associated integral. The latter appears as a natural regularization for the singular denominator in the r.h.s of Eq. (\[psik\]) and implies that the scattering part of the two-body wave function $\psi(k)$ is real and corresponds to a standing wave. Note that another regularization, such as the standard replacement $k\to k\pm i\varepsilon$, leads to the same results in the leading order at small densities. Within the more accurate method [@cherny1; @cherny2], we obtain the same equation as (\[psik\]) but with the Bogoliubov denominator $2\sqrt{T_k^2+2n_0U(k)
T_k}$. The latter provides the correct values of [*both*]{} the short- and long-range behaviour of the correlator $\psi(r)=\langle\hat{\vartheta}({\bf r}) \hat{\vartheta}(0)\rangle$ \[which is the Fourier transform of $\psi(k)$\], while Eq. (\[psik\]) provides only the short-range behaviour. Indeed, in the 3D case we have $\psi(r)\sim \cos(\sqrt{2}r/\xi)/r$ at $r\gtrsim\xi$ (see below) but not $\psi(r)\sim 1/r^{2}$ as it should be.
Equation (\[psik\]) can be rewritten in the Lippmann-Schwinger form with the help of the Fourier transformation. By using the familiar property of Fourier transformation $\int \d{\bf k}\, e^{i{\bf k} \cdot{\bf r}} g({\bf k})f({\bf k})
/(2\pi)^D= \int\d{\bf r}'f({\bf r}')g({\bf r}-{\bf r}')$ (here $D$ is the dimension), we obtain the equation for $\varphi(r)=n_0+\psi(r)$ $$\label{phir}
\varphi(r)=n_0+\int\d{\bf r}'\,V(r')\varphi(r')G(|{\bf r}-{\bf r}'|),$$ where the Green function is introduced $$\label{green}
G(r)=-{\rm P.P.}\int\frac{\d{\bf k}}{(2\pi)^D}\frac{e^{i{\bf k}\cdot{\bf r}}}{2(T_k-\mu)}.$$ In the dilute limit, when the average distance between particles is much less than the coherence length, the wave function $\varphi(r)/n_0$, describing the behaviour of two particles in the condensate, should be proportional [@cherny; @cherny5] to the $s$-wave function $\varphi^{(0)}(r)$, which corresponds to relative motion of two particles with zero momentum and obeys the two-body Schrödinger equation in the center-of-mass system $$-(\hslash^2/m)\nabla^2\varphi^{(0)}(r)+V(r)\varphi^{(0)}(r)=0.
\label{twobody}$$
In the 3D case, the coefficient of proportionality is equal to unity [@bog47] in the leading order with respect to the density, provided the following boundary conditions are imposed: first $|\varphi_{\rm 3D}^{(0)}(r)|< \infty$ at $r=0$ and second, $\varphi_{\rm 3D}^{(0)}(r)\to 1-a/r$ for $r\to\infty$. In the developed formalism, this can be easily inferred from the obtained equation (\[phir\]). Indeed, direct integration in Eq. (\[green\]) gives $G_{\rm
3D}(r)=-m\cos(\sqrt{2}r/\xi) /(4\pi\hslash^2r)$, and, hence, $G_{\rm 3D}(r)\simeq -m/(4\pi\hslash^2r)$ when $r\lesssim
\xi$. Thus we have $\varphi(r)\simeq n_0\varphi_{\rm 3D}^{(0)}(r)$ within this region, and integration of Eq. (\[twobody\]) yields $U(0)=4\pi\hslash^2a/m$. For the dilute gas we have also $n_0\simeq n$, and Eq. (\[muhom\]) leads to the familiar expression for the chemical potential $\mu\simeq 4\pi\hslash^2na/m$.
In the 2D case, the low-energy behaviour of the 2D Green’s function (\[green\]) is easily calculated: $G_{\rm 2D}(r)
\simeq m/(2\pi \hslash^2) \ln[e^\gamma r/(\sqrt{2}\xi)]$ when $r\lesssim \xi$. Then it is not difficult to see from Eq. (\[phir\]) that, first, $\varphi(r)/n_0$ obeys the 2D Schrödinger equation (\[twobody\]), and, second, its asymptotics for $r\to\infty$ is $$\varphi(r)/n_0\to 1+\ln[e^\gamma r/(\sqrt{2}\xi)]mU(0)/(2\pi\hslash^{2}).
\label{bound1}$$ Hence, due to linearity of Eq. (\[twobody\]), the solution for $\varphi(r)$ should be proportional to the wavefunction $\varphi^{(0)}_{\rm 2D}(r)$ that obeys the 2D Schrödinger equation (\[twobody\]) with the following boundary conditions: first $|\varphi_{\rm 2D}^{(0)}(r)|< \infty$ at $r=0$ second, $\varphi^{(0)}_{\rm 2D}(r)\to \ln(r/a_{\rm 2D})$ for $r\to\infty$. The latter equation can be considered as the definition of the 2D scattering length [@lieb1]. Note that in the case of hard disks, $a_{\rm 2D}$ coincides with the radius of the disks. It is convenient to introduce the dimensionless parameter $u$ by the relation $U(0)=4\pi\hslash^{2}u/m$, such that $u$ is the dimensionless scattering amplitude for two bosons in a medium of other bosons. By comparing the asymptotics (\[bound1\]) with that of $\varphi^{(0)}_{\rm
2D}(r)$, we derive $$\begin{aligned}
\varphi(r)&=&2un_0\varphi^{(0)}_{\rm 2D}(r),
\nonumber\\
-\ln (a_{\rm 2D}/\xi)&=&1/(2u)+\ln(e^\gamma/\sqrt{2}).
\label{equ} \end{aligned}$$ With the help of Eq. (\[muhom\]) and the definition of $\xi$ (see above), the relation (\[equ\]) becomes a self-consistent equation for $u$ $$1/u+\ln u=-\ln(n_{\rm 2D}a_{\rm 2D}^{2}2\pi)-2\gamma,
\label{udelta}$$ where we neglect the condensate depletion in the leading order, putting $n_0\simeq n_{\rm 2D}$. By means of the latter approximation, the expression (\[muhom\]) takes the form $$\mu={4\pi\hslash^{2}n_{\rm 2D}u}/{m}.
\label{muhom2d}$$ Thus, the 2D chemical potential is given by Eqs. (\[udelta\]) and (\[muhom2d\]), which lead to the density expansion $$\begin{aligned}
\mu&=&\frac{4\pi\hslash^{2}n_{\rm 2D}}{m}\Bigg(-\frac{1}{\ln(n_{\rm 2D}a_{\rm 2D}^{2})}
+\frac{1}{\ln^2(n_{\rm 2D}a_{\rm 2D}^{2})}
\nonumber\\
&&\times\ln\bigg[-\frac{1}{\ln(n_{\rm 2D}a_{\rm 2D}^{2})}\bigg] +\ldots
\Bigg).
\label{mu2dexp} \end{aligned}$$ Equations (\[udelta\]) and (\[muhom2d\]) are in agreement with the results of Refs. [@popov; @kolom1; @ovchin] and with the more accurate scheme of Ref. [@cherny4], which yields the correction for the chemical potential $$\mu=(4\pi\hslash^{2}n_{\rm 2D}/m)(u+u^{2}+\cdots).
\label{muhom2dcorr}$$ Here, $u$ is given by the more exact relation $$1/u+\ln u = -\ln(n_{\rm 2D} a_{\rm 2D}^{2}\pi)-2\gamma,
\label{uphi2}$$ where $\gamma=0.5772\ldots$ is the Euler constant. By means of this relation, one can rewrite Eq. (\[muhom2dcorr\]) in terms of the gas parameter $n_{\rm 2D}a_{\rm 2D}^{2}$ and obtain three more terms in the expansion (\[mu2dexp\]). Note that Eq. (\[uphi2\]) differs from Eq. (\[udelta\]) by a numerical factor under the logarithm, which is essential only for obtaining these additional terms but not the terms given by relation (\[mu2dexp\]).
The inhomogeneous case {#sec:nonhom}
----------------------
### The Gross-Pitaevskii regime {#sec:gpregime}
First of all, we should verify that the equations obtained in Sec. \[sec:gpgen\] lead to the standard GP scheme in the case $R_{\rm e}\ll \xi \ll l$, where $l$ is the characteristic length of an isotropic trap. In this regime, one can expect that the pair wave function $\varphi({\bf r}_1,{\bf r}_2)$ is very close to that obtained in the homogeneous case, with the difference that the density is spatially dependent now. So, we put by definition $\varphi({\bf
r}_1,{\bf r}_2) =\phi({\bf r}_1)\phi({\bf r}_2) \widetilde{\varphi}
({\bf r}_1,{\bf r}_2)$ and $\psi({\bf r}_1,{\bf r}_2) =\phi({\bf
r}_1)\phi({\bf r}_2)\widetilde{\psi}({\bf r}_1,{\bf r}_2)$, and, hence, $\widetilde{\varphi}({\bf r}_1,{\bf r}_2)
=1+\widetilde{\psi}({\bf r}_1,{\bf r}_2)$ by Eq. (\[phidef\]). Substituting those expressions into Eqs. (\[phieq\]) and (\[psieq\]) yields $$\begin{aligned}
&&\bigg[-\frac{\hslash^2}{2m}\nabla_1^2-\mu -E_0 + V_{\rm ext}({\bf r}_1)\bigg]\phi({\bf r}_1)
+\phi({\bf r}_1)|\phi({\bf r}_1)|^{2}
\nonumber\\
&&\quad\quad\quad\times\int \d {\bf r}_2\, V(|{\bf r}_1-{\bf r}_2|)\widetilde{\varphi}({\bf r}_1,{\bf r}_2)=0,
\label{phieqsub}\\
&&-\frac{\hslash^2}{2m}(\nabla_1^2+\nabla_2^2)\widetilde{\psi}({\bf r}_1,{\bf r}_2)
+ V({\bf r}_1-{\bf r}_2)\widetilde{\varphi}({\bf r}_1,{\bf r}_2)
\nonumber \\
&&\quad\quad\quad=[f({\bf r}_1)+f({\bf r}_2)]\widetilde{\psi}({\bf r}_1,{\bf r}_2),
\label{psieqsub}\end{aligned}$$ where we use the condition $R_{\rm e}\ll\xi$ in the first equation and introduce the notation $$\begin{aligned}
\label{eqn:f}
f({\bf r})=\int\d {\bf r}'\,|\phi({\bf r}')|^{2}V({\bf r}-{\bf r}')\widetilde{\varphi}({\bf r},{\bf r}')
+\frac{\hslash^2}{m}\frac{\nabla_{\bf r}\phi({\bf r})}{\phi({\bf r})}\cdot\nabla_{{\bf r}},\end{aligned}$$ with the last term being a differential operator. Since $\varphi({\bf r}_1,{\bf r}_2)\simeq\phi({\bf r}_1)\phi({\bf
r}_2)$ at the distances of order of the correlation length, we have $\widetilde{\varphi}({\bf r}_1,{\bf r}_2)\simeq 1$ at these distances. Consequently, the l.h.s. of Eq. (\[psieqsub\]) remains finite when the density tends to zero, while the r.h.s. becomes small. Indeed, the first term of Eq. (\[eqn:f\]) is of order of $\hslash^2an/m$. The second term is less than $\hslash^2/(m\xi^{2})$ because the characteristic scale of the order parameter cannot be smaller than $\xi$ in the case $\xi\ll l$ and the same applies to $\widetilde{\psi}$. Hence, in the leading order we can completely neglect the r.h.s. of Eq. (\[psieqsub\]), which leads to the standard Schrödinger equation (\[twobody\]) for $\widetilde{\varphi}$. Thus, we come to the approximation $$\varphi({\bf r}_1,{\bf r}_2) \simeq \phi({\bf r}_1) \phi({\bf r}_2) \varphi_{\rm 3D}^{(0)}(r).
\label{gpphi}$$ Using the well-known relation for the 3D scattering length $$4\pi\hslash^2a/m=\int\d^{3}r\,V(r)\varphi_{\rm 3D}^{(0)}(r),
\label{iden}$$ we can rewrite Eq. (\[phieqsub\]) in the standard GP form with the coupling constant $g=4\pi\hslash^2a/m$. Note that, nevertheless, the equilibrium value of the energy (\[efunc\]) differs from that of the GP value (\[gpfun\]) by the terms arising from the condensate depletion because the second term in the r.h.s. of Eq. (\[onebody\]) is not equal to zero. We will discuss these corrections to the energy in Secs. \[sec:kinpot\] and \[sec:vir\].
### 2D regime {#sec:2dregime}
Here we consider the Bose gas confined only in $z$-direction by the trapping potential $V_{\rm ext}=V_{\rm ext}(z)$. The system is homogeneous in the $x$-$y$ plane and assumed to be infinitely large. Physically this means that the $x$-$y$ size of the system is much larger than the characteristic radius of the trapping potential $l_{z} \equiv
\sqrt{\hslash/(m\omega_z)}$. The order parameter $\phi$ now becomes independent of $x$ and $y$, and the two-body function depends on the relative distance $\rho=|\bm{\rho}_1-\bm{\rho}_2|$ between points $\bm{\rho}_1=(x_1,y_1)$ and $\bm{\rho}_2=(x_2,y_2)$, so $\varphi({\bf
r}_1,{\bf r}_2)=\varphi(z_1,z_2,\rho)$. The 2D regime is provided by the condition $l_z \ll\xi$. Moreover, the condition $R_{\rm e}\ll\xi$ is fulfilled in most experiments. As was discussed in Sec. \[sec:intro\] the density profile is then governed by the ground state solution $\phi_0(z)$ of the one-particle Schrödinger equation $$\big[-\hslash^2\nabla^2/(2m)-E_0 + V_{\rm ext}(z)\big]\phi_0(z)=0,$$ because the second term in Eq. (\[phieq\]) can be treated as a small correction. Thus, we can put in the leading order $\phi(z)\simeq\sqrt{n_{\rm 2D}}\phi_0(z)$; $\phi_0(z)$ is normalized to unity. By analogy with standard perturbation theory, the chemical potential, as the first correction to $E_0$, can be found with the unperturbed eigenfunction $\phi_0$. So, multiplying Eq. (\[phieq\]) by $\phi_0(z_1)$ and integrating by $z_1$ yield $$\mu=n_{\rm 2D}\int \d\bm{\rho}\, \widetilde{U}(\rho),
\label{mu2d}$$ where by definition $$\begin{aligned}
\widetilde{U}(\rho)&\equiv&\int \d z_1\d z_2\,
V(\sqrt{\rho^{2}+(z_1-z_2)^{2}}) \nonumber\\
&&\quad\times\varphi(z_1,z_2,\rho)\phi_0(z_1)\phi_0(z_2)/n_{\rm 2D}.
\label{deftilu}\end{aligned}$$ In the same manner, one can multiply Eq. (\[psieq\]) by $\phi_0(z_1)\phi_0(z_2)$ and carry out the integration by $z_1$ and $z_2$, which results in the equation $$2\big[\hslash^{2}\nabla_{\rho}^{2}/(2m)+\mu\big]\widetilde{\psi}(\rho)=\widetilde{U}(\rho)
\label{tilpsiu}$$ for the function $$\widetilde{\psi}(\rho)=\int \d z_1\d z_2\, \psi(z_1,z_2,\rho)\phi_0(z_1)\phi_0(z_2)/n_{\rm 2D}.
\label{tilpsi}$$ Thus, we arrive at the system of equations (\[mu2d\]) and (\[tilpsiu\]), which coincides with that of (\[muhom\]) and (\[psik\]) in homogeneous case if we put $U(k)= \int\d\bm{\rho}\, \widetilde{U}(\rho)e^{-i{\bf k} \cdot \bm{\rho}}$ and perform the Fourier transformation of Eq. (\[tilpsiu\]). By the same method as in Sec. \[sec:hom\], we obtain the asymptotics for sufficiently large $\rho$ \[physically, for $R_{\rm e}\ll\rho\ll\xi$, when only the first term dominates in Eq. (\[tilpsiu\])\] $$\widetilde{\varphi}(\rho)\simeq 1+\ln[e^\gamma \rho/(\sqrt{2}\xi)]m\mu/(2\pi\hslash^{2}n_{\rm 2D}),
\label{tilphiasymp}$$ where by definition $$\widetilde{\varphi}(\rho)=\int \d z_1\d z_2\, \varphi(z_1,z_2,\rho)\phi_0(z_1)\phi_0(z_2)/n_{\rm 2D}
=1+\widetilde{\psi}(\rho).
\label{tilphi}$$ The latter relation is due to Eqs. (\[phidef\]) and (\[tilpsi\]).
In order to obtain the chemical potential in terms of the 3D scattering length $a$ and the length $l_{z}$ of the trapping potential, we use the following approximation [@kagan] $$\varphi(z_1,z_2,\rho)=C\varphi^{(0)}_{\rm 3D}(r)n_{\rm 2D}\phi_0(z_1)\phi_0(z_2)
\label{phiapp}$$ in the region $r\ll l_{z}\ll\xi$, where $r=\sqrt{\rho^{2}+(z_1-z_2)^{2}}$, and $\varphi^{(0)}_{\rm 3D}(r)$ denotes the 3D solution of the Schrödinger equation (\[twobody\]) with asymptotics for $r\gg R_{\rm e}$ $$\varphi^{(0)}_{\rm 3D}(r)\simeq 1-a/r.
\label{phi3dasymp}$$ Here the crucial point is that the constant $C\not=1$, which determines the 2D behaviour of the system. If we substitute Eq. (\[phiapp\]) into Eq. (\[tilphi\]) and take the integral, we arrive at a new expression for $\widetilde{\varphi}(\rho)$. This should be expanded with respect to the dimensionless variable $\rho/l_{z}$ and compared with Eq. (\[tilphiasymp\]). Since the main contribution in that integral comes from the asymptotics (\[phi3dasymp\]), one can use it instead of the function $\varphi^{(0)}_{\rm 3D}(r)$ itself. By performing this procedure for the harmonic trapping potential with $\phi_0(z)=\exp[-z^2/(2l_{z}^2)]/\sqrt{l_{z}\sqrt{\pi}}$, we have $$\widetilde{\varphi}(\rho)\simeq C+\frac{2Ca}{l_{z}\sqrt{2\pi}}\ln[e^{\gamma/2}\rho/(2\sqrt{2}l_{z})].$$ Comparing this relation with Eq. (\[tilphiasymp\]) yields $$C=\sqrt{2\pi}\,l_{z}u/a,
\label{cpar}$$ and the chemical potential is given by Eq. (\[muhom2d\]) with the dimensionless parameter $u$ obeying the equation $$1/u+\ln u = \sqrt{2\pi}\,l_{z}/a-\gamma-\ln(16\pi n_{\rm 2D}l_{z}^{2}).
\label{u2dquasi}$$ This result for $\mu$ is well consistent with relations (\[mu2d\]) and (\[deftilu\]). Indeed, substitution of Eq. (\[phiapp\]) with constant (\[cpar\]) into Eq. (\[deftilu\]) leads to Eq. (\[muhom2d\]) provided that the relation (\[iden\]) is employed in conjunction with the approximation $\exp[-(z_1-z_2)^{2}/(2l_{z}^{2})]\simeq1$ due to the integration with the short-range potential with $R_{\rm e}\ll l_{z}$. In the leading order at small 2D densities, expressions (\[muhom2d\]) and (\[u2dquasi\]) result in $$\mu=\frac{4\pi\hslash^{2}n_{\rm 2D}}{m}\frac{1}{\sqrt{2\pi}\,l_{z}/a-\gamma-\ln(16\pi
n_{\rm 2D}l_{z}^{2})} .
\label{muourapp}$$ This differs from the result [@shlyap2D] of Petrov, Holzmann, and Shlyapnikov only by the additional numerical term $-\gamma-\ln2=-1.2703\ldots$ in the denominator. We note that the healing length in two dimensions takes the form $\xi=1/\sqrt{4\pi n_{\rm 2D}u}$, which differs from that in three dimensions $\xi=1/\sqrt{4\pi na}$. Due to the criterion $1/\sqrt{n_{\rm 2D}}\ll\xi$, the obtained results relate to sufficiently small densities, for which $u\ll 1$.
### 1D regime {#sec:1dregime}
Contrary to the 3D and 2D non-ideal Bose gases, there is no Bose-Einstein condensate in one dimension [@popov; @hoh] in the thermodynamic limit, because the long-wave fluctuations of the phase break the off-diagonal long-range order. Nevertheless, one can speak about the quasi-condensate [@shlyap1D] if a size of the of the 1D system is sufficiently small. Indeed, at zero temperature the phase fluctuations are suppressed if $\ln(L_{z}/\xi)\ll
n_{\rm 1D}\xi$ [@shlyap1D; @ho], which can be fulfilled only at finite number of particles. Here $L_{z}$ stands for the size in $z$-direction.
All calculations concerning the 1D quasi-condensate in the case $R_{\rm e}\ll l_{\rho}\ll\xi$ can be done in complete analogy with the 2D inhomogeneous Bose gas considered in the previous subsection. The gas is strongly confined in the $x$-$y$ plane by the harmonic trapping potential in $V_{\rm ext} =m\omega_\rho^2 \rho^2/2$ with the length $l_{\rho}=\sqrt{\hslash/(m\omega_\rho)}$, and remains homogeneous in $z$-direction. In the regime involved, we can put $\phi(\rho)=\sqrt{n_{\rm 1D}}\phi_0(\rho)$, $\phi_0(\rho)=\exp[-\rho^2/(2l_{\rho}^2)]/l_{\rho}\sqrt{\pi}$ is the ground state solution of the one-particle Schrödinger equation with the energy $E_{0}=\hslash\omega_{\rho}$. Reasoning by analogy with Sec. \[sec:2dregime\], we obtain $$\mu=n_{\rm 1D}\int \d z\, \widetilde{U}(z),
\label{mu1d}$$ where we introduce the even function $\widetilde{U}(z)=\widetilde{U}(-z)$ $$\begin{aligned}
\widetilde{U}(z)&=&\int \d\bm{\rho}_1\d\bm{\rho}_2\, V(\sqrt{(\bm{\rho}_1-\bm{\rho}_2)^{2}+z^{2}}) \nonumber\\
&&\quad\times\varphi(\bm{\rho}_1,\bm{\rho}_2,z)\phi_0(\rho_1)\phi_0(\rho_2)/n_{\rm 1D}.
\nonumber\end{aligned}$$ The 1D analogue of Eq. (\[tilpsiu\]) is the equation $$2\bigg[\frac{\hslash^{2}}{2m}\frac{\d^{2}}{\d z^{2}}+\mu\bigg]\widetilde{\psi}(z)=\widetilde{U}(z)
\label{tilpsiu1}$$ for the function $$\widetilde{\psi}(z)=\int \d \bm{\rho}_1\d \bm{\rho}_2\, \psi(\bm{\rho}_1,\bm{\rho}_2,z)
\phi_0(\rho_1)\phi_0(\rho_2)/n_{\rm 1D}.$$ Equation (\[tilpsiu1\]) can be rewritten in the Lippmann-Schwinger form at $\mu\to0$ (see discussion in Sec. \[sec:hom\]) $$\widetilde{\varphi}(z)=1+(m/\hslash^{2})\int \d z'\,\widetilde{U}(z')|z-z'|/2
\label{lipshw1}$$ for the function $\widetilde{\varphi}(z)$, defining as $$\widetilde{\varphi}(z)=\int \d \bm{\rho}_1\d \bm{\rho}_2\, \varphi(\bm{\rho}_1,\bm{\rho}_2,z)
\phi_0(\rho_1)\phi_0(\rho_2)/n_{\rm 1D}.
\label{tilphi1}$$ Equations (\[mu1d\]) and (\[lipshw1\]) give the asymptotics for $R_{\rm e}\ll z\ll\xi$ $$\widetilde{\varphi}(z)\simeq 1+m\mu|z|/(2n_{\rm 1D}\hslash^{2}).
\label{asymp1}$$ On the other hand, in the region $r\ll l_{\rho}\ll\xi$ we can use the analogue of Eq. (\[phiapp\]) $$\varphi(\bm{\rho}_1,\bm{\rho}_2,z)=C\varphi^{(0)}_{\rm 3D}(r)n_{\rm 1D}\phi_0(\rho_1)\phi_0(\rho_2),
\label{phiapp1}$$ which leads to the asymptotics after the integration in Eq. (\[tilphi1\]) $$\widetilde{\varphi}(z)\simeq C-C\big(\sqrt{\pi/2}-|z|/l\big)a/l_{\rho}.
\label{asymp11}$$ Comparing Eqs. (\[asymp1\]) and (\[asymp11\]) yields $$\begin{aligned}
C&=&1/(1-\sqrt{\pi/2}\,a/l_{\rho}),
\label{cpar1}\\
\mu&=&\frac{2\hslash^{2}n_{\rm 1D}}{m}\frac{a}{l_{\rho}^{2}}\frac{1}{1-\sqrt{\pi/2}\,a/l_{\rho}},
\label{mu1d1}\end{aligned}$$ which differs from Olshanii’s result [@olshanii] through the numerical factors $\sqrt{\pi}=1.772\ldots$ in the denominator instead of the constant $1.4603\ldots$ introduced by him. We note that in the paper [@olshanii] $a_{\perp}=\sqrt{2\hslash/(m\omega_\rho)}=\sqrt{2}\,l_{\rho}$ in our notation. One can see that the criteria of applicability of the obtained results $l_{\rho}\ll\xi$ and $1/n_{\rm 1D}\ll\xi$ impose the following restriction on the 1D density $$\frac{a}{l_{\rho}^{2}}\ll n_{\rm 1D}\ll\frac{1}{a},
\label{restr1d}$$ since $\xi\simeq l_{\rho}/\sqrt{2an_{\rm 1D}}$ in one dimension.
The kinetic, interaction, and external field energy of the trapped Bose gas {#sec:kinpot}
===========================================================================
The simplest method for obtaining the values of the interaction energy (\[eint\]), the kinetic energy (\[ekin\]), and the energy of interaction with an external field (\[eext\]), is to apply the variational theorem. The latter can be formulated in general as follows. If a function $f(x)$ obeys the functional equation $$\delta F[\{f(x)\},\lambda]/\delta f(x)=0
\label{fan}$$ with the functional $F$ depending on the function $f(x)$ and the parameter $\lambda$, then the solution of Eq. (\[fan\]) $f(x)=f_0(x,\lambda)$ is also dependent on $\lambda$. Nevertheless, when calculating the derivative of the stationary value of the functional with respect to $\lambda$, we can take into consideration only the explicit dependence on this parameter $$\d F[\{f_0(x,\lambda)\},\lambda]/\d \lambda=\partial F[\{f_0(x,\lambda)\},\lambda]/\partial \lambda.
\label{fander}$$ This is obvious due to Eq. (\[fan\]).
The variational theorem (\[fander\]) is still valid if the functional contains two or more functions. In our case, the functions can be associated with $\phi(x_1)$ and $\psi(x_1,x_2)$ involved in the energy functional (\[efunc\]). Considering ${\cal
N}$ as the parameter of the variational theorem, we come to the standard thermodynamic relation $\partial E/\partial {\cal
N}=\mu'=\mu+E_0$. One can rewrite this derivative in terms of the energy per particle $\varepsilon=E/N$ and the density of particles $\partial E/\partial {\cal N} =\partial(\varepsilon n)/\partial n$, which gives the relation $\varepsilon=(1/n)\int_{0}^{n} \d
n'\,\mu(n')+E_0$. Then relations (\[muhom2d\]) and (\[u2dquasi\]) lead to $$\varepsilon_{\rm 2D}\simeq{2\pi\hslash^{2}n_{\rm 2D}u}/{m}+\frac{\hslash^2}{2ml_{z}^{2}}
\label{e2d}$$ with $u$ given by Eq. (\[u2dquasi\]). In the same manner, we obtain from Eq. (\[mu1d1\]) [@note2] $$\varepsilon_{\rm 1D}=\frac{\hslash^{2}n_{1D}}{m}\frac{a}{l_{\rho}^{2}}\frac{1}{1-\sqrt{\pi/2}\,a/l_{\rho}}
+\frac{\hslash^2}{ml_{\rho}^{2}}.
\label{e1d}$$ Equations (\[e2d\]) and (\[e1d\]) give us the equilibrium value of the energy (\[efunc\]) per particle in the 2D and 1D cases, respectively. In order to calculate the interaction energy with the help of the variational theorem, one can replace $V\to\lambda V$ and differentiate $\varepsilon$ with respect to $\lambda$ at $\lambda=1$. All we need to know is the derivative of the 3D scattering length, which reads [@cherny1; @mur] $$\lambda \frac{\partial a}{\partial \lambda}=
m\frac{\partial a}{\partial m}
=\frac{m}{4\pi\hslash^2}\int\d^{3}r\,[\varphi_{\rm 3D}^{(0)}(r)]^{2}\lambda V(r).
\label{varth}$$ It is convenient to introduce one more characteristic length [@cherny1], the positive parameter $b$, $$\begin{aligned}
b=a-\lambda\left.\frac{\partial a}{\partial \lambda}\right|_{\lambda=1}
=\frac{1}{4\pi}\int\d^{3}r\,\bigl|\nabla\varphi_{\rm 3D}^{(0)}(r)\bigr|^{2}.
\nonumber
%\label{bdef}\end{aligned}$$ So, we have $$\begin{aligned}
\varepsilon_{\rm 2Dint}&\simeq&\frac{2\pi\hslash^{2}n_{\rm 2D}}{m}u^{2}\frac{\sqrt{2\pi}\,l_z}{a}\bigg(1-\frac{b}{a}\bigg),
\label{eint2d}\\
\varepsilon_{\rm 1Dint}&\simeq&\frac{\hslash^{2}n_{1D}}{m}\frac{a}{l_{\rho}^{2}}\bigg(1-\frac{b}{a}\bigg),
\label{eint1d}\end{aligned}$$ where we use the approximation $u^2/(1-u)\simeq u^2$ in Eq. (\[eint2d\]) and restrict ourselves by the leading order in Eq. (\[eint1d\]).
With the same method, replacing $V_{\rm ext}\to\lambda V_{\rm
ext}$ (which is equivalent to $l\to l/\sqrt[4]{\lambda}$) and differentiating, we arrive at the external energy per particle $$\begin{aligned}
\varepsilon_{\rm 2Dext}&\simeq&\frac{2\pi\hslash^{2}n_{\rm
2D}}{m}\frac{u^{2}}{4}\bigg(\frac{\sqrt{2\pi}\,l_z}{a}-2\bigg)+\frac{\hslash^2}{4ml_{z}^{2}},
\label{eext2d}\\
\varepsilon_{\rm 1Dext}&\simeq&\frac{\hslash^{2}n_{1D}}{2m}\frac{a}{l_{\rho}^{2}}
+\frac{\hslash^2}{2ml_{\rho}^{2}}.
\label{eext1d}\end{aligned}$$ In the same manner, we have $\varepsilon_{\rm kin}=-m\partial\varepsilon/\partial m$, which leads to $$\begin{aligned}
\varepsilon_{\rm 2Dkin}&\simeq&\frac{2\pi\hslash^{2}n_{\rm 2D}}{m}u
-\frac{2\pi\hslash^{2}n_{\rm 2D}}{m}u^2\bigg[\frac{1}{4}\bigg(\frac{\sqrt{2\pi}\,l_z}{a}-2\bigg)\nonumber\\
&&+\frac{\sqrt{2\pi}\,l_z}{a}\bigg(1-\frac{b}{a}\bigg)\bigg]+\frac{\hslash^2}{4ml_{z}^{2}},
\label{ekin2d}\\
\varepsilon_{\rm 1Dkin}&\simeq&\frac{\hslash^{2}n_{1D}}{m}\frac{b}{l_{\rho}^{2}}
-\frac{\hslash^{2}n_{1D}}{2m}\frac{a}{l_{\rho}^{2}}+\frac{\hslash^2}{2ml_{\rho}^{2}}.
\label{ekin1d}\end{aligned}$$ One can see that sum of the kinetic, external and interaction energies equals to the total energy, as it should be. Note that the developed formalism allows us to calculate the interaction energy directly, starting from the expression (\[eint1\]) and using Eq. (\[iden\]), since we have the analytic expressions (\[phiapp\]), (\[cpar\]), (\[phiapp1\]), and (\[cpar1\]) for the short-range behaviour of the anomalous average.
We note that the ratio $b/a$ need not be small. In particular, it is of order of ten for the realistic interaction potentials of alkali atoms [@cherny3]. We stress that the term with the length $b$ appears in the mean interaction energy by virtue of the the short-range two-body correlations at the distances of order of $a$ and in the mean kinetic energy by sufficiently large momenta of order of $p\gtrsim \hslash/a$ in the momentum distribution. In the static structure factor, this region is rather difficult to be measured experimentally. The density expansion method gives the value of the release energy that is defined as [*sum*]{} of the interaction and kinetic energies $$\begin{aligned}
\varepsilon_{\rm 2Drel}&\simeq&\frac{2\pi\hslash^{2}n_{\rm 2D}}{m}u
-\frac{2\pi\hslash^{2}n_{\rm 2D}}{m}\frac{u^2}{4}\bigg(\frac{\sqrt{2\pi}\,l_z}{a}-2\bigg)\nonumber\\
&&+\frac{\hslash^2}{4ml_{z}^{2}},
\label{erel2d}\\
\varepsilon_{\rm 1Drel}&\simeq&\frac{\hslash^{2}n_{1D}}{2m}\frac{a}{2l_{\rho}^{2}}+\frac{\hslash^2}{2ml_{\rho}^{2}}.
\label{erel1d}\end{aligned}$$ As one can see, the parameter $b$ is canceled and not involved in the values of the release energy. Let us compare the values of the release (\[erel2d\]-\[erel1d\]) and total energy (\[e2d\]-\[e1d\]). The energy of zero-point oscillation is involved in the release energy with the factor $1/2$, as it should be for the harmonic trap. The other terms would coincide in the standard GP approach, but we have obvious difference due to accounting for the non-condensate contribution. In principle, the obtained corrections should be measurable in experiments.
Virial theorem {#sec:vir}
==============
The virial theorem can be obtained immediately from the energy functional (\[efunc\]) if we consider its variation in vicinity of the stationary state (ground state) with respect to the scaling transformation of the ground state functions $\phi_0$ and $\psi_0$, obeying the generalized GP equations (\[phieq\]) and (\[psieq\]). Namely, we substitute into Eq. (\[efunc\]) the functions $\phi({\bf r}_1) =\alpha^{3/2}\phi_0(\alpha {\bf r}_1)$ and $\psi({\bf r}_1) =\alpha^{3}\psi_0(\alpha {\bf r}_1,\alpha {\bf
r}_1)$. Replacing the variables in the integrals ${\bf r}_1\to\alpha
{\bf r}_1$ and ${\bf r}_2\to\alpha {\bf r}_2$, we notice that, first, the last term in the functional equals to zero for any $\alpha$, and, second, the other terms can be written in terms of its stationary values $$E(\alpha)=\alpha^{2}E_{\rm kin}+E_{\rm ext}/\alpha^{2}+E_{\rm int}\big[V(r/\alpha)\big].
\label{ealph}$$ Since the variation of the functional should be zero for any small variations of the functions, we have $\d E/\d \alpha=0$ at $\alpha=1$, which leads to $$2E_{\rm kin}-2E_{\rm ext}+E_{\rm int}[-rV'(r)]=0,
\label{virth}$$ where the terms are given by Eqs. (\[eint1\])-(\[ekin1\]). The value of the last term corresponds to the interaction energy with the potential $-rV'(r)=-r\d V(r)/\d r$. In the case of the GP approximation (\[gpphi\]), one can simplify the last item in Eq. (\[virth\]) by means of Eq. (\[iden\]) and relation [@bog47] $$\frac{4\pi\hslash^2 a}{m}=-
\int_{0}^{\infty}\d r\, 4\pi r^2 [\varphi^{(0)}(r)]^{2}\bigg(2V(r)+r\frac{\d V(r)}{\d r}\bigg).$$ The result takes a form $$\begin{aligned}
E_{\rm int}&\simeq&\frac{1}{2}\int\d{\bf R}\,|\phi({\bf R})|^{4}
\int \d {\bf r}\,[-rV'(r)]\big[\varphi_{\rm 3D}^{(0)}(r)\big]^2
\nonumber\\
&=&\frac{2\pi\hslash^2}{m}(3a-2b)\int\d{\bf R}\,|\phi({\bf R})|^{4}.
\label{gpvirth}\end{aligned}$$ If the potential is of the weak-coupling type [@classif], one can neglect $b\ll a$ and arrive at the virial theorem obtained for the $\delta$-function interaction potential [@dalfovo].
If the system is homogeneous in the $x$-$y$ plane (the 2D Bose gas of Sec. \[sec:2dregime\]) or in the $z$ direction (the 1D Bose gas of Sec. \[sec:1dregime\]), it can be considered as confined by infinite walls in associated directions. Then one should be careful when deriving the virial theorem from Eq. (\[ealph\]), as all its terms relate to the density $n_{\rm 2D}/\alpha^2$ or $n_{\rm 1D}/\alpha$ for the 2D or 1D Bose gas, respectively. For this reason, we come to $$\begin{aligned}
2n_{\rm 2D}\frac{\partial\varepsilon_{\rm 2D}}{\partial n_{\rm 2D}}\!\!&=&\!\!2\varepsilon_{\rm 2D kin}\!
-2\varepsilon_{\rm 2D ext}\!+\varepsilon_{\rm 2D int}[-rV'(r)],
\label{virth2d}\\
n_{\rm 1D}\frac{\partial\varepsilon_{\rm 1D}}{\partial n_{\rm
1D}}\!\!&=&\!\!2\varepsilon_{\rm 1D kin}\! -2\varepsilon_{\rm 1D
ext}\!+\varepsilon_{\rm 1D int}[-rV'(r)].
\label{virth1d}\end{aligned}$$ The interaction term in these equations can be easily calculated by analogy with Eq. (\[gpvirth\]) but using Eqs. (\[phiapp\]) and (\[phiapp1\]), respectively. It is not difficult to be convinced with the help of Eqs. (\[e2d\]) and (\[e1d\]) that the virial theorems (\[virth2d\]) and (\[virth1d\]) are fulfilled.
One can also find a relation between the chemical potential and the various parts of the energy. Let us multiply Eq. (\[phieq\]) by $\phi(x_1)$ and integrate over $x_1$, and multiply Eq. (\[psieq\]) by $\psi(x_1,x_2)$ and also integrate over $x_1$ and $x_2$. Summing the obtained expressions yields $$N\mu=E_{\rm kin1}+2E_{\rm kin2} +E_{\rm ext1}+ 2E_{\rm ext2}+2E_{\rm int},
\label{muenergy}$$ Here, $E_{\rm ext1}$ and $E_{\rm kin1}$ are the condensate contributions in the external and kinetic energies given by the last terms in Eqs. (\[eext1\]) and (\[ekin1\]), respectively, and $E_{\rm ext2}$ and $E_{\rm kin2}$ are associated with the non-condensate contributions, given by the residual parts of these equations. One can easily see that the relation (\[muenergy\]) is fulfilled with $E_{\rm ext1}$ and $E_{\rm kin1}$ corresponding to the last terms in Eqs. (\[eext2d\]) and (\[eext1d\]), and (\[ekin2d\]) and (\[ekin1d\]) for the 2D and 1D Bose gases, respectively. One can notice that $E_{\rm kin2}$ could be negative for the 1D Bose gas, if $b<a/2$ \[see the first two terms in Eq. (\[ekin1d\])\]. Certainly, this is not a drawback of Eqs. (\[phieq\]) and (\[psieq\]) it is but due to the choice of anzatz $\phi(\rho)=\sqrt{n_{\rm 1D}}\phi_0(\rho)$, which leads to overestimation of the quasicondensate contribution $E_{\rm kin1}$ in the 1D kinetic energy. Indeed, the Gaussian profile $n_{\rm
1D}|\phi_0(\rho)|^{2}$ relates to the [*total*]{} density of the 1D gas $\langle{\hat\Psi}^{\dag}({\bf r}) {\hat\Psi}({\bf r})\rangle$ but not to the “quasicondensate component” $|\phi(\rho)|^{2}$. The latter is difficult to define accurately in the 1D case, since there is no eigenvalue of the one-body density matrix that is proportional to the total number of particles. Nevertheless, we stress that the total value of $E_{\rm 1Dkin}$ is positive, and the results (\[eint1d\]), (\[eext1d\]), and (\[ekin1d\]) look quite reasonable.
Conclusions {#sec:dis}
===========
The main result of this paper are the generalized GP equations in the time-dependent (\[phieqt\]–\[psieqt\]) and stationary form (\[phieq\]–\[psieq\]), which allow us to determine the interaction term self-consistently for interaction potentials even containing a hard-core. The method, which can be used for homogeneous, strongly inhomogeneous quasi-low-dimensional, and cross-over regimes was derived within a general HFB framework.
The HFB method is a mean-field approximation, which generally works well only for weak-coupling potentials [@classif]. In order to extend the HFB scheme to hard-core potentials, the bare interaction potential is usually replaced by a renormalized pseudopotential $V(r)\to(4\pi\hslash^2/m)\delta^3({\bf r})$. However, such a replacement leads to an ultraviolet divergence and incorrect treatment of short-range correlations of the particles. We have shown that the appropriate renormalization can be obtained [*within*]{} the HFB scheme if, from the two-body density matrix, only the anomalous correlation function $\varphi(x_1,x_2) =\langle{\hat\Psi}(x_1) {\hat\Psi}(x_{2})\rangle$ is retained. The anomalous correlation function can be interpreted as the wavefunction of two bosons in the condensate. Its short-range behaviour is described well in the proposed scheme at the cost of loosing the correct description of the long-range behaviour. However, long-range correlations are not needed for deriving the non-linear term in the generalized GP approach, which instead is determined by short-range correlations. Methods which can describe both the short- and long-range correlations accurately were discussed in Refs. [@cherny4; @cherny1; @cherny2; @leggett1], but these methods are appropriate only for the homogeneous Bose gas. The method proposed in this paper was shown to work as well in inhomogeneous situations. Cigar (quasi-1D) and pancake (quasi-2D) geometries were considered as examples. Furthermore, it was shown that the contribution of short-range correlations to the kinetic and release energies of a tightly trapped gas can be calculated within this scheme and that they are substantial. Interesting future applications of the proposed method may include the modification of the nonlinearity in quasi-1D waveguides [@Muryshev2002a; @sinha04ep] and molecular Bose condensates in optical lattices.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to O. Sørensen and A.A. Shanenko for interesting discussions. This work was supported in part by the RFBR grant 01-02-17650.
Two-body wave functions in the Hartree-Fock-Bogoliubov approximation {#sec:pwf}
====================================================================
In general, the two-body density matrix can be expanded in a complete set of its eigenfunctions $$\begin{aligned}
\langle{\hat\Psi}^{\dagger}(x_1){\hat\Psi}^{\dagger}(x_{2})
{\hat\Psi}(x'_{2}){\hat\Psi}(x'_1)\rangle&=&
\sum_{\nu,\mu}N_{\nu,\mu}\varphi^*_{\nu,\mu}(x_1,x_2)
\nonumber\\
&&\times\varphi_{\nu,\mu}(x'_1,x'_2).
\label{rho2exp}\end{aligned}$$ The eigenfunctions can be called two-body or pair wave functions. If they are normalized to unity, it follows from Eq. (\[rho2exp\]) that $\int\d x_1 \d x_2\,\langle{\hat\Psi}^{\dagger}(x_1)
{\hat\Psi}^{\dagger}(x_{2}) {\hat\Psi}(x_{2}) {\hat\Psi}(x_1)\rangle=
N(N-1)=\sum_{\nu,\mu}N_{\nu,\mu}$, i.e., the sum of all $N_{\nu,\mu}$ is the total number of pairs. Therefore, the non-negative quantity $N_{\nu,\mu}$ can be interpreted as the mean number of the pairs in the state $(\nu,\mu)$, any pair being doubly taken.
Let us consider the homogeneous spinless Bose gas in the HFB approximation [@griffin; @hohmartin]. Within that approximation, the two-body wave functions can be easily calculated [@cherny]. The statistical average of any product of quantum operators $\hat{\vartheta}$ and $\hat{\vartheta}^\dag$ can be calculated with the Wick-Bloch-De Dominicis theorem [@bloch], since the Hamiltonian is approximated by a [*quadratic form*]{} of the Bose operators ${\hat\alpha}_{{\bf p}}^{\dagger}$ and ${\hat\alpha}_{{\bf
p}}$ connected with initial operators ${\hat a}_{{\bf p}}^{\dagger}$ and ${\hat a}_{{\bf p}}$ by the canonical Bogoliubov transformations (see Appendix \[sec:hfrel\]). Extracting the $c$-number part $\hat{\Psi} =\sqrt{n_0} +\hat{\vartheta}$ and $\hat{\Psi}^\dag
=\sqrt{n_0} +\hat{\vartheta}^\dag$ and using that theorem, one can rewrite the four-boson average in the form $$\bigg\langle{\hat\Psi}^{\dag}\Big({\bf R}+\frac{\bf r}{2}\Big){\hat\Psi}^{\dag}\Big({\bf R}-\frac{\bf r}{2}\Big)
{\hat\Psi}\Big({\bf R}'-\frac{{\bf r}'}{2}\Big){\hat\Psi}\Big({\bf R}'+\frac{{\bf r}'}{2}\Big)\bigg\rangle$$ $$\begin{aligned}
&=&\!n_{0}^{2}\widetilde{\varphi}^{*}(r)\widetilde{\varphi}(r') +\int \d^{3}p\,\d^{3}q\,
\biggl[2n_{0}\delta({\bf q}/2-{\bf p})\frac{n(q)}{(2\pi)^{3}} \nonumber\\
&&\!+\frac{n({\bf q}/2+{\bf p})}{(2\pi)^{3}}
\frac{n({\bf q}/2-{\bf p})}{(2\pi)^{3}}\biggr]\sqrt{2}\cos({\bf p}\cdot\r{})\sqrt{2}\cos({\bf p}\cdot\rp{})\nonumber\\
&&\times\exp[i{{\bf q}}\cdot({\bf R}^{\prime }-{\bf R})],
\label{rho2exphf}\end{aligned}$$ where we put by definition $\widetilde{\varphi}(r)
=1+\langle\hat{\vartheta}({\bf R}+{\bf r}/{2}) \hat{\vartheta}({\bf
R}-{\bf r}/{2})\rangle/n_0$. Because the expansion (\[rho2exphf\]) is written in the thermodynamic limit, the sum in Eq. (\[rho2exp\]) becomes an integral. The Bose-Einstein condensate manifests itself in presence of $\delta$-functions in this integral (note that the first term in the r.h.s. can be included in the integral with the help of the $\delta$-functions). By comparing the representation (\[rho2exp\]) with that of (\[rho2exphf\]), one can conclude the following:\
([*i*]{}) The quantum numbers of the pair wave functions are the relative momentum $\nu={\bf p}$ and the center-of-mass (total) momentum $\mu={\bf q}$ of two particles; all these functions belong to continuous spectrum and thus describe the scattering of two bosons in the medium of the other bosons.\
([*ii*]{}) The maximum eigenvalue $N_{0}(N_{0}-1)\simeq N_{0}^{2}$ with ${\bf
p}={\bf q}=0$ corresponds to the state of two particles in the condensate; its normalized eigenfunction $\widetilde{\varphi}(r)/V$ can be interpreted as a pair wave function of the condensate-condensate type. Thus, the anomalous average $\langle\hat{\vartheta}({\bf r})\hat{\vartheta}(0)\rangle$ can be associated with the scattering part of the two-body wave function of the bosons in the condensate [@cherny]; in particular, it is responsible for the short-range spatial correlations of two bosons in the Bose-Einstein condensate.\
([*iii*]{}) The other macroscopic eigenvalues $2N_{0}n_{q}$ with ${\bf q}=\pm 2{\bf p}$ correspond to the two-body states with one particle in the condensate and another one beyond the condensate; its eigenfunctions $\sqrt{2}\cos({\bf
q}\cdot{\bf r}/2) \exp[i{\bf q}\cdot{\bf R}]/V$ are of the condensate-noncondensate type [@note3b]. The residuary non-macroscopic eigenvalues $n({\bf q}/2+{\bf p}) n({\bf q}/2-{\bf p})$ are related to the noncondensate-noncondensate pairs with the two-body wave functions $\sqrt{2}\cos({{\bf p}}\cdot{\bf r}) \exp[i{\bf q}\cdot{\bf
R}]/V$.
Note that the wave function of the condensate-condensate type is not reduced to a product of two one-body wave functions in the condensate, which equal to $1/\sqrt{V}$ for the homogeneous Bose gas. This is obvious, as particles in the Bose-Einstein condensate interact with each other and with the other particles beyond the condensate. Another important point is that all the other two-body wave functions are symmetrized plane waves (consistent with the Born approximation) in the framework of the HFB method. This is evidently a disadvantage of the HFB scheme. As a consequence, we always arrive at divergences for a hard-core potential when evaluating the contribution of the condensate-noncondensate and noncondensate-noncondensate wave functions in the interaction energy (\[eint\]). At the same time, the contribution of the condensate-condensate “channel” should be finite in the interaction energy provided the anomalous averages are calculated in a self-consistent manner. The generalization of the expansion (\[rho2exphf\]) beyond the HFB approach and more detailed discussions can be found in Ref. [@cherny]. The pair wave function method of Ref. [@cherny] was generalized to the inhomogeneous systems in Ref. [@naidon].
Relation between the normal and anomalous two-boson averages {#sec:hfrel}
============================================================
Let us establish a relation between the normal $\langle{\hat\vartheta}^{\dag}(x_1){\hat\vartheta}(x_2)\rangle$ and the anomalous average $\langle{\hat\vartheta}(x_1){\hat\vartheta}(x_2)\rangle$ for the vacuum state, which describes the behaviour of the $N$-body system at zero temperature, in the framework of the Hartree-Fock-Bogoliubov method. We remember that the vacuum state $|0\rangle$ is defined as $\alpha_\nu|0\rangle=0$ for any $\nu\not=0$, here the quasiparticle creation and destruction operators $\hat{\alpha}^\dag_\nu$ and $\hat{\alpha}_\nu$ can be introduced through the Bogoliubov transformation ($f\not=0$) $$\begin{aligned}
\hat{a}_f&=&\sumpr_\nu\,(u_{f\nu}\hat{\alpha}_\nu+v_{f\nu}\hat{\alpha}^\dag_\nu),
\label{bogtrans_a} \\
\hat{a}^\dag_f&=&\sumpr_\nu\,(u^*_{f\nu}\hat{\alpha}^\dag_\nu+v^*_{f\nu}\hat{\alpha}_\nu),
\label{bogtrans_b} \end{aligned}$$ where $f$ and $\nu$ denote discrete (multi-)indices. The sum $\sumpr_\nu$ means $\sum_{\nu\not=0}$ and the Bose-operators $\hat{a}^\dag_f$ and $\hat{a}_f$ create and destruct a particle in the eigenstate $\phi_{f}(x)$ of the one-body matrix $\langle{\hat\Psi}^{\dag}(x'){\hat\Psi}(x)\rangle$ $$\int \d x'\, \langle{\hat\Psi}^{\dag}(x'){\hat\Psi}(x)\rangle \phi_{f}(x')
= n_{f} \phi_{f}(x),$$ normalized as $\int d x\,|\phi_{f}(x)|^{2}=1$. Note that the set of eigenfunctions including the normalized condensate function $\phi_0(x)=\langle\hat{\Psi}(x)\rangle/\sqrt{N_0}$ with $N_0=n_{f=0}$ is complete and orthogonal $$\begin{aligned}
\sum_f\phi^*_f(x)\phi_f(x')=\delta(x-x'),
\label{compl_a}\\
\int \d x\,\phi_f^*(x)\phi_{f'}(x)=\delta(f-f'),
\label{compl_b} \end{aligned}$$ where we define the “discrete" $\delta$-function as $$\delta(f)=\left\{\begin{array}{ll}
1, & f=0,\\
0, & f\not=0.
\end{array}\right.$$ From the Bose commutation relations $[\hat{a}_f,\hat{a}\dag_{f'}]=\delta(f-f')$ and $[\hat{\alpha}_f,\hat{\alpha}\dag_{f'}]=\delta(f-f')$ and Eqs. (\[bogtrans\_a\]-\[bogtrans\_b\]) we obtain at $f,f'\not=0$ $$\begin{aligned}
\sumpr_\nu\,(u_{f\nu}u^*_{f'\nu}- v_{f\nu}v^*_{f'\nu})&=&\delta(f-f'), \label{uv_a}\\
\sumpr_\nu\,(u_{f\nu}v_{f'\nu}- v_{f\nu}u_{f'\nu})&=&0.
\label{uv_b} \end{aligned}$$ By using the definition of the quasiparticle vacuum state and Eqs. (\[bogtrans\_a\]-\[bogtrans\_b\]), we can calculate the averages $$\begin{aligned}
F(f,f')&=&\langle\hat{a}^\dag_f\hat{a}_{f'}\rangle=\sumpr_\nu\,v^*_{f\nu}v_{f'\nu},
\label{fphi_a}\\
\Phi(f,f')&=&\langle\hat{a}_f\hat{a}_{f'}\rangle=\sumpr_\nu\,u_{f\nu}v_{f'\nu}.
\label{fphi_b} \end{aligned}$$ Our purpose is to find the relation between the normal $F(f,f')$ and the anomalous $\Phi(f,f')$ averages for that state. In order to simplify our calculations, we rewrite Eqs. (\[bogtrans\_a\]-\[bogtrans\_b\]) in the matrix notations $$\label{bogtransmat}
\begin{pmatrix}
\hat{a}\\
\hat{a}^\dag
\end{pmatrix}=X
\begin{pmatrix}
\hat{\alpha}\\
\hat{\alpha}^\dag
\end{pmatrix},\
X=\begin{pmatrix}
U & V\\
V^* & U^*
\end{pmatrix}.$$ Here the matrix $X$ is composed of the matrix $(U)_{ij}=u_{ij}$ and $(V)_{ij}=v_{ij}$. The columns contain the operators $\hat{a}_f$ and $\hat{a}^\dag_f$, and $\hat{\alpha}_\nu$ and $\hat{\alpha}^\dag_\nu$, respectively. We use the standard notations for the complex conjugate $(V^*)_{ij}=v^*_{ij}$, transposed $(V^\tran)_{ij}=v_{ji}$, and Hermitian conjugate matrix $(V^\dag)_{ij}=v^*_{ji}$. Then Eqs. (\[fphi\_a\]-\[fphi\_b\]) read $$F=V^*V^\tran=F^\dag,\quad \Phi=UV^\tran=\Phi^\tran,
\label{fphi1}$$ and Eqs. (\[uv\_a\]-\[uv\_b\]) can be written as $$\begin{pmatrix}
U & V\\
V^* & U^*
\end{pmatrix}
\begin{pmatrix}
U^\dag & -V^\tran\\
-V^\dag & U^\tran
\end{pmatrix}
=\begin{pmatrix}
\openone & 0\\
0& \openone
\end{pmatrix},
\label{uv1}$$ where $\openone$ denotes the identity matrix. Let us introduce the composed matrices $$\sigma_3=\begin{pmatrix}
\openone & 0\\
0 & -\openone
\end{pmatrix},\
\sigma_+=\begin{pmatrix}
\openone & 0\\
0 & 0
\end{pmatrix},
\label{defsig3}$$ and rewrite Eq. (\[uv1\]) in the form $$X\sigma_3X^\dag\sigma_3=\openone,
\label{uv2}$$ where $\openone$ stands now for the composed identity matrix, i.e. the r.h.s. of Eq. (\[uv1\]). The matrix representation (\[uv2\]) is very convenient. For example, from this equation we have $X^{-1}=\sigma_3 X^\dag\sigma_3$, and $$\begin{pmatrix}
\hat{\alpha}\\
\hat{\alpha}^\dag
\end{pmatrix}
=\sigma_3X^\dag\sigma_3
\begin{pmatrix}
\hat{a}\\
\hat{a}^\dag
\end{pmatrix}
=\begin{pmatrix}
U^\dag & -V^\tran\\
-V^\dag & U^\tran
\end{pmatrix}
\begin{pmatrix}
\hat{a}\\
\hat{a}^\dag
\end{pmatrix},$$ which reads in usual notations $$\begin{aligned}
\hat{\alpha}_f&=&\sumpr_\nu\,(u^*_{\nu f}\hat{a}_\nu-v_{\nu f}\hat{a}^\dag_\nu), \nonumber \\
\hat{\alpha}^\dag_f&=&\sumpr_\nu\,(u_{\nu f}\hat{a}^\dag_\nu-v^*_{\nu f}\hat{a}_\nu). \nonumber \end{aligned}$$ This equation together with the commutation relations leads to $$\begin{aligned}
\sumpr_\nu\,(u^*_{\nu f}u_{\nu f'}- v_{\nu f}v^*_{\nu f'})&=&\delta(f-f'), \nonumber \\
\sumpr_\nu\,(v_{\nu f}u^*_{\nu f'}- u^*_{\nu f}v_{\nu f'})&=&0, \nonumber \end{aligned}$$ which is nothing else but the matrix equation $X^\dag\sigma_3X\sigma_3=\openone$, resulting from Eq. (\[uv2\]).
Employing the idea of Ref. [@bogufn], in which the Hartree-Fock-Bogoliubov method for Fermi systems was developed, we define the matrix $K$ with the help of the notations (\[fphi1\]) and (\[defsig3\]) $$K=X^\dag\sigma_3\sigma_+X\sigma_3
=\begin{pmatrix}
\openone+F^* & -\Phi\\
\Phi^* & -F
\end{pmatrix}.$$ Due to Eq. (\[uv2\]) and the relation $(\sigma_+)^2=\sigma_+$ we have $K^2=K$. Rewriting the latter equation in terms of the matrix $F$ and $\Phi$, we obtain two independent relations $$\begin{aligned}
\Phi^*\Phi&=&F+F^2,
\label{fpfirel_a} \\
F^*\Phi &=&\Phi F,
\label{fpfirel_b} \end{aligned}$$ which read in components 0.00 mm $$\sumpr_f\!\Phi^*(f_1,f)\Phi(f,f_2)=\sumpr_f\! F(f_1,f)F(f,f_2)\!+\!F(f_1,f_2),
\label{fpfirel1_a}$$ $$\sumpr_f\! F(f,f_1)\Phi(f,f_2)=\sumpr_f\!\Phi(f_1,f)F(f,f_2).
\label{fpfirel1_b}$$ By using these equations, Eqs. (\[compl\_a\]-\[compl\_b\]), and the definition $\hat{\vartheta}(x) =\sumpr_\nu\, \hat{a}_\nu
\phi_\nu(x)$, one can rewrite Eqs. (\[fpfirel1\_a\]-\[fpfirel1\_b\]) in the coordinate representation $$\begin{aligned}
&&\int\d x\, \langle\hat{\vartheta}^\dag(x_1)\hat{\vartheta}^\dag(x)\rangle\langle\hat{\vartheta}(x)
\hat{\vartheta}(x_2)\rangle =\langle\hat{\vartheta}^\dag(x_1)\hat{\vartheta}(x_2)\rangle
\nonumber \\
&&\phantom{\int\d x\, \langle\hat{\vartheta}^\dag}
+\int\d x\, \langle\hat{\vartheta}^\dag(x_1)\hat{\vartheta}(x)\rangle
\langle\hat{\vartheta}^\dag(x)\hat{\vartheta}(x_2)\rangle,
\label{fphicord1}\end{aligned}$$ $$\begin{aligned}
&&\int \d x\, \langle\hat{\vartheta}^\dag(x)\hat{\vartheta}(x_1)\rangle\langle\hat{\vartheta}(x)\hat{\vartheta}(x_2)\rangle
=\int\d x\, \langle\hat{\vartheta}(x_1)\hat{\vartheta}(x)\rangle
\nonumber\\
&&\phantom{\int \d x\, \langle\hat{\vartheta}^\dag(x_1)\hat{\vartheta}(x)\rangle\langle\hat{\vartheta}(x)}
\times\langle\hat{\vartheta}^\dag(x)\hat{\vartheta}(x_2)\rangle.
\label{fphicord} \end{aligned}$$ If the condensate depletion is small, one can neglect the second term in the r.h.s of Eq. (\[fphicord1\]), which is of the next order. Thus, we obtain the expression $$\label{thetaapp}
\langle\hat{\vartheta}^\dag(x_1)\hat{\vartheta}(x_2)\rangle
\simeq\int\d x\, \langle\hat{\vartheta}^\dag(x_1)\hat{\vartheta}^\dag(x)\rangle
\langle\hat{\vartheta}(x)\hat{\vartheta}(x_2)\rangle.$$ Note that Eq. (\[fphicord\]) turns into identity in the approximation (\[thetaapp\]), and the same is valid for Eqs. (\[fpfirel\_b\]) and (\[fpfirel1\_b\]).
[99]{}
E.P. Gross, Nuovo Cimento [**20**]{}, 454 (1961); J. Math. Phys. [**4**]{}, 195 (1963); L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. [**40**]{}, 646 (1961) \[Sov. Phys. JETP [**13**]{}, 451 (1961)\].
A.S. Parkins and D.F. Walls, Phys. Rep. [**303**]{}, 1 (1998).
F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. [**71**]{}, 463 (1999); L. Pitaevskii and S. Stringari, [*Bose-Einstein condensation*]{}, (Clarendon, Oxford, 2003).
A.J. Leggett, Rev. Mod. Phys. [**73**]{}, 307 (2001).
C.J. Pethick and H. Smith, [*Bose Einstein Condensation in Dilute Gases*]{}, (Cambridge University, Cambridge, 2002).
N.N. Bogoliubov, J. Phys. (USSR) [**11**]{}, 23 (1947), reprinted in D. Pines, ed., [*The many-body problem*]{} (W.A. Benjamin, New York, 1961).
T.D. Lee, K. Huang, C.N. Yang, Phys. Rev. [**106**]{}, 1135 (1957); K.A. Brueckner, K. Sawada, Phys. Rev. [**106**]{}, 1117 (1957).
E.B. Kolomeisky, T.J. Newman, J.P. Straley, and X. Qi, Phys. Rev. Lett. [**85**]{}, 1146 (2000).
M. Schick, Phys. Rev. A [**3**]{}, 1067 (1971).
V.N. Popov, Teor. Mat. Fiz. [**11**]{}, 354 (1971) \[Theor. Math. Phys. [**11**]{}, 565 (1972)\]; see Eq. (3.15) at zero temperature. The $T$-matrix at low energies should be rewritten in terms of the 2D scattering length.
E. B. Kolomeisky and J. B. Straley, Phys. Rev. B [**46**]{}, 11749 (1992).
A. A. Ovchinnikov, J. Phys.: Condens. Matter [**5**]{}, 8665 (1993).
A.Yu. Cherny and A.A. Shanenko, Phys. Rev. E [**64**]{}, 027105 (2001).
E.H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev. A [**61**]{}, 043602 (2000); Commun. Math. Phys. [**224**]{}, 17 (2001); Phys. Rev. Lett. [**91**]{}, 150401 (2003); E.H. Lieb and R. Seiringer, [*ibid.*]{} [**88**]{}, 170409 (2002). A.Yu. Cherny, cond-mat/9807120.
A.Yu. Cherny and A.A. Shanenko, Phys. Rev. E [**62**]{}, 1646 (2000); [*ibid.*]{} [**60**]{}, R5 (1999).
A.Yu. Cherny and A.A. Shanenko, Eur. Phys. J. B [**19**]{}, 555 (2001).
We call the potential weak-coupling if it is small and integrable; for more discussions, see Ref. [@cherny1].
L.D. Landau and E.M. Lifshitz, [*Quantum Mechanics*]{}, (Pergamon, Oxford, 1977).
S. Cowell, H. Heiselberg, I.E. Mazets, J. Morales, V.R. Pandharipande, and C.J. Pethick, Phys. Rev. Lett. [**88**]{}, 210403 (2002).
O. Sørensen, D.V. Fedorov, and A.S. Jensen, J. Phys. B [**37**]{}, 93 (2004).
P.C. Hohenberg and P.C. Martin, Ann. Phys. (N.Y.) [**34**]{}, 291 (1965).
N. M. Hugenholtz and D. Pines, Phys. Rev. [**116**]{}, 489 (1959).
A.J. Leggett, New J. Phys. [**5**]{}, 103 (2003).
A. Griffin, Phys. Rev. B [**53**]{}, 9341 (1996).
M. J. Bijlsma and H. T. C. Stoof, Phys. Rev. A [**55**]{}, 498 (1997).
D. Hutchinson, R. J. Dodd, and K. Burnett, Phys. Rev. Lett. [**81**]{}, 2198 (1998).
M. Olshanii and L. Pricoupenko, Phys. Rev. Lett. [**88**]{}, 010402 (2002).
K. Huang and C. N. Yang, Phys. Rev. [**105**]{}, 767 (1957).
A. Görlitz, J.M. Vogels, A.E. Leanhardt, C. Raman, T.L. Gustavson, J.R. Abo-Shaeer, A.P. Chikkatur, S. Gupta, S. Inouye, T.P. Rosenband, and W. Ketterle, Phys. Rev. Lett. [**87**]{}, 130402 (2001); F. Schreck [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 080403 (2001); M. Greiner [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 160405 (2001);
B.L. Tolra, K.M. O’Hara, J.H. Huckans, W.D. Phillips, S.L. Rolston, and J.V. Porto, Phys. Rev. Lett. [**92**]{}, 190401 (2004).
B. Paredes [*et al.*]{}, Nature [**429**]{}, 277 (2004).
H. Moritz, T. Stoferle, M. Kohl, and T. Esslinger, Phys. Rev. Lett. [**91**]{}, 250402 (2003).
In this paper we use the definition $\xi=\hslash/\sqrt{\mu m}$ for the coherence or healing length as in Refs. [@shlyap2D; @shlyap1D] whereas [@dalfovo] uses $\xi=\hslash/\sqrt{2 \mu m}$.
D.S. Petrov, M. Holzmann, and G.V. Shlyapnikov, Phys. Rev. Lett. [**84**]{}, 2551 (2000); D.S. Petrov and G.V. Shlyapnikov, Phys. Rev. A [**64**]{}, 012706 (2001).
U. Al Khawaja, J.O. Andersen, N.P. Proukakis, and H. T. C. Stoof, Phys. Rev. A [**66**]{}, 013615 (2002); J.O. Andersen, U. Al Khawaja, and H.T.C. Stoof, Phys. Rev. Lett. [**88**]{}, 070407 (2002).
M.D. Lee, S.A. Morgan, M.J. Davis, and K. Burnett, Phys. Rev. A [**65**]{}, 043617 (2002).
N. P. Prokakis, K. Burnett, and H. T. C. Stoof, Phys. Rev. A [**57**]{}, 1230 (1998).
M. Greiner [*et al.*]{}, Nature [**415**]{}, 39 (2002).
S. Jochim [*et al.*]{}, Science [**302**]{}, 2101 (2003).
M. W. Zwierlein [*et al.*]{}, Phys. Rev. Lett. [**91**]{}, 250401 (2003).
M. Greiner, C. Regal, and D. S. Jin, Nature [**426**]{}, 537 (2004).
S. Dettmer [*et al.*]{}, Phys. Rev. Lett. [**87**]{}, 160406 (2001).
D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven, Phys. Rev. Lett. [**85**]{}, 3745 (2000).
T.-L. Ho and M. Ma, J. Low Temp. Phys. [**115**]{}, 61 (1999).
J. Brand, J. Phys. B [**37**]{}, S287 (2004).
M. Olshanii, Phys. Rev. Lett. [**81**]{}, 938 (1998).
A.Yu. Cherny and A.A. Shanenko, Phys. Lett. A [**293**]{}, 287 (2002).
N.N. Bogoliubov, N.N. Bogoliubov, Jr., [*Introduction to Quantum Statistical Mechanics*]{}, (World Scientific, Singapore, 1982) p. 219.
N.N. Bogoliubov, Phys. Abhandl. SU [**6**]{}, 1; [**6**]{}, 113; [**6**]{}, 229 (1962); reprinted in N.N. Bogoliubov, [*Lectures on quantum statistics*]{}, Vol. 2 (Gordon and Breach, New York, 1970), p. 1.
For simplicity we use here the approach of broken global gauge symmetry. Approaches based on number conservation have been shown to be feasible \[see, e.g., Y. Castin and R. Dum, Phys. Rev. A [**57**]{}, 3008 (1998)\] and should yield the same results.
Note that the anomalous averages $\langle{\hat\Psi}(x_1) {\hat\Psi}(x_{2})\rangle$ describes the [*bound*]{} state of two fermions in a superfluid Fermi system. See Ref. [@bogquasi] and A.Yu. Cherny and A.A. Shanenko, Phys. Rev. B [**60**]{}, 1276 (1999).
J. Brand, I. Häring, and J.-M. Rost, Phys. Rev. Lett. [**91**]{}, 070403 (2003).
W.J. Mullin, J. Low Temp. Phys. [**106**]{}, 615 (1997).
U.R. Fischer, Phys. Rev. Lett. [**89**]{}, 280402 (2002).
A.Yu. Cherny, A.A. Shanenko, Phys. Lett. A [**250**]{}, 170 (1998).
E.H. Lieb and J. Yngvason, J. of Stat. Phys. [**103**]{}, 509 (2001).
Yu. Kagan, B.V. Svistunov, and G.V. Shlyapnikov, Sov. Phys. JETP [**66**]{}, 480 (1987).
P.C. Hohenberg, Phys. Rev. [**158**]{}, 383 (1967).
Due to criterion (\[restr1d\]) the expression for the chemical potential (\[mu1d1\]) is not valid formally at very low densities. However, we arrive at the same result if we calculate first the interaction energy directly from Eqs. (\[eint1\]), (\[phiapp1\]), and (\[cpar1\]), and then integrate over $\lambda$ using the relation (\[varth\]).
V.S. Popov, A.E. Kudryavtsev, and V.D. Mur, Zh. Eksp. Teor. Fiz. [**77**]{}, 1727 (1979) \[Sov. Phys. JETP [**50**]{}, 865 (1979)\].
S. Sinha and J. Brand, to be published (2004).
A. E. Muryshev [*et al.*]{}, Phys. Rev. Lett. [**89**]{}, 110401 (2002).
C. Bloch and C. De Dominicis, Nucl. Phys. [**7**]{}, 459 (1958).
N.N. Bogoliubov, Uspekhi Fiz. Nauk [**67**]{}, 549 (1959) \[Sov. Phys. Uspekhi [**2**]{}, 2 (1959)\].
The noncondensate fraction was referred to as “supracondensate” in Ref. [@cherny; @cherny1; @cherny2].
T. Köhler and K. Burnett, Phys. Rev. A [**65**]{}, 033601 (2002).
P. Naidon and F. Masnou-Seeuws, Phys. Rev. A [**68**]{}, 033612 (2003).
|
---
abstract: |
We study the covering radii of $2$-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.
*Key words:* covering radius; $2$-transitive permutation group; unitary group; Suzuki group; Ree group
*MSC2010:* 05C70, 05A05
address:
- |
Centre for Mathematics of Symmetry and Computation\
School of Mathematics and Statistics\
University of Western Australia\
35 Stirling Highway\
Crawley 6009, Australia.Email: [john.bamberg@uwa.edu.au]{}
- |
Centre for Mathematics of Symmetry and Computation\
School of Mathematics and Statistics\
University of Western Australia\
35 Stirling Highway\
Crawley 6009, Australia.Email: [cheryl.praeger@uwa.edu.au]{}
- |
School of Mathematics and Statistics\
The University of Melbourne\
Parkville, VIC 3010, Australia.Email: [binzhoux@unimelb.edu.au]{}
author:
- 'John Bamberg, Cheryl E. Praeger, and Binzhou Xia'
title: 'The covering radii of the $2$-transitive unitary, Suzuki, and Ree groups'
---
Introduction
============
Let $n\geqslant2$. Define the *Hamming distance* $d_n$ on the symmetric group ${\mathrm{S}}_n$ by letting $$d_n(g,h)=n-|{\mathrm{fix}}(gh^{-1})|$$ for any $g,h\in{\mathrm{S}}_n$, where ${\mathrm{fix}}(x)$ denotes the set of fixed points for $x\in{\mathrm{S}}_n$. Equipped with Hamming distance ${\mathrm{S}}_n$ is then a metric space. As usual, the distance of a point $v$ from a subset $C$ in ${\mathrm{S}}_n$ is $$d_n(v,C):=\min\{d_n(v,c)\mid c\in C\},$$ and the *covering radius* of $C$ is $${\mathrm{cr}}_n(C):=\max\{d_n(v,C)\mid v\in{\mathrm{S}}_n\}.$$ Covering radii of subgroups of ${\mathrm{S}}_n$ were first studied in a seminal paper of Cameron and Wanless [@CW2005], where an upper bound was given in terms of the transitivity:
*([@CW2005 Proposition 16])* If $G$ is a $t$-transitive permutation group of degree $n$, then ${\mathrm{cr}}_n(G)\leqslant n-t$.
In light of this result, the authors of [@CW2005] were then interested in the covering radius of $2$-transitive permutation groups. For example, they showed that ${\mathrm{cr}}_{q+1}({\mathrm{PSL}}(2,q))=q-\gcd(2,q)$ for all $q$, and determined ${\mathrm{cr}}_{q+1}({\mathrm{AGL}}(2,q))$ and ${\mathrm{cr}}_{q+1}({\mathrm{PGL}}(2,q))$ for $q\not\equiv1\pmod{6}$. For $q\equiv1\pmod{6}$, a determination of ${\mathrm{cr}}_{q+1}({\mathrm{AGL}}(2,q))$ and ${\mathrm{cr}}_{q+1}({\mathrm{PGL}}(2,q))$ was accomplished in [@WZ2013] and [@Xia2017], respectively. In determining the covering radii of these groups, the key was to find a large lower bound on the covering radius, which turns out to be ad hoc for each group. To shed light on the covering radii of $2$-transitive permutation groups, especially on establishing their lower bounds, we apply a general idea of “field automorphism" construction to obtain lower bounds of the other $2$-transitive permutation groups that are simple groups of (twisted) Lie rank one: ${\mathrm{PSU}}(3,q)$, ${\mathrm{Sz}}(q)$ and ${\mathrm{Ree}}(q)$. This leads to the following main results of this paper, whose proof is at the end of Sections \[sec1\], \[sec2\] and \[sec3\], respectively.
\[BoundPSU\] Let $q=p^f$ with prime $p$. Then $$q^3-p\leqslant{\mathrm{cr}}_{q^3+1}({\mathrm{PGU}}(3,q))\leqslant{\mathrm{cr}}_{q^3+1}({\mathrm{PSU}}(3,q))\leqslant q^3-\gcd(2,q).$$ In particular, if $q$ is even then $${\mathrm{cr}}_{q^3+1}({\mathrm{PGU}}(3,q))={\mathrm{cr}}_{q^3+1}({\mathrm{PSU}}(3,q))=q^3-2.$$ Moreover, for the field automorphism $h$ defined in , $d_{q^3+1}(h,{\mathrm{PGU}}(3,q))=q^3-p$.
\[BoundSz\] Let $q=2^{2m+1}$. Then $q^2-4\leqslant{\mathrm{cr}}_{q^2+1}({\mathrm{Sz}}(q))\leqslant q^2-2$. Moreover, for the field automorphism $h$ defined in , $d_{q^2+1}(h,{\mathrm{Sz}}(q))=q^2-4$.
\[BoundRee\] Let $q=3^{2m+1}$. Then $q^3-27\leqslant{\mathrm{cr}}_{q^3+1}({\mathrm{Ree}}(q))\leqslant q^3-1$. Moreover, for the field automorphism $h$ defined in , $d_{q^3+1}(h,{\mathrm{Ree}}(q))=q^3-27$.
Cameron and Wanless [@CW2005] mentioned a geometric interpretation for their work, namely a link between the covering radius of ${\mathrm{PGL}}(2, q)$ and the classical Minkowski planes. However they did not give many details. In Section \[sec4\] we explain this link in detail and then establish a similar link between the covering radius of ${\mathrm{PGU}}(3, q)$ and a certain finite geometry $\mathcal{U}_{2,2}$. We also pose the problem of characterising the classical ovoids of $\mathcal{U}_{2,2}$ via incidence geometry.
Unitary groups {#sec1}
==============
Let $q=p^f$ with $p$ prime, and $V$ be a $3$-dimensional vector space over ${\mathbb{F}}_{q^2}$ equipped with the unitary form $$\label{eq5}
x_1y_3^q+x_2y_2^q+x_3y_1^q$$ for any vectors $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ in $V$. Denote by $\Omega$ the set of totally isotropic one-dimensional subspaces of $V$ and $\varphi$ the quotient map of ${\mathrm{GU}}(V)$ modulo its center. Then $|\Omega|=q^3+1$ and ${\mathrm{GU}}(V)^\varphi={\mathrm{PGU}}(3,q)$ is a permutation group on $\Omega$. For any $\langle(x_1,x_2,x_3)\rangle\in V$, let $$\label{eq6}
\langle(x_1,x_2,x_3)\rangle^h=\langle(x_1^p,x_2^p,x_3^p)\rangle.$$ Then $h$ is a permutation of $V$, and we have ${\mathrm{PGU}}(3,q)\rtimes\langle h\rangle={\mathrm{P\Gamma U}}(3,q)$.
\[SystemPSU\] For each $\gamma\in{\mathbb{F}}_{q^2}^\times$ and $\delta\in{\mathbb{F}}_{q^2}$ such that $\delta^{q+1}=1$, the system of equations $$\label{eq3}
\begin{cases}
a^{p-1}=\gamma^{q+1}\\
b^p=b\gamma^q\delta\\
a+a^q+b^{q+1}=0
\end{cases}$$ has at most $p-1$ solutions $(a,b)\in{\mathbb{F}}_{q^2}\times{\mathbb{F}}_{q^2}$.
We divide it into three cases by the value of $\gamma^{(q^2-1)/(p-1)}$.
$\gamma^{(q^2-1)/(p-1)}=-1$. Let $(a,b)\in{\mathbb{F}}_{q^2}\times{\mathbb{F}}_{q^2}$ be a solution of . Then by the first line of we have $$\begin{aligned}
a^q+a&=&a\left((a^{p-1})^{(q-1)/(p-1)}+1\right)\\
&=&a\left((\gamma^{q+1})^{(q-1)/(p-1)}+1\right)=a\left(\gamma^{(q^2-1)/(p-1)}+1\right)=0,\end{aligned}$$ which together with the third line of yields $b=0$. Moreover, the first line of has at most $p-1$ solutions for $a$. Hence there are at most $p-1$ such pairs $(a,b)$.
$\gamma^{(q^2-1)/(p-1)}=1$ and $\gamma^{(q^2-1)/(p-1)}\neq-1$. Note that this condition indicates $p>2$. Let $(a,b)\in{\mathbb{F}}_{q^2}\times{\mathbb{F}}_{q^2}$ be a solution of . Then by the first line and third line of we have $a\neq0$ (since $\gamma\neq0$) and $$\begin{aligned}
\label{eq4}
2a=a\left(\gamma^{(q^2-1)/(p-1)}+1\right)&=&a\left((\gamma^{q+1})^{(q-1)/(p-1)}+1\right)\\
&=&a\left((a^{p-1})^{(q-1)/(p-1)}+1\right)=a^q+a=-b^{q+1}.\nonumber\end{aligned}$$ In particular, $b\neq0$. It then follows from the second line of that $b^{p-1}=\gamma^q\delta$, which has at most $p-1$ solutions for $b$. Thus, since $a$ is determined by $b$ as shows, we conclude that there are at most $p-1$ such $(a,b)$.
$\gamma^{(q^2-1)/(p-1)}\neq\pm1$. In this case, any solution $(a,b)\in{\mathbb{F}}_{q^2}\times{\mathbb{F}}_{q^2}$ of would satisfy (using the first and third lines of ) $$\begin{aligned}
b^{q+1}=-a(a^{q-1}+1)&=&-a\left((a^{p-1})^{(q-1)/(p-1)}+1\right)\\
&=&-a\left((\gamma^{q+1})^{(q-1)/(p-1)}+1\right)=-a\left(\gamma^{(q^2-1)/(p-1)}+1\right)\neq0\end{aligned}$$ and hence (using the second line of ) $$\begin{aligned}
1=b^{q^2-1}&=&(b^{p-1})^{(q^2-1)/(p-1)}\\
&=&(\gamma^q\delta)^{(q^2-1)/(p-1)}=\left(\gamma^{q^2+q}\delta^{q+1}\right)^{(q-1)/(p-1)}=(\gamma^{q+1})^{(q-1)/(p-1)},\end{aligned}$$ contrary to the condition $(\gamma^{q+1})^{(q-1)/(p-1)}=\gamma^{(q^2-1)/(p-1)}\neq1$. This completes the proof.
\[FixPSU\] For each $g\in{\mathrm{PGU}}(3,q)$ we have $|{\mathrm{fix}}(gh^{-1})|\leqslant p+1$, where $h$ is defined as in .
Suppose $g$ is an element of ${\mathrm{PGU}}(3,q)$ such that $gh^{-1}$ fixes at least $2$ points of $\Omega$, say $\langle u\rangle$ and $\langle v\rangle$. Note that $\langle e_1\rangle$ and $\langle e_3\rangle$ lie in $\Omega$ by , where $e_1=(1,0,0)$ and $e_3=(0,0,1)$. Since ${\mathrm{PGU}}(3,q)$ is $2$-transitive on $\Omega$, there exists $x\in{\mathrm{PGU}}(3,q)$ such that $\langle u\rangle^x=\langle e_1\rangle$ and $\langle v\rangle^x=\langle e_3\rangle$. Accordingly, $x^{-1}gh^{-1}x$ fixes $\langle e_1\rangle$ and $\langle e_3\rangle$.
Let $X=\langle{\mathrm{PGU}}(3,q),h\rangle={\mathrm{PGU}}(3,q)\rtimes\langle h\rangle={\mathrm{P\Gamma U}}(3,q)$ and $$Y=\left\{
\begin{pmatrix}
\gamma^{q+1}&&\\
&\gamma^q\delta&\\
&&1
\end{pmatrix}
\ \middle|\ \gamma,\delta\in{\mathbb{F}}_{q^2}^\times,\ \delta^{q+1}=1\right\}.$$ Observe that $Y$ preserves the form in . We have $Y^\varphi\leqslant{\mathrm{PGU}}(3,q)$ and $Y^\varphi\rtimes\langle h\rangle\leqslant X_{\langle e_1\rangle\langle e_3\rangle}$. Define a map $$\theta:(\gamma,\delta)\mapsto
\begin{pmatrix}
\gamma^{q+1}&&\\
&\gamma^q\delta&\\
&&1
\end{pmatrix}
^\varphi$$ from ${\mathbb{F}}_{q^2}^\times\times\{\delta\in{\mathbb{F}}_{q^2}^\times\mid\delta^{q+1}=1\}$ to $Y^\varphi$. Then $\theta$ is a group empimorphism, and $$\begin{aligned}
|\ker(\theta)|&=|\{\gamma,\delta\in{\mathbb{F}}_{q^2}^\times\mid\gamma^{q+1}=1,\delta=\gamma^{-q},\delta^{q+1}=1\}|\\
&=\{\gamma\in{\mathbb{F}}_{q^2}^\times\mid\gamma^{q+1}=1\}=q+1.\end{aligned}$$ Consequently, $|Y^\varphi|=(q^2-1)(q+1)/|\ker(\theta)|=q^2-1$. Since $X$ is $2$-transitive on $\Omega$, it follows that $$|X_{\langle e_1\rangle\langle e_3\rangle}|=\frac{|X|}{|\Omega|(|\Omega|-1)}=\frac{|{\mathrm{P\Gamma U}}(3,q)|}{(q^3+1)q^3}=2(q^2-1)f=|Y^\varphi\rtimes\langle h\rangle|.$$ This implies that $X_{\langle e_1\rangle\langle e_3\rangle}=Y^\varphi\rtimes\langle h\rangle$.
Now as $x^{-1}gh^{-1}x\in X_{\langle e_1\rangle\langle e_3\rangle}$, there exists an integer $i$ such that $$x^{-1}gh^{-1}x=yh^i$$ for some $y\in Y^\varphi$. Since ${\mathrm{P\Gamma U}}(3,q)={\mathrm{PGU}}(3,q)\rtimes\langle h\rangle$, we deduce that $h^i=h^{-1}$, taking both sides of the above equality modulo ${\mathrm{PGU}}(3,q)$. Hence $yh^{-1}=x^{-1}gh^{-1}x$ has the same number of fixed points as $gh^{-1}$. Write $$y=\begin{pmatrix}
\gamma^{q+1}&&\\
&\gamma^q\delta&\\
&&1
\end{pmatrix}
^\varphi$$ with $\gamma\in{\mathbb{F}}_{q^2}^\times$, $\delta\in{\mathbb{F}}_{q^2}^\times$ and $\delta^{q+1}=1$. To complete the proof, we show that $|{\mathrm{fix}}(yh^{-1})\setminus\{\langle e_1\rangle,\langle e_3\rangle\}|\leqslant p-1$.
For any fixed point $\alpha=\langle(\alpha_1,\alpha_2,\alpha_3)\rangle$ of $yh^{-1}$ other than $\langle e_1\rangle$ and $\langle e_3\rangle$, we have $$\label{eq1}
\alpha_1\alpha_3^q+\alpha_2^{q+1}+\alpha_3\alpha_1^q=0$$ by . If $\alpha_1=0$ or $\alpha_3=0$, then would imply $\alpha_2=0$, contrary to the assumption that $\alpha\neq\langle e_1\rangle$ or $\langle e_3\rangle$. Consequently, $\alpha=\langle(a,b,1)\rangle$ for some $(a,b)\in{\mathbb{F}}_{q^2}\times{\mathbb{F}}_{q^2}$ with $a\neq0$. It follows from that $$a+a^q+b^{q+1}=0,$$ and $$\label{eq2}
\langle(a\gamma^{q+1},b\gamma^q\delta,1)\rangle=\langle(a^p,b^p,1)\rangle$$ since $\alpha^y=\alpha^h$. Note that is equivalent to $$\begin{cases}
a^{p-1}=\gamma^{q+1}\\
b^p=b\gamma^q\delta\\
\end{cases}$$ as $a\neq0$. We then conclude by Lemma \[SystemPSU\] that there are at most $p-1$ such $\alpha$, completing the proof.
\[FixPGU\] $d_{q^3+1}(h,{\mathrm{PGU}}(3,q))=q^3-p$, where $h$ is defined as in .
By Lemma \[FixPSU\], $d_{q^3+1}(h,g)\geqslant q^3+1-|{\mathrm{fix}}(gh^{-1})|\geqslant q^3-p$ for each $g\in{\mathrm{PGU}}(3,q)$. Hence $d_{q^3+1}(h,{\mathrm{PGU}}(3,q))\geqslant q^3-p$. To complete the proof, we only need to show the existence of $g\in{\mathrm{PGU}}(3,q)$ such that $d_{q^3+1}(h,g)\leqslant q^3-p$, or equivalently, the existence of $g\in{\mathrm{PGU}}(3,q)$ such that $|{\mathrm{fix}}(hg^{-1})|\geqslant p+1$. Let $u=\langle(1,0,0)\rangle\in\Omega$ and $v=\langle(0,0,1)\rangle\in\Omega$. Note that any element of $\Omega$ other than $u$ and $v$ has form $\langle(a,b,1)\rangle$ for some $(a,b)\in{\mathbb{F}}_{q^2}\times{\mathbb{F}}_{q^2}$ with $a\neq0$ and $a+a^q+b^{q+1}=0$.
First assume that $p=2$. Then $u$, $v$ and $\langle(1,0,1)\rangle$ are all elements of $\Omega$ fixed by $h$. Hence $|{\mathrm{fix}}(h)|\geqslant3$, as desired.
Next assume that $p>2$. Let $\omega$ be a generator of ${\mathbb{F}}_{q^2}^\times$, and $$g=\begin{pmatrix}
\omega^{(q+1)(p-1)/2}&&\\
&\omega^{q(p-1)/2}&\\
&&1
\end{pmatrix}
^\varphi\in{\mathrm{PGU}}(3,q).$$ Then it is straightforward to verify that $u$, $v$ and $$\langle(\omega^{i(q^2-1)/(p-1)+(q+1)/2},0,1)\rangle$$ with $1\leqslant i\leqslant p-1$ are all elements of $\Omega$ fixed by $hg^{-1}$. Hence $|{\mathrm{fix}}(hg^{-1})|\geqslant p+1$, as desired. This completes the proof.
*Proof of Theorem $\ref{BoundPSU}$.* Since ${\mathrm{PSU}}(3,q)$ is a $2$-transitive subgroup of ${\mathrm{S}}_{q^3+1}$, we deduce from [@CW2005 Proposition 16] that ${\mathrm{cr}}_{q^3+1}({\mathrm{PSU}}(3,q))\leqslant q^3-1$. Moreover, equality cannot be attained if $q$ is even according to [@CW2005 Theorem 21]. Hence $$\label{eq11}
{\mathrm{cr}}_{q^3+1}({\mathrm{PSU}}(3,q))\leqslant q^3-\gcd(2,q).$$ Let $h$ be as defined in . Then $${\mathrm{cr}}({\mathrm{PSU}}(3,q))\geqslant{\mathrm{cr}}({\mathrm{PGU}}(3,q))\geqslant d_{q^3+1}(h,{\mathrm{PGU}}(3,q)).$$ This together with Lemma \[FixPGU\] and verifies Theorem \[BoundPSU\].
Suzuki groups {#sec2}
=============
Let $q=2^{2m+1}$, $\ell=2^{m+1}$, and $$\Omega=\{(a,b,c)\in{\mathbb{F}}_q^3\mid c=ab+a^{\ell+2}+b^\ell\}\cup\{\infty\}.$$ Clearly, $|\Omega|=q^2+1$. Let $$\label{eq7}
\infty^h=\infty\quad\text{and}\quad(a,b,c)^h=(a^2,b^2,c^2)\text{ for any }(a,b,c)\in{\mathbb{F}}_q^3.$$ Then $h$ is a permutation of $\Omega$, and $X:={\mathrm{Sz}}(q)\rtimes\langle h\rangle$ is a permutation group on $\Omega$. We shall prove that $d_{q^2+1}(h,{\mathrm{Sz}}(q))\geqslant q^2-4$, which gives a lower bound for the covering radius of ${\mathrm{Sz}}(q)$.
\[SystemSz\] For each $\kappa\in{\mathbb{F}}_q^\times$, the system of equations $$\label{eq9}
\begin{cases}
\kappa a=a^2\\
\kappa^{\ell+1}b=b^2\\
\kappa^{\ell+2}c=c^2\\
c=ab+a^{\ell+2}+b^\ell
\end{cases}$$ has exactly four solutions $(a,b,c)\in{\mathbb{F}}_q^3$.
From the first three lines of we deduce that $a=0$ or $\kappa$, $b=0$ or $\kappa^{\ell+1}$ and $c=0$ or $\kappa^{\ell+2}$. This gives eight candidates for $(a,b,c)$. Verifying the fourth line of for these eight candidates we conclude that $$(0,0,0),\quad(0,\kappa^{\ell+1},\kappa^{\ell+2}),\quad(\kappa,0,\kappa^{\ell+2}),\quad(\kappa,\kappa^{\ell+1},\kappa^{\ell+2})$$ are all solutions of and are the only ones.
\[FixSz\] $d_{q^2+1}(h,{\mathrm{Sz}}(q))=q^2-4$, where $h$ is defined as in .
Suppose $g$ is an element of ${\mathrm{Sz}}(q)$ such that $gh^{-1}$ fixes at least two points of $\Omega$, say $u$ and $v$. Let $o=(0,0,0)\in\Omega$. Since ${\mathrm{Sz}}(q)$ is $2$-transitive on $\Omega$, there exists $x\in{\mathrm{Sz}}(q)$ such that $u^x=\infty$ and $v^x=o$. Accordingly, $x^{-1}gh^{-1}x$ fixes $\infty$ and $o$.
For any $\kappa\in{\mathbb{F}}_q^\times$ denote $y_\kappa$ the permutation of $\Omega$ defined by $$\infty^{y_\kappa}=\infty\quad\text{and}\quad(a,b,c)^{y_\kappa}=(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c),$$ and let $Y=\{y_\kappa\mid\kappa\in{\mathbb{F}}_q^\times\}$. Then $Y$ form a subgroup of $X_{\infty o}$ [@DM1996 Page 250]. Since $X$ is $2$-transitive on $\Omega$, it follows that $$|X_{\infty o}|=\frac{|X|}{|\Omega|(|\Omega|-1)}=\frac{|{\mathrm{Sz}}(q)|(2m+1)}{(q^2+1)q^2}=(q-1)(2m+1)=|Y\rtimes\langle h\rangle|.$$ This implies that $X_{\infty o}=Y\rtimes\langle h\rangle$.
Now as $x^{-1}gh^{-1}x\in X_{\infty o}$, there exist an integer $i$ and $\kappa\in{\mathbb{F}}_q^\times$ such that $$x^{-1}gh^{-1}x=y_\kappa h^i.$$ Since $X={\mathrm{Sz}}(q)\rtimes\langle h\rangle$, we deduce that $h^i=h^{-1}$, taking both sides of the above equality modulo ${\mathrm{Sz}}(q)$. Hence $y_\kappa h^{-1}=x^{-1}gh^{-1}x$ has the same number of fixed points as $gh^{-1}$. For any fixed point $(a,b,c)$ of $y_\kappa h^{-1}$ other than $\infty$, we have $$c=ab+a^{\ell+2}+b^\ell$$ since $(a,b,c)\in\Omega$, and $$\begin{cases}
\kappa a=a^2\\
\kappa^{\ell+1}b=b^2\\
\kappa^{\ell+2}c=c^2
\end{cases}$$ since $\alpha^{y_\kappa}=\alpha^h$. It then follows from Lemma \[SystemSz\] that $|{\mathrm{fix}}(y_\kappa h^{-1})\setminus\{\infty\}|\leqslant4$, and so $|{\mathrm{fix}}(gh^{-1})|=|{\mathrm{fix}}(y_\kappa h^{-1})|\leqslant5$. This implies that $|{\mathrm{fix}}(gh^{-1})|\leqslant5$ for any $g\in{\mathrm{Sz}}(q)$, which yields $$d_{q^2+1}(h,{\mathrm{Sz}}(q))\geqslant(q^2+1)-5=q^2-4.$$ Moreover, Lemma \[SystemSz\] implies that $$d_{q^2+1}(h,y_\kappa)=q^2+1-|{\mathrm{fix}}(y_\kappa h^{-1})|=q^2+1-(1+4)=q^2-4$$ for any $\kappa\in{\mathbb{F}}_q^\times$. We then conclude that $d_{q^2+1}(h,{\mathrm{Sz}}(q))=q^2-4$, as the lemma asserts.
*Proof of Theorem $\ref{BoundSz}$.* Since ${\mathrm{Sz}}(q)$ is a $2$-transitive subgroup of ${\mathrm{S}}_{q^2+1}$, we deduce from [@CW2005 Proposition 16] that ${\mathrm{cr}}_{q^2+1}({\mathrm{Sz}}(q))\leqslant q^2-1$. Moreover, the equality cannot be attained according to [@CW2005 Theorem 21]. Hence $${\mathrm{cr}}_{q^2+1}({\mathrm{Sz}}(q))\leqslant q^2-2.$$ This together with Lemma \[FixSz\] leads to Theorem \[BoundSz\].
Ree groups {#sec3}
==========
Let $q=3^{2m+1}$, $\ell=3^{m+1}$, and $$\Omega=\{(a,b,c,\lambda_1(a,b,c),\lambda_2(a,b,c),\lambda_3(a,b,c))\mid(a,b,c)\in{\mathbb{F}}_q^3\}\cup\{\infty\},$$ where $$\begin{aligned}
\lambda_1(a,b,c)&=a^2b-ac+b^\ell-a^{\ell+3},\\
\lambda_2(a,b,c)&=a^\ell b^\ell-c^\ell+ab^2+bc-a^{2\ell+3},\\
\lambda_3(a,b,c)&=ac^\ell-a^{\ell+1}b^\ell+a^{\ell+3}b+a^2b^2-b^{\ell+1}-c^2+a^{2\ell+4}.\end{aligned}$$ Clearly, $|\Omega|=q^3+1$. Let $\infty^h=\infty$ and $$\begin{aligned}
\label{eq10}
&(a,b,c,\lambda_1(a,b,c),\lambda_2(a,b,c),\lambda_3(a,b,c))^h\\
=&(a^3,b^3,c^3,\lambda_1(a^3,b^3,c^3),\lambda_2(a^3,b^3,c^3),\lambda_3(a^3,b^3,c^3))\nonumber\end{aligned}$$ for any $(a,b,c)\in{\mathbb{F}}_q^3$. Then $h$ is a permutation of $\Omega$, and $X:={\mathrm{Ree}}(q)\rtimes\langle h\rangle$ is a permutation group on $\Omega$. We shall prove that $d_{q^3+1}(h,{\mathrm{Ree}}(q))\geqslant q^3-27$, which gives a lower bound for the covering radius of ${\mathrm{Ree}}(q)$.
\[FixRee\] $d_{q^3+1}(h,{\mathrm{Ree}}(q))=q^3-27$, where $h$ is defined as in .
Suppose $g$ is an element of ${\mathrm{Ree}}(q)$ such that $gh^{-1}$ fixes at least two points of $\Omega$, say $u$ and $v$. Let $o=(0,0,0)\in\Omega$. Since ${\mathrm{Ree}}(q)$ is $2$-transitive on $\Omega$, there exists $x\in{\mathrm{Ree}}(q)$ such that $u^x=\infty$ and $v^x=o$. Accordingly, $x^{-1}gh^{-1}x$ fixes $\infty$ and $o$.
For any $\kappa\in{\mathbb{F}}_q^\times$ denote by $y_\kappa$ the permutation of $\Omega$ defined by $\infty^{y_\kappa}=\infty$ and $$\begin{aligned}
&(a,b,c,\lambda_1(a,b,c),\lambda_2(a,b,c),\lambda_3(a,b,c))^{y_\kappa}\\
=&(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c,\lambda_1(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c),\lambda_2(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c),\lambda_3(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c)),\end{aligned}$$ and let $Y=\{y_\kappa\mid\kappa\in{\mathbb{F}}_q^\times\}$. Then $Y$ forms a subgroup of $X_{\infty o}$ [@DM1996 Page 251]. Since $X$ is $2$-transitive on $\Omega$, it follows that $$|X_{\infty o}|=\frac{|X|}{|\Omega|(|\Omega|-1)}=\frac{|{\mathrm{Ree}}(q)|(2m+1)}{(q^3+1)q^3}=(q-1)(2m+1)=|Y\rtimes\langle h\rangle|.$$ This implies that $X_{\infty o}=Y\rtimes\langle h\rangle$.
Now as $x^{-1}gh^{-1}x\in X_{\infty o}$, there exist an integer $i$ and $\kappa\in{\mathbb{F}}_q^\times$ such that $$x^{-1}gh^{-1}x=y_\kappa h^i.$$ Since $X={\mathrm{Ree}}(q)\rtimes\langle h\rangle$, we deduce that $h^i=h^{-1}$, taking both sides of the above equality modulo ${\mathrm{Ree}}(q)$. Hence $y_\kappa h^{-1}=x^{-1}gh^{-1}x$ has the same number of fixed points as $gh^{-1}$. For any fixed point $(a,b,c,\lambda_1(a,b,c),\lambda_2(a,b,c),\lambda_3(a,b,c))$ of $y_\kappa h^{-1}$ other than $\infty$, we have $$\label{eq8}
\begin{cases}
\kappa a=a^3\\
\kappa^{\ell+1}b=b^3\\
\kappa^{\ell+2}c=c^3
\end{cases}$$ since $\alpha^{y_\kappa}=\alpha^h$. As each line of has most three solutions, has at most $27$ solutions $(a,b,c)\in{\mathbb{F}}_q^3$. It follows that $|{\mathrm{fix}}(y_\kappa h^{-1})\setminus\{\infty\}|\leqslant27$, and so $|{\mathrm{fix}}(gh^{-1})|=|{\mathrm{fix}}(y_\kappa h^{-1})|\leqslant28$. This implies that $|{\mathrm{fix}}(gh^{-1})|\leqslant28$ for any $g\in{\mathrm{Ree}}(q)$, which yields $$d_{q^3+1}(h,{\mathrm{Ree}}(q))\geqslant(q^3+1)-28=q^3-27.$$ Moreover, has exactly $27$ solutions $(a,b,c)\in{\mathbb{F}}_q^3$ when $\kappa=1$. Hence $y_1h^{-1}$ fixes exactly $28$ points. This implies that $${\mathrm{cr}}_{q^3+1}(h,y_1)=(q^3+1)-28=q^3-27.$$ We then conclude that $d_{q^3+1}(h,{\mathrm{Ree}}(q))=q^3-27$, as the lemma asserts.
*Proof of Theorem $\ref{BoundRee}$.* Since ${\mathrm{Ree}}(q)$ is a $2$-transitive subgroup of ${\mathrm{S}}_{q^3+1}$, we deduce from [@CW2005 Proposition 16] that $${\mathrm{cr}}_{q^3+1}({\mathrm{Ree}}(q))\leqslant q^3-1.$$ This together with Lemma \[FixRee\] leads to Theorem \[BoundRee\].
Geometric interpretations for the covering radii of ${\mathrm{PGL}}(2,q)$ and ${\mathrm{PGU}}(3,q)$ {#sec4}
===================================================================================================
Let us first describe in detail how the covering radius problem for ${\mathrm{PGL}}(2,q)$ relates to certain configurations in an associated *Minkowski plane* that was alluded to by Cameron and Wanless [@CW2005]. Consider the projective line ${\mathrm{PG}}(1,q)$ for which ${\mathrm{PGL}}(2,q)$ has its natural sharply $3$-transitive action of degree $q+1$. Consider the Segre variety $\mathcal{S}_{m,n}(q)$ lying in ${\mathrm{PG}}((m+1)(n+1)-1,q)$ obtained by taking the simple tensors of pairs of homogeneous coordinates of points of ${\mathrm{PG}}(m,q)$ and ${\mathrm{PG}}(n,q)$. So for example, $\mathcal{S}_{1,1}(q)$ is the *hyperbolic quadric* $\mathsf{Q}^+(3,q)$ in projective $3$-space, and $\mathcal{S}_{1,2}(q)$ is the *Segre threefold* in ${\mathrm{PG}}(5,q)$. We refer to [@HT2016 Chapter 4] for more on finite Segre varieties and their properties. Consider $\mathcal{S}_{1,1}(q)$. The *Segre map* in this instance is given by: $$\begin{aligned}
{\mathrm{PG}}(1,q)\times{\mathrm{PG}}(1,q)&\to \mathcal{S}_{1,1}\\
\left((X_1,X_2),(X_1',X_2')\right)&\mapsto (X_1,X_2)\otimes (X_1',X_2')=\begin{bmatrix} X_1X_1' & X_1X_2'\\ X_2X_1'& X_2X_2' \end{bmatrix}.\end{aligned}$$ We can identify the output with a row vector $(X_1X_1',X_1X_2',X_2X_1',X_2X_2')$; the homogeneous coordinates for a point of ${\mathrm{PG}}(3,q)$. Moreover, each element of $\mathcal{S}_{1,1,}$ is a zero of the following quadratic form on ${\mathbb{F}}_q^4$: $$Q(X)=X_1X_4-X_2X_3, \quad\text{for }X = (X_1,X_2,X_3,X_4)\in {\mathbb{F}}_q^4.$$ The quadratic form $Q$ gives rise to a hyperbolic quadric $\mathsf{Q}^+(3,q)$ whose points are precisely those of $\mathcal{S}_{1,1}$. Consider a permutation $\pi$ in ${\mathrm{Sym}}({\mathrm{PG}}(1,q))$. Then we can identify $\pi$ with a set of ordered pairs in ${\mathrm{PG}}(1,q)\times{\mathrm{PG}}(1,q)$: $$\pi\leftrightarrow \{ (X, X^\pi) \colon X \in {\mathrm{PG}}(1,q)\}.$$ We can then apply the Segre map above to obtain $$\pi \rightarrow {\mathsf{Graph}}(\pi):=\{ X \otimes X^\pi \colon X\in {\mathrm{PG}}(1,q)\}.$$ Geometrically, ${\mathsf{Graph}}(\pi)$ is an *ovoid* of $\mathcal{S}_{1,1}$ because two points $X\otimes X^\pi$ and $Y\otimes Y^\pi$ are orthogonal under the associated bilinear form[^1] if and only if $X=Y$. In other words, ${\mathsf{Graph}}(\pi)$ is a set of $q+1$ points, no two orthogonal in $\mathcal{S}_{1,1}$. Now we can easily distinguish the permutations that lie in ${\mathrm{PGL}}(2,q)$. Recall that elements of ${\mathrm{PGL}}(2,q)$ are Möbius transformations of the form $$\mu_{a,b,c,d}:=t\mapsto \frac{a t+b}{ct +d},\quad a,b,c,d\in{\mathbb{F}}_q, ad-bc\ne 0.$$ So $$\begin{aligned}
{\mathsf{Graph}}(\mu_{a,b,c,d})&=\{ (ct+d,at+b,ct^2+dt,at^2+bt) \colon t\in {\mathbb{F}}_q\cup \{\infty\} \},\end{aligned}$$ which is a conic section[^2] of $\mathsf{Q}^+(3,q)$. The converse is also true. So in the bijection above, we have that ${\mathrm{PGL}}(2,q)$ is distinguished amongst all permutations by taking conics amongst all ovoids of $\mathcal{S}_{1,1}$. Therefore, we have $${\mathrm{cr}}_{q+1}({\mathrm{PGL}}(2,q))=\max_{ \mathcal{O}\in\text{ovoids of }\mathsf{Q}^+(3,q)}\left(
\min_ {\mathcal{C}\in\text{conics of }\mathsf{Q}^+(3,q)} |\mathcal{O}\cap \mathcal{C}|\right).$$ Alternatively, we could use the language of *Benz planes*. The *classical Minkowski plane* $\mathcal{M}(q)$ consists of three types of objects: *points*, *lines*, and *circles*. The points of $\mathcal{M}(q)$ are just the points of $\mathsf{Q}^+(3,q)$, the lines are the generators of $\mathsf{Q}^+(3,q)$, and the circles are the non-tangent plane sections (i.e., conics) of $\mathsf{Q}^+(3,q)$. Incidence is inherited from the ambient projective space. So to compute ${\mathrm{cr}}_{q+1}({\mathrm{PGL}}(2,q))$, we need to consider sets of points of $\mathcal{M}(q)$ that meet every line once and every circle in at most $s$ points, and we seek to minimise $s$.
We will consider a special set of points lying in the Segre variety $\mathcal{S}_{2,2}(q^2)\subseteq {\mathrm{PG}}(8,q^2)$. Suppose we have an Hermitian form $\beta$ from ${\mathbb{F}}_{q^2}^3$ to ${\mathbb{F}}_{q^2}$. Define $\beta\otimes \beta$ to be the form on ${\mathbb{F}}_{q^2}^9$ defined by $$(\beta\otimes \beta)(u_1\otimes v_1,u_2\otimes v_2) := \beta(u_1,u_2)\beta(v_1,v_2)$$ and extend linearly. This form is Hermitian. The set of totally isotropic points of $\beta\otimes\beta$ is a Hermitian variety which we will denote by ${\mathsf{H}}(8,q^2)$.
Now consider the totally isotropic points of $\beta$; an *Hermitian curve* ${\mathsf{H}}(2,q^2)$ of $q^3+1$ points of ${\mathrm{PG}}(2,q^2)$. This variety gives us the natural $2$-transitive action of ${\mathrm{PGU}}(3,q)$. This time, we apply the Segre map to pairs of elements of the Hermitian curve: $$\begin{aligned}
{\mathsf{H}}(2,q^2)\times{\mathsf{H}}(2,q^2)&\to \mathcal{S}_{2,2}\\
\left(u,v\right)&\mapsto u\otimes v.\end{aligned}$$ This map is injective, but not surjective. There is no natural name for its image, but we will call it $\mathcal{U}_{2,2}$.
Let $(P_1,P_2)\in {\mathsf{H}}(2,q^2)\times{\mathsf{H}}(2,q^2)$. Then the set of points orthogonal with or equal to $P_1\otimes P_2$ (in $\mathcal{U}_{2,2}$) is $$\{ P_1\otimes X \colon X\in {\mathsf{H}}(2,q^2) \} \cup \{ X\otimes P_2 \colon X\in {\mathsf{H}}(2,q^2) \}.$$
Suppose $X\otimes Y$ is orthogonal to $P_1\otimes P_2$ with respect to $\beta\otimes \beta$. Then $$\begin{aligned}
\beta(X,P_1)\beta(Y,P_2)&=(\beta\otimes \beta)(X\otimes Y, P_1\otimes P_2)=0.\end{aligned}$$ However, no two distinct points of the Hermitian curve are orthogonal, and so we must have either $X=P_1$ or $Y=P_2$.
Now consider a permutation $f\in {\mathrm{Sym}}({\mathsf{H}}(2,q^2))$. Let ${\mathsf{Graph}}(f)$ be $$\{ X\otimes X^f \colon X \in {\mathsf{H}}(2,q^2) \}$$ as a subset of $\mathcal{U}_{2,2}$. Then:
For each $f\in {\mathrm{Sym}}({\mathsf{H}}(2,q^2))$, the set ${\mathsf{Graph}}(f)$ is a set of $q^3+1$ totally isotropic points of ${\mathsf{H}}(8,q^2)$, no two distinct elements orthogonal in ${\mathsf{H}}(8,q^2)$.
Suppose two elements $P\otimes P^f$ and $Q\otimes Q^f$ are orthogonal with respect to the form $\beta\otimes \beta$. Then $$\beta(P,Q)\beta(P^f,Q^f)=(\beta\otimes \beta)(P\otimes P^f, Q\otimes Q^f)=0.$$ and hence $P=Q$.
The converse is also true.
Let $\{ P_i \otimes Q_i \colon i=0,\ldots, q^3\}$ be a set of points of $\mathcal{U}_{2,2}$, no two distinct elements orthogonal. Then the map $P_i \to Q_i$ is a bijection.
Suppose $Q_i=Q_j$ for some $i,j \in \{0,\ldots, q^3\}$. Then $$(\beta\otimes\beta)( P_i\otimes Q_i, P_j\otimes Q_j)=\beta(P_i,P_j) \beta(Q_i,Q_j)=0$$ and hence $i=j$. Therefore, the map $P_i\to Q_i$ is a bijection.
So we define an *ovoid* of $\mathcal{U}_{2,2}$ to be a set of $q^3+1$ elements, that are pairwise non-orthogonal. For example, the Frobenius automorphism $h$ that we used in the proof of Theorem \[BoundPSU\] gives rise to the ovoid $\{ P\otimes P^h\colon P\in{\mathsf{H}}(2,q^2) \}$ of $\mathcal{U}_{2,2}$.
For each $f\in {\mathrm{PGU}}(3,q)$, the set ${\mathsf{Graph}}(f)$ spans a projective $5$-space of ${\mathrm{PG}}(8,q^2)$, and its perp with respect to ${\mathsf{H}}(8,q^2)$ is a plane $\pi_f$ which does not intersect $\mathcal{S}_{2,2}$. Moreover, $\pi_f$ is totally isotropic when $q$ is even, and non-degenerate when $q$ is odd.
The diagonal $D:=\{ P\otimes P\colon P\in {\mathrm{PG}}(2,q^2)\}$ pertains to the projective points arising from the symmetric tensors of the vector space $V:={\mathbb{F}}_{q^2}^3$. The dimension of the symmetric square of ${\mathbb{F}}_{q^2}^3$ is ${4\choose 2}=6$, and therefore, $D$ spans a projective $5$-space. Moreover, as the Hermitian curve spans ${\mathrm{PG}}(2,q^2)$, we also know that ${\mathsf{Graph}}(1)$ spans the same $5$-space, where $1$ denotes the trivial element of ${\mathrm{PGU}}(3,q)$. Now ${\mathrm{PGU}}(3,q)$ acts on $\{{\mathsf{Graph}}(f)\colon f\in{\mathrm{PGU}}(3,q)\}$ in the following way: $${\mathsf{Graph}}(f)^g={\mathsf{Graph}}(fg).$$ Note that since ${\mathrm{PGU}}(3,q)$ acts transitively on itself by right multiplication, we see that ${\mathrm{PGU}}(3,q)$ also acts transitively on $\{{\mathsf{Graph}}(f)\colon f\in{\mathrm{PGU}}(3,q)\}$. So every ${\mathsf{Graph}}(f)$ will span a space projectively equivalent to the space spanned by ${\mathsf{Graph}}(1)$.
Now let us consider the perp $\pi$ of ${\mathsf{Graph}}(1)$. Take the alternating tensors $A$ and construct projective points: $$A=\langle X\otimes Y - Y\otimes X\colon X,Y\in {\mathrm{PG}}(2,q^2) \rangle.$$ Then for all $P,X,Y\in {\mathrm{PG}}(2,q^2)$ we have $$\begin{aligned}
(\beta\otimes \beta)( X\otimes Y - Y\otimes X, P\otimes P)&=(\beta\otimes \beta)( X\otimes Y, P\otimes P) -(\beta\otimes \beta)(Y\otimes X, P\otimes P) \\
&=\beta(X,P)\beta(Y,P)-\beta(Y,P)\beta(X,P)\\
&=0.\end{aligned}$$ Therefore, $A\subseteq \pi$. However, we know that $A$ has (algebraic) dimension ${3\choose 2}=3$ and hence $A=\pi$. If $q$ is even, then $A$ is a subspace of $D$, and hence totally isotropic. Otherwise, $A\cap D$ is trivial and $A$ is non-degenerate.
In analogy with the ${\mathrm{PGL}}(2,q)$ example from before, we will call ${\mathsf{Graph}}(f)$, where $f\in {\mathrm{PGL}}(3,q)$, a *classical ovoid* of $\mathcal{U}_{2,2}$. So in the injection above, we have that ${\mathrm{PGU}}(3,q)$ is distinguished amongst all permutations of ${\mathsf{H}}(2,q^2)$ by taking classical ovoids amongst all ovoids of $\mathcal{U}_{2,2}$.
Find an incidence geometry characterisation of classical ovoids of $\mathcal{U}_{2,2}$.
We would then have a meaningful way to determine the following: $${\mathrm{cr}}_{q^3+1}({\mathrm{PGU}}(3,q))=\max_{ \mathcal{O}\in\text{ovoids of }\mathcal{U}_{2,2}}\left(
\min_{\mathcal{C}\in\text{classical ovoids of }\mathcal{U}_{2,2}} |\mathcal{O}\cap \mathcal{C}|\right).$$
Acknowledgments {#acknowledgments .unnumbered}
===============
The third author’s work on this paper was done when he was a research associate at the University of Western Australia supported by the Australian Research Council Discovery Project DP150101066. This work was inspired by the 2017 Centre for the Mathematics of Symmetry and Computation Research Retreat.
P. J. Cameron and I. M. Wanless, Covering radius for sets of permutations, *Discrete Math.*, 293 (2005), no. 1-3, 91–109.
J. D. Dixon and B. Mortimer, *Permutation groups*, Springer-Verlag, New York, 1996.
J. W. P. Hirschfeld and J. A. Thas, *General Galois geometries*, Springer Monographs in Mathematics, London, 2016.
I. M. Wanless and X. Zhang, Transversals of Latin squares and covering radius of sets of permutations, *European J. Combin.*, 34 (2013), 1130–1143
B. Xia, The covering radius of ${\mathrm{PGL}}_2(q)$, *Discrete Math.*, 340 (2017), no. 10, 2469–2471.
[^1]: In terms of tensors, the bilinear form can be written as $B(u_1\otimes u_2, v_1\otimes v_2)=
\beta(u_1,v_1)\beta(u_2,v_2)$, where $\beta$ is defined by $\beta(x,y)=x_1y_2-x_2y_1$, and then extend $B$ linearly. So $0=B(X\otimes X^\pi, Y\otimes Y^\pi)=\beta(X,Y)\beta(X^\pi,Y^\pi)$ implies $X=Y$ or $X^\pi=Y^\pi$, which are equivalent as $\pi$ is a bijection.
[^2]: In fact, the (hyper)plane of intersection has dual homogeneous coodinates $[b, -d, a, -c]$ and it is not difficult to calculate that this plane is non-degenerate with respect to the form $Q$.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.