寡妇惩罚不起作用

Question

请始终发布完整的文档，以显示问题，而不仅仅是一个片段。将寡妇惩罚设置为 3000 会使使用寡妇线中断变得不太可取，但即使你将其设置为 10000 以使在那里中断变得非常糟糕，TeX 仍可能会这样做，如果它没有更好的选择。如果你有一整页的文本，页面上没有可拉伸的空间，并且\flushbottom实际上有没有选择对于分页，TeX 将在填满页面的行后分页。如果分页惩罚过高，您将在日志中收到警告。

如果你想避免寡妇，仅仅增加违反那里你通常需要减少违反规定的惩罚别的地方通常是在某处添加一些胶水。

\setlength\abovedisplayskip{0pt plus 3pt minus 9pt}

允许空间为负数，这真的是您想要的吗？

不要\\在文档中使用，标记段落分隔符的方式是空白行。相反不在等式前添加空行。尽管评论中有对话，但您的文档中没有数字或其他浮动。

基本上，您的文档中的标记没有给 LaTeX 任何机会找到任何合理的断点。

特别是如果你去

.....aaa\\
\\
bbb....

那么这不是两个段落一段落有两个强制换行符，因此之前的分页符aaa不会产生寡妇，因此不会插入寡妇惩罚。

如果你走的话

.....aaa

bbb....

那么 bb 就是一个新段落的开始，因此在 aaa 之前或 bbb 之后的分页符也会受到惩罚可拉伸在段落分隔符处插入粘胶，有助于分页。

\documentclass[a4paper,12pt]{article}

\usepackage{amsmath,graphicx,fullpage,microtype,hyperref,subfig,hypcap,amsfonts}

\DisableLigatures{encoding=*,family=*}
\numberwithin{equation}{section}
\usepackage{parskip}
\widowpenalty=300
\clubpenalty=300
\hyphenpenalty=300
\interfootnotelinepenalty=300
\hypersetup{colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black}
\setlength{\skip\footins}{5mm}

\begin{document}

If there is a region which has a larger density of points, this region has a larger volume. This is how curvature occurs in the causal-set. This instance would be similar to the surface of sphere, where the inner shells have a higher volume relative to the outer shells, corresponding to a positive curvature. But what is the coversion rate between the number of elements and volume? Several arguments exist which mostly point to the same answer. For example, when examining the entropy of a black hole horizon, it seems there is one bit of entropy per plaquette of size $8\pi G\hbar$. This leads to a discreteness scale $l=\sqrt{8\pi G\hbar}$ around the Planck length \cite{Sorkin2003}. % speculates about size of element

This means that there are two forces which determine the relative position of causal-set elements. First of all, there is a force which tells each element to 'take up' around a Plack volume of spacetime. You can visualize this as a set of elements which grow like soap-bubbles until they are all around the same size. Each bubble touches its surrounding bubbles such that the whole set makes a surface in the hyperspace.

The second force which acts between the elements contrains those bubbles which are linked to each other. The step from one element to the next should also be about a Planck length long. Whenever two elements are causally linked, they are contrained to be around this distance away from eachother within the hyperspace.

Given these two forces, we can try to think of aln algorithm which is able to gives coordinates to each of the causal-set elements within the hyperspace. The idea is quite similar to Henson's algorithm discussed in section~\ref{sec:Obtaining the Metric} above. He proposed using the time-like distance to the top of a causal-diamond and the distance to the bottom to determine and element's relative position \cite{Henson2006}. % Very similar method for finding coordinates for causal-set elements
By looking at the repulsive forces of the growing soap-bubble, while contraining them by Planck-sized links, a similar effect occurs.

Let us now imagine an algorithm which starts off by placing the causal-set elements in a random position. For each element, it determines the position to every one of the other points and calculates the repulsive force $f_r(i, j)$ acting on it from every other element. It also looks at all the elements which are linked to it and calculates the attractive force $f_a(i, j)$ running only through those links. 
\begin{subequations}
\label{eq:Forces}
\begin{align}
f_r(i, j) & = -\frac{K^2}{\parallel x_i - x_j\parallel} & i \neq j \hspace{8mm} i, j \in G \\
f_a(i, j) & = \frac{\parallel x_i - x_j\parallel^2}{K} & i \leftrightarrow j
\end{align}
\end{subequations}

These forces can then be used to displace the element slightly, where the process can start over again until a steady-state is reached. In principle, the two forces acting between all elements gives the causal-set a particular energy. The energy is minimised when the elements are in the 'right' position. This is what the algorithm does.

Although we are looking at a Lorentzian manifold, the algorithm just described already exists for Euclidean and Riemanian manifolds. In fact, there is a large body of work done on so called 'graph-drawing algorithms', and in particular on 'spring embedders' where links are replaced by springs to form a mechanical system \cite{Kobourov2007}. % Review of spring-embedders and pseudo-code for Riemanian geometries
The spring forces on the elements move the system to a minimal energy state.

In Riemanian manifolds, the algorithm is slightly more complicated in that distances are defined in terms of geodesics. In this case, the metric of the hyperspace into which you are embedding the elements is needed. More precisely, the algorithm computes the forces acting on an element $n$ at position $x$ in its tangent space $T_xM$ where $\tau_x$ is the map from the manifold $M$ to this tangent space \cite{Kobourov2002}. % Introduces spring-embedded graphs in Riemanian geometries
The pseudo-code for this algorithm is the following:
\begin{equation}
\label{eq:Riemanian algorithm}
\begin{array}{l}
\text{generate initial layout}(G)\\
\text{while not done do}\\
\hspace{5mm} \text{foreach } n \in G \text{ do}\\
\hspace{10mm} x := \text{position}[n]\\
\hspace{10mm} G' := \tau_x(G)\\
\hspace{10mm} x' := \text{force directed placement}(n, G')\\
\hspace{10mm} \text{position}[n] :=\tau_x^{-1}(x')\\
\hspace{5mm} \text{end}\\
\text{end}
\end{array}
\end{equation}

In theory, the Riemanian metric can be extended to a pseudo-Riemanian one, such as for Minkowski space. In practice, determining the map $\tau_x$ to the tangent space is the hardest step.

These algorithms have also been translated to $Mathematica$ code \cite{Hu2005}. % Mathematica codes for spring embedders and also suggestion for multi-level drawing based on coarse-graining. 
There are two different algorithms, one which has both repulsive and attractive forces running through every link, such that each link has an optimal Planckian size. The other has its repulsive force between every element such that they all take up about a Planck volume. For our causal-sets, a combination of these two algorithms may be most appropriate.

\end{document}

Answer 1