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Characterizations of Forman curvature - Jürgen Jost, Florentin Münch
基本定义
胞腔复形与图
给定胞腔复形$X=\bigcup X_k.$
取$\delta:X_k\rightarrow X_{k+1}$为关联(incident)算子,
$\delta v(z)=\pm 1$或$0.$ 要求胞腔复形满足如下性质:
$\dim(z)\ge 1\Rightarrow \,\exists\,x\prec z;$
$|\{w\in X_0:\delta w(x)=1\}|=|\{w\in X_0:\delta w(x)=-1\}|=1,$
$\,\forall\,x\in X_1;$ (每边两顶点)
$\dim(z)-\dim(v)=2,$ $\{x:v\prec x\prec z\}\neq \varnothing$
$\Rightarrow$ $|\{x:v\prec x\prec z\}|=2$ 且
$\pm 1\in \{\delta v(x)\delta x(z):x\in X\}.$ (diamond性&
$\delta^2=0$)
(定向)连通性;
$\{x:x\prec z\}=\{x:x\prec z'\},$ $\dim z\ge 1\Rightarrow z=z'.$
(简单图)
$\,\forall\,x,y\in X,$
记$x\sim y$若$\,\exists\,v\prec x,y$或$x,y\prec z.$ 这诱导了距离:
$$d(x,y)=\inf\{n:x=x_0\sim\cdots\sim x_n=y\}.$$
注意在$X_0$上这并不是一般的路径距离. 在复形维数较高时,
点到点可以凭借高维胞腔更快速地转移.
定义带权内积 $$\left<{}f,g\right>=\sum_x f(x)g(x)m(x).$$
将$x$视为$1_x,$ 线性扩张$\delta:C(X)\rightarrow C(X).$
定义伴随$\delta^\ast :C(X)\rightarrow C(X),$
$$\delta^\ast z(x)=\frac{1}{m(x)}\left<{}\delta^\ast z,x\right>=\frac{1}{m(x)}\left<{}\delta x,z\right>=\frac{m(z)}{m(x)}\delta x(z).$$
这与Forman定义中的$\partial$有区别,
$$\partial z(x)=\delta x(z)=\frac{m(x)}{m(z)}\delta^\ast z(x).$$
Hodge Laplacian
定义Hodge Laplacian $$H=\delta \delta^\ast +\delta^\ast \delta.$$
计算$Hy(x),$ 分别计算如下式子:
$$\delta\delta^\ast y(x)=\frac{1}{m(x)}\left<{}\delta^\ast y,\delta^\ast x\right>=\frac{1}{m(x)}\sum_{v\prec x,y}m(v)\frac{m(y)}{m(v)}\delta v(y)\frac{m(x)}{m(v)}\delta v(x)=\sum_{v\prec x,y}\frac{m(y)}{m(v)}\delta v(x)\delta v(y),$$
更简单的有:
$$\delta^\ast \delta y(x)=\sum_{x,y\prec z}\frac{m(z)}{m(x)}\delta x(z)\delta y(z).$$
注意与Forman定义不同, 维数较小的胞腔权重总是在分母上.
当$X=X_0\cup X_1$为赋权定向图时, 记$m(x)=m(v,w)=m(w,v),$ $x\in X_1,$
$v,w$为两顶点. $H$作用在$C(X_0)$上为:
$$\begin{aligned}
Hf(v)=\delta^*\delta f(v)&=\sum_w\sum_{v,w\prec x}\frac{m(x)}{m(v)}\delta v(x)\delta w(x)f(w)\\
&=\sum_{v\prec x}\frac{m(x)}{m(v)}\sum_{w\prec x}\delta v(x)\delta w(x)f(w)\\
&=\sum_{w\neq v}\frac{m(v,w)}{m(v)}(f(v)-f(w))
\end{aligned}$$
即$H_0$就是带权图的Laplacian.
对$x\neq y\in X_1,$ 第一个和中至多有一项非零,
第二个和中非零项符号均一致. 当两个和中均出现非零项时,
有$\delta v(x)\delta v(y)=-\delta x(z)\delta y(z).$ 因此
$$|Hy(x)|=\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|.$$
容易得到
$$Hx(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}.$$
对(任意)自伴算子$H,$ 有分解 $$H=\Delta+D,$$ $\Delta$为极小对角占优算子,
$D$为对角阵. 显然有 $$Dx(x)=Hx(x)-\sum_{y\neq x}|Hy(x)|.$$
对Hodge Laplacian做分解, 将得到的$D$称为Forman曲率. 对$x\in X_1,$
$$F(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}-\sum_{x\neq y}\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|.$$
若$m\equiv 1,$
$$F(x)=\#\{v\in X_0:v\prec x\}+\#\{z\in X_2:x\prec z\}-\#\{\text{parallel neighbors of } x\},$$
与Forman曲率确实是一样的.
但事实上得到$F$的过程与Forman曲率并不完全一致, 对一般的$m$二者并不相同.
Forman在分解Hodge Laplacian前, 先对$H$做了对称化, 即在常权重下是对称阵.
不过根据步骤的不同, 可以转化两种不同的Forman曲率.
对偶关系
对复形$K=(X,\delta,m),$ $K'=(X',\delta,m'),$ 记$K'\le_k K,$
若满足如下条件:
$X_j=X_j',$ $\,\forall\,j\le k;$
$X_j'\subset X_j,$ $\,\forall\,j>k;$
$m(x)=m'(x),$ $\,\forall\,\dim x\le k;$
$\delta'v(x)=\delta v(x),$ $\,\forall\,x,v\in X'.$
也就是$K'\subset K$且它们的带权定向$k$-骨架相同.
定理 1.1. $K=(X,\delta,m),$ $x\in X_k,$ $k\in \mathbb{N}.$ 那么有
$$\max_{K'\le_k K}F'(x)=\min_{
\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
\delta h\cdot \delta x\le 0
\end{subarray} }\delta\delta^*h(x)$$
Ollivier曲率
对$x\in X_1,$ 其Ollivier曲率可写为
$$\kappa(x)=\inf_{\begin{subarray}{c}
\delta f(x)=1\\
|\delta f|\le 1
\end{subarray} }\delta\delta^*\delta f(x),$$
事实上只需考虑$f\in C(X_0).$
显然有如下转化:
$$\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
h\in \delta(C(X_0))
\end{subarray} }\delta\delta^*h(x),$$
定理 1.2. $K=(X,\delta,m),$ $x\in X_k,$ $k\in \mathbb{N}.$ 若所有含$x$的长度不超过$5$的圈都是$2$-胞腔, 那么有:
$$\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
\delta x\cdot\delta h\le 0
\end{subarray} }\delta\delta^*h(x),$$
两者联系
推论 1.3. $G=(X,\delta,m)$为$1$维胞腔复形, 那么$\,\forall\,x\in X_1,$
$$\kappa(x)=\max_{G\le_1 K}F_K(x),$$
证: 令$G\le_1 X,$ $X$包含所有可能的$2$-胞腔,
自然全体长度不超过$5$的圈都是$2$-胞腔. 那么:
$$\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
\delta x\cdot\delta h\le 0
\end{subarray} }\delta\delta^*h(x)=\max_{K\le_1 X} F_K(x)=\max_{X^{(1)}\le_1 K}F_K(x)=\max_{G\le_1 K}F_K(x).$$
也就是可以通过粘贴合适权重的$2$-胞腔,
使得新图的Forman曲率与原有的Ollivier曲率相同.
推论 1.4. $K=(X,\delta,m),$ $x\in X,$ 那么:
$$F(x)\le\kappa(x).$$
证:
$$F(x)\le \max_{K^{(1)}\le_1 X}F_X(x)=\kappa_{K^{(1)} }(x)=\kappa(x).$$
调整距离
基本性质
定义Ollivier曲率时可以选取其它的距离. 赋予图另一权重$\omega,$
定义路径距离$d_\omega:X_0\rightarrow [0,\infty),$
$$d_\omega(v,w):=\inf\{\sum_{k=1}^n \omega(v_k,v_{k-1}):v=v_0\sim\cdots\sim v_n=w\}.$$
总是假设$\omega$是非退化的, 即边总是顶点间的最短距离. 定义关于$\omega,$
或称关于$d_\omega$的Ollivier曲率:
$$\omega(x)\kappa_\omega(x):=\inf_{\begin{subarray}{c}
\delta f(x)=\omega(x)\\
|\delta f|\le \omega\end{subarray} }\delta\delta^*\delta f.$$
非退化的要求实际上意味着在$X_1$上Ollivier曲率是有限的.
为了令Forman曲率与新的Ollivier曲率相容,
改造其为关于$\omega$的Forman曲率:
$$F_\omega:=Hx(x)-\sum_{y\neq x}\frac{\omega(y)}{\omega(x)}|Hy(x)|.$$
$x\in X_1$时, 即为
$$F_\omega(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}-\sum_{x\neq y}\frac{\omega(y)}{\omega(x)}\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|.$$
事实上可以使它与之前得到Forman曲率的过程一致:
$$\omega F_\omega=D(\omega H)=\omega(x) Hx(x)-\sum_{y\neq x}|\omega(y)H_y(x)|.$$
对偶性质
类似地有:
定理 1.5. $K=(X,\delta,m,\omega),$ $x\in X_k,$ $k\in \mathbb{N}.$ 那么有
$$\max_{K'\le_k K}\omega(x)F_\omega'(x)=\min_{
\begin{subarray}{c}
h(x)=\omega(x)\\
|h|\le \omega\\
\delta h\cdot \delta x\le 0
\end{subarray} }\delta\delta^*h(x)$$
定理 1.6. $K=(X,\delta,m,\omega),$ $x\in X_k,$ $k\in \mathbb{N}.$ 若所有含$x$的极小圈都是$2$-胞腔, 那么有:
$$\omega(x)\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=\omega(x)\\
|h|\le \omega\\
\delta x\cdot\delta h\le 0
\end{subarray} }\delta\delta^*h(x),$$
推论 1.7. $G=(X,\delta,m,\omega)$为$1$维胞腔复形, 那么$\,\forall\,x\in X_1,$
$$\kappa_\omega(x)=\max_{G\le_1 K}F_{K,\omega}(x),$$
推论 1.8. $K=(X,\delta,m,\omega),$ $x\in X,$ 那么:
$$F_\omega(x)\le\kappa_\omega(x).$$
Forman定义的关于权重$w$的曲率为:
$$\frac{F_o(x)}{w(x)}:=\sum_{v\prec x} \frac{w(v)}{w(x)}+\sum_{x\prec z}\frac{w(x)}{w(z)}-\sum_{y\neq x}\left|\sum_{v\prec x,y}\frac{w(v)}{\sqrt{w(y)w(x)} }-\sum_{x,y\prec z}\frac{\sqrt{w(x)w(y)} }{w(z)}\right|,$$
$x\in X_1.$ 可以看出去掉极小对角占优算子前, 算子$H$是已经对称化过的.
文中定义的是:
$$F_\omega(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}-\sum_{x\neq y}\frac{\omega(y)}{\omega(x)}\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|,$$
不难发现当$m=\frac{1}{w},$ $\omega=\sqrt{w}$时,
有$F_\omega(x)=\frac{F_o(x)}{w(x)}.$
文章最后更新于 2021-10-23 11:13:29
Characterizations of Forman curvature - Jürgen Jost, Florentin Münch
basic definition
Cell complex and diagram
Given cell complex $X=\bigcup X_k.$
Take $\delta:X_k\rightarrow X_{k+1}$ as the incident operator,
$\delta v(z)=\pm 1$ or $0.$ requires the cell complex to satisfy the following properties:
$\dim(z)\ge 1\Rightarrow \,\exists\,x\prec z;$
$|\{w\in X_0:\delta w(x)=1\}|=|\{w\in X_0:\delta w(x)=-1\}|=1,$
$\,\forall\,x\in X_1;$ (two vertices on each side)
$\dim(z)-\dim(v)=2,$ $\{x:v\prec x\prec z\}\neq \varnothing$
$\Rightarrow$ $|\{x:v\prec x\prec z\}|=2$ and
$\pm 1\in \{\delta v(x)\delta x(z):x\in X\}.$ (diamond sex&
$\delta^2=0$)
(directional) connectivity;
$\{x:x\prec z\}=\{x:x\prec z'\},$ $\dim z\ge 1\Rightarrow z=z'.$
(simple picture)
$\,\forall\,x,y\in X,$
Note $x\sim y$ if $\,\exists\,v\prec x,y$ or $x,y\prec z.$ which induces distance:
$$d(x,y)=\inf\{n:x=x_0\sim\cdots\sim x_n=y\}.$$
Note that this is not a general path distance on $X_0$. When the complex dimension is higher,
Point to point can be transferred more quickly with the help of high-dimensional cells.
Define weighted inner product $$\left<{}f,g\right>=\sum_x f(x)g(x)m(x).$$
Treat $x$ as $1_x,$ linearly expand $\delta:C(X)\rightarrow C(X).$
Define accompanying $\delta^\ast :C(X)\rightarrow C(X),$
$$\delta^\ast z(x)=\frac{1}{m(x)}\left<{}\delta^\ast z,x\right>=\frac{1}{m(x)}\left<{}\delta x,z\right>=\frac{m(z)}{m(x)}\delta x(z).$$
This is different from $\partial$ in Forman’s definition,
$$\partial z(x)=\delta x(z)=\frac{m(x)}{m(z)}\delta^\ast z(x).$$
Hodge Laplacian
Define Hodge Laplacian $$H=\delta \delta^\ast +\delta^\ast \delta.$$
To calculate $Hy(x),$, calculate the following formulas respectively:
$$\delta\delta^\ast y(x)=\frac{1}{m(x)}\left<{}\delta^\ast y,\delta^\ast x\right>=\frac{1}{m(x)}\sum_{v\prec x,y}m(v)\frac{m(y)}{m(v)}\delta v(y)\frac{m(x)}{m(v)}\delta v(x)=\sum_{v\prec x,y}\frac{m(y)}{m(v)}\delta v(x)\delta v(y),$$
The simpler ones are:
$$\delta^\ast \delta y(x)=\sum_{x,y\prec z}\frac{m(z)}{m(x)}\delta x(z)\delta y(z).$$
Note that unlike the Forman definition, the weight of cells with smaller dimensions is always in the denominator.
When $X=X_0\cup X_1$ is a weighted directed graph, record $m(x)=m(v,w)=m(w,v),$ $x\in X_1,$
$v,w$ is two vertices. $H$ acts on $C(X_0)$ as:
$$\begin{aligned}
Hf(v)=\delta^*\delta f(v)&=\sum_w\sum_{v,w\prec x}\frac{m(x)}{m(v)}\delta v(x)\delta w(x)f(w)\\
&=\sum_{v\prec x}\frac{m(x)}{m(v)}\sum_{w\prec x}\delta v(x)\delta w(x)f(w)\\
&=\sum_{w\neq v}\frac{m(v,w)}{m(v)}(f(v)-f(w))
\end{aligned}$$
That is, $H_0$ is the Laplacian of the weighted graph.
For $x\neq y\in X_1,$, at most one of the first sums is non-zero,
The signs of the non-zero terms in the second sum are the same. When non-zero terms appear in both sums,
There is $\delta v(x)\delta v(y)=-\delta x(z)\delta y(z).$ therefore
$$|Hy(x)|=\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|.$$
easy to get
$$Hx(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}.$$
For the (arbitrary) self-adjoint operator $H,$, there is a decomposition $$H=\Delta+D,$$ $\Delta$ into a minimal diagonally dominant operator,
$D$ is a diagonal matrix. Obviously there is $$Dx(x)=Hx(x)-\sum_{y\neq x}|Hy(x)|.$$
Decompose the Hodge Laplacian and call the obtained $D$ Forman curvature. to $x\in X_1,$
$$F(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}-\sum_{x\neq y}\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|.$$
If $m\equiv 1,$
$$F(x)=\#\{v\in X_0:v\prec x\}+\#\{z\in X_2:x\prec z\}-\#\{\text{parallel neighbors of } x\},$$
It is indeed the same as Forman curvature.
But in fact, the process of obtaining $F$ is not completely consistent with the Forman curvature, and they are not the same for the general $m$.
Before decomposing the Hodge Laplacian, Forman first symmetrized $H$, that is, it is a symmetric matrix under constant weights.
However, depending on the steps, two different Forman curvatures can be converted.
Duality
For the complex $K=(X,\delta,m),$ $K'=(X',\delta,m'),$, write $K'\le_k K,$
If the following conditions are met:
$X_j=X_j',$ $\,\forall\,j\le k;$
$X_j'\subset X_j,$ $\,\forall\,j>k;$
$m(x)=m'(x),$ $\,\forall\,\dim x\le k;$
$\delta'v(x)=\delta v(x),$ $\,\forall\,x,v\in X'.$
That is, $K'\subset K$ and their weighted orientation $k$-skeleton are the same.
Theorem 1.1. $K=(X,\delta,m),$ $x\in X_k,$ $k\in \mathbb{N}.$ Then there are
$$\max_{K'\le_k K}F'(x)=\min_{
\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
\delta h\cdot \delta x\le 0
\end{subarray} }\delta\delta^*h(x)$$
Ollivier curvature
For $x\in X_1,$, its Ollivier curvature can be written as
$$\kappa(x)=\inf_{\begin{subarray}{c}
\delta f(x)=1\\
|\delta f|\le 1
\end{subarray} }\delta\delta^*\delta f(x),$$
In fact just consider $f\in C(X_0).$
Obviously there are the following transformations:
$$\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
h\in \delta(C(X_0))
\end{subarray} }\delta\delta^*h(x),$$
Theorem 1.2. $K=(X,\delta,m),$ $x\in X_k,$ $k\in \mathbb{N}.$ If all cycles containing $x$ whose length does not exceed $5$ are $2$-cells, then there is:
$$\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
\delta x\cdot\delta h\le 0
\end{subarray} }\delta\delta^*h(x),$$
Contact between the two
Corollary 1.3. $G=(X,\delta,m)$ is the complex shape of $1$ dimensional cell, then $\,\forall\,x\in X_1,$
$$\kappa(x)=\max_{G\le_1 K}F_K(x),$$
Certificate: Let $G\le_1 X,$ $X$ include all possible $2$-cells,
Naturally, all cycles whose length does not exceed $5$ are $2$-cells. Then:
$$\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=1\\
|h|\le 1\\
\delta x\cdot\delta h\le 0
\end{subarray} }\delta\delta^*h(x)=\max_{K\le_1 X} F_K(x)=\max_{X^{(1)}\le_1 K}F_K(x)=\max_{G\le_1 K}F_K(x).$$
That is, you can paste the $2$-cell with appropriate weights,
Make the Forman curvature of the new graph the same as the original Ollivier curvature.
Corollary 1.4. $K=(X,\delta,m),$ $x\in X,$ Then:
$$F(x)\le\kappa(x).$$
Certificate:
$$F(x)\le \max_{K^{(1)}\le_1 X}F_X(x)=\kappa_{K^{(1)} }(x)=\kappa(x).$$
Adjust distance
basic properties
You can choose other distances when defining Ollivier curvature. Give the graph another weight $\omega,$
Define path distance $d_\omega:X_0\rightarrow [0,\infty),$
$$d_\omega(v,w):=\inf\{\sum_{k=1}^n \omega(v_k,v_{k-1}):v=v_0\sim\cdots\sim v_n=w\}.$$
It is always assumed that $\omega$ is non-degenerate, i.e. an edge is always the shortest distance between vertices. Define with respect to $\omega,$
Or called Ollivier curvature about $d_\omega$:
$$\omega(x)\kappa_\omega(x):=\inf_{\begin{subarray}{c}
\delta f(x)=\omega(x)\\
|\delta f|\le \omega\end{subarray} }\delta\delta^*\delta f.$$
The non-degenerate requirement actually means that the Ollivier curvature is finite on $X_1$.
In order to make Forman curvature compatible with the new Ollivier curvature,
Transform it into the Forman curvature about $\omega$:
$$F_\omega:=Hx(x)-\sum_{y\neq x}\frac{\omega(y)}{\omega(x)}|Hy(x)|.$$
When $x\in X_1$, it is
$$F_\omega(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}-\sum_{x\neq y}\frac{\omega(y)}{\omega(x)}\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|.$$
In fact, it can be made consistent with the previous process of obtaining Forman curvature:
$$\omega F_\omega=D(\omega H)=\omega(x) Hx(x)-\sum_{y\neq x}|\omega(y)H_y(x)|.$$
Dual properties
Similarly there are:
Theorem 1.5. $K=(X,\delta,m,\omega),$ $x\in X_k,$ $k\in \mathbb{N}.$ Then there are
$$\max_{K'\le_k K}\omega(x)F_\omega'(x)=\min_{
\begin{subarray}{c}
h(x)=\omega(x)\\
|h|\le \omega\\
\delta h\cdot \delta x\le 0
\end{subarray} }\delta\delta^*h(x)$$
Theorem 1.6. $K=(X,\delta,m,\omega),$ $x\in X_k,$ $k\in \mathbb{N}.$ If all minimal cycles containing $x$ are $2$-cells, then we have:
$$\omega(x)\kappa(x)=\inf_{\begin{subarray}{c}
h(x)=\omega(x)\\
|h|\le \omega\\
\delta x\cdot\delta h\le 0
\end{subarray} }\delta\delta^*h(x),$$
Corollary 1.7. $G=(X,\delta,m,\omega)$ is the complex shape of $1$ dimensional cell, then $\,\forall\,x\in X_1,$
$$\kappa_\omega(x)=\max_{G\le_1 K}F_{K,\omega}(x),$$
Corollary 1.8. $K=(X,\delta,m,\omega),$ $x\in X,$ Then:
$$F_\omega(x)\le\kappa_\omega(x).$$
The curvature of weight $w$ defined by Forman is:
$$\frac{F_o(x)}{w(x)}:=\sum_{v\prec x} \frac{w(v)}{w(x)}+\sum_{x\prec z}\frac{w(x)}{w(z)}-\sum_{y\neq x}\left|\sum_{v\prec x,y}\frac{w(v)}{\sqrt{w(y)w(x)} }-\sum_{x,y\prec z}\frac{\sqrt{w(x)w(y)} }{w(z)}\right|,$$
$x\in X_1.$ It can be seen that the operator $H$ has been symmetrized before removing the minimal diagonal dominant operator.
The text defines:
$$F_\omega(x)=\sum_{v\prec x}\frac{m(x)}{m(v)}+\sum_{x\prec z}\frac{m(z)}{m(x)}-\sum_{x\neq y}\frac{\omega(y)}{\omega(x)}\left|\sum_{v\prec x,y}\frac{m(y)}{m(v)}-\sum_{x,y\prec z}\frac{m(z)}{m(x)}\right|,$$
It is not difficult to find that when $m=\frac{1}{w},$ $\omega=\sqrt{w}$,
Yes $F_\omega(x)=\frac{F_o(x)}{w(x)}.$
The article was last updated on 2021-10-23 11:13:29
